Finite volumes for complex applications: problems and perspectives. Vol. 2

Finite Volumes for Complex Applications I1 Problems and Pwspectives

Finite Volumes for Complex Applications I1

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0 HERMES Science Publications, Paris, 1999 HERMES Science Publications 8, quai du MarchC-Neuf 75004 Paris Serveur web : http://www.hermes-science.com ISBN 2-7462-0057-0 Catalogage Electre-Bibliographie Finite Volumes For Complex Applications I1 - Problems and Perspectives Vilsmeier, Roland* Benkhaldoun, Fayssal* Hkel, Dieter Paris : Hermks Science Publications, 1999 ISBN 2-7462-0057-0 RAMEAU : elkments finis, mkthode des analyse numkrique DEWEY : 5 15 : Analyse mathkmatique

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Finite Volumes for Complex Applications I1 Problems and Perspectives

editors

Roland Vilsmeier Fayssal Benkhaldoun Dieter Hanel

cience

Second International Symposium on Finites Volumes for Complex Applications Problems a n d Perspectives

July 19-22, 1999, Duisburg, Gemzany Internet adress : http://www.vug.uni-duisburg.de/FVCAIUcontents.html

Organizing Institutions Institut fix Verbrennung und Gasdynamik (IVG), University Duisburg, Germany INSA de Rouen, France

Scientific Committee F. Benkhaldoun, LMI, INSA de Rouen, France R. Borghi ESM2, IMT-Technopole, Marseille, France A. Dervieux, INRIA Sophia Antipolis, France T. Gallouet, UniversitC Aix-Marseille I, France D. Hanel, IVG, University Duisburg, Germany D. KrCjner, Institute f. Angewandte Mathematik, University Freiburg, Germany I. Toumi, CEA, Saclay, France J.-P. Vila, INSA de Toulouse, France R. Vilsmeier, IVG, University Duisburg, Germany N. P. Weatherill, University of Swansea, UK G. Wittum, IWR, University Heidelberg, Germany

Invited Keynote Lectures R. Abgrall, Universitk de Bordeaux 1, France F. Coquel, CNRS, Paris, France G. Degrez, von Karman Institute, St-Genesius-Rode,Belgium R. Klein, Konrad-Zuse-Zentrum f. Informationstechnik, Berlin, Germany R. Lazarov, Lawrence Livermore National Laboratory, USA J. M. Ghidaglia, ENS de Cachan, France S. Noelle, Institute f. Angew. Mathematik, University Bonn, Germany

Contents

Editors preface ....................................................................................................XI11 Invited speakers ...................................................................................................

1

Construction of some genuinely multidimensional upwind .............................................................. distributive schemes - R. ABGRALL

3

A Roe-type Linearization for the Euler Equations for Weakly Ionized Gases F. COQUEL,C. MARMIGNON ...........................................................................

11

Multidimensional Upwind Residual Distribution Schemes and Application H. DECONINCK, G. DEGREZ.............................................................................

27

Overcoming mass losses in Level Set-based interface tracking schemes Th. SCHNEIDER, R. KLEIN................................................................................4 1 Coupling mixed and finite volume discretizations of convection-diffusionreaction equations on non-matching grids .......................................... 5 1 R.D. LAZAROV, J.E. PASCIAK, P.S. VASSILEVSKI Numerical computation of 3D two phase flow by finite volumes methods ...................................................... using flux schemes - J. M. GHIDAGLIA

69

The MOT-ICE : a new high-resolution wave-propagation algorithm based on Fey's Method of Transport - S. NOELLE................................................... 95 Numerical Analysis ............................................................................................. 1 15 Error estimate for a finite volume scheme on a MAC mesh 117 for the Stokes problem - P. BLANC.............................................................. Convergence Rate of the Finite Volume Time-explicit Upwind Schemes for the Maxwell System on a Bounded domain Y. COUDIERE, P. VILLEDEU............................................................................ 125

VI

Finite volumes for complex applications

Flux vector splitting and stationary contact discontinuity - F. DUBOIS............. 133 Analysis of a Finite Volume Solver for Maxwell's Equations F. EDELVIK........................................................................................ . 141 A result of convergence and error estimate of an approximate gradient for elliptic problems - R. EYMARD, T. GALLOUET,R. HERBIN..................... 149 Finite volume approximation of elliptic problems with irregular data T. GALLOUET, R. HERBIN............................................................................ 155 Analysis of a finite volume scheme for reactive fluid flow problems A. HOLSTAD,I. LIE.......................................................................................... 163 Convergence of a finite volume scheme for a nonlinear convectiondiffusion problem - A. MICHEL............................................................... 173 Convergence analysis of a cell-centered FVM .......................................... H.P. SCHEFFLER, R. VANSELOW.

....... . . .

18 1

Error estimates on the approximate finite volume solution of convection diffusion equations with boundary conditions T. GALLOUET,R. HERBIN,M. H. VIGNAL............................. . ............... 189 The limited analysis in finite elasticity - I.A. BRIGADNOV .................................197 Entropy consistent finite volume schemes for the thin film equation G. GRUN,M. RUMPF........................................................................................

205

Convergence of finite volume methods on general meshes for non smooth solution of elliptic problems with cracks P. ANGOT,T. GALLOUET, R. HERBIN............................................................

215

Application and analysis of finite volume upwind stabilizations for the ...... 223 steady-state incompressible Navier-Stokes equations - L. ANGERMANN A new cement to glue non-conforming grids with Robin interface conditions Y. ACHDOU,C. JAPHET,F. NATAF,Y. MADAY .........................................23 1 Finite Volume Box Schemes - J.-P. CROISILLE ..................................................239 On nonlinear stability analysis for finite volume schemes, plane wave instability and carbuncle phenomena explanation - M. ABOUZIAROV .......... 247 Innovative schemes.............................................................................................. 253

A comparison between upwind and multidimensional upwind schemes for unsteady flow - P. BRUFAU,P. GARC~A-NAVARRO ...............................255 Reformulation of the unstructured staggered mesh method as a classic finite volume method - B. PEROT,X. ZHANG........................................... 263 A mixed FE-FV algorithm in non-linear solid dynamics - S.V. POTAPOV ......... 27 1

Contents

VII

An Euler Code that can compute Potential Flow - M. RAD, P. ROE................... 279 Finite volume evolution Galerkin methods for multidimensional hyberbolic problems -M. LUKACOVA-MEDVIDOVA, K.W. MORTON,G. WARNECKE ......... 289 Nonlinear anisotropic artificial dissipation - Characteristic filters for computation of the Euler equations T. GRAHS,A. MEISTER,T. SONAR...............................................................297 Nonlinear projection methods for multi-entropies Navier-Stokes systems F. COQUEL................................................................................. 307 C. BERTHON, About a Parallel Multidimensional Upwind Solver for LES D. CARAENI, S. CONWAY,L. FUCHS............................................................... 3 15 A higher-order-accurate upwind method for 2D compressible flows ......................................... 323 on cell-vertex unstructured grids - L. CATALANO A New Upwind Least Squares Finite Difference Scheme (LSFD-U) for Euler Equations of Gas Dynamics -N. BALAKRISHNAN, C. PRAVEEN........ 33 1 A finite-volume algorithm for all speed flows F. MOUKALLED, M. DARWISH ............................................................................339 Preserving Vorticity in Finite-Volume Schemes - P. ROE, B. MORTON............ 347 On Uniformly Accurate Upwinding for Hyperbolic Systems P. ROE........................................................357 with Relaxation - J. HITTINGER, Implicit Finite Volume approximation of incompressible multi-phase flows using an original One Cell Local Multigrid method ......................................................................367 S. VINCENT, J.P. CALTAGIRONE New classes of Integration Formulas for CVFEM Discretization of ConvectionDiffusion Problems - E. P. SHURINA, T.V. VOITOVICH ................................ 377 Fields of application ............................................................................................ 385

Analysis of Finite Volume Schemes for Two-Phase Flow in Porous Media ..........................................387 on Unstructured Grids - M. AFIF, B. AMAZIANE A preconditioned finite volume scheme for the simulation of equilibrium 395 two-phase flows - S. CLERC.......................................................................... Transient flows in natural valleys computed on topography-adapted mesh S. SOARESFRAZAO,J. LAUMANWAI,Y. ZECH.............................................403 A mixed Finite Volume/Finite Element method applied to combustion in multiphase medium -N. GLINSKY-OLIVIER, E. SCHALL.......................... 41 1 Turbulence Modeling for Separated Flows - L.J. LENKE,H. SIMON.................. 419

VIII

Finite volumes for complex applications

Simulation of unsteady Flow in a Vortex-Shedding Flowmeter .................................................. 429 S. PERPEET,A. ZACHCIAL, E. VON LAVANTE A Finite Volume Scheme for the Two-Scale Mathematical Modelling of T i c Ignition Process - A. AOUFI,V. ROSENBAND ................................... 437 Two Perturbation Methods to Upwind the Jacobian Matrix of Two-Fluid Flow Models - A. KUMBARO, I. TOUMI,J. CORTES...................................... 445 Finite volumes simulations in magnetohydrodynamics - M. HUGHES, L. LEBOUCHER, V. BOJAREVICS, K. PERICLEOUS, M. CROSS..........................

453

Finite Volume Method for Large Deformation with Linear Hypoelastic ............................................. 459 Materials - K. MANEERATANA, A. IVANKOV[C A finite volume formulation for fluid-structure interaction ........................................ 467 C.J GREENSHIELDS, H.G. WELLER,A. IVANKOVIC Boundary Conditions for Suspended Sediment V. B o v o ~ mL. , TAGLIALATELA ....................................................................

475

Second order corrections to the finite volume upwind scheme for the 2D Maxwell equations - B. BIDEGARAY, J.-M. GHIDAGLIA ........................ 483 A MHD-Simulation in the Solar Physics ........................................................... 49 1 A. DEDNER,C. ROHDE,M. WESENBERG A Zooming Technique for Wind Transport of Air Pollution .......................... 499 P.J.F. BERKVENS, M.A. BOTCHEV,W.M. LIOEN,J.G. VERWER Computational Solid Mechanics using a Vertex-based Finite Volume Method G.A. TAYLOR,C. BAILEY,M. CROSS............................................................. 507 Control volumes technique applied to gas dynamical problems in ............................................................... underground mines - E. VLASSEVA

5 17

Simulation of salt-fresh water interface in costal aquifers using a finite volume , OUAZAR,I. ELMAHI ........... 525 scheme on unstructured meshes -B. B o u z o u ~D. Progress in the flow simulation of high voltage circuit breakers X. YE, L. MULLER,K. KALTENEGGER, J. STECHBARTH .................................. 533

.................................................543 River valley flooding simulation - F. ALCRUDO Modelling vehicular traffic flow on networks using macroscopic models ............................................................55 1 J.P. LEBACQUE, M.M. KHOSHYARAN Finite Volume method applied to a solid/liquid phase change problem 0.MAZHOROVA ............................................. 559 M. EL GANAOUI, P. BONTOUX, Integrating finite volume based structural analysis procedures with CFD software to analyse fluid structure interactions M.A. WHEEL,A. OLDROYD, T.J. SCANLON, P. WENKE..................................567

Contents

IX

A generalized parcel method for the spray dispersion computation B. NKONGA ...................................................................................................... 575 Finite Volume Methods for Multiphysics Problems - C. BAILEY, M. CROSS,K. PERICLEOUS, G.A. TAYLOR,N. CROFT,D. WHEELER,H. LU........ 585 A Finite Volume discretization and multigrid solver for steady viscoelastic fluid flows - H. AL MOATASSIME, S. RAGHAY, A. HAKIM........................... 595

Complexity, Performance and Informatics................................................... 605 Various CG-type methods applied to finite volume schemes 0. SCHMID,A. BUBMANN, E. VON LAVANTE, M. MOCZALA ........................... 607 A Newton-Relaxation Finite Volume Scheme for Simulation

of Dynamic Motion - B.A. JOLLY,M. RIZK.................................................615 An Attempt to Develop a Multi Purpose FAS Multigrid Algorithm L. FOURNIER, 0. CLOTH..................................................................................623 On Higher Order Accurate Implicit Time Advancing for Stiff Flow Problems C. VIOZAT,E. SCHALL,A. DERVIEUX, D. LESERVOISIER ............................... 63 1 Numerical Solution of Steady 2D and 3D Impinging Jet Flows ...................................................................639 K. KOZEL,P. LOUDA,J. PR~HODA Triangular, Dual and Barycentric Finite Volumes in Fluid Dynamics J. FELCMAN, M. FEISTAUER ............................................................................. 647 Concepts for parallel numerical solution of PDEs - G. BERTI............................ 655 Performing parallel direct numerical simulation of two dimensional heated jets S. BENAZZOUZ, V.G. CHAPIN,P. CHASSAING ..........................................

663

Two-Dimensional Riemann Problems. Assessment Tests for Upwind Methods for Multi-Dimensional Supersonic Flow Problems J. VAN KEUK,J. BALLMANN ............................................................................ 671 Robustness and accuracy on unstructured grids. Numerical experiments .............................683 on finite volume schemes - E.A. MEESE,S.E. HAALAND A validation of an efficient numerical method for 3-D complex flows R. VERZICCO,P. ORLANDI ................................ 693 E.A. FADLUN,S. LEONARD],

Comparison of Two Finite Volume Methods for 3D Transonic Flows through Axial Cascades -J. FORT,J. FURST,J. HALAMA,K. KOZEL........... 701 An efficient and universal numerical treatment of source terms in turbulence modelling - B. MERCI,J. STEELANT,E. DICK.........................709 Comparison of numerical solvers for a multicomponent, turbulent flow E. XEUXET,A. FORESTIER,J.M. HERARD .......................................................7 17

X

Finite volumes for complex applications

Parallel Overlapping Mesh Technique for Compressible Flows J. ROKICKI, D. DRIKAKIS, J. MAJEWSKI, J. ZOLTAK ........................................

725

A comparison of Finite Volume and Higher-Order Finite Difference Schemes for the Solution of the Navier-Stokes and Euler equations M. MEINKE,Th. RISTER,R. EWERT................................................................ 733 Simulation of 3D turbulent flow through steam-turbine control valves V.G. KOLYVANOV, V.V. RIS, B.N. AGAPHONOV, V.D. GORYACHEV, E.M. SMIRNOV, D.K. ZAITSEV.....................................................................

743

Adaptivity, Tracking and Fitting ....................................................................... 75 1

An Adaptive Hybrid Object-Oriented Code for CFD-Applications-Adhoc3D U. TREMEL,H. BLEECKE,G. BRENNER, G . GREINER ......................................753 Adaptive mesh refinement for single and two phase flow problems ................................................................761 in porous media - M. OHLBERGER Parallel solution of hyperbolic PDEs with space-time adaptivity S., SODERBERG ........................................................................... 769 P. L O T S ~ D T Dynamic mesh generation with grid quality preserving methods A. WICK,F. THIELE......................................................................................... 777 A Finite Volume Method for Steady Hyperbolic Equations M. J . BAINES,S.J. LEARY,M.E. HUBBARD .................................................... 787 Moving grid technology for finite volume methods in gas dynamics B.N. AZARENOK, S.A. IVANENKO ................................................................... 795 Numerical Simulation of Lifted Turbulent Methane-Air Diffusion Flames M. CHEN,N. PETERS....................................................................................... 803 The application of a conservative grid adaption technique to 1D unsteady P. GARC~A-NAVARRO .................................... 809 problems - M. CASTRO-D~Az, Application of mesh adaptive techniques to mesh convergence in complex CFD D. LESERVOISIER, A. DERVIEUX, P.L. GEORGE,0 . PENANHOAT .................... 817 Multidimensional Fully Adaptive Finite Volume Schemes for the Numerical Simulation of Stiff Combustion Front Propagation in Condensed Phase A. AOUFI..........................................................................................................825 Mathematical and numerical modeling of a two-phase flow by a Level Set method -S. ROUY,P. HELLUY..................................................................... 833 Multiresolution analysis on triangles: application to conservation laws A. COHEN,S.M. KABER,M. POSTEL............................................................... 841 A local level set method for the treatment of discontinuities D. HANEL........................... 849 on unstructured grids - L. TRAN,R. VILSMEIER,

Contents

Varia .....................................................................................................................

XI

857

A Stabilized Version of Wang's Partitioning Algorithm for Banded Linear Systems -- V . PAVLOV ............................... . ......................................... 859

On Jeffreys Model of heat conduction -- M . DRYJA.K . MOSZYNSKI .................. 867 Investigation of some method for cavitating jet S. OCHERETYANY. V.V. PROKOFIEV ................................. . .......................... 875 Index des auteurs ............................................................................................... 887

Editors preface Finite Volume methods are methods directly related to the numerical solution of conservation laws. Systems of such conservation laws govern wide fields of physics and the efficiency of corresponding solution methods is an essential requirement from basic research and industry. Since the efficiency of any method must be measured by the quality of the result compared to the computational cost to spend for, corresponding developments are widely spread, ranging from very fundamental numerical analysis up to the efficient use of modern computer hardware. Although in the past the numerical methodology has made large progresses, many problems and difficulties remain, requirering further intensive research. New fields of application as well as the coupled simulation of different physical phenomena become accessible due to improved solution techniques and growing computer capacities. However, these new possibilities introduce again new physical and mathematical problems to be solved. The development of new methods as well as the extension of existing ones requires intensive and critical investigations and careful validation. One of the aims of this conference is therefore to bring together people working in theory and practice for fruitful and critical discussions about methods, their advantages and drawbacks and related experiences from arbitrary applications. The present proceedings summarize the contributions to be presented a t the second international symposium on Finite Volumes for Complex Applications Problems and Perspectives. The first symposium of this series was held summer 1996 at INSA de Rouen in France. Based on the success of this first conference, the symposium in Duisburg has again received an unexpected high attention in the numeric community. After a critical review of the submitted contributions, 98 papers by authors from 20 countries are presented in this volume. In a rough estimation, about half of the contributions can be assigned to analysis and numerics of different methods whereas the other half is essentially concerned with application and computational aspects of methods. We would like to thank all persons, who contributed to the conference and to this book of proceedings. First of all, we want to mention all the authors as well as the other members of the scientific committee, for the work of writing the papers as well as selecting these in remarkably short times. We would like to extend our thanks, acknowledging the help from the numerical staff, secretaries and students a t the IVG in Duisburg and at INSA de Rouen, keeping their good mood whenever. Finally we want to thank the following organizations for the financial support: Deutsche Forschungsgemeinschaft, Ministerium fiir Wissenschaft und Forschung Nordrhein-Westfalen, Duisburger Universitatsgesellschaft, Hewlett Packard and Sun Microsystems.

Fayssal Benkhaldoun, Dieter Hanel and Roland Vilsmeier

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Invited speakers

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Construction of genuinely multidimensional upwind distributive schemes

Re'mi Abgrall Universite' Bordeaux I Mathe'matiques Applique'es 351 Cours de la Libe'ration 33 405 Talence Cedex, France

We present a construction of a class of upwind residual schemes for the Euler equations of fluid mechanics. It naturaly generalises the PSI scheme of Deconinck, Roe and Struijs. We show some of its theoretical properties. Numerical illustrations indicates obvious advantages over the now classical finite volumes upwind schemes. ABSTRACT

Key Words

: Upwind

schemes, Residual schemes, Unstructured meshes.

1. Introduction

Most of the "modern" industrial codes that are used to simulate complex compressible fluids are written according to the ideas developed in the 80's by van Leer, Harten, Osher, Roe, and many others. These codes are versatile, robust and rather accurate. However, in some situations, the results are somewhat disapointing : the numerical viscosity of these schemes is still too high, despites some attemps to reduce it. It is still difficult t o accurately compute the lift and drag of an airfoil. These schemes are also very mesh dependent : the formulation deeply rely on the shape of the control volume. Their definition is many times not related to the physics of the problem under consideration. For these reasons, since a few years, some researchers have tried to find alternative formulations of the problem that could - in principle - lead to less dissipative and mesh independant schemes. Among the first, one might quote [Da] who tried to represent the fluxes in term of Riemann problems in direction related to the fluid. A very promissing formulation is certainly that of upwind residual schemes, who were pioniered by P.L. Roe, the H. Deconinck and their coau-

4

Finite volumes for complex applications

thors. These schemes are also related to the SUPG schemes of Hughes, or the streamline diffusion method by Johnson. If the design principle of the upwind residual schemes are rather clear for scalar convection equations, this is not anymore true for systems, and in particular the Euler equations of fluid mechanics. The aim of the present paper is to give the status of a research toward that goal. The paper is divided into four parts. In the first one, we recall the upwind residual formulation and the design principle for scalar equations. Then we reformulate the PSI scheme of Deconinck, Roe, Struijs and Sidilkover in a way that is easier t o generalise in multiD. More precisely, it is seen as a blending between the N scheme and the LDA scheme. In the third part, we discuss design principle for the Euler equations, and propose a class of schemes. Then numerical examples are given. 2. Upwind Residual schemes 2.1 The Euler equations

We consider the Euler equations for fluid mechanics with initial and boundary conditions

aw

- + div $(w) = 0

at W(xlO) = WO(X) inflow and/or wall conditions.

XER,t>O XER

(1)

In (1), the vector of conserved variables is W = (p, pu, pv, E ) T ,the x-component of the flux is = (pu, P ~ 2 p, puv, u ( E p))T. Its y-component has a similar expression. In the problem of interest, the pressure p is related to W via 1 p = (y - 1 ) ( E - -p(u2 v2)) with 7 = 1.4. The solutions of (1) has to fullfill 2 the second law of thermodynamics : we have to have

+

FZ

+

+

dS at

-

+ div (SG) 5 0 in R

where S = -ps (with s = log 2 ) is the mathematical entropy. In [Ta], E. (07) Tadmor has shown that the solutions of (1-2) satisfy the following minimum principle

Invited speakers

5

2.2 Numerical schemes

2.2.1 Generalities The discretisation of (1) is carried out on a mesh made of triangles. The list of nodes is {Mi)i,l,,,. The generic name of a triangle is T , its vertices are denoted by Mi,, Mi,, Mi,, or 1, 2, 3 when there is no ambiguity. The schemes for (1) are written as

In (4), WF is an approximation for W(Mi, nAt), (Ci 1 is the area of the dual cell associated to node M t l The residual must satisfy div

ghdz := aT

h.li € T

In this conservation relation ghis an approximation of that has to be continuous across the edges of the triangles. Under classical asssumptions, we have a Lax Wendroff-like theorem [AMN] : the scheme, if it converges, converges to weak solutions of (1).In this paper, we follow the approach of Roe-DeconinckStruijs [SDR] via the parameter vector Z = (fi, f i u , f i v , f i H j T : Z is piecewise linearly interpolated on the mesh, and we get 2h.In [AMN] we show that other approximations of 2 enable to put the finite-volume schemes in this residual framework. We introduce Sl ( I = 1 , 2 , 3 ) the inward normal to T opposite to the node - Mi, and A, B the linearised Jacobian matrices computed at the average state Z = Zi, Z i 2 Zi3 . The relation ( 5 ) becomes

+

+

3

where 2Ki = (ni), 2 has real eigenvalues.

+ (ni), B.One can show that K i is diagonalisable and

2.2.2 Design principles The schemes are constructed follwing three design principles -

the scheme is upwind : if Ki only has negative eigenvalues,

'1n particular, we have (Ci( =

5 CT,M,ET (TI.

=0

6

Finite volumes for complex applications

-

the scheme must be linear preserving : if 1,2,3.

aT = 0 then

@T = 0 for i =

the scheme must provide monotone solutions, i.e. without oscillations. The first property is a translation of the physics of the problem. The second one enable to have second order of accuracy a t steady state . This is true for example of the SUPG and streamline diffusion schemes, and not true for standard finite volume schemes. This property can be understood by local truncation analysis on regular meshes [SDR] or on finite element like meshes by variational arguments [Ab]. The last one is really obscure because very intuitive. It is clear for scalar equations, and for system we will translate it formally as a discrete version of (3) : assuming a discrete version of (3), we get a maximum principle on the numerical solution because s is concave. But this has to be understood formally, for now. -

2.2.3 Two example : t h e s y s t e m N and LDA schemes The schemes have been introduced by Deconinck and van der Weide in [DvW] after their scalar versions. We have -

System N scheme

-

. The choices of and N = K;) and N are unique thanks to the conservation constraint (5). System LDA (low diffusion advection) scheme

@?= K+(z, - 2)

(EL,

-1

2

In the definition of these schemes, the matrix N appears. It is the inverse of K,-. This matrix is not always invertible. However, for the Euler equations, one can show [AMN] that it is always invertible except at stagnation points. In any case, one can always give a meaning to K+Z or N a T because the Euler equations are symetrizable, see [AMN]. Hence, there is no problem in the definition of @?or @ f D A . Both schemes are clearly upwind. The LDA scheme is linear preserving contrarily to the N scheme. The N scheme is monotone. This is very obvious for its scalar version because in that case we have

c:=,

Invited speakers

7

>

with c z j 0. The matrix equivalent formulation involve terms like K:NK~T that are difficult t o handle. In all numerical experiments (with a large variety of geometries and flow conditions) seem to indicate that the system N scheme also satisfies a discrete version of (3). Barth [Ba] has shown that the N scheme, for a linear symetric hyperbolic system is locally dissipative. In [Ab], we show that the LDA scheme is also locally dissipative for a linear symetric hyperbolic system. 3. A positive linear preserving scheme for scalar equation : the PSI scheme revisited

We consider the scalar versions of the N and LDA schemes and the following blended scheme @i = 1@N (1 - l)@fDA.

+

The firt remark is that this scheme satisfies the conservation constraint ( 5 ) whatever I E R. This scheme is upwind by construction since the N and LDA schemes are upwind. We now consider the positivity issue with the same technique as D. Sidilkover, ai = I@:

+ (1- 1)@fDA= 1 + (1 - 1)@ ):;

to (6), the positivity is obtained if 1

( @yA'

+ (1 - 1)- @N

"

. Thanks

,

2 0 for i = 1 , 2 , 3 Ifwe set

'

q D A +

rz = , a solution is given by

@F

1 2 max cp(ri) r,,q'#O

where cp(r) = 0 if r

r 2 0 and cp(r) = else. Simple algebraic manipulations

r -1 shows that if 1 is chosen with the = sign in (7), the scheme is positive and linear preserving. In fact, it is identical to the PSI scheme. 4. A scheme for the Euler equations

Following the same ideas, we consider a scheme written as

where 1 is a matrix. In order t o illustrate the design principles, we consider 1 = 1 = Id where I E R, but a more sophisticated method is developped in IAbI.

8

Finite volumes for complex applications

Conditions TOP Bottom

p 1.4 0.7

p 1

0.25

Mach number 2.4 4

TAB.1: Conditions for the interaction of 2 parallel supersonic flows Let us denote by v = Vws. From

and thanks to the numerical experiments, we assume that the system N scheme has a local minimum principle for the specific entropy. Following the same arguments as for the scalar PSI scheme, the conditions are 1 + (1 - l)ri 2 0, for i = 1,. . . 3 TaLDA

where here ri = vi ' . One can show that this scheme is also locally dissivT@N pative [Ab]. 5. Numerical experiments

We present some results obtained in two different test cases. The first one is the iteraction of two parallel supersonic flows. The conditions are given in Table 1 The mesh is given on Figure 1-a The isovalues of the density are presented on Figure 1-b. A very clear improvement of the results can be observed. The new scheme give monotone results that are more accurate than those of the finite volume scheme (MUSCL extrapolation on conserved variables). The second test case is a GAMM test case : Naca0012, Mach number : 0.85, angle of incidence : 1 degree. We show the isolines of the Mach number (Figure 2-a) and the isoline of the reduced entropy (Figure 2-b). It is clear that the slip line out of the leading edge is improved as well as the entropy profiles. 6. Conclusions We have sketched the construction of upwind residual schemes that are also linear preserving. Some numerical example indicate that these new schemes are more accurate than the now classical finite volume schemes.

Invited speakers

9

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[Abl [AMN]

Pal [Dal

[SDRl

lTa1

IDv wl

R. ABGRALL.Upwind residual schemes on unstructured meshes, in preparation. R. ABGRALL, K. MER, A N D B. NKONGA. A Lax-Wendroff type theorem for residual schemes. In M. Hafez and 5.3. Chattot, editors, Proceeding of a conference for P.L. Roe's 60th birthsday. Wiley, to appear.. T.J. BARTH.Some working note on the n scheme. Private communication, 1996. S.F. DAVIS.A rotationaly based upwind difference scheme for the Euler equations. J. Comp. Phys., 56 :65-92, 1983. R . STRUIJS,H. DECONINCK, A N D P . L. ROE. Fluctuation splitting schemes for the 2d euler equations. VKI L S 1991-01, Computational Fluid Dynamics, 1991. E. TADMOR.The numerical viscosity of entropy stable schemes for systems of conservation laws. Math. Comp., 49 :91-103, 1987. E. VAN DER WEIDE A N D H. DECONINCK. Positive matrix distribution schemes for hyperbolic systems. In Computational Fluid Dynamics '96, pages 747-753. Wiley, 1996.

(a)

FIG. 1: (a)-Mesh for the interaction of 2 parallel supersonic flows ; (b)- Density isolines for the supersonic flows. Top-left : N scheme p E [0.7,1.4], top-right : second order MUSCL finite volume scheme p E [0.689,1.403] , bottom-left : LDA scheme p E [0.615,1.427] ,bottom-right : present scheme p E [0.698,1.402]

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Finite volumes for complex applications

FIG.2: (a)-Mach number isolines for the Naca0012 test case. Top-left : N scheme M t [0.05,1.394] , top-right : second order MUSCL finite volume scheme M E [0.05,1.422], bottom-left : LDA scheme M E [0.05,1.425], bottom-right : present scheme M E [0.04,1.522]; (b)-Reduced entropy C = V N scheme C E [0,0.038],second order MUSCL finite volume scheme [-0.009,0.039], LDA scheme C E [-0.009,0.091], present scheme X t [-0.0003,0.032]

5-

A Roe-type Linearization for the Euler Equations for Weakly Ionized Gases

Fre'de'ric C O Q UEL L A N C N R S Tour 55-65 5ieme, U.P. M. C. 75252 Paris Cedex 05 Claude M A R M I G N O N O N E R A , BP 72 92322 Chitillon Cedex

This paper is devoted to the numerical approximation of the discontinuous solutions of the Euler equations for weakly ionized mixtures of reacting gases. The main dificulty stems from the non conservative formulation of these equations due to a widely used simplifying assumption. W e show how to derive a well-posed conservative reformulation of the equations from the analysis of the associated convective-diffusive system. We then propose an exact Roe-type linearization for the equivalent system of conservation laws. Our results can be seen as an extension of the classical Roe average, for nonlinearities that cannot be recast under quadratic form.

ABSTRACT

Iiey Words: Convective-diffusive systems. Nonlinear hyperbolic systems. Non conservative products. Shock solutions. Roe-type linearization.

1. Introduction

This work treats the numerical approximation of the solutions of a convectivediffusive system, we write for short as

~ , u + A ( ~ ) ~ , u - ~ , ( D ( ~ ) ~ , u ) ~=>~O( , uZ )E , R .

(1)

This system governs ionized mixtures of reacting gases in thermal nonequilibriurn. Such plasma are studied here in the context of large Mach number flows.

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Finite volumes for complex applications

The solutions we are interested in, are thus mainly driven by the underlying first order system. The main properties of the extracted first order system

are reported below. This nonlinear system will be seen to be hyperbolic so that its solutions are known to develop, generally speaking, discontinuities in a finite time. But when dealing with discontinuous solutions of (2), a major difficulty arises : there does not exist a flux function, say f , such that A(u) = ~ , f ( u ) . In other words, the hyperbolic system (2) is under non conservation form. It is known that the non conservative products involved in A(u)d,u have no classical sense at the location of a shock since they cannot be given a unique definition within the standard framework of distributions. For this reason, it must be recognized that an additional information is required in order to specify the definition, e.g. the value, of the non conservative product d(u)d,u at shocks. This difficulty has motivated some recent works. We refer in particular to the work by LeFloch [8], DalMaso-LeFloch-Murat [6] where non conservative products are defined on the basis of a fixed family of paths in the phase space : atu [ A ( ~ ) d , u =]0, ~ t > 0, x E R. (3) After LeFloch [8] and Sainsaulieu [ll],the choice of a particular family of paths is dictated by the additional informations brought by the full second order convective-diffusive system (1) (see below for a brief survey). The key feature is that the definition of shock solutions heavily depends on the shape of the diffusive tensor 2) (u) which is modeled in agreement with the physics. These definitions provide us with a relevant setting for defining the discontinuous solutions of the non conservative hyperbolic system (2). Once defined, the first order system is well-posed and its numerical approximation could be tackled. However, two difficulties arise in that way. First, a close formula for shock solutions is in general not available. Furthermore, even when explicitely available, we have illustrated [4] that the error in the discrete capture of shock solutions unacceptably grows with the strenght of the shock. We refer to LeFloch-Liu [9] for an error analysis devoted to the scalar case. At this stage, these two difficulties make the numerical approximation of the (strongly) discontinous solutions of (3) to be virtually untractable. To overcome these two difficulties on the same time, we propose to study the existence of a conservative formulation for system (3) that is compatible with the diffusive tensor 2). That is to say, we ask for the existence of (at least) one change of variables v = v ( u )that brings the non conservative second order

+

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system (1) (with w (u) = 0) to a fully conservative convective-diffusive system

Let us emphasize that not only the first order system must find a conservation form ( P l ) but also that the second order operator must stay under conservation form (P2). Actually, these two requirements (Pl),(P2) ensure the equivalence and V are compatible and the shock of the shock solutions of (3) where solutions of ~ , F ( v )= 0 , t > 0, x E R . (5)

atv +

We refer t o Sainsaulieu-Raviart [12] for a proof. The benefit of such an equivalence is twofold. In a first hand, the shock solutions of (3) are now explicitely given by the Rankine-Hugoniot jump relations associated with (5). In a second hand, Riemann solvers under conservation form can be applied to (5) in order to approximate the equivalent weak solutions of (3). As reported below, a specific family of change of variables turns out to fulfill both (PI) and (P2). For the associated equivalent systenls of conservation laws, we then show how to derive an exact Roe-type linearization.

2. Analysis of the extracted first order system

In this section, we focus ourselves on the definition of the extracted first order system (2), the precise shape of the diffusive tensor V will be discussed later on. We treat mixture of gases made of electrons and n heavy species, ni, 1 5 ni n of them being ionized. All the species we consider are described with the same mean velocity 11. To account for the smallness of the mass ratios M e / M i << 1, i E (1, ..., n ) , the electron gas is endowed with a temperature Te distinct from the temperature T of the heavy species mixture. Moreover, n u , 1 5 n u 5 n , molecular species have their own vibrational temperature T,,j, j E (1, ..., n u ) . Neglecting a t this stage the thermo-chemical relaxation terms, the extracted first order system (2) writes : dtpY, +a,pY,v = 0, 1 i n , (6a)

<

< <

+

d t ~ u 8, ( p v 2

dtpYeEe -k atpYje,,j

+ P + pe) = 0,

ax ( p y e E e + p e ) -~ N e q e f v = 0 , + a s p Y j e , jv = 0, 1 5 j 5 n u .

(66)

(64 (Be)

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Finite volumes for complex applications

The main properties of (6) stated in the next section will enlighten several relationships with the usual Euler system. Here, Y , , i E ( 1 , ..., n) denotes the ith species mass fraction. The conservation laws (6b),(6c) respectively govern the momentum and total energy E of the mixture of the heavy species plus the electron gas. Conservation laws (6e) refer to the vibrational energies pYjeWf of the nu molecular species that are assumed t o be in thermal nonequilibrium. These ( n nu 2) conservation laws are supplemented by a balance equation (6d) : the expected conservation law for the electron gas energy pY,E, must be, in fact, balanced by the work of the electric field E. The required closure equations are as follow. The electron gas pressure p, obeys : 1 P (7- E - 1 ~ e ~ u 2 ) ye , E 11,31! (7)

+ +

while the pressure law p for the mixture of heavy species is defined by :

with yt, E ]1,3]. Here, e;(T) refers to the energy of the internal modes of species i a t equilibrium with the temperature T and hr denotes its heat of formation. The last closure equation t o be specified deals with the electric field E. The expected closure equation for E should be the Poisson law, writ,ten here in a dimensionless form :

In (9), E refers to a parameter proportional to the Debye lenght. This parameter actually yields a rough estimate of the spatial resolution that is required to approximate (9) : namely U ( E - 1 ) . However in our applications, this parameter turns out to be extremely small (see [14] for instance) To make the numerical approximation tractable, we are lead to let E goes to zero. Doing so, the Poisson law degenerates t o the so-called local charge neutrality condition while the closure equation for E is classically given by :

Here, zi refers t o the electric charge of the ith ionized species. The stricking feature of the closure equation (10) stays in that it involves partial derivatives

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and in particular makes E to be, generally speaking, a measure at the discontinuities of u ! The work of the electric field NeqeE x v in (Gd) is clearly a product involving first order and zero order terms and furthermore, it cannot be put under divergence form. Let us stress that the product NeqeE x v cannot be understood, as it is commonly handled (see [2] for instance), as a source term, e.g. a function depending solely on zero order terms. On the contrary, such a product is a constitutive part of the first order system.

2.1. Basic properties of the extracted first order system

In this section, we state some of the main properties of the first order system (6). The results given below intend to provide a deeper insight into the system under consideration and in particular to shed light on its relationships with the standard 3 x 3 Euler equations. The phase space Q associated with (6) is the following subset of RP, p = n n u 3 :

+ +

We begin with the following result devoted to the smooth solutions of (6). Lemma 1. Let u : R x R + + fl be a C1 solution of (6). Then, the closure equation on the electric field E reads :

Furthermore, u satisfies the following balance equations in non conservation form : dtYi vd,Y, = 0, 1 5 i 5 n - 1, (13~)

+

Dropping the electron gas pressure p, in (13c), the three equations (13b) to (13d) are easily recognized to coincide with the ones governing the smooth solutions of the 3 x 3 Euler system. Here, the heavy species mixture and the electron gas equally contribute to the velocity balance equation : each with

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Finite volumes for complex applications

their own pressure. The pressure balance equations (13d) and (13c) read the same but the key point stays in their complete decoupling. We indeed have : P r o p o s i t i o n 2. z) The first order system (6) is hyperbolic in the phase space R and A ( u ) : R + M ~ ~ ( R admits P) the following three real eigenvalues : v - c, v , v

+ c,

c2 = ( y p

+ yepe)lp,

(14)

+ +

where the eigenvalue v has an order of multiplicity ( n n v 1). i i ) Under standard thermodynamic assumptions, the fields associated with the eigenualues v f c are genuinely nonlinear. iii) The fields associated with the eigenvalue u are linearly degenerate. Their Riemann invariants are given by v and ( P ~ e ) . iv) The closure equation (12) for N,q,E never involves a product of a Dirac mass against an Heaviside function and is thus well-defined. v) The non conservative product NeqeE x v has no classical sense for the shock solutions of (6). According to the statement v), the shock solutions of (6) stay unknown within the standard framework of distributions. The nonconservative products involved in A(u)d,u can nevertheless receive a definition in the recent setting proposed in [6]. Indeed, such products can be defined thanks to a given fixed family of paths, denoted @, that are subject to some consistency and smoothness conditions (see [6] for the details). For a shock discontinuity separating , non conservative product is defined by : the two states UL and u ~the

+

Note that since A is not a jacobian matrix, the definition (16) entirely depends on the choice of a. After LeFloch [8] and Sainsaulieu [ l l ] , the relevant choice of @ comes from the study of the full second order convective diffusive system (1). Indeed, the shock solutions of (6) can be defined as the limit when the diffusion is neglected of the travelling waves solutions of (1). More precisely, a function u : R x R+ + Q is a travelling wave solution of (1) connecting the states UL and UR with speed a , if u is a bounded function in the form u ( x , t ) = u ( x - at) with d
lim ii(<) = U L ,

lim u(<) = UR.

5++m

<'-w

Next let be given a small parameter

6

>0

(17b)

and let us consider the function

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( 7 ) . It can be easily seen that this function is also a travelling wave solution but of the convective-diffusive system (1) with vanishing viscosity : -ad
+ A(ii,)d[u,

lim uc(() = U L ,

- ~ d ~ ( V ( i i , ) d ~ i= i ,0, )

lim u,(() = U R . E-t+m

(18~)~ (18b),

Furthermore, since d E i i , E L1(R) with IJdEU,IIL~ = JJdc~JIL wel , deduce that. {u,),,~ tends almost everywhere as E goes to zero to the step function u ::= U L + (UR-uL) H ( 0 ) while for any given 6 > 0, the following jump like conditions hold true :

Motivated by the above calculations, LeFloch [8] (see also Sainsaulieu [ I l l ) has defined the discontinuous step function u to be a s h o c k s o l u t i o n of the first order extracted system (2) which is c o m p a t i b l e w i t h t h e diffusive t e n s o r

D. It is essential to notice that by contrast with conservative hyperbolic systems, the travelling wave solutions and therefore the step function u depend on the shape of V. Indeed, two non proportional diffusive tensor generally yield distinct shock solutions for the underlying convective system. Such an issue is actually illustrated by the Proposition 8 stated below. The first order extracted system (6) can be closed according to the above framework and its numerical approximation could be tackled. However and as explained in the Introduction, two difficulties arise when dealing with the numerical approximation of the discontinuous solutions of (6). In order to circumvent these two difficulties simultaneously, we have put forward the need for admissible change of varaibles that recast (6) in the full conservation form (4). We indeed have : P r o p o s i t i o n 5 (Sainsaulieu-Raviart [12]). Let be given a C1 diffeomorphism Q : R + R. Assume that for any given smooth solution u : R x R+ + R of the non conservative convective diffusive system :

the change of variables v = Q(u) yields a solution of the following SLC :

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Finite volumes for complex applications

Then, the shock solutions of the extracted first order system of ( Z l ) , e.g. the ones compatible with 'D, do coincide with the shock solutions of the extracted conservative first order system of (22) :

T h e benefit of this equivalence is twofold : first the shock wave solutions to ( 6 ) are now explicitely given by the Rankine-Hugoniot relations associated with

a t v + a , r ( ~ )= o.

e.g. - c ( v ( u ~- v)( u L ) )

+

- F ( ~ ( ~ ~= )O. ) ) (24)

and in a second hand, upwind methods in conservation form can be applied t,o ( 2 4 ) in order to approximate the equivalent weak solutions of ( 6 ) that are compatible with V . Motivated by these strong benefits, we have laid in a related work [3] the foundations for a systematic characterization of all the C1diffeomorphisms Q that bring ( 2 2 ) from ( 2 1 ) . The starting point stems from the following observation : the equivalence stated in (23) requires in fact the fulfilment of two distinct conditions. Namely, not only the first order system must meet a conservation form ( P I ) but also that the second order operator must stay under conservation form ( P 2 ) . In what follows, we study for existence the adinissible changes of variables Q that satisfy both requirement (PI) and

(P'L).

In order to exhibit the C1-diffe~mor~hisms that satisfy the requirement ( P I ) , we propose t o characterize in a first step all the additional scalar conservation laws satisfied by the smooth solutions of ( 6 ) . Each additional conservation law indeed yields an obvious change of variables by substitution with the single equation in non conservation form (6d). We specifically have : Theorem 6 Let u : R x R+ -+ R be a C1 solutzon of (6). Then u satzsfies all the addztzonal (non trzzizal) scalar conservation laws :

where g : Q -+ R denotes an arbitrary (say of class C1) function of the common Rzernann invarzants of the two genuinely nonlinear fields : e.g.

a # 0, then the C1 d i f S e o r n o r p h i Q g : R + R , u -+ Assume that ap,igpyc Q s ( u )= ( { p ~ } pv, , p E , p g ( u ) , p ~ j e , , ~yields ) the smooth solutions of (6)

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to be equivalent smooth solutions of the following hyperbolic system in conservation form : d , p ~ , v = 0, 1 5 i 5 n , (27~)

atpx+ dtpv + a x (pv2 + + pe) = 0, JJ

+ dxpg (u)v = 0 ,

at pg ( u )

dtpYje,

+ d X p Y j e u j j v= 0,

1<j < nv.

(27b)

(274 (27e)

The above family of C1-diffe~mor~hisms that satisfy ( P I ) for the smooth solutions of (6), might look rather large. However, the next statement shows that all these ~ o s s i b l ecandidates for the fulfilment of both ( P I ) and ( P 2 ) are indeed closely related. Corollary 7. i ) All the additional conservation laws specified in (26) can be recovered from the following finite set of transport equations (since d t g ( u ) va,g(u) = 0) dtyi+vdxyi=O, l
+

+

d t l n ( p l p Y ) v d X l n ( p l p Y )= 0,

(28b)

zi) The functions g : R + R are Raemann invariants for the two genuinely nonlinear fields and thus stay constant across the associated rarefaction waves. iii) Assume that one of the change of variables Qg specified i n (28) meets the requirement (P2). Then, necessarily the related function g ( u )stays continuous for (so constant across) the shock solutions of (6) that are compatible with D. The consequences of properties i ) to iii) are briefly discussed in the next section since they will serve as a natural guideline for investigating for validity ( P 2 ) with all the possible candidates Q g . If one of them turns out to satisfy ( P 2 ) , the resulting conservative system (28) and the extracted first order system (6) will not only describe the same smooth solutions but will also admit the same discontinuous solutions : namely, (24) will hold true. This issue is addressed below. 2.2.3. Fulfilling ( P 2 ) . The existence of a C1-diff'e~mor~hism that obeys ( P 2 ) obviously heavily depends on the shape of V . In view of iii), Corollary 7 , its shape must be compatible with the conservation across the shock-solutions characterized by

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Finite volumes for complex applications

V of a least one function g ( u )in the form (26). According to the Definition 4, the characterization i), Corollary 7, leads us to derive the travelling wave equations that govern l n ( p / p y ) , l n ( p , / p Y c ) , % e V j ) in the presence of I? and to study the nonlinear combinations in the form (26) that yield a fully conservative equation. The analysis [4] shows that none of the Qg given in (27) can achieve such a requirement for a general diffusive tensor 2). However, this negative result can be bypassed provided that a physical assumption is made. This is the matter of the next statement. Proposition 8 . i) Let the diffusive tensor V (u) be defined by

(x,

where u, K respectively denote the viscosity and the thermal conductivity of the heavy species mixture, K , the thermal conductivity of the electron gas and ~ , , j the one of the j t h molecular species in vibrational nonequilibrium. Then, the associated travelling wave u satisfies for all 4 E R, with M = p(<)( v ( 0 - u ) = PL(VL - 8) = P R ( ~ R - a):

Furthermore, for arbitrary (v, K , K , , n V , j ) , none of the C 1 - d i f f e ~ r m o ~ h i s m s specified in (27) does satisfy the property (P2). ii) Assume that K , = 0, e.g. that the diffusive tensor V ( u )is now given by :

x)

Then, all the C1 diffeomorphisms Qg based on g ( u )= ( p e / p ~ e , do satisfy P 2 ) . . . According to the property iii) given in Corollary 7, the above statement ii) implies that the specific entropy p,/pYe for the electron gas stays constant accross the shock waves that are compatible with (30). We underline that such a consequence is indeed compatible with the thermodynamic second principle : the heavy species specific entropy p/pY indeed strictly increases accross shock waves (see also ii) below). The physical interpretation of these admissible C1diffeomorphisms can be found in Zeldovich-Raizer [14] (vol. 2). Proposition 8 provides us with a whole family of Qg that fulfill both ( P I ) and (P2).

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3. A Roe-type linearization

In this section, we propose a Roe method for the conservative hyperbolic system (33) in two space-variables. Since these equations stay invariant under rotations, the scheme makes use of the classical dimensional splitting and is thus derived for the associated quasi-1D system, we write with clear notations :

In what follows, we show how to derive an exact Roe-type linearization on the basis of an original Lemma for averagings. This Lemma will turn out to be central in the handling of the highly nonlinear pressure law p e . We just recall that a Roe-type linearization [lo], we denote B ( v L ,v R ) , has to satisfy the following three requirements: B ( v L ,v R ) is diagonalizable with real eigenvalues,

(37~)

In (37), the jacobian matrix V,T(V): C2 + at (RP) of the exact flux can be expressed in terms of the following set of arguments :

The precise shape of V v F ( v ) is given in a companion paper [4]. Motivated by recent works [I], [13] devoted to mixtures of neutral gases, e.g. without ionization effects ; we seek for a linearization considering the following averaged form for v V 3 ( v ) :

We have to define the set of averages involved in (39) so that the three conditions (37a)-(37c) hold true. Besides the hyperbolicity condition (37a), (37b) requires that such averages have to be derived so that the jump A F ( v ) , which is of course nonlinear in v , can be reconstructed from the linearized form (39) of V v 3 ( v ) times the jump A v . It is worthy to recall that for the 3 x 3 Euler system, the classical Roe average (see below) basically stems from the existence of a set of variables, the so-called parameter vector [lo], for which all the

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Finite volumes for complex applications

nonlinearities involved in the system can be recast under quadratic form. Furthermore, within this framework, this Roe average turns out to be the unique average that yields (37a)-(37c). In our setting, some of the scalar conservation laws involved in (35)-(36) actually exhibit the same type of quadratic nonlinearities. We are thus lead to endow the related variables with the Roe averaging procedure : namely, we choose

(40) But the pressure laws we have to deal with, bring a different type of nonlinearities. Averaging their partial derivatives indeed requires a specific treatment such that, besides (37a), the following consistency conditions hold true :

In fact, within the framework of neutral gases, V v p has been already given suitable averaged forms. We refer in particular to the works by Abgrall [I] and Shuen-Liou-VanLeer [13]. When ionized species occur, we have shown [4] that the first identity in (41) can be derived from the framework of neutral gases provided that the second identity is valid. Our Roe-type linearization tacitly makes use of one of these relevant averages for V v p . From now on, we focus our attention on the derivation of an averaged form for V v p e which is consistent with the three requirements (37a)-(37b). In that aim, we state the following first result : Proposition 11. i ) The averaged gradient V v p e can be recontructed from a set of three averaged partial derivatives :

-

& ap'={ ! ! 5 6 i o n ) x -+{I+-4 apx

where

Mi

sfon

G } x --,

~ e ; , ,

i ~ { ..., l ,n )

and

G

-,

8% (42) = 1 if the ith species is ionized and 0 otherwise. Next, assume that

a~ye

Mi

8P

-

( H ) : B ( v ~ v, R ) (39) is associated with

and V v p derived from

{E, 8, 1,p, 2,~,e,,)in (40) P

[I] or [13]. Then, assumption (H) yields : ii) the Roe consistency condition (37b) is met iff for any given ( v L ,v R ) E R 2

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iai) Whatever are these three averages, B ( v L , v R ) admzts the set of eigenvalues (6

-

fi,6 , G +

a).

that for any given true :

(7-

(VL,

Moreover, the real number b stays positive provided V R ) E R 2 , the following, where (y , ye) E ] 1,312,holds

&

% -

YePe 1) + ( y e - ? ) { ~ a+-+"" ~ (re - 1 ) ~ P Y ~ dp

P

-

*) dse

>

0.

(45)

In order to enforce the validity of the two conditions (44) and (45), we now consider the following easy but key lemma for averagings (see also Abgrall [I], Godlewski-Raviart [15]). A v e r a g i n g L e m m a . Let be given ( x L , xR), ( y L , yR) two pairs of real numbers. Let ( T L , T R ) be a n y g i v e n pair of real numbers such that TL TR # 0 . Let U S define the following unsymmetric T-averaging operators :

+

Then, the following identity holds true :

Let us apply for instance the above averaging Lemma to the pairs of interest ( p , X) with the @-averagingwe define by e L = f i l QR = & ,. We easily get from (46) and (47) the well-known Roe identity, we write with classical notations :

Q2 X

X,

fie p =p=

JPLPR

with ApX =

+ -pAX.

(48)

Turning back to the general case, we emphasize that the identity (47) is indeed valid for a n y g i v e n pair (rL,rR). Taking advantage of such a degree of freedom, we introduce below a (wide !) family of unsymmetric averagings that makes always valid the required consistency condition (44). Equipped with these families, we next turn studying how to enforce the hyperbolicity condition (45). To that purpose, the specification of the underlying averaging operator is clearly the central issue. The statements, given below, summarize our main results. They are intended to shed some light on the application of the averaging Lemma we have introduced in order to enforce the validity of both (44) and (45).

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Finite volumes for complex applications

3.1. Enforcing the Roe consistency condition (44).

In view of the electron gas pressure law (34), the required discrete identity (44) writes :

The above consistency condition involves increments in the conservative variables pYe, p and se while the underlying nonlinearities merely occur in terms of the primitive variables Ye,p and s e / p . To make the needed linearization tractable, we specify in the next statement a convenient family of averages for (42) that shifts (44) from the conservative to the primitive variables. Proposition 12. Let us consider the following averaged forms for the partial derivatives (42) of the electron gas pressure law :

dpe 8P

-

a s s S + E x {yepye-1h(2) - A x p ~ e - l h ' ( ~ ) )(50b) , P P P

-

0

Se dpe = Ye x pye-lhl(-). -

ase

P

Then, for any given pair of states (vL,vR), the consistency condition (44) is p is defined i n (48) : equivalent to the following identity, where -

In order to satisfy (51) and thus (44), it remains to define the unspecified mean values in (50). This is the matter of the next statement. Proposition 13. W i t h the notations of the Averaging Lemma, let us define the averaged forms i n (50), using respectively an arbitrary u-averaging operator :

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and an arbitrary C-averaging operator : C 't

h ( F ) x APrelAp,

if A p

#0, (53~)

C 't

h ( $ ) x yepre-', x

o p~e-lhl(:)

=

otherwise,

'C x A h ( ? ) / ~ h pr.

P

P

i f A$

# 0,

'

(53b) otherwise.

Then, the identity (51) and thus the Roe consistency conditions (37'b)-(37'~) are always valid. In what follows, we shall assume for convenience that both a and ( stay non negative. Equipped with this wide family of relevant averagings, we now turn studying for validity the hyperbolicity condition ( 4 5 ) .

3.2. Enforcing the hyperbolicity condition (45)

The main result of this second step is as follows : Proposition 14. Let us respectively define the a and ( averaging operators i n (521, (53) by

= ( 0 , l ) , if Ye L

< Ye R ,

CL=C(PL), CR=C(PR)

with

((TL, UR)

( 1 , O), otherwise;

(54)

C(p)=pYe-3.

(55)

Then, (54), (55) provide us with the unique pair of averaging operators such that the hyperbolicity condition (45) holds true for any given ( v L ,v R ) E f12. We have the following final statement. Theorem 15. Let us consider the following averaged form of V V 3 ( v ):

{g,

with 5,w,H , s e l f , % e U f } given in (do), with VTPderived from [1] o r [13] and with V v p , constructed from (do), (50), (52)-(53). ) three real Then, for any given pair of states ( v L ,V R ) E R 2 , B ( V L , v ~ admits eigenvalues with a complete set of right eigenvectors. Moreover, this matrix satisfies the Roe consistency conditions (37b) and (37c). Therefore, B ( V L , V R )

-

26

Finite volumes for complex applications

is an exact Roe-type linearization for the system (35)-(36). We emphasize that the averaging operator built on (55) provides one with an extension of the classical Roe-average (48). Indeed, let us compare two of the nonlinearities that arise in the pressure laws : namely pYe x { h ( y ) ) with

(7)').

p x {f Now, it is sufficient to apply to the latter nonlinear expression our formula (55) with the associated exponent a = 1 in place of ye to get &i.

111

R. ABGRALL, La Recherche Agrospatiale, vol 6, 31-43, 1988.

[2]

G . CANDLER 88-0511, 1988.

[41

F . C O Q U E L A N D C . MARMIGNON, Work in preparation.

151

S. C O R D I E R

[61

G . DALMASO, P . G . LEFLOCH A N D F. MURAT, J . Math. Pure Appl., vol 74, 483-548, 1995.

[71

B. LARROUTUROU, Computational methods in applied

AND

ET

R . W . MACCORMACK, AIAA paper

al., Asymptotic Analysis, vol 10, 1995

sciences, ECCOMAS, Eds Ch. Hirsch, Elsevier, 1992. [81

P.G. LEFLOCH, IMA Preprint series No 593, University of Minnesota, 1989.

[9~

P . G . LEFLOCH

[lo]

P . L . R O E , J. Comp. Phys., 357-372, 1981.

11

AND

J. L I U , Math. of Comp., 1994.

L. SAINSEAULIEU, SIAM J . Appl. Anal., 1995.

[I2]

P . A . RAVIART A N D L. SAINSEAULIEU, Mathematical Methods and Models in Applied sciences, 1995.

[I31

J .S. SHUEN

[I41

YA. ZEL'DOVICH A N D Yu. P . RAIZER, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomen, vol 1 and 2, Academic Press, 1966.

[I51

E. GODLEWSKI A N D P. A. RAVIART, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer (Eds), Applied Mathematical Sciences, vol. 118, 1996.

ET

al., NASA TM 100856, 1988.

Multidimensional upwind residual distribution schemes and applications

H. Deconinck and G. Degrez von Karman Institute for Fluid Dynamics Sint-Genesius-Rode, Belgium

ABSTRACTA review is made of upwind residual distribution schemes(RDS)for hyperbolic systems for application to compressible and incompressibleflows. First, the design principles of RDS are explained for the simplest case of linear scalar advection. Extension to linear hyperbolic systems is described next. Then, their application to compressible and incompressible pows is discussed, and illustrative examples of applications arepresen fed. Key Words: jnite volume method, finite element method, compressible paws, incompressible po ws

1. Introduction

Multidimensional upwind residual' distribution schemes, which were first introduced by P. L. Roe [ROE 871, have been developed on ideas borrowed from both the finite volume and finite element methods to become nowadays an attractive alternative to either one [PA1 971. The initial motivation for their development was a discontent about some drawbacks of the state-of-the-art finite volume solvers based on 1-D approximate Riemann solvers, namely D by D first order upwinding is very diffusive,

1D Riemann solvers do not capture the real multidimensional flow physics, higher order schemes use wide stencils. The starting point for the development of these schemes was a reinterpretation of 1D finite volume schemes based on the concept of fluctuation [ROE 821. Considering the continuous piecewise linear data representation classically used in finite element methods rather than the discontinuous cell-wise reconstructions used in finite

28

Finite volumes for complex applications

volume methods, the flux difference between two nodes (Fi+, - Fi) is reinterpreted as the flux balance (else fluctuation or residual) over the interval (element) [ i ,i + 1 ] (= J, d F / d x dx), which is to be distributed to the vertices of the element. This interpretation can be carried over to several dimensions, provided the fluctuation is now the flux balance over a simplex (trianglettetrahedron) to which is associated a continuous piecewise linear data representation. Because the elementary discretisation unit is an element rather than an edge or face for the finite volume method, the multidimensionality of the problem can genuinely be taken in to account: in 2D, the fluctuation can be distributed to the three vertices of the triangle rather than to the two neighbouring cells of an edge. Based on this idea, schemes were constructed, which combine a number of attractive features: a much lower cross-diffusion than their finite volume counterparts, due to the genuinely multidimensional upwinding they incorporate, a positivity property which ensures the satisfaction of a discrete maximum principle and consequently the absence of spurious oscillations, (almost) second order accuracy on compact stencils. In addition, the compact discretisation stencil allows for the development of efficient implicit iterative solution strategies [ISS 961 and for an easy parallelisation [ISS 98, vdW 991. The paper starts with the presentation of upwind residual distribution schemes for linear hyperbolic scalar equations and systems, successively. This is followed by a discussion on their application to non-linear problems, specifically to the compressible and incompressible flow equations. Finally, the paper concludes with a few illustrative computational examples. 2. Linear equations 2.1. Scalar advection

2.1.1. Design principles The residual distribution schemes (RDS) have been designed for an optimal discretization of the steady state convection equation

on PI finite element meshes, i.e. triangular (resp. tetrahedral) meshes with a piecewise linear solution representation. Evaluating the residual or fluctuation over an element

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29

Figure 1: Triangular element with scaled inward normals

T, defined as the integral over this element of the differential operator, i.e.

d

= C , k i u i ,where since both A and V u are constant over the element, one obtains + ki = A . Ziis called the inflow parameter, Ei being the scaled inward normal of the edge opposed to node i (see Fig. 1) and d the dimension of the problem. The method consists in distributingfractions of 4T to the vertices of the element. The resultingdisCrete equations therefore express that the nodal residuals Ri,sum of all contributions from neighbouring elements, vanish, i.e.

3

=Z@T=CpiT@T =O

Ri

T

(3)

T

in which 3'j are the distribution coefficients. On each element T, these distribution coefficients must sum up to one for consistency and conservativity. The different schemes, corresponding to different ways of computing the distribution coefficients, have been designed to satisfy several properties making them optimal: UPWIND CHARACTER (q): No fraction of the element residual is sent to upstream

nodes or pf = 0 when

ki5 0.

POSITIVITY (9): The scheme does not create local extrema or, if we write the con-

tribution to the element residual as

0 'v'j#i.

4:

= p:4T

= C j c i j u j ,we impose cij

<

30

Finite volumes for complex applications

Scheme N LDA SUPG

PSI

Table 1: Residual distribution schemes Pi1 type % k + ( @ ' CJ . kI : ) - ' ~ ~ k j ( . u ; - u ~ ) linear J (Cjkf)-'k: linear J

+

1 h

.r=--

llXll max(0, ~ N ) / ~ ~ m a xPN) (o, 7

- .

2

9 29 J

J d

linear non-linear

J

J

J

LINEARITY PRESERVATION (99): The distribution coefficients PT remain bound-

ed as # T + 0 which is equivalent to zero cross-diffusion (second order accuracy) on a regular mesh. CONTINUITY (g): The distributioncoefficients must be continuous functions of both

the advection and the solution-gradient directions. Several schemes satisfying some or all of these design criteria have been developed both in 2D and 3D. A complete summary is given in [PA1 971. The schemes used in the present study are summarized in Table 1 (formulas valid both for 2D and 3D), where k' = max(O,ki), k:, = min(0,k;). Note that as a result of a generalization of Godunov's theorem, only non-linear schemes can satisfy both the 9 and 29 properties.

2.1.2.

Finite element interpretation

Residual distribution schemes can be viewed as generalised Petrov-Galerkin finite the weighting function associated to node i element schemes. Indeed, denoting of element T, the Petrov-Galerkin discretisation of the convection equation (1) can be written F / ~ C O , % V ~= ~o .R (4)

COT

Now, from the definition of #T and from the linear approximation of u over T, as+ surning a constant advection vector A,A . Vu is constant over T and equal to # T / ~ T where nTis the surface of the triangle T (volume of the tetrahedron T in 3D). Hence, the previous equation becomes d

For the Petrov-Galerkin discretisation to be identical to the residual distribution discretisation (3), the weighting functions should satisfy the following relation :

COT

,

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31

This equation is not sufficient to uniquely associate a weighting function w? to a residual distribution coefficient PiT unless the functional form of the weighting function is prescribed and depends on a single parameter. Prescribing the weighting function to be the sum of the finite element shape function cpi and a piecewise constant contribution similar to SUPG weighting functions [DEM 971, i. e. COT= cpi + aTf, where a? is the upwind bias coefficient and is the piecewise constant function equal to 1 on element T and 0 elsewhere, the equivalence condition (6) yields

2.2.

Hyperbolic systems

Considering now the non-commuting linear hyperbolic system

the formal extension of the residual distribution discretization (3) is

xi

where the fluctuation @T is now a vector whose expression is aT= KiUi with Ki being now an inflow matrix expressed as Ki = Atni,e,and where is a distribution matrix. The design criteria for distribution matrices are the same as for distributioncoefficients for the scalar advection equation, plus the additional requirement of invariance under a similarity transformation, i. e.

PT

PT

UPWIND CHARACTER: = 0 when Ki 5 0 (where Ki 5 0 means that all eigenvalues of Ki, which are known to be real due to the hyperbolic nature of the

system, are negative). POSITIVITY: With

@T

=

pT@T = C j C i j U j ,the positivity condition becomes C i j 5

0 'dj#i. LINEARITY PRESERVATION: The distribution matrices PT remain bounded as aT--t

0. CONTINUITY

32

Finite volumes for complex applications

INVARIANCE: Under the similarity transformation dU = TdQ, the hyperbolic sys-

. discretization tem (8) transforms into A Q & 2= 0 where AQr = T - ' A ~ T The of the transformed equation being

we require that

pii = T-'P:T

It results that if the system is diagonalisable, i. e. if the matrices A, commute, then taking T as the diagonalising transformation, the transformed distribution is the diagonal matrix of scalar distribution coefficients for each matrix decoupled equation.

Pii

Many of the schemes listed in Table 1 can be formally extended to systems. For example, the system-N scheme is defined by

whereas the system-LDA scheme is defined by

The non-linear PSI scheme on the other hand proves to be more difficult to generalize. Considering the scalar PSI scheme as a limited N-scheme, its formal generalization to systems is

However, the distribution matrix of the N-scheme pT)N is not explicitly defined by Eqn. 11. For diagonalizable systems, the condition of invariance under similarity S I For general nontransformations suffices to define pT>Nand hence P ~ ~ P uniquely. commuting systems, the invariance condition is no longer sufficient. The system PSI scheme used in the numerical applications is based on one particular definition of PF)N which satisfies this condition. Further details are given in [PA1 971. The construction of non-linear positive and linearity preserving schemes for systems is still an ongoing research topic. New developments, based on the reinterpretation of the scalar PSI scheme as a blended NILDA scheme are discussed in R. Abgral1's invited paper in this conference [ABG 991.

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3. Compressible flows 3.1.

Conservative linearisation

The residual distribution schemes presented in the previous section have been defined for linear problems. To apply them to non-linear problems such as the Euler equations, an essential ingredient is a conservative linearization, which consists in finding on each triangle T an average state U such that

where

au

-

-

I

=

ax, a,

dn,

Un,dr

so that, locally, the non-linear system of conservation laws dF,/ax, = 0 is approximated by the linear system A e d ~ / d x = e 0. Let's show that, with such a linearization, conservation is indeed satisfied. Summing up the nodal residuals over the domain 0, we have

and the contributions of internal edges cancel out (telescoping property). For the Euler equations, a conservative linearization is easily obtained as a multidimensional extension of Roe's linearization [DEC 931. Indeed, assuming a linear variation of Roe's parameter vector Z [ROE 811 and since both the vector of conserved variables U and the fluxes Fb are homogeneous functions of degree 2 in the components of Z,

where Z =

3.2.

& CverticesZiis the proper average state to ensure conservation.

Transformations and preconditioning

The matrix distribution schemes presented above are invariant under a similarity transformation. It is nevertheless useful to apply a similarity transformation to the lin-

34

Finite volumes for complex applications

earized hyperbolic system in order to achieve maximum decoupling for the following reasons. On one hand, this reduces the computational cost thanks to the considerable simplification of the distribution expressions and on the other hand, this allows to select different distribution schemes on the various decoupled systems/scalar equations. Specifically, with the similarity transformation dU = T a Q , the discrete equations (9) are rewritten T PQiQa T T =o R~= (18)

CT T

where 0;= C i K Q i Q i . Partial decoupling can be achieved if the flux jacobians Al have common eigenvectors. For the Euler equations, the flux jacobians At have one common eigenvector so that it is possible to decouple one scalar equation from the original system, leaving a coupled 3 x 3 system and one decoupled scalar equation du,dpin 2D. The transformation to symmetrizing variables defined as d Q =

(s,

a2dp)' accomplishes this task. The decoupled scalar equation is nothing else than the entropy advection equation, which is well-known to derive from the Euler equations. As shown in [PA1 971, additional decoupling may be achieved by preconditioning, namely the system of equations is rewritten as1

and the residual distribution method is applied to the preconditioned system between brackets. The optimal preconditioning was found to be the van Leer-Lee-Roe preconditioning [vLE 911, which allows to decouple one additional equation, namely the total enthalpy advection equation.

3.3.

Viscous terms

Viscous terms are discretised using the finite element interpretation of the schemes presented in section 2.1.2.. Specifically, the space discretisation of the divergence of the stress tensor V . ? is

Now, since rn; differs from the PI shape function qi only by a piecewise constant, VoT = Vqi, so that, for interior points, the discretisation reduces to a Galerkin finite element discretisation. 'The original conservative variables are used here, but the same procedure can be (and actually is in practice) done for the symmetrizing variables.

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35

4. Incompressible flows For application to incompressible flows, two avenues have been explored. The first approach, followed by Waterson & Deconinck [WAT97], is based on the fact that the incompressible Navier-Stokes equations are a symmetric advective-diffusive system [DEM 971, which can therefore be discretised using the system distribution schemes presented in section 2.2. using a collocated P1 finite element representation for all variables. The collocated arrangement is made possible by the inherent pressure stabilisation effect of the system discretisation. The second approach, followed by Bogaerts et al. [BOG 981, is based on a stable finite element representation of the velocity and pressure fields (PlisoP21Pl element) which therefore does not require any pressure stabilisation. It results that scalar residual distribution schemes can be used to discretise the momentum equations, i.e. using the finite element interpretation, the discretised equations are

where q.,kis the residual distribution weighting function associated to the velocity node i and to the kth component of the momentum equation, and y j is the shape function associated to the pressure node j. Note that, because of the upwind component of the weighting function upwinding is introduced in the discretisation of the pressure gradient. This turns out to be essential for the accuracy of the method [BOG 981.

5. Applications Capabilities of upwind residual distribution schemes are now illustrated by a couple of computational examples. 5.1.

Znviscid transonicJlow over the M6 wing

The ONERA M6 wing is a well documented testcase for three-dimensional flows from subsonic to transonic speeds [AGA 941. The selected transonic case is M , = 0.84, a = 3.06". The grid consists of 316275 nodes and 1940182 tetrahedra. The far-field boundary is half a sphere with a radius of 12.5 root-chord lengths. The computation of this testcase with the present multidimensional upwind method employed the van Leer-Lee-Roe preconditioning, allowing a hybrid discretization. The decoupled

36

Finite volumes for complex applications

advection equations for entropy and total enthalpy are discretized with the scalar PSIscheme, while the acoustic subsystem is discretized with a blended system LDAINscheme. To give an indication of how the multidimensional upwind method performs compared to standard finite volume methods, the same testcase has also been computed with the Jarneson-Schmidt-Turkel (JST) artificial dissipation scheme, a matrix dissipation (MATD) scheme and the Roe upwind scheme with Venkatakrishnan's reconstruction. Plots for the pressure coefficient Cp and the relative total pressure loss, defined as 1 - p,/p,,, in the 80% span cross-section are shown in Fig. 2. From these figures it is clear that the multidimensional upwind method is much less dissipative in the leading edge region and that it has a better shock capturing. 5.2. Znviscid transonic flow over a complete aircraft

To demonstrate the applicabilty of the present multidimensional upwind method to a complete aircraft configuration, the transonic flow over a generic model of the Falcon 2000 executive jet has been computed. The grid for a half model consists of 45387 nodes and 255944 tetrahedra. The selected testcase corresponding to the cruise condition is M, = 0.84, a = 3.06". The solutions computed with monotone first and second-order multidimensional upwind schemes are compared. For the first order computation the system N-scheme is applied directly to the full Euler equations. The second order computation employs the van Leer-Lee-Roe preconditioning, where the advection equations for entropy and total enthalpy are discretized with the scalar PSIscheme and the acoustic subsystem is discretized with the blended system LDA/Nscheme. Figures 3-4 compare the Mach number and entropy isolines for these two solutions. The better shock capturing and lower spurious entropy generation in the leading edge region of the wing and the engine pylon are clearly observed.

5.3. Incompressible turbulentflow over a backward-facing step We consider now the incompressible turbulent flow over a backward-facing step, experimentally studied by Kim [KIM 781. Calling H the step height, the inlet channel width is 2H, so that the outlet channel width is 3H. The flow Reynolds number is 1.41 10' based on the inlet centreline mean velocity Uo and the outlet channel width. The standard k - E turbulence model [JON 721 is used with wall function boundary conditions. The grid is a triangulated stretched Cartesian grid extending from 3H upstream of the step to 27H downstream. Along all solid walls, the computational domain boundary is set at a distance h = 0.025H from the wall. The grid contains 6247 P1 nodes distributed along 107 vertical grid lines, where each vertical grid line contains 41 P1 nodes in the inlet channel and 61 P1 nodes in the outlet channel. The inlet boundary conditions for u , v and k are taken from the experiment and E

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37

ym=.s

MDHR ----MD

.-..- - - - .

OW5

-

0

I

0

-1

- , - - _ _-- - -\I ----_ I ,

I

I

0.25

I

I

0.5

I

,

I

I

1

0 75

X

-

0

MDHR

----Roe -0 01 I

0

0.25

,

,

.

l

l

0.5

,

,

,

,

l

0.75

,

,

l

l

J

t

Figure 2: M6 wing: Cp and total pressure loss distributions at 80% span, upwind residual distribution schemes versus finite volume schemes

38

Finite volumes for complex applications

(a) system N-scheme

(b) blended system LDAM-scheme Figure 3: Falcon: Mach number contours

FALCON (453?7 nodpp) M1.05 an1 full EuMr N-schme

(a) system N-scheme

FALCON (45386 nodes) M.35 a=l p m n d i m d Euler (WE).

PSI +

L

D

-ma~

(b) blended system LDm-scheme

Figure 4: Falcon: Mach number contours is evaluated using a mixing length model. At the outlet, p is set to 0 and homogeneous Neumann (fully developed flow) boundary conditions are specified for all other variables. Along the wall function boundaries, a homogeneous Neumann condition is used for k and a Dirichlet condition is used for E (= ~ 3 , / ~ k ' / ~ /The ~ h wall ) . shear stress is derived from the law of the wall. Fig. 5 show the flowfield pattern calculated using the LDA scheme. The reattachment length is 6.3H, a value closer to the experimental value of L,/H = 7.0 than

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39

Streamlines

Figure 5: Turbulent flow over a backward-facing step: mean flow streamlines values obtained by other numerical computations using the same turbulence model, which fall in the range (5.7 L,/H 6.0 [DEM 971). The present method is also seen to capture a small secondary recirculation region at the foot of the step.

<

<

Acknowledgements This paper summarises over 10 years of research at VKI, which was mainly carried out by a series of Ph.D. candidates: R. Struijs, H. Paillbre, E, van der Weide, J.-C. Carette, E. Issman, N. Waterson, S. Bogaerts, K. Sermeus. The ONERA M6 wing and the generic Falcon computations were carried out within the collaborative IMT project IDeMAS funded by the European Commission. The reference computations of the ONERA M6 test case were performed by Daimler-Chrysler Aerospace. The grid for the generic Falcon configuration was provided by Dassault Aviation. References [ABG 991 R. Abgrall. Construction of genuinely multidimensional upwind schemes. 2nd International Symposium on Finite Volumes for Complex Applications, July 1999. [AGA 941 A Selection of Experimental Test Cases for the Validation of CFD codes. AGARD Advisory Report No 303, 1994. [BOG 981 S. Bogaerts, G. Degrez, and E. Razafindrakoto. Upwind residual distribution schemes for incompressible flows. Fourth European ComputationalFluid Dynamics Conference,Athens, Sep. 1998. [DEM 971 T. De Mulder. Stabilizedjnite element methodsfor turbulent incompressible singlephase and dispersed mo-phasefZows. PhD thesis, K . U. Leuven, Leuven, Belgium, 1997. [DEC 931 H. Deconinck, P. L. Roe, and R. Struijs. A multidimensional generalization of Roe's

40

Finite volumes for complex applications flux difference splitter for the Euler equations. Computers and Fluids, 22:215-222, 1993.

[ISS 981

E. Issman and G. Degrez. Non-overlapping preconditioners for a parallel implicit Navier-Stokes solver. Future Generation Computer Systems, 13:303-3 13, 199711998,

[ISS 961

E. Issman, G. Degrez, and H. Deconinck. Implicit upwind residual-distribution Euler and Navier-Stokes solver on unstructured meshes. AIAA Journal, 34(10):20212029,1996.

[JON 721 W. Jones and B. Launder. The Prediction of Larninarization with a Two-Equation Model of Turbulence. Int. Journalof Heat and Mass Transfer, 15:301-304,1972. [KIM 781 J. S. Kim. Investigation of separation and reattachment of a turbulent shear layer: Flow over a backwardlfacing step. PhD thesis, Stanford University, Stanford, Ca, 1978. [PA1 971

H. Pailltre, H. Deconinck, and E. van der Weide. Upwind residual distribution methods for compressible flow: an alternative to finite volume and finite element methods. VKI LS 1997-02,1997.

[ROE 811 P. L. Roe. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of computational physics, 43:357-352,198 1. [ROE 821 P. L. Roe. Fluctuations and signals, a framerwork for numerical evolution problems. In K. W. Morton & M. J. Baines, editor, Numerical Methods for Fluid Dynamics, pages 219-257. Academic Press, 1982. [ROE 871 P. L. Roe. Linear advection schemes on triangular meshes. CoA Report 8720, Cranfield Inst. of Tech., 1987. [vdW 991 E. van der Weide, H. Deconinck, E. Issman, and G. Degrez. A parallel, implicit, multidimensional upwind residual distribution method for the Navier-Stokes equations on unstructured grids. Computational Mechanics, 23(2): 199-208, 1999. [vLE 911 B. van Leer, W. T. Lee, and P. L. Roe. Characteristic time stepping or local preconditioning of the Euler equations. AIAA Paper 91-1552-CP. [WAT 971 N. P. Waterson and H. Deconinck. A fully-implicit multidimensional upwind approach for the incompressible Navier-Stokes equations. In C. Taylor, editor, Numerical methods in laminar and turbulentjIows, volume X . Pineridge Press, 1997.

Overcoming mass losses in Level Set-based interface tracking schemes

Th. Schneider and R. Klein Konrad-Zuse-Zentrum fur Informationstechnik Berlin, Germany F B Mathematik & Informatik, Freie Universitat Berlin, Germany

A n extended level set method is presented for tracking material interfaces i n incompressible two-phase flow that ensures conservation of mass. Inconsistencies, between mass transport and interface (level set) motion due to truncation errors are reconciled by two correction steps. Step 1 involves a redistribution of mass from cells that are not intersected by the tracked front but carry unphysical intermediate densities. Step 2 controls deviations between level set and density based partial volume fractions by introducing a small correction velocity in the level set transport equation.

ABSTRACT

K e y Words: level set, VOF, interface tracking, incompressible two-phase flow

1. Introduction Among the approaches to compute incompressible flows with material interfaces such as level set methods, volume of fluid methods (VOF) and interface tracking schemes, level set methods currently attract considerable attention. Since level set methods only require solution of a scalar hyperbolic transport equation, they are simple from an algorithmic point of view, their implementation is straight forward and the computational effort to solve the scalar transport equation is negligible in comparison t o the effort an incompressible method requires. Furthermore, level set methods naturally support interface distortion as well as topological changes of the interface. The major drawback of level set methods is that they do not ensure conservation of mass by construction. Volume of fluid methods consist of two parts : reconstruction of the interface and advection of the volume fraction. The first part implies a considerable algorithmic effort since the only available information concerning the position of

42

Finite volumes for complex applications

the front is hidden in the distribution of the volume fractions. By construction volume of fluid methods conserve mass; in fact for piecewise constant densities, the volume fraction transport equation is equivalent to mass conservation. This is not true in more general cases, though.

2. Level set methods In the framework of tracking material interfaces in incompressible flows the underlying idea of the level set approach is to identify the phases by the sign of a scalar field G . In other words : areas where G > 0 are occupied by phase I, whereas areas where G < 0 are occupied by phase 11. Therefore, the location and the topology of the interface are described by the zero level set of G. Since the interface is advected along particle paths the evolution equation for G is:

S u s s m a n et al. [Sus 941 were the first to use a level set formulation to represent interfaces in incompressible flow. They performed computations of incompressible two-phase flows with density ratios up to 1000:l; they observed considerable mass losses. In order to reduce mass losses they reinitialized G after each time step. Their reinitialization ensures that G remains a distance function, i.e. IlVGllz = 1. This is achieved by performing a pseudo time integration of

until a steady state is reached. The crucial information that is needed from the distribution of G is the zero level set that represents the interface. In areas where G > 0 or G < 0 the scalar field G is, up to its sign, physically meaningless. This implies that any change may be applied to G in those areas as long as the sign of G re'mains unchanged. Adalsteinsson et al. [Ada 991 proposed a numerical method to solve the transport equation (1) of G by constructing appropriate extension velocities such that G remains a distance function. Chang e t al. [Cha 961 used another reinitialization that ensures conservation of area and by that conservation of mass. They reinitialize G by integrating

until a steady state is reached. Herein e is a small positive constant, K, is the local curvature of the front. Even though this ansatz overcomes the key

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problem of loss of volume when the advecting velocity field is divergence free, it is insufficient for our purposes: The ultimate goal of the present efforts is t o construct a flow solver for general zero Mach number variable density flows with phase interfaces. Thus we must combine the level set description with a discretization of mass conservation. This is not achieved in [Cha 961. 2. Volume Of Fluid M e t h o d s

A detailed description of numerous variants of VOF methods can be found in Rider et al. [Rid 951, [Rid 95/21 and [Rid 981. All VOF methods have in common that they consist of two parts: 1. Reconstruction of the interface : Since no direct information on the topology (normal vector 2,curvature K ) of the interface is available a t each time step the interface needs to be recon~t~ructed based upon the distribution of the volume fractions f in order to allow the advance in time.

2. Advection of the volume fraction f: The volume fraction f is advected according to the following equation:

The flux @ is constructed using the information of the interface motion that is obtained in the first part; V is an arbitrary control volume, e.g. grid cell. The numerical method that is used to solve the advection f 1. Therefore, if this must ensure that f remains bounded : 0 latter condition is violated mass will be redistributed to mixed cells in the neighbourhood that can absorb it.

< <

3. A coupled Level-Set/Volume-Of-Fluid A l g o r i t h m

Bourlioux [Bou 951 presented a coupled level set/volume of fluid method. Her basic idea was to solve an advection equation for the volume fraction f in a VOF manner and the scalar G a t the same time. The geometric information of the interface needed to advect f is obtained from a scalar field G; therefore, a reconstruction of the interface from the VOF step function data is not required. During the reinitialization of G she applied an additional correction t o

44

Finite volumes for complex applications

G in order t o minimize the difference of the volume fraction f and the volume fraction fG that results from the integration of the distribution of G within a cell according t o (5) H ( G ) denotes the Heavyside function. 4. A coupled Level Set

/

mass conservation Scheme

Our proposed method is different from Bourlioux's work in many respects even though the basic idea remains the same. The advection equation of the volume fraction f can be written as follows (the volume fraction f is associated with the phase to which the positive sign of G is assigned) :

H(G) denotes the Heavyside function. If the control volume V is a grid cell with n, cell interfaces a first order discretization of the above advection equation of the volume fraction f is :

Where

@:+'

is a second order approximation for :

p+'

<

Equation (7) does not automatically ensure that remains bounded 0 5 1 which results from truncation errors. Therefore, a correction A f needs t o be added to In order t o conserve mass A f n f l is introduced as correction of the fluxes in equation (7) according t o :

Pf1

p+'.

The equation above is a discrete Poisson equation for the scalar 4 that establishes an implicit coupling between all mixed cells along the interface. Using

Invited speakers

this correction (f n+l = J"+'

45

+ A f " + l ) equation ( 7 ) changes to:

A second correction is applied to the scalar field G in order t o control the

fGf'.

The idea is to locally difference of the partial volume fraction f n f l and move the front normal t o itself in order t o minimize this error. To determine a local correction velocity s* a piecewise linear reprensentation of the front is assumed. The local correction velocity is set to s* = ( h - h G ) / A t (see sketch below) : This local correction velocity s* is used to perform one time step of the

Figure 1: Piecewise linear representation of the front within a mixed cell

5. Results 5.1 Advection of a circle

To test the convergence rate of our proposed extended level set method we have chosen a circular density jump which is passively advected within a

46

Finite volumes for complex applications

parallel flow. The unit square domain is resolved by 32 x 32, 64 x 64 and 128 x 128 grid cells. Periodic boundaries are imposed. The L1 error el of the density is measured taking the difference of a coarse grid and fine grid solution according to :

n h is the number of grid cells of the fine grid in each direction, P 2 h the density of the coarse solution and ph the density of the fine grid solution. The rate of convergence pl based upon two L1 error approximations e l , 6 4 , 3 2 and e l , l 2 8 , 6 4 is given by :

The initial data are : radius 0.15, centered at (xo,yo) = (0.5,0.5), density ratio 1000:1, u, = 0.25, v, = 0.25. When integrating from t=O.O to t=0.5 a convergence rate of 1.8 was achieved. 5.2 Rising bubble

The governing equations for the considered variable density zero Mach number regime are (using the pressure decomposition p = Po + M ~ ~ (where ~ ) , $PO = 0, since the flow is considered as incompressible):

H denotes a smoothed Heavyside function. The numerical method for zero Mach Number variable density flow is described in [Sch 981. A rectangular domain with a height to width ratio of 2 was chosen consisting of 128 x 256 grid cells. At the boundaries slip conditions were set. This problem is characterized

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by the density ratio of liquid and gas C1, the diameter t o width ratio C2, the viscosity ratio C3, the Reynolds number Re, the Froude number F r and the Weber number W. The characteristic number and their definitions are listed below:

Surface tension is implemented in the same way as explained in [Sus 941. The initial shape of the interface is circular. The interface (thick line), streamlines (thin lines) and velocity arrows are plotted for times t=O.O, 1.0, 2.0, 3.0 below. In figure (3) the relative mass error (mass according to the distribution of G) is plotted versus time.

Figure 2: Interface (thick line), streamlines (thin lines) and velocity arrows a t times t=O.O, 1.0, 2.0, 3.0 6 . Concluding Remarks

The present scheme for fluid interface tracking methods avoids mass losses by incorporating a finite volume mass conservation equation. The inconsistencies between level set transport and mass conservation are eliminated by controlling volume fraction deviations through a suitable "penalty" - method

48 Finite volumes for complex applications

-3e-05 0

0.5

1

2

1.5

2.5

3

3.5

time

Figure 3: Relative mass error resulting from the distribution of G versus time applied t o the level set transport. Unphysical intermediate densities in nonmixed cells which can arise due t o the capturing features of the mass conservation discretization are removed by a mass redistribution step. In our present approach the thickness of the intercase is zero. Artifical smearing of densities is neither necessary nor does it occur during a computation.

Acknowlegdments This work is supported by the Deutsche Forschungsgmeinschaft through project KL-61115 - 1,2, in the framework of the CNRS-DFG Programme "Numerische Stromungssimulation" . Helpful discussions with Dr. Anne Boulioux (Montreal) and Dr. Raz Kupferman (Jerusalem) are appreciated.

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Bibliography

[Ada 991

Adalsteinsson, D. und Sethian, J . A., The Fast Construction of Extension Velocities in Level Set Methods, Journal of Computational Physics, 148, 2-22, 1999

[Bou 951

Bourlioux, A., A Coupled Level-Set Volume-Of-Fluid Algorithm for Tracking Material Interfaces, 3rd Annual Conference of the CFD Society of Canada, June 25-27, 1995, Banff, Alberta, Canada

[Cha 961

Chang, Y. C., Hou, T. Y., Merriman, B. und Osher, S., A level Set Fomnulation of Eulerian Interface Capturing Methods fo Incompressible Fluid Flows, Journal of Computational Physics, 124, 449-464, 1996

[PUC971

Puckett, E. G., Almgren, A. S., Bell, J . B., Marcus, D. L. and Rider, W. J.,A High-Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows, Journal of Computational Physics, 130, 269-282, 1997

[Rid 951

Rider, W. J., Kothe, D. B., Mosso, S. J., Cerutti, J.H. and Hochstein, J.I. Accurate Solution Algorithms for Incompressible Flows, 33rd Aerospace Sciences Meeting and Exhibit, January 9-12, 1995, Reno, NV

[Rid 95/2

Rider and W. J., Kothe, Stretching and Tearing Interface Tracking Methods, 12th AIAA CFD Conference, June 20, 1995, San Diego. Paper number AIAA-95-1717

[Rid 981

Rider, W. J . and Kothe, D. B., Reconstructing Volume Tracking, Journal of Computational Physics, 141, 112-152, 1998

[Sch 981

Schneider, T, Botta, N , Geratz, K. J . and Klein, R., Extension of finite volume compressible flow solvers to multidimensional, variable density zero Mach number flow, submitted to the Journal of Computational Physics, preprint available a t http: //www .zib.de/thomas .schneider/pub.html

[Set 961

Sethian, J. A., Level Set Methods, Cambridge University Press, 1996

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Finite volumes for complex applications

[Sus 941

Sussman, M., Smereka, P. und Osher, S., A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow, Journal of Computational Physics, 114, 146-159, 1994

[Sus 991

Sussman, M., Almgren, A. S., Bell, J., Colella, P., Howell, L. H. und Welcome, M., An Adaptive Level Set Approach for Incompressible Two-Phase Flows, Journal of Computational Physics, 148, 81-124, 1999

Coupling mixed and finite volume discretizations of convection-diffusion-reactionequations on nonmatching grids

Raytcho D. Lazarov, Joseph E. Pasciak, and Panayot S . Vassilevski Department of Mathematics, Texas A#M University, College Station, TX 77843 and Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, C A 94550, U.S.A.

I n this paper, we consider approximation of a second order convectiondiflusion problem by coupled mixed and finite volume methods. Namely, the domain i s partitioned into two subdomains, and i n one of t h e m we apply the mixed finite element method while o n the other subdomain we use the finite volu m e element approximation. W e prove the stability of this discretization and derive a n error estimate.

ABSTRACT

K e y Words: combined mixed and finite volume methods, non-matching grids.

I . Introduction

Coupling different numerical methods applied to different parts of the domain of interest is becoming an important tool in numerical analysis, scientific computing, and engineering simulations. In the coupling process several important mathematical issues arise that have t o be addressed. First problem is to find what natural and stable mathematical formulation will lead to a good computational scheme. In the case of different methods used in different parts of the domain this means t o find a stable way of gluing together the solutions in the subdomains. Secondly, we have t o find an approximation of the mathematical formulation which is stable, convergent, and accurate. And finally, we have to construct and study efficient solution methods for the resulting algebraic problem.

52

Finite volumes for complex applications

We shall consider the following homogeneous Dirichlet boundary value problem for the convection-diffusion-reaction equation: .Cp = f (x), x E

a,

p(x) = 0, x E 8 0 ,

(1)

where C = L o + C, Cop = -V . aVp is the diffusion operator, and Cp s V . (pb) cop is the convection-reaction operator. Here 0 is a bounded polygon in Rd,d = 2 , 3 with a boundary 8 0 , a = a(x) = {aif(x)) is a d x d symmetric and uniformly in fi positive definite matrix, and f = f (x) is a known function Also b = b(x) = (bl , . . - ,bd) is a given vector field and co = c,, (x) is in L2(a). a given function. We assume that b(x) and co(x) are uniformly bounded in R and satisfy the condition

+

This in turn guarantees the coercivity of the operator C in L 2 ( 0 ) and the existence and uniqueness of its solution in the Sobolev space Hi(Q).This problem is a prototype of mathematical models in heat and mass transfer, diffusion-reaction processes, flow and transport in porous media, etc. In this paper we propose and study numerical methods for this problem when in different parts of the domain different discretizations on independent meshes are used. Namely, we consider mixed finite element approximation in one part of the domain and finite volume element method in the rest of the domain. It is important to note that coupling mixed finite element and finite volume or Galerkin finite element approximations does not require any auxiliary (mortar) space on the interface of the subdomains. This is due to the fact that the Dirichlet boundary conditions are natural for the mixed formulation, while the Neumann boundary conditions are natural for the standard weak formulation of a second order elliptic problem. In the recent years there has been growing interest in the finite volume method (called also control-volume method or box-schemes). This interest is mostly due to the requirement of many applications of having locally conservative discretizations. This is a discrete variant of the property of the continuous model which expresses conservation of certain quantity (mass, heat, momentum, etc) over each infinitesimal volume. The finite volume method has been combined with the technique of the finite element method in a new development which is capable of producing accurate approximations on general simplicial and quadrilateral grids (see, e.g. [4, 5, 6, 7, 8, 131). For a collection of theoretical results and various applications we refer [2]. The main advantages of the finite volume method are compactness of the discretization stencil, good accuracy, and discrete local conservation, which for many applications is a very

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desirable feature of the approximation. Also, this method has well developed approximation schemes for convection and convection-dominated problems. The structure of the paper is the follow. In Section 2 we introduce all necessary notations and the weak form of the problem (1) in a two domains setting. In the subdomain where the mixed formulation is used we apply the more general concept of the discontinuous Galerkin method for second order equations in mixed form. In the case of convection-diffusion problems the pressure should be smoother than just L~ SO we use the space H&. Further, we study the stability and derive an a priori estimate for the solution. Section 3 is the central part of the paper. Here we introduce and study the coupling of the mixed finite element and the finite volume element method. Further, in Subsection 3.2 we discuss the coupled mixed and finite volume approximation of convection-diffusion-reaction equations. Finally, in Subsection 3.3 we prove the unconditional stability of the discrete scheme and derive an estimate for the error. 2. Variational formulation

In this section we first introduce all necessary notations for splitting the = n1 U 1 2 and using two domain of the problem (1) in two subdomains different formulations in each subdomain. The weak mixed formulation in 01 is derived when the pressure p is in the space H:,,(RL). In 0 2 we use pressure space H 1(02). We prove that the coupled mixed/Galerkin formulation is stable and derive an a priori estimate which is the prototype of estimates for the approximations schemes established further in the paper. 2.1. Two-subdomain coupled formulation

We partition S1 into two subdomains with an interface boundary I?, i.e. S1 = 01 U r U 0 2 (see Figure 1) and use the standard notations for Sobolev spaces of functions defined on R1 and 0 2 : H(div, 01), L2(R,), i = 1,2 and Ht (02, aS12 \ I?). Here the last space denotes the functions defined on R2 having generalized derivatives in L2(R2)and vanishing on dR2 \ I?. The inner products in these spaces are denoted correspondingly by

+

~ '= (v2,w2) (Vv2,V W ~which ) ~ in turn define norms denoted and (v2,~ 2 ) (Rz) by llvllH(div,nl)l Ilvill0, ni and l(v2ll1,~ 2 .

54

Finite volumes for complex applications

Figure 1: Domain partitioning: f l = R1 U I' U 02.

Whenever possible we skip the subscript in L2-norms. The dual space of

Hi ( a 2 , dR2 \ I?) is denoted by H-I ( a 2 ) , the space of the traces of the normal

component of the vector-functions in H(div, al) is denoted by H-'/~(I'), and the space of the traces of functions in Hi(O2, d o 2 \ I') is denoted by H;,/~(F). The trace spaces are equipped with the standard Sobolev norms. Finally, we shall use the notation < ., . >r for the duality pairing between H,',/~(I') and H - ' / ~ ( F ) . Further, we denote by n i the unit vector normal to aRi for i = 1 , 2 and pointing outward to the domain. Finally, we split the interface boundary r = r- U r+ where r- = {x E : b(x) . n l < 0) and I?+ = {x E r : b ( x ) . n l 2 0). Note, that this splitting is with respect to the vector n l . An illustration of these notations in 2-D is given on Figure 1. In R1 we use a mixed setting of the problem (1). That is, we introduce the new (vector) variable u = -aVp. To distinguish the solutions in the subdomains we denote by pl = plnl and p, = pl,,. The composite model will impose different smoothness requirements on the components pl and p2. More specifically, we will require that u E H(div, Stl), pl E L2(SZ1), and p2 E Hd (Stz, a0 \ I'). Note that pa is required to vanish on dR2 \ l?. Testing the equation a c l u Vpl = 0 by a function v E H(div, a),using integration by parts, the zero boundary conditions for pl on dR1 \ r , and the fact the trace of pl on I? is the same for the trace of pa on I?, one ends up with the equation,

+

Further, in order to describe the weak form of the equation

V-u

+ V . (bpi) + cop1 = f (x)

in R1

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we need to allow discontinuous functions pl from the space Hfoc: vl E L2(R1) : H,',,(fll)

=

there is a partition K of R1 such that vlIK E H1(K) for all K E K

The functions in H:,,(Rl) have traces from both sides of the interfaces of the subdomains K. Namely, for a given function pl E Htoc(Cll) we denote these traces by p? and pi, where "a" stands for the outward (with respect to K ) trace and respectively, "i" stands for the interior trace. Next, we give the weak form of the above equation. We borrow this formulation from the discontinuous Galerkin methods (see, e.g. [lo], pp. 189-196) by testing the equation by a function wl E H:oc(R1). We note, that this setting is quite similar to the mixed finite element method for convection-dominated convection-diffusion-reaction equations (see, e.g. [12]). Since the functions from Hkc(R1) are piece-wise smooth with respect to the partition K we shall integrate over each K E K and then sum the results. Following [lo] we find first the contributions of the advection-reaction operator Cpl by introducing the bilinear form CK(pl, wl) for any subdomain K E K :

Here n is the outer unit normal vector to d K . Next, we integrate by parts in each subdomain K and sum over all K E K. Thus, for pl , wl E H:oc (01) we get:

Note that this bilinear from is well defined for both continuous and discontinuous functions with respect to the partition K. From this expression we see that if the subdomain K has a sideiface on r- then the trace pi' should be replaced by its counterpart from R2, namely by p2(x). Also on J?- we have w", wl and on dR1- \ r- we take py = 0. Further, for a given function t(x) we denote by t- = min(0, t) and t+ = max(0, t) . Thus, we get the following weak form of the second equation valid for all wl E Htoc(Rl):

Finite volumes for complex applications

56

where

PI

b . V W Idx + ( % P I ,

wl)

for P I :

wl

E H:oc(Q~)

KEK

(6)

and

Finally, testing the equation (1) by a function w2 E H i ( R 2 ; an2\I?), using integration by parts, the zero boundary condition for w2 on d o 2 \ I',and the fact that u . n l = - a V p l . n l = aVp2 . n2 on I',one arrives at,

for all w2 E HA ( R 2 ; an2 \ r). There are various ways one can take into account the influence of the problem in the domain 0 1 on the problem in Q 2 . One of the possibilities, which we shall use further, is to try to make a formulation, which is stable for small diffusion coefficient (or even for vanishing diffusion). In this case it is very important to formulate correctly the boundary conditions. Namely, at the "inflow" part of the interior boundary the solution should be specified from the "outside" data. Taking into account that I?+ is the "inflow1' part of r for the subdomain Q 2 , we add Sr+P I w2 b . n2 d s and subtract its equal Sr+p2w2 b . n2 ds since on I' we have pl = pa. Thus, we get the following form of the last equation:

for all

w2

E H,1 ( Q 2 ; aQ2 \

I?), where

Thus, the coupled system for the three unknowns u E H(div, Q l ) , pl E a02 \ I ') consists of the equations (3), ( 5 ) , and (9)

Hk,(S11) and p2 E H;(n2;

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summarized as:

for all v E H(div, R1), wl E H&,(R1), and w2 E Hi(R2; aR2\I?), respectively. The bilinear forms aij (., .) are defined by (6), (7), (lo), and (1I ) , respectively. 2.2. Well-posed-ness of the composite problem

Here we verify the existence and uniqueness of the solution of problem (12) and its stability in an appropriate norm. For this we shall need some additional notations. Let & = { e ) be the set of edges/faces of the subdomain O1 from K and Lo the set of interior for R1 edges/faces. Recall, that nl and nz are the outward unit normal vectors t o R1 and R2, respectively. For any edge e E Lo denote by n, a fixed unit vector normal t o e and let K,f and KT be the two adjacent to e subdomains from the partition K. For edgeslfaces that are on dRl we shall always assume that n, = n l . Further, denote by [vl] and El the jump and the average of the discontinuity of vl, respectively, along any edge e. More precisely, this is the difference and the arithmetic mean of the traces vl J K a and vl l K ; taken from both sides of e:

Further, we use the following natural norm for vl E H&,(fl~) and v2 E H;(R2; 8 0 2 \I?):

+

(aVv2, Vv2).

(13) All terms in the expression on the right are nonnegative and this defines a norm on the space Hk,(O1) x H;(R2; dR2 \ I?). Note, that under certain conditions on the vector field b this is a norm even if yo = 0.

58

Finite volumes for complex applications

The stability of the composed problem (12) is based on the following theorem:

Theorem 1 T h e solution of the problem (1.2) satisfies the a priori estimate:

II~11~2(n~) + II~lllq,nl + Il~2llf,0, 5 cilf IlE, n l

(14)

where the * - n o r m i s defined b y (13). The proof of the above theorem is based on the following lemmas.

Lemma 1 T h e bilinear form (6) defined for vl, wl E H,',,(Rl) can be transformed t o the following form:

Furthermore,

+-2 Ian,,

v; b . nl d s - -

2 Ian,-

v; b . nl ds.

Lemma 2 For all vz € H,'(R2, df12 \ F)

3. Coupling mixed and finite volume approximations of the convectiondiffusion equation Our approximation strategy is based on the finite volume method in the framework studied by Cai [4],Cai, Mandel, and McCormick [6] and also by Bank and Rose [I].

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3.1. An outline of the finite volume element method

We first outline a finite volume discretization method for the case of pure diffusion problem posed on R2,

Here f E L 2 ( n 2 ) is given and q ~ for, the time being is assumed given in the space L 2 ( r ) . The finite volume method under consideration uses two different finite dimensional spaces: a solution space W 2 and the test space W;. The Wz is the standard conforming space of piecewise linear functions over the triangulation E = {T) of f12 into triangles in 2-D and tetrahedra in 3-D (we call them simplices). To introduce the test space W,' we need a dual partition Vz of the domain into the finite (control) volumes V. Let N denote the set of all vertices (nodes) of the trianglesltetrahedra from 72 and let No be a subset of those vertices that are not on the Dirichlet part of the boundary df12 \ I?. In each simplex T E 72 one selects an interior node XT. Next, in 2-D one links XT with the midpoints of the sides of the triangle. In this way the triangle is split into three quadrilaterals. In 2-Dl one can select XT t o be the orthocenter of the finite element T and then the edges of the volume V(x) will be the perpendicular bisectors of the finite element edges (see the right Figure 2). With each vertex x E N of a simplex from 72, we associate a volume V = V(x) that consists of all quadrilateral/polyhedra having x as a vertex (see Figure 2 for finite volumes in 2-D). The splitting of f12 into finite volumes V forms the partition V2 (see, Figure 3). Consider now the test space W2*spanned by the characteristic functions of the volumes V E V2 and that vanish at the nodes N \ No on the boundary 8 0 2 \ r. If one defines the piecewise constant interpolant I; with respect to the volumes V E 112, then the space W; is actually equal to I,*W2 because they have the same degrees of freedom (associated with the vertices x E ni).

The L2(Rz) and H1(R2) norms in W2 are defined in a standard way. We shall need also discrete variants of these norms for functions in W;. First, we define the interpolation operator Ih : Wh+ I-+ Wh by the following natural rule: Ihv2+is the piece-wise linear interpolant of vd over each finite element T E 72. Then we define I l ~ f l l = ~ , (IIhvfl(l,n, ~ This norm is essentially formed by the squared differences of the values of vf a t the vertices of each finite element.

60 Finite volumes for complex applications

Figure 2: A finite volume V associated with a vertex from the primal triangulation. Left: the vertices of V interior t o the triangles T are arbitrary, whereas those on the edges of T are midpoints.

Figure 3: In we use the lowest order Raviart-Thomas spaces over the finite elements T; in R2 we use a solution space Wz of continuous functions that are linear over the finite elements T and a test space W,* of pice-wise constant functions over the volumes V.

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Next, we define the following, in general nonsymmetric, bilinear form on w 2

x

w;:

a2,h(v2,~;) = xENo

=-

/

w;(x)

/

aVp2,h'nds

~ V ( X )

aVp2,h - n w; ds, for all v2 E W2, w; E W;

xEN~av(x) Then the finite volume approximation of (15) is: find v2 E W2 which satisfies the following identity for all w,' t W,';

We note that the integrals over dV for V = V(x) a volume corresponding to a Neumann node x (i.e., if x E I?), contain only the interior (to a2)part of d V . We assume that the triangulation 72 is aligned with the possible jumps of the coefficient matrix a(x), i.e. over each finite element T E 72 the matrix a(x) has smooth elements. Therefore, there is a constant Co > 0 such that for all

where &(x)=

a(s)ds/meas(T), x t T. IT

These inequalities of two d x d matrices with real elements are understood in the sense of inequalities for the corresponding bilinear forms, i.e. a ii(x), iff tTa[ 2 JTii(x)t, V 6 E Rd. Also, the above equality of the matrices a(x) and E(x) is understood in element-by-element sense, i.e. the elements of ii(x) are the mean values over T of the corresponding elements of a(x). Obviously, in case of piece-wise constant coefficients a(x) G(x) and Co = 0. The well posed-ness of the finite volume element approximation follows from the weak coercivity of the bilinear from a2,h(v2,wa)for sufficiently fine partitions 5.We have:

>

-

Lemma 3 Let the partition 72 be so fine that h < l/Co, where the constant Ca is determined in (17). Then the following inequality holds true SUP

zu;EW,'

a 2 , h ( v 2 2 .W')

IlwZflll,h

2 ~ l l v ~ l lfor ~ ,all ~ v2 ~ ,t

with a constant C independent of h.

~2

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Finite volumes for complex applications

In R1 we use the lowest order Raviart-Thomas spaces. Thus, V as a subspace of H(div; Ol), and Wl is the space of piece-wise constant functions with respect t o the partition 71 and therefore a subspace of Hio,(R1) for K = 5 . Thus, the advection term Cpl will be discretized using the pressure space Wl of piecewise constant functions on the triangulation 71. For the discretization in O2 we will use the space W2 c HA(02, an2 \ r) of continuous piecewise linear functions on 72. Further, in the finite volume setting we use as a test space W,+of piecewise constant functions on V2. Thus, applying equation (4) consecutively for (vl, wl) E Wl x W2, and (712,w;) E W2 x W;, respectively, we get the mixed finite element and finite volume approximations, respectively, of the bilinear form corresponding to the first order term. However, like in the standard Galerkin finite element method this approximation of the operator C will lead to central differencing, which in turn will lead to a conditionally stable (only for sufficiently small step-size h) scheme. In order to derive a unconditionally stable scheme we shall use upwind approximation in 0 2 . 3.2. Derivation of the coupled method

Since both vl and wl are discontinuous piece-wise constant functions with respect t o the triangulation 71 the formula (4) is applied in a straightforward manner for K = 71 so we get the following approximations at, and a:, of the forms all and al2, respectively:

+(covl, wl) for vl E WI, wl E Wl and at2(v2,w1) =

J,-

Iiv2

WI

b

e

nl ds for

v2 E W2,

WI

E

Wl.

(19)

Now we find the contributions of the the operator C from 0 2 and we define the approximations of the bilinear forms a21 and a22. We shall simply rewrite (4) for lC = V2:

Since the functions in W2 are continuous then C(v2,w;) is well defined for all v2 E W2 and w; E W2f. Taking into account that the functions in W,* are

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Figure 4: The shaded area is the volume V centered at the vertex x" the doted boundary of V denotes the "inflow" boundary, while the solid one is the "outflow" part; on the various pieces of the boundary dV we have the following approximation of the convection term: on (a, b) , (m, c) , (f,g ) and (9,a): v2(x) = vz (xi); on (b, m) : v2(x) = v2(xO)= v2(C); on (c, d) : vz (x) = 2r2(5O) = vZ(D); on (d, e) : vz(x) = vz(xO) = 'uz(E); on (e, f ) : vz(x) = t12(x0) = v ~ ( F ) . constant over each finite volume V E V2 then the contributions from each finite volume V E V2 are:

Since v2 is continuous then obviously, we have v,O = vi = VZ(X). On the boundary r+ the values v,O are not defined (this is the inflow boundary for 0 2 ) and we shall take them from the corresponding counterpart in 01, i.e. as vl (x). Thus, we split the integrals over dV into two parts and get

Unfortunately, the exact calculation of the first integral in (21) will lead to central differences and therefore to a scheme which is stable only for sufficiently small step-size h. The limitation of the step-size h will depend on the magnitude of the convection coefficient b relative to the diffusion coefficient (matrix) a . For problems with dominating convection this will lead to prohibitively small h. In order to avoid this conditional stability we introduce an up-wind approximation of the integrals. This approximation is done in the following way. We denote by

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Finite volumes for complex applications

V(xi) a finite volume centered at the vertex xi and by V(xo) any of neighboring volumes centered at the vertices xO. Theintegralover dV \I?- is split into subintegrals over the boundaries of V y = V(xi) n V(xO)n T with its neighboring finite (control) volumes and contained in the finite element T. We assume that over each y the function 6(x). n does not change sign (i.e, is either nonnegative or negative). Then on y we use upwind approximation of the following form (for a 2-D illustration, see Figure 4): b ~ n v ~ ( x ) ~ ( b - n ) + t ~ 2 ( x ~ ) + ( b ~ n ) for - v 2 z( ~~~ )y, .

Note, that in the finite volume V(xi) we have I;tvz(x) = v2(xi). Similar equalities are valid for the neighboring volumes V(xo) as well. Thus, roughly speaking the function v2(x) has been replaced by its interpolant in the space of discontinuous functions W,* and then taken the appropriate (up-wind or in the opposit direction of the vector-field b(x)) values at the finite volume interfaces. A particular finite volume in 2-D is shown on Figure 4. Summing for all V E V2 we finally get the following form by taking also into account the diffusion term (16):

for all v2 E W2, wa E W; and the form (ul, W;) =

l+

v1 w;

b - n2 d~ for ui

E Wl, w; E W;.

(23)

The coupled mixed finite elementlfinite volume approximation of the composite problem (12) reads as: find uh E V, pl,h E Wl, and p2,h E W2, such that (a-luh, V)

- ( ~ l , hv , 'v)

-(v

-a:l@l,h,

'

uh, wl)

+ < PZ,h,V

'

n l >I?=

0,

~ 1 )-42(I;tp2,h, wl) = -(f, wl),

(24)

for all v E V, wl E Wl, and wz* E Wz*,respectively. 3.3. Stability of t h e coupled scheme and e r r o r e s t i m a t e

An important feature of the described above discretizations is that the corresponding operator is coercive in an appropriate norm and the method is stable.

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65

For proving the stability we shall follow the same argument as in the case of original setting. Let as before E = { e ) be the set of edgeslfaces of the elements from 3 and let Eo be the set of interior edgeslfaces. Similarly, = {y) is the set of edgeslfaces, with each y being the boundary of two adjacent volumes Vl E V2 and Vz E V2 contained in a finite element T, i.e., y = V1 n V2 n T. This splitting can be also used in the computational procedure, since it will lead t o element-wise contributions of the convection term to the stiffness matrix. Note, that all edgeslfaces y are in the interior of R2. For the coercivity of the coupled problem we need the following discrete variant of the norm (13):

Here ti is a piece-wise constant matrix with respect t o the partition by (17).

5 defined

Theorem 2 Let h < l/Co, where Co is defined by (17). Then the solution of the problem (12) satisfies the a priori estimate:

where the (*, h ) - n o r m is defined by (25) with respect to the partitions

5 and

72. Proof: As in the continuous case, by testing (24) with v = uh, wl = -pl,h, and w2 = -Ilp2,h we get the equation:

('-luh,

~ h +) a ? ~ ( ~ l ,~h1, , h + ) a:2(pl,hr Iip2,h)

+ a z h l ( ~ ~ ~ 2~, hl , h+) a,h,(pz,h, I i ~ 2 , h = ) (f,

+ (f, Ilp2,h).

(27)

~ 1 , h )

Further, the estimate (14) is a consequence of the simplified form (28) of a?, and (29) of ag2, which are established in the lemmas below. Lemma 4 For any edge/face e denote by n, a fixed unit vector normal t o e and let T: and T; be the two adjacent elements t o e. Similarly, for any

66

Finite volumes for complex applications

edge/face y E denote b y n, a u n i t vector normal t o y pointing t o one of the neighboring volumes V: and V;. Also, [vl] denotes the jump of a function across a n underlined boundary (here we have either e o r y) and Zl denotes the arithmetic m e a n of the jump (introduced in Section ). T h e n ,

=,c 1

J[vl][wlll~.nlds+ c ~ P . ~ , [ ~ I ] s I ~ s

~ c ( ~ I , w I )

e€Eo

+(cOvl,W I )

e€&o

+

S

I?. nl vlwl ds for v l , W I

E Wl.

(28)

Similarly, for all v2 E W 2 , w,* E W,* the following identity is valid (to simplify the expressions we have used the notation v,' = Ilvz):

Proof: The proof of (28) and (29) essentially repeats the arguments of Lemma 1. There is a small difference in the proof of (29) where the integrals over each y E G have been computed by using up-wind approximation.

Lemma 5 T h e following identity i s valid for all vl E W l :

Similarly, for all Iiv2):

7.12 E W2

(here i n order t o simplify we use the notation v,* =

Finally, we have the following error estimate:

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T h e o r e m 3 Assume that the solution p(z) of the problem ( I ) is H2-regular i n 0. Then the solution (uh,pl,h,p2,h) of the coupled mixed discontinuous finite element and finite volume methods (24) converges to the solution (u, PI, p2) of the composite problem (12) and the following error estimate holds true:

The constant C does not depend on h but may depend on the ratios

2 and 2.

4. Bibliography

R.E. BANK A N D D.J. ROSE, Some error estimates for the box method, SIAM J. Numer. Anal. 24(1987), 777-787. F. BENKHALDOUN A N D R. VILSMEIER (EDs), Proc. First Intern. Symposium on Finite Volumes for Complex Applications, July 15 18, 1996, Rouen, France, Hermes, 1996. F. BREZZIA N D M. FORTIN, Mixed and ~ y ' b r i dFinite Element Methods, Springer-Verlag, 1991.

Z. CAI, On the finite volume element method, Numer. Math., 58 (1991) 713-735. Z. CAI, J . J . JONES, S.F. MCCORMICK,AND T.F. RUSSELL, Control-volume mixed finite element methods, Computational Geosciences, 1 (1997) 289-315. Z. C A I , J . MANDEL,AND S . F . MCCORMICK, The finite volume element method for diffusion equations on general triangulations, SIAM J . Numer. Anal. 28 (1991), 392-402.

S.H. CHOUA N D P.S. VASSILEVSKI, A general mixed co-volume framework for constructing conservative schemes for elliptic problems, Math. Comp., 68 (1999). S.H. CHOU, D.Y. KWAK, AND P.S. VASSILEVSKI, Mixed covolume methods for elliptic problems on triangular grids, SIAM J . Numer. Anal. 35 (1998), 1850-1861.

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Finite volumes for complex applications

T. IKEDA, Maximum Principle in Finite Element Models for Convection-Diffusion Phenomena, Lecture Notes in Numer. Appl. Anal., Vol. 4, North-Holland, Amsterdam New York Oxford, 1983. C. JOHNSON, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987. R.D. LAZAROV,J.E. PASCIAK, A N D P.S. VASSILEVSKI, Iterative solution of a combined mixed and standard Galerkin discretization methods for elliptic problems, (submitted to Numer. Linear Alg . APP~.).

M . LIU, J. WANG,A N D N.N. YAN, New error estimates for approximate solutions of convection-diffusion problems by mixed and discontinuous Galerkin methods, Univ. of Wyoming, Preprint, 1997. I.D. MISHEV,Finite volume methods on Voronoi meshes, Numerical Methods for Partial Differential Equations, 1 4 (1998), 193-212.

P.A. RAVIARTAND J . M . THOMAS, A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods (I. Galligani and E. Magenes, eds), Lecture Notes in Math., Springer-Verlag, 606 (1977), 292-315.

P . A . RAVIARTAND J.M. THOMAS,Primal hybrid finite element methods for second order elliptic equations, Math. Comp., 31 (1977), 391-396. M. TABATA, A finite element approximation corresponding t o the upwind finite differencing, Mem. Numer. Math., 4 (1977) 47-63. C. WIENERSAND B . WOHLMUTH,Coupling of mixed and conforming finite element discretizations, American Mathematical Society, Cont. Math. 218 (1998), 547-554.

Numerical computation of 3D two phase flow by finite volumes methods using flux schemes

J.-M. Ghidaglia Centre de A4athe'matiques et de Leurs Applications E N S de Cachan and C N R S UMR 8635 61 aerenue du Pre'sident Wilson 94235 Cachan Cedex France [email protected] http://www.cmla.ens-cachan.fr/Njmg

We propose here a general class of cell centered finite volume methods specially designed for the discretization of complex models of partial differential equations like those occuring in 3D two phase flow. After a b r k f introduction to these models (the so called averaged models in Eulerian formulatzon), we develop all the tools which are needed in order to arriue to a fully discrete scheme suitable for coding. Hence we discuss conserz~atiuesystems, non conservatzve ones, time discretization, discretization of source terms, of diffusion operators, of boundary conditions, . . . We also briefly dzscuss non conformal meshes. We strongly rely on the concept of flux scheme ,which is, according to us, very well suited for the problems considered here. ABSTRACT

ICey Words : T w o phase pow, Flux schemes, Finite volumes, non hyperbolic convection, Source terms, Footbridges.

1. Introduction Our goal in this contribution is to discuss a class of cell center finite volume methods, on unstructured conformal or non conformal meshes, designed for conlplex models (such a goal was also addressed by T. Gallouet [GAL 961

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Finite volumes for complex applications

in the previous version of this symposium). We are mainly motivated by the computation of two phase flows occuring in several fields of the nuclear power industry (of course there are many other contexts where multiphase models occur), that is flow of water and its vapor. Models pertaining to this field encompass many difficulties, whose non exhaustive list is follows : (i) the solutions might develop strong gradients, (ii) the equations might involve stiff source terms, (iii) non conservative products might be present in the equations, (iv) the thermodynamics could be highly complex (e.g. phase transition), (v) the total convection operator might be non hyperbolic, (vi) in some part of the flow low Mach numbers can occur, (vii) (viii) and last but not least : there is no "universal model" to work with. Some of these features are present in one phase Computational Fluid Dynamics (lph-CFD), other are typical in combustion but 2ph-CFD often involves toghether all of them. In this paper we shall propose a very general framework for the discretization of such systems. Generality is essential in this context according to (viii). And passing from one model to an other (which might implies even the variation of the number of scalar equations to be considered) should cause only minor changes in the code. Moreover one should ask almost no a priori conditions on the model (besides the fact that it is build upon physical considerations). One of the major by-product of this stategy is that a code derived according to this philosophy can be a tool for the validation of the physical models, and this point is one of the main issues in 2ph-CFD at the present time. Most of the results presented here were inspired by a very fruitful colaboration with the DCpartement Transfert Thermique et Akrodynamique (TTA/D R&D/EDF) and with the Service de Simulation des Systemes Complexes et de Logiciels (SYSCO/DRN/CEA).

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71

2. A s h o r t discussion of t h e m o d e l s We consider the motion of two non miscible fluids, like e.g. air and water or water and steam. Since we are concerned with macroscopic dynamics, we shall consider a continuum model. For single phase flows, except for very hight speed rarefied flows, Navier-Stokes Equations provide a hightly satisfactory continuous model. For two phase flows, a first approach could consists in writting for each phase a Navier Stokes model completed with jump relations across interfaces between the two fluids. This is of course possible for very very simple situations and cannot be envisaged in general. We shall consider here so called "two fluid" models (in Eulerian formulation) obtained t,hrought an ad hoe averaging process. That is at each point in space and time, the modelization assumes that both fluids are present and a parameter (which is also t o be determined) represents the proportion of each fluid. Such a model can be derived by an averaging process and we refer to e.y. Ishii [ISH 751, Ransom [RAN 891, Drew and Wallis [DRE 941 for more details. The models rely on the three usual balance equations (mass, momentum and energy) for each fluid. Denoting with subscripts v and 1 the quantities attached to one of the two fluids (e.g. density, velocity, energy,..) and by a,, a1 the volume fraction of each fluid : a , a[ = 1, these equations reads as follows ( k E {v, 0).

+

Balance of mass.

where pk is the density of the fluid k, uk the velocity of the fluid k and rk the density of the mass transfert to the fluid k resulting from interfacial exchanges with r, + ri = 0. B a l a n c e of m o m e n t u m .

where P is the pressure in the mixture, T k the viscous stress tensor, g the gravity field, Ik the volumic density of the momentum transfert to the fluid k resulting from interfacial exchanges (after substraction of the pressure contri-

72

Finite volumes for complex applications

bution) and I,

+ Ii = 0

Balance of total energy

where e k is the specific internal energy of the fluid k , h k the specific enthalpy of the fluid k , a k Q k the volumic heat transfered to the fluid k , a k q k the heat flux in the bulk of the fluid k , IIk the volumic density of the energy transfered 111= 0 . to the fluid k resulting from interfacial exchanges with II,

+

Once this set of differential equations written, it remains to close the system by adding enought closure relations in order to obtain the same number of equations and independent unknowns. At the present time, as emphasized in the Introduction, it is an open problem. However by using various methods it is indeed possible t o obtain, in definite physical contexts, some closed models from the equations above. These models will in general leave free a few number of constants that are t o be determined by experiments (correlations). Hence the numerical modelisation of two fluid flows must be strongly coupled with experimental programs. Basic models

We give below three typical models (that can be written in 1, 2 or 3 space dimensions). These models are very simple in the sense that they do not involve complicated physical correlations in their right hand side. Actually these models are used in the context of numerical benchmarks designed in order to check that the numerical method used is able to capture some of the important features of the flow considered and also in order to evaluate the numerical dissipation of the method. We refer t o [MIM 991 for a collection of numerical benchmarks.

2.1. A 3 equation model This is the so called homogeneous 3 equation model that we write for the sake of simplicity in 1D : ap ~ ( P u=) 0 , at

-+- ax

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Here p represent the density, u the average velocity, u, the relative velocity a heat flux (source between phases, c the quality, L the latent heat and term). See also Example 1.

2.2. A

4

equation model

This two-fluid model reads as :

Here p k represents the density of the phase k , u k its velocity, p is the pressure, and a k denotes the volume fraction of the phase k , ( a , a t = 1 ) . T h e relative velocity is simply u, = u, - ue. Here, the source term is a drag force which is in g equations (9) and ( l o ) , where R b is given by the correlation k = gcrupe the bubble radius and CD the drag coefficient.

+

2.3. A 6 equation model In the previous two fluid model, no energy balance equations were written. This corresponds t o isentropic flows. Here we want t o consider the full set of balance equations (mass, momentum and energy) and this leads to a so called six equations model (which lead in 3D t o a set of 10 scalar evolution partial differential equations). Let us consider the following system of equations :

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Finite volumes for complex applications

~(@,P,E,) at

a(aepe at

+ diu (a,p,H,u,)

- p-

8%

at

+ d i u (arpeHeui) - p-aae at

-

- agp,y.u,,

=a~p~g.uj.

(16)

We still have to describe the thermodynamics of this "one-pressure" model. To this extent, we assume that two pressure laws are given, one for the "gas" and one for tlie "liquid" phase :

Here e k , denotes the internal phase energy and Ek f ek + $ IUk

12,

Hk = E k + P. Pk

3. The finite volume approach

The goal of this section is to describe the program that lead to the full discretization of the models of Section 2. We shall not present numerical results refering to the bibliography for that purpose : [AGT 991, [BOU 981, [GI
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3.1. The conservative case Althought almost all the models encountered in two phase fluid flow are non conservative, we are going to deal first with conservative ones since they are convenient in order t o introduce the basic notations. Let us consider a system of m balance equations : ( v = (vl , . . . , v,) E Rm))

~ yy z , ~

here V . F ( v ) = where F j maps W m into Rm. This equation is posed in a nd-dimensional domain R ( n d = 1 , 2 or 3 in practice).

to (6) enters in this category. AsExample 1 The homogeneous model (4) . . suming that a thermodynamic law p = P ( p , e ) is given, and when u, = 0 , one recovers the usual Euler equations. In the case where u,. # 0 , we have also to prescrzbe the quality c = C ( p , e ) and in this case, we cun write (4) to (6) into the form (18) provived we set v l = p, va = pu, u~ = p E , F,' = pu Fi =p ( 7 + ~ ~c(1 - c)u:) + p, F i = p ( u H + u,.c(l - c ) ( L + 21u, + - 2c)u;)). Here E e + $ luJ" H = E +:. It sho.uld be noticed that even when u, ,is a fixed constant and even for the physical laws of water, this system is not necessarily hyperbolic [TAJ 981. ,

,

(4

We assume that the computational domain R is decomposed into smaller volumes (the so called control volumes) Ii' : R = U K E T K and, in a first step, we assume that R = U K E T K is "conformal" 2.e. that it is a finite element triangulation of 0 . In practice one can use triangles for nd = 2 and tetrahedrons for nd = 3. The cell-center finite volume approach for solving (18) consists in approximating the means

where ciol(Ii) denotes the nd-dimensional volume of I< and ui.eu(A) stands for the ( n d - 1)-dimensional volume of an hypersurface A. Integrating (18) on I< makes the norinal fluxes, F&, appear

where i ) I i is the boundary of I<, u ( u ) the unit external normal on d I i and du denotes the (nd-1)-volume elenlent on this hypersurface. Indeed, we have

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Finite volumes for complex applications

according to (18)

-+dt

~Z'K

1 FIK= 0 vol (K)

The heart of the matter in finite volume methods consists in providing a formula for the normal fluxes FaVKin terms of the { v ~ ) LAssuming ~ ~ . that the control volumes I< are polyhedra, as it is most often the case, the boundary dIC is the union of hypersurfaces IC n L where L belongs to the set ni(lil),the set of L E 7 ,L # Ii', such that Ii' n L has positive (nd - 1)-measure. We can therefore decompose the normal flux as a sum :

where ( v K , points ~ into L)

Motivated by the case where (18) describes wave propagation phenomena, it is natural to look for an approximation of (23) in terms of v ~ ( t and ) v ~ ( t :)

F K ,=~aarea(Ii' n L) @ ( v K , v ~Ii',; L) ,

(24)

where @ is the numerical flux to be described. At this level of generality @ must satisfy 2 structural properties : (i) consistency : @(w,w; Ii',L ) = F(w) . VK,L, (ii) conservation: @(v,w; Ii',L) = -@(w, v; L , Ii'). 3.2. The characteristic flux approach ([GIi'L 961)

Let us consider the equation

where v E R m and f : IRm A(v) the jacobian matrix according to (25),

++

IRm: (2.e. (18) ) when nd = 1 ) . We denote by then and observe that since A ( V ) % =

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This shows t h a t the flux f ( v ) is "convected" by A ( v ) (like v since we also have av A ( v ) % = 0 ) .

+

Let us assume t h a t (25) is smoothly hyperbolic that is t o say : for every v there exists a smooth basis ( r l ( u ) ,. . . , r,(u)) of R m made with eigenvectors of A ( u ) : 3 X k ( v ) E such t h a t A ( u ) r k ( v ) = A k ( ~ ) r k ( u )I.t is then possible t o construct (11 ( u ) ,. . . ,lm ( u ) )such that t A ( v ) l k(v) = Xk(v)lk(u) and l k ( v ) . r p ( u )=

dk,P. T h e ~lumericalflux Q, represents t h e flux a t an interface. Using a mean value p of v a t this interface, we replace ( 2 6 ) by the linearization :

It follows t h a t setting f k ( u ) characteristic flus, we have

l k ( p ) . f ( u ) , whic11 is ternled as the k-th

Solving exactly this last equation (see also Remark 7), leads to t811efollowing formula :

Remark 1 Non hyperbolic convection operators In the complete physical nzodels, the equations for the balance of m,omenttinz and energy (see (2) and (3)) invol.ue terms with second order gradients like e.g. molecular and tllrbulcrzt dissipation and dzffusion. Here we are o d y d~isczissing the conoectrire part, i . ~ . terms with first order gmdients, see Section 9.8 co~zcerizingthe discretizutio~z of second order gradzents. It might happerz (this is very often the case when dealing with two phase flow, [STE 8411,that this conuectior2 operator is riot hyperbolic (see also Example 1). That is the matrix 24(.u) nzlght have co~nples ezgenvalues. When it is the case, we keep formula (29) unchanged but this tiine, X k ( v ) denotes the real part of the k-th eigenualue of A ( v ) . The physical meaning of this is that we still identify these numbers as physically meaningful wave speeds.

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Finite volumes for complex applications

3.3. Flux schemes

We einphasize the fact that formula ( 2 9 ) has the following reinarkable property : the numerical flux at an znterface @ is a linear combination of the fluxes f ( v ) and f ( w ) on both sides of the interface. This has led to the definition of flux schemes ([GHI 981).

Definition 1 The numerical flux @ corresponds to a flux scheme when there exists a matrix U ( v ,w; IC,L ) such that

The extension of formula ( 2 9 ) to the lnultidimensional case is therefore given throught the following definition.

Definition 2 The nunzerical flux of the "VFFC" nzethod is obtuined by formula (30) when we take : U ( v ,w ;I<, L ) = sgn where p ( v , w; I<, L ) is a mean between v~ and v~ which only depends on the geometry of Ii and L :

Remark 2 In formula (31) the sign of a matrix M is the matrix which has the sume generalized eigenspaces as M but whose eigenvalues are the sign ( - 1 , 0 or. + 1 ) of the corresponding eigenvalues of M . When the ezgenstructure of Dl is not enplicitely known, this matrix can be rzumerically coii~putedin two diflerent ways, . First one can use a numerical package that determines thzs eigenstructure. Second one can use a very eficient iterative method (a variant of the Newton-Schultz algorithm, see [ALO 991) which produces a polynomial sequence of matrices converging rapidly to the matrix s g n ( M ) . Remark 3 Other flux schemes Many other numerical schemes ure flux schemes (see (GHI ,981). They have in general a very similar behavior as fur as numerical accuracy is concerned. For example Roe's scheme is a flux scheme. For complex models like those considered here, determinzng Roe's average can be a

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dificult challenge while our VFFC scheme involrie a simple geometric average. We note also that Stadke et a1 [SFW 971 have proposed also u flux scheme (for a particular system) with a simple average. Concerning Roe's scheme, Toumi [ T O U 921 has derived very eficient procedure in order to use this scheme in the context we consider here. Later Halaoua [HAL 981 has generalized Roe's scheme t o non hyperbolic convection operators. We particularly recommend the paper by Toumi, lilurnbaro and Paillere [ T I P 991 for a very nice review on these questions (see also [GLT 991. Proposition 1 Combining ( ' I ) , (221, (24), (30)' (31) and (321, the finite uolume approszmation of (18) is the following system of 0.d.e. 's :

3.4. The non conservative case

Let us now consider a system of m balance equations which reads as :

and which therefore generalizes ( 1 8 ) . The terms ~ , ( v ) % are called non conservative products, and they introduce new difficulties both from theoretical and practical point of view. Often, such terms come from modelisation (e.g. approximations, closure relations,. . . ) procedures and according to the way they are dicretized, one can obtain non physical solutions or even numerical divergence phenomena. Such terms are present in almost all the models encountered in t w o phase flow Auid dynamics and they represent one of the major difficulty in this area. Example 2 The two fluid model (7) to (10) enters in this category when k = 0 (otherwise we just have a source term on the right hand side of (34)). Here we take vl = a , p V , v2 = aepe, vs = ~ ~ , p , u , , v4 = aepeue, F: = ~ U ~ U U U = aepptie, F'; = a v ( p , ~ p~) , F: = a t ( p e ~ ; p) and the matrix Cl (v) is

+

+

,

80 Finite volumes for complex applications

Remark 4 Of course the form 34 is not unique since one can add an arbitrary function of v, G, t o F and substract V,G(v) t o the matrix C ( v ) . Since the function F which appears has a physical meaning (the so called physical flux), zts form is chosen on physical ground rather than on an abstract mathematical principle. For example the choice made i n Example 2 is dictated by the property that when we have only one phase which is present (e.g. when a , G 0 ) one recovers the classical fluxes of the Euler equation. Actually, other non conservative products can occur, and the general system of equations reads as :

Example 3 For the 3-dimensional case ( n d = 3 ) , we can recast system (11) to (16) under the form (35) with rn = 10 ; vl = a,p,, vz = aepe, vs = avpuuv,l,' ~ 4= aipiUe,i, v5 = awpvuv,z, = aepeue,z,v7 = ~ U P U ~ . L , Sus , = aepeue,s, v g = a,p, E, and vlo = aepeEe.Then one gets the F; like i n Example 2 (they are given expilcitely i n [BOU 981).

+

Assuming t h a t t h e matrix I ~ ( v is) invertible (which is the case in practice), we can transform easily ( 3 5 ) into (34). Hence we can only discuss t h e discretization of this last equation. In the one dimensional case, equation (34) reads as :

Denoting this time by J ( v ) the jacobian matrix

we have instead of (26),

+

where A ( v ) J ( v )( I C ( v )J ( v ) - l ) . Hence, in the multidimensional case, formula ( 3 1 ) is replaced b y

and A,(v) C ( v ). v G

+

J,(v)(I C,(v) J,(v)-') where J,(v) = vj.Cj(v).

nd

a F v .v

and Cv(v) =

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Remark 5 Of course when C

81

0 both formulas (31) and (38) agree.

Now that we have a formula for computing the numerical flus at an interface Ii n L , we have to discuss the discretisation of the non conservative products. Integrating equation (18) on a volume control I<, leads to the question of approximating the integral JK ~~~1 C j (v) dx. Following [BOU 991 (see also [BOU 98]), we take :

where E K , L is the following m x m matrix :

--

E K , L C V ( v J~, () ~ ) - ' , u = Y K , L and p = ~

K , L

Proposition 2 In the non conservative case, the finite volume crpproaimatzort of (34) is the following system of 0 . d . e . ' ~:

3.5. Fully discrete schemes

In the previous Section, we arrived to systems of 0 . d . e . ' ~((33 and (41)). In order to actually compute the solutions, we have to perform a time discretization. This discretization can be either explicit or implicit. As it is well understood, the explicit discretization of the convective flux will lead to a stable scheme only under a Courant-Friedrichs-Lewy (C.F.L.) condition. This condition will impose on the time step a very restrictive constraint. Namely that this number must be smaller than Ax/c where Ax is a measure of the smallest length in the space discretization and c a measure of the largest speed of propagation appearing in the physical problen~(note that in liquid and vapor flows this speed can be of the order of 2000 ms-l). Hence when there are long transients, explicit methods are not effective in this context.

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Finite volumes for complex applications

We consider (34). The previous Section was devoted to the space discretization of this equation and we arrived to (41). The Euler explicit scheme for this equation reads :

As already said, this scheme is stable only for small time step. An alternative is then Euler implicit schemes. We solve the following equation for v;+' :

Althought of interest, this method can be too diffusive since all the characteristics are made implicit regardless the magnitude of the characteristic speed. We have proposed in Ghidaglia, Kumbaro, Le Coq and Tajchman [GI
Definition 3 In the multidimensional case, the numerical flux o f t h e "implicitexplicit VFFC" scheme is given by

where the m x rn matrix U,, a E { 1 , 2 , 3 , 4 ) are defined as follows

with fi defined by

the eigenelenaents (1,r and A ) refer to the jacobian nlatrix the w,Q are given according to the following cases.

w:

= 1, w; = 0 for

(Y

E {2,3,4).

dF(Ua)vV~.L l u Z p

and

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(22) When - K d L , ~ / ( t n + l- t n ) 5 X k ( p ) < 0 . w ~ = l , w t = Of o r a ~ { 1 , 3 , 4 ) . (iii) When

Xk

( p )= 0. w: = w; =

(iv) When

Xk

(48)

-,1 2

W;

= 0 for a E ( 3 ' 4 ) .

(49)

( P ) > ~ d K , ~ / ( t n+ tl n ) . K~K,L

n+l

-

,Wk

tn)Ak(~)

-l -

- w ~ = ,O ~for ~a E { 2 , 4 ) .

(50)

Above, bK,L denotes a characteristic lenght in It' which depends also on L and K is a nondzn~ensionalnumber between 0 and 1 t o be specified (see Remark 6).

Remark 6 We can take for example (nd 2 2)

and K is determined thanks to the stability limit of the explicit scheme, see Boucker [BOU 981 (for example K x 0.3 for ~ z d= 3).

Remark 7 the one dimensional case When nd = 1 , formulas (44) to (51) are changed as follows : s j ( v ; , v ; + l , ~ ; + l , v j ; : l ) = Ul f ( v 7 )

+ U 2 f ( . ~ : y ++~ )

+

~ 3 f ( ~ J n + l ulf(vjn++:) ) 1

(53) where the m x m matrix U,,

(Y

E ( 1 ' 2 , 3 , 4 ) are defined as follows :

with p is defined by

and w t according to the following conditions

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Finite volumes for complex applications

(2) When 0

< Xk (P) 5

( I j1/2(tn+l - i n ) . w i = 1 , wf = 0 f o r a E { 2 , 3 , 4 ) .

(56)

(ii) When -lIj+1)/2(tn+l - t n ) I h ( p ) < 0 . w: = l , w ; = O f o r c u ~ { l , 3 , 4 ) .

(57)

(iii) When X k ( p ) = 0. w; = w i =

1 2

-, wf = o for a

E {3,4).

(iv) When h ( p ) > IJjl/2(tn+l - i n ) .

In the case where the C.F.L. conditions :

are satisfied, formula (29) follows from these considerations.

3.6. O n the discretization of source terms

Instead of ( 2 5 ) , we want to solve

by the numerical scheme :

(58)

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85

Equation ( 6 3 ) can be obtained by integration of ( 6 2 ) on the rectangle [t,, t,+l] x [ ~ j - ~ /z2 ~, + ~and / ~this ] leads t o the following form for Cy :

Actually, this formula is not suitable when the source term is large and one must modify ( 6 4 ) according to the expression of the numerical flux gj where f7+1/2= g j ( ' ~ : + l ,vJjL) and

where U ( . ,.) is a inatrix t o be specified.

Due to t,he fact that the ilurllerical flux is not centered (upwind bias) one can observe large errors on the permanent solutio~lunder investigation. Let us introduce the notion of enhanced consistency as follows. Denoting by

we say that C: satisfies, with respect to the numerical flux g , the enhanced consistency property when we have : if a t some time-step, v; is such that

then tryt1 given by ( 6 3 ) must be equal t o 11;. This condition can be equivalei~t,ly formulated as

cj"= f J n t l / 2

- f?-l/:!

A xj

(68)

if vjn satisfies ( 6 7 ) . In Alouges, Ghidagliaand Tajchmari [AGT 991, the following result is sllow~t.

the nlatris Theoren1 1 Let ( g j ) be given zn the form (65) and denote b y U j + + ( u 7 ,21j"+~).T h e enhanced consistency (2.e. (68)) will be satzsfied if we discretrze the forcing t e r m S according t o the following formula

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Finite volumes for complex applications

Observe that from the computational point of view, formula (69) induces almost no extra cost since Uj++ has already been computed in order to construct the numerical flux. In order to illustrate formula ( 6 9 ) , let us show what it means in the case of a single linear equation vt c v, = S where e.g. c > 0 and v E R. Here the usual upwind scheme amounts to take Ujn = 1, i.e. g ( v , w )= c v and ( 6 9 ) reads as :

+

3.7. On the boundary condztions

So far we have not discussed the implementation of boundary conditions. This is a very important matter since they actually determine the solution. Let us consider the space discretization of system ( 1 8 ) by our cell centered finite volume method. We have found the sytem of o.d.e.'s (33). Of course this evolution equation is not valid when I< meets the boundary of 0. When this occurs, we have to find the numerical flux @(vK,I{, da).In practice, this flux is not given by the physical boundary conditions and moreover, in general, ( 1 8 ) is an ill-posed problem if we try to impose either v or F(v) . v on dR. This can be simply understood on the following linearisation of this system :

where v represents the direction of the external normal on jacobian matrix :

K n dR, J v is the

and 2 is the state around which the linearisation is performed. When ( 1 8 ) is hyperbolic, the matrix +J is diagonalizable on R and by a change of coordinates, this system becomes an uncoupled set of m advection equations :

Here the ck are the eigenvalues of -J and according t o the sign of these numbers, waves are going either into the domain 0 (ck < 0 ) or out of the domain S2 (ck > 0 ) . Hence we expect that it is only possible to impose p conditions on

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87

Ii' n dR where p z #{k E ( 1 ) . . . , m) such that ck < 0 ) . Let us consider now a control volume Ii' which meets the boundary dR. We take q = V K and write the previous linearisation. We denote by x the coordinate along the outer normal so that (71) reads :

which happens to be the linearisation of the 1D (i.e. when nd = 1) system. First we label the eigenvalues c k ( q ) of J-, by increasing order :

(i) The case p = 0. In this case all the informations come from inside R and therefore we take :

In the Computational Fluid Dynamics litterature this is known as the "supersonic outflow" case. (ii) The case p = m. In this case all the informations come from outside and therefore we take :

R

where are the given physical boundary conditions. In the Computational Fluid Dynamics litterature this is known as the "supersonic inflow" case. (iii) The case 1 5 p I m - 1. As already discussed, we need p informations coming from outside of R. Hence we assume that we have on physical ground p relations on the boundary :

Remark 8 The notation gl(v) = 0 means that we have a relation between the components of v. However, in general, the function gl is not given explicitely i n terms of v . For example gl(v) could be the pressure which is not, i n general, one of the components of v .

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Finite volumes for complex applications

Since we have to determine the m components of @ ( v K Ii', , d a ) , we need m - p supplementary informations. Let us write them as

In general (78) are termed as "physical boundary conditions" while (79) are termed as "numerical boundary conditions". Then we take : @(vKK , , dQ) = F ( v ) . V K , (80) where v is solution to (78)-(79)(see however Remark 10 and ( 8 6 ) ) . Remark 9 The system (78)-(79) for the m unknowns v E G is a m x m nonlinear system of equations. We are going to study its solvability, see Theorem 2. Let us first discuss the numerical boundary conditions (79). By analogy with what we did on an interface between two control volumes K and L, we take (recall that g = v K ) :

In other words, we set hk ( v ) 5 lk ( u ~ ) . ( F ( v ) . v ~( )v -~~) ~. ( F ( v ~ ) .We v ~ have ). denoted by ( I l ( g ), . . . , l,(g)) a set of left eigenvectors of J- : tJ-lk ( g ) = cklk ( 2 ) and by ( r l ( 2 ), . . . , r , ( g ) ) a set of right eigenvectors of & : &rk (2)= ckrk (21). Morover the following normalization is taken : l k ( g ) . r p ( g )= Sk,p. According to Ghidaglia [GHI 991 we have the following result on the solvability of (78)-(79). Theorem 2 Assume that

With the choice (81) the nonlinear system (78)-(79) has one and only one solution for v - 2 sufjciciently small. Remark 10 In practice, (78)-(79) are written in a parametrized manner. We have a set of m physical variables w (e.g. pressure, densities, velocities,. . .) and we look for w satisfying :

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89

and then we take : @ ( v KI<, r afl) = a . The system (83)-(84)-(85) is then solved by Nezuton's method.

3.8. On the discretization of viscous fluxes

Our aim is to introduce an algorithm for the dicretization of second order elliptic operators in the context of finite volume schemes. The technique consists in matching t o a finite volume discretization based on a given mesh, a finite element volume representation on the same given mesh. An inverse operator is also built. This numerical algorithm, which was introduced in [ G P l 991,(see also [GP2 991,[GP3 991 and [PAS 991, is based on footbrzdges between the finite volume representation of functions and a finite element one, both defined on the same mesh. We are going to give some details in the two dimensional case, refering to the aforementioned references for a more complete account and for the general nd-dimensional case (which is in fact similar to the 2D case). Let S2 he the computational domain that we assume being triangularized :

S2 =

U T.

Let ; y be ~ the piecewise-constant characteristic function defined

TET

on each triangle T. For each edge A, let us define the piecewise-linear function (PA such that c p ~ ( M= ) 1 if M is the center of A and c p ~ ( M=) 0 if M is the center of an edge different from A. Let us denote by V, the finite volume approximation space, spanned by the X T , and by E , the nonconforming piecewise-linear finite element approximation space, spanned by the P A . T h e dimension of V is equal to the number of triangles, N E , and the dimension of E is equal to the number of edges, N F . To each v in V, we associate an element in E denoted by u = n E v u which is the solution to the least square problem : - DII

2

dx = min

Similarly, to each u in E , we first construct v = n v E u in V solution of :

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Finite volumes for complex applications

T h e finite element representation allow to discretize easily a viscous operator t.hanks t o a variational formulation and the footbridge U V Eallow t o resume to a finite volume discretization well suited for applying upwinding. But H v E o UEv # I d v , then in the case of a zero diffusion parameter, the solution is modified by the use of the two footbridges. We construct another footbridge II from 8 into V such that H o nEv= Idv. For that purpose, it is sufficient t h a t for each u E F in & t o find v V F in V solution of the system :

Our first observations are that IIEv is injective and that (89) has no solution in general. Let B be the N F x N E matrix such that Bj, = Jn xa pjdx and bEF the N F vector given by b f F = Ja u E F(pjdx, for j = 1, . . . , N F . T h e system ( 8 9 ) can then be written in the following way : BuvF = bEF. I t is a rectangular linear system with N E unknowns and N F equations. As asymptotically, N S N F 3 N E / 2 , a least square formulation :

y,

-

find u V F E V minimizing

1 1 ~ ~ -~ bEF[12 ' ,

(90)

is used t.o solve this system. Since the rank of B is N E , (90) has a unique solution u V F = p v E u E F . Its value, ( t ~ ~ ) t- ~l b E F has , low computational cost since t B B is a sparse matrix and a Cholesky factorization can be applied. Hence we get PVE o IIEV = I d V .

3.9. Non conformal meshes

Nor1 conformal mesh techniques were investigated in two situations when partitionning the whole domain into several sub-domains, allow a non conformal junction between meshes of adjacent sub-domains (see figure 1.b); use local refinments, keeping the structured type of the main discretization (see figure 1.a). Flux schemes have interesting features in these situations :

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I

I

I

I

a. non conformal refinment

1

91

/

b. non conformal junction

Figure 1: Non conformal mesh examples

First, they are based on a finite volume, cell-centered, discretization (i.e. no need to interpolate node values). Second, the main part of the algorithm consists in computing numerical fluxes passing through interfaces. So, we have to build once at the beginning an explicit (unstructured cases) or an implicit (structured cases) list of interfaces. At each timestep, the main loop will use this list t o compute numerical fluxes and put flux contributions to interface neighbours. For unstructured meshes, the same algorithm can be applied to conforn~alor non conformal meshes. For structured meshes with local refinments, a treelike structure has been tested with simulated recursive procedures (see figure 2 which represents the tree associat,ed to figure 1.a). We refer to Tajchman and Freydier [TAJ 961 for more information.

4. Bibliography

[ALO 991

ALOUGES F., Matrice signe et systkmes hyperboliques, in Aspects the'oriques et pratzques de la simulation nume'rique de quelques problemes physiques, 249-257, Mhmoire d'Habilitatior1, Universite Paris-Sud, 1999.

[AGT 991

ALOUGES F . , GHIDAGLIA J . M . 4 N D TAJCHMAN M . , On the interaction of upwinding and forcing for nonlinear hyperbolic

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Finite volumes for complex applications

12 x 3 sub-mesh Figure 2: Tree representation of a local refinment

systems of conservation laws, Prepublication du CMLA and article to appear, 1999.

[BOU 981

BOUCICER M . , Mode'lisation nume'rique multidimensionnelle d'e'coulements diphasiques liquzde-gaz en r6gim.e~transitoires et permanents : me'thodes et applications, Thhse, ENSCachan, France, 1998.

[ROU 991

BOUCI<ER M . A N D GHIDAGLIA J.M., 011 the discretization of non conservative terms in the context of multidimensional flux schemes, in preparation.

[ D R E 941

DREWD . A . A N D WALLIS G . B . , Fundamentals of Two-Phase Flow Modeling, in Multzphase Science and Technology, Volume 8, Hewitt a1 Editors, Begell House, Inc., New-York, 1994.

[GAL 961

GALLOUET T., Rough schemes for complex hyperbolic systems Proceedings of the j r s t symposium on Jnite volumes for complex applications 15-18 july 1996, 1996, Editions Hermes Paris.

[GHI 981

GHIDAGLIA J .M., Flux schemes for solving nonlinear systems of conservation laws, Proceedings of the meeting in honor of P.L. Roe, Chattot J . J . and Hafez M. Eds, Arcachon, July 1998, to appear.

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93

[GHI 991

GHIDAGLIA J . M . , O n the numerical treatment of boundary conditions i n the context of flux schemes, In preparation.

[GKL 961

GHIDAGLIA J . M . , KUMBARO A. ET LE COQ G . , Une mCthode volumes-finis 2 flux caract6ristiques pour la rCsolution numtrique des systkmes hyperboliques de lois de conservation, C.R.Acad. Sc. Paris, 1996, 322, I, 981-988. GHIDAGLIA J . M . , KUMBARO A , , LE COQA N D TAJCHMAN M . , A finite volume implicit method based on characteristic flux for solving hyperbolic systems of conservation laws, Proceedings of the Conference otz : Nonlinear evolution equations and infinite-dimensional dynumical systenzs (Shangal, June 1995), Li Ta-Tsien Ed., 1997, World Scientific, Singapore. Ghidaglia J.M., Le Coq G . and Toumi I., Two flux-schemes for computing two-phases flows throught multi-dimensional finite volume methods, Nuclear REactor Thermal Hydraulics (NURETH 9), October 1999 A N D PASCAL F., Passerelles volumes finis C.R.Acad. Sc. Paris , I , 328, 711-716, 1999.

GHIDAGLIA J.M. - e'lkments finis,

GHIDAGLIA J . M . A N D PASCAL F., Passel-elles uolun~esfinls - e'le'ments finis, nze'thocles et applicattons, Rapport EDF/DER/TTA/HT-33/99/002/A, Chatou, France, 1999. [GP3 991

F., Footbridges between finit.e GHIDAGLIA J . M . A N D PASCAL volumes and finite elements, Article to appear.

[HAL 981

HALAOUAI<., Quelques solveurs pour les opPrute,urs cle convection et leurs applications a la ~ne'caniqnedes Jluzdes diphasiques, These, ENS-Cachan, France, 1998.

[ISH 751

ISHIIM., Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris, 1975.

[KUM 971

KUMBAROA., Me'thode VFFC, application au problenze du robinet de Ransom, Rapport EDF/DER/TTA/HT33/97/021/A, Chatou, France, 1999.

[MIM 991

MIMOUNIS., Quelques cas tests nume'riques pour la simulation des 6coulements diphasiques, Rapport EDF/DER/TTA/HT-33/98/031/13, Chatou, France, 1999.

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[PAS 991

PASCAL F., Passerelles Volumes finis - itlitments finis, Proceedings of the 1lhe seminaire sur les e'coulements compressibles et la me'canique des juides nume'rique, CEA Saclay, Janvier 1999.

[RAN 891

RANSOMV. H ., Numerical Modeling of Two-Phase Flows, Cours de 1'Ecole d'kte d'Analyse Numitrique - CEA-INRIAEDF, 12-23 juin 1989. STADICEH . , FRANCHELLO G . A N D WORTH B., Nilmerical simulation of multi-dimensional two-phase flow based on flux vector splitting, Nucl. Eng. Design, 1997, 1 7 7 , 199-213.

[STE 841

STEWARTH . B . , WENDROFFB ., Two-phase flows : models and methods, J. Comput. Phys., 1984, 5 6 , 363-409.

[TAJ 961

TAJCHMAN M. A N D FREYDIERP . , Me'thode Volumes Finis 6 Flux Caracte'ristiques. Application ci un calcul bidimensionnel sur un maillage non conforme, Note, Ditpartement Transferts Thermiques et Aitrodynamique, Direction des Etudes et Recherches, Electricit6 de France, EDF HT-30/96/004/A, May 1996.

[TAJ 981

TAJCHMANM . A N D FREYDIERP., Sche'ma VFFC : upplication ci l'e'tude d'un cus test d'e'bullition en tuyuu dl-oat repre'sentant le fonctionnement en bouillotte d'un coeur REP. Note E D F HT-33/98/033/A, ditcernbre 1998.

[TOU 921

TOUMI I., A weak fornlulation of Roe's approximate solver, J . Comput. Phys., 1992, 102, 360-373.

[TKP 991

TOUMI I., KUMBAROA.

A N D P A I L L ~ H., R E Approximate Riemann solvers and flux vector splitting schemes for twophase flow Note presented at the 3oth VKI CFD Lecture Series, 8-12 March 1999.

The MOT-ICE:a new high-resolution wave-propagation algorithm based on Fey's Method of Transport

Sebastian Noelle Institut fur Angewandte Mathematik Universitat B o n n Wegelerstr. 10 531 15 Bonn, Germany [email protected]

The numerical solution of multi-dimensional systems of conservation laws is dominated by upwind-schemes based on one-dimensional Riemannsolvers. Such schemes neglect the physical directions of wave-propagation and replace them by the grid-directions. One of the algorithmic alternatives is Fey's Method of Transport (MOT). Similarly to its one-dimensional fore-runner, the Steger- Warming-Scheme, the M O T suffers from an inconsistency at sonic points. Here we derive a new version of Fey's multi-dimensional flux-vector-splitting and a globally consistent second-order-accurate characteristic scheme based o n Interface-Centered-Evolution, the MOT-ICE. Numerical experiments show the stability, accuracy and efficiency of the new scheme. ABSTRACT

K e y Words: Systems of Conservation Laws, Multi-dimensional Flux- VectorSplitting, Fey's Method of Tranport.

1. Introduction

Since the work of Godunov, Van Leer, Harten-Lax and Roe, the numerical solution of systems of hyperbolic conservation laws is dominated by Riemannsolver based schemes (see [HIR 901 for the classical references). These onedimensional schemes are extended to several space-dimensions either by using dimensional-splitting on cartesian grids or by the finite-volume approach on un-

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structured grids. For both approaches, convergence and error estimates have been established for multi-dimensional scalar conservation laws, see for example the results and references in [CCL 94, KNR 95, NOE 96, WN 991. Naturally, there are no comparable results for multi-dimensional systems, since no existence and uniqueness of the p.d.e.'s is known in this case. The first systematic criticism of using one-dimensional Riemann-solvers for multi-dimensional gas-dynamics goes back to Roe himself [ROE 861: the Riemann-solver is applied in the grid- rather than the flow-direction, which may lead to a misinterpretation of the local wave-structure of the solution. A description of a number of failings of exact and approximate Riemann-solvers for the two-dimensional Euler-equations of gasdynamics may be found in Quirk's paper [QUI 941. LeVeque and Walder [LW 921 present difficulties of Godunov's scheme for strong two-dimensional shockwaves arising in astrophysical flows and propose the use of rotated Riemann-solvers. In [ROE 91, NOE 941 Roe and Noelle study oscillations generated by dimensional-splitting-schemes for a prototype linear system. Since the mid-eighties, Roe, Deconinck, Van Leer and many others developed the so-called fluctuation-splitting schemes for the equations of gasdynamics. In these schemes the divergence of the multi-dimensional flux-vector (called fluctuation) is split according to a grid-independent wave model (see [VNL 921 for references). In [NOE 941, the author developed a fluctuationsplitting-scheme for a 2x2 model problem which cures the directional effects exhibited by dimensional-splitting-schemes. However, this fluctuation-splittingscheme seems t o be very dissipative for non-stationary problems. Another multi-dimensional approach is the Corner-Transport-Upwind (CTU) scheme of Colella [COL 901. Here waves are not only propagated to the neighbors which join a common side with a cell, but also to the corner cells. LeVeque's wave-propagation-algorithm, which forms the backbone of the publicly available software-package CLAWPACK, splits one-dimensional fluxes in the transverse directions and propagates them to the corner cells (see [LEV 971 and the references therein). In [BT 971, Billet and Toro introduce weightedaverage-flux (WAF) schemes with good multi-dimensional upwinding- and stability-properties. In this paper we follow the flux-vector-splitting approach, which was first proposed by Steger and Warming [SW 811 in one space-dimension. Instead of decomposing the divergence of the flux-vector, as in the fluctuation-splitting schemes of Roe et.al., the conservative variables and the flux-vector themselves are split (see [HIR 901, Ch.20.2). In his dissertation, Fey developed a multidimensional version of the Steger-Warming-scheme, and called it Method of Transport (MOT). This scheme integrates the acoustic waves over the entire

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Mach-cone (see also [LMW 971 for recent related progress). Subsequently Fey, Jeltsch et.al. simplified the MOT and expanded it in various directions (for references, see [FJMM 97, FEY 98a, FEY 98bl). The starting point of this work are the papers [FEY 98b, FJMM 971. In these papers, the MOT takes the following form: Step 1. A multi-dimensional wave-model leads to a reformulation of the conservation law as a set of nonlinearly coupled advection equations. Step 2. At the beginning of each timestep, the system is linearized and decomposed into a set of scalar advection equations with variable coefficients. Step 3. The solution of each scalar advection equation a t the end of the timestep is computed using a characteristic scheme. Step 4. The solution is projected back onto the conservative variables using the wave-model of Step 1. Our own contribution to Fey's method may be summarized as follows: Given a multi-dimensional wave-model, we simplify Step 2 by distinguishing clearly between linearization and decomposition error. This makes it possible to write down a general second-order-correction term for the decomposition error in a single line. Then we discuss the first-order version of Fey's characteristic scheme (Step 3 ) and show an inconsistency of the numerical scheme on the scalar level. Subsequently, we derive new first- and second-order characteristic schemes which are globally consistent. Due to the simpler second-ordercorrection term for the decomposition error and a particularly simple choice of numerical transport-velocities our second-order scheme needs only 2.2 times the CPU-time of the first order scheme, and is very efficient. Fey's transport algorithm might be called MOT-CCE, since his scalar scheme uses Cell-Centered-Evolution. We call our new scheme, which is based on Interface-Centered-Evolution, MOT-ICE. In Section 2, we discuss the multi-dimensional wave-models and the linearization and decomposition steps. In Section 3, we show the inconsistency of the MOT-CCE and present the MOT-ICE. In Section 4, we give numerical experiments. Finally, in Section 5, we discuss topics for future research. This is a report on work in progress. The proofs of the theoretical results in Sections 2 and 3 as well as several important implementation details in Section 4 will be given in the forthcoming paper [NOE 991. The author would like to thank Dr. Michael Fey and Christian von Torne for stimulating discussions on the Method of Transport. This work was supported by DFG-SPP ANumE. The hospitality of Dr. Fey, Prof. Jeltsch and his group during a visit a t ETH Ziirich is gratefully acknowledged. 2. Decomposition of multi-dimensional hyperbolic systems

into scalar advection equations In this section, we recall Fey's advection form and give a general secondorder-accurate linearisation and decomposition of systems which can be written in advection form.

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2.1. A general framework for multi-dimensional flux-vector-splitting

Consider a multi-dimensional system of n conservation laws in d space dimensions, where U : R~ x R+ -+ Rn is the state-vector and f : Rn -+ R~~~the fluxvect,or. The following framework for multi-dimensional flux-vector-splitting (FVS) has been developed by Fey, Jeltsch and collaborators [FJMM 971: Definition 1 A wave-modelfor (1) is a set of L 2 1 mappings

which satisfy the consistency conditions

and

Given a wave-model, the conservation law (1)may be rewritten in the following advection form: L

C

(at

Sl (U) + V . (Sl ( U ) a1 ( W T ) )= 0.

(4)

1=1

Wave-models for the wave-equation, the equations of isentropic and non-isentropic gas dynamics and the equations of ideal magneto-hydrodynamics may be found in [FJMM 971. Common to all of them is a finite set of acoustic waves which approximate the Mach-cone (typically four waves in two spacedimensions). For the Euler-equations one adds an entropy-wave, and for the MHD-equations, the two Alfvbn and possibly some slow magneto-acoustic waves.

2.2. Decomposition into scalar advection equations

Let us see how the advection form can be used when advancing the solution for a single timestep k = At. Ideally, one would like to set each summand in (4) to zero seperately. As we shall see below, this will lead to an error of 0(Ic2) during each time-step, limiting the overall accuracy of the scheme to first-order. The following linearisation will be helpful in order to obtain a second-orderaccurate decomposition:

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Lemma 2 (linearisation) Let U be a smooth solution of ( I ) , k = At

99

> 0. Let

satisfy

k

G = a l ( U ( t n ) )+ p

at

a r ( u ( t n ) )+ o ( k 2 )

and

Then V ( t n + k ) - U ( t n + k ) = (3(Ic3) The proofs of this and the following results will appear in [NOE 991. Note that the approximate transport velocities Zl are now prescribed coefficients which depend on space but not on time. System ( 5 ) is still nonlinear in V . It is, however, linear in the components S 1 ( V ) ,since V = Sl(V). Next let us decompose the linearized advection form ( 5 ) by setting each summand to zero seperately: Let

EL,

satisfy

Then

W ( t n+ k ) - V ( t n

k +k) = 2 V.

where Tl = Tl( U ( t n ) ,V . U ( t n ) ) is given by

Therefore, this naive ansatz leads to a first-order decomposition error. Taking a closer look a t the leading part of the error, we note that

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since

This can be used in order to derive a second-order-accurate decomposition of (5) respectively (4).

Theorem 3 (second-order-accurate decomposition) Let

satisfy at

2,

+ v . ( Z , GT) = 0

(tn) = St (U(tn)) + $ Ti (U(tn),V . U(tn)). Then z(tn

+ k) - u ( t n + k) = 0 ( k 3 ) .

For the Euler equations and the shallow-water equations, Morel, Fey and Maurer have also derived a second-order-accurate decomposition into linear advection equations. Instead of evaluating the velocities at the half-timestep, as done in Lemma 2, they freeze them at the original timestep. As a consequence, the linearization- and decomposition-errors are not seperated, and the secondorder-correction terms become more involved. 3. Solving the scalar advection equations by characteristic schemes

The linearisation and decomposition given in the previous section leads us, at the beginning of each timestep, to a set of scalar transport equations of the form

at cp + V . (cp aT) = 0.

(6)

For the rest of this section, we consider the velocity-vector a : R~ x R+ + R~ to be a given function of x and t. The function cp : Rd x R+ + R is the unknown, and initial data are prescribed at t = tn := nk,

These equations may be solved by introducing characteristics z ( r ;x, t) by

Since the flux-vector cpaT is always parallel to the characteristics, we have the following

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Figure 1: Backwards characteristic transport of cell K i j . The region bounded by the curved lines is z(t,; K i j ,tn+l), and the dashed lines are backwards characteristic curves issuing from the corners of cell Kij at time tn+1.

Lemma 4 For all K C Rd, t ,T E

R+

3.1 Characteristic schemes

We would like to use Lemma 4 to construct numerical methods for solving (6). For simplicity, we restrict the analysis from now on to two spacedimensions, and use the notation x = (x, y ) E R2 for the space variable and a = (a, b) E R2 for the velocity field. Let Kij C R2 be the cells of a uniform cartesian grid with mesh-size h = Ax = Ay, and recall that k = At is the timestep. Due to Lemma 4,

Note that KiJ3,n z(t,; K i j ,tn+1) is that part of Kitj, that will be mapped to Kij by the characteristic flow from time t, t o time tn+1 (compare Figure 1 ) .

Definition 5 A characteristic scheme for (6) is given b y

where c p is ~ a piecewise smooth spatial reconstruction of p(.,t,), and Kt!j, an approximation of Kit j f n z ( t n ; Kijlt,+l).

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For the piecewise smooth spatial reconstruction of p(.,t,) we will either use piecewise constant or piecewise linear functions. We will label the corresponding schemes P O resp. PI. For the moment, let us consider schemes using piecewise constant reconstructions and focus on the approximation of the characteristic flow. 3.2 Cell-Centered-Evolution: the Mo T- CCE

In a series of papers (see [FEY 98b, FJMM 971 and the references therein), Fey and collaborators have used the following approximation of the characteristic flow, which we will call Cell-Centered-Evolution: In each cell K i t j , ,consider the local characteristic flow defined by

Then set

..

Ki?j, ::= Kitj, n z;(t,;

K i j ,tn+l).

In one space-dimension this leads t o the following algorithm:

where X := k / h and the subscripts f denote the positive resp. negative part of a quantity. Fey calls his scheme MOT ("Method of Transport"), and therefore we will use the acronym MOT-CCE-PO. Let us discuss the consistency of the MOT-CCE-PO with the differential equation ( 6 ) . If a is constant, then (7) coincides with the first-order upwind scheme. For variable coefficients the situation is more complex:

Example 6 Let us consider the case

corresponding to a compressive wave (V . a

< 0). For constant initial data,

the exact solutaon remains constant in space but grows exponentially in time,

The approximate solution produced by the MOT-CCE-PO 2s

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As Ic ten.ds to zero with X fixed, this solution converges to

Thus (7) is inconsistent with the differential equation (6) at the "sonic" point x = 0, where the transport velocity a changes sign. The cell-centered-evolution leads to an inconsistent approximation of V . a, which takes the value 2 instead of 1. An analogous inconsistency occurs for the case of an expansive wave, V . a > 0. We omit the details. This difficulty was already described by Steger and Warming [SW 811 in 1981: the numerical flux produced by their splitting is not continuously differentiable at sonic points for the equations of gas-dynamics. This results in so-called "glitches" at sonic points. Subsequently, Van Leer developed a splitting with continuously differentiable fluxes (compare [HIR 901, Ch. 20.2.3). In her dissertation, Morel [MOR 971 also observed glitches a t sonic points for two-dimensional shallow-water-computations carried out with the MoT-CCEPO, and generalized the Van-Leer-flux-vector-splitting to two space-dimensions in order t o remove these numerical artefacts. However, this slows down the algorithm considerably, and Morel herself remarks that her method does not seem to be generalizable to second-order-accuracy. Finally, let us remark that in Example 6 above, the MOT-CCE-PO diverges in the Loo-norm, but converges in L1. Moreover, this inconsistency does not occur when one approximates the velocity field a by piecewise linear functions. 3.3 Interface-Centered-Evolution: the Mo T-ICE

In the following we will derive a consistent alternative to the cell-centeredevolution discussed above. Since the new scheme will approximate the transport velocities a t the interfaces between the cells, we will call it M O T - I C E for Interface-Centered-Evolution. Instead of using a cell-centered approximation of the local characteristic flow to decompose Kij, we now define auxiliary transport velocities ii;+4,j on the vertical interfaces

resp.

Fj+;

on the horizontal interfaces

between the cells. Let X = klh be fixed and suppose that the CFL-condition

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Figure 2: Auxiliary transport velocities and approximate characteristic decomposition of a two-dimensional cell K , via Interface-Centered-Evolution. I

k 2 kb

is satisfied. In order to approximate the backwards characteristic flow for one timestep, we shift the vertical interfaces Ii,j+; horizontally by (-k ;iy+,, .) and 2

3

the horizontal interfaces Jij++ vertically by (-k % j + 4 ) . Depending on the signs of the velocities 2 and $, each cell K i j will be divided into one up to nine sub-rectangles, leading to a natural subdivision

(compare Figure 2). For example,

and so on. 3.3.1. The MOT-ICE-PO In order to define the MOT-ICE-PO, we approximate cp by piecewise constants and define the auxiliary transport velocities by

-

where the remainders Zi+;,j and bij++ are supposed to be lipschitz-continuous.

Theorem 7 Suppose that X = k / h is fixed, and that the CFL-condition (8) is satisfied. Then for any given smooth velocity field a : R2 x R+ + R ~the , MOT-ICE-PO is consistent of order one with the differential equation (6).

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Note that the theorem states global first order consistency. The difficulty is to control the truncation error near points where the characteristic velocities change sign. See [NOE 991 for the full proof.

3.3.2. The MOT-ICE-PI Let us next construct a scheme based on piecewise linear reconstructions, the MOT-ICE-PI. We have to define the auxiliary transport velocities a t the interfaces and the piecewise linear reconstruction of the solution cp. We denote the piecewise linear reconstruction of cp by c p ~ :For (x, y) E Ki, let

A rather involved computation leads to the following choice of auxiliary transport velocities at the interfaces which guarantees second order consistency for smooth solutions:

where Z and b are supposed t o be lipschitz-continuous. Let us give a geometrical interpretation of these terms: the factors (a+ $ a t ) resp. (b i b t ) simply reflect the fact that we wish to evaluate the transport velocities a resp. b in the center of the space-time interfaces Ii+;,jx [tn,tn+l] resp. Ji,j++x [t,, t,+l] in order to achieve second-order accuracy in time. The

+

factor (1 - $(a,

+ by)) which appears both in 2 and g,is an approximation of

at t = k / 2 . This approximates the evolution of cp along the characteristics Z(T; X, t), where (6) gives

The correction terms $ (ab), resp. $ (ab), account for fluxes across corners of the grid, see [NOE 991 for a more detailed discussion. Let us now state the consistency of the MOT-ICE-P1:

Theorem 8 Let a : R2 x R+ - + R2 be a gaven smooth velocity field and let cp : R2 x R+ -+ R be a smooth solution of (6). Suppose that there are lipschitz-continuous functEons & and such that ~ in (9) satisfies the piecewise linear reconstruction c p defined

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Let the auxiliary transport velocities iii+3,j and bij+; be given by (10) ( l l ) , let X = Iclh be fixed and suppose that the CFL-condition (8) holds. Then the M O T - I C E - P i is consistent of order two with the dzfferential equation (6). Once more, we refer t o [NOE 991 for the proof, which requires a careful study of the truncation errors of the piecewise linear reconstruction ( P R . Let us stress again that we do obtain second-order consistency a t all points, including those where the transport-velocities change sign.

4. Numerical experiments In this section we present numerical experiments which confirm the accuracy and stability of the new MOT-ICE for smooth and discontinuous solutions. For the piecewise-linear reconstruction, we choose a central version of the W E N 0 (Weighted Essentially Non-Oscillatory) reconstruction [JS 961. Details may be found in [NOE 991.

4.1 Scalar advection with periodic coefficients In order t o illustrate the failure of consistency of the MOT-CCE-PO (cellcentered evolution) and the consistency of the new MOT-ICE (interface-centered evolution), we consider the one-dimensional scalar advection equation

over the interval [-I, 11 with

and cpo(x) r 1. Note that a t x = 0, cp(0, t ) satisfies the ordinary differential equation

SO

Similarly, and these are the maximum and minimum of the exact solution. We compute the solution at time T = l o g ( 2 ) / ~so ,

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Table 1: EOCs for one-dimensional advection with periodic coefficients ix 40 80 160 320 640 ix 40 80 160 320 640 ix 40 80 160 320 640

MOT-CCE-PO EOC L1 Lm 5.456309e-02 3.439066e-01 3.210264e-02 0.77 4.215299e-01 1.731594e-02 0.89 4.617376e-01 8.996724e-03 0.94 4.813319e-01 4.587514e-03 0.97 4.908176e-01 MOT-ICE-PO EOC L1 Lm 8.834546e-02 3.075886e-02 1.599486e-02 0.94 4.741382e-02 8.114853e-03 0.98 2.471667e-02 4.094363e-03 0.99 1.263470e-02 2.055908e-03 0.99 6.379015e-03 MOT-ICE-P1 with WENO-limiter EOC L1 LW 2.193228e-03 5.123679e-03 5.330401e-04 2.04 1.532804e-03 1.281400e-04 2.06 3.878244e-04 2.993688e-05 2.10 7.604266e-05 5.934229e-06 2.33 1.523297e-05 -

EOC -

-0.29 -0.13 -0.06 -0.03 EOC -

0.90 0.94 0.97 0.99 EOC -

1.74 1.98 2.35 2.32

We choose a CFL-number of 41og(2)/~,roughly 0.88. In Table 1, we list the experimental orders of convergence (EOCs) both with respect to the L1 and Lw norm. In the L1-norm, the EOC of the MOT-CCE-PO starts at 0.77 and increases towards 1. However, the method diverges in Lm, as it should have been expected from Example 6 in Section 3.2. Contrary to that, the MOT-ICE converges uniformly (i.e. in L1 and Lm) to the expected orders. Comparing the L1 errors of the MOT-CCE-POand the MOT-ICE-PO (piecewise constant reconstructions), one sees that the MOT-ICE converges with a better rate (especially on the coarser grids) and produces roughly half the error of the MOT-CCE. The convergence rates of the MOT-ICE-P1 (piecewise linear reconstructions) are even better than 2 on the finer grids, both for the unlimited and the limited version (we omit the table for the unlimited scheme). The error of the scheme using WENO-limiter is only slightly larger than that of the unlimited scheme and it is of course orders of magnitude smaller than that of the first order MOT-ICE-PO.

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4.2 Rotating Smooth Hump

Our next test problem is the two-dimensional scalar equation (6) with

in the domain [-I, 112. Note that on this problem, the MOT-CCE-PO and the MOT-ICE-PO will produce identical results, since a, E by r 0, so

and

.-.

bij+; = bij+; = bij =xi for all i.

We compute one full rotation, i.e. T = T, and fix X = A t l A x = ~ / 6 Assuming . that the maximal transport velocity is this corresponds to a CFL-number of roughly 0.74. First, we consider smooth initial data. Let xo := 0.5, yo := 0, ro := 0.3,

a,

Table 2: EOCs for smooth rotating hump. MOT-ICE-PO EOC EOC Height L, 9.528519e-01 7.129756e-01 0.287 0.454 6.479957e-01 0.56 5.459218e-01 0.39 3.994583e-01 0.70 3.690224e-01 0.56 0.631 0.777 2.277596e-01 0.81 2.228099e-01 0.73 0.876 1.228174e-01 0.89 1.239005e-01 0.85 MOT-ICEP1 with unlimited central differences ix EOC EOC Height LI LCC 0.794 2.056520e-01' 40 2.596250e-01 0.961 80 5.505040e-02 2.24 4.491245e-02 2.20 160 1.091899e-02 2.33 8.892373e-03 2.34 0.995 320 2.426323e-03 2.17 1.967944e-03 2.18 0.999 640 5.822946e-04 2.06 4.673955e-04 2.07 1.000 MOT-ICE-P1 with WENO-limiter ix EOC Height EOC L1 LW 4.74975le-01 40 4.351708e-01 0.525 0.767 80 1.201747e-01 1.86 2.327933e-01 1.03 0.898 160 3.337591e-02 1.85 1.018161e-01 1.19 320 6.350393e-03 2.39 4.022132e-02 1.34 0.960 640 1.030164e-03 2.62 1.230235e-02 1.71 0.988 ix 40 80 160 320 640

LI

-

-

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For the MOT-ICE-PO the EOCs increase slowly towards unity both in L1 and LCO (see Table 2). The error is very large on the coarser grids, and the convergence is initially slow. This is natural, since the initial data are not well resolved on the coarse grids, where they appear rather as a sharp peak than a smooth hump. For the MOT-ICE-P1 with unlimited central differences the EOCs are better than two both in L1 and LCO and they converge towards two as the grids are refined. For the MOT-ICE-P1with WENO-Limiter, the EOCs in L1 are slightly below two in the beginning. However, as the grid is refined, the EOCs increase drastically and well beyond two. In Lm, the EOCs start slightly above unity on the underresolved coarse grids, but show a similar dramatic increase as the grids are refined. In our experience, this behavior is typical for the central WENO-limiter.

4.3 Rotating Cylinder

Next, we consider the rotating cylinder:

Since the solution is discontinuous, we only give the experimental orders of convergence in L1 (see Table 3). We also display the maximal height of the cylinder, in order to see if it is excessively smeared, or whether there are overshoots in the numerical solution. For the MOT-ICE-PO, the EOCs tend towards 0.5 as expected for a linear problem and a scheme based on piecewise constant reconstructions. The maximal height of the cylinder increases towards 1.0 as the grid is refined. For the unlimited MOT-ICE-P1 (we omit the table), the EOCs are decreasing from 0.78 towards 0.70. As should be expected for a

Table 3: EOCs for rotating cylinder. ix 40 80 160 320 640

,

MOT-ICE-PO EOC LI 7.012658e-01 5.046964e-01 0.47 3.618134e-01 0.48 2.583896e-01 0.49 1.837913e-01 0.49

Height 0.868 0.968 0.990 0.995 0.997

MOT-ICE-P1,WENO-limiter EOC Height LI 0.993 3.758964e-01 1.000 2.278516e-01 0.72 1.000 1.398394e-01 0.70 1.000 8.588859e-02 0.70 1.000 5.270885e-02 0.70

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Figure 3: Rotating cylinder. MOT-ICE-PO and MOT-ICE-P1, 160 x 160 points.

discontinuous solution computed with unlimited piecewise-linear reconstructions, there is an overshoot of 9 to 11 percent of the height of the cylinder. The MOT-ICE-P1 with WENO-limiter converges a t rate 0.70, produces no overshoots and only slightly larger L1-errors than the unlimited scheme. As can be seen from the maximal height especially on the coarser grids, the computation is much less smeared than the one with the MOT-ICE-PO. The error on the finest grid is a factor 3.5 smaller for the limited MOT-ICE-P1 than for the MOT-ICE-PO. Figure 3 shows that the cylindrical shape of the solution is nicely preserved by both versions of the scheme.

4.4 Shallow- Water-Equations

In her dissertation [MOR 971, Morel reports glitches a t sonic points for a radially symmetric explosion for the shallow-water equations computed with the MOT-CCE-PO. Results using the new MOT-ICE, both PO and P1, show no such glitches, and produce almost perfectly radially symmetric solutions (see Figure 4). From these pictures, the results of the MOT-ICE-P1 seem to

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Figure 4: Explosion problem for the shallow water-equations. MOT-CCE-PO, MOT-ICE-PO, MOT-ICE-P1. Grid of 160 x 160 Points. Left Column: 25 Contours of water-height. Right Column: cross-sections along the x-axis and the diagonal " x".

"+"

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be of the same quality as those computed by Morel using CLAWPACK (see [NOE 991 for crucial implementation-details for the MOT-ICE-P1 for systems of conservation laws). 4.5 Comparison of eficiencies

Let us give a first comparison of efficiencies. Morel reports that the MoTCCE-PO (which is inconsistent at sonic points), Van Leer's flux-vector-splitting and CLAWPACK T110(using the first-order Roe-solver without transverse wave-propagation) all use the same amount of cpu-time (say one time unit). CLAWPACK T1ll takes 1.5 units, and the first-order fix at sonic points proposed by Morel 2.9 units. Our preliminary experience with the MOT-ICE is the following: the MOT-ICE-PO, which is consistent at sonic points, takes 0.9 units and is hence as fast as standard first order schemes. The MOT-ICE-P1 takes 2.2 units, which is the sames as the second order CLAWPACK T 2 - 2This . compares favorably with the MOT-CCE-PI,which is consistent a t sonic points, but needs 10.5 units of cpu-time. 5. Discussion

The new Method of Transport with Interface-Centered-Evolution (MoTICE) is consistent a t sonic points, second-order-accurate for smooth solutions and nonoscillatory a t discontinuities. In all test-calculations carried out by the author so far it produces high-resolution approximations of multi-dimensional phenomena with almost no grid-orientation effects. It needs about the same CPU-time as a second order version of LeVeque's wave-propagation algorithm CLAWPACK which, according to numbers given by Morel, is 4 to 5 times faster than the original second-order-accurate MOT-CCE-PI. In ongoing joint work with von Torne, we plan t o develop the MOT-ICE into a fully adaptive code, capable of handling general geometries. The multidimensional core of the Method of Transport are Fey's wavemodels. Both for theoretical and practical reasons, it would be desirable to establish a systematic approach for their derivation. Are there wave-models for any hyperbolic system of conservation laws, or do they require special symmetries, which are satisfied by the fluid-dynamical equations for which wavemodels have been found so far? Which properties should a good wave-model have, besides the formal consistency-requirements (2)-(3)? Can one analyze the interplay between properties of the wave-model and the stability and efficiency of the resulting wave-propagation algorithm? Let me make one final remark: I would very much like t o see a more widespread continuation of the "Great Riemann-Solver Debate" started by Quirk [QUI 943. Are the failures of Riemann-solver-based schemes reported by Quirk of a cosmetic nature, visible only in the "picture-norm", or can they

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be quantified? Do the schemes diverge or converge when the grid is refined? How do the so-called "genuinely multi-dimensional" schemes perform in these situations? 6 . Bibliography

For lack of space, we do not attempt to give a well-balanced (much less a comprehensive) bibliography. However, the following list should contain lots of useful information:

[BT 971

BILLET,S. A N D TORO, E., O n WAF-type schemes for multidimensional hyperbolic conservation laws. J . Comput. Phys. 130 (1997), 1 24. -

[CCL 941 COCKBURN, B., COQUEL,F. A N D LEFLOCH,P., Convergence of the finite volume method for multidimensional conservation laws. SIAM J . Numer. Anal. 32 (1995), 687 - 705.

P., Multidimensional upwind methods for hyperbolic con[COL 901 COLELLA, servation laws. J . Comput. Phys. 87 (1990), 171 - 200. [FEY 98a] FEY, M. Multidimensional upwinding. I. The method of transport for solving the Euler equations. J . Comput. Phys. 143 (1998), 159 - 180. [FEY 98b] FEY, M. Multidimensional upwinding. II. Decomposition of the Euler equations into advection equations. J. Comput. Phys. 143 (1998), 181 - 199. [FJMM 971 FEY, M., JELTSCH,R., MAURER,J . A N D MOREL,A.-T., The method of transport for nonlinear systems of hyperbolic conservation laws i n several space dimensions. Research Report No.97-12, Seminar for Applied Mathematics, ETH Ziirich (1997). [HIR 901

HIRSCH,C., Numerical computation of internal and external flows. Vo1.2, Wiley-Interscience 1990.

[JS 961

JIANG, G.-S. A N D SHU, C.-W., Eficient implementation of weighted E N 0 schemes. J . Comput. Phys. 126 (1996), 202 - 228.

[KNR 951 KRONER,D., NOELLE, S. AND ROKYTA,M., Convergence of higher order upwind finite volume schemes o n unstructured grids for scalar conservation laws i n several space dimensions. Numer. Math. 71 (1995), 527 - 560. [LEV 971 LEVEQUE,R.J., Wave Propagation Algorithms for Multidimensional Hyperbolic Systems. J. Comput. Phys. 131 (1997), 327 - 353.

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[LW 921

LEVEQUE,R.J. AND WALDER,R., Grid alignment effects and rotated methods for computing complex flows i n astrophysics. eds. J.B. Vos, A. Rizzi and I.L. Rhyming, Proceedings of the Ninth GAMM Conference on numerical methods in fluid mechanics. Notes Num. Fluid. Mech. 35 (1992), 376-385.

M., MORTON,K . A N D WARNECKE, G. [LMW 971 LUKACOVA-MEDVIDOVA, Evolution Galerkin methods for hyperbolic systems i n two space dimensions. Report 97-44, Univ. Magdeburg, Germany (1997). [MOR 971 MOREL, A.-T., A genuinely multidimensional high-resolution scheme for the shallow-water equations. Dissertation, ETH Ziirich Diss. No. 11959 (1997). [NOE 941 NOELLE,S., Hyperbolic systems of conservation laws, the Weyl equation, and multi-dimensional upwinding. J . Comput. Phys. 115 (1994), 22 - 26. [NOE 961 NOELLE,S., A note o n entropy inequalities and error estimates for higher order accurate finite volume schemes on irregular families of grids. Math. Comp. 65 (1996), 1155 -1163. [NOE 991 NOELLE, S., The MOT-ICE: a new high-resolution wavepropagation algorithm for multi-dimensional hyperbolic systems of conservation laws based on Fey's Method of Transport. To be submitted for publication, 1999. [QUI 941

QUIRK,J., A Contribution to the Great Riemann Solver Debate. Int. J . Numer. Meth. Fluid Dyn. 18 (1994), 555 - 574.

[ROE 861 ROE, P., Discrete models for the numerical analysis of timedependent multidimensional gas dynamics. J . Comput. Phys. 63 (1986), 458 - 476. [ROE 911 ROE, P., Discontinuous solutions to hyperbolic systems under operator splitting. Numer. Meth. Part. Diff. Eq. 7 (1991), 207.

[SW 811

STEGER,J. AND WARMING,R.,Flux vector splitting of the inviscid gas-dynamic equations with applications to finite difference methods. J . Comput. Phys. 40 (1981), 263 - 293.

[VNL 921 VAN LEER, B., Progress i n multi-dimensional upwinding. ICASE Report 92-43 (1992). [WN 991

WESTDICKENBERG, M. A N D NOELLE,S., A new convergence proof for finite volume schemes using the kinetic formulation of conservation laws. Accepted for publication in SIAM J. Numer. Anal. (Feb. 1999).

Numerical Analysis

Error estimate for a finite volume scheme on a MAC mesh for the Stokes problem Philippe Blanc CMI, Universite' de Provence Technopsle de Chdteau Gombert, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France email: [email protected] Abstract

W e consider a finite volume scheme on MAC mesh for the Stokes equations. Under regularity assumptions o n the solution, we prove an error estimate of order 2 i n the L' n o r m for the pressure and H1 n o r m for the velocity.

Keys words

Finite volume scheme, MAC mesh, Stokes equations.

1. I n t r o d u c t i o n

In this paper, we study some error estimate for the velocity in norm H 1 and for the pressure in the L2 norm for the Stokes equations, discretized by a finite volume method on the "Marker and Cell" (MAC) mesh. This mesh was introduced by Harlow and Welch [HAR 651 in the middle of the sixties. Since then, the convergence of different schemes constructed on the MAC mesch has been studied for several methods. One of the first results for the convergence of the MAC scheme was found by Porshing [POR 781 in 78. More recently in 1997, Shin and Strikwerda found an Inf-Sup condition [SHI 971 and an error estimates [SHI 961 for finite difference approximations. Two other methods are used on a MAC mesh: the mixed finite element method and the finite volume method. An example of mixed finite element method is given by Farhloul and Fortin in an article [FAR 971 in 1997. And in 1998, Han and Wu [HAN 981 proved a first order error estimate for both the velocity and the pressure. There exist several finite volume methods, some examples have been given, in 1983, in the book of Peyret and Taylor [PEY 831. One of these has been studied by Nicolaides [NIC 921 in 1992 : the covolume method. This method

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allows the use of some results proved for "div-curl" systems. And in 1997, Nicolaides and Wu [NIC 971 proved a first order error estimate for the velocity and the pressure. In 1998, Chou and Kwak [CHO 981 introduced a covolume method for the generalized Stokes problem, and proved a first order estimate for the velocity and the pressure, provided that the exact velocity is in H 2 and the exact pressure is in H1. We choose here t o directly discretize the velocity and pressure in the Stokes equations by a finite volume method, which we describe in the second part. Finally in the third part, we will prove a second order error estimate for the velocity and the pressure, provided that the exact velocity is in C 3 and the exact pressure is in C 2 .

2. The finite volume scheme

The purpose of this work is the study of a finite volume scheme for the discretization of the following Stokes problem:

where v is the viscosity, (u, v) is the velocity, p is the pressure and ( f ,g ) is the given force. And we assume: Assumption 1

(i) (ii) (iii)

R is an open bounded polygonal convex subset of lR2, v > 0,

f E L~(R) and g E ~ ~ ( 0 )

We consider a MAC mesh on R (cf figure 1) satisfying: Assumption 2 There exists C > 0 such that for all i and all j , we have:

(i) (ii)

hi > i h , kj>Ch,

(iii)

hi+1/2 > Ch,

(iv)

kj+l/2

> Ch.

Numerical analysis

Figure 1: MAC mesh Then we write the finite volume scheme:

where

and for all i = 1,. . . ,n - 1; j = 1 , . . . , m:

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and for all i = 1,. . . , n; j = I , . . . , m - 1

This scheme [2]-[6] has a solution, which is unique up to an additive constant for the pressure (see e.g. [EGH 971). 3. E r r o r estimates

3.1. Notations

Let (u, v,p) be the solution of the Stokes equation satisfying:

and we assume: Assumption 3 u E C2(O),v E C 2 ( n ) andp E C1(n). Then we denote: Notation 1

Let (u,, 21, ,p,) be the solution of the scheme [2]-[6] satisfying:

then we denote the error: Notation 2

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Remark 1 We have: n

m

Notation 3 For a mesh 7, X ( 7 ) denotes the space of functions which are constant on each cell o f 7 .

3.2. Error estimates

W e prove here the following error estimate: Theorem 3.1 Under the assumptions 1 and 2, i f u E C2 u E C2 and p E C 1 ( f i )there , exists C depending only o f the exact solution ( u ,u , p ) and R such that:

(a),

(a),

P
(n),u E C3(n),p 6 C 2(a)and i f the mesh is uniform,

E'uthermore i f u E C3 then we have:

where Ileylll,ru and lley 111, 7 , are the errors for the velocity in the discrete Hd norm (see [EGH 971 or [EGH 991 for the exact definition) and Ile? 11 is the error for the pressure in the L2-norm. 3.2.1. Error estimate for the velocity

Lemma 3.1 Under the assumption 1 and 3, and with the notations 2, there exists C1 depending only on ( u ,v,p) and on R such that, for all a > 0, we have:

Furthermore i f u E C 3 ( n ) v, E C 3 ( n ) p, E C 2 ( n )and if the mesh is uniform, then we have:

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To prove the first part of this lemma, we replace in the scheme by (ey+,,,,j, eG+,/,, e4,)i then we introduce some second order terms. Then we multiply each equation respectively by ey+l12,j , ey,j+l12 and eT,j, we sum over i and j , and we add them. We obtain, using a discrete inetegration by part:

thanks to Young's inequality and the regularity of u and v, one has:

and for all a E R, we have:

(Y

> 0, thanks t o Young's inequality and the regularity of p,

Taking back [ll]and [12] in [lo], we obtain the result. For the second part, we use the regularity of (u, v,p) and of the mesh t o replace the second order terms by some third order terms, changing all "h2" by "h4". 3.2.2. Error estimate for the pressure Lemma 3.2 Under the assumptions 1, 2 and 3, and with the notation 2, there exists C2 depending only on (u, v,p) and R such that:

If u 'LI C3(0), v E C 3 ( 0 ) , p E C2(O) and if the mesh is uniform, then:

To prove this lemma, we use the following theorem proved by Shin and Strikwerda [SHI 971. Theorem 3.2 Under the assumption 2, there exists Cf depending only of R such that for all q E X ( T ) , there exists (21, a) E X(7,) x X(T,) satisfying:

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Applying this theorem to e$ we obtain a l'velocity" (G, V) E X(7,) x X(7,). Then we replace ( ~ i + ~ vi,j+l/z,pi / ~ , ~ , ,j)in the first and the second equations of the scheme by (ey+l12,j, ey,j+lln,eT,j), we multiply each respectively by iii+l/2,j and by ui,j+l/2, and we sum each equation on i and j and we add both to obtain:

with, thanks to the Cauchy-Schwarz inequality:

where C1 only depends on (u, v,p) and R. Hence from [16],[17], [18] and [15] and thanks to Theorem 3.2, there exists C > 0 such as:

so by Lemma 3.1, with a = &:

the result. For the second part of the lemma, we replace again all "h2" by "h4" in the preceding reasoning. 3.2.3. Proof of theorem 3.1 The error estimate for the pressure is given by Lemma 3.2. For the velocity, we take back the value of a and C2 in the estimate [8], and we obtain:

Therefore the theorem is proved with C = max(C2, C4).

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References [CHO 981 CHOUS.H. AND KWAKD.Y ., A Covolume method Based on Rotated Bilinears for the Generalized Stokes Problem. S I A M J. Numer. Anal., Vol. 35, No 2, p. 494-507, April 1998. [EGH 991 EYMARDR., GALLOUET T. AND HEREINR., Convergence of finite volume schemes for semilinear convection diffusion equations. Numer. Math., Vol. 82, p. 91-116, 1999.

R., GALLOUET T . AND HERBINR., Finite Volume Meth[EGH 971 EYMARD ods. Preprint NO 97-19 L A T P , Aix-Marseille 1, to appear i n Handbook of Numerical Analysis, P.G. Ciarlet, J.L. Lions e d ~ . [FAR 971 FARHLOUL M . AND FORTIN M . , A New Mixed Finite Element for the Stokes and Elasticity Problems. S I A M J. Numer. Anal., Vol. 30, No 4 , p. 971-990, August 1997. [HAN 981 HAN H. AND W U X . , A New Mixed Finite Element Formulation and the MAC Method for the Stokes Equations. S I A M J. Numer. Anal., Vol. 35, No. 2, p. 560-571, April 1998. [HAR 651 HARLOWF.H. AND WELSHJ . E . , Numerical calculation of timedependant viscous incompressible flow of fluid with free surfaces. Phys. fluids, Vol. 8 , p. 2181-2189, 1965. [NIC 921

NICOLAIDES R.A., Analysis and Convergence of the MAC Scheme I. The Linear Problem. S I A M J. Numer. Anal., Vol. 29, p. 15791591, 1992.

[NIC 971

NICOLAIDES R.A. A N D W U X . , Covolume Solutions of ThreeDimensional Div-Curl Equations. S I A M J. Numer. Anal., Vol. 34, No 6, p. 2195-2203, December 1997.

R. A N D TAYLORT . D . , Computational Methods for Fluid [PEY 831 PEYRET Flow. Springer- Verlag, New York, 1983. [POR 781 PORSHING T.A., Error Estimates for MAC-Like Approximations to the Linear Navier-Stokes Equations. Numer. Math., 29, p. 291-306, 1978. [SHI 961

SHIND. AND STRIKWERDA J.C., Convergence Estimates for Finite Difference Approximations of the Stokes Equations. J. Aust. Math. Soc. Ser B, 38, p. 274-290, 1996.

[SHI 971

SHIN D. AND STRIKWERDA J.C., Inf-sup Conditions for Finite Difference Approximations of the Stokes Equations. J. Austral. Math. Soc. Ser. B, Vol. 39, p. 121-134, 1997.

Convergence Rate of the Finite Volume Timeexplicit Upwind Scheme for the Maxwell System on a Bounded domain

Yves Coudihre INSA, cornplexe scientifique de Rangueil, 31077 Toulouse Cedex 4, France P. Villedieu O N E R A Toulouse, 2 Avenue Edouard Belin, 31055 Toulouse Cedex, France

ABSTRACT : W e derive an O ( h l / ' ) error estimate for the upwind, explicit in time, finite volume scheme for Friedrichs systems. Explicit schemes in that case can not be seen as standart time-space finite element ones. Our demonstration is the generalisation on bounded domains of the ideas of Vila and Valledieu for the Cauchy problem i n EXd. It is applied to the case of Maxwell's equations. Key Words: hyperbolic system, Maxwell's equations, finite volumes, error estimates.

1. I n t r o d u c t i o n

We are interested in the approximation by finite volume means of the Friedrichs systems of the form d

dtu+CaAiu=O i=l

(]O,T[xR),

(An - M ) u = 0

4 0 , .) = uo

(10, T [ x d R ) ,

(a),

(1)

d

where Ai are some symmetric matrices,An = x ~ (n ~ being nthe~ unit nori=l ma1 outward to R ) . (1) belongs to the class of hyperbolic systems first introduced by Friedrichs [FRI 581. For M such that ker (An - M ) is maximal positive, it have a unique solution in L 2 ( ] 0 ,T [ x R ) [RAU 851.

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Finite volumes for complex applications

We shall treat, as an example, the case of the two dimensional Maxwell's equations i n Transverse Magnetic Mode. Our aim is to derive an 0 ( h 1 I 2 ) error estimate for the upwind, explicit in time, finite volume scheme for ( I ) , under very general assumptions on the mesh and minimal regularity hypothesis on the continuous solution u . The discontinuous Galerkin method, applied to the approximation of solutions of (1) is a space-time finite element method, which can be interpreted as a finite volume method, but with an implicit in time discretisation [JOH 871. However, finite volume schemes with time-explicit discretisation can not be interpreted in general as discontinuous space-time finite element schemes associated to a coercive bilinear form. Derivation of error estimates of order h'I2 for such schemes in the case of the Cauchy problem on EXd was first obtained by Vila and Villedieu [VIL 97, VIL 991, using a new technique of demonstration. On bounded domains, the additional difficulty is the discretisation of the flux A n u on the boundary of S1. Here, we propose a general form for the numerical flux on dS1, which guarantee the consistency and the stability of the corresponding scheme. We also prove an error estimate of order h1I2 (as for time implicit schemes). In section 2, we present the Maxwell's equations under the form (1). The scheme is defined in section 3. The actual results are stated in section 4. Sketches of the proofs can be found in section 5. 2. The Maxwell system

As an example, we consider the bidimensional Maxwell system in Transverse Magnetic mode (TM waves). It may be written as a Friedrichs system of the 2 c R2,with form (1) in f

T

The unlcnown is u = [ E H z H , ] (Electric an Magnetic Fields; supposing that c = 1 is the light speed). We shall suppose that R is the bounded domain between an obstacle and an outer boundary. Classical boundary conditions are - on the obstacle (dR1), the metallic condition: E

= 0,

Numerical analysis -

127

on the outer boundary (i3R2),the lznearized absorbing boundary condition of Silver Muller: H A next = E.

These conditions may be stated like in (1) by taking

where the outward normal to 8 0 is next=

[

a

lT

,l?

3. The numerical scheme

Let Th be a mesh of R , composed of polyhedral cells I(. In order to avoid any local degeneracy of the mesh, we assume that there exists some positive constants a and b such that

m(Il'), m(dli'), diam(Ii') denote, respectively, the measure in Eld of the cell K, the measure in I W ~ -of~ the boundary of K, the diameter of K. Let At be a time step, and t n = nAt. We shall approximate the solution of (1) by a piecewise constant function vh such that

The values of vh are calculated according to the following scheme:

B+ and B- denote, respectively, the positive and the negative parts of a symmetric matrix B.

g k e are some numerical fluxes, defined below;

3.1 The interior numerical flux

Let Sj: be the set of the interfaces interior to R. We take the natural upwind scheme on such interfaces:

3.2 The numerical flux on the boundary

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Finite volumes for complex applications

Let dSh be the set of the edges of the boundary dR. We propose to take the following general form for the numerical flux on the boundary:

VIc E Th, Ve E dSh n dI<,

gEe =

1

5 ( A n , + M e ) v$ + ye IIAn,

-

Mell QevE.

1 1 Since A n = - ( A n M ) - ( A n - M ) and ( A n - M ) u = 0 on the boundary, 2 2 a natural choice is ye&, = 0. But it yields an non stable scheme in general.

+

+

Q , is a stabilization term that does not modify the consistency of the flux. We shall take: Q e is the perpendicular projection on ker ( A n , - M e )I ye > 0 is a parameter measuring the importance of the stabilization term. It shall have an effect on the CFL condition.

3.3 Case of the Maxwell system

For the maxwell system given by the matrices ( 2 ) , and the boundary conditions (3), a easy calculation shows that, on a boundary edge e:

Metallic boundary condition:

Absorbing boundary condition:

We point out that classical boundary numerical fluxes fall in this class for some particular choices of y [PIP 991. Metallic boundary condition: with the mirror state technique, the flux is given by taking (5) with v k e = [ -E H z H y (if the interior state i s v t = [ E H, H,

1'):

1 2

-(An

lT

+ M ) + y llAn - MI1 Q is replaced by

which is exactly the previous one, for y = llAn - Mll 2 .

Numerical analysis

129

Absorbing boundary condition: the flux is given by taking v;"c, = 0 in (5) (no incoming waves) :

1 (An 2

-

+ M) + y []An- Mil Q is replaced by ~

n + ,

which is equivalent to the formulation above, only for the limit case y = 0.

4. M a i n R e s u l t s

Under some regularity assumptions on the uo, the Ai and a, (1) admits a unique solution u E V = CO([OlTI, (H1(Q))m) n C1([O, TI, ( ~ ' ( 0 ) ) " ) [RAU 851. Theorem 1 ( C o m p a r i s o n ) For T such that

> 0,

suppose that v E L2(]0, T[xS1) is

there exists p E V' (the dual space of V) such that VII, E V,

there exists a measure v such that V4 E C1([O, TI x

then

IIu

2

- v l l p ( l o , ~ [ x n5) (v, 4 )

n),

- 2 ( ~ 4, ~1 )

with 4 ( t , x) = T - t.

Theorem 2 ( C o n v e r g e n c e Rates) Under the following CFL conditions: on the interior interfaces, on the boundary edges,

Atm(dK) m(K)

Atm(dIi') m(h')

(1

I1 - &,

(i

1 1 ~ n .-eM ~ I I +27:)

5 27e - & ,

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Finite volumes for complex applications

the approximate solution vh given by the finite volumes scheme (4) converges to the exact solutiori u of ( I ) , and we have

5. Proofs

5.1 The Theorem of Comparison

An easy calculation yields, for any on 10, T [ x R ,

)

-

(

,

)

4 E C1([O, T] x a)such that $ ( t , z ) 2 0

(sinceM+~~>O,and4>0).

The theorem 1 is obtained by taking 4 ( t , x) = T - t (dtd = -1, and did = 0 ) . 5.2 L2 Stability

The scheme is expressed in a non conservative form, and then as a convex combination of interfaces contributions: At yn+l K -21;-(gke - A n ~ e v k )m ( e ) = n ~ m(e) V + (~e ) , K e€aK

where

1

I vg v>+l(e) =

(XK =

X K ( - A ~ K (~v g) e - v E ) (interior case), - AK ( M e - A n e IIMe - A n , 11 Q e v k (boundary case) 2 -

vk

+

Atm(8Ii')

). Calculating lv>"

m(h') Interior case:

'1

v l ( )

5

IU>

I2

(e)

12,

we find that

- AK ( v z T (-An;.) T

V&

- v k eT (-An;.)

- E X K ( V ~ - v k e ) ( - A ~ K( ~v k) - WE,) , using the CFL condition in a classical way.

vke)

Numerical analysis

13 1

Boundary case:

1 vKn + l ( e l l 2 5

I

V

~ -I ~ ~

K u ~ ~ ~ e v g ,

where

Me = 2Be

-

XKB

~ ~ B(Be~=,

Me - An, 2

+ Y llMe - Anell

Qe)

Summing over the e E dl<, and coming back to the conservative form, we find that the following inequality should be satisfied on the boundary of 0:

It is the case under the CFL condition of theorem 2 (and because Qe is the I

projection on ker (An, - Me) ). Remark that, for the absorbing boundary condition, the form of Me enables to conclude even if ye = 0 (ie for the classical numerical flux). Finally, summing over the cells Ii' E Th and the time steps n = 0 . . . N (T = (n + l ) A t ) , we have

5.3 The Errors of Conszstency

We start by stating the expressions of the p h and vh that will be used in the theorem of comparison (with some obvious notations):

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Finite volumes for complex applications

+

x GT

(Me - An,)

(4;

- Q ~ ( t ~ + m(e)At l))

(13)

n,K,aKnaS,

+:

($2 and are average values on Itn, tn+l[xe, and 4 K , $K are average values on K ) . The final estimates are deduced from the inequality of comparison of theorem 1. We have to calculate ( v h , $) - 2 ( p h , u$) for d ( t , x) = T - t . ( 7 ) and (11) are easily estimated if uo 6 H 1 ( 0 ) . (8), (9) and (12), (13) are combined and bounded by the L2 norms of v h and its discrete derivatives, using Cauchy Shwarz inequalities and a suitable Young inequality. (10) is bounded as wanted because Qeu2 = 0 (u is exact solution). See [COU 991 for the details. [COU 991

Y. Coudiere. Analyse de sche'mas volumes finis sur maillages non structure's pour des probl2mes line'aires hyperboliques et ellaptiques. PhD thesis, Universite Paul Sabatier, 12 janvier 1999.

[FRI 581

K.O. Friedrichs. Symmetric positive linear differential equations. Comm. Pure and Applied Math., 1958.

[JOH 871

C. Johnson. Numerical solution ofpartial differential equations by the finite element method. Cambridge University Press, 1987.

[PIP 991

S. Piperno. L2-stability of finite volumes schemes for the Maxwell system in two and three dimensions on arbitrary unstructured meshes. submitted, 1999.

[RAU 851

J . Rauch. Symmetric positive systems with boundary characteristic of constant multiplicity. Trans. of the A M S , 1985.

[VIL 971

J.P. Vila and P. Villedieu. Convergence de la methode des volumes finis pour les systkmes de Friedrichs. C.R. Acad. Sci. Paris, 1997.

[VIL 991

J.P. Vila and P. Villedieu. Convergence of the Time-Explicit Finite Volume Scheme for Linear Symmetric systems. submitted to SIAM Journal of Numerical Analysis, 1999.

Flux vector splitting and stationary contact discontinuity Franqois D u b o i s Conservatoire National des Arts et Mktiers 15 rue Marat, F-78 210 Saint Cyr llEcole, France. [email protected]

Abstract In order t o capture in a stable and accurate way a boundary layer with an upwind finite volume scheme, the numerical analysis of a stationary contact discontinuity problem shows that under natural symmetry hypotheses, a flux splitting generates a numerical viscosity proportional to the difference of densities whereas the numerical viscosity is null for a flux difference splitting approximating all the waves of the Riemann problem. Introduction. We study the Euler equations of gas dynamics in one space dimension. They take the form of an hyperbolic system of conservation laws between 1)

state W and physical flux f (W) :

W

=

aw + a at

-

f ( W ) = 0 where state

W ( t ,x) E IR3 represents a volumic density of mass, momentum and

W = (p, pu, p E = pe flux f (W) E IR3 is given by

energy :

1 t + -pu2) . 2

The algebraic form of physical

f (W) = (PU,pu2 + P, PUE+ P U ) ~; (1) it uses pressure, which is a function of density p and of internal energy e parameterized for a polytropic perfect gas by ratio y > l of specific heats : P = (7 - 1)Pe. In order to approach numerically the solutions of the gas dynamics equations, we introduce the so-called finite volume method ; space is discretized with a grid j Ax ( j E Z?) and time by multiples n At ( n E IN) of time step At. We search an approximate value WjlZ of the field W(*, 0 ) at particular vertex j Ax and discrete time level n At thanks to the family of numerical n+1/2 ( j E Z, n E IN) (see e.g. Harten, Lax and Van Leer [HLV83]) : fluxes j'j+lj2 1 1 n+1/2 n+1/2 fj-1/2 ) = 0 . In this note, we restrict At (w;" - WjlZ) + Ax ( fj+l12 ourselves to a two-point numerical flux function that is explicit and first order n+1/2 accurate in space and time, e.g. of the form : fj+l12 - @ ( w T IWy+l). -'

134

Finite volumes for complex applications

l We distinguish between two types of numerical flux functions depending of two arguments : on one side, exact or approximate solutions of the Riemann problem ("flux difference splitting") between states WT and W>l (see e.g. Godlewski-Raviart [GR96] for mathematical and numerical context) with the numerical fluxes proposed by Godunov [Go59], Roe [Roe811 and Osher [Os81] and on the other side flux decompositions ("flux vector splitting"). A flux vector spliting, with Sanders-Prendergast [SP74], Van Leer [VL82], Bourdel, Delorme and Mazet [BDM89] or Perthame [Peg11 among others, suppose that f (W) E IR3 explicited in (1) has been the physical flux function IR3 3 W written under the form

-

f ( W ) = f + ( W ) + f -(W) (2) with a set of constraints on functions f + (0) and f - ( 0 ) detailed for example in the book of Godlewski and Raviart [GR96]. For modelling upwinding, the numerical flux admits the following very simple form : @(Wl,Wr) = f+(Wl) + f - ( W r ) . In the context of a stationary aerodynamics problem, Van Leer, Thomas, Roe and Newsome [VTRN87] compare the Van Leer flux vector splitting [VL82] and the Roe scheme [Roe811 that uses a Riemann problem for a linearized equation. They show that in order to give a correct prediction of skin friction and heat flux on the boundary with a relatively course grid, it is possible with the Roe flux difference splitting while it is not with the Van Leer flux vector splitting of the type (3) for convective part of fluid flow. Their conclusion is to reject flux vector splitting methodology if the objective is to predict more than the simple pressure field. l In fact, the problem occurs in the boundary layer. Along the direction x normal t o the boundary, normal velocity u is very small. Then it is natural to study the evolution of a flux vector splitting (3) for the very simple model of a stationary contact discontinuity, i.e. a boundary layer with infinitesimal thickness. I t is a particular problem of decomposition of discontinuity where the given states Wl and Wr define on one hand a velocity field identically null composed by ul for x < 0 and by ur for x > 0 : (4) Ul = ur = 0 , and on the other hand a pressure field denoted respectively by pl for x < 0 and pr for x > 0 without discontinuity : (5) P l = P r = P . The physical solution of such a stationary contact discontinuity does not depend on time : density jump is maintained at the interface x = 0 as long as time variable is increasing and it is the addition of viscous term or of geometrical perturbations like in Kelvin-Helmholtz instability that modify the interface, which is crucial for a correct capture of boundary layers and shear instabilities. In this note, we prove that in a general way if a flux vector splitting satisfies very natural hypotheses of left-right invariance (section 2), then the associated scheme for gas dynamics contains a numerical viscosity essentially proportional t o the jump of density, then of the order zero relatively t o space step (section 3). (3)

l

Numerical analysis

135

2)

Left-right invariance. We consider transformation a of state W obtained by changing the sign of velocity : t (6) f f ( p , p u , p ~=) ~( P , - P ~ , P E ) . Taking into account the particular algebraic form of state W and relation (I), we observe that

(7) f ( f f W ) + Of(W) = 0 ; when we change the sign of velocity, we change the sign of mass flux and of energy flux but we do not the sign of momentum flux. a Because changing the sign of velocity is equivalent to changing the sign of space direction x, it is useful1 t o introduce the normal unitary vector n ( p (u n) , (pu2 p) n , t o this direction (n E { - 1, I ) ) , to set g(n, W)

--

+

+

and also o n = -n. If we change both signs of velocity ( p E p) (u n) ) and of space direction, the mass and energy fluxes remain unchanged but the sign of momentum flux is changed. We have in consequence : f f W ) = ff g(n, W) . Natural extension of this left-right invariance property to the numerical flux can be formalized by setting : if n = +1 @(WL,W,) (9) Q W l , n , W,) = -@(Wl, W,) if n = -1. The left-right invariance for numerical flux consists to remark that if we exchange both left and right states, the sign of their velocity and the normal direction, we only change the sign of momentum flux. (8)

-

Definition 1. Left-right invariance property. @(Wl,W,) satisfies the left-right The numerical flux function (Wl, W,) invariance property if the function 9(., a, a) defined in (9) and operator a defined a t relation (6) and by a n = -n satisfy the condition (10)

9 ( a W , , a n , a w l ) - uQ(Wl, n , W,) = 0 . We remark that consistency condition @(W,W) = f (W) can be translated for the pair (9,g) by the relation 9 ( W , n, W ) = g(n, W ) and in this particular case, relation (8) shows that *(OW, a n , OW) - uQ(W, n , W ) = g(an, a W ) - a g ( n , W ) = 0. This remark establishes a particular case of relation (10) when Wl = W, = W. Left-right invariance of a flux vector splitting.

Proposition 1.

A flux vector splitting (2) associated with a numerical flux function (3) satisfies the left-right invariance property if and only if we have f+(UW) af-(W) = 0 VW. (11) P r o o f of P r o p o s i t i o n 1. We introduce the representation (3) inside relation (10) when n = +1 :

+

9 ( u W r , a n , awl) - o 9(Wl, n , W,.) = = -@(gWr, gW1) - aQ(W1,W,)

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Finite volumes for complex applications

for each pair (Wl,W,). If we make the choice of two independent states Wl and W, the preceding relation states clearly the relation (11) and when we explicit the action of the operator a on a vector (see relation (6)), we obtain the detail of the algebra for each component, i.e. relation (11). l On the other hand, if condition (11) is satisfied, then the relation (10) is correct for n = 1 ; it remains true for n = -1 because the left member is an odd function of variable n due to relation (9). In consequence the proposition is established. l Proposition 2. Particular case of classical flux vector splittings. Van Leer flux, Sanders Prendergast flux and Boltzmann schemes satisfy the left-right invariance property.

Proof of Proposition 2. l Van Leer flux satisfy the following relation f + ( W )

=

f (W) if M

= -u

2

<

1 and f - ( W ) = f ( W ) if M -1. If the Mach number M of state W is greater or equal t o 1, then the Mach number of state OW is lower or equal to -1 ; then f + ( a W ) = 0 = - 0f - ( W ) and relation (11) is established in this case. If on the contrary M 5 -1, then f + ( a W ) = f (OW) and f - (W) = f ( W ) and relation (11) is in this case a simple re-writing of relation (7). l When / M I 5 1 , Van Leer flux vector splitting satisfies the relation

and relation (11) is clear. Consistency condition (2) is not obvious ; the proof is an algebraic calculus introduced in the original work [VL82]. In the case of a Boltzmann scheme, we write the state W under the particuv-UI 1 lar form W = 1 v 12) t dv where ~ ( o is) a positive l

function that defines the numerical scheme and T is the temperature. The flux +m v t is simply evaluated by f ( W ) = (v, 1 v I Z , 5 I v 12) dv. x('

/

-00

-)

'

To take into account the fact that flux f + represents the action of all the particles going from the left t o the right, we set

and in an analogous way, due t o (2) :

Numerical analysis

137

After having made the change of variable v ++ -v inside the integral (13), we v-UI v deduce f f ( o W ) = andrelation (-vl I v 12, -- 1 v 2 (11) est clear. l We consider now the case of Sanders and Prendergast splitting ; we just replace function x of relation (13) by a linear combination of Dirac measures a t particular points u - c , u , u c in velocity space, where c is the sound waves celerity, in order to satisfy the following relations +m +m +m v dp(v) , pu2 p = v2d~(v). (15) P = dp(v) 1 P U

+

- 1,

1,

+

1,

Then total energy p E is decomposed under the form pE = +a

( )

dp(v) and the particular algebraic form that controls the mea-

-

sure dp(v) [SP74] allows to deduce

In an way analogous t o the other Boltzmann schemes, the flux vector splitting results from an integration on the interval 10, +m[ t o evaluate f + and on the is a consequence of opposite interval ] - oo,O[ for f -. Even parity of parity of energy decomposition in (15) whereas odd parity of f: and f$ is a consequence of imparity of impulse in (15) and of relation (16). This result establishes relation (11) for Boltzmann schemes and Proposition 2 is proven.

fif

Numerical viscosity on a stationary contact. Definition 2. Numerical viscosity. Numerical viscosity V(Wl, W,) of a two-point numerical scheme of the type n+1'2 = @(W;, W;,) is defined by the relation fj+1/2 1 1 ww1, WT) = (f (Wl) f (W,)) - ,V(Wl, Wr) . (17) 3)

+

Proposition 3. Numerical viscosity of a flux vector splitting. Let @(ele) be a flux vector splitting of the type (3). Then numerical viscosity V(Wl, Wr) satisfies the relation V(W1, WT) = (f+(W,) - f-(Wr)) (18) Proof of Proposition 3. It results from the following calculus :

+

- (f+(Wl) - f - ( w ) ) .

v(w11 WT) = f(w1) f(w,) - 2 @ ( w , w r ) due t o (17) = (f+(WO + f - ( w l ) ) + (f+(W,) f-(wv)) - 2 ( f f ( W ) f-(Wr)) = (f+(WT) - f-(wT)) - (f+(Wl) - f - ( w ~ ) ) and relation (18) is established.

+

+

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Finite volumes for complex applications

Proposition 4. Stationary contact discontinuity. Let W be a state with a velocity equal to zero. It satisfies in particular (19) OW = W . Then if the flux vector splitting a ( * , 0) defined in (2)-(3) satisfies the left-right p(p, p) E invariance property, there exists two functions 10, +oo [' 3 (p,p) IR and 10, +CCI[~ 3 (p,p) ~ ( p , pE) IR in order t o satisfy l

-

-

Moreover, for a stationary contact discontinuity (4)-(5),the numerical viscosity satisfies 1 t V ( W ~WT) > = 5 (P(PT,P) - P(PL> P) 7 O , '(PT~P) - ' ( ~ 1 ,P) ) (21) '

l

In the case of Van Leer flux vector splitting, relations (12) show

and for a Boltzmann scheme, we have, taking into account (13) and (14), +oo foo pB(p,p)=Jd x(-$!~);.u and r B ( p , p ) = l ( I U ' ) f2d v .

Xx

Proof of Proposition 4. l Relations (19) and (11) show that ff(W) +ufP(W) = 0, aW = W . (22) Joined with relation (2), we have from relation (22) f t ( W ) - a f + ( W ) = f (W) if u W = W and relation (20) is established for function f +(.). The entire relation (20) is a direct consequence of (22). The detail of the computation of numerical viscosity is a consequence of the relations (18) and (20). l Proposition 5. Residual numerical viscosity. If one of the functions p(o, e) and '(0, 0) explicitly depends on density, i.e.

a'

# 0 or -(p, p) # 0 , then the numerical viscosity 80 a0 of a flux vectdr splitting scheme is not infinitesimal for a stationary contact discontinuity, whatever be the size of the mesh. Proof of Proposition 5 is an immediate consequence of Proposition 4 and in particular of the relation (21). l Proposition 6. Approximate Riemann solver. Let a ( * , 0) be one of the three exact or approximate Riemann solvers proposed by Godunov [Go591 (exact solution of the Riemann problem), Osher [Os81] (approximate solver containing only rarefaction waves or contact discontinuity) and Roe [Roe811 (approximate solver containing only contact discontinuities). Then numerical viscosity V(Wl, WT) of such a numerical scheme is null if given states Wl and WT satisfy the particular conditions (4)-(5) of a stationary contact discontinuity. if we have

*(p, p)

Numerical analysis

139

P r o o f of P r o p o s i t i o n 6. In the case of Godunov and Osher schemes, conditions (4)-(5) state that the solution of the Riemann problem is effectively only composed by a stationary contact discontinuity. Due to Rankine-Hugoniot relations for a stationary discontinuity, physical fluxes of the two states Wl and W, are equal and we t have f* == (0, p, 0) = f ( W L )= f (W,). Taking into account relation (7) in Definition 2, the result is established in this case. If we use the Roe flux. we c o m ~ u t ein a first step - /Roe811 intermediate L

velocity u* of a mean state : u*

F

Jpr~+ l 6% = 0 due to relation ,fi " . + ,hz * '

(4) and we evaluate also total enthalpy of this mean state before the calculus of Roe matrix A* that satisfies, taking into account previous expression of f * : A* (Wr Wl) = f (W,) - f (Wl) = 0. The difference (W, - WL) is an eigenvector of matrix A* relatively to the eigenvalue u* = 0 due t o the expression of intermediate velocity u*. In consequence, when we decompose discontinuity Wr - WL on the basis of eigenvectors for matrix A*, we observe that this difference is non null only for the linearly degenerated wave, i.e. on the contact discontinuity itself. Conclusion is then exactly the one done previously for Godunov and Osher fluxes. -

Conclusion. In order t o capture numerically a boundary layer with a finite volume scheme, numerical analysis of the problem of stationary contact discontinuity shows that classical flux vector splitting schemes satisfying the left-right invariance property generates a numerical viscosity of order one relatively t o the jump of densites whereas it is not the case if we use an exact or approximate decomposition of the Riemann problem. This fact founded on very simple algebra shows that Van Leer at a1 conclusion can be extended to all flux vector splittings referenced in this note : flux vector splitting satisfying the left-right invariance is incompatible with viscous computations. The previous remark conducted us during the time of development of the Navier Stokes solver Ns3gr to include the Osher flux whereas the initial choice was the Sanders and Prendergast flux vector splitting. This choice has been performing, even for the resolution of the Euler equations of gas dynamics in the particular case of capturing shear stationary waves (see [DM92]). In an analogous wa,y, the parabolized version Flu3pns (Chaput et a1 [Chgl]) of Flu3c computer software has required the introduction of the Osher flux decomposition in order to simulate flows with a precise evaluation of viscous effects. 4)

5) References. [BDM89] F. Bourdel, P. Delorme, P. Mazet, Proceedings of the 2th International Conference on Nonlinear Hyperbolic Problems, Notes on Numerical Fluid Mechanics, vol. 24, p. 31-42, Vieweg, 1989. [Ch91] E. Chaput, F. Dubois, G. MoulQs,D. Lemaire, J.L. Vaudescal. A Three Dimensional Thin Layer and Parabolized Navier-Stokes Solver Using the MUSCL Upwind Scheme, AIAA Paper no 91-0728, january 1991.

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Finite volumes for complex applications

(DM921 F. Dubois, 0 . Michaux. Solution of the Euler Equations Around a Double Ellipsoydal Shape Using Unstructured Meshes and Including Real Gas Effects, Workshop on Hypersonic Flows for Reentry Problems (DksidBri-Glowinski-Pkriaux Editors), Springer Verlag, vol. 2, p. 358373, 1992. [Go591 S.K. Godunov. A Finite Difference Method for the Numerical Computation of Discontinuous Solutions for the Equations of Fluid Dynamics, Mat. Sbornik, vol. 47, p. 271-290, 1959. [GR96] E. Godlewski, P.A. Raviart. Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118, Springer, New York, 1996. [HLV83] A. Harten, P.D. Lax, B. Van Leer. On Upstream Differencing and Godunov-type Schemes for Hyperbolic Conservation Laws, SIAM Review, vol.25, no 1, p. 35-61, january 1983. [Os81] S. Osher. Numerical Solution of Singular Perturbation Problems and Hyperbolic Systems of Conservation Laws, in Mathematical Studies n047 (Axelsson-F'ranck-Van der Sluis Eds.), p. 179-205, North Holland, Amsterdam, 1981. [Peg11 B. Perthame. Second Order Boltzmann Schemes for Compressible Euler Equations in One and Two Space Variables, SIAM Journal of Numen'cal Analysis, vol. 29, p. 1-19, 1991. [Roe811 P. Roe. Riemann Solvers, Parameter Vectors, and Difference Schemes, Journal of Computational Physics, vol. 43, p. 357-372, 1981. [SP74] R.H. Sanders, K.H. Prendergast. The Possible Relation of the 3kiloparsec Arm t o Explosions in the Galactic Nucleus, The Astrophysical Journal, vol. 188, p. 489-500, march 1974. [VL82] B. Van Leer. Flux-Vector Splitting for the Euler Equations, Proceedings of the ICNMFD Conference, Aachen 1982, Lectures Notes in Physics, vol. 170, p. 507-512, Springer Verlag, 1982. [VTRN87] B. Van Leer, J.L. Thomas, P. Roe, R.W. Newsome. A Comparison of Numerical Flux Formulas for the Euler and Navier Stokes Equations, AIAA Paper no 87-1104, AIAA 8th CFD Conference, 1987.

Analysis of a Finite Volume Solver for Maxwell's Equations

Fredrik Edelvik Department of Scientific Computing, Uppsala University Box 120, S-751 04 Uppsala, Sweden [email protected]

A finite volume solver for Maxwell's equations is analyzed. The solver shows excellent dispersion characteristics on three difierent uniform triangular grids. Long term stability is achieved for general unstructured grids using a third order staggered Adams-Bashforth scheme for the time discretization. The solver has been hybridized with a finite difference solver and the resulting hybrid solver is shown to be second order accurate for a 2 0 cylinder scattering case. ABSTRACT

Key Words: Finite volumes, Maxwell's equations, Dispersion, Hybrid grid

1. Introduction

The most popular numerical method for solving electromagnetic problems is the Finite-Difference Time-Domain (FDTD) method [TAF 951. It is normally used on a Cartesian grid and the variables are staggered both in time and in space. The main disadvantage with FDTD is its inability t o model complex shaped geometries producing staircasing errors. One way of solving this problem, but still taking advantage of the efficiency of the FDTD method, is to use a hybrid grid solver. A Finite-Volume Time-Domain (FVTD) solver on an unstructured grid is used in the near vicinity of the object and this solver is coupled to an FDTD solver on a structured Cartesian grid that is used in the outer region. This approach was proposed by Riley et al. [RIL 971 and it is also used in the Swedish Computational Electromagnetics project GEMS (General Electromagnetic Solvers). GEMS is a collaborative research and code development project between Swedish industry and academia. This paper con-

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Finite volumes for complex applications

centrates on the finite volume solver. The solver is described in Section 2 and its dispersion characteristics and stability properties are analyzed in Sections 3 and 4, respectively. In Section 5 some numerical results obtained with the hybrid solver are presented and the last Section summarizes the paper and some conclusions are drawn for future work. 2. The Finite Volume Solver

The finite volume (FV) solver has so far been implemented in 2D for solving the transverse magnetic (TM) Maxwell's equations,

for a linear, isotropic, non-conducting and non-dispersive material. Here Hx and H, are the x- and y-component of the magnetic field, E, is the z-component of the electric field, p is the magnetic permeability and 6 is the electric permittivity. The basis for the FV solver is the following integral form (in 2D) of Maxwell's equations

=-&nx eJA g

d

Edl,

=fin ~ x Hdl,

where A is an arbitrary area and the line integral is taken along the path I' that encloses A and n is a unit normal. In 2D the computational domain is discretized using a staggered unstructured grid, consisting of a primary grid of triangles and a dual grid (cf. Fig. 1). For T M mode, Hz and H, are stored at the nodes, whereas E, is stored at the barycenters of the primary cells. By joining barycenters of neighboring cells the dual grid is constructed during the pre-processing stage. The area integrals are evaluated by the assumption that the electric and the magnetic field components remain constant over a primary and a dual cell, respectively. Simplifying the two integrands in the line integrals implies

where H and E, denote average fields and the sums are taken along the edges with unit vector t of the respective cells. Ad and A, are the areas of the dual

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143

and primary cells. The evaluation of the first line integral is performed by the assumption that the electric field is piecewise constant along each dual edge. Following Riley et al. [RIL 971, the second line integral is evaluated according to

where HL is the FDTD component in the direction nl orthogonal to the dual edge crossing the primary edge t , H i and H I ,are the magnetic field in the nodes of the edge t . Hl is incorporated in the edge projected field value to guarantee that the divergence is preserved on a local cell level. This is very important, since not fulfilling the two divergence relations of Maxwell's equations could cause spurious modes that would destroy the solution. The time integration method that we use is a third order staggered AdamsBashforth (ABSS) scheme proposed by Fornberg et al. [FOR 991 for scalar wave equations. For the TM Maxwell's equations the scheme takes the form

where A and B are operators taking care of the space discretization 3. Dispersion Analysis on Triangular Grids

A frequently used technique t o characterize the errors of a numerical scheme is to use Fourier analysis. Assuming that the fields are periodic in space and neglecting boundary conditions, we make the following ansatz for the three unknown field components

where k, and Icy are the x- and y-components of the numerical wavevector, respectively. The numerical wavenumber k will in general differ from the physical wavenumber k defined by k = w / c , where w is the angular frequency and c is the speed of light. k depends on how well the wave is resolved in the grid, the type of grid and the angle of propagation in the grid. The difference between and k gives rise to numerical phase and group velocities that depart from the exact values, which cause numerical errors that accumulate with distance. In figure 1 the different uniform grids used in the analysis are shown. Note that the normals and edge vectors align in the equilateral and diamond grid. Thus, only the FDTD correction Hl survives expression ( 6 ) , so for those two grids we do not have t o take the magnetic field in the nodes into account. For

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Finite volumes for complex applications

the one directional grid, however, we have to consider both the magnetic field at the nodes and the FDTD correction. Since the approximation of the dispersion relation is in the midpoint of the respective building blocks shown in figure 1, in the analysis, the E, components are interpolated from the barycenters to the midpoint of the building blocks.

Figure 1: T h e different u n i f o r m triangular grids used in t h e analysis, the equilateral grid, t h e one directional grid and the diamond grid. T h e respective dual grids are indicated by dashed lines. Let us first look at the errors introduced by the space discretization for the equilateral grid. Substituting (9) into (8) and assuming that no error is introduced by the time discretization results in

4 jwEz = -[ s i n ( - k y A / 2 h ) H I €A

+ s i n ( i X A / 4+ , k , A / 4 6 )

HH,

+ s i n (-kXA/4 + ~ A / 4 h H) H ~. ]

(10)

Similarly, from (7) we have for the three edges (see Fig. 1)

jwH1 = 2fi'Ez ---s i n ( - k , ~ / 2 ~ j ), PA

Inserting (11)-(13) into (10) leads to the numerical dispersion relation

Letting A

-+

0 we obtain

i.e. the dispersion relation is approximated to second order accuracy. The same analysis on a square grid with edge length A, where the solver is identical to the

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145

&

FDTD scheme, gives an error constant equal to [TAF 951. The dispersion relation is approximated to second order accuracy also for the grids with right triangles, but since these two expressions are somewhat lengthy they will be omitted here. The dispersion error characteristics can be presented in terms of the error in the phase speed of waves of different grid resolution and propagation angle. Let k, = k cos(cu) and i,= sin(cr), where cr is the angle of propagation relative to the x-axis. Substituting these two expressions into the dispersion relation (14) and solving the nonlinear equation for i,results in a relation between the numerical wave speed, up, and c , since vp/c = k/&. Figure 2 illustrates the numerical phase velocity as a function of propagation angle and grid resolution. The results obtained by the FV solver are in excellent agreement with the true wave speed for all three grids even for a moderate grid resolution of 10 points per wavelength. The dispersion characteristics are clearly better for triangular grids than for square grids.

Equilateral tri grid - U10

Equilateral lri grcd - N10 Diamond tri grld N10 One directionaltrl grid - N10

-

0.901

0 975

0

30

60

W

120

150

180

O'OgO

30

60

W

120

150

180

Figure 2: Variation of numerical phase velocity for diflerent grids as a function of grid resolution and propagation angle. Discretizing the time derivative with ABS3 we obtain the following fully discrete dispersion relation for the grid with equilateral triangles

The time step is chosen according to the stability limit for ABS3 (see next Section) as cAt = &A/7. As in the semi-discrete case we solve (16) for k . Since ABS3 is a third order accurate scheme, the errors from the space discretization will dominate and a plot of vp/c shows almost identical results as in figure 2. If we use the Leap-Frog scheme for the time-stepping we get a small improvement in the dispersion characteristics since the scheme is only second order accurate and thus the errors from the space and time discretization cancel to some extent.

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Finite volumes for complex applications

4. Stability Analysis of t h e FV Solver

An important issue for explicit solvers across many disciplines is how to achieve long term stability without adding too much artificial dissipation. The stability region for ABS3 in the scalar case is given in figure 3.

,

3

Figure 3: Stability region for third order staggered Adams-Bashforth. The scheme is stable between f12/7 along the imaginary axis compared to the Leap-Frog scheme, which is stable between f2. That implies that we have to use a shorter time step for ABS3. However, the main disadvantage with Leap-Frog is that it is only stable on the imaginary axis and becomes unstable as soon as we have eigenvalues with a nonzero real part, which we are likely to have on unstructured grids and when boundaries come into play. Let us analyze how ABS3 behaves in our case. After some straightforward algebra (7) and (8) can be written in matrix form as

The plane wave ansatz

leads to the following generalized eigenvalue problem

where A, B and 6 are the matrices in (17), respectively. We begin by looking a t the uniform triangular grids used in the analysis of the dispersion error. For the grid consisting of equilateral triangles we get the matrices A and B from (10)-(13). A stability condition for the Leap-Frog scheme on the three triangular grids is easily derived and equals

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147

where the first part in the right-hand side is the stability condition on Cartesian is equal to the shortest edge in the unstructured region. grids and minlength When we solve the generalized eigenvalue problem (18) on these grids we observe that stability is obtained if the time step suggested by (19) is decreased by a factor 6/7, which is exactly the relative difference in stability along the imaginary axis (see Fig. 3) between the two methods. To analyze the eigenvalues for a general unstructured grid including boundaries we can no longer use Fourier analysis. Instead, let

and after some straightforward rearrangements we are able to write (7) and (8) on matrix form as in+'= P ( A , B)Zn. Analyzing the eigenvalues of the iteration matrix P, for the grid shown in figure 4, reveals that if we choose the time step for ABS3 in the same manner as above we get the eigenvalue spectrum shown in figure 4, where all eigenvalues are within the unit circle. If we use the Leap-Frog scheme with the same time step the largest eigenvalue is of the order 1.0003. Hence the Leap-Frog scheme is unstable even for a tinie step well within the stability limit along the imaginary axis.

Figure 4: Eigenvalues of the iteration matrix P using ABS3 for a PMC cylinder scattering case. Primary grid around the cylinder is shown to the right. The long term stability using ABS3 on general unstructured grids around a PMC and a PEC cylinder has also been verified by feeding the FV solver along one of the outer boundaries with a narrow square wave pulse for ten million time steps without any signs of instability. 5 . Numerical Results

The FV solver has been hybridized with a finite difference solver and the resulting hybrid solver has been tested on two scattering cases, a PMC and a PEC cylinder. In both cases a plane Gaussian shaped wave is impinging on the cylinders. In figure 5 the results for different resolutions using the hybrid

148

Finite volumes for complex applications

solver are compared to a well resolved FDTD solution. An analysis of the results reveals that the hybrid solver is second order accurate for both cases. The hybrid code has been verified to be long term stable in the same manner as the stand alone FV solver.

Figure 5: Errors for hybrid solutions of different resolution. PMC cylinder to the left and PEC cylinder to the right. 6. Conclusions

We have demonstrated that our FV solver has excellent dispersion characteristics on three different uniform triangular grids. A third order staggered Adams-Bashforth scheme is used instead of the normally used Leap-Frog scheme. Indications on how to choose the time step for ABS3 on general unstructured grids are given. The solver has been shown to be long term stable for two cylinder test cases. The implementation of the FV solver in 3D is ongoing and the hybrid technique will be used there as well. 7. Acknowledgment

The author would like to thank Ulf Andersson and Gunnar Ledfelt at KTH, Stockholm, who participated in the hybridization of the code and performed the numerical tests in Section 5. 8. Bibliography

[TAF 951

TAFLOVEA., Computational Electrodynamics, The FzniteDifference Time-Domain Method, Artech House, Norwood, 1995.

[RIL 971

RILEY,D. J. et al. , (rV0LMAX:A Solid-Model-Based, Transient Volumetric Maxwell-Solver Using Hybrid Grids n, IEEE Antennas Propagat. Magazine, N" 39, 1997, p. 20-33.

[FOR 991

FORNBERG, B. et al. , ((Staggered Time Integrators for Wave Equations B, submitted to SIAM J. Sci. Comput.

A result of convergence and error estimate of an approximate gradient for elliptic problems

Robert ~ ~ r n a r ,d 'Thierry Gallou#, Raphadle ~ e r b i n ~ Ecole Nationale des Ponts et Chausse'es, Marne-la- Valle'e, France Universite' de Provence, Marseille, France

--

Using a classical finite volume piecewise constant approximation of the solution of a n elliptic problem i n a domain R , we build here a n approximate gradient of the solution. It is then shown that this approximate gradient converges i n Hdi,(R). A n error estimate is given when the solution to the continuous problem belongs to H 2 ( R ) .

ABSTRACT

K e y Words: elliptic equations, finite volumes, gradient, convergence, error estimate.

1. Introduction

As a paradigm of elliptic problems, we consider the Laplace equation -Au(x) = f (x), for a.e. x E R ,

(1)

with Dirichlet boundary condition: U(X)

= 0 , for a.e. x E aS2,

where we make the following assumption. Assumption 1 1. 0 is a n open bounded polygonal subset of R d ,d = 2 or 3,

2. f E L 2 ( R ) .

(2)

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Finite volumes for complex applications

Here, and in the sequel, "polygonal" is used for both d = 2 and d = 3 (meaning polyhedral in the latter case). Note also that "a.e. on do" is a.e. for the d - 1-dimensional Lebesgue measure on 80. Finite volume methods for Problem (1),(2) have been intoduced by many authors (see [EGH 991 and references therein), with some proofs of convergence and error estimates. Since the approximate solution constructed with a classical cell-centered finite volume scheme is piecewise constant, an approximation of the gradient of the solution may be seen to be more complex than with a finite element method. Indeed, the convergence of a reconstructed gradient has been shown in [CVV 971, for certain quadrangular meshes using a nine point scheme. It has also been shown on certain meshes by rewriting the finite volume scheme as a finite element scheme [ABM 951, [VAS 981, [AWY 971 or a Petrov-Galerkin scheme [DUB 971. Here we show that one may construct an approximate gradient on all "admissible" meshes by using some mesh functions which generalize those used in mixed finite element theory (see e.g. [ROT 911). 2. A p p r o x i m a t e of u a n d Vu

The following definition of admissible meshes for the finite volume scheme includes a large variety of meshes, such as, in two space dimensions, triangular meshes (with a four-points scheme), cell-centered or vertex-centered rectangular meshes (with a five-points schemes) and Voronoi meshes. Definition 1 (Admissible meshes) Let R be an open bounded polygonal subset of R d ,d = 2, or 3. A n admissible finite volume mesh of 0 , denoted by 7 ,is given by a family of "control volumes", which are open polygonal convex subsets of 0 (with positive measure), a family of subsets of contained i n hyperplanes of IR~, denoted by E (these are the edges (ZD)or sides (3D)of the control volumes), with strictly positive (d - 1)-dimensional measure, and a family of points of 0 denoted by P satisfying the following properties (in fact, we shall denote, somewhat incorrectly, by 7 the family of control volumes): (2)

The closure of the union of all the control volumes is

a;

(ii) For any K E 7 ,there exists a subset EK of E such that d K = ?T \ K = .UnEzKu.Let E = U K ~ ~ E K . (iii) For any (K, L) E 72 with K # L, either the (d- 1)-dimensional Lebesgue measure of R n is 0 or R n = i? for some a E E, which will then be denoted by KIL. (iv) The family P = ( X K ) K €is~such that X K E K (for all K E 7 )and, if a = K I L , it is assumed that the straight line DK,L going through X K and X L is orthogonal to KIL.

Numerical analysis

15 1

I n the sequel, the following notations are used. T h e mesh size is defined by: size(7) = sup{diam(K), K E 7). For any K E 7 and a E E, m(K) i s the d-dimensional Lebesgue measure of K (i.e. area if d = 2, volume i f d = 3 ) and m(a) the (d - 1)-dimensional measure of a . T h e set of interior (resp. boundary) edges i s denoted by Eint (resp. Eext), that is Lint = {a E E ; a (f d o ) (resp. EeXt = { a E E; a C dR)). For any K E 7 and a E EK, we denote by dK,, the distance from XK t o a and set TK,, = and we denote by n ~ , , the u n i t vector normal t o a outward t o K.

2,

Under Assumption 1,let 7 be an admissible mesh in the sense of Definition ~ u~ be defined by 1 and let ( U K ) ~( u~, )~, ~, and

A proof of the convergence in L2(R) of the approximate solution u~ to the unique variational solution u E H i ( 0 ) of Problem ( I ) , (2) and error estimates are given for example in [EGH 991. We now build an approximate of Vu, the gradient of the solution of the continuous problem. To this purpose, we introduce, for K E 7 and a E EK the solution $K,,, E H1(K) of the following Neumann problem .d A 4 ~ ,(5) a =4 m ( ~ ,)

for a.e. x E K,

J,

4K,v(x)dx = 0, V ~ K , , ( Y.)nK,,, = 1, for a.e. y E a , V ~ K , , ( Y-) n ~ ,=s 0, for a.e. y E a , @E E K , #~a .

(6)

The functions V@K,, generalize on every finite volume mesh the RaviartThomas low degree finite element. Using the finite volume approximate of the fluxes, these functions allow to build the approximate of the gradient GT, given by uu - U K

V @ K , u ( ~for ) , a.e. x E K.

UEEK

3. Convergence t h e o r e m

The following convergence theorem states the convergence of the approximate solution UT to the continuous one u in L2(R) and the convergence of GT to Vu in Hdi,( 0 ) .

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Finite volumes for complex applications

Theorem 1 Under Assumption 1, let 7 be an admissible mesh (in the sense of Definition 1) . Let u~ be defined by (3)-(5) and let u E H i ( R ) be the unique variational solution of Problem ( I ) , (2). Then UT + u in L2(R) as size(7) tends to 0. Furthermore, assume that there exist two fixed values C > 0 and M such that the inequalities dK,, 2 Cdiam(K) and M 2 card(EK) hold for any control volume K E 7 and for any a E E K . Let GT be the approximate gradient defined by equations (6)-(7). Then GT converges in Hdiv(R) to the gradient of the unique variational solution u E H i ( Q ) of Problem ( I ) , (2) as size(7) -+ 0 . Sketch of the proof (the complete proof of this theorem is given in [EGH]). First note that for all K E 7 and for a.e. x E K .

and therefore divGT -+ -f in L2(R) as size(7) tends to 0. Assuming only u E H; (R), let E > 0 and cp E C,"O(R) such that 2 I l u - ~ l l ~ l ( n ) E. Using the variational formulation of (6) in each K E 7, one proves the existence of some Fl (M, 5) > 0 and of some F2(R,f , cp, M, C, 7 ) > 0 which tends to zero as size(7) -+ 0 such that

<

with

With expressing the consistency of the fluxes, and following the steps of the ~ ~ (error 0 )estimate proof, one then proves the existence of some F3(R, f , p, 7 ) > 0 which tends to zero as size(7) -+ 0 such that

Using the triangular inequality, one has 116- - V

+

(12)

+ 0.

Using (S), we get

U I I L ~ ( I~ )2 ~llGr - ~ c p l l ~ 2 ( n 2 l ~E.

Choosing size(7) small enough such that F2(a, f , 9 , M , 6 , 7 ) + Fi (M, C)F3(0, f , cp, 7)I E, equations (9)-(12) lead to

which shows that GT -+ V u in L2(R) as size(7) GT -+ Vu in Hdiv(R) as size(7) + 0.

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153

4. Error estimate

Theorem 2 Under Assumption 1 let > 0 and M > 0 be given values and 7 be an admissible mesh (in the sense of Definition 1) such that the inequalities ~ K , C> - Cdiam(K) and M >_ card(EK) hold for all control volume K E 7 and for all u E E K . Let u~ be given by equations (5')-(5). Then there exists c, only depending o n R , u and C, such that

llu - U T I I L ~ ( R )

i c size(7-I

(14)

Furthermore, let GT be defined by equations (6)-(7). Assume that the unique variational solution, u, of Problem ( I ) , (2) belongs to H 2 ( R ) . Then there exists C > 0 which only depends o n R , u, and M such that

<

The proof of this theorem entirely follows the steps of the proof of Theorem 1, replacing cp by u E H2(R) (it is given in detail in [EGH]).

Bibliography [ABM 951

AGOUZAL, A.,BARANGER, J.,MAITRE,J.-F. and F. OUDIN (1995), Connection between finite volume and mixed finite element methods for a diffusion problem with non constant coefficients, with application to Convection Diffusion, EastWest Journal on Numerical Mathematics., 3, 4, 237-254.

[AWY 971

ARBOGAST,T., WHEELER,M.F. and Y o ~ o v ,I.(1997), Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, S I A M J. Numer. Anal. 34, 2, 828-852.

[CVV 971

COUDIERE, Y., VILA,J.P. and VILLEDIEU, P. (1997), Convergence Rate of a Finite Volume Scheme for a Two Dimensionnal Convection Diffusion Problem, accepted for publdcatdon i n M2AN.

[DUB 971

DUBOIS,F. (1997) Quels sch6mas num6riques pour les volumes finis ?Neuviime se'minaire sur les fluides compressibles, C E A Saclay, France.

[EGH 991

EYMARD,R., GALLOUET, T. and HERBIN,R. (1999), Convergence of finite volume schemes for semilinear convection diffusion equations, Num. Math., vol. 82, p. 91-116.

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Finite volumes for complex applications

LEGH]

EYMARD,R., GALLOUET, T. and HERBIN,R . , Finite volume approximation of elliptic problems and convergence of an approximate gradient, submitted to publication.

[ROT 911

ROBERTS,J . E . and THOMAS,J.M. (1991), Mixed and hybrids methods, in Handbook of Numerical Analysis I1 (NorthHolland, Amsterdam) 523-640.

[VAS 981

VANSELOW,R., SCHEFFLER,H.P. (1998), Convergence Analysis of a Finite Volume Method via a New Nonconforming Finite Element Method, Numer. Methods Partial Dzfferential Eq.,14, 213-231.

Finite volume approximation of elliptic problems with irregular data Thierry Gallougt and Rapha6le Herbin Universzte' d'Aix-Marseille 1, Centre de mathe'matiques et informatique 39 rue Jolzot Curie, 13453 Marseille, France [email protected], [email protected].

abstract We prove here the convergence of a cell-centered finite volume scheme for the discretization on a non-structured grid of the Laplace equation with irregular data towards the weak solution of the equation. Keys words

Finite volumes scheme, non-structured mesh, diffusion equation,

irregular data.

1. Introduction We are interested here in proving the convergence of the finite volume method in the case of the following model equation: -Au = p ,

in

a,

with Dirichlet boundary condition:

where

Assumption 1 1. R is an open bounded polygonal subset of lFtd,d = 2 or 3, 2. p E LP( R ) for p E [I, +m] or p is a signed bounded measure. Such problems arise for instance when modelling heat transfers in the presence of electric current in which case the heat term due to ohmic loss writes

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Finite volumes for complex applications

p = aV@V@where a

E LCo(R)is the electric conductivity and @ E H 1 ( R ) is the electric potential; hence p E L 1 ( R ) (see e.g. [ F H 941). Another field where such a problem arises is in oil reservoir simulation, where the dimension of the well is often small enough with respect to the size or the domain of simulation so that it is modelled by a Dirac measure in the two-dimensional case ( d = 2). The purpose of the proposed presentation is to show that the finite volume method is well adapted to this type of problem; we can show in particular that the analysis tools recently developped by Boccardo, Gallouet et al. [BG 891 for the study of nonlinear partial differential equations with measure data can be adapted to show the strong convergence as the size of the mesh tends to 0 of the approximate finite volume solution in for any p E [dl towards a weak solution of (1)-(2) which is a function u from St to IR satisfying:

wtsP

A[

Remark 1 The Laplace operator is considered here for the sake of simplicity, but more general elliptic operators are possible to handle, for instance operators of the form - d i v ( a ( u ) V u ) with adequate assumptions on a. A by product of the convergence analysis which is presented here is the existence of a solution of (3).

2. The finite volume scheme The finite volume scheme is found by integrating equation (1) on a given control volume of a discretization mesh and finding an approximation of the fluxes on the control volume boundary in terms of the discrete unknowns. Let us first give the assumptions which are needed on the mesh.

Definition 1 (Admissible meshes) Let S1 be an open bounded polygonal subset of I R ~ .A n admissible finite volume mesh of R , denoted by 7, is given by a family of "control volumes", which are polygonal convex subsets of St (with positive measure), a family of subsets of 2 contained in hyperplanes of IR~, denoted by C (these are the edges of the control volumes), with strictly positive (d - 1)-dimensional measure, and a family of points of R denoted by 7' satisfying the following properties (in fact, we shall denote, somewhat incorrectly, by 7 the family of control volumes): (i) the set of all control volumes is a partition of R ;

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157

(ii) For any It' E 7 ,there exists a subset ZK of & such that dIi' = i\ Ii' = U U E ~ , a Let . E =UjyETE~. (iii) For any (Ii', L ) E T 2 with Ii' # L , either the (d- 1)-dimensional Lebesgue measure of n is 0 or il n = a for some a E &, which will then be denoted by Ii'l L . (iv) The family P = ( x K ) ~ E 7is such that X K E (for all Ii' E 7 ) and, if a = Ii'lL, it is assumed that X K # X L , and that the straight line D K , ~ going through X K and X L is orthogonal to Ii'lL. In the sequel, the following notations are used. The mesh size is defined by: s i z e ( 7 ) = sup{diam(Ii'), Ii' E 7 ) . For any K E 7 and a E E , m(Ii') is the d-dimensional measure of Ii' and m ( a ) the (d - 1)-dimensional measure of a . The set of znterior (resp. boundary) edges is denoted by Eint (resp. l e X t ) ,that is Eint = { a E &; a !$ d R } (resp. lext = { a E &; a c d R ) ) . The set of neighbours { L E 7;3u E E K , T = of I< is denoted by N ( I i ' ) , that is N ( I i ' ) For any K E 7 and a E ZK we denote by dK,, the Euclidean distance between X K and a . For any u E &, we define d, = dK,, dL,, if a = K l L E Iint (in which case d, is the Euclidean distance between X K and x L ) and d, = d ~ , if, a E Eextn&K. For any a E E ; the "transmissibility" through a is defined by r, = m ( a ) / d , if d, # 0 and r, = 0 if d, = 0. In some results and proofs given below, there are summations over a E go, with Eo = { a E E ; do # 0). For simplicity, (in these results and proofs) & = Eo is assumed.

-

n}.

+

W e may now introduce the space o f piecewise constant functions associated norm for this space. T h i s with an admissible mesh and some "discrete witP" discrete norm will be used t o obtain some estimates on the approximatesolution given by a finite volume scheme. Definition 2 (Discrete norm) Let R be an open bounded polygonal subset of R d ,d = 2 or 3, and let 7 be an admissible mesh. Define X ( 7 ) as the set of functions from R to IR which are constant over each control volume of the mesh. For u E X ( 7 ) , and p E [ l ,+m), define the discrete w;lpnorm by

where, for any a E 7 , D,u = IuK - uL1 if u E lint, a = KIL, Dou= (UK(i f u E l e x t n l ~ , where U K denotes the value taken by u on the control volume Ii' and the sets I , Grit, ZeXt and IK are defined in Definition 1.

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Finite volumes for complex applications

Let 7 be an admissible mesh. Let us now define a finite volume scheme to discretize (1)-(2). Let ( U K ) K € ~denote the discrete unknowns associated with the control volumes Ii' E 7. In order to describe the scheme in the most general way, one introduces some auxiliary unknowns namely the fluxes FK.,, for all IC E 7 and a E EK, and some (expected) approximation of u on an edge a, denoted by u,, for all u E E. For I< E 7 and a E CK, let nK,, denote the normal unit vector to u outward to Ii' and VK,, = j", v ( x ) . n ~ , , d y ( x ) . Note that dy is the integration symbol for the (d - 1)-dimensional Lebesgue measure on the considered hyperplane. We may now write the finite volume scheme for the discretization of Problem (1)-(2) under Assumption 1 as the following set of equations:

where F K ,is~defined by

and u,

= 0, va E Cext

Note that the values u, for a E Eint are auxiliary values which may be eliminated so that (5)-(8) leads to a linear system of N equations with N unknowns, namely the ( u K ) K ~with ~ , N = card(7). 3. E x i s t e n c e a n d e s t i m a t e s f o r t h e a p p r o x i m a t e s o l u t i o n

Let us first prove the existence of the approximate solution and an estimate on this solution. This estimate will yield convergence thanks to a compactness theorem which we recall below. L e m m a 1 (Existence and e s t i m a t e ) Under Assumptions 1, let 7 be an admissible mesh in the sense of Definition 1, and let: dK,o C = min min K E T U E E K d, '

then there exists a solution ( u K ) ~ to € the ~ system of equations (5)-(8). Furthermore, let p E [ 1 , and let u~ E X ( 7 ) be defined by u7(x) = UK for a.e. x E I<, and for any I< E 7; there exists C E IR, only depending on S1, C, p and p , such that

A),

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159

PROOF of Lemma 1 The existence and uniqueness to the solution of the scheme was proved in e.g. [H 951. Let us now turn to the estimate. For B E ( l , + m )let , p be the bounded function from IR to IR defined by p ( s ) = Multiplying (5) by p ( u K ) and summing over Ii' E 7 yields

&.

By a discrete integration by part and from the fact that

one obtains:

+

where a, = So p l ( u K t ( u L - u K ) ) d t if u = IirlL E Eint, and a , = So p l ( ( l t ) u ~ ) dift D E Ee,t n E K . Now for 1 5 p < 2, by Hijlder's inequality and from (11) 1

1

Let us reorder the summation over the control volumes in the right-hand-side This yields the existence and remark that by definition of (, one has do j of C1 E IR+ depending only on /I, R , 0 , p and (, such that

9.

Using a discrete Sobolev inequality (the proof of which is similar to the one proved in [EGH 971 or [CGH 98]), there exists C z E IR+ depending only on p such that:

5 C211u~111,p,~.

(13) From ( 1 2 ) and ( 1 3 ) , there exists C3 E IR+ depending only on B,p and ( such that IIuIIP*

llull;*

A,

5 C3(1+ l l U 7 1 1 ~ ) .

&

2-P

Hence, for p < one has < p", so that one may choose B E ( 1 , 2 ) such that 5 p*. Since p > f p , from (14) and (12), there exists C depending only on p , S1, p and (, such that:

$

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Finite volumes for complex applications

4. Convergence

Let us now show the convergence of approximate solutions obtained by the above finite volume scheme when the size of the mesh tends to 0. One uses Lemma 1 together with the Kolmogorov compactness theorem given at the end of this chapter to prove the convergence result. In order to use the Kolmogorov compactness theorem, one needs the following lemma. L e m m a 2 ( E s t i m a t e o n t h e s p a c e t r a n s l a t e s ) Let R be an open bounded set of IFtd, d = 2 or 3. Let 7 be an admissible mesh and u E X ( 7 ) . One defines ii by ii = u a.e. on R, and ii = 0 a.e. on IFtd \ a . Then there ezists C > 0, only depending on S1, such that

The proof of this lemma is an easy adaptation of the proof which is available in [EGH 971 of [EGH 991. We are now able to state the convergence theorem. T h e o r e m 1 (Convergence) Under Assumption 1, let 7 be an admissible mesh. Let ( U K ) K ~ Jbe- the solution of the system given by equations (5)-(8). Define u~ E X ( 7 ) by uT(x) = ZLK for a.e. x E I<, and for any li' E 7. Let (7n)nEN be a sequence of admissible meshes such that size(7,) -+ 0 as n -+ +co and such that dK,o C = inf min min > 0, (16) ~ E IK NE T , OEEK d,

then there exists a subsequence of converges in LP(S1) for p < to Problem (3) as size(7) + 0 .

o

still denoted which weak solution u E nl
R e m a r k 2 In the case of the uniqueness of d = 2, or if d = 3 and R is convex (see e.g. ( u ~ tends ~ to) the~ solution ~ ~ of (3), and size(7) -+ 0 under the condition that there 2 C for all 7. minKE7 rnin,,~,

a solution to (3), for instance i f [G 97]), then the whole sequence therefore u 7 + u i n LP(R) as exists zeta > 0 such that =

PROOF of Theorem 1 Let Y be the set of approximate solutions, that is the set of functions u 7 as defined in Theorem 1 where T is an admissible mesh which satisfies (16). Thanks to Lemma 1, for any p < there exists C1 E IR, only depending ~ for ~ ~ all( u~~ E~ Y. Then, thanks on R, p,Co and p, such that I I u ~ 5~ C1

A,

Numerical analysis

16 1

to Lemma 2 and to the Kolmogorov compactness result (see e.g. [EGH 991 or [EGH 971 for the case p = 2), the set Y is relatively compact in LP(R). Now by Lemma 1, we know that for any q E [1, &), there exists C2 E IR, only depending on a,p , (0 and q , such that )lu7111,q,~ C 2 Hence, adapting a result of [EGH 971, one may show that any possible limit (in LP(R)) of a sequence (u7")nE1N C Y (such that size(%) + 0) belongs to w;"(R). Therefore, there remains to prove that if ( u T n ) n EC~ Y converges towards some u E w,"'(o) in LP(R) and size(7,) -+ 0 (as n + co),then u is a solution to (3). We prove this result below, omiting the index n , that is assuming u~ + u in LP(R) as size(7) -+ 0.

<

Let $ E C,Q3(R)and let size(7) be small enough so that $(x) = 0 if x E I< and li' E 7 is such that 8 K n d R # 0. Multiplying ( 5 ) by $(xK), and summing the result over K E 7 yields

Since the set {K, I< E 7) is a partition of 52,

where qT is defined from R to IR by G7(x) = $ ( x ~ )for x E K ; since $ E C,DO(R),by the Lebesgue dominated convergence theorem, one has:

Now, using the same technique as in the variational framework (see [EGH 971 or [EGH 99]), one has:

) Hence, letting size(7) + 0 in (17) yields that u E npE[,' d~- 1) W ; ' ~ ( Rsatisfies

which, in turn, yields (3) thanks to the fact that u E w ; ~ ~ ( R ) ,and t o the density of C,00 (0)in w ~ ' ~ ( R ) . This proves that u7 + u in LP(R) as size(7) + 0, where u is a solution (in npEII,A)w;"(R)) to (3) and concludes the proof of Theorem 1.

162 Finite volumes for complex applications

References [BG 891

BOCCARDO L . and T. GALLOUET, Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal., 87, 1, , 149-169, 1989.

L., T. GALLOUET, J . L . VAZQUEZ, nonlinear elliptic [BGV 931 BOCCARDO equations in Rn without growth restrictions on the data, Journal of Differential Equations, Vol. 105, & 334-363, 1993. Y., T . GALLOUET and R. HERBIN, Discrete Sobolev [CGH 981 COUDIERE inequalities and LP error estimates for approximate finite volume solutions of convection diffusion equations, Prepublication 98-13 du LATP, Marseille, submitted. [G 971

GALLOUET T, Problgmes elliptiques et paraboliques non lineaires, Cours de DEA de I'ENS Lyon, 1997.

and R. HERBIN, The finite volume [EGH 971 EYMARDR., T. GALLOUET method, Prepublication 97-19 du LATP, Marseille, to appear in Handbook of Numera'cal Analysis, P.G. Ciarlet and J.L. Lions eds. and R. HERBIN, Convergence of a [EGH 991 EYMARDR . , T . GALLOUET finite volume scheme for semilinear convection diffusion equations, Numer. Math., 82, 91-116, 1999. [FH 941

FIARD,J . M . , R. HERBIN(1994), Comparison between finite volume finite element methods for the numerical simulation of an elliptic problem arising in electrochemical engineering, Comput. Meth. Appl. Mech. Engin., 115, 315-338.

and R. HERBINM.H. Vignal, Error estimates for the [GHV 991 T. GALLOUET approximate finite volume solution of convection diffusion equations with Dirichlet, Neumann or Fourier boundary conditions, Pripublication 99-05 du LATP, Marseille, submitted. HERBINR. (1995), An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh, Num. Meth. P.D.E. 11, 165-173.

Analysis of a finite volume scheme for reactive fluid flow problems

Astrid Holstad Institute of Energy Technology P.O.Box 40 N-2007 Kjeller, Norway

Ivar Lie Norwegian Meteorological Institute P. 0.Box 43 Blindern N-0313 Oslo, Norway

-

ABSTRACT We analyze a mathematical and numerical model for reactive multiphase fluid flow problems focusing on chemical reactions and transport of solutes. The discretization consists of a cell-centered finite volume scheme in space and an implicit Runge-Kutta scheme in time. Convergence results are presented for the system of governing equations consisting of both timedependent PDEs and ODES.

Key Words: PDEs, Reactive fluid flow, Time integration

1. Introduction

Most of the geochemical and biogeochemical phenomena of interest to Earth scientists are the result of the coupling of some combination of fluid, heat and solute transport with chemical reactions. Examples of applications where these phenomena are studied are pollution of the atmosphere and the ocean, groundwater contaminant transport, remediation processes, geochemical weathering, ore deposition and diagenesis of sedimentary rocks. In the above applications the chemical reactions and the fluid flow may take place in a very heterogeneous medium under varying temperature and pressure conditions with different time scales. Many different types of chemical reactions take place in a geochemical system:

164

Finite volumes for complex applications

Precipitation and dissolution of solids, dissolution and exsolution of gases in to the aqueous phase, ion exchange, redox reactions, decomposition of organic material, radioactive decay, isotope fractionation and biological processes. Because of the complexity of reactive transport processes in porous media, modeling their dynamic behaviour is a challenging task. Many models of different geochemical processes have been proposed over the years, see [HOL 99aJ for a brief review. In this paper we analyze a mathematical and numerical model for reactive fluid flow in porous media focusing on efficient integration of the equations over long time periods. As an example, studying diagenetic processes in sedimentary rocks often requires simulation periods of 10 - 100 million years. The model is discussed in detail in [HOL 99a] and [HOL 99bI. The rest of the paper is organized as follows: In section 2 we present a simplified system of the governing equations which discretization is discussed in section 3. The error analysis of this discretization is then discussed in section 4. 2. The governing equations

The transport of solutes in a porous medium under the assumption of multiphase flow is given by

where fi is the molal concentration of the ith species, pi is a function to correct for porosity 4, phase saturation S, and phase density p,, Ji denotes the flux, I, denotes the rate of the rth chemical reaction, vi, denotes the stoichiometric coefficient and qi denotes any general source term. Let c = [pl f i , . . . ,pNfN]. For kinetically governed reactions,

The complete equation system describing multiple species, multiphase and nonisothermal reactive fluid flow becomes when classifying the species into primary (p), secondary (s) and kinetic (k) species,

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165

where I/ and w denote matrices derived from the stoichiometric coefficients in (I), [c, c , ~ k =] [ p l f l , . . . ,pN fN], denotes the mass action equation for the r t h equilibrium reaction, a,,, and ai denote species activities, i.e. the concentrations corrected for nonideal behaviour and fiir denotes an element of the matrix p. By splitting the flux into a convective and a diffusive part we can write (3a) in component form,

where Crn = 1 for mobile species and 0 for immobile species. Immobile primary or secondary species may occur in applications involving sorption and ion exchange reactions. The convective and the diffusive flux terms in (4) cannot be simplified due to the difference in the phase specific parameters. In applications involving only a single moving phase and no immobile species (4) simplifies to

when defining

3. T h e discrete s y s t e m

In this section we discretize the simplified equation (5) in space using FVM and in time using the Runge-Kutta formulas

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Finite volumes for complex applications

By augmenting the Butcher matrix A = {a,,) with the vector { b s ) as the bottom row, the formula (7b) is included in (7a) if we take r = 1,.. . ,P + 1 and CP+I = 1. In the following we will use only (7a) with this convention. A comprehensive treatment of RK methods can be found in [HWN 871 and [HW2 961. Discretization of (4) in the general case is treated in [HOL 99aI.

Integration of (5) in terms of a finite volume V gives the following ODE,

with convection term

and diffusion term

We integrate the terms in (8) over a time interval h = t,+l - t, and obtain the following expressions: Accumulation,

convection,

diffusion,

Numerical analysis

source term, using

fi =

167

& ( ( I - 4 ) p sf i ) and Qj = Bat Q 37 .

+

We combine corresponding terms and obtain (P 1 ) . Np generalized concentration equations and (P 1) . Rkin mineral equations:

+

- (4'j)tn) =

m(V)((4'j)tn+,,h P -

h

C

arsuav(tn

+ c s h , Qj,tn+csh)

s= 1 P

+

hC

+

a r s W ~ v ( t n c s h r Qj,tn+c,h)

s= 1 Rkin

+

m(v)C G j i ( I ( l - d ) ~ s f i l ~ , , +-~[ (~I h- d ) p sf i I t n ) i=l

+

m(v)(Qj,tn+c,h

- Qj,tn)7

The concentrations of the primary and the secondary species are found from the algebraic system:

For details in solution of the equation system (17a), (17b),see [HOL 981. 4. Error analysis

In this section we give some error estimates for the finite volume RK discretization presented in previous section. For the transport equations the analysis will be based on the results in [EGH 991, in which references to related

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Finite volumes for complex applications

analysis based on different techniques, e.g. the finite element techniques, also can be found. For the mineral ODEs the analysis will use the theory of numerical methods for ODEs. The transport equations are given in (5) with the definition (6). The dissolution and precipitation of the minerals is assumed governed by first order kinetics as in (2):

~ . relation between Qj, j = 1 , . . . , N, and c, when setting yi = p, f ~ , + The is given by the equations (17a) and (17b). For our purpose, it is sufficient to assume the existence of a sufficiently smooth (at least continuous) mapping such that c, = +(Q1,. . . , QNP). We also impose a boundedness restriction on Igi(cp)l = Igi(a(Q1,.. . , QN,))~I M, (19) which means that the right hand side of (18) is Lipschitz continuous. The error analysis will consist of deriving a bound on the sum of the discretization errors of Qj and yi since the equations involved are coupled. We will use the following notation for Qj: The space and time continuous quantity appearing in (5) is denoted by Qj(x, t). We can project this function onto the space of piecewise constant functions on the cells, Po(T), where T denotes the set of cells covering R , and this projection on a cell V is denoted by Qjv(t). The fully discrete value on a cell V at timepoint tn is denoted by Sjnv.The total error on V at t, is then

but it may also be useful to consider the discretization errors in space and time separately, Qjv(tn) - Q>. v -v In the above error expressions x v denotes the representative point on cell V, e.g. the barycenter. The immobile species are computed only on the cells, so the discretization error on a cell V at timepoint tn is IYiv(tn) - YYvIFor the error analysis of the numerical solution of ODEs by Runge-Kutta methods we use the B-convergence theory, see e.g. [HW2 961 for details. 4.1 Estimates usdng the theory of Gailouet et a1

We will use the analyses in [EGH 991, and in order to do that we partly decouple the equation system ( 5 ) , (18). We insert the relation ii= $((l-+)yi) into (5) and move the resulting term to the left hand side. We can then write,

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169

where L denotes the flux operator, i.e. L(!Pj) = V . (u!Pj - 4DV!Pj).

~2;" i;rjiyi. This is a 1-1 mapping between u j and Qj,

Let aj = @Dj - (1 - 4) j = 1 , . . . , N, provided q5

# 0. Thus we have

since L(yi) = 0. The decoupled PDE system (20) is of the type considered in [EGH 99, ch.41. We equip (20) with an initial condition uj (x, 0) = ujo (x) and a nonhomogeneous Dirichlet boundary condition u j = G j on the boundary 6%. Consider first the backward Euler finite volume discretization of (20),

where

and L~(u,'$') is the linear operator representing the finite volume discretization of r JaV ( U U~ @Dogj)- n d s by the upstream scheme for the first term and the VF9 scheme for the second term. We then have the following convergence theorem, Theorem 1 Let R be a polygonal open subset of R d , T > 0, u j E C 2 ( a x R , R ) , ujo E C2(fi, R ) , Qj :j C2( a x R , R ) , and let 7 be an admissible mesh in the sense of Definition 3.1 i n [EGH 991. Then there exists a unique vector ( ~ j v ) satisfying " ~ ~ (21) such that

Furthermore, let en, = uj (xv, t n ) - uyv for V

E

7. Then

where k = size(7), the size of the largest cell i n 7 . Proof: See [EGH 99, theorem 4.1, pp.97-991..

We will now extend this result to higher order time integration methods. It is to be emphasized that this is done primarily to be able to solve the stiff system (18) with a more robust method, including local error control and variable timesteps. The higher accuracy in time is not considered important to reduce

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Finite volumes for complex applications

the the total discretization error. The space error will be of first order because of the projection onto the space of constant functions on the cells, and this error will probably dominate in the total error except for very fine meshes. There is a fairly straightforward extension of theorem 1 to the use of the Backward Differentiation Formulas (BDF) class of methods, see [HWN 87, ch. 111.11 for details. The result is: Corollary 1 Let uj E C(p+')(fi x R , R) and ujo E ~ ( p + l ) ( fR). i , Otherwise the assumptions are as i n theorem 1. If we apply a BDF method of order p to (20), the error satisfies

( C( e ~ ) ~ m ( ~ )<) C(k + h P ) . VET

Proof: See [HOL 99a, Sect.51. w

We now proceed to the implicit RK methods. Corollary 2 Let uj E ~ ( p + ' ) ( f ix R , R ) , ujo E C(pf ' ) ( a , R),and make the remaining assumptions as in theorem 1. If we apply a P-stage implicit RK method of order p to (20), the error satisfies

( C ( e ~ ) ~ m ( v ) ) 5 C ( k + hP). VET

Proof: See [HOL 99a, Sect.51. w Because of the simple relation between Qj and uj it follows that the above results, theorem 1, corollary 1 and corollary 2 also apply to Qj.

Remark 1: If the exact solution is less smooth than stated in the above results, the convergence order will be correspondingly reduced. If crj E c('+'),r 5 p the convergence order will be r . 4.2 Semigroup estimates

We may also prove convergence results with a more abstract approach using the fully discrete approximation of semigroups acting on a Banach space X. The classical semigroup approach can be used if the generator of the semigroup (the spatial part of the differential operator) is time-independent. The extension to time-dependent differential operators is discussed in [FAT 83, ch.71. If we start with the Cauchy problem for the transport equations,

Numerical analysis

17 1

we can construct approximations to the semigroup generated by L. The operator generates a semigroup provided it satisfies the conditions in the Hille-Yosida theorem, see e.g. [FAT 831. In our case it can be shown that the operator L generates a Co-semigroup, i.e. a semigroup S(t) which satisfies the basic esMewt, t 0, see [EVA 98, p.4211 . Hence the solution timate IIS(t)lltcxl is a$, E X. aj(t) = S(t)ajo,

<

>

We introduce some notation for approximation of the semigroup S ( t ) , referring to [MBI 981. Let X N denote the space of discrete values, in our case this is the space Po(T). Note that in view of theorem 1, X may be taken as C2. NOW define restriction and prolongation operators,

and from the finite volume approximation we have

Let r ( z ) denote a rational complex approximation to ez, such that r(Lt) approximates the semigroup. The approximation is said t o be A-acceptable if r(z) is analytic for R(z) < 0 and Ir(iy)l 5 1, y E IR. Let e be the maximal value such that r ( z ) is analytic in the set {t : R ( t ) < E ) . We then have the following convergence theorem, Theorem 2 Let r(z) be an A-acceptable approximation to eZ of order p, and assume that ajo E D ( L ~ ) . Then Ilaj(tn) - ujnllllx I c t ,e-p(e)hp(e)ew~tnI I L ~30. , 11x + O(k)r (30)

for

e = 0,1, . . .,p + 1, C # (p + 1)/2 and C-i 2~ p+l

provided 0

when when

e< 9 e>9


Proof: The first term representing the error in time is proved in [MBI 981. A complicated second term representing the error in space is also derived in [MBI 981. In our case this error can be written

The operators p~ and r N are assumed to be uniformly bounded. The expression Ilr(hA)"II is also bounded since Ilr(hA)nllL(x) 5 e"K(nh),see [MBI 981. From (29) the last term in the above space error is O(k). rn Remark 2: Initial boundary value problems can be solved by introducing the boundary condition in the definition of D(L), for example we may take D(L)= ~ ~ n ( H,'(fi). 0 )

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Finite volumes for complex applications

T h e non-homogeneous case can b e treated with t h e same tools as we applied in theorem 2. Consider again (28) . T h e solution is

where * is t h e convolution operator in time. Denote by 5j t h e solution of t h e homogeneous problem, then a j ( t )- e j ( t ) = S ijj(t)if ajo = 5j0. Let t h e discrete solutions be given by ayv - 5yv = pNrN(S * ijj(t)). Hence by subtraction,

*

T h e last t e r m on t h e right hand side can be bounded by O(k) in t h e X-norm, a n d t h e error in t h e non-homogeneous problem is also given by theorem 2.

5 . References [CRM 841

CROUZIEX M, MIGNOTA,, Analyse numerique des equations differentielles, Masson, Paris, 1984.

[EGH 991

EYMARD R, GALLOUET TI HERBINR., Finite Volume Methods, In Handbook of Numerical Analysis, Vol VIII, F.G.Ciarlet, J.-L. Lions eds., North-Holland, Amsterdam, 1999.

[EVA 981

EVANS,L C . , Partial Differential Equations, American Mathematical Society, Providence, 1998.

[FAT 831

FATTORINI H O., The Cauchy Problem, Addison-Wesley, London, 1983.

[HOL 981

HOLSTADA,, Numerical solution of nonlinear equations in chemical speciation calculations, To appear in Computational Geosciences, 1998.

[HOL 99a]

HOLSTADA,, A mathematical and numerical model for reactive fluid flow systems, Submitted to Computational Geosciences, 1999.

[HWN 871

HAIRERE , WANNERG , N ~ R S E T TS, P., Solving Ordinary Differential Equations I, Nonstiff Problems, Springer Verlag, Berlin ,1987.

[HW2 961

HAIRERE , WANNERG., Solving Ordinary Differential Equations 11, Stiff and Differential-Algebraic Problems, Springer Verlag, Berlin, 1996.

[MBI 981

B J ~ R H UMS, LIE I., Fully discrete rational approximation to semigroups, preprint, DNMI, 1998.

[HOL 99b]

HOLSTADA., Numerical solution of the reactive fluid flow equations, Preprint, IFE, Kjeller, Norway, 1999.

Convergence of a finite volume scheme for a nonlinear convection-diffusion problem Anthony MICHEL, C.M.I. Marseille Email : [email protected] ABSTRACTW e construct a time explicit scheme for a nonlinear convection diffusion problem which is Lm stable under a C.F.L. condition. W e also obtain discrete estimates that allow us t o apply I(o1mogorov's theorem. T h e n u p t o a subsequence we get the convergence of the scheme t o a weak solution of o u r problem. In the last part, we give some elements of a simple proof of uniqueness i n a special case by a dual method. K e y Words: finite volumes, parabolic nonlinear equations , Kolrnogorov's theorem.

1. Problem

We consider the following nonlinear parabolic degenerate problem in a bounded polygonal domain S1 of I W q :

This problem arises in the study of a two-phase flow in a porous medium. The unknown u is the saturation of the first phase so it takes its values in the interval [O, 11. The global flux vector v is a given function in cl(Qx [O,T]). In the same way, f and p are given in C1([O,I]). The only assumptions that we make are

(H3) cp is strictly increasing but p' may vanish (in which case the equation degenerates) and f is nondecreasing.

2. Scheme Let 7 a cell-centered unstructured mesh (see for instance [EGH 971, [EGH 991, or [YNS 971 for definition of an admissible mesh) and St a time step. We will always suppose for the sake of simplicity that there exist N E W such that (N 1)dt = T. We construct an approximate piecewise constant solution a.e. on fl x [0,T) by

+

where { u ; ) is defined by an "upwind" finite volume scheme which we now describe.

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Finite volumes for complex applications

For a control volume Ii' E 7 we take

For Ii' E 7 and n E 10, N ] we define ufdKtl from { u g ) K E T by

KnL and u k $ is an upwind choice of u on the interface between Ii' and L (it equals U; if v Y & , ~ 2 O and u t otherwise). As a consequence of (Hl),(H2), we have

vEtL= Jo v ( . ,n d t ) . n K , L if u = i? n i,TKPL =

vz

v;,, = 0. LEN(K) If we denotes by x- = max(0, - I ) , equation [3] is therefore equivalent to the following non-conservative form Proposition 1

By using monotony of the scheme we then easily prove the following lemma Lemma 1 Let @ = max ) p l ( x ) )and F = max If l ( x ) \ . Assume that ~€[0,11

XE[Q,11

dt

+

T K r L Q v&,; F 5 1 for all I( E 7. [51 m(It') LEN(K) Then the scheme is LW stable, i. e. if u o E [ A ,B] a.e. then u7,dt E [ A ,B] a. e.. 3. D i s c r e t e e s t i m a t e s on p ( u ) and f ( u ) Lemma 1 give an LW estimate on u7,dt. But it is not enough to deal with the nonlinear terms. We now prove the following inequalities which are crucial for the proof of convergence

P r o p o s i t i o n 2 Let @ = max Ipl(x)l and F = max I f l ( x )1 . On condition that

"

x

XE[O,~I

x€[O,~I

T K T L+ @v>,LF 5 1 - d < 1 for all Ii' E 7 ,

m(K) LEN(K) then there exist C(E,u o ) > 0 such that

N

and

E a t

1C

C

vZ,;

( f ( u k )- f

( ~ 2 )< )CF. ~

[61

[8I

Proof. We use again the non conservative form [4]. We multiply the equation by 6tu;i and sum over li' E 7 and n E [0, N ] . We get E l + E2 + E3 = 0, where

Numerical analysis

n=O

175

KET

N

n=O

KET

and E 3 = x d t x n=O

x

V~,~-(~(U"K-~(U~))U&.

K E TL E N ( K )

Let give now a useful1 result (see for instance [EGH 971)

Lemma 2 (technical lemma) Let g(z) = J : t f ( t ) d t where f E C' ( [ a PI) , is monotone then if we denote b y F = max If l ( x ) 1, for all a and b € [ a ,PI, xE[a,PI

By lemma 2,

xat5x x N

E3

2

n=O

1

KET L E N ( K )

But v g j L = - v F S K . So by using proposition 1, we get

and finally

1

v;,,(f(uk)-f(~2))~~

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Finite volumes for complex applications

It remains to estimate

R1 =

C -1 C rn(~<)(u;+~

-

ua2. By [4],

n=O 2 K E 7

Therefore by using condition [6], we get

rn(~)(uk)~5

Moreover,

Then by collecting the previous inequal-

-

KG7 -

1 ll~011;2

ities, we complete the proof with C ( [ ,uo)= 5

*) .

4. Translates estimates As a consequence of estimate [7], by classical technics that can be seen for instance in [EGH 971, we get the two following inequalities Corollary 1 (Space-translate estimate) [Jnder the assumptions of proposition 2,

where h = max diam(K) and RE = R 0 R-[. K€T

Corollary 2 (Time-translate estimate) Under the assumptions of proposition 2, there exist C'(E,p, f , v , uo,R ,T ) > 0 such that

5. Cornpacity and Convergence We handle now the first part of the conclusion under the convergence of the scheme. For that we need t o use the classical following compactness theorem.

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177

Theorem 1 (Kolmogorov)

Let 3 a bounded subset of L ~ ( I wsuch ~ ) that

Then for every R C C IRd, F is relatively compact in L ~ ( R ) . We are now able to state the convergence theorem

(T,, 6tm) a sequence of meshes and time steps that satisfies assumptions of proposition 2. Assume moreover that h , tends to zero when m tends to co (which implies that s t , tends to zero ) . Then there exist u E L" ( R x ( 0 ,T ) )such that p ( u ) E L2 ( 0 ,T ,H 1 ( a ) )and up to U T ~ =, u ~ for~ L~ M ( a x ( 0 ,T ) ) weak star topology and a subsequence, lim,,, in Lp(R x (O,T)),V p < a.

Theorem 2 (Convergence theorem, part 1) Let

Elements of proof. Let extend uT,at on iRq+l by zero out of x ( 0 , T ) . From corollary 1 and corollary 2, we directly deduce that for every (J, s ) E RQ+l, llp(ur,at(.+ t, .

+ 2C's + ( 2 ~ l t l m ( a a+) 2 m ( ~ ) M,', s)

+ t ) )- c p ( u ~ , a t ('))lli2(Rm+l) ., i

where M, = max l p ( x )1.

2Cltl(lEI+ 2h)

[O,11

This inequality allow us to apply theorem 1 and we obtain regularity on the limit by looking at accroissements taux which converge to the derivatives in D'. (see for example [EGH 971) 6. Convergence, part 2 Theorem 3 (Convergence theorem, part 2) W e suppose that the assumptions of

theorem 2 are satisfied, and we assume also that there exists B > 0 such that for all mesh T,, the followzng regularity property is satisfied :

Then the function u given i n theorem 2 as solution of [ I ] in the following sense : V$ E Ctejt = {q E C2"(C? x [0,TI) such that V q . n = 0 and q(., T ) = O),

Proof. The convergence of u, to u is strong which implies that f ( u , ) and cp(u,) converge to f ( u ) and p ( u ) . So it suffices to show that

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Finite volumes for complex applications

Let m fixed and 7 = 7,,St = St,. Let then multiply [4] by St*;, where = $ ( x K ,n S t ) and sum over I( E '7- and n E 10, N ] . We get Tl T2 T3 = 0 , where

+ +

Q :

n=O

KET L E N ( K )

where Bk$ is equal to Q E if given by

and S3 = /a

v k I L5 0 and 42 otherwise.

We compare ?;: to Si

jnvf ( u ) . v*

Classically, (see for instance [YNS 971, [EGH 971) because II, is a regular function and $ ( T )= 0 we get lim (S1- TI) = 0. m--too

Numerical analysis

179

By using estimates [S], [7] and regularity condition [9] we get also

and

ITz-SZI

5 J -

CI($) h .

Then lim IT2 - SS/= lim IT3 - S3J= 0, and the proof is complete m+m

m+w

7. Uniqueness

Let u l et u2 be two solutions of [lo]. We denote by ud = ul - 212. For all w E Cte,t, we have

where F =

f ( ~ 1-) f (

~ 2 )and

u1 - u2 attention with the dual ~ r o b l e m:

= p ( u l ) - p ( u z ) . SO it is natural to pay 211- u2

Fro111 [LSU 681, we can state the following result

Theorem 4 (Existence t o the regularized dual problem) Let F , v and @ be CCO functions under fi x [O,TI, and assume that there exists 6 > 0 such that @ ( x , t )2 6 . Then for every y E C,00 (Q x (0, T)) there exists an unzque solution to [I21 Moreover, we have also the following estimates

Proposition 3 Let $ a solutzon t o the regularized dual problem with second naembey xand M Y , ,%fa, Mv and MF some upper bounds for 1x1, @, IvI et IFI. Then there exzst C ( y , M a , M v , M F , R , T ) > 0 such that

and

Elements of proof. [I31 is a direct consequence of the maximum principle for parabolic equations. For [14] and 1151, we multiply the equation by A$ and integrate over d ! x (0, T). Because of [13], IIV+IIL~(nx(O,T)) is controled by dllA$llLz(nx(o,~)). We complete then the proof by using time and space integrates by part and Young inequalities. -

-

We are now able to give the main result of this secbion

180

Finite volumes for complex applications

Theorem 5 Assume cp-l is an holder continuous function with exponent there exist an unique solution to [lo].

$.

Then

Proof. By theorem 3, there exist a t least one solution to [lo]. We now turn to the study of uniqueness. Let x E C r ( R x (0, T ) )and 6 > 0 (6 5 MG). Oh = max(6, O) is again in Lm(M x (0, T ) )and 6 is a lower bound for it. We don't have regularity hypothesis on Qa and F but we can construct G,, F, and v, some sequences of regular functions on fi x [0, TI that converge to Oa, F and v in LP(M x (0, T)) for p < rn and such that For every n , by theorem 4 there exist a solution $, in C2,'(!,?x [0, TI) to the dual . . problem associated to G,, v,, F, and X . Because the upper bounds of G,, v,, F, and the lower bound 6 of G, are independant from n , estimates on A$, and V$J, are also independant from n , so we get rT r

But Oa - @ 5 6 Ilio,s) because @ and @a are equal on {O 2 6). Then, if we denote by As = { u d # 0) n {O < 61, we get

Because cp-l is an hijlder continuous function with exponent Moreover m(Aa) tends to zero, so that

i, ud 5 6 on As.

Since that is true for every regular function X, the proof is complete. [EGH 971

R. EYMARD, T. GALLOUET, R. HERBINFinite volume methods, Prebublication 97-19 LATP Marseille, to appear in Handbook of numerical analysis, Ph. Ciarlet & J.L. Lions ed., 1997.

[EGH 991

R. EYMARD,T. GALLOUET,R. HERBIN Convergence of finite volume schemes for semilinear convection diffusion equations. Numerische Mathematik 82 : 91-116, Springer Verlag, 1999.

[YNS 971

Y. NAIT SLIMANE Me'thodes de volumes finas pour des probldmes de dzffusion-convection non line'aires. Th6se de l'universite Paris 13, 1997.

[LSU 681

O.A. LADYZENSKAJA, V.A. SOLONNIKOV, N.N. URAL'CEVA. Linear and quase-linear equations of parabolic type. Transl. of math. monographs 23, American Mathematical Society, 1968.

Convergence analysis of a cell-centered FVM

Hans-Peter Scheffler *, Reiner Vanselow " "Institut fur Analysis **Institut fur Numerische Mathematik TU Dresden, Mommsenstr. 1 3 D 01062 Dresden -

A well-known cell-centered F V M with Voronoi boxes for discretizing the Poisson equation is analyzed. To achieve this purpose, a nonconforming F E M is constructed, such that the system of linear equations obtained by using the nodal basis coincides completely with that for the F V M . I n this way, convergence properties of the FEM, which are formulated i n terms of function space norms, can be transformed t o the F V M .

ABSTRACT

K e y Words: cell-centered finite volume method, Voronoi boxes, convergence analysis, nonconforming finite element method.

1. Introduction Finite Volume Methods (FVMs) are standard methods for finding numerical solutions of partial differential equations. Like Finite Element Methods (FEMs) they can be applied to a wide class of problems over arbitrary domains and allow local refinements of the domain partition. For a given f E L2(fl) we consider the Poisson equation -divgradu= f i n f l c R 2 , u = O onI'=dfl. (1.1) In order to simplify the presentation we restrict ourselves to open, convex and bounded polygonal domains R. The convergence proof of the FVM is based on the following two steps: Description of a nonconforming FEM such that the system of linear equations coincides completely with that for the FVM. Proof of convergence for the corresponding FEM.

182

Finite volumes for complex applications

In distinction to other authors (cf. e.g. [HAC 891) we use a nonconforming FEM basing on dual Voronoi boxes. The dual Voronoi boxes in combination with the choice of special discrete function spaces are well suited for our aim. The linear convergence of the FEM with respect to some energy norm is proved under the assumption that the solution u of (1.1) belongs to H2(R). As usual, some geometrical properties have to be satisfied for the partitions of 0 . A more detailed representation of this subject is given in [VAS 981. Here we give an improved version for the estimation of the consistency error term and discuss convection-diffusion problems, too.

2. The FVM and the corresponding nonconforming FEM

2.1. Box and dual box partitions

In the following, let M = {P) with P E cl (R) be an arbitrary finite set of points. Further, we use the notations Mi = M n R and Ma = M n I?, where m = card (Mi) > 0 and card (Mb) > 1 have to be satisfied. Let IP - Q ( denote the Euclidian distance between two points P and Q.

Definition 1. For P E M the Voronoi box bp is defined by bp = { Z E cl (R) : IZ - PI IZ - QI VQ E M ) . The set B = { b p } of all Voronoi boxes is called box partition.

<

If for different points P, Q E M the intersection bp n bQ is non-empty, then the corner points are denoted by E1(P, Q) and E2(P, Q), i.e. it holds EI(P,Q)E2 (P,Q) = bp n b~ . Now, for P E Mb we define the set N E ( P ) = {Q E Mb : P and Q are neighbours on I?), and for P E Mi we use the notations N ( P ) = {Q E M : Q # P, bp n b~ # 01, PPQ = IP- QI and YPQ = IEi(P,Q) - E2(P,Q)I VQ E N ( P ) , N N ( P ) = {Q E N ( P ) : y p > ~ 0 ) and N N i ( P ) = N N ( P ) n R . To define the FEM, which is used for the convergence analysis of the FVM, we need another partition of the domain R which is dual to the box one.

Definition 2. For P E Mi and Q E N N ( P ) the dual Voronoi box dbpQ is defined by

d b p ~= A P E 1 (P, Q)E2 ( P ,Q) U AQEi (P, Q)E2 (P, Q ) . The set d B = {dbpQ) of all dual Voronoi boxes is called dual box partition.

Numerical analysis

183

For the further considerations, the following property is assumed to be satisfied: For all P E Mb and Q E N E ( P ) it holds b p which is obviously equivalent t o f dl?= 1 J dl? V P E Mi

n bQ # 0 ,

(A)

Q€NN(P) b p n b ~

dbp

2.2. The Finite Volume Method

If we integrate both sides of (1.1) over the Voronoi box b, E B we obtain by applying Green's formula the equations - f [n(bp)lT grad u dl? = J"J f dR =: D ( P ) V P E Mi, (2.1) abp

bp

where the vector n(bp) denotes the outer normal direction of b p . Further, on the straight lines bp n bQ the outer normal n(bp) coincides with the vectors

Now, if in (2.1) the arising integrand [epQITgrad u is substituted by the constant finite difference approximation 4 Q ) - ,(PI PPQ and if (A') is used, then we obtain the following well-known cell-centered FVM for the Poisson equation (1.1): Find u V = u V ( M ) E Rm such that L V u V= bV (2.3) , , where the matrix L' and the right-hand side bV are given by ^iPQ -for Q E NNi(P) PPQ and b: = D ( P ) k,Z = 1,..., m. = YPR for = k [ePQITgrad u x

.

LL

I

C,,

R€NN(P)

0 otherwise The point P belongs to the index k and the point Q to the index 1.

2.3. The corresponding Finite Element Method

A weak formulation of the boundary value problem (1.1) reads as follows: Find u E V = HA(R) such that Vv E V. a(u,v) := J'(gradu)Tgradv dR = JJ fwd0 =: d(v) (2.4) R

n

Finite volumes for complex applications

184

For the FEM we define a finite-dimensional space Vh by Vh = {I E L2(n) : 1 i n t (dbPp)€P(P, Q ) Q ~ ~ PE Q dB v is continuous in P E M and v(P) = 0 Q P E Mb) , where P(P, Q) with P = (xp, yp)T and Q = (XQ,YQ)Tdenotes the space F ( p , Q) = span (1, [(XP - XQ)(X- XP) + (YP - YQNY- YP)]), and the function values at the points P E M iare choose as degrees of freedom. For the convergence analysis, we consider the nonconforming FEM: Find uh = u h ( M ) E Vh such that

with

x

ah (uh , vh) := d

JJ ([epQITgraduh) ([epQITgradvh) d o ,

(2.6)

b € d~B dbpq ~

epQ defined by (2.2) and D ( P ) defined by (2.1). The bilinear form ah is also defined on [V @ Vh] x Vh and, because of grad wh = Wh(Q)- wh(P) epQ vwh E P(P, Q) , DpQ which results in [grad vITgrad wh = ([epQITgradv) ([epQITgradwh) VV E H1(dbpQ),W h E P(P, Q) . This implies ah (v, wh) = JJ" [grad .uITgrad wh d o Vv E V, wh E Vh . (2.8)

x

dbpqEdB dbpq

Using the nodal basis functions and denoting the vector of the function values of the solution uh of the FEM (2.5) in the points P E Miwith uE, a linear system of equations arises, which has the form LEuE = bE. (2.9) The stiffness matrix L ~ the , vector uE = uE(M) E Rm and the right-hand side bE are given by = ah(@Q,@p), U ; = uh(P) and bf = dh(@p) k,1 = 1,...,m. Here, the indices are analogously used to Section 2.2. From the special form of the right-hand side dh it follows (cf. [VAS 981)

LE

Theorem 1. The problems (2.3) and (2.9) are equivalent, i.e. the vectors uV and uE coincide.

2.4. Convergence concept for the FVM

The solution of a FVM is a vector in Rm, whose entries can be considered as approximations of u(P), P E Mi, where u solves (1.1).Nevertheless and

Numerical analysis

185

in contrast to some other authors (cf. e.g. [LAM 96]),we prove convergence results for the FVM (2.3) in terms of function space norms. For that reason we need a bijective correspondence between the vector uV E Rm that solves (2.3) and the function uh E Vh which is the solutions of (2.5). Theorem 1 supplies such one additionally satisfying the interpolation property u h ( P ) = u! for all P E Mi. Let now a norm 11.11 on Vh be given that is a seminorm on V $ Vh . Definition 3. For a sequence { M h ) of sets of points satisfying the assumpti) corresponding sequence of approximate ons in Section 2.1 let { U ~ ( M ~be)the solutions defined by (2.3). W e say that the FVM (2.3) is convergent with respect t o ll.llh , i f f llu - uhllh approaches 0 for the solution u of (2.4) and the sequence { u h ) = { u h ( M h ) ) defined by the FEM (2.5).

3. The convergence result

3.1. The first step

To prove convergence of the nonconforming FEM (2.5) the well-known second Strang Lemma is used, which leads to an estimation of the form

with a positive constant C independent of h (cf. [VAS 981). In our application we choose the energy norm 2 ll"llh

= (3.2) with ah given by (2.6),such that the assumptions of the second Strang Lemma are satisfied.

3.2. The second step

To obtain an error estimation it is necessary to deduce bounds for the terms on the right-hand side of (3.1). For the approximation error term inf

vhEVh

1126

- Uh1Ih

we introduce the interpolation operator IIdB : H2(51) n H1(51) C V which is defined by (&B U ) ( P ) = u ( P )

VP E M ,

-+

Vh,

Finite volumes for complex applications

186

and take advantage of inf

IIu

~ h E v h

- vhllh

I 1121 - ~ B u I I ~ .

Standard techniques like that one in [CIA 911 lead to

with a positive constant c independent of h, if it holds u E H2(fl)n Hh (0)and if the partitions of fl satisfy some geometrical properties, which are comparable with the minimal angle condition for a corresponding Delaunay triangulation. The problem, that it holds can be overcome by appropriate spaces and seminorms like and

II[~PQ]

T

Iqliil(dbpQ)= gradqIIO,dbpQ in place of Sobolev spaces like H1(dbpQ) as well as a slight modification of Theorem 15.3 of [CIA 911. Obviously, in that theorem the assumption can be substituted by the weaker one I(I - W(~>l,,,,n = 0 VP E Pk(fl). For the consistency error term

we have to estimate dh (u, wh) = ah(^, wh) - dh (wh) which was done in [VAS 981 in the following way: At first, we obtain

1

+I [wh(P) - wh(Q)] J

b p nbq

[ e p ~ l ~ g r a dd ru.

(3.4)

Thereby, it is used, that the standard bilinear form ah, which is given by the right-hand side in (2.8), is substituted by that one in (2.6). Then IddbpQ(u,wh)l was estimated by applying Theorem 33.1 of [CIA 911, which under the same assumptions as above leads to

Numerical analysis

187

again with a positive constant c independent of h But, if we use (2.7), in place of (3.4) we get

hpQ ( u , w h ) = { w h ( Q ) - w h ( P ) ) v d p Q (21) with

Now, lrldpQ( u ) [can be estimated by applying the well-known BrambleHilbert lemma (cf. e.g. Theorem 28.1 of [CIA 911). Together with

this results in the same inequality (3.5), but gives a shorter proof. 3.3. The third step

Altogether (3.1), (3.3) and (3.5) lead to the following convergence result. Theorem 2. If the solution u of (2.4) belongs to H 2 ( R )n H i ( R ) and if the partitions of R satisfy some geometrical properties (for the details cf. [VAS 98]), then there exists a positive constant C independent of h such that it holds

llu - uhlih where

5Cd

max

b p E~d B h

PPQ Iul2,n.

is defined by (3.2) and uh is the solution of the FEM (2.5).

Because of Definition 3 the convergence properties of the FEM (2.5) can be transformed to the solution of the FVM (2.3).

4. Discussion

In [HAC 891 the convergence of a FVM like that, which is given by (2.3), is proved by using triangles for the partition of Cl and the well-known conforming linear FEM. It is an advantage of our approach, that the analysis can extended to convection-diffusion equations of the form div {- E grad u + b u ) = f , where E is a positive parameter and b is a given constant vector. One possibility is the use of a full-upwind technique with the approximation [epQIT{- E grad u bu) E PPQ ~ i: { K ( T [ ~ P Q IU (~P ~) -) K ( [epOITb)U ( Q )

+

PPQ

with the function function K defined by

13P0 E

}

188

Finite volumes for complex applications

K ( t )=

{

ifz20

z

otherwise, which altogeth'er leads to a FVM like that in [MIS 981. Another one is to use exponential fitting with the approximation [epQIT{- E grad u b u) 1 PPQ T % B(- @!2 eTb) U(Q) - B(T e b) u(P) ~

+

-

PPQ

{

E

}

(cf. e.g. [BBF go]), where the Bernoulli function B is defined by

5 . Bibliography

[BBF 901

BANK,R.E., BURGLER, J.F., FICHTNER, W., SMITH,R.K., <Some Upwinding Techniques for Finite Element Approximations of Convection-Diffusion Equations,, Numerische Mathematik, 58, 1990, p. 185-202.

[CIA 911

CIARLET,P.G.,
[HAC 891

HACKBUSCH, W., , Computing, 41, 1989, p. 277-296.

[LAM 961

LAZAROV,R.D., MISHEV,I.D., in , in Finite Volumes for Complex Applications, (1996), p. 231-240.

[MIS 981

MISHEV,I.D., &Finite Volume Methods on Voronoi Mesh e s ~ Numerical , Methods for Partial Differential Equations, 14, 1998, p. 193-212.

[VAS 981

VANSELOW, R., SCHEFFLER, H-P., , Numerical Methods for Partial Differential Equations, 14, 1998, p. 213-231.

Error estimates on the approxinlate finite volunie solution of convect ion diffusion equations with boundary conditions

Thierry Gallouet, Raphakle Herbin Universite' de Protlence, CMI, 39 rue Joliot Curie, 13453 iZilarseille Cedez 13, France;

Marie Hklkne Vignal Lab. MIP, UFR MIG, 118 route de Narbonne, 91062 TouEouse, Cedex 4, France; W e study here the convergence of a finite volume scheme for a diffusionconvection equation on an open bounded set of IRd (d = 2 or 3) for which we consider Dirichlet, Neumann or Fourier boundary conditions. W e consider unstructured meshes which include Voronoi' or triangular meshes; we use for the diflusion term a 'Ifour-point" finite volume scheme and for the convection term an upstream finite volume scheme. Assuming the ezact solution at least i n H z we prove error estimates in a discrete HA norm of order the size of the mesh. Discrete Poincare' inequalities then allow to prove error estimates i n the L2 norm.

ABSTRACT.

K E Y WORDS :

boundary conditions, convection, diffusion, error estimate, finite volume

schemes.

1. Presentation of the problem

Let R be an open bounded subset of IRd (d = 2 or 3) which is assumed to be polygonal if d = 2 and polyhedral if d = 3. We denote by OR its boundary and by n the unit normal t o OS2 outward to R . We consider the following convection-diffusion-reaction problem:

with different boundary conditions and where

Assumption 1 f E L 2 ( R ) , b E R,v E C1(n,Etd) such that divv/2

+ b 2 0.

In this paper, we consider three different types of boundary conditions. The S 2order ) t,o obtain error first one is a Dirichlet condition. Let g D E ~ ~ / ~ ( d(in estimates), then: u(z)=yD(z), X E ~ R , (2) The second one is a Neurnann condition. assuming Assumption 2:

Vu(z).n(z)= g N ( z ) , z E OR,

(3)

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Finite volumes for complex applications

A s s u m p t i o n 2 b = 0 and divv = 0 on R , v . n = 0 on dR and g N E H ' / ' ( ~ Q ) satisfies the follotving compatibility relation: Jan g N (2) d y ( x ) Sn f ( x )dx = 0.

+

Finally the last type we consider is a Fourier condition, under Assumption 3:

+ >

A s s u m p t i o n 3 g F E ~ l / " ( a R ) , X E R such that v . n / 2 X 0 on dQ. Furthermore, zf v ( x ) . n ( x ) / 2 X = 0 for all x E dR then one assumes the such that zts d-dimensional measurp m ( O ) # 0 and such exzstence of 0 C that div(v)/2 b # 0 on 0 .

+

a

+

R e m a r k 1 Assunzpttons 1 and 3 glue the coercttzvzty of the elliptic operator assoczated t o the varzatzonal equality of ( I ) , (4). It does not need a conzpatzbzlzty wlatzon, so we conszder that, euen i f X = 0 , thls case I S not a Neumann condztzon but a Fourier condztlon. This elliptic problem is then discretized with a finite volume scheme: a "four-point" scheme is used for the diffusion tern1 and an upst,rea~llscheil~efor the convection one. A discrete system is obt,ained for each type of boundary condition. Exist,e~lceaiid uniqueness (for tlie Neumann's boundary c o ~ ~ d i t i o n , the uniqueness is up to a constant like i11 t.he continuous case), of t,lle approxilllat,e solution is proven. If the exact solutioii is assumed t~obe a t least in H 2 ( R ) , one may then esta.blish the convergence of the scheme by proving error estimat,es; a first est,imate in a discrete HA norm is obt.ained. An error estimat,e in the L' nor111 follows with t8hehelp of discrete PoiiicarC inequalit.ies. The convergence of the method for Neumann and Fourier condit,ions requires seine additional work compared t o t,llat of the Dirichlet case. I11 the case of Neuinann boundary condit,ions, a "discrete mean PoincarP" inequality needs to be proven in order t.o obtain an L' error estii11at.e. In t,he case of the Fourier condition, it is interest,ing t o not,e that an art,ificial upwinding has t,o be introduced in the treat,ment of tmheboundary condition in order for the scheine t o be well defined with no additional condition on the mesh. Finite volume schemes for a diffusion convection equation with homogeileous boundary conditions were studied in e.g. [LMSSG], [He931 and [VPLS'L] wit,h different assumptions on the d a t a and the mesh.

2. D i s c r e t i z a t i o n In order to discretize the problem, first we define the mesh D e f i n i t i o n 1 ( A d m i s s i b l e m e s h e s ) A finite volume mesh of R, denoted by

7, zs gzven by a famzly of "control volumes", which are open polygonal (or polyhedral) convex subsets of R (wzth posztive measzlre), a fanzzly of szibsets of 2 contazned in hyperplanes of K t d , denoted by f (these are the edges (zf d = 2) o r szdes (zf d = 3) of the control tloltimes), wzth strzctly posztzue ( d - 1)dzmenszonal measure, and a famzly of points of denoted by P . The finite

Numerical analysis

191

zlolume mesh is said t o be admissible if the properties (i) to (iv) below are satisfied and restricted admissible if the properties ( i ) t o (v) below are satisfied. ( i ) The closure of the unzon of all the control volumes is

It;

(12) For any I i E 7 ,there exzsts a subset EK of E such that d l i = U o E z Aa . Let E = U K ~ ~ E K .

r\I<=

(212) For any ( I i , L ) E 'T2 wzth li # L , ezther the ( d - 1)-dzrnenszonal Lebesgue measure of n zs 0 01- n = a for some a E E , whzch wzll then be denoted by IiIL. (22)) The farnrly P = ( x ~ 2s such ) ~that X~ K E~ (for all I< E 7 )and, zf a = IiI L , zt zs assumed that X K f X L , and that the strazght lzne D K , L golng through X K and X L is orthogonal to I
= = K , o n a.

In the sequel, the following notations are used. The mesh size is defined by: s i z e ( 7 ) = sup{diam(li), Ii E 7 1 , where diam(1i) is the diameter of Ii E 7. For any II' E 7 and a E 8 , m ( K ) is the d-dimensional Lebesgue measure of Ii, m ( a ) the ( d - 1)-dimensional measure of a , and n K , , denotes the unit normal vector to a outward to I i . The set of interior- (resp. boundary) edges is denoted by E,,t (resp. Eext), that is Ei,t = { a E £; a (t dR) (resp. EeXt = {a E E ; a c a n ) ) . If a = I i I L , we denote by d, or d K I L the Euclidean distance between X K and X L (which is positive). If a E EK n Ee,t, let d , denote the Euclzdean distance between X K and yo. Adinissible (or restricted admissible) meshes include, for instance, meshes made with triangles and rectangles in two space dimensions, and also Voronoi meshes: the latter consist in building a mesh using the orthogonal bisectors from a given family of points (for more details see [EGH]).Admissible meshes will be used for the Neumann boundary conditions. Property ( v ) of the restricted admissible meshes is needed for the Dirichlet and Fourier boundary conditions. Now we discretize the problem. The approxinlate finite volume solution which is sought here is constant on each cell of the mesh. The discrete unknowns . of classical finite volume schemes is to are denoted by ( u K ) ~ T~h e~principle integrate the equation on each cell of the mesh in order t o obtain an equation which is sometimes called the balance equation, for each control volume. Let I i E 7 ,using Green's formula, one has: [-AU(X)

+ div (V ( X )

) +b

U(X)

I

U(X)

dx =

192

Finite volumes for complex applications

where dy is the integration symbol for the (d-1)-dimensional Lebesgue measure on the considered hyperplane. Since the approximate solution is constant on each cell of the mesh, one nlay approximate bu(x) dx by bm(I<) u ~ For . all a E E K , one denotes by FK,, (respectively VK,,) the numerical diffusion (respectively convection) flux. Then the considered finite volume scheme is defined by the following equation:

SK

First, one gives the fluxes for interior edges. Let a E f K f ~ E i , t and let us denote by L the cell in 7 such that a = K I L, one approximates the diffusion flux using a "four-point" finite volumes scheme and for the convective numerical flux, one uses an upstream scheme, that is

where uu,+

=

UK

u~

if J, V ( X ) . ~ K dy(x) ,, otherwise.

> 0,

Now let us give the discretization for boundary terms. Let a E £K n &,t, we consider the three cases corresponding to the different boundary conditions Dirichlet b o u n d a r y condition As u is known on the boundary, we set

and the numerical convective flux VK,, is defined using ( 6 ) with

uu,+ =

uK U,

So

if v ( x ) . n K , u~ Y ( x2) 0, otherwise,

and y, defined in Definition 1. Remark that the definition of the diffusion flux on a boundary edge (8) allows d, = 0, in this case one has U K = g D ( y u ) and FK,, becomes an unknown. N e u m a n n b o u n d a r y condition Since v . n = 0 on dR the numerical convection flux equals the exact one, that is 0. We do the same for the numerical diffusion flux, we set

Fourier b o u n d a r y condition

Numerical analysis

193

I11 this case the discretization of boundary terms is performed with the help of some auxiliary unknowns which are defined on the edges of the boundary. These may be eliminated when solving the linear system. We shall however keep them throughout our study because they simplify several expressions in the error estimate. Hence in this section the discrete unknowns are ( u ~U ( u , ) , ~ E , , ~ . Then ones defines F K , by ~ (8) and V K , by ~ (6) with u,,+ given by ( 9 ) . There reniains to give the equations associated with the boundary unkiiowns ( U , ) ~ ~ E , These , ~ . are obtained by discretizing ( 4 ) . The discretization which we choose involves the upstream valued u,,+ in order for the scheme to be well defined with no additional condition on the mesh. It writes:

(12) Remark 2: In order to dzscretzze the boundary condztzon on an edge u E Ee,t of I\' E 7 ,we zase a decentered scheme summzng and substractzng v . nd y ( z ) . Thzs chozce zs performed zn order to proue exzstence and untqueness for X < 0 wzth no restrzctzon on the mesh; (see Remark 3). In fact, zt would be more natural to dzscretzze the boundary condztzon as follows:

So

However, if X < 0 , the proof of ezistence, uniqueness and convergence towards the exact solution requires more restrictive assumptions on the mesh. Hence, (12) will be preferred for the discretization of the boundary condition so as to be able to handle negative values of X with no additional condition on the mesh.

3. Existence and uniqueness of the approximate solutions

Dirichlet boundary condition Let us begin with the approximate solution associated to ( 1 ) with the Dirichlet boundary condition. The proof of existence and uniqueness of the approximate solution is performed by establishing a discrete maximum principle Proposition 1 Under Assumption 1 and assuming g D E H ~ / " ( ~ Q )let , 7 be an admissible mesh in the sense of Definition 1. Let ( u , ) , ~ ~ , , ~be, defined by (10). If f ( x ) dx 0 for all Ii' E 7 , and u, 2 0 , for all u E E,,I, then if ( U K ) K € 2s ~ a solution t o (5), (6), (?'),(a), (9), u~ 0 for all Ii' E 7.

SF

>

>

The proof of this maximum principle may be found in [EGH]or [He931 using a strong formulation and in [EGH99]using a weak formulation. It immediately yields the existence and uniqueness of the solution to the finite volume scheme. We may therefore now define the approximate finite volume solution u 7 by: ~

~

= 5U K ) if x E K, Ii' E 7 ,

(

(14)

)

~

194

Finite volumes for complex applications

where ( u K ) ~ E 7is the unique solution t o ( 5 ) , (6), (7), (8) arid (9). N e u m a n n boundary condition One proves the following result which gives the existe~lceof the approximate solution and its uniqueness u p t o a coristant like in the continuous case. P r o p o s i t i o n 2 Under Assumptions 1 and 2, let 7 be an admissible mesh in the sense of Definition 1. Then, there exists a solutlon u~ to ( I d ) , (5), (6), (7) and (11). This solt~tionis unique t ~ pto a constant.

T h e proof of this result can be found, for instance, in [GHV] or [Vi97]. Fourier b o u n d a r y c o n d i t i o n One proves the following result: P r o p o s i t i o n 3 Under Assumptzons 1 and 3, let 7 be an admissible mesh 111 the sense of Definztzon 1. Then there exzsts a unique solutzon ( u K ) K E TU ( ~ U ) C ~ E E , , , to (51, (6), ( Y), (81, (9) and (12). Idea of t h e proof: Assuming JK f ( x )d x for all I< E 7 and S , f ( z ) d y ( x ) = 0 for all u E Eext and using the numerical scheme (for more details see [GHV]) one proves t h a t

+ Jan (A + V ( x )2 n ( x ) )

(nT(x))' dx

5 0,

where DCuj- = IuK - uLI if a 6 Eint, u = A I L and D,uT = l,uK - ual if u E GXt n EK, Ii' E 7 . Thanks t o Assumptions 1 and 3, this result gives u~ = 0 for all I< 7 and u, = 0 for all a E EeXt. This proves uniqueness and therefore existence since the dimension of the space is finite. R e m a r k 3: If the discretization (13) is used instead of (12), computations similar t o those of the above proof yield:

where v ~ , = , J, V ( X ) . n ( x ) d y ( x ) . SO if X 2 0 , this inequality gives Proposition 3 but if X < 0 one must assume some more restrictive assumption on the mesh as we already mentionned in Remark 2; for instance one could assume m(a)X $ v ( x ) . n ( x ) d y ( x ) Xd,m(u) 2 0. We may now define the approximate solution by

+ So

+

u T ( x ) = U K if x E I<, I< E 7 and u r ( x ) = u , if x E u, a E

re,,.

(15)

Numerical analysis

195

4. Error estimates

One gives in this section the three results corresponding to the three different boundary ~ondit~ions.

Theorem 1 (Dirichlet boundary condition) Under Asstimytzon 1 and assnn,lng y D € H ~ / ? ( ~ Qlet) 7 , be a restrzctzzle admzsszble mesh zn the sense of ~am(h-) Definztton 1 a71d let = m i n K ~ Tm i n U E E ,d. Let u~ be defined by (Id), d" (5), (6). (7). (8) and (9). ilsstime that the zinzque uarzatzonal solutzon u to (1) and (2) belongs t o H 2 ( R ) . Let eT be defined by e T ( x ) = eK = ~ ( x K -) u~ zf .c E I i , I i E 7 Then, there exzsts C ' , only dependzng on u , v , b, Q , d and C, such that

<

Theorem 2 (Neumann boundary condition) linder Asszlmytzons 1 and 2, let 7 be an adn~zsszble mesh tn the sense of Definztzon 1 and let C = d~am(K) tlllllh-~T1 1 1 1 1 1 , ~ ~~ . One assumes that the unzque e1arzatzonal solutzon dm u E H 1 ( Q ) , such that ~ i ( zdx) = 0 , of Problem (I), (3) satzsfies u E H 2 ( Q ) . Let U T be the solutzon t o (Id), /,5). (6), (7) and (11) such that C K E 711(Ji) T ( L K = Ch-ET rn(I<)t / ( x K ) ,where x~ zs defined tn Definztzon 1. Let eT be defined by e T ( x ) = eK = t ! ( x ~-) u h t f x E I<, I< E 7 . Then, t h e w exzsts C', only rlependzng on u , v , b, d , Q and i,such that:

h2

where DueT = ( e K - e L ( i f a = I i ( L and the set

Elntzs

defined zn Definztzon 1.

Theorem 3 (Fourier boundary condition) Under Assumytzons 1 and 3, let 7 be (1 restrtcttzre adnlzsszble mesh zn the sense of Definztzon 1 and let d~arn(K) C = ~ l l i l minoE,-, l ~ ~ ~ . et 1 1 7 be the solzltzon t o (1,5), (5), (6), (7), (8), d" (9) and (12) Asstime that the unzqve uarzatzonul solutzon u of Problem (I), (4) satzsfies ti belongs t o H 2 ( Q ) . Let €7 be defined by e T ( c ) = e~ = t / ( x K )- PLK z f z E I<, I i E 7 a n d e T ( x ) = e, = u(y,) - u , t f x E a , a E EeXt. Then, there extsts C', only dependrng on u , v, b, A, 12 and C, such that:

196

Finite volumes for complex applications

Proof of Theorem 3: Using the numerical scheme, the fact that F ~ , is, a consistant approxm 0) imation of ;;i?;T - V u ( x ) . n ( x )d y ( z ) for all a E £ and the conservativity of the numerical and exact flux through a given edge, one proves the first estimate of (16), which can be seen as an error estimate in a discrete H i norm. In ordre to establish the second one, one uses the following discrete PoincarC inequalities: Lemma 1 Let 7 be an admissible mesh in the sense of Definition 1 and that u be a function which is constant on each cell of 7 and each edge of i s u ( x )= u i~ f x E I<, I< € 7 a n d u ( x ) = u o i f x E E , a E l e x t . Let r C a R such that its ( d - 1)-dimensional measure m ( r )# 0 and 8 c R such that 2ts d-dimensional measure m ( 0 ) # 0. Then there exzsts C , only depending on S1, such that

So

where for all

a

E E , D,u is defined in Theorem 3.

5. Bibliography

[EGH] R . EYMARD, T. GALLOUET and R. HERBIN, The finite volume method, to appear in Handbook of Numerical Analysis, J.L. Lions and P.G. Ciarlet eds. [EGH99] R . EYMARD,T. GALLOUET and R. HERBIN, Convergence of finite volunle schemes for semilinear convection diffusion equations, accepted for publication in Numer. Math. (1999). [GHV] T. GALLOUET, R. HERBIN and M.-H. VIGNAL, Error estimates on the approximate finite volume solution of convection diffusion equations with Dirichlet, Neumann or Fourier boundary conditions, submitted. [He931 HERBINR., 1993, An error estimate for a finite volume scheme for a diffusion convection problem on a triangular mesh, Num. bleth. in P.D.E. [LMV96] R.D. LAZAROV, I.D. MISHEVand P.S. VASSILEVSKI (1996), Finite volume methods for convection diffusion problems, SIAM J. Nzimer. Anal. 33, 31-55. [VPL92] P . S . VASSILEVSKI, S.I. PETROVA and R.D. LAZAROV(1992), Finite difference schemes on triangular cell-centered grids with local refinement, SIAM J. Scz. Stat. Comput. 13, 6, 1287-1313. [Vi97] M.H. VIGNAL(1997), Schkmas Volumes Finis pour des Cquations elliptiques ou hyperboliques avec conditions aux limites, convergence et estimations d'erreur ThGse de Doctorat, Ecole Normale Supkrieure de Lyon.

The limited analysis in finite elasticity

Igor A. Brigadnov D e p a r t m e n t of I n f o r m a t i c s N o r t h - W e s t e r n Polytechnical I n s t i t u t e 5, M i l l i o n n a y a Str., St. Petersburg 191 186, R u s s i a E - m a i l : [email protected]. r u

T h e variational formulation of the elastostatic bounary-value problem for hyperelnstic materials is considered. For elastic potentials having the linear growth i n modulus of the distortion tensor, the limited analysis problem is formulated. I n the framework of this problem the strength of nonlinear elastic solid is estimated. From the mathematical point of view this problem is non-correct and, therefore, needs a relaxation. T h e partial relaxation is descibed for the limited analysis problem. It is based on the special discontinuous finite-element approximation with functions having breaks of the sliding type. The numerical results show that this technique has qualitative advantages over standard continuous finite-element approximations. ABSTRACT

ICey Words: elastostatics BVP, the limited analysis problem, partial relaxation, discontinuous FEA.

1. Introduction The solution of elasticity boundary-value problem (BVP) is of particular interest in both theory and practice. At present there are many models of elasticity in the framework of the finite deformations theory (GRE 75, BAR 76, CIA 88, LUR 90). Adequacy and the field of application of every model must be found only by the correlation between experimental data and solutions of appropriate BVPs. Therefore, the analysis of mathematical correctness and the treatment of numerical methods for these problems is very important (BRI 93-98).

198

Finite volumes for complex applications

In this paper the finite elasticity BVP is formulated as the variational problem for the displacement (CIA 88). For materials with ideal saturation the elasu E R3 tic potential @(x,Vu) of which has the linear growth in IVul, where x is the map, the existence of the limited static load (such external static forces with no solution of BVP) and discontinuous maps with breaks of the sliding type was proved by the author (BRI 93-98). From the physical point of view these effects are treated as the destruction of a solid. The limited analysis problem for nonlinear elastic solid is formulated. In the framework of this problem the estimation from below for the limited static load is calculated. As a result, the shape optimization problem is also formulated for the nonlinear elastic solid of the maximum strength. It is demonstrated, using the simplest example, that the limited analysis problem has discontinuous solutions with breaks of the sliding type. From the mathematical point of view the limited analysis problem is noncorrect (EKE 76, FUC 80, TEM 83, GIU 84) and, therefore, needs a relaxation. We use the partial relaxation which is based on the special discontinuous finiteelement approximation (FEA) (REP 89, BRI 98). After this discontinuous FEA the limited analysis problem is transformed into the non-linear system of algebraic equations which is badly determined, because the global stiffness matrix has lines with significantly different factors. Therefore, for the numerical solution the decomposition method of adaptive block relaxation is used (BRI 96c-98). Its main idea consists of iterative improvement of zones with "proportional" fields by special decomposition of variables, and separate calculation on these variables. The numerical results show that for the estimation of the limited static load, the proposed technique has qualitative advantages over standard continuous finite-element approximations.

2. The l i m i t e d analysis p r o b l e m i n nonlinear elasticity

Let a rigid body in the undeformed reference configuration occupy a domain c R3. In the deformed configuration each point z E E moves into a position u(x) = u(x) x E R3, where u and u are the map and displacement, respectively. Here and in what follows we use the Lagrangian coordinates. We consider locally invertible and orientation-preserving maps u : + R3 with gradient (the distortion tensor) Q ( u ) = Vu : R -+M3 such that det(Q) > 0 in R (GRE 75, CIA 88, LUR go), where the symbol M~ denotes the space of real 3 x 3 matrices. The finite deformation of materials is described by the energy pair (Q, C), where C = {Cia} is the first non-symmetrical Piola-Kirchhoff stress tensor. It is known that the Cauchy stress tensor a = {aa@) has the components = ( d e t ( ~ ) ) - ' C q ~ ! . Here and in what follows the Roman lower and Greek upper indeces correspond to the reference and deformed configurations, respectively, and the addition over repeating indeces is used.

+

a

Numerical analysis

199

Elastic materials are characterized by the response function (the stressstrain relation) C = C(x, Q) such that C(x, I) = 0 where I = Diag(1, 1 , l ) and 0 is the zero matrix. For hyperelastic materials the scalar function (elastic potentzal) W : Cl x M 3 + R+ exists such that CF(x, Q) = d W ( x , &)/a&?for every Q E M3 and almost every x E R. If a material is incompressible, t.hen det(Q) = 1, but for a compressible material IC(x, Q) I -+ m , W ( x ,Q) -+ +m as det(Q) + +O. We consider the following boundary-value problem. The quasi-static influences acting on the body are: a mass force with density f in R, a surface force with density F on a portion I" of the boundary, and a displacement vo of a portion I" of the boundary is also given. Here I" U I" = dS1, T1 n r2= 0 and area(rl) > 0. For hyperelastic materials the finite elasticity BVP is formulated as the following variational problem (CIA 88, BRI 93-98) v* = arg (inf{I(v) : v E V ) ) ,

(1)

where @(x,Vv(x) + I ) dx - A(v), R

Here I/' = {v : --+ R3; v(x) = vo(x), x E I") is the set of admissible displacements, (*, v) is the specific and A(v) is the full work of the outside forces under the displacement v. It must be marked that even for "dead" forces, i.e. f , F = const(v, Vv), the specific work has the form (g, v)(x) = ga (x) va (x) only in the Descartes coordinates (CIA 88, BRI 97). According to the general theory (EKE 76, FUC 80, TEM 83, GIU 84), for potentials of linear growth t.he set of kinematically admissible displacements is a subset of non-reflexive Sobolev's space W1ll (R, R3)

We remind the definition of the limited static load (BRI 93). For this reason we introduce the set of addmissible "dead" outside forces for which the functional I(v) is bounded from below on V and, therefore, a solution of the problem (1) exists B = { ( f ,F) E LW(Cl,R3) x

L ~ ( IR" ~, :) inf{I(v)

:v E

V)

> -co) .

This set is non-empty because for small outside influences the problem (1) is transformed into the classical variational problem of linear elasticity (LUR 90) which always has a solution (CIA 88; BRI 96a, 97).

200

Finite volumes for complex applications

Definition 1. For outside forces ( f ,F) E B we examine the sequence of forces which are proportional to the real parameter t 2 0. The number t, 2 0 is named the limited parameter of loading and t,(f, F ) is named the limited static load, i f t ( f , F ) E B for 0 < t 5 t , and t ( f , F ) $ B f o r t > t , . The limited analysis problem is the investigation of the set of positive parameters t , for which the functional

is bounded from below on the set of admissible displacements (2). In practice the estimation from below for the limited static load is interesting because this information is sufficient for estimation of the strength of nonlinear elastic solid. Statement 2. For the limited parameter of loading the following estimation from below is true

where

According to the sense, cp(x) is the function of saturation, it being known that cp(x) > 0 and for a homogeneous solid cp = const(x). From the definition 1 it follows that for t - < 1 the elastostatic variational problem (1) can have no solution. From the physical point of view this effect is treated as the destruction of solid.

3. Discontinuous FEA and the partial relaxation

From the previous author's results (BRI 96-98) it follows that for elastic potentials of linear growth the appropriate limited analysis problems need a relaxation (TEM 83). For variational problems with the multiple integral functional of linear growth the expanded space equals the BD space of vector-functions with bounded variations and generalized derivatives as the bounded Radon's measures (TEM 83, GIU 84). In the numerical analysis only finite dimension subspace of BD is used. Therefore, for the limited analysis problem (3) we will use the partial relaxation which is based on the special FEA with functions having breaks of the sliding type along ribs of simpleces (REP 89, BRI 98). Here we examine the plane limited analysis problem. Let Cl C R2 and Clh = UTh such that area(Cl\Clh) + 0 and length(dQ\dQh) + 0 as h 4 SO, where

Numerical analysis

201

Th is the triangle and h is the characteristic step of the regular approximation (CIA 80). Every FEA is described by the set of nodes { x ~ } ~and = ~ the set of ribs Gh = {rap [xa, xp]) including inside ribs and ribs on a portion rj!, of the boundary dfih. For the displacement the following spacial piecewise continuous approximation is used "h (x) = UapQap(x) where a,,8 = 1 , 2 , ..., msuch that rap E G h , UaP is the component of the displacement in the node xa which is perpendicular to the rib rap, qap : 52h + R is the piecewise linear discontinuous function such that 9,p(xY) = day (a,Dl -y = 1 , 2 , ..., m) and Qap # Qp,. The supp (4,p) = supp (qp,) consists of two triangles having the rib rap as common. If a rib r a p E I'k then the supp (qap)consists of the only triangle. In this case the subspace V c W1ll(fi, R2) is approximated by the subspace Vh C BD(St,R 2 ) which is isomorphous to R ~where ~ M, is the number of ribs in the set G h . The described FEA possesses the following properties. The component of the displacement, which is perpendicular to an appropriate rib, is continuous; but the tangent projection on this rib has a finite break. As a result, we have the special FEA with functions having breaks of the sliding type along ribs of triangles. The relaxated problem for the limited analysis problem (3) has the following form th = min{Kh(v) Rh(v) : v E Vh, Ah(") = 1 1 , (4)

+

where

"+"

and "-" correspond to the displacement and the function of Here indeces saturation on the triangles and TL having the common rib rap, index r corresponds to the tangent projection of displacement on this rib, and for ribs on ri the outside displacement is fixed, for example, v; = vi. Functions ( v i , fh, Fh)are the standard spacial piecewise linear continuous FEAs of outside influences. According to the properties of FEA (CIA 80) and the results of paper (REP 89) we have th \ t- as h -+ +O regularly. From the computational point of view the functional in the problem (5) is singular because it has no the classical derivative. Therefore, in this problem we use the simplest approximation of the modulus 1 1 m ( z 2 c 2 )'I2 with the regularization parameter E << 1 (BRI 98).

~h+

+

202

Finite volumes for complex applications

It was proved that for potentials of linear growth the FE approximating algebraic systems can be badly determined (BRI 96c-98). As a result, for the numerical solution the decomposition method of adaptive block relaxation is used, because it practically disregards the condition number of the global stiffness matrix. The main idea of this method consists of iterative improvement of zones with "proportional" deformation by special decomposition of variables and separate calculation on these variables.

4. N u m e r i c a l r e s u l t s

In the numerical experiments, the following boundary value problem was considered: a finite round rod is axial symmetrically stretched in the test machine by a given axial force P. In this case, the map is described by the following relation in the reference cylindrical coordinates U(P,

+

P, 2) = x (UP a r(p, z), p, lz

+[ ~ ( p ,

2))

,

where p E [O,l], p E [O,2n), z E [ O , l ] , a and 1 are the radius of section and semilength of the rod, respectively; 9 = l/a is the geometrical parameter. The material of rod is subject to the elastic potential (BRI 96-98)

where p > 0 is the shear modulus under small deformations. For the limited stretching force P, the estimation from below P, 2 a p 7 r a 2 t - is true, where the parameter of loading t - is the solution of the following limited analysis problem

where

According to the convexity of domain, axial symmetry of the problem (6) and continuity of the axial component of dispalcement, the minimizer may have a break of the sliding type along the only line z = 1. This break is defined

Numerical analysis

203

by a finite break of the function r ( p , 1 ) . Therefore, in the set Vl the condition r ( p , 1) = 0 is ignored, but in the functional the appropriate penal item is used

In the computational experiments the regular N x N triangulation of the domain ( O , 1 ) x ( 0 , l ) and the regularization parameter E = 10-I were used. In Figure 1 the experimental relations 77 I-+ -,tiare shown. Lines 1, 2 and 3 correspond t o the continuous FEA with N = 10, N = 20 and N = 40, respectively. Line 4 corresponds t o the discontinuous FEA with N = 10. It is easily seen that continuous solutions converge to the discontinuous solutiori wit.h increase of domain's discretization. The decrease of the regularization parameter E until practically does not improve either continuous or discorltinuous solutions.

[BAR 761

BARTENEVG.M., ZELENEVY u . ~ . ,A course i n the physics of polimers, Chemistry, 1976 (Russian).

[BRI 931

BRIGADNOV I.A., On the existence of a limiting load in some problems of hyperelasticity. Mech. of Solids, No 5, 1993, p. 46-51.

[BRI 96a]

BRIGADNOV I.A., Existence theorems for boundary value problems of hyperelasticity. Sbornak: Mathematics, Vol. 187(1), 1996, p. 1-14.

[BRI 96b]

BRIGADNOV I.A., On mathematical correctness of static boundary value problems for hyperelastic materials. Mech. of Solids, No 6, 1996, p. 37-46.

[BRI 96c]

BRIGADNOV I. A., Numerical methods in non-linear elasticity. In: Numerical Methods i n Engineering'96. Proc. 2nd E C C O M A S Conf. (19961, Wiley, p. 158-163

[BRI 971

BRIGADNOV I.A., Mathematical Methods for Boundary Value Problems of Plasticity and Non-Linear Elasticity. D.Sci. Thesis, St. Petersburg State University, 1997 (Russian).

[BRI 981

BRIGADNOV I.A., Discontinuous solutions and their finite element approximation in non-linear elasticity. In: A C O M E N ' 9 8 Advanced Computational Methods i n Engineering. Proc. 1st Int. Conf. ACOMEN'98 (1998), Shaker Publishing B.V., p. 141-148.

[CIA 801

CIARLETPH.G., The Finite Element Method for Elliptic Problems, North-Holland Publ. Co., 1980.

[CIA 881

CIARLET PH.G., Mathematical Elasticity. Vol. 1: Dimensional Elasticity, North-Holland Publ. Co., 1988.

Three-

204

Finite volumes for complex applications

[EKE 761

EKELAND I., TEMAMR., Convex Analysis and Variational Problems, North-Holland Publ. Co., 1976.

[FUG 801

FUCIKS., KUFNERA . , Nonlinear Difirentaal Equations, Elsevier Sci. Publ. Co., 1980.

[GIU 841

GIUSTIE., Minimal Surfaces and Functions of Bounded Variations, Birkhauser, 1984.

[GRE 751

GREENA . E . , ZERNAW., Theoretical Elasticity, Oxford University Press, 1975.

[LUR 901

LURIEA.I., Nonlinear Theory of Elasticity, North-Holland Publ. Co., 1990.

[REP 891

REPIN S. I., A variational-difference method for solving problems with functionals of linear growth. U.S.S.R. Comput. Math. and Math. Phys., Vol. 29(5), 1989, p. 693-708.

[TEM 831

TEMAMR., ProblBmes Mathe'matiques en Plasticite', GauthierVillars. 1983.

Figure 1. The experimental relations between the geometrical parameter and the limited parameter of loading for difleereent FEAs.

Entropy consistent finite volume schemes for the thin film equation Giinther Griin and Martin Rumpf Uniuersitat Bonn, Institut fzj:r Angewandte Mathematik ABSTRACT W e present numerical schemes for fourth order degenerate parabolic equations that arise e.g. i n lubrication theory for the time evolution of thin films of viscous fluids. It turns out that a finite volume ansatz is the right approach to gain estimates o n energy and entropy of discrete solutions. The latter are the k e y estimates t o ensure nonnegatiuity of discrete solutions i n a natural way. Another important feature is the question of tracing the solution's free boundary eficiently. This is achieved by a timestep control that makes use of a n explicit formula for the normal velocity o f the free boundary. Finally, we present some recent numerical experiments which indicate that also for fourth order degenerate parabolic equations a waiting time phenomenon occurs. K e y words: fourth order degenerate parabolic equations, free boundary problem, finite volumes, adaptiuity i n time

1. Introduction

In this contribution, we will present new numerical schemes of finite volume type for fourth order degenerate parabolic equations of the form

+

ut d i v ( M ( u ) ~ A u ) = 0 a u = =aA U = o 4 0 , .) =

in L! x (0, T) c 1Ftd+l, on x (0, T), in S1.

(1)

Equation (1) is obtained as lubrication limit from the Navier-Stokes equations and models the height of thin films of viscous liquids that - driven by surface tension - spread on plain, solid surfaces. Assuming a no-slip boundary condition a t the bottom of the thin film, the mobility becomes M ( u ):= 1uI3, whereas tlie assun~ptionof various slip boundary conditions leads to mobilities of the form M ( u )= el lul" cCgIuIPwith positive numbers cl, c~gand P E (0,3). From the analytical point of view, this initial boundary value problem shows a rather peculiar behaviour:

206

Finite volumes for complex applications

in contrast t o solutions t o nondegenerate fourth order parabolic equations, initially nonnegative solutions t o (1) preserve nonnegativity (cf. [BF90], [Gr95], and [DPGGSS]), if M ( u ) = t i n with 12 E ( 0 , 3 ) , solutions to (1) exist that exhibit for t > 0 a zero contact angle a t the contact line between liquid, solid and gas (cf. [BergGa]. [Ber96b], and [BDPGGSS]). Moreover, this contact line propagates with finite speed, i.e. we are dealing with a free boundary problem. However, if initial d a t a have a nonzero contact angle, the propagation speed may be singular for 1 = 0. no maximum or conlparison principles are known. Those aforementioned issues also mean a great challenge in designing efficient numerical tools. A natural approach t o guarantee nonnegativity of discrete solutions is t o develop a numerical scheme that allows for discrete counterparts of the relevant estinlates i. e. the energy and entropy estinlate - known from the continuous setting. In section 2, we will introduce a n implicit finite volume scherile which gives the perfect framework t o realize this concept. Having presented in section 3 the relevant a priori-estimates in order t o obtain compactness of sequences of discrete solutions, we will show in section 4 that a certain kind of harmonic integral means is the right choice for a n entropy consistent numerical flus that allows for nonnegative discrete solutions. In section 5 we will introduce our method of tinlestep control which is based on a new, esplicit fornlula for the velocity of the free boundary. This allows a tracing of the free boundary reminiscent of laws. On the other hand, the the tracing of shocks in hyperbolic ~onservat~ion very formula for the velocity of the free boundary suggests that for sufficiently smooth initial d a t a a t u a i t t ~ z gtime p h e n o m e n o ~ zoccurs, i.e. there is a slight delay in the onset of spreading. This phenomenon is well know11 for solutions t o second order degenerate parabolic equations, like the porous media equation. We will present numerical silnulations which give strong evidence that it also happens in the case of fourth order degenerate equations.

'

-

2. Deriving the entropy consistent finite volume scheme T h e two major classes of discretizations for evolution problems - finite volume and finite element schemes - have both significant advantages. Finite volume schemes very easily lead tto conservative schemes and illcorporate fluxes on cell faces in a natural way, whereas finite element schemes correspond t o a Galerki~i discretization of the continuous problem and therefore carry strong provisions concerning a convergence analysis. In general, it is unusual to apply finite volume schemes t o fourth order parabolic problems. But. due t o the peculiar diffusive structure of the elliptic term in ( I ) , the thin film equation plays a n

'For a different ansatz to ensure nonnegativity , based on variational inequalities, we refer to [BBG].

Numerical analysis

207

esceptionary role - as d o other related higher order equations like the CahnHilliard equation with degenerate mobility, too. A suitable mixed finite volume - finite element discretization is the starting point t o derive a conservative and entropy consistent numerical flux with a n intuitive interpretation of t,he construction. In particular, the entropy consistency guarantees nonnegativity of the resulting numerical solution independent of the spatial resolution. Later on we will derive from the original finite volunle scheme a finite element scheme. which will turn out t o he preferable concerning further investigation in the numerical analysis. For simplicity we assume C2 t o be a n interval in 1D or a polygonally bounded domain in 2D. respectively. We suppose St t o be subdivided into cells. On these cells we suppose the discrete height U to be piecewise constant. We denote the negative Laplacian of the height which physically has the interpretation of pressure by p = - A u . Finite elements allow a straightforward discretization of this Laplacian. Thus we have t o find a suitable finite element mesh and function space for the pressure. We choose linear finite elements on the mesh dual to the finite volume mesh. To be more precise, we start with a simplicia1 grid Th on Q consisting of subintervals, respectively triangles E, on which the discrete pressure P will be defined a s a function in the corresponding linear finite element space vh, where h indicates the chosen grid size. Then a dual mesh is built of open dual cells D,, again intervals, respectively polygonally bounded cells, corresponding t o the vertices x of the primal mesh (cf. Figurel). We define a single dual cell by

D, := {y E St : dist(y, x)

< dist(y, E ) , .i is vertex of the

mesh)

.

In t,he following, discrete functions will be denoted by uppercase letters, in constrast t o lowercase letters for arbitrary functions in the nondiscrete function spaces. T h e discret.e height U will be defined spatially constant on these dual cells. Figure 1 shows a n example of such dual triangulations. To st,art with the

FIGURE 1. A 2D finite element triangulation whose edges are outlined in black and the corresponding dual finite volume mesh indicated by dashed lines. discussion of finite volurne schemes, let us consider a cell D of the dual grid. On this subvolun~ewe can rewrite equation (1) in conservation form

208

Finite volumes for complex applications

where p = -Au and v is the out.er normal on 8D. The right hand side describes the inflow a t the boundary, and M ( u ) G p is the corresponding flux. Thus, a numerical mobility M and a numerical pressure gradient are the main ingredients of a spatial discretization. As already mentioned, with the dual grid a t hand the latter requirement is easy to fulfill. For given U we define P = -Ah[/ on vh,i.e. P is the unique function in vh with

where Zh : C0(!2)-+ vh is the nodal projection operat.or and (-, .)h indicates the well-known lumped mass scalar product corresponding to the integration formula (O, Q ) h :=Jnzh(OQ) . Gradients of P are by construction piecewise constant on elements E and thus almost everywhere on faces F of dual cells D. T o pay account to the fact that the values Uf = U(X&EV)may be different due to the discontinuity of U across cell boundaries, we suppose the discrete mobility M t o be a function M : 1Fi2 -+ 1R; ( U t , U p ) I-+ M ( U + , U-) , where U+, U - are the outer, respectively interior values of U a t the corresponding face. Finally, we can formulate our semi-discrete finite volume scheme

where Q ( U + , U-, V P ) = M ( U + , U - ) V P is the corresponding numerical flux. In our case we suppose that the discrete mobility M ( U t , U - ) is a symmetric which is positive semidefinite and piecewise constant on E E f i . matrix in lfLdxd T h e resulting scheme is known t o be conservative [Kr97] if this flux is symmetric, i.e. Q ( U + , U-, V P ) = Q ( U - , U+. V P ) respectively

Thus, the inflow on F corresponding t o D should coincide with the outflow with respect t o the adjacent element a t the face F. This immediately implies the conservation of mass J"aU dx . Furthermore the flux should be consistent with the continuous flux q = M ( u ) V p , i. e.

where the second term on the right hand side vanishes for decreasing grid size. There is still a great flexibility in selecting a numerical mobility. Let us recall that in case of hyperbolic conservation laws upwind discretization and entropy consistency conditions on the numerical fluxes, i. e. certain monotonicity properties, select the right entropy solution and guarantee moreover that neither artificial oscillations nor nonphysical shocks occur. These ideas carry over to the discrete modelling of thin films. For the trivial choice M ( U + , U-):=M(-)1d nonnegativity of the numerical solution can no longer be guaranteed. Entropy consistency will ensure discrete nonnegativity independent of the selected grid size. In fact, we will be lead t o some type of harmonic integral mean as an appropriate choice. This can

Numerical analysis

209

also be interpreted as a suitable type of upwinding. In the continuous setting an entropy is defined by

Choosing its derivative as a test function in the continuous problem we find that J'G(u) is decreasing with time. This is in analogy to hyperbolic problems, where entropy estimates can be derived by testing the viscous approximated problem accordingly. These entropy estimates carry over to the discrete case provided we define if U+ # Uotherwise. This numerical mobility can be regarded as a function M ( Z h U ) on the primal grid. For the generalization to arbitrary dimensions, we refer t o Section 4. Finally, the semidiscrete scheme can be discretized in time implicitly or explicitly. Therefore suppose [0, T ]t o be subdivided in intervals Ik= (tk, k k + l ] with tk+l = t k r k for time increments r k > 0 and k = 0,. . - , N - 1. We will use backward difference quotients with respect t o time which we shall henceforward denote by d;, respectively. Because of the significant stiffness of our problem we choose an implicit discretization. Otherwise a CFL-type condition r 5 C h4 would entail very small timesteps. In Section 5, we will discuss the selection of appropriate timesteps in detail. These ideas to construct entropy consistent finite volume schemes can be carried over t,o an appropriate pure finite element discretization. Therefore, we consider P and U both as functions in Vh and obtain the following finite element formulation of equation (1) with fully implicit, backward Euler discretization in time:

+

For given UO E vh such that

vh find a sequence (uk,pk)for k = 0, - .. ,N -

1 with

uk,pk E

for all O, Q E V h . Thus, a solution of (3) is obtained solving a nonlinear system of q = d i m v h equations for each time step. Let us define by M h , Lh the standard lumped mass, respectively stiffness matrix and by L h ( W ) the matrix corresponding to the degenerate quadratic form, i. e.

(L,(w)u,v) =

M(W)VUVV~~.

Here we denote the nodal value vector for a function V E Vh by V , and with a slight misuse of notation rewrite L ~ ( w )for L h ( W ) . Then for given Uk E IRq we search U k + l E IRq such that F(Uk+') = 0 for

210

Finite volumes for complex applications

Due t o the absence of Dirichlet boundary conditions, Ah is not injective, i. e. kerAh = { r c-, CIC E IR) . This corresponds t o the observation t h a t AhU =

0. In contrast t o the finite volunle interpretation, the replacement of the exact mobility M ( . ) by a certain quantity M ( U )may now be interpreted as the choice of a specific quadrature t o integrate the elliptic term numerically. For a certain class of grids in 2D (cf. [GRSS]), a tedious but straightforward computation proves the equivalence of the finite volunle and finite element approaches. 3. Existence, stability and compactness of discrete solutions

In this section, we recall the main results concerning existence and regularity. T h e proofs can be found in [GR98]. T h e key estimate for numerical analysis is the following energy estimate:

lT

+

W ( U )1 ~

:s, / v u o ( ~ ) ( ~

~ 1 d' x d f = -

dr.

In 1D a direct consequence of this estimate is a result on Holder continuity w.r.t. time for spatial averages of discrete solutions U which can be cornhilled with the energy estimate t o yield the following pointwise Hijlder regularity:

+

Lemma 3.1: Assume d = 1 and that for integer I , k > 0 with 1 k < N the relation k r 2 h4 holds. Then for u discrete solution (UThr PTh)with 11 M(U,h)(l, M I independently of T,h , there exists a constant C depending such that only on

< ll~~]l

Iu"~(~) forx E

- U"(.)l

< c(kr):

a.

As a consequence, convergence of discrete solutions to a solution in the continuous setting can be proven. For the quite different techniques t o be used in higher space dimensions, we once more refer t o [GR98]. 4. Entropy estimates dimensions

- discrete nonnegativity in arbitrary space

In the case of space dimensions d > 1, the discrete mobility can no longer be given by a scalar valued quantity. This is due t o the observation t h a t on each cell of the dual mesh numerical fluxes coming from different directions have t o be treated differently. It turns out t h a t the right approach is t o choose the mobility a s a field of elementwise constant, symmetric positive semidefinite

Numerical analysis

21 1

d x d-matrices which depends continuously on the discrete function U E v h . To make the mobility matrix consistent with the entropy function, an additional admissibility condition has t o be satisfied: . ~ I ~ ( I : ) o z ~ G ' (=~ V ~U ) , where G'(s) := y ( r ) d r with y(s) = rn(r)-'dr.

Ji

Jl

Here, m is a n appropriate approximation for the continuous mobility JM (for its explicit form depending on the snloothness of M, we refer to [GR98]). Not,e that C: is nonnegative and convex by construction. For nondege~~erat~e reference simplices E(,,, ... ,,,):=convex hull(0, cylel, ,~rded), we verify i~nmediatelythat

(L(~, ['(ate,)

if1 =

(iilij)

i . j = l .... , d

with i f i j =

1 -ds)

4 s )

-1

dij

satisfies the asiollls above. For U ( n k e k ) = I/(()) the definition simplifies to ,TIkk = nz(/[(O)). For elements E which can be mapped onto a reference element E by a rigid transformation x H j. = xo + A - l x , A an orthogonal matrix, the matrix M := AAI.il-' satisfies conditions (ii), (iii). Since A is orthogonal, M is symmetric and positive sernidefinite; hence condition (iv) is satisfied, too. For the general case, we refer to [GR98]. This ~orlst~ruction allows to obtain the following discrete analogue of the entropy est,imate: Lemma 4.1: Let (/I, P) be a solution to the system of equations (3)-(4), and assunre that ( M ,G') is an adnxissible t-ntropy-mobility pair as described above. Then, for arbitrury T = I i r , Ii E IN, the following estimate holds:

As a consequence, the following result on nonnegativity of discrete solutions can be obtained which is in fine accordance with related results in the continuous setting (cf. [BF90] and [Gr95]):

lJs)

Theorem 4.2: (Existence of nonnegative discrete solutions Let Th be an admissible triangulation of S1 and let n > 0 be the growth coefficient of JW in zero. Assume that the mobility .M is monotoneously increasing and z~anzsheson IR- U (0). For arbitrary E > 0, there exists a positive control 0 parameter a 0 which only depends on d , n, E , h and the initial datum uo

>

212

Finite volumes for complex applications

such that: For every 0 < a < a0 discrete entropy-mobility pairs (G,, M,) can be constructed having the property that the corresponding discrete solutions Uyh of equation (3)-(d), W' 0 , satisfy: U$ > - & i f u 0 > O a n d O < n < 2 . > - E if u0 2 uOa n d n = 2. Uyh > a/2 if u0 2 uo and n > 2 . For a proof of this theorem, we refer t o [GR98]. Let us remark that L. Zhornitskaya and A.L. Bertozzi [ZB] who studied finite difference schemes for growth 2 obtained quite similar results on positivity of discrete solucoefficients n tions.

>

+

FIGURE 2. Numerical approximation of the solution t o ut div(u . V A u ) = 0 for initial d a t a given as the characteristic function of a nonconvex set

5. Implementation, timestep control, waiting time phenomenon One of the most important questions with respect t o numerical sinlulat,ions of wetting phenomena is how t o trace the solution's free boundary in an efficient way. In order t o describe the arising difficulties, let us first consider questions of implementation. In each timestep, we have t o solve a nonlinear system (cf. section 2 ) . In fact, we first consider a related semi-implicit system, given by

+

~ = ~ Luk. ~ ) ( ~ d rkM ; ~ L ~ ( W ) A u For the solution of the fully implicit scheme, we apply a n appropriate fixedpoint iteration t o satisfy the original problem with w = (for det,ailscf. [GR98]). Now observing that in the semi-implicit scheme the numerical free boundary

Numerical analysis

213

cannot propagate Inore than a distance h in each timestep, it. is reasonable to choose the time increment r smaller than the quotient -where speed(t) stands for the maximum normal velocity of t,he numerical free boundary a t time t. As a consequence of this special choice of time increment, only a very small number of iterations is necessary to obtain the solution of the fully implicit scheme. Formal considerations - performed in the continuous sett,ing - indicate that the normal velocity V,([(t)) of the free boundary in a point [(t) can be related t o spatial derivatives of u in [(t) according to the following formula: ~ , ( < ( t )= ) lim

~ - r ~ ( t u(t, ) x)

A dv

x E supp(u(t, .)).

,

(7)

This formula has been proved for self-similar source-type solutions in [GR98]. In the framework of the algorithm studied in this paper, we formulate a discrete counterpart of formula (7) in the following way: In a timestep tk, we first determine on each E E Thnumbers

-

M:%lE' (zbllar.

v(tk, E ) :=

A h

1)

if uThIE 2 0 and otherwise.

(%)E

> 0 (8)

with y E (0, 1).

(9)

Then, we define the time increment by the formula rk

:=

0.01

+ m a yhx ~ ~~r (, t kE, ) '

If n 2 1, the results on Holder continuity in space for discrete solutions allow to h-512. give a robust, but coarse upper bound: m a x E ~ T ,v(i, E ) 5 This implies for the time increment:

cIIuIIZZ~~~)

Hence, the assumption r longer.

> h4 in Lemma 3.1 does not mean any restriction any

Fornlula (7) indicates that for sufficiently smooth initial data the velocity of the free boundary vanishes. So let us take R = ( 0 , l ) and as initial d a t a the 5

function uo(x) = ([cos ( $ a x ) ] + ) . We choose M ( u ) = u2 in equation (1) and obtain for t E [O,1] a solution u as shown in the six diagrams on the left of figure 3. From top left linewise to bottom right, they represent six snapshots of u ( t , -) for increasing t,imes t . To have a closer look a t the behaviour a t the free boundary for small t, we depict on the right the function v(t, 2 ) =

log (1020 . u(t, x))

if u(t, x ) > 10WZO otherwise

a t four different times ranging from t = 0.0 in the background t o t = 2.5- lo-" in the foreground. It turns out that for t < 2.6 the free boundary does not move, whereas for larger times the support monotoneously increases. This

2 14

Finite volumes for complex applications

FIGURE 3. Delayed onset of spreading for solutions to the = 0 and sufficiently smooth initial equation ut ( u 2 - u,,,), data(number of gridpoints: 500)

+

gives very strong evidence that also for the thin filrrl equation a waiting time phenomenon occurs. For other simulations illustrating the variety of phenomena encountered in thin film flows, we refer the reader to [GR98] and [Gr99].

J. Barrett, J. Blowey, and H. Garcke. Finite element approximation of a fourth order nonlinear degenerate parabolic equation. to appear in Numer. Mathematik. [BDPGG98] M. Bertsch, R. Dal Passo, H. Garcke, and G . Griin. The thin viscous flow equation in higher space dimensions. Adv. D i n . Equ., 3:417-440, 1998. F. Bernis. Finite speed of propagation and continuity of the interface for thin [Ber96a] viscous flows. Adv. in Diff. Equations, 1, no. 3:337-368, 1996. [Ber96b] F. Bernis. Finite speed of propagation for thin viscous flows when 2 5 n < 3. C . R . Acad. Sci. Paris; Se'r.1 Math., 322, 1996. F. Bernis and A. Friedman. Higher order nonlinear degenerate parabolic equa[BF90] tions. J. Diff. Equ., 83:179-206, 1990. [DPGG98] R. Dal Passo, H. Garcke, and G . Griin. On a fourth order degenerate parabolic equation: global entropy estimates and qualitative behaviour of solutions. SIAM J. Math. Anal., 29, 1998. [Gr95] G. Grun. Degenerate parabolic equations of fourth order and a plasticity model with nonlocal hardening. Z. Anal. Anwendungen, 14:541-573, 1995. [GR98] G . Griin and M. Rumpf. Nonnegativity preserving convergent schemes for the thin film equation. 1998. Preprint No. 569, SFB 256 University of Bonn. [Gr99] G . Grun. On the numerical simulation of wetting phenomena. 1999. t o appear in Proceedings of 15th GAMM-Workshop, Kiel. [Kr97] D. Kroner. N u m e ~ i c a lSchemes f o r Conservation Laws. Wiley and Teubner, Chichester and Stuttgart, 1997. L. Zhornitskaya and A.L. Bertozzi. Positivity preserving numerical schemes for [ZBl lubrication-type equations. SIAM Num. Anal. submitted, 1998. [BBGl

Convergence of finite volume methods on general meshes for non smooth solution of elliptic problems with cracks Philippe Angot Universite' de la Me'diterrane'e, I.R.P.H.E. Chiteau-Gombert, 38 rue F. Joliot Curie, F-13451 Marseille Cedex 20 - E-mail : [email protected]

Thierry Gallouet and Raphakle Herbin Universite' de Provence, C.M.I. - L.A. T . P . , 39 rue F. Joliot Curie, F-13453 Marseille Cedex 20 - E-mail : [gallouet,herbin]@gyptis.univ-mrs.fr

ABSTRACT A model of insulating cracks for elliptic problems is presented and proved to be well-posed. The solution is indeed discontinuous. A finite volume scheme o n general polygonal meshes is introduced to solve such problems. Since n o unknown is required at the fracture interface, the scheme is as cheap as more standard schemes for the same problems without cracks. With weak regularity assu.mptions, uJe establish for discrete norms some error estimates in O(h) where h is the maximum diameter of the control volumes of the mesh.

Key Words: elliptic problems, insulating cracks, discontinuous solutions, finite volumes, fracture resistances, error estimates.

1. Introduction The concept of contact resistance is sometimes introduced empirically for diffusion problems (Fick's law) with imperfect contact, e.g. thermal (Fourier's law) or electrical (Ohm's law) contact resistance, or also hydraulic resistance of fissure for flows in porous media described by the Darcy's law. The objective is t o take account of fault lines or too thin layers cornpared to the largest scale under study. Hence, from this macroscopic scale the solution at the interface is indeed discontinuous. In previous works e.g. [ANG 891, we generalized this concept and formulated for such elliptic problems a mathematical model with discontinuous coefficients which includes a jump transmission condition linking the divergential flux with

216

Finite volumes for complex applications

the jump of the solution a t the interface. It ensures the well-posedness of the associated elliptic or parabolic problems and we proved in [ANG 98, ANG 98b] the global solvability within a variational framework. Besides, we have shown how to use the imperfect transmission problem for fictitious domain modelling with immersed boundary conditions imposed by a penalty method. In that case, we performed the convergence analysis and derived the associated error estimates as functions of the penalty parameter; see also [ANG 971. For the numerical solution on a rectangular mesh, we proposed in [ANG 89, ANG 89b] an original finite volume method, either cell-centered or vertex-centered, and based on the introduction of "fracture resistances" a t the faces of the control volumes. For this scheme, some error estimates in O ( h ) are established in [ANG 98, ANG 98b] and various numerical results have illustrated the capabilities and the efficiency of this methodology. In the present work, we generalize the model in two ways; see also [ANG 991. First, we do the mathematical analysis in the case where the fracture interface C is an open surface strictly included in the bounded domain fl, e.g. without any connection with its boundary = dfl. We prove the existence and uniqueness of the solution for a diffusion-reaction problem. This case is more difficult because the open domain is no longer located locally on one side of its boundary since the fault interface does not divide fi into two disjointed subdomains. Second we extend, in the case of a polygonal interface, the finite volume scheme t o general polygonal meshes, as considered in [EYM 971 or [HE1 87, SHA 961, e.g. triangular [HER 951 or VoronoY meshes; see also [COU 96, HER 96, LAZ 961. The construction of a general admissible mesh is made in such a way that the discontinuity lines of the operator coefficients, and/or the polygonal fracture interface lie on faces of some control volumes. Then, we construct a finite volume scheme including fracture resistances at faces of control volumes, well-suited t o the numerical approximation of the imperfect transmission problem. We show how t o satisfy both conservativity and consistency of the numerical fluxes. Indeed, the numerical scheme is locally conservative by construction. Let us notice also that our numerical scheme inherently involves the locally conservative approximation of the immersed jump condition (3) without using unknowns located on the interface C. Hence, it only uses a four-point stencil for triangular finite volumes in 2-D, or a six-point one in 3-D. This means that the solution cost is as cheap as for a more classical finite volume scheme without any fracture interface, and hence cheaper than with the "double-node" finite difference scheme proposed in [SAM 781.

2. Well-posed elliptic model for insulating cracks Let the domain fi c Rd (d = 2 or 3 in practice) be an open bounded polygonal set, ?= dfl being its boundary, which includes a polygonal interface C C Rd-'.Let us define the open bounded set R such that fi = 0 U C and its boundary F _= d R = f' U C. I t is always possible to prolong C within a

Numerical analysis

21 7

polygonal interface 2 > C which divides the domain fi into two disjointed subdomains R- and R+ such that fi = 0- U 2 U R+. We denote by X - and X + the characteristic functions of R- and a+, respectively. Let n be the normal unit vector on C oriented from R- to R+. For the data f E L2(R), we consider the second-order elliptic problem for the real-valued function u defined in R :

where the symmetric second-order tensor of diffusion a ( ~ i ~ ) l < i , and ~ < dthe reaction coefficient b are measurable and bounded functions verifying classical ellipticity assumptions : a E ( L ~ ( R ) ) ;~ ' l a o

> 0,

V< E lRd, a(x)-

<.< 2 a0 1<12

a.e. in R.

(Al)

For the sake of clarity, we restrict ourselves to homogeneous Dirichlet boundary conditions on r, although other kinds of boundary conditions can be considered by using the results in [EYM 971, as well as more general or nonlinear elliptic or parabolic problems. In the usual transmission problem, the perfect transmission conditions on C are considered, i.e. continuity of the traces of both the solution ulc and normal component of the flux vector (~(u). nlC -(a. Vu). n(C on C, and the solution of the perfect transmission problem is the unique solution u E H; (fi) satisfying (1) over the whole domain fi. 2.1. Jump boundary condition at the interface

We now investigate the case of an imperfect contact on C with the model for "insulating cracks", see [ANG 981. We consider the following jump or fracture boundary condition on C which assumes the normal diffusive flux cp(u). nlc t o be continuous and linked to the jump of the solution through C by :

-cp(u). nIc

-- (a- Vu)-• nlc = (a- VU)+-nlc,

- - ( ~ ( u ) - n l=~ P [ u ] ~- g z P (u+ - u-)lc - 9,

o n C, (3) on

where [u]lc denotes the jump of the solution on C oriented by n , g is given in L2(C) and P is the transfer coeficient on C satisfying the following assumption:

The inverse of P can be defined as the fracture resistance p = 1/P through 0 on C, the condition (3) degenerates into the the interface C. When P non-homogeneous Nedmann boundary condition on C: cp(u)-nlc = g. The

21 8

Finite volumes for complex applications

-

condition ( 3 ) written with p instead of /3 yields the perfect transmission condition when p 0 on C , see the proof in [ANG 98b] and also Sec. 3.1. A particular case of the condition ( 3 ) when g 0 is also considered in [SAM 781.

2.2. Global solvability of the imperfect transmission problem

We study within a variational framework the global solvability of the previous problem (1-3) with the assumptions ( A l ) , ( A 2 ) , ( A 3 ) , g E L2((C and f E L 2 ( R ) . Obviously, the solution u $ ~ , ' ( f l )when [ U ] ~# 0. We begin by some technical results for the traces of functions of H 1 ( R j on dR= U C.

r

Lemma 1 (Traces on and C in H1(Cl)) If fi i s a n open bounded polygonal subset of lRd including a polygonal surface C C Rd-I such that f l = R U C and d R = afi u E, hereafter called the configuration hypothesis ((311, then we can define the following trace applications T , y+, y- for all v i n H 1 ( R ) by: T ( v ) r vlr, y+(v) v+Ic, y ( v ) 5 V-I=. They are linear and continuous from H 1 ( R )into L ~ ( I ? ) or L 2 ( C )respectively, i.e. 3E(R),c+(R),c p ( R )> 0 such that Vv E H 1 ( R ) : 11vlfllL2(f)5 E(R) I l v l l ~ l ( f i )I ,l v + l c l l ~ ~ (5c )~ ' ( I I~v I I1H 1 ( f i ) and

Ilv-lcll~2(5 ~ )c - ( 0 ) I I ~ I I I I ~ ( ~ ~ ) . SKETCHOF PROOF. For a given choice of the subdomains R+ and 0- for the partition of 0, we define the set of regular functions R = { c p E c m ( f l \ C ) ; pin+ E em(=), cpln- E C m ( F ) ) . Recall that cp E C o 0 ( E \ X) if there exists $I E cW(lRd\ C ) such that $I = cp in fi \ C , cp E Cm(W)if there and similarly in 0-. We first state exists $I E c m ( R dsuch ) that cp = that R is dense in H 1 ( R ) . Hence, the trace application yo for the regular functions of R can be prolonged by a linear continuous trace operator from H 1 ( R ) into ~ ~ (Uf C), ' still denoted by yo. Then for v E H 1 ( R ) , the trace operators above are defined as follows: ?(v) = (yov)li., y+(v) = (yovln+)jc, and y - ( v ) = (yovln-)lE. The key point is to show that these trace operators are independent on the choice of R+ or R-; This can be proved at least for a polygonal boundary of 0, i.e. if ('fl)holds; see [ANG 991. Lemma 2 (Jump of traces on C in H 1 ( R ) ) If ((31) holds, then :

PROOF. We get for all v E H 1 ( R ) ,using the continuity of traces in Lemma 1 I I [ v I c l l ~ 2 (=~ )II(v+ - ~ - ) C I I L Z ( ~ )

I maz ( c - ( R ) , c + ( R ) ) I

I V I I ~ ~ ( ~ )

=

We now define the Hilbert space W = { v c ~ ~ ( ffliH) 1 ( R ) ;? ( v ) vIF = 0 ) equipped with the usual inner product and associated norm in H 1 ( R ) . Then we prove the following proposition by considering the variational formulation of the problem (1-3) in R as below : Find u E W such that Vv E W ,

Numerical analysis

219

Theorem 1 (Existence and uniqueness for the crack model) If the assumptions (X), ( A l ) , ( A 2 ) and ( A 3 ) hold, the problem (1-3) with g € L 2 ( C ) and f E L Z ( f l )has a unique weak solution u E W satisfying (4) Vv E W , such that:

PROOF.The bilinear symmetric form in the left-hand side of (4) is clearly coercive in W. By using Lemma 2, we easily show the continuity of both this bilinear form in W x W and the linear form, i.e. the right-hand side of (4), in LV. Then we apply the Lax-Milgram theorem which completes the proof.

3. Finite volume approximation on general meshes For the sake of simplicity, we only consider here the case of an isotropic tensor of diffusion, i.e. a scalar coefficient variable in space a ( x ) satisfying ( A l ) . The general case of a diffusion matrix a ( x ) requires an additional condition of quasi-regularity of the mesh, e.g. [EYM 971. Definition 1 (Admissible meshes) Let fi = R U C be an open bounded polygonal subset of E d , d = 2, or 3 satisfying (7-1) in Lemma 1. A n admissible finite volume mesh of 0 , denoted by 7, is given by a family of "control volumes", which are open polygonal convex subsets of f l , a family of subsets of fi contained i n hyperplanes of Rd,denoted by £ (these are the edges (twodimensional) or sides (three-dimensional) of the control volumes), with strictly positive (d - 1)-dimensional measure, and a family of points of R denoted by P satisfying the following properties (in fact, we shall denote, somewhat incorrectly, by 7 the family of control volumes): -

(i) The closure of the union of all the control volumes is

0.

(ii) For any K E 7,there exists a subset EK of £ such that 8K = K UuEcKa . Furthermore, E = UKG7fK.

\K

=

(iii) For any ( K ,L ) E T2with K # L , either the (d- 1)-dimensional Lebesgue measure of K n Z is 0 or ?7 n 21 = a for some o E E, which will then be denoted by K I L . (iv) The family P = ( x ~is such ) that ~ X~K E ~K (for all K E 7 ) and, if a = K I L , it is assumed that X K # X L , and that the straight line V K , L going through x~ and X L is orthogonal to KIL. (v) There exists Ex

c&

such that

C=UuE~zZ.

(vi) For any a E & such that CJ c dR E dfi U C, let K be the control volume such that a E E K . If X K @ a , let Z ) K , ~be the straight line going through X K and orthogonal to g , then the condition Z ) K , ~ n a # 0 is assumed; let YU = D K , n ~ CJ.

220

Finite volumes for complex applications

I n the sequel, the following notations are used. The mesh size is defined by: hT = sup{diam(K), K E 7 ) . For any K E 7 and a E E, m ( K ) is the ddimensional Lebesgue measure of K (it is the area of K in the two-dimensional case and the volume in the three-dimensional case) and m ( a ) the (d - 1)dimensional measure of a. The set of interior (resp. boundary) edges is denoted by Lint (resp. Eext), that is Eint = {a E E; a d o ) (resp. Ecxt = { a E E; a C dl?)). The set of neighbours of K is denoted by N ( K ) , that is N ( K ) = {L E 7; 3 0 E EK, (T = K n If a = KIL, we denote by d, or d K I Lthe Euclidean distance between XK and XL (which is positive) and by d ~ , ,the distance from X K to a; hence we have d, = d ~ d, ~ , ~, . If a E EK f l Eext7let d, denote the Euclidean distance between XK and y, (then, d, = d ~ , , ) .

z).

+

For each K E 7 or a E EX, define the discrete values of the coefficients by their mean values over K or a respectively, i.e. a~ = m f ~ .) f ~a(x) dx, bK = 1 1 J K b(x)dx, f~ = m ( K ) JK f (2) dx, and PC = P(X)ds(x), gu = J, g(x) d ~ ( x )Or , simply by aK = ~ ( x K )bK , = ~ ( x K ).fK , = f (xK), pu = p(y,) or g, = g(y,,) if the coefficients are regular enough to define these quantities. Define X ( 7 ) as the set of functions from R t o R which are constant over each control volume of the mesh. For v E X ( 7 ) such that VK denotes the value taken by v on any control volume K E 7, define the discrete H i ( 6 ) norm by :

& J,

where D,v =

IVK

-

vLI for a = KIL E

lint

and D,v = 1~x1 for

E Eext

n EK.

3.1. Finite volume method with fracture resistances As in [ANG 89, ANG 981 for a rectangular mesh, the scheme is derived with respect to pic instead of /?Ic because it is more practical since the case when pic 0 and glc 0 in (3) allows t o recover directly the perfect transmission conditions. This is also used t o write the numerical flux at any cell face in $2 that does not lie on C and where the latter conditions are indeed verified. The finite volume scheme is only written with respect t o the discrete unknowns supposed t o be approximations of the values u(xK) of the exact solution u at the nodal points XK of the mesh. The key point to derive the scheme is t o introduce some auxiliary interface unknowns, say u, for a 6 &\ or u:, u; for a 6 Ex, t o write by finite differences the numerical flux FK,,through the face a of any control volume K E 7 . Then, these auxiliary unknowns can be eliminated using the local conservativity property, i.e. F L , ,= ~ -FK,, for all a = KIL E Eint and writing for the discrete quantities, either the perfect transmission condition u, -= u; = u,: for a = KIL E Eint\Ex, or the jump boundary condition (3) u: -u; = p, (g, -FK,,), for CT = K I L E Ex.

--

--

Numerical analysis

22 1

For u E Eext, the Dirichlet boundary condition is written. Finally, the numerical scheme reads in the following synthetic form: for all K E 7,

a, =

a, and 1+P,z

a, =

du aK aL dt,, U K + d ~ , U, L

f o r u = K I L E Eint,

(10)

3.2. Convergence analysis and error estimate Theorem 2 (Error estimate of the FV scheme) If ('H) and the assumptions ( A l ) , ( A 2 ) , (A3) hold, let u E W be the unique weak solution of the problem (1-3) with g E L 2 ( C ) , f E L 2 ( 0 ) .Let 7 be an admissible mesh for the discretization of that problem, i n the sense of Definition 1. Then, there exists a 2 ~ to the ~ finite ) ~volume ~ scheme ~ (6-7-8-9-10-11). unique discrete solution ( Moreover, assume piecewise suficiently regular data: V K E 7 ,f 1~ E CO(K), b 1 E~ c l ( K )and alK i n C 1 ( f 7 ) ,and V a E E X , pi, and gl, in C1(?F),and such that the solution u E H 1 ( R ) satisfies u l E~ C 2 ( F ) ,for any K E 7 .Define the error e 7 E X ( 7 ) by e7(x) = u ( x K ) - U K for a.e. x E K , and for all K E 7 . Then, there exists C > 0 only depending on u , a , b, p and R such that the following error estimates hold for the discrete H t and L2 norms:

SKETCHOF PROOF. It is inspired from [EYM 971. With the conservativity property, a discrete coercivity lemma is proved which ensures both the existence and uniqueness of the discrete solution via a discrete maximum principle and the stability in the sense of finite differences. By using Taylor expansions and the regularity assumptions, the weak consistency in L?(h7)of the approximate volumic terms and the numerical fluxes, specially for (3) on C, is proved. Then it yields the error estimate for the discrete HA norm. Finally, the L 2 estimate follows from a discrete Poincarb inequality. Remark 1 (Convergence and weaker assumptions) With no more regularity on the solution than u E H 1 ( R ) , it is possible to prove directly the convergence of the scheme by using some compactness results and a discrete trace

222

Finite volumes for complex applications

lemma established i n [EYM 971. Besides, error estimates in O ( h T ) like above, also hold for weaker regularity assumptions on the solution, i.e. u E H1(fl) and U I K E H 2 ( K )for any K E 7 . It requires the use of Taylor expansions with integral residuals to prove the weak consistency. See [ANG 991 for the details. References [ANG 891 [ANG 89b]

[ANG 971

[ANG 981

[ANG 98b]

[ANG 991 [COU 961

[EYM 971

[HE1 871

ANGOTPH., Ph.D. Thesis, University of Bordeaux I, april 1989. ANGOTPH., Modeling and visualization of thermal fields inside electronic systems under operating conditions, IBM Technical Report, TR-47095, 70 p., april 1989. ANGOTP H . , Analysis of singular perturbations on the Brinkman problem for fictitious domain models of viscous flows, M'AS Math. Meth. i n the Appl. Sci., Preprint 1997 (to appear). ANGOTP H . , Finite volume methods for non smooth solution of diffusion models; Application to imperfect contact problems, in Advances i n Numerical Methods and Applications, O.P. Iliev et al. (Eds), Proceedings 4th Int. Conf. N M A '98, Sofia, World Scientific Pub., 1999. ANGOTP H . , Mathematical and numerical modelling for a fictitious domain method with penalized immersed boundary conditions, Preprint H D R Thesis - Univ. Me'diterrane'e, 1998 (submitted). ANGOTPH., GALLOUET T H . , HERBINR . , in preparation. COUDIERE Y., VILAJ.-P., VILLEDIEU P . , Convergence of a finite volume scheme for a diffusion problem, in Finite Volumes for Complex Applications - Problems and Perspectives, F. Benkhaldoun and R. Vilsmeier (Eds), Hermks, p. 161-168, 1996. EYMARD R., GALLOUET T H . , HERBINR., Finite Volume Methods, in "Handbook of Numerical Analysis", P.G. Ciarlet and J.L. Lions (Eds), North-Holland, Preprint 1997 (to appear). HEINRICH B . , Finite Difference Methods on Irregular Networks, Int. Series Numer. Math. 82, Birkhauser Verlag, Basel, 1987.

[HER 951

HERBINR., An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh, Num. Meth. for P.D.E., 11:165-173, 1995.

[HER 961

HERBINR . , Finite volume methods for diffusion convection equations on general meshes, in Finite Volumes for Complex Applications - Problems and Perspectives, F. Benkhaldoun and R . Vilsmeier (Eds), HermBs, p. 153-160, 1996.

[LAZ 961

LAZAROV R . D . , MISHEVI.D., Finite volume methods for reactiondiffusion problems in Finite Volumes for Complex Applications Problems and Perspectives, F . Benkhaldoun and R . Vilsmeier (Eds), Hermks, p. 233-240, 1996. SAMARSKII A.A., ANDREEV V . B . , Me'thodes aux Diffe'rences pour ~ ~ u a t i o Elliptiques, ns Editions MIR Moscou, 1978. SHASHKOV M., Conservative finite-difference methods on general grids, CRC Press New York, 1996.

[SAM 781 [SHA 961

Application and analysis of finite volume upwind stabilizations for the steady-state incompressible Navier-Stokes equations

Lut2 Angernzann lnstitut fur Analysis und Numerik Fakulttt fur J/lathentatik Otto-won- Guericke- lfniuersitiit hIagdeburg PF 4120 D-39016 fifugdeburg. Germany

Among the various dificulties in the numerical solution of NavierStokes equations, the case of high Reynolds numbers requires extra care in the choice of a concrete discretization scheme. The specific treatment of the nonlinear convective term by means of finite-volume-based approaches within the framework of conforming or nonconforming finite element methods forms one class of practically releuant stabilization techniques. The aim of the contribution is t o describe a general approach to the design of those methods in multidimensional situations and t o extract the basic principles of their analysis. ABSTRACT

K e y Words: finite volume upwind method, incompressible .Wavier-Stokes equutions

1. Introduction Let c > 0 he a real number, f2 C R d with d = 2 or d = 3 be a bounded, polyhedral domain with Lipschitz continuous boundary and f : R + R d be a 0

given vector field. With V := w ~ ( Q and ) ~ Q := L $ ( R ) := {q E L 2 ( R ) : (q, 1 ) = 0). the following weak Navier-Stokes problem is considered: Find ( u , p ) = ( u l , . . . , u ~ , E~V )x Q such that it holds:

224

Finite volumes for complex applications

where the trilinear form n : V 3 -+ R is defined as n ( w , u , v ) := ( ( w - V ) u , v ) and denotes the inner product of L2(R) or L 2 ( 0 ) d . NOWlet Vh,Qh be members of two families of finite elenlent spaces approximating, in certain sense, the elements of V, Q. Here, the discrete spaces need not be subspaces of I/; Q. While it is not difficult to replace the forms ( V u , V v ) and (p, V - v ) by their "broken" counterparts on the underlying partitions Th of 0 ( . , a )

the trilinear form n has t o be defined in a more careful way. In the book [RST 96, Sect. IV.21, the authors pointed out that the following : -+ R are sufficient conditions to three properties of the discrete form nh : V establish convergence of the numerical method:

>

Semidefiniteness: n h ( w h ,vh, v h ) O7 Lipschitz-continuity: Inh(wh, u h , vh) - nh(zh, uh, vh)l 5 Cllwh - ~hllhll~hllhllvhllh, Consistency: I ~ ( w , U, Vh) - nh(Ihw, I h u , vh)l Chllw112,2,nll~l12,2,nllvhllhr u h ,v h ,w h , z h E vh, U , w E ~ ~ ~n v,( 0 ) ~

<

where 11. [Ih is a norm on Vh and Ih: w $ ( Q ) ~n V + Vh denotes some interpolation operator. The following general approach for the design of the discrete trilinear form satisfying the above conditions is based on three different partitions of C2. One of the partitions is denoted by Th and is either a triangulation (i.e. it consists of d-simplices) or a block-partition (i.e. it consists of convex quadrilaterals ( d = 2) or convex hexahedra (d = 3)). It is assumed that Th is admissible in the usual sense, i.e. two elements of the partition are allowed to have in common either a vertex or an edge or, if d = 3, a face. Using the notation T for the elements of Th,the parameter h of the partition is defined as usual: If T has the diameter hT, then h := max hT. Notice that this partition is related TETh

to the approximation u h of the unkonwn U ; in certain situations the partition for the discrete unknown ph may differ from Th. Next, on each partition Th a (not necessarily conforming) finite element space is defined, the elements of which are piecewise polynomially of maximal degree 1 E NU{O). In particular, the polynomial space on T may be incomplete, especially for quadrilateral/hexahedral elements. Given some finite element space, the corresponding set of functionals (global degrees of freedom) naturally splits into Langrangian functionals and others, where a Langrangian functional is defined via point values of its argument. Therefore, considering these Langrangian functionals, a collection of (global) nodes can be associated in a natural way. This collection of nodes can be subdivided into the class of nodes lying on element boundaries and the class of nodes belonging t o the interior of some element. For example, the second

Numerical analysis

225

class is empty for all serendipity elements, but for the mini element it contains all elenlent barycenters. Let Kg denote the set of indices of all the nodes from the first class. The subset Ag c Kg contains, by definition, the indices of interior (w.r.t. R ) nodes only. Finally, declare ang := A, \Ag and let A g C~ Kg contain the (global) indices of the nodes belonging to the element T. Due t o the boundary conditions, the above finite elenlent space is restricted to elements satisfying a discrete boundary condition, i.e. we set

T h e distance between t,wo nodes xi, x j ( i ,j E x g ) is denoted by dij. Now, an important observation is t h a t the index set Kg can be decomposed - -A into two disjoint subsets A, A such that Kg = ; iu and ; ifl KA = 0. Such a decomposition can be generated, for example, by a hierarchical decomposition of the finite element space Shl into a "lower degree part" and its linear to complement. Then K corresponds t o the nodes of the first part and the complement's nodes. Obviously, this decomposition induces similar ones of Ag, aA, and AgT, respectively. If Shlis a space of elements of low degree, then it is allowed that the decomposition is trivial, i.e. the complement may be the is empty by trivial space consisting of the zero element only. In this case, definition. As an example, in the case when Sh2is built by means of conforming P2-elements (Taylor-Hood elements, d = 2), we have the decon~position

xA

xA

where Shlis the space of conforming PI-elements (Courant elements) and S&is the space of so-called "bump functions", i.e. the set of the piecewise quadratic nodal basis functions associat.ed with the edge midpoints. T o describe the discretization of the trilinear form n, a further partition of R is needed. This second partition 7; consists of subdomains Qi C Q, the boundary part R n doi of which is the union of subsets of (d - 1)-dimensional hyperplanes. An incidence relation between these two partition is defined by the help of the nodes of Th, i.e. each Qi should correspond to one node xi and vice versa. Further, for all indices i , j E K, j # i, let rij := n Dj and mij := measd-1 ( r i j ) . Then it makes sense t o introduce the index set

ni

As a consequence of these definitions, the following representation of aRi is valid: a R i = rjj, i E A. Moreover, there are the obvious symme-

U

j€Az

try relations dji = dij, rji = rij, mji = mij. The boundary parts rij can be structured in a finer way: I?$ := rij n T for T E Analogously, r n c := measd-1 (r;) . Now we formulate a set of geometrical assumptions.

x.

226

Finite volumes for complex applications

( A l ) There exists a constant C di5j C'mij.

<

> 0 such that, for all

(A.1) There exists a third partition subdomains 2 ': (i) i2: (ii)

{a:)

i,j E

x.it holds

i,jE ai+j,T,,~,o of R

such that the

have the following properties:

=R ,:

-

rc c Q,:

(iii) Each Q:

xi E

q,x j E q.

can be decomposed into a finite number 1; of pairwise

T1 disjoint open d-simplices Qi/ such that -T Rij =

I -T,l U12, Rij

and, for

is the image of a fixed (reference) simples any 1 E [1,1c]~, T under a regular affine transformation, where the pre-image i' of -T 1 I'Z1 := riinni; does not depend on the particular values of i, j, T ,1. (iv) On each

rZ1,the unit outer

The diameter of the simplex

dij

(

is constant.

Furthermore, we set

> 0 such that, for all i, j E x,it

ma.

T E 7 h : m e >O

(A4) There exists a constant C [ 1 , 1 $ ] ~ ,it holds

denoted by h:'.

S " R :

(A3) There exists a constant C

(w.r.t. Qi) normal "u:

rnz1dij

>0

holds

l€[l,le]~ such that, for all i, j E

< Cmeasd (a:')

x,T E %, 1 E

.

(A5) There exists a constant C > 0 such that, for all i , j E [I, it holds ( h 2 ' i 4 5 Cmeasd (a:') .

x,T E x,1 E

A dimensional analysis of the quantities appearing in the Assumptions ( A l ) , (A3), (A5) easily shows that these conditions are not very restrictive.

2. Discretization (Treatment of the trilinear form n ) d

Because of n ( w , u,v ) :=

X((W. v ) u l , u l ) , the description of the discretizaIT1 0

tion can be restricted t o the scalar case. So let w E w ~ ' ( R ) ~be such that

Numerical analysis

227

0

G - w = 0 and define, for u, v E w ~ ( Rthe ) , form

It is not difficult t o see t h a t n, can be represented equivalently a s

Taking into considerat.ion the condit,ion V . w = 0, it is not so far t o omit the last. term, i.e. we get n,(w,

1

U , 21)

1

= -[(V . ( w ~ )v), - ( u , V . ( w , ~ ) = ) ] ?[b(w, u, v) + b(-w, v, u)],

-

2

1 - ( ( V - w ) u , v). This is the starting point 2 for the discretization. Suppose there is some control function r : W -+ [O, 11 satisfying the following properties: where b(w. u ,

11)

:= ( O

- ( W T L )v) ,

-

r ( : ) is isotone for all z ,

343)

is Lipschitz-contilluous on the whole real axis.

l..

Furthermore, set yij := m ~ '

u

-

. w ds and r i j

?. . d . . := r ( 7 ) . We mention

..I

t.hat yij is ant,isyinmetric, i.e. yji = -yij. Moreover, in the definition of yij, the value of J,--u - w ds can be replaced by certain approximation which has to satisfy, among natural error est,imates, the above antisymmetry condition. Then, by standard arguments in the derivation of finite volunle discretizations (cf. [ANG 95]), we can write z,

Thus we get

:C C

n,, ( w , u,V ) M -

iciz

jeA,

[(rij - i ) ( U i -

+

1

- U 8 V3) 1 73. m V. -. uj)(vi - ~ j ) -(ujvi 2

228

Finite volumes for complex applications

sfl,

Now, redefining for wh E Vh := uh, uh E Shl the quantities Y.. .- m- 1 u . wh ds as well as rij, we set 23 'tj TETh :m:>O 11

C

lT

Returning t o the original form n , we set for wh, uh, vh E lTh d

n ( w , u , v ) 63 nh(wh, uh, vh) :=

-

n , s h ( ~ h~,' h~, l h ) 1=1

COROLLARY 1 If the control function r satisfies (P5), then it holds

1

-

Typical examples of the control functions are r ( z ) = ;[sign z

I

scheme), r ( z ) = 1 -

expz - 1

+

+ 11 (full upwind

(exponentially fitted scheme), r ( z ) =

[ i I] (Samarskij's scheme). 2 2 - t 121 Finally, some discrete forms and operators have to be introduced. For u, u E 0

( R ) + S h l , we set ( V u , V U ) ~where ( V u , v u ) := ~

( V u , V V ) ~:= TETh

The extension of these definitions to the case of va valued functio~lsis obvious and will not be denoted separately. By Ih: W;(Q) t Shl, an interpolation operator is denoted, whereas Lh : C ( 1 ) t L,(Q) stands for a so-called lumping procedure. T h a t is, the image of Lh is the subspace consisting of functions being constant on the elements of the secondary partition 7;.Concrete properties of these operators are collected in the subsequent assumptions.

3. Properties of the discrete forms

In order to verify the required properties of Lipschitz-continuity and consistency of n,h, we formulate further assumptions.

Numerical analysis

229

(A6) There exists a constant C > 0 independent of h such that, for all 'uh E Shl,

(A7) For arbitrary p E [I, 61, there exists a constant C vh E Shl,it holds IlvhllO,p,n Cllvhllh

> 0 such

that, for all

<

> 0 such

(A8) There exists a constant C all u h E S h l , it holds

-A

Since A

that, for arbitrary p E [I, 61 and for

is non-empty, in general, IILhvhllO,p,nis only a seminorm on Shl.

(A9) There exists a constant C > O such that for all vh E Shl, it holds

> 0 such

(A10) There exists a constant C

that

(i) for all v E W22(R),it holds 11Ihu110,m,n (ii) for all v E w ~ ( R )and all T E 11h"-'11,2,T

<

x,it holds < C llu112,2,~,

(iii) for arbitrary p E (d, 61, v E W:(R)

> 0 such

( A l l ) There exists a constant C (i) for arbitrary p E (d, 61,

21

11v112,2,n,

and all T E Th, it holds

that

E Wi(C2) and all T E

Th,it holds:

I(I- I ~ ) U I ~ 5 , ~C, ~T; - ' I I U I I I , ~ , T , 1 = 0 or 1 = 1 , (ii) for all v E W22(R) and all T E Th, it holds

<

[ ( I- Ih)vI1,2,~ ~ h $ - ' l l v 1 ( 2 , 2 , ~ , 1 = 0 (A12) There exists a constant C T E Th,it holds

>

01:

1 = 1.

0 such that, for all u E W2??(C2)and all

230

Finite volumes for complex applications

Now, the folllowing results can be proved (for details, see [ANG 981).

LEMMA 1 Suppose (Al), (A?), (A3), (Ad), ( A j ) , (A6), (A?'), 648). Then, for arbitrary wh, z h E LTh and u h , u h E Shlthe estimate

holds, where C

>0

is a constant which does riot depend on h.

L E M M A 2 Suppose (A?), (Ad), (.46), (A?'), ('48), (AQ), (.410), ( A l l ) , ( A ) . Then, for any w E b ~ ' z ( R )n ~ V satisfying O . w = 0, any u E 0

w ( R ) n bVi(s2) and any element

holds, where C

>0

tjh

E

Shl

the estimate

is a constant tvhich does not depend on h .

4. Application

The above approach call be used to give an alternative proof of the convergence properties of Schieweck's fanlily of nonconfornling quadrilateral/l~esal~edral elenlents [SCH 971 which find successful applicat,ioll in parallel Navier-Stokes codes. Det,ails for the case of the so-called PI-parametric element are described in [ANG 981. 5. Bibliography

[ANG 951

L. Error estimates for the finite-element soANGERMANN, lution of an elliptic singularly perturbed problem. Iilf-4 J. Numer. Anal., 15, 1995, p.161-196.

[ANG 981

ANGERMANN, L. Error analysis of upwind-discret,izations for the steady-stat,e incolnpressible Navier-Stokes equations. Fakultat fiir Mathematik, Otto-von-Guericke-U~liversitat Magdebnrg, Prepri~itNr. 33, 1998.

[RST 961

ROOS, H.-G., STYNES, M. A N D TOBISKA,L. ivumerical methods for singularly perturbed differential equations. Springer- Verlag, Berlin-Heidelberg-New York, 1996.

[SCH 971

SCHIEWECK,F. Parallele Losung der stationken inkompressiblen Navier-St,okes Gleichungen. Habilitationsschrift,, Fakultat fiir Mathernatik, Otto-von-Guericke-Universit,at Magdeburg, 1997.

A new cement to glue non-conforming grids with Robin interface conditions

Yves Achdou Insa Rennes, 20 Av. des Buttes de Coesmes 35043 Rennes, France

Caroline Japhet, F'rederic Nataf CMAP, Ecole Polytechnique 91128 Palaiseau, France

Yvon Maday Laboratoire d'Analyse Nume'rique, Universite' Pierre et Marie Curie

4, place Jussieu, 75252 Paris Cedex 05, fiance

We propose and analyse a domain decomposition method based on Schwarz type algorithms that allows for an extension to optimized interface conditions on nonconforming grids. We consider the convection-diffusion equation discretized b y a finite volume method. The nonconforming domain decomposition method is proved to be well-posed and the error analysis is performed. Then we present numerical results that illustrate the method.

ABSTRACT

Key Words: Domain decomposition methods, optimized artificial interface conditions, non-conforming grids, convection-diffusionproblems, finite volume methods, parallel computing, High Performance Computing.

1. Introduction

The goal of our project is to design domain decomposition methods based on the use of optimized interface conditions on non-matching grids. The original Schwarz algorithm is based on a decomposition of the domain 0 into overlapping subdomains and the solving of Dirichlet boundary value problems in the subdomains. It has been proposed in [L 891 to use more general boundary conditions for the subproblems in order to use a non-overlapping decomposition of

232

Finite volumes for complex applications

the domain. The use of exact artificial boundary conditions as interface conditions leads to an optimal number of iterations, see [HTJ 881, [NRdeS 951. As these boundary conditions are pseudo-differential, "low frequency" approximations of these conditions have then be proposed, see [D 931, [NR 951, [GGQ 961. In [J 961 approximations which minimize the convergence rate of the algorithm are proposed, and increase dramatically the convergence speed of the method. The mortar method, first introduced in [BMP 891, which enables the use of non-conforming grids, can't be used easily with optimized interface conditions in the framework of Schwarz type methods. The goal of our work is to design and study a non-conforming domain decomposition method which allows for the use of Robin interface conditions (&f a ) for Schwarz type methods. We consider the convection-diffusion equation

discretized by a finite volume method where q and v are positive but arbitrary small and a is the vector field. We first consider the symmetric definite positive case ( a = 0) and then in fj 5 the convective case ( a # 0). 2. Domain Decomposition at the continuous level

Let R be a bounded domain in Rd for d following problem: Find u such that

(-A

+ 7)( u )= f

> 1 and q > 0.

We consider the

in R

u = 0 on a R The domain R is decomposed into N non-overlapping subdomains, 0 = UI j i 5 N f i i . Let a > 0, the above problem is reformulated as a domain decomposition prob) ~-< ~such < N that lem: Find ( u ~ -

An iterative method for solving the above domain decomposition method is:

Numerical analysis

233

The well-posedness and convergence of the above problems and algorithm have been studied in [L 891. We are interested in the discretization of (4) by a finite volume scheme with non matching grids on the subdomains's interface. 3. F i n i t e volume discretization

The scheme is taken from [H 951. On each domain Ri let Z be a set of closed polygonal subsets of Ri such that f i i = U K K ~and ~Eni the set of edges associated with Z , i.e. a set of closed subsets of dimension d such that for any (K, K') E with K # K', one has either K n K' = 0,dim(K n K') 5 d - 1 and K n K ' E Eni. In this case, d K n dK' is denoted by [K, K']. We also assume that no edge intersects both dRi/aR and dRi n d o . We shall use the following notations: Let ~i be an edge of Eni located on the boundary of Ri, K(ei) denotes the control cell K E such that E, E K . & i is~ the set of edges such that dR n d o i = UEE&iDE. Let us recall that a Dirichlet boundary condition is imposed on this part of the boundary. Ei is the set of edges such that dRi/dR = UeEdiE. Let us recall that a Robin interface condition is imposed on this part of the boundary. &(K) denotes the set of the edges of K E EiD(K) = & ( K )n & i is~ the set of the edges of K E Z which are on dR n dRi. Ei(K) = &(K)nEiis the set of the edges of K 6 7;: which are on aRi/dR. N i ( K ) is the set of the control cells adjacent to K : Ni(K) = {K' E Z/ K n K ' € En,). We make the following

x.

A s s u m p t i o n 3.1 W e assume that there exist points (ye)cEEnion the edges (y, E E ) and points (xK)K~-T,inside the control cells such that for any adjacent control cells, K and K', the straight line [xK,xK!] i s perpendicular to the edge [K, K'] and [ X K , XK'] fl [K, K'] = { Y [ ~ , ~ 'and ] ) , for any edge E E &i U &a, the straight line [xK(<), ye] is perpendicular to E. It is then possible to write a finite volume scheme for the equation (4). We shall use the primary unknowns ( u ~which ) aim ~ at~ being ~ approximations to ~ ( x K ) The . scheme is obtained by integrating (4) over each control volume --

This relation is discretized by

234

Finite volumes for complex applications

where rneas([K,K ' ] )is the measure of [ K ,K'] and p, is a discretization (defined below) of the normal derivative du/dnK on the edge [ K ,K']. For an edge [ K ,K'] common to two control volumes K and K ' ,

We have then the useful property that p ~ , =~ - Pt K I , K . For an edge 6 on the boundary 80, the homogeneous Dirichlet boundary condition (5) is taken into account by

When there is no domain decomposition, this scheme has been analyzed in [H 951 in the more general case of discontinuous coefficients and is proved to be of order one for a discrete H1-norm. 3.1 Discretization of the interface conditions On each interface edge t E Ei of a control volume K = K ( t ) , we introduce p, and u, related by the relation

Then, the interface condition (6) on an interface edge c E K = K ( E )is discretized by

Liof a control volume

It will be useful in the error analysis to interpret (15) as a L2 projection. Indeed, let P O ( a R i / a R be ) the set of functions from d R i / d n into R which are piecewise ~ ~associate , constant on the interface edges. To any discrete values ( v , ) , ~we T ~ ( ( V ~ ) ~E ~ P &O ; )( d R i / d R )its natural piecewise constant extrapolation:

The L2 projection on PO(dRi/dR)is denoted by Pi. With these notations, (15) is equivalent to

Numerical analysis

235

For simplicity, (18) will be (sometimes) denoted by abuse of notation Pt

+ Qui = p i ( C ( - P j

f

~uj)l[sn.nan,])

(19)

j#i

Theorem 3.2 The finite volume discretization defined by (11)-(12)-(13) -(Id)(15) is well-posed. 4. Error analysis

For the error analysis, we need the following additional assumption on the interface edges (this assumption is relaxed in [AJNM 991):

Assumption 4.1 a) For any i and any E E Ei, y, is the barycenter of E. b) For any i, j , dRi n dRj = U E E ~ci C , a ~ ji.e. ~ , dRi n d R j can be written as the union of edges of Ei and of Ej. Let

iik

= u(xK) for any control cell K in T(Ri) and for any E E EiD, let = %(ye). For any interface edge E E Ei, let 13: (resp.

21: = u(y,) = 0 and

&

2L:) be the mean value of (resp. u). With the notations of § 3.1, this can & ,pi(&) ) and T~((uZ;),~&,) = Pz(u). Let FK be be rewritten as T ~ ( ( ~ ! ) , ~ = an approximation of order IKlO(h) of the integral of f on the cells K . The solution of the finite volume discretization with FK as a right hand-side is denoted by u& for any control cell K in T(Ri) and p%on any edge E E Ei U &in and u: for any edge E E Ei. We shall estimate the discrete errors e& = u& e: = uf - ii; and q: = p%As for a , we take a = P/hY with y 2 0.

iik,

Theorem 4.2 We assume that the solution u of (2) is in C2(fi). Let us consider a family of admissible meshes T?iIN indexed by k which satisfy Assumption 4.1 and: 3C1> 0 s.t. V i , Vk,V&E Ek, yE E E, d2u d(y,, xK(,)) 2 C' m e a s ( ~ ) ~ m e a s ( d Rmax i ) [-?I an; a n i Let hk = s ~ p , < -, < ~ { d z a m ( K K ), E -

(20)

Kk} Then, 3 C (independent of k) s.t. ).

236

Finite volumes for complex applications

5. Advection-diffusion problems

We consider now the advection diffusion problem with a continuous velocity field a and a viscosity v :

Theorem 5.1 Under the same assumptions as i n theorem 4.2, and provided that a E C(a) satisfies V . a = 0, we have the error estimate

+

N

i=1

v e & m e a s ( ~5) c(hZp7 + h). KET,

6. Numerical results In order to illustrate the use of Robin interface conditions on non matching grids, 2-D test problems were performed with the finite volume scheme analyzed above. We solve the following problem:

with 7 and v positive constants. The stopping criterion of the algorithm is that the max norm of the jump of the Robin interface condition is smaller The numerical solution is compared to an exact solution. We than choose u(x, y) = x3 y2 sin(xy). The right-hand side is obtained by applying the operator t o u and using an approximate integration formula, the Gauss quadrature rule with four by four nodes per control volume. Thus the error due t o inexact integration will be small with respect to the scheme error. A decomposition into 2 x 2 subdomains is considered. The L, and discrete H l (Theorem 5.1) norms are given for successively refined grids:

+

Case 1:

v = 1, a = (0,0), v = 1. Initial grid:

9 x 9-8 x 8-7 x 7-6 x 6.

Case 2: 7 = 1, a = (y, -x), v = le-2. Initial grid: 9 x 9-8 x 8-7 x 7-6 x 6.

Numerical analysis

237

For case 1 (diffusive case), the first order in the discrete H i norm as expected from the theory seems to be attained only asymptotically for rather fine meshes. The error reduction between the finest two meshes is 1.88. The error reduction factor in the L, norm is also improved with mesh refinement and seems better than O(h). On the contrary in case 2 (convective case), the error reduction factor deteriorates as the mesh is refined and seems to converge to the expected value

JZ.

Table 1: Error vs. mesh refinement

-

Table 2: Error vs. mesh refinement

No convection

-

Convection

7. Bibliography [AJNM 991

ACHDOU Y., J A P H E T C . ! NATAF F., MADAY Y., A new cement to glue non-conforming grids with Robin interface conditions, RI No 419, CMAP Ecole Polytechnique, may 1999.

238

Finite volumes for complex applications

[BMP 891

BERNARD1 C., MADAY Y. A N D PATERA A., A new nonconforming approach t o domain decomposition: the mortar element method, Nonlinear Partial Differential Equations and their Applications, eds H. Brezis and J.L. Lions (Pitman, 1989). DESPRES B. Domain decomposition method and the Helmholtz problem. II, Kleinman Ralph (eds) et al., Mathematical and numerical aspects of wave propagation. Proceedings of the 2nd international conference held in Newark, DE, USA, June 7-10, 1993. Philadelphia, PA: SIAM, 197-206 (1993).

[HTJ 881

HAGSTROM T., TEWARSON R.P. A N D JAZCILEVICH A,, Numerical Experiments o n a Domain Decomposition Algorithm for Nonlinear Elliptic Boundary Value Problems, Appl. Math. Lett., 1, NO 3 (1988), 299-302. HERBIN R., A n error estimate for a finite wolume scheme for a diffusion-convection problem o n a triangular mesh, Numer. Methods Partial Differential Equations 11 , no. 2, 1655173, (1995). J A P H E T C., Optimized Krylow-Ventcell Method. Application to Convection-Diffusion Problems, DD9 Proceedings, John Wiley & Sons Ltd (1996). LIONS P.L., O n the Schwarz Alternating Method III: A Variant for Nonoverlapping Subdomains, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM (1989), 202-223.

[NR 951

NATAF F . A N D ROGIER F., Factorization of the ConvectionDiflusion Operator and the Schwarz Algorithm, M ~ A s ,5 , n l , 67-93 (1995).

[NRdeS 951

NATAF F., ROGIER F. A N D DE STURLER E., Domain Decomposition Methods for Fluid Dynamics, Navier-Stokes Equations and Related Nonlinear Analysis, Edited by A. Sequeira, Plenum Press Corporation, pp367-376 (1995). GASTALDI F., GASTALDI L. AND QUARTERONI A , , Adaptative Domain Decomposition Methods for Adwection dominated Equations, East-West J . Numer. Math 4, 3, p 165-206 (1996).

Finite Volume Box Schemes

Jean-Pierre CROISILLE De'partement de Mathe'matiques Universite' de Metz F-57045 Metz Cedex 01 [email protected]

W e present the numerical analysis on the Poisson problem of a mixed Petrov-Galerkin Finite Volume scheme for equations i n divergence form div p ( u , V u ) = f , which has been introduced i n [CoC 981. A s the original box scheme of Keller, this scheme uses face centered degrees of freedom for the primal unknown u and for the flux 9. The underlying Finite Element spaces are the non-conforming space of Crouzeix-Raviart for the primal unknown and the div-conforming space of Raviart-Thomas for the flux. Optimal order error estimates are derived for the Poisson problem. ABSTRACT:

Key Words: Finite-Volume method, Box Scheme, Box Method, Mixed Method.

1. Introduction

The name of "box-scheme" is a generic denomination for several numerical schemes of different origins. It has been introduced primitively by H.B. Keller in the '70 on the 1-D heat equation, [Ke 711. Generally speaking, the discrete equations are defined in a box-scheme from some kind of averages of the continuous equations on "boxes". Therefore, they are conservative schemes, i.e. schemes which guarantee, for equations in divergence form, an exact conservation of the flux a t the level of the box. At least two variants of box-schemes are known in the litterature. The first one has been introduced in the '80 in compressible Computational Fluid Dynamics. As in Keller's scheme, the basic idea is that locating the degrees of freedom a t the center of the faces instead a t the center of the cells, could be more interesting for an accurate evaluation of

240

Finite volumes for complex applications

conservative fluxes, [CDH 831, [CM 861, [Co 921. Note that this kind of schemes has received much less attention in the CFD Finite Volume communauty than the cell-centered Finite Volume schemes. The second kind of box-schemes is known under the name of LLbox-method"or "finite volume element method". For the Laplace equation -nu = f , it consists of an approximation of u in a finite-element P1 or Q1-space. The discrete equations are defined by averaging the equation on a dual box surrounding each vertex of the mesh, [BR 871, [Ha 891, [CMM 911, [TS 931. In this kind of scheme, two meshes are used. The primal one as support of the trial functions, and the dual one for designing the boxes for the discrete conservative equations. This design is in fact similar t o the one of the cell-vertex Finite Volume method in CFD, [FS 891. Concerning the numerical analysis of Finite Volume cell-centered methods, beside the exhaustive direct analysis of [EGH 971, there is by now an attempt in the FEM communauty for interpreting mixed Finite Elements methods as Finite Volume methods, [BMO 961, [Du 971, [YMAC 991. The scheme presented here can be more or less attached t o this kind of study. 2. The Finite Volume Box Scheme

In [CoC 981, we introduced a new kind of box-schemes for equations in divergence form. As in box-schemes with finite-differencing interpretation, [Ke 711, [Co 921, the degrees of freedom are located a t the center of the faces of the mesh. Nevertheless, the discretization is interpreted here as a Finite Element approximation, allowing to use Finite Element theory for the numerical analysis. Let us consider the 2D Poisson problem in mixed form with Dirichlet homogeneous boundary conditions d i v p i f = 0 in R p-Vu=O i n 0 u = 0 onto dR Suppose given a triangulation 7j, of the domain St C R2 by triangles K. The number of triangles is N E . The number of internal edges, boundary edges are N A i , NAb and the total number of edges is NA = NAi N A b . The Finite Element spaces that are used are u-space: The non-conforming Crouzeix-Raviart space with homegeneous boun-

+

dary conditions Vh = PA,,o, equipped with the mesh dependent norm

Numerical analysis

241

p-space: The div-conforming Raviart-Thomas space of least order Qh = RTO, equipped with the continuous norm

IIphIIdiv = (l~hl;,,

+ ~divphli,,)"~

Recall that these spaces are Vh = {vh/V K E '&, vh I K E P' (K), uh is continuous at the middle of each edge, vh = 0 at the middle of each edge on d o )

The scheme reads: find (uh,ph) E Vh x Qh such that

In (2), the number of unknowns is 2NA, since the global degrees of freedom for uh, ph are scalars located at the center of the faces of the mesh. The number of equations is clearly 3NE+ NAb. A simple count of the faces (the edges) proves that in fact we have

Let us mention that coupling these two spaces is not standard in the mixed finite element methods, because they do not verify the Babuska-Brezzi condition, [Ba 711, [Bre 741. 3. Numerical Analysis 3.1. Reformulation as a mixed Petrov-Galerkin method

We consider the following mixed formulation of problem (1):find (u,p) E H,' x Hdiv such that for any (v, q) E L2 x (L2)2

B [ ( uP); , (v,q)l = (P, q) + (div P, v) - (021, q) = -(f, v)

(4)

or equivalently (divp+f,v)=O ( p - vu,q) = 0

VvgL2 vq E (L2)Z

(5)

242

Finite volumes for complex applications

Applying the general Babuska theorem [Ba 711 onto mixed formulation we find easily that ( 5 ) is a well posed problem, whose solution is (u, Vu) , u E H,' nH z being the unique solution of the original problem (1).The scheme (2) appears now as a Petrov-Galerkin non conforming approximation of (5). Calling Po the space of constant functions in each triangle, it can be rewritten: find (uh,ph) E Pi,,o x RTOsuch that for any (vh,qh) E Po x

or equivalently

Applying now the theory for mixed Petrov-Galerkin approximations, [Ni 821, [BCM 881, [BMO 961, [Cr 991, we get the following result: Theorem 1 (i) The scheme (7) does possess a n unique solution ( u h , p h ) E Pi,,, x RTO verifying

(ii) We have the error estimate

3.2. Dual scheme

Another mixed form of the Poisson problem linked with the bilinear form

B is dual from (1):find (v, q) E L2 x (L2)2such that for any (u,p) E H; x Hdiv

or equivalently

Numerical analysis

243

Again by the Babuska theorem, problem (11) does possess an unique solution (v, q) = (u, Vu). The corresponding Petrov-Galerkin scheme is: find (vh,qh) E PO x such that

As precedingly, we get the following result Theorem 2 (i) The dual scheme (12) does have an unique solution (vh, qh) E Po x verifying

lvhlo,n + Iqhlo,n

C If

10.0

(ii) We have the error estimate

Note that (12) defines a non standard cell-centered Finite-Volume scheme for computing both the unknown and the gradient from the knowledge of the laplacian.

3.3. Second order error estimate

Using the error estimates for the primal and dual schemes ( 7 ) , (12), allows to derive a second order error estimate in the L2 norm for the unknown uh in (7). The proof uses an Aubin-Nitsche like argument. Theorem 3 The solution ( u h , p h ) of Scheme (7) verifies the optimal error estimate

3.4. Further remarks

A natural question is to ask whether there is the link between this scheme and the family of Finite Element mixed methods, [RT 771, [AB 851, [BDM 851, [BF 911. In fact, it can be proved that the gradient part ph in (7) coincides with the mixed gradient ph in [RT 771. However, the .iih part in the mixed method

244

Finite volumes for complex applications

is only a Po approximation of u. The optimal error estimate is therefore only of first order in the L2 norm. They are several methods in order t o interpolate a posteriori .iih to an higher order approximation, [AB 851. All these methods introduce a three variables problem (Uh,ph, Ah), involving a new degree of freedom (a Lagrange parameter) Ah at the interfaces of the mesh. It can be proved, [Cr 991, that one of these interpolations coincides in fact with the scheme (7). Moreover, contrary t o the standard mixed method in its original formulation, the degrees of freedom for uh and ph are decoupled, allowing to solve only an O ( N A ) system in uh, which is in addition symmetric definite positive. We refer to [CoC 981 for details onto the implementation of (7), and t o [Cr 991 for the proofs of the results described here. 4. Conclusion

Several works are devoted to the a posteriori interpretation of mixed Finite Element methods as Finite Volume ones for the primal unknown iih. In other works, they are attempts to compute more easily this unknown, by introducing additional degrees of freedom. The advantage of the scheme (7) is that it is basically designed as a true Finite Volume scheme on a single computational cell, for uh and ph. In addition, it gives a natural decoupling between the unknowns uh and ph. Moreover, it has an optimal order of accuracy both for uh and ph, without any post-processing or a posteriori interpretation. Note that this kind of schemes is not restricted t o triangular meshes, or to the dimension 2. We think that the formulation of this scheme as a Petrov-Galerkin method, combining the advantages of mixed and Finite Volume methods, can be particularly interesting for computations involving complex fluxes. Moreover, it can be of some help for a better understanding of the link between mixed and cell-centered Finite Volume methods. Higher order extensions are currently in progress. Acknowledgments: The author acknowledges friendly B. Courbet, F. Dubois, and A. Debussche, J. Laminie, for stimulating discussions and encouragements.

Numerical analysis

245

References

[AABM 981 B . ACHCHAB, A. AGOUZAL, J. BARANGER, J - F . MA~TRE, Estimateur d'erreur a posteriori hikrarchique. Application aux klkments finis mixtes Numer. Math. 80, 1998, 159-179. F . BREZZI, Mixed and non-conforming finite elements [AB 851 D . N . ARNOLD, methods: implementation, postprocessing and error estimates, Math. Model. and Numer. Anal. 19,1, 1985, 7-32. [Ba 711 I. BABUSKA,Error-Bounds for Finite Elements Method, Numer.Math., 16, 322-333. [BR 871 R.E. BANK,D . J . ROSE,Some error estimates for the box method, SIAM J. Numer. Anal., 24,4, 1987, 777-787. [BMO 961 J. BARANGER, J . F . MA~TRE, F. OUDIN, Connection between finite volume and mixed finite element methods, Math. Model. and Numer. Anal., 30,4, 1996, 445-465. [BCM 881 C . BERNARDI,C . CANUTO,Y. MADAY,Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem SIAM J. Numer. And., 25,6, 1988, 1237-1271. [Bre 741 F. BREZZI,On the existence, uniqueness and approximation of saddlepoint problems, arising from lagrangian multipliers R.A.I.R.O. 8, 1974, R-2, 129-151.

[BF 911 F. BREZZI,M . FORTIN,Mixed and Hybrid Finite Element Methods, Springer Series in Comp. Math., 15, Springer Verlag, New-York, 1991. L.D. MARINI,Two families of Mixed [BDM 851 F . BREZZI,J. DOUGLAS, Finite Element for second order elliptic problems, Numer. Math., 47, (1985), 217-235. The finite volume element [CMM 911 Z. CAI, J. MANDEL,S. MCCORMICK, method for diffusion equations on general triangulations, SIAM J. Numer. Anal., 28,2, 1991, 392-402. [CDH 831 F . CASIER,H. DECONNINCK, C. HIRSCH,A class of central bidiagonal schemes with implicit boundary conditions for the solution of Euler's equations, AIAA-83-0126, 1983.

[CM 861 J . J . CHATTOT,S. MALET,A "box-scheme" for the Euler equations, Lecture Notes in Math., 1270, Springer-Verlag, 1986, 52-63.

246

Finite volumes for complex applications

[Co 921 B. COURBET,Schkmas boite en rkseau triangulaire, Rapport technique 18/3446 EN, (1992), ONERA, unpublished. [Co 911 B. COURBET,Etude d'une famille de schkmas boites & deux points et application & la dynamique des gaz monodimensionnelle, La Recherche Adrospatiale, no 5, 1991, 31-44. [CoC 981 B. COURBET,J . P . CROISILLE, Finite Volume Box Schemes on triangular meshes, Math. Model. and Numer., 32,5, (1998), 631-649. Finite Volume Box Schemes and Mixed Methods, [Cr 991 J-P. CROISILLE, Preprint, Universitk de Metz, 1999. [Du 971 F. DUBOIS, Finite volumes and mixed Petrov-Galerkin finite elements; the unidimensional problem. Preprint 295 du C.N.A.M., 1997.

[EGH 971 R. EYMARD,T. GALLOUET, R. HERBIN,Finite Volume Methods, in Handbook of Numerical Analysis, vol. 5, Ciarlet-Lions eds., (1997). B. STOUFFLET, A class of implicit upwind schemes for Eu[FS 891 L. FEZOUI, ler equations on unstructured grids, Jour. of Comp. Phys., 84, 1989, 174-206. [Ha 891 W. HACKBUSCH, On first and second order box schemes, Computing, 41, 1989, 277-296. [Ke 711 H.B. KELLER,A new difference scheme for parabolic problems, Numerical solutions of partial differential equations, 11, B. Hubbard ed., Academic Press, New-York, 1971, 327-350. Existence, uniqueness and approximation for ge[Ni 821 R . A . NICOLAIDES, neralized saddle point problems, SIAM J. Numer. Anal., 19,2, 1982, 349-357. [RT 771 P . A . RAVIART,J.M. THOMAS,A mixed finite element method for 2nd order elliptic problems, Lecture Notes in Math., 606, SpringerVerlag, 1977, 292-315. [TS 931 T. SCHMIDT,Box scheme; on quadrilateral meshes, Computing, 51 1993, 271-292. A new for[YMAC 991 A . YOUNES,R . MOSE, P. ACKERER,G . CHAVENT, mulation or the Mixed Finite Element Method for solving elliptic and parabolic PDE, Jour. of Comp. Phys., 149, 1999, 148-167.

On nonlinear stability analysis for finite volume schemes, plane wave instability and carbuncle phenomena explanation

M. Abouziarov Research Institute of Mechanics Loba chevskey university, Nizhnii Novgorod, Russia e-mail [email protected]

1. Introduction

Courant restriction of integration time step, which follows from stability analysis of linearised Euler equations, does not provide the stability of some numerical methods for nonlinear problems . Some of these problems and attempt of analysis of the origin of these instabilities were made in [I]. In this report a procedure of the instability analysis for finite volume methods for the nonlinear equations and a method to correct the numerical schemes to avoid these instabilities are presented.

2. Nonlinear stability analysis

Assume for a 2D case a cell number ( i j ) is integrated, where integer indexes i ,j ( X and Y direction respectively ) are related with parameters (i.e. grid, pressure, density, velocities, energy) at the cell centers and half-

248

Finite volumes for complex applications

pressure and velocity perturbations are small, but there can be the density discontinuity. For the calculation process to be stable the perturbation of the new velocity is to be equal to the old one or this velocity perturbation is to be diminished step by step. *

n

I\

.

.

n

u 1J. = u + u , v .ZJ- = v + v , p IJ. . = p + p a n d ~ ' ~ = v + v , , (5) and

Assume:

respectively

and etc.

(values with covers are perturbations of the appropriate values). For the original G o d u n o v ? ~method, using linearised Riemann?~solution with the different left and right densities and sonic velocities, the following simple estimation can be obtained:

A

A similar formula is for U n ~For ~ the . second order modification [3], the formulas

A,j = b'j -

- Pi-l,j

are analogous. For linearised Euler equations, when

, it is

obvious from ( 6 ) that the Courant restriction is enough for stability. But when there is the density discontinuity and the change of the density in the cell during one step is big, to provide the calculation stability the time step is to be stronger restricted o r the scheme is to be reconstructed to correct these additional instability terms; without it even small perturbations of the velocity may to increase. It can be shown that on the back of the shock wave instability estimations are similar (6) but with stronger possible instability. For example, such instability situations are possible on the back of a strong shock wave in a wind tunnel (plane wave instability) o r near the head of the shock wave, generated by the sphere or other body in the supersonic flow ?carbuncle phenomena?. In the ?carbuncle phenomena?, at the beginning of the iterations, the situation is very similar to the plane wave instability case - a strong

Numerical analysis integer indexes show the parameters on the cell boundaries.

Eq.1 represents the mass conservation law for this cell. For simplification (without any restrictions) the grid is assumed to be stationary and regular.

vo,S. Af

are

respectively volume , surface area of a boundary of this cell and integration time step; u and v are velocities for X and Y directions respectively; and the values with the upper indexes indicate the new time step level.

Eq.2 represents the momentum conservation law in Y direction for this cell (similar for X direction). Using the expression for

V, * ( / Y ) j J

=

" J

from e q I . it follows from Eq.2 ,

vo* p* v i j +[<j.-ln - f J . + m ] * ~*ts +

(3)

Eq.4 is a formula for obtaining the velocity on the new time step based on the velocity for the previous one. The formulas for velocity in the X direction and for pressure derived from the energy equation have a similar form. These formulas are nonlinear and follow from the conservation laws. Using different ways for obtaining the fluxes on the cell boundaries, different numerical schemes can be constructed. Let us consider the behavior of (4) for the plane wave in the X direction in the vicinity of the density discontinuity for widely used G o d u n o v ? ~method. Assume that our flow is near to equilibrium in its pressure and velocity and the values of

249

250

Finite volumes for complex applications

shock wave is separated from the body and goes ahead.

3. A possible way of correction

A possible way to correct this instability follows from formula (6): it is necessary to eliminate the perturbations which appear in addition to the standard Courant perturbations. These corrections should have the following logic: corrections are to have higher order accuracy than that of the numerical scheme, they are not to change the approximation error; they are to correct only the instability terms to avoid the increase of the perturbations. One way to make the scheme stable is to use appropriate corrections of instability terms after integration, but in this case it is necessary to analyze the other kind of discontinuity problem more accurate (shock wave) too, which is rather difficult. The second way to eliminate instability terms is to correct such terms in the fluxes [2]. For the momentum equations (3) the correction term is as follows:

-

r n

n

Pi.j-112 - Pij-+1/2

*

+ (w)i,,2.j* ( ; i - l t , -

a.j)

-(P),+,,,

where the values with the covers (respective errors) are to have the second order accuracy in the appropriate directions. For example:

Numerical analysis

The momentum fluxes for the X direction and energy fluxes are corrected in a similar way. With these corrections, the original Godunov method and its second order accuracy modification [3] become stable with standard Courant stability restrictions for those schemes and d o not have the plane wave instability and ?carbuncle phenomena? troubles. In (5) it was assumed that near the pressure and velocity equilibrium state the absolute values of the appropriate perturbations are the same for different cells; it is reasonable because the relative truncation error for all values is the same and depends only on the type of a computer. From our density discontinuity analysis it follows that the velocity and pressure perturbations do not compensate each other after integration in formula (4) because the flux perturbations are proportional to the appropriate densities, which are different. For a shock wave; the situation is more coniplicated - absolute perturbation values are different and are related with the absolute values of the respective fields.

4. Conclusions

Thus, it may be concluded that: in constructing numerical methods and analyzing their stability for computer calculations in the regions of possible big gradients, it is necessary to take into account that only relative truncation error is the same for all numbers in the computer, whereas absolute values of perturbations are proportional to this relative truncation error and to those absolute values.

5. Bibliography

[ I ] J. QUIRK, "A contribution to the great Riemann solver debate," International Journal for numerical methods in fluids 18 (1994), pp. 555-574. [2] ABOUZIAROV M. Nonlinear stability analysis of Euler equations for predictor

corrector schemes. /I Godunov?~method for gas dynamics: Current Applications and Future

25 1

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Finite volumes for complex applications

Developments, A Symposium Honoring S.K. Godunov, May 1-2, 1997, The University of Michigan, Ann Arbor, Michigan [3] ABOUZIAROV M. On accuracy increasing of Godunov's method for nonlinear problems of continuum mechanics. 11 Godunovls method for gas dynamics: Current Applications and Future Developments, A Symposium Honoring S.K. Godunov, May 1-2, 1997, The University of Michigan, Ann Arbor, Michigan

Innovative schemes

A comparison between upwind and multidimensional upwind schemes for unsteady flow

P. Brufau and P. Garcia-Navarro Centro Polite'cnico Superior 5001 5 Zaragoza, Spain

A n ideal scheme for the solution of multidimensional non-linear systems of partial diflerential equations governing fluid motion has not yet be found despite years of research effort and many scientific contributions. I n the context of finite volume approximations based on flow dependent schemes, discussion is opened as to whether upwind or multidimensional upwind schemes are preferable for solving 2D problems. I n this work we consider the use of upwind and multidimensional upwinding techniques for 2 0 shallow water flows, i n particular, to the simulation of dam break flows. The basis of the two numerical methods is stated and the particular adaptation to the shallow water system is described. A s test cases, laboratory experimental data supplied by partners of the Working Group on D a m Break Flow Modelling ( C A D A M ) are used. ABSTRACT

K e y Words: upwind schemes, shallow water, dam break.

1. Introduction

Upwind schemes[l, 91 are accepted as accurate methods for the numerical solution of systems of conservation laws in one dimension. Attempts of applying these techniques in higher dimensions have generally relied on essentially one-dimensional algorithms combined with some form of operator splitting. Indeed, the standard extension of the finite volume methods is still to solve onedimensional Riemann problems created at the interfaces of the finite volumes by the discontinuities in the reconstruction stage of the algorithm. When extended in this way many features of the flow, particularly if they are not aligned with the grid, can be completely misinterpreted by the numerical scheme. Discussion is open as to whether the schemes based on 1D Riemann solvers

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Finite volumes for complex applications

are the most suitable choice for multidimensional calculations because they seem inadequate for capturing 2D flow features. Recently, a class of upwind methods has emerged which attempts to model the equations in a genuinely multidimensional manner[2,3, 5 , 6 ] . These schemes are designed to monitor the average time evolution of the approximation to the solution within a complete grid cell rather than concentrating on the activity at the interfaces. Multidimensional upwind schemes were developed initially for the approximation of steady state solutions of the two-dimensional Euler equations on unstructured triangular grids, although they could be applicable to any system of hyperbolic conservation laws. One such case is given by the shallow water equations[4]. In this paper the performance of an upwind and a multidimensional upwind technique for 2D shallow water flows is described in first order accuracy. In the next sections, the basis of the numerical methods is stated and the application to the simulation of 2D dam break flows is presented. 2. Governing equations

Depth averaging of the free surface flow equations under the shallow water hypothesis leads to a common version of the 2D shallow water equations which, in conservative form is,

au

-+V.(F,G)

at

=H

(1)

with

where U represents the vector of conserved variables (h depth of water, hu and hv unit discharges along the coordinate directions), F and G are the fluxes of the conserved variables across the edges of a control volume. They consist of the convective fluxes together with the hydrostatic pressure gradients. H is the source term. In addition, u and v are the velocities in the x, y directions respectively, g is the acceleration due to the gravity, So,, SOYare the bed slopes and Sf,, Sfy the friction terms in the x, y directions. For the friction term, the Manning equation has been used.

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257

3. Numerical models

3.1 Upwind method

A cell centered finite volume method is formulated for equation (1) over a triangular control volume where the dependent variables of the system are represented as piecewise constants. The integral form of (1) for a fixed area S

and, applying the divergence theorem to the second integral, we obtain

where C is the boundary of the area S , and n is the outward unitary normal vector. Given a computational mesh defined by the cells (volumes) of area Si, where i is the index associated with the centroid of the cell in which the cellwise constant values of U are stored, equation (4) can be represented by

"sdti

+

h,

( F , G ) . n d C = Hisi

where a mesh fixed in time is assumed. The contour integral is approached via a mid-point rule, i.e., a numerical flux is defined at the mid-point of each edge, giving NE

(F,G ) - n d C =

( F , G):& . nwkdCwk k=l

where wr,represents the index of edge k of the cell, N E is the total number of edges in the cell. The vector nwkis the unit outward normal, d e w , is the length of the side, and (F, G ) j , is the numerical flux tensor. The evaluation of the numerical flux in equation (6) is based on the Riemann problem defined by the conditions on the left and right sides of the cell edges. An important feature of the 1D upwind schemes for non-linear systems - is exploited here. This is the definition of the approximated flux of equations jacobian, Ai+;, constructed at the edges of the cells and satisfying special conditions[7]. The 2D numerical upwind flux in equation (6) is obtained by applying the expression of the numerical flux across the interface i of a cell in a 1D domain to each edge wk of the computational cell in a 1D form. The 1D philosophy is followed along the normal direction to the cell walls, making use of the normal numerical fluxes, so that

+

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Finite volumes for complex applications

R and L denote right and left states respectively at the wk edge, ( A ~ L ) represents the approximate Jacobian of the normal flux. As suggested by Roe[7] the matrix iiRL has the same shape as A, being

but is evaluated at an average state[l]. We can now substitute (7) into ( 6 ) , so that (5) \ , can be written as NE d U-i 1 - -(F,G):~ . nw,dC,, Hi dt Si k = l

C

+

which is an ordinary differential equation and can be integrated by standard methods such as a forward Euler time integration procedure. (10) The stability criterion adopted has followed the usual in explicit finite volumes [1] di 6 t <_ min (11)

1

where d,j is the distance between the centroid of the cell i and its neighbours. 3.2. Multidimensionad upwind method

The application of this technique to the shallow water system requires the equations to be written in terms of the conserved variables, so that (1) can be expressed in its non-conservative form as

in which only the homogeneous part has been considered. It is useful to express the equations in terms of the primitive variables V =(h,~,v)~ (13) in a non-conservative way, as

In the conservative formulation, the fluctuation is defined as

innovative schemes

259

We can use the relation between the two sets of variables to define new matrices R and S,

so that

F,

+ G,

= RV,

+ SV,

Provided that the variables V are linear over the cells T , the gradients, V, and V,, are constant, and we can write the fluctuation as 4~ = -ST[fivX f SVy] (I8) with the definitions:

We approximate R,

3 by

S = S(V)

R =( V ) where the averaged variables are simply

Y=

( )=J

hl

+ h2 + h3 + v2 + v3

(u1+u1+u3) Vl

(21)

summing over the nodal values at the vertices of the triangle T . Note that with this definition of R, S we are only approximating equation (18). As a consequence, we lose conservation in the numerical evaluation of the fluctuation. The next step consists on computing the fluctuations and distributing them to the vertices of every cell by means of an advection scheme[3]. This last step requires the description of wave models[8] and the choice of an advection scheme. 4. Numerical results

Results obtained with first order upwind and multidimensional upwind approximations on unstructured Delaunay triangular meshes for the experimental test case proposed by Prof. Zech (Civil Engineering Dpt., UCL Belgium) from the Working Group on Dam Break Flow Modelling are going to be presented. The test combines a square shaped upstream reservoir and a 45O bend channel (see Fig. 1). The flow will be essentially two-dimensional in the reservoir and at the angle between the two straight reaches of the 45' bend channel. Two features of the dam break resulting flow are of special interest: the damping effect of the corner, and the upstream moving hydraulic jump which is formed by reflection at the corner.

260

Finite volumes for complex applications

The initial conditions are water a t rest with the free surface 25cm in the upstream reservoir and lcm water depth in the channel. All boundaries are solid non-slip walls except the outlet which is considered free. The Manning coefficient is nb = 0.0095 for the bed. The number of elements used in the mesh is 15397. Nine gauging points were used in the laboratory t o measure water level in time. Their location is shown in Fig. 1. The measurements a t these stations are compared to the numerical results and displayed in Fig. 2. In Fig. 3 the free surface is described a t time 10s and finally, Fig. 4 represents snapshots of the free surface a t time 18s. In general, the figures indicate a good performance of the two numerical schemes. The arrival times of the main shock fronts is better captured by the upwind method. Some differences are noticeable in P 3 and P4 as the reflected shock front celerity is concerned. This may be attributed to the treatment of the boundary conditions. The great difference between the results obtained with bot
An upwind and a multidimensional upwind scheme for the solution of the 2D shallow water equations has been applied in first order accuracy for dam break modelling. The numerical results have been validated by comparison with experimental data in one test case. Differences on the results obtained with both techniques do not follow a clear tendency and it is difficult to establish the superiority of one over another. In the test case presented, both results fail to reproduce exactly the arrival times of the reflected wave (P2 and P3 gauge points). This suggests that the reflection at the corner may require an improved numerical treatment. Moreover, differences can be noticed a t the free surface plots. Reflected waves from the walls are clearly observed in the multidimensional upwind results and perhaps a good test could be performed taking measures along the channel walls to give a new point of view in the numerical analysis.

6. Bibliography

[I]

BRUFAUP . , An upwind scheme for the 2D shallow water equations with applications , Num. Anal. Rep. 11/97, University of Reading, England (1997).

PI

BRUFAUP . AND GARC~A-NAVARRO P ., Two dimensional dam break flows in unstructured grids, Hydroinformatics98,

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261

Copenhague, Denmark (1998).

DECONINCK H. et al., Multidimensional Upwind Methods for Unstructured Grids, Unst. Grid Met. for Advec. Domin. Flows, AGARD, 787 (1992).

G A R C ~ AAVARRO -N P . et al., Genuinely Multidimensional Upwinding for the 2D Shallow Water Equations, J. Comp. Phys., 121 (1995), p. 79-93. HUBBARDM .E., Multidimensional Upwinding and Grid Adaptation for Conservation Laws, PhD Thesis, University of Reading, England, 1996. PAILLERE H., Multidimensional Upwind Residual Distribution Schemes for the Euler and Navier-Stokes Equations on Unstructured Grids, PhD Thesis, University of Brussels, Belgium, 1995.

ROE P.L., A Basis for Upwind Differencing of the TwoDimensional Unsteady Euler Equations, Num. Met. Fluid Dyn. I1 (1986).

RUDGYARD M . , Multidimensional Wave Decompositions for the Euler Equations, VKI Lecture Series, Comput. Fluid Dynam. (1993). SLEIGHP.A. et al., An Unstructured Finite Volume Algorithm for predicting flow in rivers and estuaries, Comp. and Fluids (1997).

7. Figures

Figure 1: Plane view of the channel.

262

Finite volumes for complex applications

I uur Ip*lmd

Figure 2: Water depth history at points a) P3, b) P4, c) P5 and d) P9.

Figure 3: Free surface a t time t=lOs with upwind (a) and multidimensional upwind schemes (b).

Reformulation of the unstructured staggered mesh method as a classic finite volume method Blair Perot & Xing Zhang University of Massachusetts 219 Engineering Laboratory Amherst. MA 01003

ABSTRACT A generalization of the Harlow & Welch (1965) staggered mesh method to twodimensional unstructured meshes is presented. With certain choices of the interpolation operators, it is shown that this method can be recast as a classic finite volume method using a single set of non-overlapping control volumes and collocated variables. When rhe divergence form of the Navier-Stokes equations are discretized using the unstructured staggered mesh method the resulting equations are equivalent to a classicfinite volume method for the velocity vector. When the rotational form of the Navier-Stokes equations are discretized using fhe unstructured staggered mesh method the resulting equations are equivalent to a classic finite volume method for the vorticiry vector.

Key Words: unstructured, stannered mesh, reformulation, Navier-Stokes equations.

1. Introduction The Cartesian staggered mesh method has a number of mathematical properties that make it a popular choice for simulations of incompressible fluids. In particular. the method does not have spurious 'pressure modes' and does not require stabilization or damping terms to control unphysical small-scale pressure fluctuations. In addition, the method is known to conserve mass, momentum, total energy, kinetic energy and vorticity. The latter two conservation properties are not found in generic control volume approaches and are particularly important in direct and large eddy simulations of turbulence where the cascade of turbulent kinetic energy (or enstrophy) from large to small scales (or vice versa) is critical to the overall predictions of the turbulence behavior.

264

Finite volumes for complex applications

The success of the Cartesian staggered mesh method originally developed by Harlow & Welch [HAR 651 has motivated the search for generalizations of the method to unstructured meshes. While such a generalization is a non-trivial exercise, the unstructured staggered mesh methods of Porsching [AM1 811, and Nicolaides [NIC 931 have demonstrated many of the attractive properties of the Cartesian staggered mesh method by taking advantage of the fact that every unstructured mesh has a locally orthogonal dual mesh - the Voronoi tesselation. Chou [CHO 971 has shown the connection of the unstructured staggered mesh methods to nonconforming finite element methods. In this paper we discuss the direct connection with classic finite volume methods. It is via this connection with classic finite volume methods that the hitherto uninvestigated conservation properties of unstructured staggered mesh methods can be evaluated.

Mesh

- Dual Mesh. Figure 1. Two dimensional unsfructured mesh and the dual Voronoi mesh.

2. Analysis of the Divergence Form

The unstructured staggered mesh discretization is simply a way of forming discrete difference operators. It is actually independent of the equations to which it is applied. Hence, different discretizations of the Navier-Stokes equations are possible depending on which form of the equations are discretized. In this section, we will look at unstructured staggered mesh discretizations of the divergence form of the Navier-Stokes equations.

Discretizations based on the divergence form of the equations are of interest because they are able to discretely conserve momentum. While momentum conservation is a trivial consequence of a classical finite volume method, it is not an obvious trait of staggered mesh methods. This is because the staggered mesh methods only updates the normal velocity components at cell faces, tangential

Innovative schemes

265

velocity components are interpolated not evolved. It will be shown that with certain choices of the interpolation operators, the staggered mesh update of face normal velocities is directly equivalent to a classic finite volume method which updates the velocity vector at cell centers.

2.1 Discretization of the Divergence Form The normal vector at each face is assumed to point from cell C1 to cell C2. At boundary faces the normal vector is assumed to point out of the domain and cell C2 is a virtual cell located at the domain boundary. The discrete equation for the evolution of the normal velocity component is then given by,

cell

la~r*

where cL =$ ZufdAf is a conservative discretization of the convection term cell

faces

evaluated in each cell,

d, = $ Zv(Vu + VuT) .iifA ,

is a conservative

discretization of the diffusion term evaluated in each cell, VL is the volume of each cell, A, is the face area, W, is the distance between neighboring cell circumcenters, and W,' is the distance between the face circumcenter and the cell circumcenter. Note that u = u . n , is the normal velocity component at each cell face and G is the normal velocity component that points out of a particular cell. Similarly ii, is the normal vector pointing out of a particular cell.

2.2 Reformulation as a Classic Control Volume Scheme The reformulation of the divergence form is accomplished by multiplying each evolution equation for the face normal velocity component (Eqn. [ 2 ] ) , by the face normal vector, and summing over all the faces in the computational domain. This results in the following equation,

The goal is then to recast this into a form that looks like a summation over control volume cells. Recognizing that W, = W;, + W;, and that W;, = 0 at

Finite volumes for complex applications

266

boundary faces, and also noting that at boundary faces pcZ= p i , rewritten as a summation over cells. cell

cell

cells

Z Z

+?ec

niWiAf cell

ninfW:Ai cell face\

boundary

[41

z p, z P,A, -fxk ~ , P , A+Y , dc z niniWiAi

cells

=

[3] can be

face

This can be further simplified using three identities derived from Gauss' divergence theorem. Gauss' Divergence Theorem for an arbitrary bounded volume and a vector quantity f is,

is the boundary of the volume with unit normal where C2 is the volume and vector P oriented outwards from the volume. We are actually interested in convex polygonal volumes where Gauss' Theorem simplifies to, cell

..~

fses

If f is a nonzero constant vector then 1;,A, = 0 . If f = (x .a)b where a and b are nonzero constant vectors and x is the position vector with an origin located at the cell circumcenter then it follows f ~ o m [6] that in two-dimensions el1

iaca

C n i n fW,'A

= VcI where I is the identity matrix. Finally, if f = (a.x)u where u

is the velocity vector and a is an arbitrary nonzero constant vector, then Gauss' Theorem gives, cell

The gradient of the position vector is the identity matrix ( x , , = 6, ), and since a is an arbitrary vector

Innovative schemes

267

where G is the outwards normal component of the velocity at the cell faces. This is an exact equation for polygonal volumes. If we assume that the velocity field u is a constant function (a first order approximation), then the second term will be zero and the integrals can be evaluated. In two dimensions, this gives the relation,

The first two identities are geometric and they are exact. The last expression (Eqn. [9]) is really not an identity, it is a first order approximation for the cell velocity vector given the normal velocity components at the cell faces. With these geometric identities and this definition for the cell velocity vector, [4] becomes, ntl

CCII,

Uc

-Uc

n

houndary

+ycLVL= - Z facer

+yd L V L

pfifAf

This equation is true for a collection of cells, but it is also true for a single mesh cell. The preceding analysis makes no distinction as to the number of cells. Applying the previous definitions for c, and d, we can therefore write that

This is true for each mesh cell, and has the form of a classic control volume scheme for the velocity vector in the mesh cells. It is important to note however, that despite the apparent similarity there remains a subtle distinction from classic control volume schemes. In the staggered mesh scheme, the normal velocity component u is the primary unknown and u, is a derived quantity. In classic control volume schemes, u, is the primary velocity unknown and the normal velocity component at faces is derived.

3. Analysis of the Rotational Form In this section, we will look at unstructured staggered mesh discretizations of the rotational form of the Navier-Stokes equations.

where u is the velocity vector, w is the vorticity, pd = p + j u . u is the specific dynamic pressure, a n d v i s the kinematic viscosity. This equation assumes that viscosity is constant, but it is otherwise equivalent to other forms of the incompressible Navier-Stokes equations. Variable viscosity diffusion can be still be

268

Finite volumes for complex applications

represented in rotational form but the extra term (involving second derivatives of viscosity) complicates the analysis unnecessarily. This particular form of the Navier-Stokes equations is of interest because it N2 appears to be inherently suited to the staggered mesh discretization. The classic t staggered mesh method can be rearranged to look like a discretization of [12]. It C 1~ will be shown that in two dimensions the staggered N1 mesh update of face normal velocities is directly Figure 2. Notation for a cell face in relation equivalent to a classic finite to neinhborinn cells and nodes. volume method which updates the vorticity at nodes and where the control volumes are the dual mesh Veronoi polyhedra. ""

3.1 Discretization of the Rotational Form Using the rotational form of the Navier-Stokes equations, the normal component of the face velocity is discretized as,

where a, is the vorticity at a node in the direction out of the two-dimensional plane. The face tangential points from node N1 to node N2 and is oriented 90 degrees counterclockwise to the face normal vector. The tangential velocity at the nodes in the convection term is given by v n = u, . t, .

3.2 Reformulation as a Control Volume for Vorticity The reformulation of the rotational form is accomplished by dividing each normal velocity evolution equation (Eqn. [13]) by the face area and then multiplying by -1 if the face normal points clockwise with respect to the node in question, and finally summing over all the faces touching a specific node. The result will be shown to be a control volume equation for the vorticity. In mathematical notation we start with the following equation,

Innovative schemes node

U"'

faces

node f,

-Un

I W rAt

269

- C +(wn0vno+w",~"i)W,

node

node

faces

faces

where the normal vector at each face has been chosen to point in a direction counterclockwise with respect to the node in question. In addition, node no is the node around which the summation is occurring and node ni is the other node adjoining that face. In this case, we use a discrete version of Stokes Curl Theorem to simplify the equations. Stokes theorem says that for an arbitrary bounded surface and vector quantity f,

where S is the surface with normal z, aS is the boundary. of the surface, and the integration takes place in a counterclockwise direction around the boundary with respect to the face normal. We are actually interested in the planar polygonal Veronio regions surrounding each node, it which case Stokes Theorem simplifies to node

If we let f equal the velocity vector and make the first order assumption that the velocity is constant in the Veronio cell then we obtain, nude ta~es

w,,A,, =

c

UW,

[171

where A nis the area of the Veronio cell surrounding the node, and the face normal vectors are assumed to point in a counterclockwise direction around the node. In conjunction with [17] it is clear that for interior nodes, the pressure term is identicaily zero. The net result is that [14] can be written as, node

"*I

On

An

- On

At

node

+ 'y (vw) I, W, = '5(vniwn,-vnowno)Wi/ A ,

270

Finite volumes for complex applications

z,

where the face vorticity flux is given by ( v o ) I, = +(w,,u,, + o,,u,, ) . and i, points out of the Veronio cell. This is a discrete version of the continuous two-dimensional vorticity evolution equation,

Again, it is important to note that despite the apparent similarity there remains a subtle distinction from classic control volume schemes. In the staggered mesh scheme, the normal velocity component u is the primary unknown and on is the derived quantity. In a classic control volume schemes, w, would be the primary unknown. S o the staggered mesh scheme differs from a standard vorticitystreamfunction or vorticity-velocity formulation in the fact that (often complex) boundary conditions on the vorticity are not required.

4. Discussion The primary result of the current work is that staggered mesh methods are not just control volume methods applied on staggered control volumes, but are directly equivalent to classic collocated control volume methods. It was shown that unstructured staggered mesh discretizations of the divergence form of the NavierStokes equations are equivalent to classic control volume method for the velocity vector in mesh cells. Likewise, unstructured staggered mesh discretizations of the rotational form of the Navier-Stokes equations are equivalent to classic control volume method for the vorticity vector at mesh nodes (in Veronio cells). These equivalencies imply that the method possesses certain conservation properties.

References [HAR 651

HARLOW, F. H. & WELCH,J. E., Numerical calculations of time dependent viscous incompressible flow of fluid with a free surface, Phys. Fluids, 8 , 1965, p. 2182.

[NIC 931

NICOLAIDES, R. A., The covolume approach to compuling incompressible flow, Incompressible Computational Fluid Dynamics, M . D. Gunzburger & R. A. Nicolaides, eds., Cambridge University Press, 1993, p. 295-234.

[AM1 811

AMIT, R., HALL,C. A. & PORSHING, T. A., An Application of Network Theory to the Solution of Implicit Navier-Stokes Difference Equations. J. Comput. Phys. 40, 1981, p. 183.

[CHO 971

CHOU,S. H., Analysis and convergence of a covolume method for generalized Stokes problem, Math. of Comput. 66 (217), 1997, p. 85.

A mixed FE - FV algorithm in non-linear solid dynamics

Serguei'V. Potapov EDI.; Research and Development Division Acoustics & Vibration Mechanics Branch I , avenue d u Gknkral-de-Gaulle, B.l? 408 92141 Clamart Cedex France

ABSTRACT This article presents an extension to the transient non-linear solid dynamics of a mixed Finite Element-Finite Volume algorithm initially proposed for simulation of hydrodynamic problems with compressible~uids.This algorithm is based on an Arbitrary Lagrangian Eulerian (ALE)formulation of motion. By using a fractional step method, the ALE problem is divided in two phases : a Lagrangian one and a transport one. The Lagrangian phase dealing with spatially symmetric terms is approximated using a conventional Galerkin Finite Element formulation. To deal with the transport phase, whose operator is not symmetric, the Finite Volume formulation is applied. Using two kinds ofjnite volume cells warrants a compatibility of spatial approximations of the Lugrangian and transport phases and allows to perform correctly the transport of all unknown variables.

Key Words : non-linear solid dynamics, finite element,finite volume, explicit scheme

1. Introduction The ALE formulation is well established in fluid dynamics where it is applied to the transient analysis of flows on moving meshes. The capacity of the ALE kinematics to manage strong fluid flows on dynamically deforming meshes is attractive to use in the non-linear solid mechanics dealing with large elastic-plastic deformations. However, this extension is not trivial because the stress tensor of solids is not only function of instantaneous quantities (as for fluids) but depends on the specific history of each material point. In order to evaluate the stress state on a current step, the constitutive equation must be integrated in time. Since in the ALE formulation

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Finite volumes for complex applications

the computational grid and the material move independently, the quadrature points at which the stresses are evaluated coincide throughout the deformation process with different material points, having in general different deformation histories. Thus, the stress transport procedure is needed to overcome the numerical difficulties due to relative motion between grid nodes and material particles. Furthermore, the same update procedure must be applied to each internal variable of the elasto-plastic analysis, e.g. to the yield stress in isotropic hardening or to the back stress in kinematic hardening. Direct time integration of the ALE equations is not easy because they contain the spatial derivative terms corresponding to very different physical phenomena. So, we use a fractional step method to integrate in time such equations. The ALE problem of each time step is split into two phases. The first one is so called Lagrangian phase, in which the computational grid is assumed to follow the material points, what cancels all transport terms. The second phase is a convective one in which only the transport terms are taken into account. The resolution of ALE problems on unstructured grids is generally made via the weak formulation suited by the finite element approximation. However, the standard Galerkin finite element approach produces spurious numerical oscillations when applied to non-symmetric operator of the transport phase. Therefore, the upwind technique must be used to overcome these instabilities. In [HUE 951, two distinct strategies, borrowed from the fluid dynamics experience, are presented. A Lax-Wendroff scheme and a Godunov algorithm are implemented in finite element context to perform the transport of stresses and stress-related variables. To stabilise the momentum equation some ad-hoc donor-element upwind technique is used. Here, we use the finite volume technique to perform, in unified and conservative way, the convection of all unknown variables [POT 971.

2. A mixed approach using Finite Element and Finite Volume formulations First, we rewrite the standard ALE equations under a conservative way reinforcing the momentum, the energy and the solid time dependent constitutive equations by the mass conservation equation. Then, the ALE system is split into two sub-systems. All symmetric terms like local strains of the continuum and diffusion terms are grouped in the first sub-system which corresponds to the Lagrangian phase. The second subsystem contains only purely convective terms describing the motion of the material through the ALE computational grid. It corresponds to the transport phase. Such a splitting is very general and allows to consider both geometric and material nonlinearities related to the problem. Furthermore, it enables us to choose for each phase the appropriate integral formulation and method of resolution.

2.1 F i n i t e Element formulation of the Lagrangian phase The equations of the Lagrangian phase possess a self-adjoint (symmetric) spatial operator. To guarantee the best approximation, we apply to them a conventional Galerkin integral formulation with symmetric weigh functions wr,, , I;Zk. Due to the fact that all equations of the ALE formulation contain convective terms, we must

w,,

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consider the mass density p, the material velocities vj, the internal energy eint and Cauchy stresses aij as main variables of the Lagrangian phase and solve together all following equations :

mass equation

monlentum equation

internal energy equation

constitutive law

In the constitutive relations, Cijklis the material response tensor which relates any frame-invariant rate of the Cauchy stress to the velocity strain tensor Dij ; the tensor Sijklacting on the spin tensor w k l assures the objectivity of the Cauchy stress tensor. It should be noted that pressure gradients describing the propagation phenomenon (acoustic waves) and evaluated usually in fluid dynamics algorithms by approximate Riemann solver together with pure convection fluxes are included here to the Lagrangian phase. The reason for it is that in non-linear solid materials, the propagation phenomenon is more complex then in Newtonian fluids because of coupling of constitutive equations. Thus, it is problematic to construct a very general algorithm based on the approximate Riemann solver capable to deal with any non-linear solid material. A finite element formalism is then used. The computational domain R C R3 is approximated by a discretisation Ph using tetrahedrons or brick elements Pe E Ph. The following discrete spaces are also introduced :

Uh

={

~h

I ~h

E

[La(SZ)]yuh Ipe=

U,

= const , VPe E Ph }

where PI is the space of polynomials in three variables and of degree 1 ; v h is the approximated value of the velocity on the discretization Ph ; uh is the m-dimensional vector containing approximate values of p, eint,stresses and internal variables.

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2.2 Finite Volume formulation for the transport phase After splitting, the sub-system of the transport phase can be written in the following conservative vector form :

with

V=

peint pui j

) [ Fk=

~(vk vk) pvi(vk-ik)

peht(vk - v k ) paij(vk - c k )

)

(i,j,k=l,2,~)

where V E Rmx]O,tf,,,l] -+Rm,F ( V ) E Rm + Rm. The spatial operator of the transport phase sub-system is non-symmetrical. Therefore, an upwind-like procedure must be applied . Dealing with unstructured grids, we use the finite volume method because of its ability to solve hyperbolic problems in a conservative form using flux computations at the cells boundaries. The direct application of a finite volume technique with only one kind of cells (like cell or node centred cells) is not optimal for our ALE algorithm. On one hand, the difficulty lies in the way the physical variables are represented. In the Lagrangian phase some of them are represented with a constant approximation per element like stresses whereas others have a linear approximation such as velocities and displacements. Therefore, a complete use of a cell-centred technique in the transport phase requires a double averaging operation to pass from the nodal representation to the constant per element one and vice-versa, which gives too much diffusion. On the other hand, the necessity to govern the motion of the deformable computational grid does not enable us to use the same spatial location for velocities and element-centred variables, like used in conventional Computational Fluid Dynamics (CFD) on fixed grids. It is due to the fact that the ALE grid rezoning techniques usually deal with nodes attached quantities (for instance, when solving the Laplace equation to regularise the ALE grid). In order to avoid such difficulties, we use two kinds of cells (cell and node centred cells) in a combined finite volume algorithm of the transport phase. The momentum is localised on node-centred cells built around finite element nodes of the discretisation Ph,whereas the other conservative quantities are integrated using cells coinciding with the finite element mesh P,. To approximate the momentum, the third discrete space is introduced :

a.

where CI are node centred cells forming an other covering of the domain A variational formulation of the transport phase can be written in general form as follows : Find V h that satisfies :

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Choosing whin an appropriate way in the spaces Uh or W h and using Green's formula, we can rewrite (2) on a local level as follows : momentum equation

with VI =

(

Pvi

),

F l k

=(

pui (uk - 6 k )

) , on the node-centred covering and

other quantities

integrated using cell-centred cells coinciding with the finite element mesh P,. Such integral forms present an advantage to be compatible with the spatial approximation adopted in the Lagrangian phase. Moreover, they allow to deal with moving and deformable grids. To evaluate the convective flux integrals at the interfaces between neighbour polygonal cells, we suit the standard procedures described by many authors (see for example [NKO 941). However, due to the original split technique presented above, the construction of numerical convective fluxes is simpler than in standard CFD calculations. Indeed, the solution of the Riemann problem is quite simple because the transport phase system is an hyperbolic degenerated system. Thus the Roe scheme based on characteristic variables cannot be applied. Since the upwind direction is uniquely determined by the quantity (v-8).n denoting the scalar product between a convective velocity across the cell boundary and an outward normal to this boundary, we can use the well known full upwind scheme which respects the flow direction.

3. Numerical examples The efficiency of the mixed convection technique presented above has been evaluated on two solid dynamics problems dealing with elasto-plastic material.

3.1 S t r e s s w a v e p r o p a g a t i o n p ~ o b l e m The first example is a 1D elastic wave propagation problem first considered in [LIU 861. A stress wave is generated in a long rod by a pressure pulse applied at one end starting at t = 0. The rod is discretised in 400 constant-width elements over a length sufficient to avoid wave interactions with the unloaded end of the grid during the whole simulation. The ALE grid is kept fixed until t = 24 see, then suddenly moved with a positive constant velocity until the final time t = 32 sec. The definition of the problem is following :

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To provide a severe test of the proposed transport algorithm, the grid velocity is taken equal to as much as one fourth of the sound speed (c = I), whereas the material velocity (of the order of 0.01) is almost negligible compared with assumed grid velocity (v = 0.25). The analytical and numerical solutions of this problem are superimposed in the Figure 1. Only the spatial interval 24 5 z 5 34 is shown in order to closely examine the wave fronts. As may be observed, the implemented transport technique captures well the discontinuities of the analytical solution. Performing correctly the convection of all main variables, the algorithm introduces no high frequency damping because no average operation is used. That provides the good conservation properties of the presented algorithm. We can also note that the algorithm is almost insensitive to the grid motion because very close results are obtained with the fixed grid. To reduce numerical oscillations due to the centred representation of the propagation phenomenon in the Lagrangian phase the artificial damping is used.

x Irn)

r (m)

Figure 1. Liu's stress wave propagation test, a,, profile

3.2 T a y l o r bar i m p a c t p r o b l e m To validate our mixed FE-FV algorithm in multidimensional case we consider a non-linear solid mechanics problem of a cylinder impact against a rigid wall known as Taylor bar impact problem. This test is typically used to validate fast-transient dy-

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narnics computer codes because the final length and shape of the cylinder are very sensitive to the constitutive behaviour of its material. The geometrical and material data of the problem are shown in the Figure 2. The cylinder impacts a rigid, frictionless wall with an initial velocity of 227 m/s. The material is supposed elasto-plastic with isotropic hardening. The final time of the simulation is 80ps corresponding to a moment when all kinetic energy of cylinder is consumed by the plastic deformation process. As the original problem is axisymmetric we discretise only a quarter of the

Figure 2. Taylor cylinder bar impact cylinder using 1050 hexahedrons (with 50 elements in the length). Because of a large elasto-plastic material flow this test is particularly severe for the purely Lagrangian codes. Due to a fast impact velocity, the computational grid, when attached to the material particles, undergoes so strong local compression that element volumes become negative and calculations numerically "explode". In order to achieve the Lagrangian calculation we use a coarse grid near the impact zone. However, when we use the ALE formulation with a regularisation condition, the computational grid is automatically maintained uniform during all calculation. The Figure 3 shows the evolution of the computational grid in Lagrangian and ALE cases. In spite of a great difference between the grid and material velocities in ALE case, involving a high non uniform convective velocity, the algorithm manages successfully with the transport phase. The values of the final height Hf and radius R of the cylinder obtained by our mixed transport technique are in good agreement with the results computed by various 2D and 3D codes, as can be seen in the following table.

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Finite volumes for complex applications

Figure 3. Comparison of the Lagrangien (left) and ALE (right) calculations

4. Conclusion A mixed ALE algorithm using Finite Element and Finite Volume techniques has been presented and tested on solid mechanics problems with non-linear material behaviour. One can note that this algorithm initially proposed for simulation of hydrodynamic problems dealing with compressible fluids is able to treat fast solid dynamic problems. Capable of dealing separately with compressible fluids and nonlinear solids, the presented approach will be applied to the numerical study of strong fluid-structure interaction problems.

Bibliography [HUE 951 A.HUEWA,ECASADEI and J.DONEA, ALE Stress Update in Transient Plasticity Problems, 4th International Conference on Computational Plasticity - Fun damentals and Applications, Barcelona, 3-6 April, 1995. [POT 971 S.POTAPOV, A Fast Dynamics ALE algorithm based on a mixed Finite Element Finite Volume approach, Ph.D. Thesis, Ecole Centrale Paris, 1997. [NKO 941 B.NKONGA and H.GuILLARD, Godunov type method on non-structuredmeshesfor three-dimensional moving boundaryproblems, Comp. Meths. Appl. Mech. Engrg. 113, 1994, p.183-204. [HAL 861 J.O.HALLQUIST and D.J.BENsoN,DYNA-3D :User's Manual, Univ. Of California, Lawrence Livermore National Laboratory, Report UCID-19592, 1986. [LIU 861 W.K.LIU,T.BELYTSCHKO and H.CHANG, An Arbitrary Lagrangian Eulerian Finite Element Methodfor Path-Dependent Materials, Comp. Meths. Appl. Mech. Engrg. 58, 1986,p.227-246. J.DONEA and A.HUERTA, Arbitrary Lagrangian Euterian Finite [CAS 951 ECASADEI, Elements in Non-Linear Fast Transient ConrinurunMechanics, JRC, European commission, Report EUR 16327 EN.

An Euler Code that can compute Potential Flow

Mani Rad and Philip Roe Department of Aerospace Engineering The University of Michigan Ann Arbor, Michigan 48109-2140 U.S.A. Centre pour Mathe'matiques et leurs Applications Ecole Normale Supe'rieure de Cachan, France

ABSTRACT A n e w approach is t a k e n t o the computation of the elliptic part of the Euler equations. I n each cell of a n unstructured triangular grid, o n which the solution is stored at the vertices, the residual is decomposed i n t o purely elliptic and purely hyperbolic contributions. T h e elliptic part is minimised i n a n o r m suggested b y a n earlier analysis of linearized potential flow. T h e minimit.ation is carried o u t by projecting the direction of steepest descent i n t o a surface o n which both the entropy and enthalpy are constant. W h e n the method is applied t o subcritical flows for which potential solutions should be obtained, the enthalpy is constant t o machine zero, and the entropy is cons t a n t t o a n extremely high degree of accumcy. I i e y Words: Euler equations, potential flow, fluctuation splitting.

1.

Introduction

This paper contributes further t o the development of computational methods for the Euler equations (and eventually for other problems that share their structure) t h a t reflect genuinely multidimensional physics. These methods aim t o retain the benefits of a physical aproach t o strongly discontirluous flows while avoiding the defects of upwind methods applied t o almost incompressible flow. There has recently been considerable success in extending compressible flow

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Finite volumes for complex applications

codes to the low Mach number limit by means of matrix preconditioning. The present paper takes the approach of equation decomposition, which is a very closely related concept. We continue with the analysis in [ROE 99-11 where we showed that the decomposition of the residual into elliptic and hyperbolic parts, and the update procedure are sufficiently simple that explicit formulae can be given for them. This enables the singularities of the procedure, near sonic or stagnation point,s, to be displayed and percieved harmless. In that paper the update for the elliptic part was performed by a straightforward method of steepest descent. Here we constrain the descent direction so that the elliptic part of the update does not affect the entropy or the enthalpy variables. In this way our Euler code is able to compute with very high precision potential flows for which those variables should be exactly constant.

2.

Forms of t h e Euler Equations

Different aspects of the Euler equations are most readily expressed by choosing different sets of unknowns. For computing compressible flow, the most fundamental choice is probably the set of conserved quantities

since a weak solution of this form of the equations captures shocks that satisfy the Rankine-Hugoniot conditions. Associated with these variables is the flux tensor F = (pv, pv @ v p i , v ( f ~p)It (2) There is computational convenience in the parameter vector

+

+

where h is the total specific enthalpy (E p ) / p , on account of the property that all components of u , F are simply bilinear in terms of z. This allows the having conservation properties, both in the construction of local 1inear:zations one-dimensional [ROE 811 and multi-dimensional [DEC 931 cases. Also important, at least to the present approach, are what have been called [HAY 631 the natural variables

where S is entropy, since it is uniquely in these variables that the Euler equations can be decoupled into maximally independent subsystems [HAY 63][ROE 96-l][ROE 96-31. This enables the essentially different hyperbolic and elliptic behaviour t o be computed independently with the minimum of "crosstalk". In this paper we are particularly concerned with the solution of the elliptic subsystem, which contains only the derivatives of pressure and flow angle.

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3.

281

Fluctuation Splitting

In this approach [ROE 94][DEC 951, the computational domain is divided into elements in an unstructured way, with the unknowns stored at its vertices. Iteration toward a steady solution from an in intial guess takes place by computing an average residual for each cell (the fluctuation dT in element T), and then changing the current solution at each node of the cell T by an amount proportional to dT;

Suj = ~ c r T 4 ~ (if node j belongs to cell T)

(5)

where w is a relaxation factor and the weight a; is a matrix t o be determined. Our aim is t o create a method of this kind that exploits the different properties of the different sets of unknowns. We represent the solution in terms of the parameter vector, and find the conservative residual by integrating over the cell (of area A ~ )

where the matrices C = a g / a z and C = a F / d z are locally constant because =Y of the quadratic property [DEC 931. Our aim is then to reduce this residual in some suitable norm. The norm is chosen by splitting the residual into its elliptic and hyperbolic parts, which is uniquely possible in the natural variables [HAY 63][ROE 96-11

where s, n are streamline and normal coordinates. The first two equations represent the elliptic part and involve only the two unknowns of pressure and flow direction; the second pair are the hyperbolic part. In [Roe 991 the elliptic part was solved by minimising that component of the residual, whereas the hyperbolic part was solved as a pair of scalar advection equations. In this paper the elliptic part of the update is constrained not to interfere with the hyperbolic part and vice versa.

4.

The Update Matrix

This gives a convenient way t o think about Fluctuation-Splitting schemes. Let the initial state of a cell (in two dimensions) whose vertices are (a, b, c) be

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Finite volumes for complex applications

represented by the twelve scalar quantities

where w is whatever quantity we have decided to store at the vertices. The update can be represented as a matrix multiplication

where U is some matrix that is constant within the cell and w is a relaxation constant. For example, suppose we intend t o update the solution by a steepest-descent minimisation of

where Q is some symmetric matrix representing a local norm. If we write Q = PtP this is equivalent t o an L2 minimization of P d T , which is some weighted combination of the residuals. We will treat dT as a linear function R ~ W of the ~ vertex values because it arises from a linear process to find the local derivatives, followed by multiplication by matrices that are frozen during the update. In general R is a rectangular matrix formed from three 4 x 4 blocks (see the examples below). Hence the quantity to be minimized is

The gradient of this is

and so the update procedure within each triangle

where w is some global constant, implements the required procedure. If we had chosen t o store the natural variables x it is easy to show that the four residuals 4, in (7) are given by

where, using standard methods t o evaluate the derivatives,

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283

+

with unl = [ u ( A , ) , - v ( A ~ ) and ~ ] U S ] = [ v ( A y ) j u ( A z ) , ] with (A,),) the vector representing the side opposite vertex j . Thus s,n now represent directions along and normal t o an edge. The first two components of this vector comprise the elliptic part of the problem, if M < 1, so the quantity to be minimized is

R j x j where D = diag(1,

m. 0.0)

j=a,b,c where the relative weighting of the continuity and vorticity residuals follows [ROE 96-31, Then the update matrix is

Since in fact we are storing the parameter vector z we must estimate the quantity to be minimised as

and the update becomes

This was the process studied in [ROE 99-11, merely as a first step. It has the disadvantage that minimising the elliptic part of the residuals changes the convected quantities, entropy and enthalpy. We replace it here by a constrained minimization, in which the elliptic part of the residual is reduced as quickly as possible subject to the constraint that this part of the procedure should not change the convected quantities.

5.

Constrained Update Procedure Following [ROE 991 the unconstrained update matrix can be written

where from now on the matrices R incorporate the change of variable dx/dz For example Ra is the matrix

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Finite volumes for complex applications

where

and

After exploiting some coincidences each block of the update matrix becomes

Since we intend complete decoupling a t the update level between the elliptic and hyperbolic parts, the elliptic part of the residual should only contribute to changes in pressure and flow angle and not to entropy and enthalpy. Conversely the entropy and enthalpy residuals should change only the entropy and enthalpy. For each natural variable, the direction of a vector along which only that variable changes is

An arbitrary change in the parameter vector dz, coming from the unconstrained minimisation can be expressed in this basis by solving dz, = P a where cu = ( a s ,ah, a p la g ) tis a vector of coefficients and P = [rs r h r , r e ] is the direction matrix. T h e constrained changes due to the elliptic part are then found by supressing c r ~ , bThis is to say,

dz, = P(P-ldz,)

(13)

where the overbar indicates taking the last two components. The constrained update matrix is therefore

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285

and carrying out the matrix multiplications reveals

The unconstrained steepest descent has the property that changes due t o any triangle sum to zero, thereby ensuring conservation. To preserve this property it is necessary t o prqject each triangle residual onto some local set of directions P and then use these for the changes induced by that triangle at each of its nodes. Close t o M = 0 these directions degenerate; r s , rh and rp become nearly parallel. Under these circumstances we found that the code would not converge. To achieve convergence we needed to use a nonconservative version in which the changes at each node were projected onto the allowable directions. However, at higher Mach numbers the conservative technique worked well.

6.

Computational Examples

The first figure presented is a potential flow calculation on a symmetrical airfoil at zero incidence, where the least-squares method is used only in the subsonic region. Where a supercritical region exists, the hyperbolic part of the problem is handled using the PSI advection scheme. No special procedure was used to match the calculations across the sonic line or across the shock. The last two figures show Euler computations on a cylinder in a subcritical case ( M , = 0.35) and in the incompressible limit ( M , = 0.01) using the residual decomposition scheme described in this paper. The elliptic part of the problem was handled using the constrained least squares scheme as presented in sections 3,4 and 5 while the LDA upwind advection scheme was used on the hyperbolic part. Results shown are for a grid consisting of about 3544 cells and it can be seen that a very small amount of spurious entropy is generated very close to the cylinder wall and especially in the stagnation regions. The rate of entropy generation is measured by

for and is measured to be 1.0 x 10W9for the incompressible case and 1.0x the subcritical case. Not shown in the figures but of certain importance is that

286

Finite volumes for complex applications

enthalpy stays everywhere constant and equal t o its initial farfield value. These results reveals that the projection ideas discussed in section 5 are completely successful for the enthalpy and satisfying for the entropy. In any case, it seems that the present formulation provides a solid framework for treating sensitive regions around the wall boundary. The accuracy of the scheme was measured by comparing the numerical solution to the analytical solution in the incompressible case. Error is defined as ( v is velocity vector)

for a coarse grid of 860 cells and 2.3 x and was measured t o be 9.0 x for a finer grid of 3544 cells. In other words, the scheme's convergence behavior is very close t o second order.

7.

Concluding Remarks

The present contribution is still not yet intended to provide a practical recipe for solving the Euler equations. The main defect is that steepest descent is, by itself and even with the improvements derived here, a very slow way to solve nonlinear equations. And applied to a linear elliptic system it amounts only to a point Jacobi relaxation. However, it forms the starting point for many other (mainly Newton-like) methods that are extremely efficient, and which have the same fixed points. However, the objective here is entirely to find an iterate that has a good fixed point, not one that reaches some fixed point quickly. It also remains t o be proved that an analogous method will work in three dimensions. There is no problem formally. All the formulae given here extend straightforwardly. The issue is whether the streamwise vorticity that becomes coupled to the potential flow in three dimensions [ROE 96-21 will be well captured. It is significant that discretisations exist [ROE 99-21 for which correct growth of the discrete vorticity is automatic, in the sense that a discrete Kelvin Theorem is guaranteed. Future work will determine whether this is possible.

8.

References

H., ROE, P. L., STRUIJS,R. J., CA multidimensional gen[DEC 931 DECONINCK, eralisation of Roe's flux difference splitter for the Euler equations >, Computers and Fluids, 2 2 , p215, 1993.

[DEC 951 DECONINCK, H, PAILL~RRE, H., ROE, P. L.,
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[HAY 631 HAYES,W , PROBSTEIN, R. F., Hypersonic Flow Theory, Academic Press, 1963. [ROE 811 ROE, P . L., <Approximate Riemann solvers, parameter vectors and difference schemes w , J. Comput. Phys., 43, pp357-372, 1981. [ROE 941 ROE, P. L., <Multidimensional upwinding, motivation and concepts w , von Karman Institute Lecture Series 1994-04 [ROE 96-11 ROE, P . L., MESAROS,L. M., <Solving steady mixed conservation laws by elliptic/hyperbolic splitting w , 13th International Conference on Numerical Methods in Fluid Dynamics, Monterey, July, 1996. [ROE 96-21 ROE, P. L., TURKELE.,
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Finite volumes for complex applications

Figure 1. Subsonic ( M , = 0.36) and transonic (Mm = 0.85) N A C A 0012, Mach contours.

Figure 2. Cylinder pow i n the incompressible limit ( M , = 0.01), pressure and entropy contours. Euler soluta'on.

Figure 3. Cylinder pow in the subcritical range ( M , = 0.35), Mach and entropy contours. Evler solution.

Finite volume evolution Galerkin methods for multidimensional hyperbolic problems M. LukGovA- M e d ~ i d o v & K. l ~ ~W. , Morton2, G. Warneckel

Institut fiir Analysis und Xumerik, Otto-von- Guericke- lTniversitit Magdeburg, PSF 4120, 39 106 Magdeburg, Germany, 2Department of Mathematical Sciences, University of Bath, Bath B A 2 7AY, United Kingdom (also Oxford University Computing Laboratory), Institute of .Mathematics, Faculty of Mechanical Engineering, Technical University Brno, Technicka' 2, 61639 Brno, Czech Republic

The finite volume evolution Galerkin method couples a finite volume formulation with an approximate evolution Galerkin operator, which takes into account all of the infinitely many directions of propagation of bicharacteristics for multidimensional systems. Piecewise linear recoztery yields second order accuracy even with the first order approximate evolution operator. Numerical comparisons of the evolution Galerkin schemes with the commonly used finite volume methods for the wave equation system and for the Euler equations are presented.

ABSTRACT

It'ey Words: genuinely multidimensional schemes, hyperbolic systems, wave equation, Euler equations, evolution Galerkin schemes, finite volume methods

It is our belief that the most satisfying methods for approximating evolutionary PDE's are based on approximating the corresponding evolutionary operator. In order to construct a genuinely multidimensional numerical scheme for hyperbolic conservation laws all of the infinitely many directions of propagation of bicharacteristics have to be taken into account.

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Finite volumes for complex applications

This is the main idea of the evolution Galerkin (EG) methods, which evolve the initial data using the bicharacteristic cone and then project them onto a finite element space. In the recent paper Luk6Eov6, Morton and Warnecke [LMW 991 three new first order evolution Galerkin schemes for a system of hyperbolic equations, and particularly for the wave equation system are derived and analysed. It is shown that the evolution Galerkin scheme (denoted in the paper [LMW 991 as the EG3 method), which is based on the general theory of bicharacteristics for hyperbolic conservation laws gives the most accurate numerical scheme. This is a base for the second order scheme, which will be constructed using a piecewise linear recovery. In the recent years the most commonly used numerical shemes for hyperbolic conservation laws were the finite volume methods, which are based on a type of directional splitting and on an approximate solution of one-dimensional Riemann problems. Their popularity is particularly due to the simplicity of their formulation as well as implementation. However, it is a known fact that these methods can produce structural deficiences in the solution for some special multidimensional problems (see, e.g., [FEY 921, [LEV 971, [LMW 981, [LMW 991). The finite volume evolution Galerkin method combines advantages of both approaches: the simplicity of the finite volume formulation and the multidimensionality of the evolution Galerkin schemes. 2. Finite volume evolution Galerkin methods

Consider a general hyperbolic system in d space dimensions

where & = E k ( U ) , k = 1,.. .,d represent given flux functions and the un= ( u l , . . ., ~1,)~ E Em.Let us denote by E ( s ) : known functions are ( ~ I ~ ( R -+~ )( )H~~ ( I R ~the ) ) exact ~ evolution operator associated with a time step s for the system ( I ) , i.e.

U(-, t

-

+ s ) = E ( s ) U ( - ,t).

(2)

We suppose that S i is a finite element space consisting of piecewise polynomials of order p 0. Let En be an approximation in the space S[ t o the exact solution U(.,t,) a t a time t , > 0 and take E, : S;I -+ ( H ~ ( ( I Rt o~ be ) ) a~ suitable approximation t o the exact evolution operator E ( T ) ,r 2 0. We denote by Rh : S: --+ Si a reconstruction operator, r > p 0 . In the present paper we

>

>

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291

shall limit our con~iderat~ion to cases of co~lst~ant time step At, i.e. t , = n A t , and of a uniform mesh consisting of d-dimensional cubes with a uniform mesh size h,. Definition 1. Starting from some initial valtre (1' at time t = 0 , the finite volume evolution Galerkin method ( F V E G ) is reczlrsirlely defined by nteans of

where the central di'erence v ( x + h / 2 ) - v(x - h / 2 ) is denoted by 6,v(x) and SXk&(LJn+*) represents an approximation to the edge flu2 difference. The cell boundary zlalue is evolved using the approximate evolution operator E, to t, r and averaged over 0 r 5 A t and along the cell boundary, i.e.

+

un+*

<

where xij is the characteristic function of anij. There are several advantages to this formulation. The most important is that the first order accurate approximation E, to the evolution operator E ( r ) yields an overall second order update from Un to To obtain t,his second order approximation in the discrete scheme it is only necessary to carry out a recovery stage a t each time level t o generate a piecewise linear approximation - = RhUn E S i , from the piecewise constant Un E SE, to feed into the calculation of the fluxes. In the next section we illustrate this procedure for the wave equation system in two space dimensions.

un+'.

on

3. Wave equation system

The wave equation can be written down as a first order hyperbolic syst,enl

+

with the unknown functions 4, u, v. Consider a cone with the apex P = (x, y, t A t ) and the base points Q = Q(8) = (XSCAS cos 8, y+cAt sin 0, t ) parametrized . by P' = (x, y, t ) the center of the base of the by the angle 0 E [O, 2 ~ 1 Denote cone. The lines from Q(8) to P generating the mantle of the so-called bicharacteristic cone are called bicharacteristics, see, e.g., [LMW 991 for more details. Using the theory of bicharacteristics it can be shown that the solution (4,u, v ) a t the point P is determined by its values on the base as well as on the mantle of the characteristic cone and the exact evolution formulae can be derived. In the

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Finite volumes for complex applications

recent papers LukSEovS, Morton and Warnecke [LMW 981, [LMW 991 several approxinlate evolutioii operator for the wave equation system were analysed. I t was show11 t h a t the following approxinlate evolution operator leads t o the best first order schellle in terms of accuracy, see [LMW 981, [LMW 991.

3.1. First order approximate evolution operator

eQ- 2.uQ cos8 - 2vQ sin Ode + o ( A ~ ~ ) ,

up

=

I 2

-up1

+-

-24Q sin 8

+ 3zlQ sin 8 cos 8 + uQ(3sin20

(GI

-

l ) d ~

Denote by Ph L2 - projection onto a space of piecewise co~lstantfunctions in R< then we obt,ain the first order scheme = PhEatUn, which is in [LMW 991 referred t o a s the EG3 scheme. Space integrals coliling from the projection step are computed exactly, i.e. no numerical quadrature is used. T h e finit.e difference forn~ulationcall be found in [LMW 993, where the coefficients of the schellle are given explicitely.

zn+'

3.2. Second order reconstruction

t second order FVEG scheme we take the first In order t o c o n s t r ~ ~ cthe order accurate approximate evolution operator (6) - ( 8 ) and define a bilinear reconstruction Rh. There are many possible recovery schemes, which could be used. For our computation we choose a discontinous bilinear recovery using a four point averages a t each vertex, but others can be used a s well. It is taken t o be conservative and given a s

where Aolv(t) = (v(x + h) - u(x - h ) ) = (v,+l - v,-I), an analogous notation is used for Aoy. For the computation of fluxes through cell edges the time averaged value Un+*has t o be known, see (4). Instead of exact time integration

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the second order midpoint rule is used, i.e.

Two dimensional space integrals of bilinear functions RhUn with respect to 9 and S which occur in (9) are computed exactly without any numerical quadrature and thus all of the infinitely many directions of propagation of flow information are taken explicitely into account. The above construction leads to the overall second order scheme, which gives in regions of smooth solution very accurate results even on coarse grids, see Table 1 below.

3.3. Numerical results

We consider the space periodic problem for the wave equation system (5) with the initial data 1 d(c, 0) = -(sin 2 r x + sin 27ry), u ( g , 0) = 0 = v(g,0). c In this case the exact solution can be easily computed [LMW 991. In Table 1 we compare errors of several second order schemes, namely the second order FVEG method, the Lax-Wendroff method (LW) and the standart finite volume fluxvector directional splitting method (FV-FVS), which uses a MUSCL technique for the flux comput.ation and a second order Runge-Kutta t,ime approximation. For the details on t,he latter one see, e.g., [KRO 971. We use meshes of 20 x 20, 40 x 40,. . ., 640 x 640 cells and con~putealso the experimental order of convergence (EOC) from two meshes of sizes iV1 and Nz

EOC = 111

ll&~,

- Ukl

I1

/ l u ~ -~!2k211 (~)

In all cases the results are for a CFL-number v of 0.45 and an end time of T = 0.2. Experiments for several other values of v and T confirm the second order accuracy of the FVEG schenle.

I N / lli(T) unll I -

20

FVEG

LW

I

I

EOC

I

I FV-FVS I

1 0.008647 1 0.065500 1 0.057748 1

1

1.94

1

I

1.99

1

.2.00

1

Table 1. Comparison of accuracy of the FVEG method, the Lux- WendrofJ method and the F V flux-vector splitting method

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4. Euler equations

The system of Euler equations describing the motion of compressible flow in two space dimensions can be written in the form of hyperbolic system (1) with d = 2; the definition of and F i ( u ) is well-known and can be found e.g. in [KRO 971. If R = (r, , r2, 3,-n , r4 ,) denotes the matrix of right eigenvectors we can decompose thk-vei~orof conservative variables / for any direction 14 in the following way 9-

4

u

&&-I(' - = --

-- C okrk,,.

(10)

k=l

A general procedure for derivation of the exact evolution operator for hyperbolic problems can also be applied to the Euler equations. For linear problems this procedure works with the characteristic variables W = Now, instead of W the vector 9 is used. The following approximate evolution operator for the Euler equation was derived in [OST 971

&-'u.

c ( g ,t

+ At) = 1'1

1

kakrx((r(Qk(K7E)))d0

Okx1

where 0 denotes the unit sphere, Q l ( & , n= ) (g - A t ( g - cn),t), Q 2 ( g , n ) = Q3(g, E ) = (& - Atg, t), Q4(g,R) = (g - At (g cn), t), c is a local speed of sound and g represents the fluid velocity. After L~ - projection onto a space of piecewise constants we obtain the EG scheme for the Euler equations. It was shown by Ostkamp [OST 971 that Fey's method of transport [FEY 921 for the Euler equations can be reinterpreted as the above evolution Galerkin scheme.

+

4.1 Numerical results

We take the well-known test problem, namely the two-dimensional Sod's problem and compare the behaviour of the evolution Galerkin method (11) and the LeVeque wave propagation algorithm [LEV 971, which is available as a public domain software package called CLAWPACK. In order to avoid a discussion of the limiters we compare here only first order schemes. Note however that in LeVeque's wave propagation algorithm also first order correction terms for xy-cross derivative are included. The computational domain is the square [-1,1] x [-1,1]. To ensure the CFL stability condition, the CFL number is taken 0.8. We choose periodic boundary conditions and the following initial data

Innovative schemes p= 1,u=O,v=O,p= 1

p = 0.125, u = 0, v = 0 , p = 0.1

if 1x1 5 0.4, otherwise.

295

(12)

Solutions a t time T = 0.2 are computed 011 a quadrilateral grid with 200 x 300 grid cells. In Figures 1 and ",he isolines of solution obtained by the evolution Galerkin method (11) and by the wave propagation algorithm, respectively are drawn. T h e y-velocity is not depicted since it is symmetric to the x-velocity. For the evolution Galerkin scheme the resolution of the flow phenomena is the same in all directions and illforlnation is moving in infinitely many directions in a circular manner. However, we can notice that the wave propagation algorithm, which makes use of an improved directional splitting, does not preserve circular symmetry in such a good lnanner as our scheme and some dependence of the solution on the grid can well be seen.

Figure 1. Evolution Galerkin scheme: density (left), r-zlelocity (middle), pressure (right)

Figure 2. I.Vat~epropagotior2 algorithm: density (left), t-tlelocity (middle), pressure (right)

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Finite volumes for complex applications

5. Acknowledgement

This research has been supported under the DFG Grant No. Wa 63316-1 of Deutsche Forschungsgemeinschaft, and partially by the Grant No. 201/97/0153 of the Czech Grant Agency and by the DAAD.

6. Bibliography

[FEY 921

FEY M . , JELTSCH R.,
[KRO 971

KRONERD., Numerical Schemes for Conseruation Laws, Wiley - Teubner, Stuttgart, 1997.

[LEV 971

LEVEQUE R.J., <Wave propagation algorithms for multidimensional hyperbolic systems,, J. Comp. Phys., 131, 1997, p. 327-353.

[LMW 981

LUKACOVA - MEDVIDOVA M . , MORTON 1C.W.. WARG., , Proceedings of ENUNECKE

MATH'98, World Scientific Publishing Company, Singapore, 1998, p. 445-452. [LMW 991

LUKACOVA- MEDVIDOVAM . , MORTONIi.W., WARNECKE G., <Evolution Galerkin methods for hyperbolic systems in two space dimensions,, Preprint University of Bath, England, 1999, submitted t o MathComp.

[OST 971

OSTKAMP S., cMultidimensional characterisitic Galerkin schemes and evolution operators for hyperbolic systems,, Math. Meth. Appl. Sci., 20, 1997, p. 1111-1125.

Nonlinear anisotro~icartificial dissi-patiGn Characteristic filters for computation of the Euler equations Thorsten Grahs, Andreas Meister and Thomas Sonar Institut fur Angewandte Mathematik U n i v e r s i t a t H a m b u r g , B u n d e s s t r . 55 20146 H a m b u r g , G e r m a n y

ABSTRACT W e employ a nonlinear anisotropic diffusion operator like the o n e s used as a means of filtering and edge e n h a n c e m e n t i n image processing, i n numerical methods for conservation laws. It t u r n s out that algorithms currently used i n image processing are very well suited f o r t h e design of nonlinear higher-order dissipative t e r m s . I n particular we stabalize a central scheme, k n o w n for its oscillating behaviour b y the construction of a nonlinear diffusion t e r m . Using i n f o r m a t i o n from the data a so called structure tensor following from ideas of anisotropic differsion i n image processing due to W e i c k e r t is constructed, containing informat i o n about the orientation and strength of the necessary diffusion. I n particular this means constructing a diflusion m a t r i x consisting of eigenvectors parallel and perpendicular t o discontinuities and eigenvalues denoting t h e a m o u n t of dissipation depending o n the local strength of the gradients. These directions are used t o steer the a m o u n t of dissipation which m e a n s surpressing d i g u s i o n across the shock front and using the perpendicular direction t o enable the necessary diffusion t o stabilize the underlying second order scheme. T h e diffusion t e r m s are based o n the artificial compression method ( A C M ) due t o Harten and the extensions o f Yee. T h e limiting takes place o n characteristic based diffusion t e r m s acting o n the characteristic velocity. Numerical results are shown for the two-dimensional Euler-equations. K e y words: low dissipation, centralfinite difference schemes, shock-capturi n g methods, anisotropic diffusion, characteristic filters, Euler equations

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1. Introduction

The construction of suitable artificial viscosity terms for stabilizing finite volume and finite difference schemes of higher order is a difficult task. In the last decade we observed therfore a strong tendency to construct numerical approximations of conservation laws without explicit knowledge of their numerical diffusion. The modern total variation diminishing (TVD) or essentially non-oscillatory (ENO) shemes belong to this class, in which a basic firstorder scheme is improved by the use of sophisticated recovery functions, see [Son97, MS991. There, are however, certain circumstances in which an approach using explicit construction of artificial dissipation would be advantageous. If we consider pseudospectral methods the concept of E N 0 recovery is very hard to apply if the degree of the polynornials used is high. Here one would like to compute shocked solutions with central schemes and post-process the numerical solution such that high frequency oscillations are filtered, and shocks are steepend and represented with high resolution. Over the years there were no general attempts to derive a constructive theory which would enable the design of suitable artificial viscosities within the CFD community. However, filtering and edge enhancement is a fundamental task in image processing and in recent years a theory of nonlinear anisotropic diffusion was created and can now be found in textbooks like [Wei98]. In a noisy picture one also would like to filter the high frequency components before detecting the edges (i.e. jumps in grey level). Then one would like to enhance the edges in order to represent the edges in high resolution. Now there is nothing which keeps us from interpreting our numerical solution corresponding to the conservation law as a photograph or picture. In the same way the photographer would very much prefer to see the contours on his picture as sharp thin lines the numerical analyst would prefer to see shocks as crispy lines instead smeared thick regions. To accomplish this, the picture as well as the numerical solution have to be denoised. After removing the high frequencies we would like to spend a dose of diffusion tangential to shocks - that is what anisotropy is all about in this context. In contrast, in the vicinity of shocks we would like to solve a kind of nonlinear anisotropic backward heat equation to enhance the structure of a shock front. Devices and algorithms satisfying exactly these requirements are ready to use if one is willing to enter the area of image processing. The aim of this paper is mainly to show how one can use ideas from image processing in the area of the numerical treatment of conservation laws. We have used suchs methods succesfully for scalar mixed Burgers-advectionequation namely stabilizing a Lax-Wendroff scheme and surpressing the oscillatory behaviour of the scheme in the vicinity of shocks (see [GMS98]). Here we would like to discuss the extension to the Euler equations. First we describe the governing equations and the basic scheme. In the following we construct the so called structure tensor to gain informations about the local strength and orientation of the shock. This information will be used to develop a diffusion

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matrix t o steer the amount of diffusion necessary to stabilize a second order central scheme.

2. Anisotropic diffusion applied to characteristic filters 2.1. Governing Equations

We consider the two dimensional time-dependent Euler equations for a compressible gas in equilibrium in conservation form, i.e.

a,u + a , ~ ( u+) ~ , G ( u )= o in which

denote the vector of conserved quantities and the flux functions in x and y, respectively. If t E R : := {t E Rlt 2 0) denotes time and x = ( x l ,~ 2 E R2 ) space coordinates, the mapping

IR2 x IR; 3

(2,t)

p'VM'H (p, u :=

( ~ 1~ ,

2P ,

El ) H )~( x ,t ~)

denote density, velocity, pressure, total energy and enthalphy, respectively. Enthaiphy is defined by

H := E + -P.

P

To close the system, an equation of state is needed. For ideal gases one uses

where y denotes the ratio of specific heats. In the case of dry air one assumes a value of y = 1.4.

2.2. Construction of the characteristic filter We discretize on a cartesian grid with mesh size A x , Ay, respectively. Since in this case finite volume and finite difference techniques are equivalent, we use in the following a typical finite difference notation. We start from an explicit central scheme written as (in+' 2 r3

= U , , j - At

(d.

-[Fi+l12,j

-

+ A1Y -

- J'i-i_l12,j]

~ [ G i , ~ +l /~zi , j - 1 / 2 I

~

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Finite volumes for complex applications

where the numerical flux functions include the filter terms and hence the cross diffusion parts and can be written as follows:

The filter operators L A ,L B depend on the Jacobian-matrix A and B of the ) some kind flux function F and G, respectively. The filter terms L ~ (models of cross diffusion and depends on a combination of the Jacobians, namely

(see [HirSO] for details). The vector n( ) := ( n i i , nil)' is chosen in direction of the cell diagonals, i.e n+ := (1, ~ ) ~and / fn-i := (1, - I ) ~ / & . The matrices are all evaluated as a Roe average ( see [RoeBl]). So one writes the corresponding filter terms as

and similar for the other terms. Here R(') denotes the matrix of the right eigenvectors corresponding to the appropriate matrices written as superscript. a ( ' ) denotes the real filter term. In the basic construction, we follow the recent paper of Yee and her co-workers [YSDSS], based on the Artificial Compression Method (ACM) of Harten and the extensions of Yee (see f.e.[Har78, Har83, Yee851). We extend these method by the integration of a directional based - that its what we denoted with anisotropic - diffusion known from image processing. So the elements of @(.I in (1) are denoted by $'('), 1 = 1 , 2 , 3 , 4 . We give here as an example the construction of the term All other terms are constructed in the natural ext,ension of this definition depending on the matrices and grid points were they are evaluated. So the eleinents of @ i( +) l / 2 , j write as

):i)l,kj.

The P q e n o t e s the weighting coefficients steering the anisotropic diffusion. The construction of this coefficient will be described in t>hefollowing section. The

Innovative schemes

301

choice of the q5i+l/2,j is exactly the same like the one described in [YSD99] with the use of the limiter functions g; as

Additional possible choices for this function and a detailt description are given in [YSD99]. The definition of 4' will be give below. The choice of the filter function depends on the characteristic splitting of the flux representation of the gradients. Thus, we consider as elements of

a i + l / 2 , j := (Rftl 2,j)-'(Ui+l J - U i , j ) .

d

The different a epend again on different matrices of the right eigenvectors depending on the gridpoint were they are evaluated. They are given as:

4 f + 1 / 2 , ~ + 1corresponds /~ to ai+l/2,j+l12 and so on The filter function writes as

Hence

with

The a:+l12,1 in (3),(4) are the characteristic speeds of the corresponding Jacobians, which are equal to their eigenvalues. This term can also be weighted with the diffusion coefficient steering the amount of dissipation corresponding to this direction.

/'?i+l12,j

2.3. The structure tensor

We start from the so called structure tensor J t j . The extensive description of its construction and properties can be found in Weickert [Wei98]. We use a

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Finite volumes for complex applications

smoothed version of the characteristic gradients R-' A U d , where Ua is a presrnoothed version of the d a t a U n . This means convolution with a Gaussian kernel whith convolution scale 6, i.e

Since in the continnous case this is equivalent with solving the heat equation we apply this t o the data UiXjwith stopping time T = i d 2 , where 6 is a parameter which has to be chosen. In order t o remove the small scale oscillations by means of this smoothing technique we define 6 depending on the grid size h := d m . Consequently, the structure tensor reads as

with

where cr1i6 corresponds t o the smoothed data U a . Therbye one can also use a smoothed version of the structure tensor (5)) i.e

which means component-wise convolution with scale u , which denotes the width of the averaging region. In practice we are solving the heat equation for each component seperately.

2.4. The diffusion matrix

After having constructed the structure tensor we are going t o compute the eigenvectors and eigenvalues of (6),

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Therbye the corresponding eigenvalues are given by

Now we construct the dissipation matrix D by introducing the ansatz due to Weickert [Wei98]:

Here V, denotes the matrix of the eigenvectors (7), (8) and A, = diag(ll,l2) represents a diagonal matrix where the diagonal elements have to be calculated in a convenient manner as described below. In order t o recover shocks (or, equivalently, in order to enhance edges) the diffusivity l I perpendicular to edges should be reduced if the contrast XI,, is high. This can be achived by an anisotropic regularization of the Perona-Malik niodel [PMSO] also adapted from Weickert :

The values of m and C, are chosen in such a way that t,he so-called flux @(s) := S ~ ( Sis) increasing in an interval s E [0, A] and decreasing in s E]X, a[. The choices depend on a one-dimensional analysis of the Perona-Malik model. In agreement with Weickert we chose m = 4 and thus C4 = 3.31488. The so-called contrast parameter A, seperating areas with forward (low contrast) from backward (high constrast) diffusion, can be chosen freely. Based on numerical experiments it turns out that for calculations concerning systems it is useful1 t o have an adaptive parameter instead of a fixed one. We calculate for each variable the maximum of the according gradients in every timestep and choose a fixed percent of this maximum to determine the parameter A. Thus, the diffusion matrix (10) can be written as

with coefficients

The superscript 1 reminds that the diffussion where V I , ~= (vllip, matrix (10) and so the structure tensor (5) resp. (6) have to be computed for every characteristic variable seperatly.

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Finite volumes for complex applications

Weickert proved that there is always a finite difference stencil which leads to a stable scheme. Moreover, he was able to show that three directions suffice to guarantee a non-negative discretization, since a negative discretization is equivalent to an ill-posed problem. The discretization of a 3 x 3 stencil reads as follows:

This gives the weighting coefficients for the dissipative fluxes which leads to a steering of the dissipation terms depending on the magnitude of the gradients as described it in the last section.

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305

3. Numerical results

As an example we use a test case given by LeVeque [Lev931 with initial d a t a

lJl for U,. for

d

m < 0.13, > 0.13.

and Ul = (2,0,0,15)T,U,. = ( 1 , 0 , 0 ,l)T in primitive variables. The solution consist of a shock running outwards followed by a rarefaction wave and a contact discontinuity. A second shock moves inwards towards the center. We use an equidistant discretization with Ax = A y = h = 0.025 and a CFL-Number = 0.4. The smoothing parameters are given with b = 0.25h and u = 0.0 which means no smoothing of the structure tensor takes place. The contrast parameter X is chosen as 0.4 max(X1;,).

Figure 1: Solution of the test case (12) a t time t=0.13

4. Concluison We have extended the characteristic filter approach by Yee, Sandham and Djomehri [YSD99] by an anisotropic directional based diffusion from image processing. Since this integration into the field of numerical conservation laws is a novel approach, the calculations presented here are preliminary results and will need further research. Overall we are quite optimistic concerning the behaviour of the suggested extension of this scheme.

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Finite volumes for complex applications

References [GMS98] Th. Grahs, A. Meister, and Th. Sonar. Image Processing for Numerical Approximations of Conservation Laws: Nonlinear anisotropic artificial dissipation. Hamburger Beitrage zur Angewandten Mathematik, Reihe F: Computational Fluid Dynamics and Data Analysis 8, 1998. (submitted to SIAM J. Sci. Cornp.). [Har78] A. Harten. The artificial compression method for computation of shocks and contact discontinuities. 111. self-adjusting hybrid schemes. Math. Comp., 32:363-389, 1978. [Hart331 A. Harten. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 49:357-393, 1983. [HirSO]

Ch. Hirsch. Numerical computation of internal and external flow, volume 2. J . Wiley & sons, 1990.

[Lev931 R.J. LeVeque. Simplified multi-dimensional flux limiter methods. In M.J. Baines and K.W. Morton, editors, Numerical Methods for Fluid Dynamics 4, pages 175-190. Oxford University Press, 1993. [MS99]

A. Meister and Th. Sonar. Finite Volume Schemes for compressible fluid flow. in press: Surveys of Mathematics in Industry, 1999.

[PM90]

P. Perona and J . Malik. Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach.Intel1, 12:629-639, 1990.

[Roe811 P.L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43:357-372, 1981. [Son971 Th. Sonar. Mehrdimensionale ENO- Verfahren. Advances in Numerical Mathematics. B.G.Teubner Stuttgart, 1997. [Wei98] J . Weickert. Anisotropic Diffusion in Image Processing. B.G. Teubner, Stuttgart, 1998. [Yee85] H. C. Yee. Construction of explicit and implicit symmetric TVD schemes and their application. J. Comput. Phys., 68:151, 1985. [YSD99] H. C. Yee, N. D. Sandham, and M. J . Djomehril. Low-dissipative high-order shock-capturing methods using characteristic-based filters. J. Comput. Phys., 150:199-238, 1999.

Nonlinear projection met hods for multi-entropies Navier-Stokes systems

Christophe BERTHON

Frkdkric COQUEL

ONERA, BP 72, 92322 Chitillon Cedex, FRANCE.

4, place jussieu,

LAN- CNRS, 75252 Paris Cedex 05, FRANCE.

ABSTRACT This paper is devoted to the numerical approximation of the compressible Navier-Stokes equations with several independent pressures. Several models derived in plasma physics or in turbulence typically enter the proposed framework. The striking novelty over the usual Navier-Stokes equations stems from the impossibility to recast equivalently the present system in full conservation form. Classical finite volume methods are shown to fail in the capture of shock layers. We propose a new method, the so-called nonlinear projection operator, for correcting the errors while preserving all the stability properties.

Key Words: Navier-Stokes equations, Non conservative products, Nonlinear projection.

1. Introduction The present work treats the numerical approximation of the solutions of the Navier-Stokes equations for a compressible fluid modelled by two independent pressures, e.g. each of the pressures comes with its own specific entropy. Despites that such models are seen t o exhibit several close relationships with the usual Navier-Stokes system, the fundamental discrepancy stays in the lack of an admissible change of variables that recasts the system in full conservation form. None of the entropy balance equations boils down t o a conservation law involving non conservative products that account for dissipative phenomena : namely the entropy dissipation rates. Such systems occur in several distinct physical settings. They arise for instance in plasma physics and they can be also recognized within the frame

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Finite volumes for complex applications

of the "two transport" equations models for turbulent compressible flows. All these models are addressed below with the assumption of a large Reynolds number. The numerical capture of the viscous shock layers is of primary importance in the present work. Since the Reynolds numbers of interest are large, these layers display the character of a shock wave in that they differ from their end states only in a small interval of rapid transition. Our purpose is to correctly capture the two end states, vr, and V R , of a given shock layer together with its speed of propagation u whitout resolving sharply the viscous layer itself. It is quite well-known that such an issue does not raise special difficulties within the standard frame of the Navier-Stokes equations, e.g. in conservation form. At the very core of this success, is the definition of v R ( q vL) stays free from the entropy dissipation rate and in this sense, its numerical capture stays also free from the rate of numerical dissipation. The situation turns out to be completely different in the setting of the extended Navier-Stokes equations. Its non conservation form makes this time the triple (u; VL,vR) to heavily depend on the precise shape of the diffusive tensor. Such a dependence stays at the basis of recent works devoted to hyperbolic systems involving non conservative products (see Lefloch [6], Dal Maso-LeFlochMurat [2], Raviart-Sainsaulieu [8]). In the setting of numerical methods, this dependence implies that the numerical viscous tensor must fit the exact last one. The negative consequences can be found in the numerical result presented below. In the present work, we propose an approach based on the analysis of the discrete dissipation rates of a given Lax entropy. It turns out that the end states require for their correct capture to prescribe explicitely the Lax entropy dissipation rate. This requirement asks the numerical methods to satisfy in turn an imposed rate of entropy dissipation. This non standard issue is precisely the main motivation of the present work. Let us underline that the results we state below extend in a straightforward way to higher space dimensions, taking advantage of the rotational invariance of the equations. 2. Mathematical model

We consider a gas with density p and velocity u, which is modelled by two independent pressure laws p and p,, associated with two constant adiabatic exponents y > 1 and y, > 1. The system of PDE1s that governs such a fluid model writes :

Innovative schemes

309

where the involved temperatures respectively read T = p/p and T, = p,/p. This convective-diffusive system can be understood as an extension of the standard Navier-Stokes equations when considering an additional PDE for governing an additional pressure. Depending on the closure relations for the viscosities p , p, and the thermal conductivities K and K , , several distinct physical models enter the present framework. In this section, all these transport coefficients are assumed to be fixed positive constants for the sake of simplificity in the discussion. The smooth solutions of system (1) obey additional governing equations as we now state : L e m m a 1 Smooth solutions of (1) satisfy the following conservation law:

+ 5 + 3. Smooth

where the total energy p E is defined by p E = solutions satisfy i n addition the following balance equations

where the specific entropies are respectively given by s := log Consequently, smooth solutions of (1) obey :

:

F,

s , := log

E.

where the right hand side follows under the assumption of two constant viscosities p and p,. The three balance equations (2), (3) and (4) can be proved t o be the only non trivial additional equations for smooth solutions. As a consequence the discrepancy stays in the lack of four non trivial conservation laws. Indeed, none of the equations (3), (4) and (5) boils down t o a conservation law and ( I ) cannot be recast in full conservation form. After the works by LeFloch [6] and Raviart-Sainsaulieu [8], the non conservation form met by (1) makes the end states of shock layers to depend on the closure relations for the coefficients p, p, and K, 6,. In order to assess this issue, let us focuse on the non standard balance equation (5) where by contrast with (3) and (4) which dissipation rates are independently imposed, (5) exhibits a c o m p a r e d rate of both the entropy dissipations. The idenity (5) continues t o play a major role in a general setting since they encode a generalized jump condition which turns out to play a central role for our numerical purpose :

3 10

Finite volumes for complex applications

Theorem 2 Assume that p, p,, rc and n, denote positive constants. Then , associated with the resulting dissipative tensor necessarily the triple (a,v ~vR) obeys the jump relations :

and necessarily satisfies the two entropy inequalities :

Under an appropriate setting (see the companion paper [I]), (7) can be specified as follow :

defining a generalized jump condition where the involved averages find a unique definition (see [I] for details). Remark 3 In view of the jump relations (6) and ( S ) , one of the two entropies, either ps or ps,, must be understood as a nonlinear function of the four remaining independent variables (p,pu, pE, .).

3. Godunov methods with nonlinear projections

For the sake of simplicity , we do not address the discretization and we set K , = 0 We refer the reader to the companion paper [I] for the required discrete formulae.

K

=

3.1. L2 projections (tn 4 tn+'lp)

An equivalent system using the conservation laws for p, pu, pE and the evolution law governing ps, is considered. Notice that the evolution law for ps can be understood as an additional law sat,isfied by the smooth solutions. Choosing consitant formulae, based on a Godunov method for instance, the discrete solution satisfies, formally, the following dissipation rates of entropie :

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311

As a consequence of the error which occurs in ( l o ) , the expected generalized jump relation (8) is systematicaly violated. The reader is referred to [l]for a rigourous proof and to the numerical results below for an illustration of the negative consequences of such a failure.

3.2. Nonlinear projection methods (tnf

'1-

+ tn+' )

We propose to add an additional step to classical L2 projection methods, the nonlinear projection step, which purpose is t o correct the errors. In order to preserve the required conservation properties, let us define :

Setting v:+' = (p;+"-, (pu):+l'-, ( p ~ ) y + l ' -(ps,):f' , ), to enforce the validity of the generalized jump condition at the discrete level, we propose to seek for ( p ~ , ) ? + ' as a solution of :

The above nonlinear problem in the unknown (ps,):+l can be shown to admit a unique solution as soon as the approximate Riemann solver involved in the first step obeys discrete entropy inequalities for the Lax pair (ps, psu). The nonlinear projection step (12) allows to prove in addition the following stability results : Theorem 4 Under the required CFL restriction, the following discrete entropy inequalities are satisfied :

+

for all strictly decreasing and concave functions and $I. The following maxim u m principles for the specific entropies s and s, are m e t :

Both the pressures p:+l and p,"+' positive.

stay positive as soon as the density pn+' is

3 12

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The proof of the above statement is detailled in [I].

3.3 Numerical results The ability of the schemes in the capture of shock layers for (1) is evaluated when testing their sensitivity in the prediction of the end states with respect to the mesh refinement. The initial data is made of two constant states prescribe by

Problems are directly motivated by the three distinct regimes that underly the flow model under consideration and that are dictated by the amplitude of the viscosity ratio p T / p . The viscosities p and /I, are assumed to be positive constants we referre to [I] for the setting of varying viscosities and non zero heat conductivities. conductivities. In all the benchmarks discussed below, the Reynolds number is set at the constant value Rey = lo5.

All the figures assess that the usual numerical strategy (see [7] or [5]concerning the details of this method) grossly fails to restore the correct end states in the three regimes. Turning considering the L2 projection method, the discrete solutions agree with the exact ones only in case A. Such a property no longer holds for problems B and C and consequently large errors occur. These two schemes furthermore suffer from a dramatic sensitivity with respect to mesh refinements for problem C in that discrete solutions do not seem to converge to a given limit function even for the finest proposed grids. By contrast and concerning benchmarks A and B, the discrete solutions stay non sensitive with respect to the mesh refinement but the "limit" function does not coincide with the expected exact solution. Turning considering the nonlinear L2 projection method, it produces approximate solutions that achieve a fairly good agreement with the exact solutions while staying almost non-sensitive with Ax in the three investigated regimes.

Innovative schemes Classical Scheme

L2 Projection Scheme

Figure 1: Problem A : p , / p

Classical Scheme

3 13

Nonlinear Projection Scheme

<< 1

L2 Projection Scheme

Figure 2: Problem B : p , / p = 1

Nonlinear Projection Scheme

3 14

Finite volumes for complex applications Classical Scheme

L2 Projection Scheme

Figure 3: Problem C : p , / p

Nonlinear Pro.iection Scheme

>> 1

Bibliography C. BERTHON A N D F. COQUEL, Nonlinear projection methods for systems in non conservation form. work in preparation. G. DALMASO,P . LEFLOCHA N D F. MURAT,Definition and weak stability of a non conservative product, J. Math. Pures Appl., 74, 483-548 1995.

E. GODLEWSKI A N D P.A. RAVIART, Hyperbolic systems of conservations laws, Applied Mathematical Sciences, Vol 118, Springer 1996. T. Y. HOUA N D P . G . LEFLOCH, Why nonconservative schemes converge to wrong solutions : error analysis, Math. of Comp., Vol 62, No 206, 497-530 1994. A N D C. OLIVIER, On the numerical appproximation of B. LARROUTUROU the K-eps turbulence model for two dimensional compressible flows, I N R I A report, No 1526 1991.

P .G . LEFLOCH,Entropy weak solutions to nonlinear hyperbolic systems under non conservation form. C o m m . Part. Dzff. Equa. 13, No 6, 669-727 (1988). B. MOHAMMADI A N D 0. PIRONNEAU, Analysis of the K-Epsilon Turbulence Model, Research in Applied Mathematics, Masson Eds 1994. P . A. RAVIART A N D L. SAINSAULIEU, A nonconservative hyperbolic system modelling spray dynamics. Part 1. Solution of the Riemann problem, Math. Models Methods in App. Sci.,5, No 3, 297-333 1995.

About a Parallel Multidimensional Upwind Solver for LES

D. Caraeni S. Conway L. Fuchs Division o,f Fluid Mechanics, Lund Institute of Technology SE-221 00 Lund, Sweden

A parallel ,pow solver has been developed at Lund Institute of Technology, ,for Large Eddy Simulations ( L E S ) of turbulent compressible jlows. The numerical algorithm is based o n the Residual Distribution scheme approach. The scheme employs a n extremely compact stencil, while still having second order accuracy, which makes it well suited ,for parallelization. I n this paper we report about the numerical scheme and about the parallel algorithm implemented i n the code. First the performance concerning the parallel scalability is addressed. Finally some results ,for the L E S of the classical channel ,flow are prese,nted.

ABSTRACT

Key Words: Residual Distribution Scheme, Large Eddy Simulations, Parallel Flow-Solver.

1. Introduction The advances made in computer technology over the last years, have led to a great increase in the engineering problems which can be simulated using CFD. The computation of flows over complex geometries a t relatively high Reynolds number is becoming more common using LES, as recent reviews on the subject have shown [PI0 981. LES for engineering applications are typically extremely expensive, requiring huge resources in terms of processor power and memory, and long computational times. When using large parallel computations, LES becomes accessible for many industrial applications. Accurate numerical algorit,hms, well suited for parallelization are needed in order to make LES available for production CFD.

316

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2. LES Equations

The expression for the LES equations, can be obtained by " tophat" Favrefiltering of the time dependent compressible Navier Stokes equations, written in conservative form:

The subgrid terms representing the subgrid stress tensor, 7;" = p [ u x j -

Ezj],

&Gj],the subgrid heat flux, H:" = PE [~; and the subgrid viscous work, G;gs = F [ r s i - GUi],a,ll require modelling. In the present simulations, the 7;;' was modeled by using a Sub-Grid Scale (SGS) model, as described below. All the other SGS terms presented above, are not explicitly modeled. 3. SGS Models In the Large Eddy simulation technique, the largest scales of the turbulence are resolved and the effect the smallest scales have on the resolved scales, is modeled. This is done by introducing a SGS model. In the present code, three SGS models have been implemented: the Smagorinsky model, Lilly's dynamic model [LIL 911 and the Dynamic Divergence model (DDM). The DDM model is a novel, anisotropic dynamic SGS model with independently determined coefficients in each coordinate direction. Note that in this case, it is the divergence of the SGS stress tensor, 7:i:,that is modeled [JHF 981 [SCF 981. For the dynamic models, the model parameter is calculated dynamically at each point from the instantaneous flow field. Negative values of this parameter enable back-scatter, which implies that turbulent energy can be transferred intermittently from the small scales to the larger ones. Local filtering, and artificial bounds, have to be used, in order to avoid large oscillations in the model parameter.

Innovative schemes

3 17

4. Numerical Algorithm

Residual Distribution (RD) or Fluctuation Splitting (FS) schemes date back to the late 80's [ROE 821, when P.L. Roe developed schemes for solving scalar advection-diffusion problems on unstructured meshes. More recently, Van der Weide and H.Deconinck [PDE 971 proposed a matrix generalization of the scalar schemes which can be applied to the solution of non diagonalizable hyperbolic systems. Wood proved [WKK 981 that RD schemes give a better accuracy of the solution, for advection diffusion problems, when compared with the classical finite-volume formulations. Though most applications of RD schemes have been limited so far to steady state computations [DSB 941, we extended their application to accurately simulate time dependent flows [CAO 99][CAB 991. Our code uses a second order in time, implicit scheme of Jameson-type [JAM 911. This is a dual-time step scheme, with sub-iterations to converge the solution at every new real-time step. Multigrid iterations are performed to accelerate the convergence of the solution in these sub-iterations. The code works on an originally developed unstructured mesh, cell-vertex data structure. A hierarchical oct-t,ree organization of the data was employed with a single tetrahedron division rule. This was shown to allow significant reductions in memory usage, [SCL 97].The discretization of the convective term of the Navier-Stokes equations is based on the matrix Residual Distribution schemes approach [PDE 971. A finite element, central-Galerkin scheme has been used to discretize the viscous part, the unstationary term and the source terms, i.e. Coriolis forces, centrifugal forces, etc. The update scheme is u,"' lvk+'- u ': l v k AUi, where the correction AUi is given by:

+

where U = (p, pu3, pE) is the conservative variables vector, j = 1..3, AT is the pseudo-time step, n - is the real-time step index, T is one of the cells (tetrahedral) sharing the node i, i = 1..4 is the node index, V , is the volume of the dual cell Oi, surrounding the i node. Here U,"+lpkis the approximation of u?" at the pseudo-time step k, ,BT is the mod.l;fied matrix distribution coefficient for the convective term, is the advective-residual over the cell T.

where Fj" = ~ j " ( ~ " + lis?the ~ ) advective flux vector, and F," = F,"(U"+ l , k ) is the viscous flux vector, both computed for un+llk,nj,i are the cell-face inward normal vectors, is the contribution of the advective part

,BT@zviscid

3 18

Finite volumes for complex applications

of the Navier-Stokes equations to the nodal residual, i(F;njsi), the contribution of the viscous part of the Navier-Stokes equations to the nodal residual, is the volume average of the unstationary term, on the (T n Oi). We used an implicit second order discretization with finite differences for

[g]jzt:jt

[g]:

A modtfied second order matrix Residual Distribution scheme has been used for the convective part of the Navier Stokes equations. It employs a dynamic correction of the artificial dissipation of the scheme based on the local flow characteristics. When doing LES, the artificial dissipation of the scheme, i.e. the LW scheme [PDE 971 in our case, is not desired and it can be shut off in regions far from shock waves. Close to shock waves, the artificial dissipation of the original scheme, is reintroduced. If strong shock waves are present, a continuous blend between the second order scheme and a positive first order scheme, i.e. the Narrow scheme, is used in the shock wave region. This enabled us to have monotone shock capturing, while still using very small dissipation in the smooth flow regions. The matrix distribution coefficients, P$, are given by:

The matrix distribution coefficients are computed, for the second order scheme, LW:

and for the positive first order, Narrow scheme:

Here v,,ll is a cell-CFL number. Ki are the Jacobians of the convective aFc flux vector relative t o face i, Ki = &nj,i. The coefficients vCe[land w e n d are corrected dynamically. When doing LES, they vanish in regions far from flow field discontinuities. More details about the matrix generalization of the Residual Distribution schemes can be found in [PDE 971. 5. Parallel Algorithm Since the Fluctuation Splitting algorithm only needs to access the first order neighbors to update the solution a t a vertex, the parallelization process is

Innovative schemes

3 19

greatly simplified. If it were necessary to access the second-order neighbors, or even higher-order neighbors, then for any vertex that lies along a boundary between processors, a very elaborate and likely expensive method for communicating the appropriate update information across these inter-partition boundaries would need to be devised. Second order classical finite volume codes require access to the second-order neighbors. However, these algorithms get around the complication of having to store the communication information to reach these second order neighbors across processors by using a two-step update procedure. This requires also two communication cycles per time step [VNK 921, whereas the Fluctuation Splitting algorithm only needs one. This makes the fluctuat,ioil splitting method very attractive for parallel implementation. The Parallel Virtual Machine (PVM) defacto standard has been used to implement the parallel algorithm in our code [CAD 991. The code runs as effectively on large distributed memory machines (Cray T3E, IBM SP2...), on the SGI Origin 2000 servers or on a heterogenous collection of platforms. The code takes advantage of its unstructured grid for realizing an almost perfect load balancing by assigning a number of volumes to compute to each processor, proportional with the processors speed. The parallelization of the code has been done using the SPMD paradigm, i.e. the same code is executed in all processors. The global data structure, i.e. grid, field variables, etc. is decomposed into a number of subdomains, equal with the number of running tasks. While any algorithm for domain decomposition can be employed, so far only one dimensional domain decomposition has been used. This is done in a separate preprocessing step and does not affect the solvers flexibility. Each task runs with its own data structure, which is a disjunct part of the global data structure. The tasks are implicitly synchronized by using blocked data exchange. The algorithm implies at each real time step sub-iterations for converging the solution. In each sub-iteration, the solver: - computes the residual, i.e. the integral of the convective, diffusive, source term and unstationary part of the Navier-Stokes equation, over one tetrahedral cell, looping through all cells. Splits the residual according with the distribution scheme in fluctuations, and sends fluctuations to the nodes, where these are collected in some special fields. - communicates using PVM, with neighboring parallel processes, the fluctuations accumulated in nodes and some other useful information, for nodes laying on the domains interface. - imposes boundary conditions, - loops through all nodes, to update the nodal values using the fluctuations stored in those special fields. As it. can be seen, the algorithm needs no communication while computing and distributing the fluctuations. This step in the solver algorithm requires the longest computational time. It requires only one back and forward communication for each sub-iteration before applying the boundary conditions and updating the nodal values. The volume size of this communication is reduced, since only nodes on the domains interface exchange information.

320

Finite volumes for complex applications

6. Numerical Results

LES simulations of complex turbulent flows in engineering applications, using the present code, have been presented in [CAO 991[CAB 991. Here we only address the parallel scalability of code and the accuracy of our LES computations.

6.1. Parallel scalability

The code has been run on an SGI Origin 2000 platform, on up to 8 processors, and also on a cluster of 2 WinNT PC's, using an Ethernet connection. The results, addressing the parallel speed-up are presented graphically in figure 1. It can be seen that the performance, when using the two WinNT PC cluster, has been close to 75%, due to poor communication while using the Ethernet connection. The code proved instead to have excellent scalability on the Origin platform (98%) [CAD 991.

6.2. LES of channel flow

Results for the classical turbulent channel flow are presented. The Reynolds number based on the bulk velocity and the channel height, is 5800. No-slip isothermal boundary conditions have been set at the walls, while periodicity boundary conditions have been used both in the strearnwise (x) and the spanwise (2) directions. Computations have been performed using the dynamic D D M model. The grid size is ( 2 7 4 (6) (2~613)where 6 is the channel height. The grid has (85)(51)(51) nodes and aprox. 1,240,000 tetrahedral cells. The grid is stretched in the y-direction (yi = 6(1 - cos(pi))/2, for pi = ~ (- 1j) / ( N - I), j = 1,2, ...., N. Here N is thenumber of grid points in the y-direction).A uniform grid has been used in the streamwise and the spanwise directions. The flow has been simulated for enough flow-through times (the domain length in the strearnwise direction divided by the bulk velocity) to obtain a statistically stationary turbulent channel flow. Data for statistics are accumulated over the last 15 flow-through times. Planar averages are calculated by averaging for all points on planes parallel to the walls and in time, and the results are presented as a function of wall-distance (y) only, and are compared with experiments (Kreplin, 1979) and with D N S data (Kim, 1987). The averaged non-dimensional friction velocity E, = 0.06156 (i.e. uT/Ub, where Ub is the bulk velocity) compares well with the D N S and the experimental (i.e. u, = 0.0643) result. Figures 2 to 5 show planar averaged time averages of the mean of the axial velocity and Root-Mean-Square of velocity fluctuations (RMS of Favre U",V", W"). The results are normalized using the friction velocity, u,. The results obtained using the DDM model compare well with the DNS and the experimental results. Figure 6 shows an instantaneous dis-

Innovative schemes

321

tribution of the skin friction coefficient, Cf.

7. Future Work Future work will include extensive testing of different dynamic SGS models. The goal is to perform parallel LES for complex engineering applications.

[PI0 981

PIOMELLI,U., Large Eddy Si~nulation:present state and ,future directions

[ROE 821

ROE, P .L., Fluctuations and signals. A framework for numerical evolution problems, K.W. Morton and M.J. Baines, editors, Numerical Methods for Fluid Dynamics, Academic Press, 1982.

[PDE 971

PAILLERE,H , et al., Upwind residual distribution methods for compressible flows : An alternative for finite volume and finite element methods, VKI 28th CFD Lecture Series, March 1997.

[DSB 941

DECONINCK,H. et al., High resolution shock capturing cell vertex advection schemes on unstructured grids, VKI Lecture Series, March 21-25 , 1994.

[WKK 981

WOOD, WILLIAMA. et al., Diffusion characteristics of upwind schemes on un,ctructured Triangulations, AIAA 98-2443, Albuquerque, NM 1998.

[JAM 911

JAMESON,A., Time dependent computations using multigrid, with applications to unsteady flows past airfoils and wings, AIAA paper, 91-1596, 1991.

[CAB 991

CARAENI,D. et al., LES of spray in compressible flows on Unstructured Grids, AIAA 99-3762, Norfolk, 1999.

[CAO 991

CARAENI,D. et al., Large Eddy Simulation of the Flow in a Bladed Diffuser, TSFP, Santa Barbara, 1999

[CAD 991

CARAENI,D. et al., Parallel NAS3D: An efficient algorithm for enggineering spray simulations using LES, pres. C1-C, International Parallel CFD'99, 1999, Williamsburg.

[SCL 971

MITRAN,S. et al., Large Eddy Simulation of rotor stator interaction in centrifugal impeller, JPC, Seattle, July 1997.

[JHF 981

HELD,J . et al., Large Eddy Simulation of separated transonic flows around a wing section, AIAA 98-0405, Reno 1998.

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Finite volumes for complex applications

[SCF 981

CONWAY, S. et al., Investigation of the ,flow across a swirl generator using LES, AIAA 98-0921, Reno 1998.

[LIL 911

LILLY,D.K., A proposed modzfication of the Germano subgridscale closure method, Phys. Fluids A, 4:633-635, 1991.

[VNK 921

VENKATAKRISHNAN V. et al., A MIMD implementation o,f a parallel Euler solver ,for unstructured grids, The Journal of Supercomputing, Vol. 6 , 1992.

Fig. 1 Parallel Speedup

. ... .... DDM

Fig 2. Planar Time Averages of U

Fig 3. Planar averages of Unrms

1 2

---

-

3-

B 06

2

D

-

0 6

Exp

0 4

. .. . ... . D

0 2

M

Exp DNS

-DDM

0

0

0.25

05 ylh

0 75

-

1

Fig 4. Planar averages of V1'rms

I

I

0

025

0 5

075

1

yln

I

Fig 5. Planar averages of W"rms

-X

Fig. 6 Skin friction distribution

A higher-order-accurate upwind method for 2D compressible flows on cell-vertex unstructured grids

L. A. Catalano Istituto di Macchine ed Energetica Politecnico di Bari Via Re David 200, 70125 Bari, ITALY E-mail: [email protected]

~~p

-

-

-

~

p

p

~

p

p

- ---

A finite-volume method for the solution of two-dimensional inviscid compressible flows on cell-vertex unstructured grids is presented. The method is based on a novel bi-linear reconstruction of the unknowns and on a standard flux-diflerence-splitting scheme. Moreover, a new approach is proposed to achieve the same higher-order accuracy also near solid walls. The method is validated by computing the inviscid flow i n a two-dimensional cascade i n subsonic and transonic conditions.

ABSTRACT

Key Words: higher-order reconstruction, upwind, unstructured.

1. Introduction

In the last decade, a great effort has been devoted by many CFD researchers to the development of higher-order-accurate upwind solvers on unstructured grids. Different approaches have been proposed, including both finite-element and finite-volume discretizations [BAR 91, DEC 92, CAT 97, HAL 97, SEL 961, the major difficulty being the discretization of the inviscid terms in the conservation equations. Concerning the finite volume discretization, most of the upwind schemes proposed to date are based on a gradient-based reconstruction of the flow variables onto the two sides of the surface which defines the finite volume built around each node. Then, an approximate Riemann solver is applied a t each interface to select the upwind contributions.

324

Finite volumes for complex applications

How to perform a higher-order reconstruction on an unstructured triangular grid is not trivial: if the unknowns are located in the center of each triangle, the finite volume is the cell itself, and the higher-order reconstruction onto its sides becomes a very cumbersome task, because of the difficulty of defining and computing the flow gradients. For such a reason, many researchers have redirect their efforts to cell-vertex discretizations: a dual mesh is first constructed by tracing the medians of each cell, so as to build a finite volume around each node [BAR 93, SEL 96, HAL 971. Clearly, the interfaces are now located at the midpoint of each side. To knowledge of the author, all of these methods base the reconstruction on a nodal definition of the flow variable gradient. Different approaches have been proposed to compute its value from the gradients in the surrounding cells, but all of them appear complex and time-consuming, see e.g. [BAR 93, HAL 971. In this paper, an alternative, much simpler approach is proposed, which also includes a new higher-order near-wall discretization. 2. Numerical method 2.1. Governing equations

The governing equations for two-dimensional, compressible, inviscid flows, are written in integral form as:

as,

U = (p, pu, pv, peO)Tis In eq. 1, n is the inward normal of the contour of S , the vector of the conservative variables and 3.n = [(pv,), (puvn+p72,), (pvv,+ pn,), (phOvn)lTis the flux entering through the unit length of dS. As usual, p is the density, p is the pressure, e0 is the total internal energy and hO is the total enthalpy. Moreover, v will denote the velocity vector, with normal and tangential components v, = v . n and v, = v . s, respectively, and with Cartesian components u and v. The system of governing equations is closed by assuming perfect gas. 2.2. Space discretization

The domain is discretized by means of an unstructured mesh composed of triangles with unknows located at each cell-vertex. In particular, the primitive variable vector Q = ( p , ~ , v , is~ chosen )~ as unknown. A finite volume is constructed around each internal node by connecting the barycenters of two neighbouring triangles, see the node i in fig. 1. An upwind discretization of the RHS of eq. 1 is then obtained as follows: a left state and a right state are reconstructed at the interface (ij) defined on each side connecting the node i

Innovative schemes

%.

325

*..----.......'i----------i' ,

Figure 1: Construction of finite volumes for internal and boundary nodes. Determination of the cell C j i . Construction of the auxiliary cells. with each sorrounding node j. Then, a standard Riemann solver is used to select the upwind contributions to the flux. The left and the right states can be accurately reconstructed by employing a bi-linear variation of Q:

In eq. 2, lji and lij are the two opposite vectors pointing from the two nodes to the intersection of the interface with the side, see fig. 1. Concerning the definition of the gradients, first consider the interface (i + 112) of a uniform one-dimensional grid; in such a case, Q k I l 2 is linearly reconstructed as:

It must be remarked that the reconstruction is based on the gradient of Q in the left-neighbouring cell, rather than on the gradient defined in the node i . Similarly, in two dimensions, the gradient ( V Q ) j i (similar arguments hold for ( V Q ) i j )must be defined in one of the cells sharing the node j , rather than in the node itself, as it suggested in [BAR 93, HAL 971. The extension of the previous arguments to two dimensions suggests to define ( V Q ) j i as the gradient in the cell Cji which contains the prolongation of the side (ji), plotted as a dot-dashed line in fig. 1. Clearly, the choice of a cell-vertex triangular grid allows to define a bi-linear variation of Q in each cell, thus defining the cell = (VQ)c,,uniquely: gradient (VQ)ji

326

Finite volumes for complex applications

In eq. 4, Q k is the primitive variable vector in the node k . Moreover, the opposite side k has length s k and inward normal n k . Standard one-dimensional limiters can also be applied to the gradient, but have not been introduced yet. The flux-difference-splitting of Roe [ROE 861 is then used to solve the Riemann problem defined at each interface. The flux is computed as

k=l

In eq. 5, ak, k = 1,...,4, are the intensities of the entropy, of the shear and of the two acoustic waves, and X k , k = 1,...,4, are the corresponding propagation velocities:

Finally, e k , k = 1 , ..., 4 are the eigenvectors which project each wave contribution onto the conservative variable vector:

In eqq. 6 and 7, 6( ) = (

)R -

(

)L,

and - denotes the Roe averages:

2.2. Boundary conditions When a side lies on a boundary, the construction of the finite volumes is completed by tracing one third of the corresponding median and by adding half of the side t o the finite-volume contours of its two nodes: as an example, fig. 1 shows the resulting finite volume associa.ted to the boundary node b. Clearly,

Innovative schemes

327

the flux through each boundary side is computed directly by means of the nodal values; moreover, solid boundaries need only to add the corresponding pressure forces to the momentum equation. Characteristic boundary conditions are then applied to the nodal residuals to ensure that physical boundary conditions are satisfied. The lack of further cells beyond the boundaries make impossible to perform a higher-order reconstruction onto the near-boundary interfaces. The reduced

Figure 2: Computational mesh. accuracy is negligible for far-field boundaries, but a significant amount of numerical entropy would be generated near solid walls. A simple procedure to overcome this problem is here proposed: a row of auxiliary cells is created by adding an isosceles triangle beyond each solid face and by connecting the corresponding auxiliary nodes, see the dashed cells in fig. 1. The states in the auxiliary nodes are updated by imposing the following conditions of Isentropic Simple Radial Equilibrium a t the mid-point of the solid face:

R, being the radius of curvature a t wall. 2.3. Time integration The state in each node is updated by means of a two-stage Runge-Kutta explicit scheme with non-optimal coefficients 0.4 and 1 and CFL number 0.35.

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Finite volumes for complex applications

3. Results

The rnethod previously described has been applied to the computation of the inviscid flow in a cascade of VKI LS-59 gas turbine rotor blades, for two different values of the outlet pressure. The blade profile has been modified at the trailing edge by adding an artificial wedge (having no load) in order to simulate the presence of the recirculation zone, which cannot be captured by the present inviscid formulation. Fig. 2 shows the grid used in both the cases analyzed, obtained by slightly modifying a structured grid with 129 x 17 nodes (97 nodes are located on the blade surface). Clearly, the resulting quality of the grid is rather poor. The Mach number contours obtained for outlet Mi,

Figure 3: Mach number contours for outlet Mi, = 0.81 ( A M = 0.05). (isentropic Mach number) equal to 0.81 are shown in fig. 3. A good agreement

0

exp. (Kiock)

- computed

Figure 4: Experimental and computed distributions of Mi, on the blade for outlet Mi, = 0.81. between the computed distribution of Mi, on the blade and the experimental one [KIO 861 is visible in fig. 4.

Innovative schemes

Figure 5: Mach number contours for outlet Mi,= 1

329

(AM = 0.05).

The transonic flow with outlet Mi,equal to 1 has been then considered: the Mach number contours in fig. 5 show the formation of two shocks on the suction side of the blade, which are both sharply captured. Small oscillations are present and can be eliminated by applying a standard limiter. The surface Mach number distribution shown in fig. 6 is again in good agreement with the experimental data of [KIO 861, except for the discrepancy on the suction side of the blade, due to the presence of a separation bubble caused by the shockboundary layer interaction. Such a phenomenon is obviously missed by the present inviscid analysis.

Figure 6: Experimental and computed distributions of Mi,on the blade for outlet Mi,= 1.

3. Conclusions

A finite-volume method for the solution of two-dimensional inviscid com-

330

Finite volumes for complex applications

pressible flows on cell-vertex unstructured grids has been presented. In particular, a novel approach to the higher-order reconstruction of the unknowns has been proposed, and extended to the near-wall regions by means of a row of auxiliary cells. The method has been validated by computing the inviscid flow in a two-dimensional turbine cascade in subsonic and transonic conditions. The extension to the discretization of the viscous-flow conservation equations can be based on state-of-the-art methodologies. 4. Bibliography [BAR 911

BARTHT. J., Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations, Lecture Series 1991-06, Von Karman Institute, 1990.

[BAR 931

BARTHT. J . , A 3-D least-squares upwind Euler solver for unstructured meshes, Lecture Notes in Physics, 414, Springer Verlag, pp. 90-94, 1993.

[CAT 971

CATALANO L. A. et al., Genuinely multidimensional upwind methods for accurate and efficient solutions of compressible flows, Euler and Navier-Stokes solvers using multidimensional upwind schemes and multigrid acceleration, Notes on Numerical Fluid Mechanics, 57 Vieweg, Braunschweig, Germany,, 1997.

[DEC 921

DECONINCK H. et al., Multidimensional upwind methods for unstructured grids, Agard R-787, May, 1992.

[HAL 971

HALLOL. et al., An implicit mixed finite-volume-finiteelement method for solving 3D turbulent compressible flows, International Journal for Numerical Methods in Fluids, 25, pp. 1241-1261, 1997.

[KIO 861

KIOCKR. et al., The transonic flow through a plane turbine cascade as measured in four European wind tunnels, Transactions of the ASME, Journal of Engineering for gas turbines and power, 108, No. 2, pp. 277-284, 1986.

[ROE 861

ROE P. L., Characteristic based schemes for the Euler equations, Ann. Rev. Fluid Mech., 18, pp. 337-365, 1986.

[SEL 961

SELMINV., FORMAGGIA L., Unified construction of finite element and finite volume discretizations for compressible flows, International Journal for Numerical Methods in Engineering, 39, pp. 1-32, 1996.

A New Upwind Least Squares Finite Difference Scheme (LSFD-U) for ~ u l e r ~ ~ u a t i of o nGas s Dynamics N. Balakrishnan

Praveen. C

Assistant Professor G r a d ~ ~ nSt~rdent te CFD Centre, Department of Aerospace Engineering Indian Institute of Science, Bangalore - 560 012 ABSTRACT A new upwind Least Squares Finite Difference Scheme(LSFD-U) has been developed. The fundamental principle underlying this method is the representation of the derivatives of the fluxes appearing in the conservation laws, using a Generalised Finite Difference strategy based on the method of least squares. This method can operate upon any kind of grid and requires only local connectivity information pertaining to a cloud of grid points around any given node. The results obtained are very encouraging and the use of LSFD-U in the computation of flows past complex configurations is extremely promising. Key Words: Finite difference, Finite volume, Upwind schemes, Hyperbolic equations, Method of Least Squares. 1. Introduction One of the remarkable progresses made in the area of CFD, in recent years is the development of Grid Free Method [I-41 for numerically solving the conservation laws encountered in fluid dynamics. The fundamental principle underlying this method is the representation of the derivatives of the fluxes appearing in the conservation laws using a generalised finite difference strategy based on the method of least squares. This method which can operate upon any kind of grid (structured. unstructured or cartesian) requires only local connectivity information pertaining to a cloud of grid points around any given node. The utility of this method in computing flow past complex configurations is extremely promising. The present work draws its inspiration from the fact that this method has been applied only in the framework of flux vector splitting schemes and not in the framework of flux difference splitting schemes. Here we have attempted to extend the applicability of this Grid Free Method to the framework of flux difference splitting schemes and in the process arrived at an entirely new methodology equally applicable to flux vector splitting schemes. In section 2, we present briefly the details regarding the Least Squares Kinetic Upwind Method (LSKUM) [I-41. In section 3, the new least squares scheme is presented with a brief review of the flux difference splitting schemes. In section 4, we present the difficulties in extending the methodology to 2D and 3D flows, and also present two variations of the scheme, which would circumvent such problems. In section 5, we present the results and discussions. Concluding remarks are made in section 6.

2. Least Squares Kinetic Upwind Method (LSKUM) The kinetic schemes for solving Euler equations of gas dynamics are obtained by exploitin,o the fact that these equations are moments of the Boltzmann equation of kinetic theory for gases. Consider the ID Boltzmann equation : f,+vf,=O (1) where f is the velocity distribution function which is a iLlaxwellian and v is the molecular velocity. The fundamental principle underlying LSKUM is that the discrete approximation to f, appearing in the Boltzmann equation is obtained using a least squares approximation given

332

Finite volumes for complex applications

I, =-

( ~ f AX) ,

(2)

11~~112 where Af,=f,-f,; Ax,=x,-x,, subscript '0' denotes the node under consideration and 'j' its neighbour. The moment of the discretized Boltzmann equation will lead to an upwind scheme for the Euler equation, if the stencil of the grid points to be used in equation (2) is chosen, taking into account the direction of signal propagation. In other words, discrete approximation to f, at any given point is obtained by using the grid points on its left if v>O and vice versa. An interested reader is referred to the papers cited above for a number of interesting developments in LSKUM, such as the use of weighted least squares in equation (2). This idea when extended to the flux vector split framework of Euler equations, given by, U , + F,'+ F,=O (3) where U is the vector of conserved variables and F is the flux vector, the discrete approximation to F,'at any given point will involve grid points to its left, and F; will involve grid points to its right.

3. Upwind Least Squares Finite Difference Method (LSFD-U) Inspired by the fact that the discrete least squares approximation to the derivative F, involves the flux difference term AF, it was thought that it would be appropriate to make use of flux difference splitting in the least squares framework. Before we discuss the details of the present least squares algorithm, we briefly discuss the flux difference splitting scheme as applied to finite volume framework. \

\

\\

I I

\

I I

j-1

m

w

I I 1

Fig 1.

I

//'

I

/

I

0.'

'+,I ../=-='

I

Left State

I

\ \ \ \ \

\

I

i

a

I

I

\\

'\

a

I'

\ \

\

*'

,=

a

j+ 1

Right State

Typical 1D Finite Volume Computational Domain

Fig 1.depicts a typical I D finite volume computational domain. In the flux difference splitting scheme[5], the total flux difference AF = FR- FL is split into a positive part (AF)' corresponding to the right running waves and a negative part (AF)' corresponding to the left running waves, based on a suitable linearization procedure, in such a way that the interfacial flux F, is given by Fj = \

3

+ (AF)

= FR - (AF)+-

1.

(4)

In a finite volume framework, an interfacial flux obtained as an average of the above two expressions is made use of in the state update formula. Now we describe the present methodology. Equation 4 given above clearly suggests that the flux difference between any fictitious interface perpendicularly intersecting the line connecting the two grid points and the points themselves, can be obtained using a suitable linearization procedure. At the heart of the present methodology is the use of such flux differences based on upwinding principle in the discrete least squares approximation to F, appearing in the Euler equations. This leads to an upwind scheme based on the least squares principle. Also, it is not necessary that the present methodology should be used only in conjunction with Fiux Difference Splitting Schemes. The very fact that an upwind estimate of

Innovative schemes

333

the flux at the fictitious interface is all that is required for the determination of flux differences involved in the present method, suggests the use of this method in conjunction with all upwind flux formula. As can be easily seen, the present methodology makes use of a global stencil of grid points in contrast to the methodology described in section 2, which requires an upwind stencil of grid points. Also, we make use of an upwind estimate of the interfacial flux for determining AF's appearing in the least squares formula, in contrast to the explicit use of the nodal values of the fluxes in the earlier framework.

4. LSFD-U in 2D Here we present some of the interesting problems we face when extending the algorithm just described for solving 2D flows. Fig.2 gives a typical 2D stencil of grid points. Let I, represent the fictitious interface drawn across the line connecting the point under consideration

'0'.

and its j'h neighbour, and t?

represent the unit vector along

'oj'.

Representing the flux along oj by

we have

where F and G are fluxes in x and ;directions respectively. Now our job reduces to recovering the information regarding the gradients of the 2D fluxes namely VF, and Vcofrom the many I D flux difference terms given in equation 6. The derivatives F, and G, thus recovered would eventually be used in the state update formula. Equation 6 represents an over determined system of equations and it appears that the straight forward way to obtain the gradients of the fluxes is to minimise E? with respect to the derivatives of the fluxes, where, J

J

=

J

- (

A

n

- G o 64j I n r j

(7)

Unfortunately, simple algebra would demonstrate that such a procedure leads to a singular system. To circumvent this problem we suggest the following two methodologies. 4.1 Metirod I Method 1 draws its inspiration from the work of Ghosh and Deshpande (21. Here we effect locally a co-ordinate transformation (x.y)-t((,q), in such a way that one of the axes (5) coincides with the streamwise direction. It is a well known fact that the fluxes normal to the streamwise co-ordinate direction involve only pressure terms and a global stencil of grid points can be used for approximating the derivatives of such fluxes without loss in stability. If

-F and

represent the fluxes along the new co-ordinate directions

derivatives

5

5

and

I;and q respectively, the

can be obtained using the least squares procedure described in 131,

making use of a global stencil of grid points. Thus the streamwise rotation of the co-ordinates leaves us with a non singular system involving and iq. A simple least squares procedure

6

described in the previous section can be adopted for the estimation of derivatives

and G,,thus

@

and

iq. The

determined are substituted in the state update formula. It should

334

Finite volumes for complex applications

be remarked that in this method, the second and third components of the fluxes represent 6 and q momentum conservation.

and

4.2 Method 2

In this method we locally rotate the co-ordinate system from (x,y) j ( < , q ) , in such a way that

Brl +a5= 0.

where

-

F

and

6

represent the fluxes along the new co-

ordinate directions 6 and q respectively. Note that the second and third components of the flux

-

-

vectors Fand G still represent the x and y momentum conservation. It can easily be demonstrated that a co-ordinate system rotated at an angle P, given by,

2

would satisfy the condition that

+

= 0 . The gradients of the fluxes used in the

estimation of p are obtained using the least squares procedure [3] making use of a global stencil of grid points. This results in an overdetermined system of equations with two unknowns

9 and G- v , which can be solved using least squares procedure. The derivatives

thus determined are used in the state update formula. 5. Results and Discussions The new least squares upwind finite difference method (LSFD-U) is validated using standard ID and 2D test problems. In the computations high resolution is obtained based on a linear reconstmction procedure [6]. Non physical oscillations in the solution are suppressed using Venkatakrishnan limiter [7]. In all the computations presented in this work the fictitious interface is always placed at the mid point of the line segment under consideration. Figure 3 gives the results obtained for the I D shock tube problem of Sod [8]. The results are obtained on a non-uniform grid generated using cosine spaclng for grid points. One hundred grid points have been used in the computation. Roe [ 5 ] flux has been used in these calculations. The grids made use of in the 2D computations are presented in Fisure 4. The grid details are given in Table 1. All the 2D computations have been made with KFVS [9] flux formula.

Configuration Cylinder NACA 00 12

Table 1 No. of nodes 4317 4733

No. of nodes on the wall 160 160

The pressure contours obtained for a low subsonic flow past a cylinder(in the incompressible limit; M,=0.1) are presented in Figures 5 and 6. The wall pressure data are compared with the exact potential flow solution in Figure 7. The Mach contours obtained for subsonic(M,=0.63, angle of attack=lU) and transonic (M,=0.85, angle of attack=lU)flows past NACA0012 airfoil are presented in Figures 8,9,11 and 12. The Cp values are compared with those obtained using the cell vertex finite volume scheme in Figures 10 and 13. The first order results obtained using Method 2 are identical to those obtained using Method 1, and therefore are not presented here. The CL and CD values obtained using LSFD-U are compared with GAMM[IO] results in Table 2.

Innovative schemes

Table 2 R.I,=0.63, AOA=Z0

Method I II GAMM

C~

CD

0.3387 0.3343 0.3335

0.00 14 0.0006 0.0000

335

M,=0.85, AOA=IO

cL

0.3772 0.3806 0.3790

CD 0.0550 0.0545 0.0576

From the results it is evident that the new LSFD-U framework is capable of capturing all features expected out of inviscid compressible flows. Also, it should b e remarked that the linear reconstruction procedure developed for the finite volume schemes, when used in conjunction with the present LSFD-U framework, can produce excellent improvement in solution resolution.

6. Conclusions A new upwind least squares finite difference method has been developed. T h e new scheme by the virtue of using a least squares framework can b e considered as a Grid Free Method. It has an added advantage of making use of a global stencil. T h e way the interracial fluxes are calculated in the new scheme resembles that of finite volume method and therefore all t h e developments that have taken place in finite volume method, like the method of reconstruction [6] can b e directly adopted. T h e use of LSFD-U in the computation of flows past complex configurations is extremely promising.

Acknowledgements The authors like to thank Mr J. C. Mathur. NAL. Bangalore. for providing the unstructured mesh generator. The authors like to thank Mr. Krishnakumar, under graduate student, IIT. Kharagpur. for his involvement in the initial phases of development of the code used in the computations, during his stay at IISc, Bangalore. as a Summer trainee. The authors also like to express their gratitude to Mr. Hiwish. R, project assistant. CFD Centre, IISc, for his invaluable help in preparing the manuscript.

References S.M. Deshpande, A.K. Ghosh and J.C Mandal. "Least Squares Weak Upwind Method for Euler Equations", 89 FM 4. Fluid Mechanics Report. Department of Aerospace Engineering. Indian Institute of Science, Bangalore. A.K. Ghosh, "Robust Least Squares Kinetic Upwind Method for Inviscid Compressible Flows", June 1996, PhD thesis. S.M. Deshpande, P.S. Kulkmi, and A.K. Ghosh, "New Developments in Kinetic Schemes", Computers Iblath. Applic., Vol 35, No 112, pp. 75-93, 1998 K. Anandhanarayanan, D.B. Dhokrikar, V. Ramesh and S.M. Deshpande. "A Grid Free Method for 2D Euler Computations using Least Squares Kinetic Upwind Method", Proceedings of the third Asian Computational Fluid Dynamics Conference, pp. 390, vol 2, December 7' - 1 lLh1998, Bangalore, India. Roe, Philip. L. "Approximate Rrimann Solvers Parameter Vectors and Difference Schemes", Journal of Computational Physics, Vol. 43, pp. 357-372, 1981 T.J.Banh " Higher Order Solution of the Euler Equations on Unstructured Grids using Quadratic Reconstruction", AUA-90-0013, 1990 V. Venkatakrishnan. "Convergence of Steady State Solutions of Euler Equations on Unstructured Grids with Limiters", JCP, vol 118. pp. 120, 1995. G.A. Sod, "A Survey of Several Finite Difference Methods for system of Nonlinear Hyperbolic Conservation Laws", JCP, vol 27, pp. 1, 1978. J.C. Mandal and S.M. Deshpande, " Kinetic Flux Vector Splitting for Euler Equations", Computers and Fluids. vol 23. No 2. pp 447-478. GAMM Workshop on Numerical solutions of Compressible Euler Flows, June 1986.

336

Finite volumes for complex applications

Fig 2. Typical 2D grid distribution for LSFD-U

Fig 3. LSFD-U applied to shock tube problem

Fig 4. Unstructured grid used for 2D computations Linear Reconstruction

Fig 5 Pressure contours obtained using Method 1 for M,=O.l

Linear Reconstruction

Fig 6 Pressure contours obtained using Method 2 for kI,=O.l

Innovative schemes

First Order

337

Linear R e s a n s t i u s t . a ~

Fig 8 Mach contours using Method 1; M,=0.63 and AOA=2' L~naar Reconstructban

:Reco~ImitC1 - MI M2.I Ore.,

x/c

Fig 9 Mach contours using Method 2 for M,=0.63 and AOA=2"

Fig 10 Pressure distribution for M20.63 and AOA=2"

338

Finite volumes for complex applications F i r i t Ordar

Fig 11 Mach contours using Method I; hI,=0.85 and AOA=1°

Linear Rsconslruc:~cn

Fig 12 Mach contours using Method 2 for M,=0.85 and AOA=1°

Fig 13 Pressure distribution for M,=0.85 and AOA=1°

A finite-volume algorithm for all speed flows F. Moukalled and M. Darwish American University of Beirut, Faculty of Engineering & Architecture, Mechanical Engineering Department, P. 0.Box 11-0236, Beirut, Lebanon. ABSTRACT. A new collocatedfinite volume-based solution procedure for predicting viscous compressible and incompressible flows is presented. The technique is equally applicable in the subsonic, transonic, and supersonic regimes. Pressure is selected as a dependent variable in preference to dens;@ because changes in pressure are significant at all speeds as opposed to variations in density which become vevy small at low Mach numbers. The newlv developed algorithm has two new features; (i) the use of the Normalized Chriable and Space Formulation methodologv to bound the convective flzixes; and (ii) the use of a HighResolz~tionscheme in calculating interface densi@ values to enhance the shock capturing proper@ ofthe algorithm. Keywords; Pressure-based method, All speedflows, High-Resolution algorithm.

1. Introduction

In Computational Fluid Dynamics (CFD) a great research effort has been devoted to the development of accurate and efficient numerical algorithms suitable for solving flows in the various Reynolds and Mach number regimes. The type of convection scheme to be used in a given application depends on the value of Reynolds number. On the other hand, the Mach number value dictates the type of algorithm to be utilized in the solution procedure. These algorithms can be classified into two groups: density-based methods and pressure-based methods, with the former used for high Mach number flows, and the latter for low Mach number flows. The ultimate goal however, is to develop a unified algorithm capable of solving flow problems in the various Reynolds and Mach number regimes. To understand the difficulty associated with the design of such an algorithm, it is important to understand the role of pressure in cotnpressible flow [KAR 861. In the low Mach number limit where density becomes constant, the role of pressure is to act on velocity through continuity so that conservation of mass is satisfied. Obviously, for low speed flows, the pressure gradient needed to drive the velocities through momentum conservation is of such magnitude that the density is not affected significantly and the flow can be considered nearly incompressible. Hence, density and pressure are very weakly related. As a result, the continuity equation is decoupled from the momentum equations and can no longer be considered as the equation for density. Rather, it acts as a constraint on the velocity field. Thus, for a sequential solution of the equations, it is necessary to devise a mechanism to couple the continuity and momentum equations through the pressure field. In the

340 Finite volumes for complex applications

hypersonic limit where variations in velocity become relatively small as compared to the velocity itself, the changes in pressure do significantly affect density. In this limit, the pressure can be viewed to act on density alone through the equation of state so that mass conservation is satisfied [KAR 861 and the continuity equation can be viewed as the equation for density. The above discussion reveals that for any numerical method to be capable of predicting both incompressible and compressible fluid flows the pressure should always be allowed to play its dual role and to act on both velocity and density to satisfy continuity. Several researchers [KAR 86, RHI 86, MAR 94, DEM 93, LIE 931 have worked on extending the range of pressure-based methods to high Mach numbers. In most of the published work the first order upwind scheme is used to interpolate for density, exception being in the work presented in [DEM 931 where a central difference scheme blended with the upwind scheme is used. The bleeding relies on a factor varying between 0 and 1, which is problem dependent and has to be adjusted to eliminate oscillation or to promote convergence. In the work presented in [LIE 931 the retarded density concept is utilized in calculating the density at the control volume faces. This concept is based on factors that are also problem dependent and requires the addition of some artificial dissipation to stabilize the algorithm (second-order terms were introduced), which complicate its use. To this end, the objective of this paper is to present a newly developed collocated pressure-based solution procedure that is equally valid at all Reynolds and Mach number values. The algorithm will have two new features. The first one is the use of the Normalized Variable Formulation (NVF) [LEO 871 andlor the Normalized Variable and Space Formulation (NVSF) [DAR 941 methodology in the discretization of the convective terms. The second one, is the use of HighResolution (HR) schemes in the interpolation of density in the source of the pressure correction equation and the convective fluxes in order to enhance the shock capturing capability of the method. The increase in accuracy with the use of HR schemes for density is demonstrated by comparing predictions, for the flow over a bump, obtained using the third-order SMART scheme for all variables except density (for which the Upwind scheme is used) against another set of results obtained using the SMART scheme for all variables including density. 2. Finite volume discretization of the transport equations

The conservation equations governing two-dimensional compressible flow problems may be expressed in the following general form:

at

+ V -(p$)= 0 .(r'v$)+Q4

111

where $ is any dependent variable, v is the velocity vector, and p, I?, and Q' are the density (=P/RT), diffusivity, and source terms, respectively. Integrating the above equation over a control volume (Fig. 1) and applying the divergence theorem, the following discretized equation is obtained: datp p $ k ~ + ( ~ e + +J,, ~ w+ J ~ ) = Q @ V

[21

Innovative schemes where J f represents the total flux of 4 across face 'f and is given by J~ = ( P V ~ - T@v() .sf

341

PI

Each of the surface fluxes Jf contains a convective contribution, J: , and a diffusive contribution, J: , hence:

J , = J;

+ J;

where

J;- bv+),.sf J:(-r4v4).s,

[51 The diffusive flux at the control volume face 'f is discretized using a linear symmetric interpolation profile so as to write the gradient as a function of the neighboring grid points. The convective flux across face f can be written as: JF = cnf [61 where Cf is the convective flux coefficient at cell face 'f. As can be seen from [6] the accuracy of the control volume solution for the convective scalar flux depends on the proper estimation of the face value $f as a function of the neighboring $ nodes values. Using some assumed interpolation profile, $f can be explicitly formulated in terms of its node values by a functional relationship of the form: ef = f(e*) [71 where $,b denotes the neighboring node I$ values. After substituting [7] into [6] for each cell face and using the resulting equation along with the discretized form of the diffusive flux, [2] is transformed after some algebraic manipulations into the following discretized equation: =

NWP)

where the coefficients a$ and aRBdepend on the selected scheme and b$ is the source term of the discretized equation. \

Figure 1. Control volume.

3. The NVSF methodology for constructing HR schemes As mentioned earlier, the discretization of the convection flux is not straightforward and requires additional attention. Since the intention is to develop a high-resolution algorithm, the highly diffusive first order UPWIND scheme [PAT 811 is excluded. As such, a high order interpolation profile is sought. The difficulties associated with the use of such profiles stem from the conflicting requirements of accuracy, stability, and boundedness. Solutions predicted with high order profiles tend to provoke oscillations in the solution. To suppress these

342

Finite volumes for complex applications

oscillations, the composite flux limiter method [LEO 881 is adopted here. The formulation of high-resolution flux limiter schemes on uniform grid has recently been generalized in [LEO 881 through the Normalized Variable Formulation (NVF) methodology and on non-uniform grid in [DAR 941 through the Normalized Variable and Space Formulation (NVSF) methodology. To the authors' knowledge, the NVSF formulation has never been used to bound the convection flux in compressible flows. It is an objective of this work to extend the applicability of this technique to compressible flows. For more details the reader is referred to [DAR 941.

4. High resolution algorithm The need for a solution algorithm arises in the simulation of flow problems because a scalar equation does not exist for pressure. Hence, if a segregated approach is to be adopted, coupling between the u, v, p, and P primitive variables in the continuity and momentum equations will be required. The segregated algorithm adopted in this work is the SIMPLE algorithm [PAT 811, which involves a predictor and a corrector step. In the predictor step, the velocity field is calculated based on a guessed or estimated pressure field. In the corrector step, a pressure (or a pressure-correction) equation is derived and solved. Then, the variation in the pressure field is accounted for within the momentum equations by corrections to the velocity and density fields. Thus, the velocity, density, and pressure fields are driven, iteratively, to better satisfying the momentum and continuity equations simultaneously and convergence is achieved by repeatedly applying the above-described procedure. The key step in deriving the pressure-correction equation is to notice that in the predictor stage a guessed or estimated pressure field from the previous iteration, denoted by P("), is substituted into the momentum equations, the resulting velocity field, denoted by v * , which now satisfies the momentum equations, will, in general, not satisfy the continuity equation. Thus, a correction is needed in order to obtain a velocity and pressure fields that satisfy both equations. Denoting the pressure, velocity, and density corrections by P', v'(u', v'), and p', respectively, the corrected fields are given by: P = P ( " ) + P ' , v = v * + v f , and p = p ( n ) + p ' [91 Combining momentum and continuity and substituting P, v, and p using [9], the final form of the pressure-correction equation is: P' P' P' P' P' P' apPi,=aEPi+awPb+aNPh+aSPi+bp where

From [11] it is clear that the starred continuity equation appears as a source term in the pressure correction equation. Moreover, in a pressure-based algorithm, the pressure-correction equation is the most important equation that gives the pressure, upon which all other variables are dependent. Therefore, the solution accuracy depends on the proper estimation of pressure from this equation. Definitely, the

Innovative schemes

343

more accurate the interpolated starred density ( p * ) values at the control volume faces are, the more accurate the predicted pressure values will be. The use of a central difference scheme for the interpolation of p* leads to instability at Mach numbers near or above 1 [KAR 86, DEM 931. On the other hand the use of a first order upwind scheme lead to excess diffusion [KAR 861. The obvious solution would be to interpolate for values of p* at the control volume faces the same way interpolation for other dependent variables is carried. That is to employ the bounded HR family of schemes for which no problem-dependent factors are required. Adopting this strategy, the discretized form of the starred continuity equation becomes:

The same procedure is also adopted for calculating the density when computing the mass flow rate at a control volume face in the general conservation equation.

5. Results and discussion The validity of the above described solution procedure is demonstrated in this section by presenting solutions to the inviscid flow over a bump. The physical situation consists of a channel of width equal to the length of the circular arc bump and of total length equal to three lengths of the bump. Results are presented for three different types of flow (subsonic, transonic, and supersonic). For subsonic and transonic calculations, the thickness-to-chord ratio is 10% and for supersonic flow calculations it is 4%. In all flow regimes, predictions obtained over a relatively coarse grid using the SMART scheme for all variables including density are compared against results obtained over the same grid using the SMART scheme for all variables except density, for which the UPWIND scheme is used. Due to the unavailability of an exact solution to the problem, a solution using a dense grid is generated and treated as the most accurate solution against which coarse grid results are compared.

5.1 Subsonic flow over a circular arc bump With an inlet Mach number of 0.5, the inviscid flow in the channel is fully subsonic and symmetric across the middle of the bump. Isobars displayed in Fig. 2(a) reveal that the coarse grid solution obtained with the SMART scheme for all variables falls on top of the dense grid solution. The use of the upwind scheme for density however, lowers the overall solution accuracy. The same conclusion can be drawn when comparing the Mach number distribution along the lower and upper walls of the channel. As seen in Fig. 2(b), the coarse grid profile obtained using the SMART scheme for density is closer to the dense grid profile than the one predicted employing the upwind scheme for density. The difference in results between the coarse grid solutions is not large for this test case. This is expected since the flow is subsonic and variations in density are relatively small. Larger differences are anticipated in the transonic and supersonic regimes.

344

Finite volumes for complex applications

(b) Figure 2. SubsonicJlow over a 10% circular bump; (a) isobars and (b) profiles along the walls. 5.2 Transonic flow over a circular arc bump

Results for an inlet Mach number of 0.675 are displayed in Fig. 3. Figure 3(a) presents a comparison between the coarse grid and dense grid results. As shown, the use of the HR SMART scheme for density greatly improves the predictions. Isobars generated over a coarse grid (63x16 c.v.) using the SMART scheme for all variables are very close to the ones obtained with a dense grid (252x54 c.v.). This is in difference with coarse grid results obtained using the upwind scheme for density and the SMART scheme for all other variables, which noticeably deviate from the dense grid solution. This is further apparent in Fig. 3(b) where Mach number profiles along the lower and upper walls are compared. As shown, the most accurate coarse grid results are those obtained with the SMART scheme for all variables and the worst ones are achieved with the upwind scheme for all variables. The maximum Mach number along the lower wall (z1.41), predicted with a dense grid, is in excellent agreement with published values [DEM 931. By comparing course grid profiles along the lower wall, the all-SMART solution is about 11% more accurate than the solution obtained using SMART for all variables and upwind for density and 2 1% more accurate than the highly diffusive all-upwind solution. 5.3 Supersonicflow over a circular arc bump

Computations are presented for an inlet Mach number value of 1.4. Mach number contours are compared in Fig. 4(a). As before, the course grid all-SMART results (58x18 c.v.), being closer to the dense grid results (158x78 c.v.), are more accurate than those obtained when using the upwind scheme for density. The Mach profiles along the lower and upper walls, depicted in Fig. 4(b), are in excellent

Innovative schemes

345

agreement with published results [NI 821 and reveal good enhancement in accuracy when using the SMART scheme for evaluating interface density values. The use of the upwind scheme to compute density deteriorates the solution accuracy even though a HR scheme is used for other variables. The all-upwind results are highly diffusive.

--

-

--

1-6

'--T

ri

r,

.,

i

I * -

L , .

-

A

-

.

.

._id

- -

X

(b) Figure 3. Transonic flow over a 10% circular bump; (a) isobars using various schemes, and (b) profiles along the walls.

(b) Figure 4. Supersonic flow over a 4% circular bump (Mi,,=1.4); (a) Mach number contours using various schemes, (b) proJiles along the walls.

346

Finite volumes for complex applications

6. Concluding Remarks A new high-resolution pressure-based algorithm for the solution o f fluid flow at all speeds was formulated. The new features in the algorithm are the use o f a HR scheme in calculating the density values at the control volume faces and the use of the NVSF methodology for bounding the convection fluxes. Results obtained were very promising and revealed good enhancement in accuracy at high Mach number values when calculating interface density values using a High-Resolution scheme.

7. Acknowlegments The financial support provided by the European Office o f Aerospace Research and Development (EOARD) (SPC-99-4003) is gratefully acknowledged.

8. Bibliography [DAR 941 Darwish, M.s. and Moukalled, F.," Normalized Variable and Space Formulation Methodology For High-Resolution Schemes," Numerical Heat Transfer, Part B, vol. 26, pp. 79-96, 1994. [DEM 931 Demirdzic, I., Lilek, Z., and Peric, M.,"A Collocated Finite Volume Method For Predicting Flows at All Speeds," International Journal for Numerical Methods in Fluids, vol. 16, pp. 1029-1050, 1993. [KAR 861 Karki, K.C.,"A Calculation Procedure for Viscous Flows at All Speeds in Complex Geometries," Ph.D. Thesis, University of Minnesota, June 1986. [LEO 871 Leonard, B.P.,"Locally Modified Quick Scheme for Highly Convective 2-D and 3-D Flows," Taylor, C. and Morgan, K. (eds.), Numerical Methods in Laminar and Turbulent Flows, Pineridge Press, Swansea, U.K., vol. 15, pp. 35-47, 1987. [LEO 881 Leonard, B.P.,"Simple High-Accuracy Resolution Program for Convective Modelling of Discontinuities," International Journal for Numerical Methods in Engineering, vol. 8, pp. 129 1-1318, 1988. [LIE 931 Lien, F.S. and Leschziner, M.A.,"A Pressure-Velocity Solution Strategy for Compressible Flow and Its Application to ShockiBoundary-Layer Interaction Using Second-Moment Turbulence Closure," Journal of Fluids Engineering, vol. 115, pp. 717-725, 1993. [MAR 941 Marchi, C.H. and Maliska, C.R.,"A Non-orthogonal Finite-Volume Methods for the Solution of All Speed Flows Using Co-Located Variables," Numerical Heat Transfer, Part B, vol. 26, pp. 293-31 1, 1994. [NI 821 Ni, R.H.,"A Multiple Grid Scheme for Solving the Euler Equation," AIAA Journal, vol. 20, pp. 1565-1571, 1982. [PAT 811 Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere, N.Y., 1981. [RHI 86) Rhie, C.M.,"A Pressure Based Navier-Stokes Solver Using the Multigrid Method," AIAA paper 86-0207, 1986.

Preserving Vorticity in Finite-Volume Schemes

Philip Roe and Bill Morton Dcpar-tment of Aerospace Enganeerzng The linz~~erszty of Mzchzgan Ann Arbor, Mzchzgan 48109-2140 ( '5,4

Department of Mathematical Sciences Universzty of Bath Bath B A 2 YAY United Iizngdorn

ABSTRACTW e discuss the fact that m a n y otherwise accurate finite-volume schemes have a tendency t o yield anomalous solutions i n certain circumstances. These are strongly linked t o the appearance of spurious vorticity. For a model problem, we show that certain finite volume methods i n fact preserve vorticity. Although these are not new schemes, they are not currently fashionable. A possibility exists t o modify t h e m so that they are are of high-resolution and upwind with respect t o acoustic waves. Ii'ey Words: Conservation laws, Euler equations, vorticity

1.

Introduction

Vorticity is a very important aspect of many fluid flows, especially in three dimensions, on account of the great times for which it persists in a highReynolds number flow following its initial creation. The velocities "induced" by vorticity are essential t o the operation of many fluid devices,and the behaviour of those regions where vorticity is concentrated are the key to the generation of sound. For computational purposes, incompressible flows are often formulated in variables t h a t include vorticity, and "vortex methods" t h a t explicitly track

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concentrated vortex cores are also widely used. For compressible flow, vorticity is hardly less important, but it is rare for it to be included in the computational formulation. Instead, the stress is placed on conservation so as to ensure the correct capturing of shocks. The consequence is that vortical structures are often badly diffused by numerical dissipation and this is a serious impediment to the prediction of complcated viscous flows. We have recently begun [MOR 991 construction of finite-volume methods that exert control over vorticity. We make no attempt to solve the vorticity evolution equations themselves, but try to find schemes such that a correct evolution of the discrete vorticity is an exact consequence of the discrete conservation laws, just as at the continuum level. Another way of expressing this is to say that whatever form of numerical dissipation is employed to stabilise the propagation of waves should have no impact on the advection of vorticity. There is a further motivation, which is that at the level of the Euler equations many otherwise accurate methods, especially those that attempt to minimise numerical dissipation by adopting upwind strategies (so that any disturbance is damped by the smallest damped compatible with its propagation speed) notoriously produce nonphysical solutions (carbuncle phenomenon) under some special (but not uncommon) circumstances. It now appears that these solutions are in fact quite legitimate solutions of the Euler equations, and could only be excluded by some selection principle over and above that of entropy-satisfaction. On the basis of examples encountered to date, all of these undesirable solutions appear to feature much greater vorticity that the anticipated solution. A very simple example is given in Section 2. There is also the related phenomenon discovered by Quirk [QUI 941; that the same schemes that give rise to carbuncles can be "destabilsed" by very small grid perturbations. In simple academic examples it is usually easy to detect that something is wrong with the solution, and to tinker with the code until it goes away, but in situations where the flow is anyway expected to be complex there may be no simple way to detect unwanted phenomena. There is therefore an urgent need to understand these anomalies and to find rneans of avoiding them.

2.

Nonunique Solutions of the Euler Equations

Consider the supersonic flow past a flat-nosed two-dimensional body (although the argument works just as well for a flat-nosed cylinder with axial symmetry). We expect the solution to feature a rather smooth detached bow shock, giving rise to an embedded subsonic region, from which the flow accellerates smoothly to regain supersonic speed. This is shown in the top half of Figure 1. In the bottom half is an alternative solution. Plane oblique shocks originate from a more-or-less arbitrary point on the line of symmetry. Behind them is a triangular region of stagnant fluid extending to the shoulder of the body. Outside of this region the flow behaves exactly as it would if flowing past a triangular wedge (or cone) of solid matter. The pressure, velocity, etc.

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Figure 1. T w o possible Euler solutions for the the flow past a flat-faced body. in that region may be found from the usual compressible tables and formulae. Prandtl-Meyer rarefaction waves return the flow to its original direction and weaken the initial shocks. Such a flow is in fact possible t o create experimentally, and is illustrated in Plate 272 of [VAN 821 which is a schlieren picture taken in a windtunnel. The unusual flow configuration is the result of a thin "splitter plate" placed along the axis AB. Presumably what happens is that the boundary layer on the plate creates vorticity that is then diffused outward. In the limit of very high Reynolds number the vorticity eventually becomes concentrated in the infinitesimal shear layers AC, AD There is no net circulation generated in the flow but each half of the stagnant region (above and below the axis) has circulation around it. This flow is a valid second solution of the Euler equations, providc that the pressure in the stagnant region matches the pressure outside it. The temperature in the stagnant region is arbitrary if we allow that an entropy layer might also exist, so there is a doubly-infinite family of solutions. All the shocks are entropy-satisfying, but under some conditions the shear layers might not be stable. One may now imagine that the corners of the body are steadily rounded off until the nose becomes a semi-circle. The alternative solutions would become the carbuncles. According to Pandolfi and L)'Ambrosio, who have made very detailed observations [PAN 991 the solutions produced in the carbuncle phenomenon are also genuine Euler solutions. One might hope that by solving the Navier-Stokes equations instead the carbuncle would automatically disappear. However, Gressier and Moschetta

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[GRE 991 report that the Reynolds number had to be reduced t o as little as 100 before this happened. It seems that the mechanisms by which the Euler code introduces the vorticity is stronger than the mechanism by which the Navier-Stokes code removes it. The Quirk phenomenon occurs when a plane one-dimensional shock is propagated along one direction of a rectangular grid. If the grid is indeed perfectly rectangular then one recovers the one-dimensional solution but if one row of grid points, parallel to the direction of motion, is very slightly perturbed, an exponentially-growing pat,tern of high-frequency disturbances is generated that eventually yields a flow pattern not unlike that seen in carbuncles. Robinet et a1 [ROB 991 attempt to relate this to a new observation concerning the linear stabilty of a plane shock. Although this is a classical flow stability problem, they have discovered a 'strange mode' overlooked by previous analysts. They find perturbations of a plane two-dimensional flow by separation of variables. This leads to an eigenvalue problem having linearly independent solutions except for certain combinations of frequency in time and wavenumber in space. Rather unexpectedly, these exceptional solu~ionsinvolve a resonance in which an acoustic mode and a vortical mode become indistinguishable. The linear algebra problem has t o be completed by a Jordan block with a generalised eigenvector. Coupling this with the equations governing the shock perturbation reveals a solution that can grow exponentially with time. Many features of this solution are shown to appear in numerical experiments on the Quirk phenomenon. Experience to date is that all numerical schemes that display the carbuncle phenomenon also display the Quirk phenomenon. There is a proof [GRE 981 t h a t all schemes capable of resolving a parallel shear layer without dissipation (a highly desirable property, and one responsible for the widespread adoption of flux-difference-splitting schemes rather than flux-splitting schemes) will in fact display the Quirk phenomenon. This collection of facts almost suggests a crisis situation. In the next section we present analysis, condensed from [MOR 991 that opens an avenue of escape.

3.

A Model Problem

The simplest model problem to combine wave propagation and vorticity is the system wave equation. We will write this in two space dimensions in the matrix form, using a notation corresponding to acoustic waves in a fluid that is stationary in the mean, with pressure p* and velocity 2 = ( u * ,u * ) , thus

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Cells Vertices

0

35 1

p, u, v p', u', v'

Figure 2. Grid definition. Here u = ( p * / ( p c 2u) ,* / c ,v * / c ) r ( p , .u, v , ), c is the sound speed in the mean flow, and

Restriction to two dimensions is merely for economy of notation; all of the analysis extends very straightforwardly to three dimensions. We study the wave equation in system rather than scalar (dttu = c 2 V 2 u )form for two reasons. Firstly because this is the form of the wave equation that is hidden inside the Euler equations, whether in their two-dimensional unsteady or in their three-dimensional supersonic steady forms. Secondly, because the scalar form automatically implies vanishing vorticity, whereas the interaction of the waves with vorticity is one of the aspects we want to study. Here the interaction is very simple, as befits a model problem. We easily deduce from (1,2) that

at< = 0 ,

where

< = d X v- a y u .

In other words, there is no interaction and any initial distribution of vorticity is preserved. Maintaining this independence at the discrete level will be our objective.

3..

Discrete Notation

In this paper we concentrate on simple finite-difference formulations on uniform square grids, such that the spacing in the x and y directions is h and the time step is At, with u:,~a discrete approximation located at ( x , y,t) =

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(ih, jh, n A t ) . The standard discrete differencing and averaging operators are defined by

The points with integer coordinates will be called cells, those with one integer and one half-integer coordinate edges and those with two half-integer coordinates vertices. The variables stored at cells will be (p, u , 7)). The same variables stored at vertices will be distinguished by primes where neccessary. The variables stored at vertical edges will be ( P , U ) and those stored at horizontal edges (Q, V). In a finite-volume interpretation the edge quantities are the fluxes. (See Figure 2) If it is recognised that each application of the above operators moves the mesh values t o a different set of grid points then arbitrarily long products of operators are allowed and all multiplications commute.

3.2.

Conservation Form

In the cell-centred finite-volume method, discrete conservation is ensured by drawing a control volume around the grid point of interest, and writing the update as an integral around this volume. In the generic case of a vector U of conserved variables, with fluxes F, G in the (z, y)-directions respectively, one has

where F*,G* are numerical fluxes evaluated from some formula to be determined. In the present case we can write, with u = cAt/h and following the notation of Figure 2,

It will usefully restrict the schemes to require that they can be written in this form. A second-order scheme of the Lax-Wendroff type follows from taking U , V, P, Q t o be estimates halfway through the time step. However, we will find subsequently that it is a rather special type of conservation form that emerges from the analysis.

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3.3.

353

Preserving Vorticity

The first step in creating a scheme that preserves vorticity is t o define a discrete vorticity, which we do as follows

This will be preserved if

and the condition p,P = p y Q will be met if we take P = p y r l ,Q = p z r l where r1 is some quantity defined at vertices. The only way to define a consistent local pressure while retaining a nine-point stencil is to take

In that case we have

To obtain second-order accuracy, r1 must now be updated to half-way through the time step. The simple formula

is the unique symmetrical formula t o achieve this without enlarging the stencil, leading to

We remark here that a general example of the Lax-Wendroff family will generate vorticity at a rate proportional to h3; this is shown in [MOR 991.

3.4.

Complete Evolution Operator

We construct the matrix operator that will update the solution, so that if

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certain elements of the matrix Ma a.re already uniquely determined by insisting that the velocities are updated with second-order accuracy while preserving the discrete vorticity (6). We have in fact, from (4) ,(5),(9),(10),

The adjoint property divh = -grad: requires that this matrix be symmetric [MOR 991 and an agument based on maintaining a compact stencil leads to the complete matrix operator

The scheme represented by this matrix has been uniquely determined by the requirements of conservation, vorticity preservation, symmetry of the solution under grid transformations, adjoint symmetry of the discrete operator, and second-order accuracy. However it is not a new scheme. It can be recognised by noting that M A can be factored as

where

and therefore can be written as a two-step scheme. The operation

gives a provisional solution at the vertices. The operation

completes the update by integrating around the vertices. This is in fact the version of Lax-Wendroff known as the Rotated Richtmyer scheme[RIC 621. It is shown schematically on the left of Figure 3. The original motivations for this scheme were compactness, computational economy and stability. In the nonlinear case, as in all two-step Lax-Wendroff schemes, one avoids any multiplication by the Jacobian matrices. The vorticity-preserving property does not seem t o have been previously noticed. It is shown in [MOR 991 that the scheme is stable for the maximum possible CFL range c A t / h 5 1.

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Figure 3. (left) The Rotated Richtmyer scheme. In the first step, symbolised by white arrows, data from the cells is used to create a half time-step solution at the vertices. In the second step (black arrows) integration round the vertices updates the central cell. (right) Ni's cell-vertex scheme. In the first step (white arrows) we zntegrate around the cells to obtain a 'cell-residual', In the second step (bluck arrows) these are distributed to the vertices. 3.5.

Duality

Since bot,h factors of M A depend only on LA they commute, and so the scheme nlay also be written as

In this form it is Ni's cell-vertex scheme [NI 821, in which the variables are usually thought of as located a t vertices of a grid, defining a bilinear interpolant over a square element. T h e first step LA is to integrate d,F dyG over this element and the second step is to distribute this to the nodes of the element. This distribution operation is described by the first factor in (17)

+

4.

Commentary

By choosing a correct definition of discrete vorticity, the results above can be extended [MOR 991 t o the linear wave equation on unstructured two- and

three-dimensional meshes, and to problems with non-constant coefficients for which vorticity should be created. In the latter case there is a discrete Kelvin Theorem giving the growth of circulation around a certain class of contour. All of this is of course only a beginning t o the design of practical schemes, but an important observation is that the vorticity-preserving property does not depend on choosing any particular expression for the quantity r' in (7). It is enough t o compute as a first step any vertex pressure, even a ridiculous one, and then to find the edge presure by averaging along the edges. This gives scope for iritroducing nonlinear methods (limiter functions) t o control the os-

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cillations around shockwaves without introducing vorticity. I t is beyond t h e scope of this paper t o discuss details, b u t i t is possible t o design schemes t h a t preserve vorticity in two o r three dimensions, a n d which reduce in one dimension t o high-resolution upwind schemes. However, t h e two-dimensional fluxes are computed from six rather t h a n two neighbouring states, which takes the schemes outside the applicability of the results in [GRE 981. Therefore schemes t h a t preserve contact discontinuities b u t avoid carbuncle-type behaviour may still be possible. We are currently investigating such schemes. For t h e fully nonlinear Euler equations, the evolution of discrete vorticity is quite complicated. However, t h e evolution of the curl of the momentum is V x (pv) is rather simple, and t o avoid unwanted solutions i t may be enough t o control this quantity. We hope t o be able t o report shortly on the outcome of numerical experiments.

5.

Bibliography

[MOR 991

MORTON,K. W.. AND ROE, P., "Vorticity-preserving Schemes of LaxWendroff Type", Submitted to SIAM J. Sci. Comp., 1999.

[ROB 991

ROBINET,J-CH., GRESSIER,J., CASALIS,G. AND MOSCHETTA, J-M., "Shock Wave Instability and Carbuncle Phenomenon- Same Intrinsic Origin?", preprint, ONERA Toulouse, submitted to J. Fluid Mech., 1999.

[QUI 941

QUIFX, J. J., "A Contribution to the Great Riemann Solver Debate", Int. J. Num. Meth. in Fluids, 18, pp. 555-574, 1994

[VAN 821

VAN

[PAN 991

PANDOLFI,M. AND D'AMBROSIO,D., "Numerical Instabilities in Upwind Methods: Analysis and Cures for the "Carbuncle" Phenomenon", preprint, Politechnico di Torino, submitted to J. Comput. Phys., 1999

[GRE 991

GRESSIER,J . AND MOSCHETTA, 3-M., "Robustness versus Accuracy in Shock-Wave Calculations", preprint, ONERA Toulouse, submitted to Int. J. Num. Meth. in Fluids, 1999.

[GRE 981

GRESSIER,J. AND MOSCHETTA, J-M., "On the Pathological Behaviour of Upwind Schemes", AIAA Paper 98-0110.

[RIC 621

RICHTMYER, R. D., "A survey of difference methods for non-steady fluid dynamics", NCAR Tech. Note 63-2, Nat'l. Center for Atmos. Research, Boulder, CO, 1962.

[NI 821

NI, R-H,, "A multiple-grid scheme for solving the Euler equations", AIAA Jnl, 20, p 1565, 1982.

DYKE,M., An Album of Fluid Motion, Parabolic Press, Stanford, CA.. 1982

On Uniformly Accurate Upwinding for Hyperbolic Systems with Relaxation

Jeffrey Hittinger and Philip Roe D e p a r t m e n t of Aerospace Engineering T h e U n i v e r s i t y of M i c h i g a n Ann Arbor, Michigan, USA 48109-2140

ABSTRACT T h e design of uniformly accurate, upwind Godunov schemes for hyperbolic systems with relaxation source t e m s is discussed. T h e goal is t o develop upwind methods whose sole time step constraints are due to the advection terms yet which obtain accurate solutions even when the relaxation t e m s are undewesolved. A n archetypal model system is considered, and analysis of the Riemann initial value problem for this model system is discussed. The ideas learned from this are re-enforced by a n asymptotic analysis of general Riemann problems, and these results are described. A strategy for developing a suitable numerical fEux function is then outlined. Key Words: hyperbolic systems, relaxation, numerical upwinding, stiff source terms

1.

Introduction

Many flow problems such as those of dilute gases or fluid mixtures have natural formulations as hyperbolic systems with relaxation source terms. These pose interesting challenges for numerical approximation. The source terms cause the system to be dispersive; generally, both the eigenvalues and eigenvectors of the system are modified by the relaxation processes which drive the system towards equilibrium. Another issue arises when the relaxation source terms operate on much smaller time scales than the advection terms; the problem is then said t o be stiff. Unfortunately, many problems simultaneously exhibit behavior in both the stiff and non-stiff limits, as well as in between. It is therefore desirable to develop high-resolution numerical algorithms which are uniformly accurate a t all scales. For example, if the data are such that

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the advection terms permit a time step large enough to bring the flow into local equilibrium, one would wish to see an accurate solution of the equilibrium problem emerge without resolving the details of the transition. In recent years, there has been much research into such uniformly accurate methods, although a clearly satisfactory procedure has yet to emerge. Pember's early work in the area [PEM 931 identified many properties desirable for Godunov-type schemes for hyperbolic systems with stiff relaxation but did not resolve the relative significance of the frozen and equilibrium wave speeds. Splitting schemes [CAF 97, JIN 961, have been proposed, but general objections to splitting were made in [ARO 96, ROE 931. These authors proposed characteristic schemes but found no natural way to make them conservative. A conservative method based on an approximate linearization was proposed in [BER 971. This uses a staggered grid t o eliminate the difficulty of resolving the Riemann problem, which no longer has a self-similar solution. However, such schemes may lose definition of linearly degenerate waves, such as contact discontinuities, which are likely to be very important in reactive or relaxing flows. Currently we are attempting to develop an approximate Riemann solver that represents the physics with uniform accuracy and whose computational costs are acceptable. The general behavior of relaxing Riemann problems is now known thanks t o the work of Liu and Zeng [ZEN99], although they have concentrated mostly on qualitative behavior of rather general systems. It is complementary t o our own work which has involved very detailed analysis of a linear model system which we feel is archetypal. This model has an exact (integral) solution for the Riemann initial value problem, which allows one to see how a simplified solution might be developed.

2.

A Model System

To gain a better understanding of the behavior of hyperbolic systems with relaxation source terms, we have studied the linear1 system

where x E R and t E E%+ are the spatial and temporal independent variables, respectively; 6 E R+ is a constant relaxation time; and r E [O,1] corresponds to the equilibrium wave speed. 1 0 n l y the linear problem is considered for the obvious reason that analysis is made tractable. However, this is not the restriction it might seem, as other authors [BER 971 have identified a suitable linearization for nonlinear hyperbolic systems with relaxation. We will demonstrate their requirements in Section 3.2.

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359

The first equation ( l a ) describes the evolution of the conserved quantity u(x, t). The flux of u, that is v(x, t), has its own evolution equation (lb), and is driven towards equilibrium over a time scale O(E) by the relaxation source term. In equilibrium, this source term vanishes (v E ru), and the system reduces to the single linear advection equation with the equilibrium wave speed r . Near equilibrium, say v = r u + O(c), it can be shown that

which is an advection-diffusion equation. In similar analysis applied to more complex problems, such as moment approximations of the Boltzmann equations, (2) corresponds to the Euler equations, and (3) t o the Navier-Stokes equations.

3. 3.1.

The Riemann Problem

The Model Problem

For upwind schemes, the Riema.nn problem is the heart of the numerical algorithm. We will describe the Riemann problem for the model system and point out behavior that is generic to all relaxation systems. Sett'ing 4, = -u and $t = v allows the system (1) to be reduced to a scalar equation, to which an exact integral solution can be constructed using classical methods. (For details see [ROE 931.) This integral solution is amenable to asymptotic analysis, particularly a t small times, and can also be evaluated numerically to provide a complete picture of the evolution of the solution from the small-time t o the long-time asymptotics. Consider the system (1) in vector form

with the piecewise constant initial conditions

u(x < 0,O) = UL

and

u(x 2 0,O) = UR.

(5)

The discontinuity in value a t the origin has a domain of influence bounded by the frozen characteristics I[) = Ix/tl = 1. Outside of the domain of influence, the initial conditions are constant, and, hence, the system of partial differential equations (4) reduces to a system of ordinary differential equations in time. This system is easily integrated t o find

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where T = t / ~ The . conserved variable u remains constant while the relaxation variable v decays exponentially from its initial value to the equilibrium value ru. This rapid adjustment of the relaxation variable forms a temporal boundary layer, as is apparent in the contour plots presented in Figure 1. Within the domain of influence, the solution is

where

with q* = t - x, <* = t

+ x, 'EJ= R(0, O), and the Riemann function

is the product of an exponential and the modified Bessel function of the first kind lo. Examination of this solution shows that, just behind the frozen waves, the solution decays exponentially to the equilibrium state ahead of each wave. That is, the strengths of the frozen waves are decaying exponentially, as can be seen in Figure 1. The actual decay rates are exp (-(1 f r ) ~ / 2 ) . For small times, T = t/c + 0 , the solution for u ( x , t ) is a perturbation about the frozen solution uo, with the O ( n ) corrections each a polynomial of degree n in the variable [:

where

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361

Riemann Solution u, E = 0.01, =0.5

Riemann Solution v, E = 0.01, ~ 0 . 5

Figure 1. Contour plots of the exact solution for the Riemann problem = (1, 5)T and LIE = (10,3)T for the equilibrium wave speed of r = 0.5 and relaxation time c = 0.01. Left-hand plots show a small-time behavior while the right-hand plots demonstrate a longer-time behavior. The dotted lines I x / t J = 1, representing the frozen waves, are drawn to emphasize the decay of these waves.

UL

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The piecewise-constant frozen solution is replaced by a smoothly varying solution as the result of decay along and interaction between the characteristics. The diagonalization of (4) explicates this behavior of the characteristics:

Here, v = & ( u+ v , u - v ) is~ the state vector of the frozen characteristic variables. On the right-hand side, the diagonal elements correspond to the previously-noted decay along the characteristics. The off-diagonal elements characterize the interaction between characteristics. For long times, T -+ CO, between the original two waves, the integral factors in TL and TR can be expanded by Laplace's method. The leading-order terms of the solution are

In this limit, the original two waves vanish and are replaced by the equilibrium wave a t E = T . In Figure 1, the rise of the equilibrium wave can easily be seen. As the small-time asymptotic behavior decays, the solution behind the frozen waves tends t o the states ahead of the waves except in the narrow region of the equilibrium wave. Here, the solution varies rapidly but smoothly from the left state to the right state. Finally, we consider the flux a t J = 0, which corresponds to a cell interface in an upwind Godunov scheme. For the same Riemann problem as Figure 1, the interface state is plotted in Figure 2 together with the small-time and large-time asymptotic expansions. Note that the interface flux is just (v, u ) ~ . The flux begins as the frozen flux, then varies smoothly but not necessarily monotonically to the equilibrium flux with this transition in the vicinity of T = 1, where the relaxation and advection are equally important. 3.2.

General Linear Systems

Consider the general linear system in the form (4), where now u E IRm and A , Q E Rmxm. We assume that rankQ = (m - n) > 0, so that Q has a null space Jd(Q), which is the equilibrium manzold in which the equilibrium solution takes place. Let Lo E Rnxmbe the row matrix of the left eigenvectors of Q which span N ( Q ) ; similarly, let Ro E RmXnbe the column matrix of the right eigenvectors of Q which span N ( Q ) . It is easy t o show that the equilibrium (Euler) equation is

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363

Interface State, ~=0.01,e 0 . 5

Figure 2. T h e interface state for the R i e m a n n problem u~ = (1,5)T and UR = (10,3)T for the equilibrium wave speed of r = 0.5 and relaxation time t = 0.01. T h e solid line is the numerical calculation of the exact solution, the , the dotted line is the small-time asymptotic expansion u p to terms O ( T ~ )and dashed line is the leading-order long-time asymptotic solution. and the late-time solution to the Riemann problem comes from the self-similar solution to this problem with Riemann data LouL,LouR. In [BER 971, it was proposed to select an average matrix Q ( u ~u ,R ) such that L~(Q)A& (Q) is the correct Roe matrix for the equilibrium problem. Because they avoided consideration of the early time Riemann problem, they had no need to select any particular linearization of A , but if this is simply a standard Roe linearization, then isolat,ed discontinuities will be correctly captured in both limits. If n = (m - I ) , the solution is only stable if the equilibrium wave speeds (the eigenvalues of LoARo) precisely interleave the eigenvalues of A [LIU87]. For this case, an equilibrium wave will grow from the smooth flow between each pair of frozen waves, and simultaneously, this equilibrium wave will be diffused. The frozen waves decay unless they carry a jump lying in N ( Q ) ;such waves will be found in both the early and late time solutions. If n < (m - l ) , are no theoretical results, but the general picture of frozen shocks undergoing exponential decay with equilibrium shocks arising from the spaces between them appears t o be correct. At early times one can obtain expansions similar to (12). The k-th frozen wave decays such that

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where lk and r k are the k-th left and right eigenvectors of the matrix A , and a k ( 0 ) is the strength of the initial jump. The solution between the waves is again piecewise polynomial of degree n at order n. 4.

C o n s t r u c t i n g an Upwind Flux Function The substitution u = e x p { Q ~ ) wtransforms (4) into

Integrating this around a cell of a regular, uniform grid with cell-centered data, and translating back to the original variables, one finds

where t' = t - nAt and f = A u . If the solutions fjf ; ( t ) are known on the interface, this is an explicit formula for the new cell average. The exponential decay in the transient solution is represented explicitly, and within the integral, the exponential factor acts as a filter. To elucidate the weighting in the integrand, let z = Lu, where L is the row matrix of left eigenvectors of Q. This transformation creates a state vector of the n conserved variables (those in N ( Q ) )and of the (m - n) relaxation variables which vanish in the equilibrium limit. Multiplying the update formula by L, one obtains

where f = LAL-'z and A is the diagonal matrix of the eigenvalues of Q. For conserved variables, the corresponding eigenvalues are zero, so the exponential factors in the update (23) are just unity. For relaxation variables, interpreting At as the time step based solely upon the advection terms, the exponential weighting in the integrand is essentially unity over the entire interval if At/&<< 1. However, if At/&>> 1, the main contribution of the integral comes from the vicinity of the end point t' = At. The question now is whether a computationally acceptable approximation can be found for the fluxes on the interfaces, arising from the Riemann problems with data uy, u s l . Clearly this is possible in the two limiting cases At/€ <( 1 and At/&>> 1, and if the majority of cells are of one of these types, there will not even be any great expense involved, because a switch simply could be used to select the appropriate formula. These are the limits on which previous work has concentrated, and the present approach deals with them very simply.

Innovative schemes

365

The difficulty is only with those cells for which A t / € = O(1). An empirical blending of the two formulae will probably not be very accurate, because of the non-monotone transition seen in Figure 2. However, Figure 2 also shows that the early-time expansion may have a sufficiently large radius of convergence that it can capture the crucial behavior. We are presently investigating such expansions in the general case. The solution for the jumps is given above, and the solution for the linear variation is also straightforward. The quadratic and higher terms appear more complicated, although there is a straightforward means t o derive them. But even if the formulae prove expensive they should only be needed a t relatively few interfaces.

5.

A Practical Example

Between the continuum (Euler) model of fluid flow and the molecular (Boltzmann) description, there lie many intermediate models. The hierarchy of models devised by Levermore [LEV 961 are of especial interest, because by design, they possess many properties required for trouble-free computation: they are hyperbolic, symmetrizable, entropic, and well-posed. Beyond the Euler equations, the next member of the hierarchy retains ten moments of the Boltzmann equation by assuming that the distribution of the random velocity c is proportional t o exp(-iOijlcicj), where Oij is a symmetric, non-negative 3 x 3 matrix closely related to the temperature. Moments of this distribution give the pressure and temperature as tensor quantities. The relaxation process is simply that the temperature tries t o relax back toward a scalar,

and the eigenstructure of the frozen problem is straightforward (See [BRO 951). We feel that although the procedures outlined in this paper may be expensive in general, they will prove relatively simple for well-motivated physical models. 6.

Conclusions and Future Work

The analysis of a simple linear model system has provided clues towards the development of a uniformly accurate upwind method for hyperbolic systems with relaxation. Specifically, the solution of the Riemann problem can be constructed and analyzed for this model system, and this analysis has identified strategies for designing upwind methods for hyperbolic systems with relaxation source terms. Currently, we are implementing and evaluating these ideas.

366

7.

Finite volumes for complex applications

Acknowledgment

This work is supported in p a r t by a U.S. Department of Energy Computational Science Graduate Fellowship.

8.

Bibliography

[ARO 961

ARORA,M., Explicit Clbaracteristic-Based High-Resolution Algorithms for Hyperbolic Conservation Laws with Stiff Source Terms, Ph.D. thesis, The University of Michigan, 1996.

[ARO 981

ARORA,M. A N D ROE, P., L ' I s ~ ~and e s Strategies for Hyperbolic Problems with Stiff Source Terms", Barriers and Challenges in Computational Fluid Dynamics, V. Venkatakrishnan et al., eds., Kluwer Academic Publishers, Norwell, MA, 1998, pp. 139-154.

[BER 971

BEREUX,F. A N D SAINSAULIEU, L . , "A Roe-type Riemann Solver for Hyperbolic Systems with Relaxation Based on Time-Dependent Wave Decomposition", Numer. Math., 77, 2, 1997, pp. 143-185.

[BRO 951

BROWN,S . , ROE, P . , A N D GROTH,C., LLN~merical Solution of 10Moment Model for Nonequilibrium Gasdynamicsl', AIAA Paper 95-1677, June, 1995.

[CAF 971

CAFLISCH, R . , J I N , S . , A N D RUSSO,G., "Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation", SIAM J. Numer. Anal., 34, 1, 1997, pp. 246-281.

[JIN 951

J I N , S., "Runge-Kutta Methods for Hyperbolic Conservation Laws with Stiff Relaxation Terms", J. Comput. Phys., 122, 1, 1995, pp. 51-67.

[JIN 961

J I N , S. A N D LEVERMORE, C., "Numerical Schemes for Hyperbolic Conservation Laws with Stiff Relaxation Terms", J. Comput. Phys., 126, 2, 1996, pp. 449-467.

[LEV 961

LEVERMORE, C., "Moment Closure Hierarchies for Kinetic Theories", J. Stat. Phys., 83, 5/6, 1996, 1021-1065.

[LILT871

T.-P. LIU, " Hyperbolic conservation laws with relaxation",

Comm.

Math. Phys, 108, 1, 1987, pp. 153-175. [PEM 931

PEMBER,R., "Numerical Methods for Hyperbolic Conservation Laws with Stiff Relaxation: 11. Higher-Order Godunov Methods", SIAM J. Sci. Comput., 14, 4, 1993, pp. 824-859.

[ROE 931

ROE,P. A N D ARORA,M., "Characteristic-Based Schemes for Dispersive Waves: I. The Method of Characteristics for Smooth Solutions", Numer. Methods Partial Differential Equations, 9, 5, 1993, pp. 459-505.

[ZEN991

ZENG,Y., "Thermal Nonequilibrium and General Hyperbolic Systems with Relaxation", preprint, University of Alabama at Birmingham, 1999.

Implicit Finite Volume approximation of incompressible multi-phase flows using an original One Cell Local Multigrid method

Stkphane V I N C E N T and Jean-Paul C A L T A G I R O N E Avenue Pey-Berland BP 108 33402 Talence Cedex France

The numerical simu~ationof multi-phase flows involving stretching and tearing of interfaces requires accurate tools, able to describe near the free surface the different scales of the flow which results from the development of instabilities. O n fixed Cartesian mesh, a n original local multigrid method, which refines the grid at the cell scale and adapts i n time and space, is proposed. A n implicit Finite Volume solver, coupled with a T V D - V O F like interface capturing method, is carried out on each grid level. The method is validated and discussed o n analytical velocity fields and Rayleigh-Taylor instabilities. ABSTRACT

Key Words: free surface flows, multigrid method, implicit Finite Volumes

1. I n t r o d u c t i o n

The numerical simulation of multi-pha.se flows with strong stresses acting on the interface is classically achieved by the implementation of fixed Cartesian rrieshes with an interface tracking method (Marker [DAL 671, VOF [YOU 821 or Level Set [SUS 971). However, due t o the memory limit of supercomputers and the computational time, the numerical simulation of non-symmetric threedimensional free surface flows is restricted t o problems where the length-scales of the phenomena occurring near the interface arc close. To limit the computation node far away from the free surface and concentrate the calculation points on the interface, an original One Cell Local Multigrid method (OCLM) is proposed. Starting on a coarse grid Go which corresponds t o the physical domain 0, a refinement criterion R, is defined t o detect the points t o be refined on

368

Finite volumes for complex applications

<

each multigrid levels G l , 1 l 5 l,,. A hierarchy of embedded subdomains G1,,, 0 5 s 5 ,,,,s is obtained, each fine calculation grid GI,, corresponding to a coarse control volume around a detected point of GlP1 cut by 3 in each space direction. An odd cutting ensures a perfect reconnection between the solutions on each GI (Fig. 1). Contrary to the classical multigrid methods such as FIC [ANG 921 or AMR [BER 891, where the fine calculation meshes contains tens to hundreds of cells, all the multigrid calculation domains have the same size (3 x 3) in the OCLM technique, which is an outstanding property of the method. Indeed, on meshes of reduced size, the numerical solvers converges quickly and requires very low memory.

Coarse grid Coarse gr~d

+

1 level of local multigr~drefinement

Refinement of a coarse control volume around a detected point

Figure 1. Local mesh refinement technique at the scale of a control volume around a coarse node. After a brief presentation of the motion equations, the numerical solver on a unique grid is explained. The various stages of the OCLM method are next described and finally, the local refinement technique is discussed on classical interfacial problems. 2. 1-fluid model and unique grid solver

The multi-phase flow is modeled by means of the dimensional Navier-Stokes equations for incompressible fluids. In Cartesian coordinates, with a bounded domain Q, we get v.u=o (1)

Innovative schemes

369

where p is the pressure, u is the velocity field, C is the phase function , g is the gravity, p is the density and p is the dynamic viscosity. The colour function C repairs the different phases of the flow standing for example C = 0 in one fluid and C = 1 in the other. The interface between these fluids is naturally defined as C = 0.5. The motion equation system (1-3) is a 1-fluid model, in which the discontinuous physical characteristics are estimated according t o the arrangement of the fluids as follows

where po, p1,po and p1 are the physical characteristics of fluids 0 and 1. Finite Volumes on a staggered grid (MAC) are investigated to discretise the Navier-Stokes equation system (1-3). In (1-2), the temporal derivatives are approximated by a Gear scheme of second order, whereas the discretisation of the spatial derivatives is achieved through a Quick scheme for the non-linear terms and a centred scheme for the diffusive one. Moreover, in the presence of discontinuous physical characteristics, a robust augmented Lagrangian numerical solver (Vincent and Caltagirone [VIN 991) is carried out t o calculate the solution of (1-2). The coupling between the pressure p and the velocity u is gone around thanks to a penalization terms added in the momentum conservation equation (Fortin and Glowinski [FOR 821). Then, the implicit discrete equation system reads

'

gu;+l - 2 4 +

At < p;+l-p;=au;+l

C,n \

1

= --vp;

P

-

At

c;

+ (u," . v)u:+' + B;(u:+' - u,) - r V ( V . u,"") 1 + -v . [p(vu;+' + vTu;+l)] + g

rv . u;+l

- B:(u;+'

--

iu;-l

- u,)

P

on the surface

r of the physical domain

+u;.vc:=o

(5) where A t is the time scale, r is a numerical parameter controlling incompressibility and n is a normal to r. The exponent n corresponds t o time (n At) and subscript 0 refers to the coarse grid Go. u, is a reference velocity and B; is a volume control parameter, which is used to impose a velocity in R (Angot [ANG 891) whereas B: is a Fourier like surface control parameter enforcing boundary conditions on r. An iterative Bi-Conjugate Gradient Stabilised algorithm (BiCgStab, Van Der Vorst [VAN 92]), preconditionned with a Modified Incomplete LU method (MILU) is investigated to solve the linear system gen-

370

Finite volumes for complex applications

erated by the discretisation of the motion equations. In the presence of a discontinuous phase function C , the hyperbolic advection equation (3) must be discretised by means of non-oscillating TVD schemes of high order (Vincent and Caltagirone [VIN 991). In this way, no spurious oscillations appear and a second order accuracy is obtained in each fluid. At the interface, the order of the TVD scheme used is decreased to ensure the monotonicity of the solution. With the implementation of a Lax-Wendroff TVD scheme, accurate solutions were obtained on complex multi-phase flows (see [VIN 991 and [VIN2 991). 3. One Cell Local M u l t i g r i d solver If it is supposed that the solution (u:,p:, Con) of (5) on the coarse grid Go at time (n At) is known, then (u,"+',~,"+', con+') can be calculated solving (5) on Go. A refinement criterion R, is defined to detect the coarse points where a multi-scale solution is necessary. For free surface flows, a local mesh refinement is built on the coarse control volumes cut by the interface. In this way, R, is expressed as Rc =11 VC II (6) If R, (M) # 0 with M E Gl-l, the control volume around M is refined and a fine calculation domain G1,, is created. For all I such that 1 5 1 5 I,,,, (ur?:, Pr?t, are prolongated on GI using a classical Q1 interpolation procedure and for all s such that 1 5 s 5 s,,,, (5) is solved on each Gl,, to The time step is then incremented and another multiget grid solving procedure is initiated. Finally, thanks t o the OCLM method, we have access to a multi-scale solution, which adapts in time and space to the interface. Each fine grid cell is a 3 x 3 discrete domain, which is extended to 5 x 5 for the treatment of the boundary conditions. It must be noticed that the main difficulty in the OCLM method is to define physical boundary conditions in such a way that a suitable solution is calculated on the fine grids. Indeed, V1/1 5 1 1 l,,, the boundaries of the fine grids Gl,, do not correspond inevitably with those of Go. The first idea was to impose the interpolated values of the coarse grid G1-l a t the boundaries of G1,, by means of Dirichlet boundary conditions (B: = +m), as in the works of Caltagirone et al. [CAL 951, Angot et al. [ANG 921 and Berger and Collela [BER 891. Nevertheless, due to the reduced size of the fine grids, the physical system is too much stressed and the solutions on the 5 x 5 domains are not physical. Indeed, the prolongation operator do not preserve the divergence free property and so the

clk+l1)

c:").

Innovative schemes

371

Dirichlet boundary condition enforces the numerical solver to converge to a non suitable solution. To relax the stresses a t the boundaries of the multigrid calculation grids, composite boundary conditions have been introduced in the Navier-Stokes equation system as follows vL/l 5 1 5 lmaz,

O
(7)

A theoretical work was developed t o impose the conservation of the shear stress between two multigrid levels (see [VINS 991). In this way, orders of magnitude of the composite boundary condition parameters and B: were estimated, included between l o 4 and 10'. Moreover, the Navier-Stokes solving on the 5 x 5 fine grids is improved thanks to an incomplete ILUD preconditionnning, more powerful than the MILU one.

BE

4. Results

4 . 1 . The vortex test

A concentration circle of radius 0.15 meter located a t point (0.5, 0.75) in a 1 meter-long square domain is stretched in the symmetrical vortex velocity field of Rider and Kothe [RID 951. The authors demonstrated that only the Marker method can reproduce the fine scale of the solution. However, this technique is very expensive. All the other methods tested in their paper destroy the fine features of the interface. The Level Set technique and the TVD scheme solution induce strong diffusion and lose precision when the interface is strongly stretched. The PLIC VOF method introduces artificial surface tension in the small scale regions of the interface which are artificially cut. To be efficient, these techniques require very fine calculation grids. Thanks to the OCLM method, coupled with a TVD interface capturing method, the problem is precisely solved with a very low number of calculation points (Fig. 2). Starting with a 70 x 70 coarse grid and applying the OCLM method on two levels, a precision equivalent to 630 x 630 is reached in interfacial regions. Thanks to the local mesh refinement, the memory costs have been divided by 4 with respect to a computation on a full 630 x 630 grid. The local and adaptative character of the OCLM method is perfectly illustrated in this problem.

372

Finite volumes for complex applications

Figure 2. Multigrid simulation of the vortex problem. Go is a 70 x 70 grid. A 3-level solution is presented after large deformations have been induced. The local mesh refinement (only 40% of the total cells are plotted for convenience) and the interface position (C=0.5) on Gz are presented.

Innovative schemes

373

4 . 2 . Rayleigh-Taylor instabilities

Figure 3. Numerical simulation of two-dimensional Rayleigh-Taylor instability by means of the OCLM method. The initial perturbation is 10% of the domain height. The viscosities are the same in the two fluids. The results are presented respectively at time 2.25 s. The distribution of the multigrid cells on the coarse grid and the free surface (C = 0.5) are plotted. The Atwood number A is 0.05 and the Reynolds number Re is 10. The Rayleigh-Taylor instability is a classical and widely studied phenomenon which underlines the competition between the surface tension forces and the viscous stresses. When a heavy fluid lays above a lighter one, any perturbation of the interface between these fluids is amplified under the action of gravity,

374

Finite volumes for complex applications

whereas the surface tension tends t o minimise the deformation of the free surface. The Rayleigh-Taylor instability problem is simulated t o highlight the abilities of the OCLM method to solved free surface flows. The Navier-Stokes equations and the interface capturing procedure are computed on all the grid levels. For example, the local mesh refinement is carried out on 2 grid levels (Fig. 3). Go is a 40 x 80 grid. The free surface evolutions are perfectly captured by the OCLM method. The local character (4 or 5 cells surrounding the interface) and the adaptative property of the method in time and space are verified. The composite boundary conditions have allowed t o solved precisely the motion equations on the fine calculation domains. Moreover, the reconnection between the hundreds of fine grids is perfectly achieved thanks to the odd cutting. In the present problem, the ratio between the number of calculation point in the OCLM solving and the number of node on an equivalent fine 120 x 240 grid is always less than 0.09. The gain in memory is so very important and the calculation time is decreased about 10 %. The difference between the memory and calculation time improvements lays in the additional operations needed t o solved on the multigrid levels and in the detection procedure.

5 . Conclusion

We have presented a new local and adaptative multigrid refinement method (OCLM) for the numerical simulation of free surface flows. The motion equations and the interface capturing are solved a t the cell scale thanks to the definition of new composite boundary conditions. The method brings important gains in memory and calculation time. The extension of the method to three dimensions is immediate. Moreover, the OCLM method is a general method independent on the equations solved as well as on the numerical solver. The method was successfully applied to the numerical simulation of natural convection in a square cavity. [DAL 671

DALYB . J., Numerical study of two-fluid Rayleigh-Taylor instability , Phys. Fluids, 10, 1967, p. 297-307.

[YOU 821

YOUNGSD. L., Time-dependent multi-material flow with large fiuid distortion, K. M. Morton and M. J. Baines, 1982, p. 27-39.

[SUS 971

SUSSMAN M. AND SMEREKA P., Axisymmetric free boundary problems , J. Fluid Mech., 341, 1997, p. 269-294.

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375

[ANG 921

ANGOTP . , CALTAGIRONE J.-P. AND KHADRAK., Une mkthode adaptative de raffinement local: la correction du flux & l'interface , C. R. Acad. Sci. Se'rie II b, 315, 1992, p. 739-745.

[BER 891

BERGERM. J . A N D COLLELAP . , Local adaptative mesh refinement for hyperbolic partial differential equations , J. Comput. Phys., 82, 1989, p. 64-84.

[VIN 991

VINCENTS. AND CALTAGIRONE J . - P . , Efficient solving method for unsteady incompressible interfacial flow problems , Int. J. Numer. Methods Fluids, 1999, to be published.

[FOR 821

FORTINM . AND GLOWINSKY R., Methodes de Lagrangien augment&. Application c i la re'solution nume'rique de probl6mes aux limites, Dunod, 1982.

[ANG 891

ANGOTP . , Contribution & l'ktude des transferts thermiques dans des systbmes complexes: application aux composants 6lectroniques, PhD thesis, University Bordeaux I, 1989.

[VAN 921

VAN DER VORSTH. A., Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems , SIAM J. Sci. Stat. Comput., 13, 1992, p. 631644. VINCENTS., CALTAGIRONE J . - P . A N D ARQUISE., Numerical simulation of liquid metal particules impacting onto solid substrate: description of hydrodynamic processes and heat transfers , J. High Temperature Material Processes, 1999, to be published.

[CAL 951

CALTAGIRONE J . - P . , KHADRA K. AND ANGOTP . , Sur une mkthode de raffinement local multigrille pour la rksolution des kquations de Navier-Stokes , C. R. Acad. Sci. Serie II b, 320, 1995, p. 295-302.

[VIN3 991

VINCENTS. AND CALTAGIRONE J.-P., One Cell Local Multigrid method for solving multi-phase flows , J. Comput. Phys., 1999, in corrections.

[RID 951

RIDERW. J . AND KOTHE D. B., Stretching and tearing interface tracking problems , AIAA paper, 95, 1995, p. 1717.

New Classes of Integration Formulas for CVFEM Discretization of Convection-Diffusion Problems E.P. Shurina and T.V. Voitovich Novosibirsk State Technical University, Novosibirsk, Russia ABSTRACT This paper is concerned with the development of FEM-like finite volume technique for discretization of convection-diffusion problems over unstructured simplicia1 grids. Proposed finite volume technique is original in it's essence - technology of approximation of convection-diffusion fluxes and sources. Polynomial interpolation is used to represent unknown variables, transfer coeflcients, thermophysical characteristics and source terms as linear combinations of local form functions at each element. Local form functions are expressed in terms of simplex barycentric coordinates. Three special classes of non-symmetric formulas for exact integration of monomials of barycentric coordinates over dual mesh lines and corresponding subregions are introduced. The first class contains formulas for exact integration of barycentric coordinate monomials along dual mesh lines lying on a simplex, and is used to express convection and diffusion fluxes in terms of discrete unknowns. The second class consists of formulas for exact integration of monomials of barycentric coordinates over simlplex subregions, corresponding to different finite volumes sharing the same element, and is used for discretization of source terms. The last class prescribes exact integration of the monomiuls along segments of the edges, approximating boundary curves, and is intended for realization of boundary conditions. Geometric interpretation of proposed integration formulas of the lowest order is presented. Described technology can be applied to a wide variety of PDE models, and with the use of volume barycentric coordinates allows uniform extension to three-dimensional case.

Key Words: control volume finite element methods; unstructured grids; dual mesh; barycentric coordinates; upwind schemes.

1.

Introduction

Adequate numerical simulation of fluid and gas dynamics processes imposes strict requirements on the modern methods, namely: exact representation of domains with complex geometry and high-order accuracy of spatial and temporal discretization schemes. Two types of grids, non-orthogonal and unstructured allow

378

Finite volumes for complex applications

exact treatment of complex domains. W e use the latter to account for additional requirements, such as adaptivity and local grid refinement realization. As for the spatial discretization, the finite volume method has some advantages, in particular, conservative properties of corresponding schemes, simplicity and ability of more natural use of the upwind schemes. Formulation of a finite volume element scheme typically involves the following basic steps 1 . A set of admissible discretization templates is chosen. 2. An arrangement of all unknowns of source nonlinear coupled system (cellvertex, cell-centered, side-centered arrangement) is specified. 3. Element based interpolation polynomials are specified for transfer coefficients, conservative variables, thermophysical variables and source terms. 4. Weighted residuals principle, with weighting functions being equal to one over each finite volume is applied. 5. For each element convection-diffusion fluxes through the dual mesh lines are expressed as linear combinations of local discrete unknowns, with coefficients of the combination being components of the local diffusion and convection matrixes. Similarly, integration of the source term interpolation polynomial over subregions of a finite element gives the local source vector. 6. Global matrixes and vectors are obtained via the element-by-element assembly. 7. Resulting sparse linear system is solved by a specified method. We are interested in the development of a finite-element like procedure for generation of discrete analogue coefficients, thus we will take a closer look at the fifth step. The main contribution of this paper lies in the development of a new uniform technique of convection-diffusion fluxes and source terms approximation, with exact integration of interpolation polynomials, which are represented with simplex barycentric coordinates. The technique is based on the three classes of special formulas introduced by the authors. These classes give exact values for integrals of barycentric coordinates monomials along dual mesh lines, simplex subregions and triangle edges segments. Alternative approach with exact integration of polynomials in finite volume discretization, in which all polynomials are expressed as a generalized Taylor series, was proposed by Yen Liu and Marcel Vinokur [LIU 971. In FE discretization of Navier-Stokes equations only a few convergent pairs for pressure-velocity interpolation functions are known. In the case of triangular finite elements linear pressure - quadratic velocity is a usual choice. It has been shown that the Ladyzhenskaya-Babuska-Brezzi (LBB) restrictions still hold for hybrid FEM/FVM techniques [SHU 971. T o our best knowledge, appearance of LBB restrictions strongly depends on the choice of a pressure-velocity coupling technique. For example, SIMPLE-like algorithms [PRA 851 for linear interpolation functions of equal order do not lead to ill conditioned matrices for the pressure correction. Finite element community also reports several methods of circumventing the LBB restrictions [ZEN 951. Thus, the triangular based linear form functions can be used to demonstrate the main features of the new approximation approach. However, the form functions of higher degrees are allowed, and this extension will be discussed later.

Innovative schemes

379

2. Proposed finite volume technique Consider the conservation equation for a scalar quantity

where p is the fluid density, u is the velocity vector,

#

r4 is the diffusion coefficient

and Sq is the volumetric source of #

Domain discretization. The computational domain is divided into triangular elements. Assume that computational points and finite element vertexes coincide ("the cell-vertex" arrangement), and both velocity components and scalar variables have the same finite volume ("the collocated" arrangement). Dual mesh is constructed on centroids of triangles and mid-points of the edges, such that each node i has the corresponding complete, or "incomplete" finite volume Q i, bounded by median segments (see Figure 1 ) . Let Si denote the boundary of Q i. Let us assume fixed global and local numeration on the set of triangulation nodes and on each finite element. The integral conservation form of equation ( 1 ) is

where J = J, + J d is the combined convection-diffusion flux of # . The proposed CVFEM technology, similarly to the known ones, uses the following principles of finite element discretization: (i) specification of elementbased form functions for conserved quantities and sources (it should be noted, that the usage of interpolation functions by FV is more flexible, because different classes of profiles can be applied to different terms of equation (1)); (ii) element-by-element assembly of discrete analogue matrices and vectors. The essence of our contribution is the use of barycentric coordinates in local representations of scalar variables,

Figure 1. Fragment of primary (FE) and secondary (FV) grids corresponding to linear interpolation and notation used for a triangle en .

380

Finite volumes for complex applications

transfer coefficients and sources and introduction of special integral formulas for barycentric coordinates monomials, much as in FEM. Consider a finite element e n . Let S,nb denote a segment of the dual mesh, lying on a median, which comes out of the triangle node nb (see Figure I), and J v n b is a value of the combined flux through the median segment S:b in the direction of the outward normal to the boundary of the finite volume f i v , J,,

D ,,b = J,, ,,b

+J

: ~= ~

Gfnb +~ ~ ~ ~ ). (see . n Figure d s 2). It will be sufficient to

approximate three of the six introduced fluxes, because J , , = - J 3 , , 532

J I 3=-J23,

=-J12.

Diffusion jluxes approximation. Using Green's formula, the diffusion flux

through a median segment S i b can be represented as follows

It is well known that FEM naturally allows representation of significantly varying transfer coefficients by an interpolation function on an element, unlike most of the FVMs. Let us demonstrate proposed technology on a problem of approximation of the diffusion flux, using linear interpolation for T@ and @ [3]:

Figure 2. Outward unit normals and corresponding notation forfluxes.

Innovative schemes

38 1

where y,n = Gm(x) are local linear basis functions (see Figure 3), T , are the local values of T$ (m=1,2,3), and coefficients a $ ,b$ ,c4 can be expressed in terms of the local discrete unknowns @1,@2,@3 via an inverse matrix of the matrix of coordinates of the triangular nodes:

Thus, approximation of the diffusion flux [3] becomes

It is clear from [7] that it is necessary to introduce special formulas for integration of the local basis functions along the median segments to complete discretization of the diffusion flux. Note, that in the case of triangular based linear interpolation, local basis functions coincide with simplex barycentric coordinates Li , Wi (x)= Li, i = 1,2,3. For every local basis function consider a pair of trapeziums and a triangle, which are constructed using finite element median segments and their images on the surface of the basis function (see Figure 3). Projections of the trapeziums and the triangle to coordinate planes gives

Figure 3. Sonte notation for the median projections and one of the local linear

functions on the triangle e n .

382

Finite volumes for complex applications

v f nb; Here I,,

l y are the lengths of the projections of the median segment :S onto the

x and y axis, respectively, with the signs corresponding to the anticlockwise ~ Taking into consideration formulas [8],[9] , we have for integration; v , n b {1,2,3}. three determining diffusion fluxes of the element:

Finally, to determine coefficients in the first, second and third row of the local diffusion matrix, combinations J 1 2 - J23, J23 - J31, J3i - J I 2 are considered in terms of conservation unknowns. Coefficients at $1, $2, in the combinations give three elements in the current row of the local diffusion matrix. Approximation of convection fluxes is performed as @ J ~

Innovative schemes

383

Special Prakash-Patankar interpolation of u is used to avoid spurious pressure oscillations in the framework of the collocated approach. An upwind scheme is used to express Gnb in terms of the local unknowns. The f l u x j pu.nds is 'nb

approximated using formulas [8,9]. Source terms approximation. Let

element

en

(n0= 1,2,3),

htb denote

an intersection of the finite

of interest and the finite volume of the node with local number nb

fl::b= Qnb n e" .

Figure 4. Notation for the finite element subregions

Formulas for exact integration of barycentric coordinates over the subregions are introduced as follows

(One can easily obtain [14] representing each of the calculated volumes as a set of truncated right prisms). Formulas [14] are used to compute local RHS vector of e n . Boundary conditions. For realization of boundary conditions (f.e. approximation of convection fluxes through outlet boundaries), one-dimensional analogues of [8,9] for integration of the local basis functions along the element edges segments are introduced. Fut~ireinvestigations. It is well known in the finite element community, that an advantage of the use of barycentric coordinates, is in the existence of integration formulas, simplifying computations of integrals along the edges of a triangular element and over an element itself, namely the following formulas by Eisenberg and Malvern [EIS 73 1:

384

Finite volumes for complex applications

It is obvious, that in the proposed FV technology formulas [8,9,14] play the same role as [15,16] in FEM, and have the lowest order. Following classes of formulas for exact integration of barycentric coordinates monomials (integration over dual mesh lines, finite element subregions and boundary edges segments, respectively) are the subjects of our investigation:

Classes of the formulas [17] are the basis of the uniform discretization technique, which allows to form functions of higher order, since these functions can be represented through barycentric coordinates. Although the capabilities of [8,9,14], are such that these formulas of the first order can be used as a basis for CVFEM discretization, we consider the technique incomplete until the exact values for [17] are be found.

3.

Concluding remarks

A new approach for finite volume discretization is developed in analogy with the FE approach, namely we use barycentric coordinates in local representation of scalar variables, transfer coefficients and sources. Final discrete analogues of diffusion-convection fluxes and sources are obtained via special classes of integration formulas introduced by the authors.

4.

Bibliography

[LIU 971

Liu Y., Vinokur M. Exact integrations of polynomials and symmetric Quadrature Formulas over Arbitrary Polyhedral grids, J.Comput.Phys., 140, 122, 1997.

[SHU 971 Shurina E.P., Voitovich T.V., Analysis of the finite element and finite volume methods based upon unstructured grids for solution of the Navier-Stokes equations, Computational Techlologies, 2,4, 1997 ( in Russian). [PRA 941 Prakash, C., Patankar, S.V., A control volume-based finite-element method for solving the Navier-Stokes equations using equal-order velocity-pressure interpolation, Numer.Heat Transfer, 8,259, 1985. [ZEN 951 Zienkievicz, O.C., Codina, R., A general algorithm for compressible and incompressible flow. Part I: The split, characteristic based scheme', lnt.j.numer.meth.fluids,20, 869, 1995.

[EIS 731 Eisenberg M.A, Malvern L.E., On finite element integration in natural coordinates, 1nt.j.numer.methods eng., 7,574, 1973.

Fields of application

Analysis of Finite Volume Schemes for Two-Phase Flow in Porous Media on Unstructured Grids

M. Afif ('1 and B. ~ m a z i a n e ( ~ ) Universite' Cadi Ayyad Faculte' des Sciences Semlalia

B. P. S. 15, 40000 Marrakech Maroc. E-Mail: [email protected] Universite' de Pau et des Pays de 1'Adour De'partement de Mathe'matiques Recherche Av. de l'Universite', 64000 Pau France. E-Mail: [email protected] (2)

This paper is devoted to the analysis of finite volume schemes on unstructured grids for a nonlinear, degenerate, convection-dz;ffusion equation arising in flow in porous media. A semi-implicit scheme is considered. We prove that this scheme is L", weak BV stable under a CFL condition and satisfy a discrete maximum principle. We then derive a convergence result. Results of numerical experiments using the present approach in the 2-D case are reported. ICey Words: finite volume method, degenerate parabolic equation, nonlinear diffusion-convection, porous media, unstructured grids. ABSTRACT

1. Introduction

Flow simulation in petroleum and groundwater reservoirs has been extensively studied using finite element methods in past years (see, e.g., [AWZ 96, CHJ 86, CHE 971 and the bibliographies therein). Also, a discretization using both finite element and finite volume methods for two-phase flow in porous media is presented in [CJR 951. However, it appears that there are few results on convergence theory for the degenerate problem [AWZ 96, CHE 971. More

388

Finite volumes for complex applications

recently, finite volume methods were developed and analyzed for immiscible two-phase flow in porous media in the case where the diffusion term is neglected, (see [EGH 98, VEV 981 and the references therein). This approach leads to robust schemes applicable for unstructured grids and the approximate solution has various interesting properties which correspond to the properties of the physical solution . These methods have been useful for advective flow problems because they combine element by element conservation of mass with numerical stability and minimal numerical diffusion. In this paper we will study a numerical approximation of two-phase flow in porous media. We focus on two-phase immiscible flow, which corresponds physically to the water flooding of a petroleum reservoir. We consider twophase water and oil flow in a porous media, using the global pressure, the total velocity and the water saturation as the primary variables. This formulation leads to a coupled system of partial differential equations which includes a nonlinear degenerate parabolic saturation equation and an elliptic pressurevelocity equation [CHJ 861. The saturation equation is convection dominated and thus special care should be taken in discretization. The diffusion term there is small but important and can not be neglected. Also irregular geological features, which suggest the use of irregular grids. We will consider a system of equations describing the flow of two immiscible incompressible fluid phases through a porous medium in which for sake of simplicity, we will neglect the effect of gravity. This system is then the following: Saturation Equation: 0 5 u(x,t) 5 1 ~ ( x$ ) ddiv (b(u)q? - div ( K ( x ) V a ( u ) )= 0 ulr, = 1 and K V a ( u ) . jilrZVr3 = 0 u(x,O) = uO(x)

+

in QT in QT on J in Q

(1)

Pressure Equation:

where u ( x , t ) is the water saturation, P is the global pressure, K ( x ) is the 2 c R2 whose boundary I? splits absolute permeability tensor of the reservoir - iup into three parts such that r = 1;;U r2U r3 and ri n rj = 0 for i # j ; rl is the part of the boundary where the water is injected, r2 is the impervious part of the boundary and r3 is the producing part of the boundary. Also Q, denotes Q x 10, r [ . @(x) is the porosity of the porous media. a ( u ) b(u) and d(u) are nonlinear functions which depend on the mobilities and the capillary pressure with diffusion coefficient a' vanishing for two values of saturation: crl(0) = a l ( l )= 0 (degeneration of the diffusion term). The main difficulties related to the convergence of the approximate solution of such systems are the coupling between the pressure and the saturation equations and the degeneracy of the saturation equation. In view of these difficulties, the aim of this paper is to construct finite volume schemes (FVS) for

Fields of application

389

the saturation equation that satisfy the discrete maximum principle and derive convergence results. More precisely, we will consider a numerical approximation of this system where the saturation and pressure equation are decoupled. We will concentrate on the study of the convergence of the FVS for t.he saturation equation taking into account the diffusion term and the anisotropic heterogeneity of the reservoir. A mixed-hybrid finite elements method is used to obtain an accurate approximation of the velocity [BRF 911. In [AFA 971, the one-dimensional problem was analyzed. In this paper, we extend these ideas to multiple dimensions. We will restrict attention to R2, however, the methodology and the analysis can be extended to problems in El3. This paper is organized a s follows. In the next section, we present the finite volume discretization of the problem (PI). An explicit approximation of the convection term combined with an implicit approximation of the diffusion term is considered. The solution is approximated by a piecewise constant. In section 3, we present the L", weak BV stability under a CFL condition and a convergence result. Numerical simulations for a 2-D example are presented in section 4.

2. Finite volume discretization Before describing the finite volume discretization of the model problem ( P I ) we give some notations. As usual, let (tn),=O ,,.,,N , be a partition of 7 with a time step A t = tn+l - t , and let A h = (7;:)i=0,,,.,Ne be an admissible triangulation of W, C h = (Mi)i=O,.,,,Nd the Donald dual mesh, where

h := m i n l ~ i 1verify ~ sup lM,l 1

such that XT =

n

i

5

6.h2, and XT the barycenter of T E

Ah

is

n

dT E M ! TnMf0 1 := d M i n d M j n T the line segment between the points X T and xij the midpoint of ( x M , ,x M , ) and let Lh = ( 1 E d M \ r , forM E &}. For an initial condition u0 E LOU(G) n B V ( R ) , we set u k = JM u0 ( x )d+. For simplicity we assume that the functions Q and I i are piecewise constants, we set @M = @ I M and KT = K I T . Let U ; be an approximation of u ( x M t,) , which will be defined precisely in the sequel Integrating ( 1 ) over the set M x [t,,t,+I] where M E C h , we obtain the following scheme d M E T . We denote by

XM

MnTf0

:=

&

where n ' ~is, the ~ outward normal to 1 E d M . Using an explicit approximation of the convection term and an implicit approximation of the diffusion term, we

390

Finite volumes for complex applications

which could be written in the following form

+---

K T ~ M111, ~

where

with Ml E C h is such that 1 E Ml n M . The advection term is approximated by an upwind Godunov scheme and the diffusion term is approximated in the following way: for T n M # 0

where L E dT such that L n M = 0,and using a piecewise linear approximation for V a , we get

) dij. where X M E PI such that X M , ( X M ,= Finally, since divq'= 0 we have the following semi-implicit scheme

where DM,l = - IT1 V X M , ,.TICTVXM,Tfor all 1 E aM\r

Fields of application

39 1

R e m a r k 1 We can use an explicit [resp. implicit] approximation for both the diflusion and the convection terms t o obtain an explicit [resp. implicit] finite volume scheme which lead generally to large CPU time computing. For more details on the comparaison of these schemes see [AFA 971. 3. LCa stability weak BV e s t i m a t e s a n d convergence r e s u l t s

In this section we shall analyze the F.V. scheme obtained in section 2. Let us state the following assumptions: -

(Al) R is a bounded open polygonal subset of R2. (A2)

-

-

E L" (R) such that 0 < a- 5

(x) 5

a+ < 1.

(A3) Ii' is a bounded, uniformly positive definite symmetric tensor on S1 (i.e. V t # 0, 0 < K- (ti25 ( I . I<(x)(') 5 I<+1[1< m a.e. on x E Q). (A4) u0 E B V (R) such that 0

< uO(x) 5 1 a.e. in a.

(A5) a E C 2 ([O, I]), such that crl(0) = a l ( l )= 0 and crl(s) > 0 V s E 10, l [ . (A6) b, d E C1 ([O, 11) such that b is a monotone increasing function, and d ( s ) 2 d- > 0.

It should be noted that a full tensor for the absolute permeability K is considered and the approximation obtained in (4) is such that 0 < D- 5 Dml15 D+ < m which is an important property for the analysis of this FVS. In fact, we can write the coefficient DM,[ in the following forme:

where L and L' E 8T such that L n M = 0 and L'

__+

n Mr = 0,hence

---+

Furthermore, we have = &xlxM . adj [I
+. adj [KT] + + ? x'x~= , [UT]X XM [UT]X XM,

X'XM

,

then for Ii'(x) = k(x)Ko in R, where k(x) > 0 and Ii'o is a constant symmetric positive definite matrix such that adj [I
392

Finite volumes for complex applications

OKo := [Uo](a)a transformation of the domain S1 by the matrix Uo, with an admissible triangulation such that

where 17 is a small parameter independent of h , then we get also 0 < DThe extension of the present technique to the case where

=

< DM.l. -

NP

U Qp

such that

p=l

Ii'(x)ln, = kp(x)Kp, with kp (I)> 0 and lipis a constant symmetric definite positive matrix, for all p = 1, .., Np, is straightforward. Let Cq defined by

where N& = card (1 E d M ) , such that sup N,& is assumed to be finite, we introduce the CFL condition defined by: C F L :=

M

At o

~ ('(3) ~

5 1l

The following results hold: Proposition 1Under the assumptions (A1)-(A'/) and the CFL condition (5) the scheme (4) is Lw stable. Moreover, the approximate solution (uG)satisfies 0

5 1 for all M € C h and n = l , . . , N ,

(6)

Proposition 2Under the assumptions (A1)-(A'/) and the CFL condition (5) ,with [CFL I 1 - €1, we have the following estimates

At lM I ( a ( u z l ) - a(u%)' n,M

+

At 111 (a(u&,) - a(u%))'

5 C ( h + At)

n,l

where E is a small parameter and C is a constant indepenclents of h and At. Let us introduce a weak solution u of the problem (PI):

0 I u(x,t) 5 1

a.e. in Q,, a ( u ) E L2(0,T; ~ l ( S 1 ) )

S SQr[ Q u v ~+ (b(u)f-- KVcr(u)) . Vn] dcdt

+Sn @uov(x, 0) dx = 0 VV E V := { V E C211(S1;[O, TI),

vlr, = 0

(7)

, v (., T) = 0 )

The data are assumed to be smooth enough to guarantee the existence and uniqueness of the weak solution. We have the following result: Theorem 1 Under the assvmptions (A1)-(AT) the approximate solution uh, gzven b y the scheme (4) and the CFL condition (5) with [ C F L 5 1 - E ] , converges to u in L2(Q,) as h and At goes to zero.

Fields of application

F i g u r e 1. Reservoir

a

F i g u r e 2.

393

Triangulation of f10 = Uo (0)

5. N u m e r i c a l s i m u l a t i o n s In this section, we present some numerical results in 2-D. An IMPES simulator which applies a mixed-hybrid finite element method for computing pressure and fluid velocity approximations and the finite volume scheme described here for the saturation was developed. Numerical results prove the effectiveness and the robustness of this methodology. We have run various simulations, herein we are presenting only one of such a simulation to illustrate our results. A simulation was performed on a heterogeneous reservoir shown in Figure 1 for a two phase flow problem. An injection well is placed at the lower left hand corner of the reservoir and a production well is placed at the top right hand corner of the reservoir. The highest value of permeability (Ii' = 10) is in the darkest shaded blocks. The lowest permeability (K = 0.1) is in the lightest shaded blocks. The intermediate value of permeability (K = 1) is the intermediate shade in the picture.

I

F i g u r e 3. Saturation contours

I

Figure 4. Total velocaty

_I

<

394

Finite volumes for complex applications

6 . Bibliography

[AFA 971

AFIF M. A N D AMAZIANE B., On convergence of finite volume schemes for one-dimensional two-phase flow in porous media, Preprint, 1997, to appear.

[AWZ 961

ARBOGASTT., WHEELERM.F. AND ZHANGN.Y., A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media, SIAM J. Num. Anal., 33, NO 4, 1996, p. 1669-1687.

[BRF 91)

BREZZIF. AND FORTINM., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.

[CHJ 861

CHAVENTG . A N D JAFFREJ ., Mathematical Models and Fznite Elements for Reservoir Simulataon, North-Holland, Amsterdam, 1986.

[CJR 951

CHAVENT G., JAFFRE J. AND ROBERTSJ . E . , Mixed-hybrid finite elements and cell-centred finite volumes for two-phase flow in porous media, in: A. Bourgeat, et al., eds., Proceedings Mathematical Modeling of Flow Through Porous Media, World Scientific, London, (1995), p. 100-114.

[CHE 971

CHEN Z. A N D EWING R.E., Fully-discrete finite element analysis of multiphase flow in groundwater hydrology, SIAM J. Num. Anal., 34, NO 6, 1997, p. 2228-2253.

[EGH 981

EYMARDR., GALLOUET T., GHILANIM. A N D HERBINR., Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes, IMA J. Num. Anal., 18, 1998, p. 563-594.

[VEV 981

VERDIERES. A N D VIGNALM.H., Numerical and theoretical study of a dual mesh method using finite volume schemes for two phase flow problems in porous media, Numer. Math., 80, NO 4 , 1998, p. 601-639.

A preconditioned finite volume scheme for the simulation of equilibrium two-phase flows

S6bastien Clerc Commissariat d I'Energie Atomique, CEA-Saclay, 91 191 Gif-sur- Yvette F R A N C E

In this paper, we report on a preconditioned finite volume technique to simulate two-phase flows in complex geometries.

ABSTRACT

Key Words: two-phase flows, low Mach number flows, preconditioning.

1. Introduction The numerical simulation of two-phase flows is a crucial problem for the design and safety analysis of heat*exchangers, especially in the nuclear industry. Generally, the fluid is in the liquid phase a t the inlet of the exchanger. Because of the heat release or a loss of pressure, vapor is created. The description of the interface between the phases would require a very fine space scale with respect to the characteristic length of the geometry. For this reason, one must consider a macroscopic model. This model should reproduce some of the characteristic features of two-phase flows. The most basic of these features is probably the compressibility effect introduced by vapor bubbles. Indeed, although the density of liquid water can usually be considered as a constant, the apparition of bubbles creates large variations of the mean density of the flow. Moreover, the dependence of this mean density with respect t o the pressure can not be neglected. In other words, if we consider the bubbly flow as a homogenized mixture, the resulting averaged speed of sound is quite low in usual conditions. In order t o study this feature, we consider the very simple homogenizedequilibrium two-phase flow model. Indeed, using more sophisticated models would introduce other difficulties which are not in the scope of this study.

396

Finite volumes for complex applications

2. T h e equilibrium two-phase flow model The homogeneous equilibrium model of liquid/vapor flows is characterized by a kinematic and thermodynamic equilibrium assumption. The evolution of the mixture can be described by the Euler equations for a single fluid:

+

+

Here E = e luI2/2 denotes the total energy, and H = h luI2/2 the total enthalpy of the mixture. To close the system, an equation of state (EOS) links the pressure p to the conservative thermodynamic variables p and pe. The pressure law must be such that the partial derivatives x and K with respect to p and pe satisfy mh x > 0.

+

The sound speed c of the fluid is the square root of this quantity. Generally, a drift flux is added to the set of equations (1)-(3) to incorporate the effects of the slip velocity between the phases. This will however not be the case in the present study.

2.1 Equation of state The equation of state consists of three zones: a liquid, a two-phase mixture, and a vapor zone. The liquid and the vapor zones are described by usual single fluid equations of state. In the two-phase zone, each fluid is supposed to be at saturation, so that the density pe, p, and the enthalpy he, h, of each phase are functions of the pressure only. The density p and the enthalpy h of the mixture satisfy : h, - hl - hv - h h - hl -----. P Pe PU

+

The transitions between the mixture zone and the two others occur along the saturation curves. The equation of state is continuous across this curve but not continuously differentiable. This fact is clearly visible in Fig. 1.

Fields of application

397

Figure 1: Specific volume l l p versus enthalpy h for the water equilibrium EOS, at pressure p = 7MPa. The liquid zone corresponds to lower values of h , and the vapor zone to the higher values of h. In the two-phase zone in-between, l l p varies linearly with h . Note that the derzuatiues of the equation of state are descontinuous across the transitions. 3. The preconditioned finite volume scheme 3.1. Some notations

We rewrite the Euler equations (1)-(3)with the compact form:

dtU

+ div F ( U ) = 0.

(4)

We consider a triangulation of the computational domain by polygonal cells. For a cell I<, we denote by Jli'l its volume and N(1i') the set of neighboring cells. If J E JZ/(K), the surface of the common interface is U K J and n K J denotes the unit normal, oriented from Ii' to J. Finally, we denote by 6t the step of the time discretization. The finite volume method for the solution of the system of con~ervat~ion laws (4) takes the form:

Here, Uk denotes the average value of the solution in the cell Ii',at time n6t. To completely determine the numerical method, we have to specify the expression of the numerical flux Q K J in terms of the cell average values. We will consider

398

Finite volumes for complex applications

numerical flux functions of the form:

The viscosity matrix Q depends on the left and right states UL and UR. The numerical flux we have used is a preconditioned version of Roe's numerical flux. We will first begin by describing the extension of Roe's flux to our system, and then int,roduce the preconditioning. 3.2. Roe's flux

Roe's scheme is obtained by chosing for Q = IAl, where A is the Roe /XiIri@ li, if A = Airi@ li is the eigen-decomposition matrix, and IAl = of A . We will denote by 1.1 the difference between the right and left values of any quantity, i.e. (.)R - (.)L. The Roe matrix has to satisfy the following properties:

Ci

xi

A, has a complete set of real eigenvectors An(UL,UR) + D F ( U ) . n when UL and UR tend to U

As is well known (see e.g. [ViMo, To 91]), a Roe matrix for the Euler equation with an arbitray EOS can be found if we can provide a linearization of the pressure jump bl = fUpl + 4IpeIll (7) such that (8) Geometrically, the solutions to this problem are located on the intersection of a line and a half plane in the (x, K ) plane. We will denote by ([XI - the vector (([p],([pel)and by Dp = (x, K ) , such that (7) becomes b] = 1x1 . Dp. Following the lines of [ViMo], we start with a point & which is some average value of DpL and DpR, but does not satisfy (7). We project it on the line, with respect to a scalar product defined by a given symmetric matrix M :

Dp= ~ + < M ( [ ~ ] with , I = bl - 1x1 . Dp M b l . UxD

We can then check a posteriori if condition (8) is satisfied. If not, we must consider another initial guess 5. It is clear that if p is continuously differentiable, stays bounded when 1x1 tends to zero. Moreover, if we choose for M z n approximate value of the Hessian as expected. For of p, then will be close to one. Finally, D p will tend to

<

Fields of application

399

our simulations at p = 5 MPa, we found that a diagonal matrix diag(1, lo-'') gave satisfactory results. However, if we are close to a point where p is not continuously differentiabb then ( will certainly blow up when [x] goes to zero. As a result, the value of D p will not tend continuously to G.This fact lies at the root of some numerical difficulties in the computation of phase transition inside the flow. It becomes particularly important when the discrepancy between the derivatives is large, i.e. for lower values of the pressure. Obviously, more research seems in order on this point.

3.3. The preconditioned flux

For low Mach number applications, it is preferable to use a preconditioned version of Roe's scheme. The viscosity matrix (see eq. (6)) is now Q = P-' IPAI, where P is a properly chosen non-singular matrix. Note that since we have only modified the viscosity matrix, the consistency of the numerical scheme is not affected. Therefore, the scheme is time-accurate and suitable for transient problems. This formulation was first introduced by [GuiVio] and further studied by [Cl 981, see also [PCV+]. In this work, we have used Turkel's diagonal preconditioner [Tu 871. This matrix depends on a single parameter /3 which was set to the local Mach number.

4. Implicit time stepping 4.1. Choice of the variables for the linearization

In low Mach number applications, an implicit time stepping cannot be avoided. First, the CFL constraint on the time step for explicit schemes becomes too restrictive when the sound velocity gets large. Moreover, for the time-accurate preconditioned scheme, the stability condition is even more restrictive than the CFL condition. One can consider the usual linearized backward Euler scheme:

+

with Un+' = U n SU. It is also possible to linearize the problem with respect to an other set of variables: setting U = U(V), one can consider

400

Finite volumes for complex applications

+

with Vn+' = Vn 6V. At steady state (6V = 0), this formulation is still conservative. The conservation is not ensured during the transient but can be restored if necessary by performing a few Newton steps within a time-step. Note that this is also true of the conservative variables formulation (9). More specifically, we advocate the use of the following set of variables: (p, p u , p ) . Including the pressure in the set of variables has two advantages in our case. First, it makes the computation of the thermodynamic variables simpler and faster. If the density and the pressure are known, we can readily know in which zone of the EOS the state is located. On the contrary, if the energy rather than the pressure is known, we must first guess in which zone it is, and modify this guess if necessary. More importantly, it can dramatically improve the convergence of steady state computations when contact discontinuities are present (see [Cl 991). Indeed, the pressure time-increment is usually very small in low Mach number flows, although the density and the energy time-increments can both be important. With a non-linear EOS such as the equilibrium EOS, the conservative variables formulation (9), the density and energy increments will automatically induce spurious large pressure increments. This phenomenon is responsible for the apparition of spurious acoustic waves of large amplitude in the solution, which prevents the use of large time steps in the implicit scheme. With formulation ( l o ) , larger time steps can be used. Although in some cases the time step can be set close to infinity, it is generally advisable to stay close to Sx/u, where Sx is the typical mesh size and u is the reference material velocity. In low Mach number flows, this time step is of course much larger than the maximum explicit time step.

4.2. Approximate Jacobian for the preconditioned flux Equation (9) (and (10)) require the computation of the partial derivatives d@/dUL and d@/dUR of the numerical flux. Due to the algebraic complexity of the numerical flux, an exact computation of these matrices is not easy, especially for a general EOS. We have thus employed the usual approximation for Roe's flux:

To obtain the derivatives with respect t o V, we left-multiply these matrices by the Jacobian of the change of variables aU/aV. Practically, it amounts to expressing the left eigenvectors of A (for Roe's scheme) or P A (for the preconditioned scheme) in terms of the pressure rather than the energy.

Fields of application

401

5. A two-phase bump c h a n n e l flow.

We briefly present a two-dimensional test-case for the equilibrium two-phase flow model. We consider a channel with a 20% sinusoidal bump (Fig. 2). A

Figure 2: Channel with bump: 20 x 80 structured mesh. pressure of 5 Mpa is imposed at outflow. A constant velocity of 10 m . s-' is imposed at the inlet, and the enthalpy is set to 1154.2 KJ. This corresponds to a liquid state close to the saturation. As the pressure drops with the restriction of the section, a small concentration of vapor appears. The Mach number jumps from to 0.4, so that the compressibility becomes important. We initialize the computation with a purely liquid steady-state computed with a higher value for the pressure. The pressure profile is displayed on Fig. 3. As expected, no discontinuity can be detected across the phase transition lines. On the contrary, the influence of the vapor is clearly visible on the Mach number profile (Fig. 3, right), which shows sharp discontinuities. For both profiles, the symmetry is quite satisfactory.

Prcssvrc (1U.eS Pa), mln = 49.5397, msx = 50.1 117

log(M). mln = -4.87074. max = 6.882866

F i g u r e 3: Channel with bump: Left: pressure, 20 isolines. Right: logarithm

of the Mach number, 80 isolines. Although the Mach number exhibits a strong discontinuity, the pressure field is continuous.

402

Finite volumes for complex applications

6 . Bibliography

[C199]

CLERCS., Accurate computation of contact discontinuities for flows with general equations of state, Computer Methods in Appl. Mech. Engrg., to appear.

[C198]

CLERCS., On the preconditioning of finite volume schemes, 7 t h Int. Conf. on Hyperbolic Problems, Ziirich, February 1998, Birkhauser.

[GuiVio]

GUILLARDH., C. VIOZAT,On the behavior of Upwind Schemes in the Low Mach number limit, Computers & Fluids, 28, 1999.

[PCV+]

P A I L L ~ RH., E CLERCS., VIOZATC . , TOUMII . , MAGNAUD J.-P., Numerical methods for low Mach number thermalhydraulic flows, Proceedings of the 4th ECCOMAS Conf. on Comput. Fluid Dynamics, Athens, 1998, John Wiley & sons.

[To 911

TOUMII., A weak formulation of Roe's approximate Riemann solver, J.C.P., 102, 1991, pp. 360-373.

[Tu 871

TURKELE., Preconditioned methods for solving the incompressible and low speed compressible equations, J. C.P., 72, 1987, pp. 277-298.

[ViMo]

VINOKURM . , M O N T A G NJ~. L~ . , Generalized flux-vector splitting and Roe-average for an equilibrium real gas, J . C.P., 89, 1990, pp. 276-300.

Transient flows in natural valleys computed on topography-adapted mesh

S. Soares Frazao, J. Lau Man Wai,Y. Zech Universite catholique de Louvain Place du Levant 1 B - 1348 Louvain-la-Neuve

The paper deals with the finite-volume computation of severe transient flows i n natural valleys. The major problem concerning those flows is to have an appropriate representation and treatment of the topographical source term. A lateralization of the intercell flux is presented here to account for the bottom slope. Computation eficiency is gained by using a topography adapted mesh. Finally, an application to a Belgian valley is shown.

ABSTRACT

Key Words: Finite-Volume, topography, source terms, mesh

1. I n t r o d u c t i o n

Severe transient flows in natural valleys occur during extreme flood events, or as a result of a dam-break, which can be considered as the worst case. Several Finite-Volume numerical schemes have been developed for the computation of such flows, mainly focusing on an accurate evaluation of the numerical flux between cells. However, in natural topographies, important source terms appear due to the shape of the valley itself, they can also have relevant effects on the wave propagation. Therefore, an important step for a reliable computation is thus an accurate representation of the valley complexity, requiring a suitable mesh generation, in order to achieve a proper calculation of the topographical source term.

404

Finite volumes for complex applications

2. Numerical scheme

A Finite-Volume numerical scheme is used to solve the Saint-Venant shallowwater equations. Those are written, in the 2D case, Ut

+ Fz ( U )+ G y( U ) = S ( U )

(1)

where subscript t and x denote time and space derivatives respectively. The vectors of dependent variables U, fluxes F ( U ) and G ( U ) , and source terms S ( U ) are defined as

u=

= (

( )

v2h

:vl +

uh ( U ) = (.2h,,,,)

igh2

)

,s(U)=

(

0 gh(Sox-Sfx) gh(Soy - S f v )

)

with h the water depth and u, v the depth-averaged velocity components. So and S f are the bottom and friction slope, respectively written for the x- and y-direction. Friction slope S f can be evaluated from the Chezy, Manning or another empirical formula. Integrating (1) on a discrete cell yields the following finite-volume scheme

where A is the cell area, L j is the length of the cell interface j and n the number of cell interfaces. The vector is the vector of rotated hydraulic variables obtained by applying the rotation matrix T to the original vector U and is aligned with the new co-ordinates axes (z',y') where x' is the direction normal t o the considered cell boundary. The numerical fluxes between two cells located at the left and right side of the interface (cells 1 and r ) are then calculated by the Boltzmann model [SOA 991

u

Fields of application

405

right

Figure 1: Lateralization of the topographical source term where F r is the Froude number. F r represents the local Froude number, calculated with the velocity normal to the interface.

3. Lateralized topographical source terms

In each discrete cell, the bottom level zb is assumed constant. To compute the concept of flux lateralization the topographical source term So, = [CAP 961 is used, and adapted to 2D problems. This means that the flux coming out of the cell located a t the left side of the interface differs from the flux entering the cell located a t the right side by a quantity gdz(hleft accounting for the change in the bottom level (see figure 1, hatched area). The hydrostatic pressure term in the flux is calculated with h being (hleft-dzb) or (bright) for the left and right cell respectively. This ensures that for water a t rest, the pressure terms and the topographical source term, which are the only non-zero terms, cancel each other out. According to Nujic's compatibility condition [NUJ 951, the water will thus stay at rest. The lateralization is a very robust method, as no special reduction of the time step (or of the CFL number) is needed. However, it must be stated that it is probably not the most accurate method, especially for 1D problems. Indeed, in some cases, influence on the wave front velocity was observed, leading to a too slow flow propagation.

-2,

2)

gg

4. Topography-adapted mesh generation

The description of the valley topography coming from a DTM (Digital Ter-

406

Finite volumes for complex applications

Figure 2: Definition of the segment density rain Modeling) usually consists of a square grid of points giving the bottom elevation. For dense grids, it is likely that the computation time becomes too important when considering all the available points. On the other hand, building a uniform and coarser mesh upon the given square grid can lead to accuracy losses and interpolation errors. In fact, rather flat areas could be represented with a lower mesh density than regions with abrupt changes in the topography and local steep slopes. A triangular mesh based on a given DTM can be constructed as follows. A first triangulation is generated only with the points defining the boundary of the computational domain. Additional intermediate points are then added one by one on existing segments, until the requested refinement has been reached. The mesh segments to be divided are selected according to their density. A segment between two points A and B (see figure 2) is said to have a high density when the error made by replacing the exact topography between A and B by a straight line is important. The error is quantified in the following way : I ~ A B , e z a c t- ~ ~ ~ , s t r a i ~ h t l d ~

(4)

After each addition of a new point, the mesh is checked locally to avoid triangles with small internal angles. This leads to a local Delaunay mesh, with well conditioned triangles, and a finer discretization where needed, according to the given topography.

5. Application to the valley of Robertville The valley of Robertville is situated in the Eastern part of Belgium. A concrete gravity dam is located at the upstream end of the computed reach. The valley is steep and narrow, and widens at the downstream end, near the city of Malmedy (see figure 3). The aim of the study that was carried out was to determine the arrival time

Fields of application

407

Figure 3: valley of Robertville - distances i n [m] of the water a t the city of Malmedy in case of a dam break. A DTM was available for the valley, the grid size being of 40m. 5.1 Estimation of the friction coefficient

A Manning friction coefficient of 0,02 sm-'I3 was estimated for the valley. It is clear that a unique friction coefficient for the complete valley is not the most accurate choice, but this was due t o the lack of information. A sensitivity analysis was thus carried out and showed the importance of this parameter on the wave propagation velocity. 5.2 Computed results

The failure of the dam is considered as instantaneous. Several computations were run, and the criterion selected to compare the results is the arrival time at the city of Malmedy. A first run with a 1D model gave a very short time (see Table 1). Computation were then run with different triangular mesh refinement levels identified by the maximum cell interface length : loom, 70m 60m. The square grid given by the DTM was also used t o compare with the other meshes. Figure 4 shows the inundated area once the water has reached the city of Malmedy. The city is located in the bottom left corner of the figure and the upstream reservoir is located a t the right side of the dam. The sharp bends and the very narrow shape of the valley clearly appear. It can be seen in Table 1that for triangular meshes, the travel time decreases

408

Finite volumes for complex applications

Figure 4: Inundated area - distances in [m] with the mesh refinement. However, it is not expected to reach the short travel time of the 1D model, even with further refinement. Indeed, the 1D model considers the valley as being straight, i.e. without any bends, which is absolutely not the case for Robertville. The influence of bends on the flow was shown in [SOA 991: important head losses occur, that slow down the wave front considerably. Sharp bends can even lead to partial reflection of the flow, with a bore travelling back in the upstream direction. The square grid appears to lead to the longest travel time. A possible explanation is the important error made on the flow path when the valley is not aligned with the mesh (see figure 5). A similar feature can be observed on mesh 1D 2D triangular (L = 60 m) 2D triangular (L = 70 m) 2D triangular (L = 100 m) 2D square

time [minutes] 6 17 21 31 50

Table 1: Arrival times at Malmedy

Fields of application

409

Real shape of the valley /

Computed valley

Figure 5: Influence of the square grid on the flow path triangular meshes, as the cell limits do not always match the exact wetted area delineation. However, with a triangular mesh, the error is less, as no preferential flow path is induced. Moreover, the fact that the mesh is generated according to the topography leads t o a denser mesh a t the steep river banks, allowing a more accurate representation of the wetted area delineation. 6. Conclusions

A lateralized treatment of the topographical source term of the 2D shallowwater equations was presented. Computations of a dam-break flow in a steep natural valley were shown on different kinds of meshes (triangular and square). The importance of an accurate representation of the topography of natural valleys clearly appears : the mesh used has a significant influence on the wave front velocity. Indeed, important differences in terms of wave propagation time were observed on the different runs. References [SOA 991

SOARESFRAZAOS., SILLEN X . , ZECHY., Dam-Break Flow Through Sharp Bends. Physical Model and 2D Boltzmann Model Validation , to appear in CADAM, Proceedings of the first expert meeting, Wallingford UK

[CAP 961

CAPARTH . , YOUNGD.-L., HUANGS.-Y., PAN J . - M . , A lateralized flux-predictor scheme for a computation of openchannel flow in arbitrary topography , Proceedings of the 20th

4 10

Finite volumes for complex applications

National Conference on Theoretical and Applied Mechanics, National Taiwan University, Taipei, China (1996) [NUJ 951

NUJIC M., Efficient Implementation of Non-oscillatory Schemes for the Computation of Free-Surface Flows , Journal of Hydraulic Research , 33 ( I ) , 1995

A mixed Finite Volume/Finite Element method applied to combustion in multiphase medium

Nathalie Glinsky-Olivier t, Eric Schall $ t CERMICS-INRIA, Sophia Antipolis, France, $ INRIA, Sophia Antipolis, France

The present study is concerned with a mixed finite volume/finite element method applicable to triangular unstructured meshes for the simulation of wildfires propagating through a fuel bed using the multiphase approach. The forest fire is represented by a gas phase propagating through solid phases representing the heterogeneous combustible medium. Attention is focused on the development of a solution method of these equations. An explicit second-order i n time and third-order i n space accurate scheme has been constructed based o n a special approximate Riemann solver applying the Roe's formulation. The radiative energy flux is calculated by the Discrete Ordinates Method. This model has been applied for simulating a wildfire through a litter of dead pine needles. ABSTRACT

Key Words: finite volume, finite element, multiphase approach, upwind scheme, combustion.

1. I n t r o d u c t i o n

The present study is concerned with simulation of wildfires propagating through a fuel bed using a multiphase mathematical model. For this reason, attention was focused on an adaption of the Roe's scheme t o our new set of equations. The source term includes many complex physical phenomena (heating, drying, pyrolysis of the fuel material and char combustion) and operates in each conservation equation.

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2. The physical model

For this study, we have applied the multiphase formulation proposed by M. Larini et al. [LAR 991. A brief description is given below. The fuel bed is supposed to be an heterogeneous medium made of solid particles of different types. A small control volume V contains N solid phases and a gaseous phase. Each solid phase is composed of particles of same thermochemical and geometrical properties (i.e. shape, size, arrangment) providing the same behavior under fire. The gas phase is a mixture of five species : CO, CO2, H20, 0 2 and N2 numbered from 1 to 5 in this order. The packing ratio of the phase k is a k = Vk/V where Vk is the volume occupied by the phase k in the total volume V. In the same way, the fractional porosity is a, = Vg/V. We then have the relationship a, C kN= l a k = 1. For the gaseous phase, we consider the conservation equations for the variable Wg Wg = ( P , P U , P pe, ~ , py1, py2, py3, ~ ~ 4 1 ~

+

where p = p, a, and p, is the density of the gas, (u, v) is the velocity of the gas, e is the specific internal energy and Yi are the mass fractions of the species with c:=, Yl = 1. This relationship allows us to solve only 4 conservation equations for the species, the fifth mass fraction being deduced from the others. The set of conservation equations has been provided by M. Larini and B. Porterie [POR 981 following the ideas in [GRI 851. It can be written in conservative form :

where the notation Wt means dW/dt. The convective fluxes F and G write :

The pressure a g p is calculated by using the perfect gas law for a gas mixture and the definition of the specific internal energy

where M~is the molar mass of the species i and T, is the temperature of the gas. By combining these relationships, the pressure can be expressed as a function

Fields of application

41 3

of y, the ratio of the specific heats for the gas mixture

The diffusive fluxes R and S write:

where 2) is the diffusion coefficient of the species. The stress tensors have the classical expressions

0, is the source term : me20

+ mp' + k S " ' f

-dragx - pg sin 0 -drag, - pg cos 8 N

N

5

where mHZ0represents vaporization, m p ' and m s " ' f are the the solid mass loss due t o pyrolysis and char combustion respectively. drag, and drag, are the two components of the drag force and g is the norm of the gravity. Qradand Q c o n d represent the heat transfer between gas and solid phases due to radiation and conduction respectively. Ah! are the formations enthalpies of the species. More details can be found in [POR 981. The solid phase is described by the variable Wk [POR 981 Wk = ( a k p k , ~ k where p k and Tk are respectively the density and the temperature of the solid phase k. The mass balance and the heat conduction equations for the solid phase k write : ( ( ~ Y ~=P -'hH20 ~ ) ~ - hP'- m S U T f a k pk C p , (Tk),= 0, (Wg,Wk)

{

)

~

414

Finite volumes for complex applications

where C ,, is the specific heat at constant pressure and Tk is the temperature of the solid phase lc. The solid phases are supposed to be at rest, then the momentum equations reduce to uk = v k = 0. These equations contain no flux, only a source term. The equations for the solid phase are solved separately from the gaseous phase equations both system being coupled by the source terms fl, and Ok which depend on Wg and Wk. For the closure of the model, we distinguish two types of boundary conditions : the open sides (right, left and top of the domain) when a special upwinding is applied, and the lower boundary of the domain which corresponds to the ground where we apply no slip conditions for the velocity u = v = 0 and adiabatic temperature dT/dn = 0. The system formed with the convective fluxes is hyperbolic i.e. for any vector (71,72), the matrix M = qldF/dWg q2dG/dWg is diagonalizable. Its eigenvalues are real and are

+

Then, we can write M = T-'(W,, if) A(Wg,fj) T(Wg,fj) where A = diag(A;) is the diagonal matrix composed of the eigenvalues A; of M and T is the invertible transformation matrix. The absolute value of M writes IMI = T-l(Wg, 7-j) JA(W,, );I T(W,, 7-j). We can notice that the two last eigenvalues A7 and As have different expressions from the classical reactive Navier-Stokes system. Their values are close to the usual eigenvalues u71 v 172 -f

+

m.

3. Spatial approximation

We give here a short description of the numerical model, details can be found in [SCH 991. Starting with a (possibly unstructured) Finite Element triangulation of the calculation domain D, dual Finite Volume cells are constructed joining successively the gravity centers of the triangles having a node i as a vertex and the middle of the sides adjacent to this node. We write the variational formulation of the gaseous system. The mixed FV/FE formulation traduces by the choice of the test functions \ki. For the temporal, convective and source terms, the test function Q i is the characteristic function of the cell Ciand for the diffusive part, Q i is the P1 Galerkin basis function a t the node i whose support is Si= UT,iETT.After integration of the different terms on their respective support, application of the Green formula and mass-lumping

Fields of application

4 15

for the temporal and source terms, we obtain

+

-

LC,

+ G ( W )n2)d o = A, C2(W)i (R(W)nx+ S ( W ) n ydo ) (R(W)('@i)x + S(W)('@i),) dzdy +

d i (Wt)i

/ L,

( F ( W )nl

where d i is the area of the cell C;, (nl,n21T is the outward normal to the cell C, and (n,, n y )is the outward normal t o the boundary of the calculation domain D. A first-order upwind approximation of the convective fluxes write

+

( F ( W )n~ G ( W )nz)do =

@ (Wi,Wj7G i j ) j neighbour of i

where rit3 is the integral normal to the interface between Ci and Cj and @ is the numerical flux through this interface. This numerical flux is calculated by applying a Roe scheme [ROE 811 adapted to our special convective fluxes F and G. For the calculation of the fluxes of the mass fractions equations, we use the multi-componenents flux proposed by B. Larrouturou [BLA 891, which allows to preserve positivity for the mass fractions. If is the numerical flux for the density variable, the flux of the species Y is :

High-order spatial accuracy is obtained via the M.U.S.C.L. method [VLE 721, [LFE 851 and a P-scheme formulation which consists in writing the numerical with for instance Wij = W, + 1/2[(1 - 2P)(Wj- W,) + flux (Wij,Wj,, 2 p d ~ i . i ;where /3 is an upwinding parameter. Choosing P = 1/3 provides a third-order accurate scheme in space. A P1 Finite Element interpolation for the approximation of the diffusive fluxes leads to

where AT is the area of the triangle and R I Tand S I Tare constant on T. More details can be found in [LFE 891. The multiphase radiative transfer equation is solved using the Discrete Ordinates Method [SAK 961. Since the soots are not included in this model, the contribution of the radiation is negligible compared to the other physical phenomena. A detailed description of this implementation can be found in [SCH 991.

416

Finite volumes for complex applications

3. Numerical results

This numerical model has been applied for describing the behaviour of a spreading wildfire through a litter of dead pine needles. The domain is l m height and 2m long and the fuel depth is 0.04m. The mesh is composed of 5151 nodes. Test conditions are the same as described in [POR 981. The fire is ignited at the middle of the litter and the ignition temperature of the solid is equal to 700K since the gas is at ambient temperature. We present the isovalues of the gas temperature, of H 2 0 and C 0 2 at two different times (0.26s and 1.08s) showing the evolution of the phenomenon. At time 1, an energy convective-conductive transfer is well observed. We also = 0.14) and the formation of notice the presence of water vapor (H20,,, (C02,,, = 0.28) while the gas temperature reaches T,,, = pyrolysis gases 1717.K) Later, at time 2, an increase in gas temperature (T,,, = 3068.K) = 0.095) and the is due to the combustion of the pyrolysis gases (COz,,, decrease of the water (H20,,, = 0.017). The propagation to the left of the flame front can be observed : the maximum values of the variables have moved towards the unburnt part of the litter. 4. Conclusions

A mixed Finite Volume/Finite Element method to solve a 2D multiphase radiative and reactive model of line-fire propagation is proposed. The different physical phenomena are well observed (heating, pyrolysis and combustion). The first results obtained with this model are very encouraging. Physically, a model including soot formation will be considered to show the importance of the radiation in such combustion problems. Numerically, this kind of unsteady problems is very time consuming. An Adapted accurate implicit solver is in progress. Aknowledgment The authors thank M. Larini and B. Porterie for having provided the physical model and the set of equations. This work has been financed by the European Economic Commission under the contract EFAISTOS No.ENV4-CT96 0299. 5. Bibliography

[LFE 851

FEZOUI,L., Resolution des equations d'Euler par un schema de van Leer en elements finis, Rapport de Recherche INRIA No.358, janvier 1985.

[LFE 891

FEZOUIL. et al., Rksolution numerique des equation de Navier-Stokes pour un fluide compressible en maillage triangulaire, Rapport de recherche INRIA No.1033, mai 1989.

Fields of application

41 7

[GRI 851

GRISHINA . M . e t al., Study of the structure and limits of propagation of the front of an upstream forest fire, translated from : Fizika Goreniya i Vzryva, 21(1), pp.11-21, 1985.

[LAR 991

LARINIM. et al., A multiphase formulation for fire propagation in heterogeneous combustible media, Int. J., of Heat and Mass Transfer (to be published).

[BLA 891

LARROUTUROU, B. e t al., On the equations of multicomponent perfect or real inviscid flow, Carasso, Charrier, Hanouzet et Joly Editors, Non linear hyperbolic problems, pp.69-98, Springer-Verlag Heidelberg, 1989.

[POR 981

PORTERIEB. et al., Wildfire propagation : a two-dimensional multiphase approach, Combustion, Explosion and Shock waves, Vo1.34, No.2, 1998 also in Fizika Goreniya i Vzryva, Vo1.34, No.2, 1998.

[ROE 811

ROE,P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J.C.P., 43, pp.357, 1981.

[SAK 961

SAKAMIM. et al., Application de la Methode des Ordonnees Discr6tes au Transfert Radiatif dans un Milieu Bidimensionnel Gris & GCometrie Complexe, Rev. Gen. Therm., Vo1.35, p p . 8 3 ~ - 9 4 1996. ~,

[SCH 991

SCHALLE. e t al., A mixed Finite Volume/Finite Element method applied to combustion in multiphase medium, INRIA Report, to appear.

[VLE 721

VAN LEER B., Towards the ultimate conservative difference scheme I : the quest of monotonicity, Lecture notes in Physics, Vo1.18, pp.163, 1972.

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Finite volumes for complex applications

Figure 1: Isovalues of the temperature - time 1 and time 2

Figure 2: Isovalues of H 2 0 - time 1 and time 2

Figure 3: Isovalues of C 0 2 - time 1 and time 2

Turbulence Modeling for Separated Flows

L.J. Lenke and H. Simon Department of Turbomachinery, University of Duisburg 0-47048 Duisburg

The influence of the turbulence modeling on viscous flow field calculations has often been discussed i n the past. For a meaningful comparison of different turbulence models the access to reliable measurement data is necessary. Therefore, the 2 - 0 flow through a symmetrical diffuser with a large asymmetrical separation is chosen to investigate the separation behavior of three different two-equation turbulence models. The k-w-, a low-Reynolds-number k f - and an expliczt algebraic Reynolds stress model are considered i n this investigation. The numerical results of the algebraic Reynolds stress model are similar to the k-w-model zn prediction of the reattachment length but are i n better agreement with experimental data downstream the reattachment. Significantly improved numerical results were obtained compared to the k--model. Furthermore, different 3 - 0 calculations of the flow within the return channel of a multz-stage centrifugal compressor are presented. Especially at off-design conditions the turbulence models deviate notably i n their prediction of separation and show great differences i n the calculation of the 3 - 0 flow structure and secondary flows. Key Words: k-w, k t , algebraic Reynolds stress model, separation, diffuser, return channel

1. Introduction

There are already a large number of two-equation turbulence models but a basic problem of these kind of models, namely, their failure to correctly predict the amount of separation in adverse pressure gradient flows, is still unresolved. Among the various models in existence, the k-€-model is currently most popular and applicable to many practical complex flows with reasonable computational

420

Finite volumes for complex applications

economy and accuracy. However, for the prediction of large flow separations and recirculations especially with regard to the prediction of the flow downstream of the separation the k-€- and k-w-model have to be abandoned in favor of the higher order turbulence models which will increase strongly the numerical effort. Another point of criticism is that the k-E-models need damping functions within the boundary layer. Recently, the explicit algebraic Reynolds stress models (ARS) which are applied in the present paper within the context of the k-€-equation extend the range of applicability of two-equation models. This will be illustrated by two test cases (2-D diffuser and return channel) involving separations at plane and curved surfaces. The 2-D flow through the diffuser is characterized by a large asymmetrical separation at one side of the diffuser. The second test case show the 3-D flow within a return channel which is typically to join the exit from one stage of a centrifugal machine to the inlet of the next stage. 2. The Numerical Scheme

In this investigation the code developed by Reichert [RE1 951 with a finite volume formulation of the full Navier-Stokes equations is used picking up elements from Roe's and Osher's scheme. F'urthermore, only steady state solutions are considered. For high convergence rates, an implicit Newton-Raphson-like iterative method is used to solve first the averaged conservation equations and then the modeled transport equations but with different CFL numbers for each set of equations. Furthermore, the local time step size is calculated using a CFL number, which is a function of the local change of the density (details are described in Lenke and Simon [LEN 97al). For the simulation of turbulent flows non linear eddy-viscosity models are an increasingly popular approach, motivated principally by the desire to combine the physical realism offered by second-moment closure with the simplicity and numerical robustness of linear eddy-viscosity models. Gatski and Speziale [GAT 931 derived an explicit algebraic stress equation for three-dimensional turbulent flows which must be solved in conjunction with two transport equations. In the present study the algebraic stress equation is solved with the standard k-equation and an &-equation which is extended by an additional production range time scale and a cross-diffusion term (ARS-model) to improve the separation behavior of the turbulence model (details are described in Lenke and Simon [LEN 97bl). This ARS-model will be compared with the low-Reynolds-number k-€-model devised by Lam and Bremhorst [LAM 811 and the k-w-model devised by Wilcox [WIL 911. 3. Boundary Conditions

At the inlet boundary, in both test cases the averaged total pressure and

Fields of application

421

total density were specified. Furthermore, the distributions of the velocity components were taken from the measurements. At the outlet, in both cases a variation of a constant static pressure was used to establish the correct mass flow. All solid surfaces were modeled as rigid, non-slip and adiabatic. The other flow quantities were extrapolated from the interior. In both cases no values about the measured turbulent kinetic energy a t the inlet are available so that k and E were extrapolated from the interior which leads to an averaged turbulence intensity of nearly 1%. 4. Numerical Results

4.1. 2-D Diffuser

The performance of the two-equation turbulence models were evaluated for the flow through a 2-D symmetrical diffuser with a generating angle of 2 0 = 14.25'. A 118x65 grid was used and the numerical results were compared with measurements taken from Gersten et al. [GER 871. The flow through the diffuser is characterized by a large asymmetrical separation at one side of the dieuser which is shown in Fig. l a . All the turbulence models calculated such a separation but there are great differences in the prediction of the reattachment point or velocity profiles. Fig. l b shows the velocity distributions near the beginning of the diffuser. At this position the ARS- and k-w-model calculate a very small backward flow which is similar t o the measurements. The k-Emodel calculates a symmetric velocity distribution which shows no similarity with the measurements. Fig. l c and Id show the velocity distributions within the separation and demonstrate the underprediction of flow separation with the k-&-model in this case. The two other models are similar and differ only within the separation. At this location the ARS-model calculates the highest negative velocities and agrees very well with the measurements. Near the reattachment and especially downstream the reattachment the results demonstrate the superiority of the ARS-model in relation t o the standard two-equation models. The k-w-model overpredicts the reattachment length and calculates downstream the reattachment great differences between the measured and predicted velocity distribution (Fig. Id and e). The ARS-model calculates the reattachment nearly correct with the highest negative velocities within the separation and the highest positive velocities downstream of the separation. 4.2. Return Channel

The calculations to be presented have been done for the flow through a return channel with a small flow coefficient and a design flow angle a3 = 26' at the inlet. The geometry of the channel and the computational H-grid are

422

Finite volumes for complex applications

shown in Fig. 2. The grid is a single block grid with y+ < 1. Only near the leading edge y+ has maximum values of 5 in a very small region. Two streamlines illustrate the flow at design point. The geometry of the channel is characterized by a constant channel width between point 3 and 4 with an increasing cross-sectional area up to point 4' and a decreasing cross-sectional area up to point 6. The Mach number distribution within the channel at design point is shown in Fig. 3. Due to the high circumferential component of the velocity at the inlet and the small channel width, the velocity distribution at the exit of the 180"-bend is uniform between hub and shroud. The secondary flow has no great influence on the flow. The differences between the turbulence models are small so Fig. 3 show only the Mach number distribution of the ARS-model. Each turbulence model calculate a large separation at the suction side which increases from hub to shroud and reattaches in the rear half of the vane. The models differ only by the prediction of the separation length with the smallest separation calculated by the ARS-model. This behavior is in contrast to the results of the 2-D diffuser which demonstrated the fact that the k-€-model usually underestimates large separations. With higher mass flow rates (a3= 33O) the differences between the turbulence models increase. Due to the incidence angle the ARS- and k-w-model calculate a separation near the leading edge at pressure side (Fig. 4). The k-Emodel calculates this separation too small with a higher pressure distribution in this region (Fig. 6). In the middle of the suction side the separation behavior of the turbulence models changes. The ARS-model calculates at suction side a smoother pressure distribution which agrees better with the measurements. This results by a smaller separation at mid span compared to the k-Eand k-w-model. In contrast to the ARS-model the k-w-model calculates always larger separations compared t o the k-€-model which agree better with the measurements at pressure but not at suction side. Within the 180"-bend and in the range of the leading edge the Mach number distributions of the turbulence models are very similar. But downstream of the separations the ARS-model calculates relatively high values of the turbulent viscosity within the whole channel. This leads to a very homogeneous Mach number distribution between the rear half of the vanes and within the 9O0bend. The k-E- and k-w-model calculate in wide areas no turbulent viscosity like a laminar flow with more vorticity especially within the 90°-bend (Fig. 5). But the great differences between the Mach number distributions have no significant influence on the static pressure distribution. 5. Conclusions

An explicit algebraic Reynolds stress model has been tested against a low Reynolds number k-e- and k-w-model for a diffuser flow which involves a large separation. The comparisons with experimental data show that the ARS-model

Fields of application

423

does provide significant improvements over the standard tw-equation models especially with regard to prediction of flow separations. For the flow through a return channel which is characterized by strong stream line curvature the comparison of the turbulence models shows minor improvements of the ARSmodel. These improvements mean not only the enlargement of separation at pressure side like the k-w-model but also the reduction of separation at suction side. Furthermore, the ARS-formulation used in the framework of k-E-equations is as robust as k-€-models. Thus, these kind of turbulence models increase the range of applicability of two-equation turbulent closure formulation at a minimal increase in cost and storage. 6. Bibliography

[GER 871

Gersten, K., Hiktl, A,, Pagendarm, H.G., Optimierung von Diffusoren beziiglich der Diffusorstriimung und der Diffusomoiinde. VDI-Verlag Diisseldorf 1987.

[GAT 931

Gatski, T. B., Speziale, C. G., O n expliczt algebraic stress models for complex turbulent flows. Journal of Fluid Mechanics 254 (1993) 59-78.

[LAM 811

Lam, C. K. G., Bremhorst, K. A., Modified Form of the k€-Model for Predicting Wall Turbulence. Journal of Fluids Engineering 103 (1981) 456-460.

[LEN 97a]

Lenke, L. J., Simon. H., Viscous Flow Field Computations for a Transonic Axial-Flow Compressor Blade Using Different Turbulence Models. ASME Paper 97-GT-207 (1997).

[LEN 97b]

Lenke, L. J., Simon. H., A n Improved Algebraic Reynolds Stress Model for Predicting Separated Flows. Beijing, Proceedings of the 7th International Symposium on Computational Fluid Dynamics, 15.-19. September 1997.

[RE1 951

Reichert, A. W., Stromungssimulationen zur optimierten Gestaltung von Turbomaschinenkomponenten. Dissertation, Duisburg, Hansel-Hohenhausen Verlag 1995.

[ROT 841

Rothstein, E., Experimentelle und theoretische Untersuchung der Stromungsvorgange i n Riickjkhrkaniilen von Radialverdichterstufen, insbesondere solcher mit geringen Kanalbreiten. Dissertation, Aachen 1984.

[WIL 911

Wilcox, D.C., Comparison of Two-Equation Turbulence Models for Boundary Layers with Pressure Gradient. AIAA Journal 31 (1993) 1414-1421.

424

Finite volumes for complex applications

i. , ,.~ ! .......... .\: ~

el .

,

'

,

>

\

#

,

I

:,

I .

,

, ,

- ARS-model ....- . k-w-model

~

k-E-model

,

,

.I

i x = 1370 T

2 -1.2

-.4

.4

1.2

YIH

Figure 1: Flow through a 2-D diffuser with 2 0 = 14.25" (a) streamlines with the ARS-model; b)-e) velocity distributions)

Fields of application

425

Figure 2: Streamlines and grid of the return channel (44x 37x 162 points)

:

,

leading edge

Figure 3: Mach number distribution at design point (a3= 26').

426

Finite volumes for complex applications

Figure 4: Velocity and Mach number distributions within the return channel (ARS-model; as = 3 3 O ) .

Fields of application

427

1 leading

Figure 5 : Mach number distributions within the return channel (kt-model; as = 3 3 O ) .

428

Finite volumes for complex applications

-.5

1 ....-a . q...........

I ...........:........... Cr

.........

-1.5

ARS-model

........

----- kr-model ---- k-w-model

-. 0.0

0.5

measurements (Rothstein [ROT 841)

1.O

Figure 6: SuP-facepressure distributions at mid span (a3= 3 3 O ) .

Simulation of unsteady Flow in a Vortex-Shedding Flowmeter

Stephan Perpe'et, Andreas Zachcial and Ernst von Lavante Institute of Turbomachinery, University of Essen, 0-45127 Essen, Germany

Most commercial vortex-shedding flowmeters rely o n known relationship between the vortex-shedding frequency and the mass flow, needing regular and well defined vortex structure as well as shedding mechanism. However, it has been observed that some designs result in rather irregular pressure signature of the vortex system, leading to problems i n the signal processing. I n the present study, the flow about the bluff body i n a vortex-shedding flowmeter was numerically investigated using a solver of the unsteady, compressible Navier-Stokes equations i n two and three dimensions. The computations were compared with experimental results obtained by ultrasonic measurements downstream of the bluff body. Several different body shapes were studied, trying to optimize the resulting pressure signature downstream of the body. Furthermore, first results of the influence of pulsations on the flow will be presented.

ABSTRACT

-

~

Key Words: Vortex-shedding flowmeter, Low Mach number, Pulsation

1. Introduction

Many aerodynamic problems require a volume- or mass-flow data for its quantitative solution. Therefore, a number of methods for flow rate measurement have been developed. In the present work, attention was paid to the so-called vortex-shedding flowmeters. Commercial flowmeters use a large variety of bluff body shapes, often restricted by the attachment of the pressure sensors or, more likely, the patent laws. In principle, the vortex-shedding flowmeters use the separation frequency of vortices behind a bluff body to measure the mean flow velocity of a fluid flow (Figure 1).Downstream of the bluff bo-

430

Finite volumes for complex applications

dy, a von KarmSn vortex street develops; it's width and distance between the vortices depends on this frequency, and therefore on the bluff body's shape. Preferably, the vortex-shedding frequency should depend linearly on the mean flow velocity for a wide Reynolds number range. The dependency of the vortex frequency f , the mean flow velocity u, and the width of the bluff body d is expressed by the dimensionless Strouhal number:

Figure 1. Principle of a vortex-shedding flowmeter Previous simulations of some current bluff body designs lead to fairly irregular pressure signatures, making them unreliable. It was, therefore, decided t o investigate a few alternate bluff body designs. In the presently used ultrasonic method, the bluff body shape is restricted only by the required minimum strength and rigidity to avoid vibrations. The design could be optimized regarding the pressure signature or the pressure drop across the body. The signal processing requires well-defined vortices at only one dominant frequency, without any secondary effects. Therefore, an economic signal processing is only feasible with an optimized shape of the vortex body. 2. Numerical Algorithm

The numerical algorithm employed uses the three-dimensional, time-dependent full Navier-Stokes equations describing the conservation of mass, momentum and energy of the flow. The divergence form in body-fitted, curvilinear coordinates is:

with Q = J - ' ( p p u pv p w e)T the vector of the conserved variables. J is the Jacobian of the coordinates transformation from physical (x, y, z , t ) to computational (<,q,5, T ) space. The program is based on the finite-volume formulation, using a cell-centered organization of the control-volumes. The spatial discretization is carried out with the help of Roe's Flux Difference Scheme,

Fields of application

43 1

a Godunov-type method providing an approximate solution of the Riemann problem on the cell interfaces. Here, the flux is [VAT 871:

+

The index i $ describes the values on the cell interfaces, R and L the values right and left from it. The Roe-averaged matrix A is given by the differentiation of the local-linearized function F ( Q ) . This scheme is formally a central difference type plus a damping term. The method has been proved to be very accurate and effective in the simulation of low Mach number viscous flows [LAV 931. Upwind-biased differences are used for the convective terms, central differences for the viscous fluxes. Starting with a constant initialization of the scalar variables and body-fitted velocity components, the integration in time is carried out by a modified explicit Runge-Kutta time stepping as well as, optionally, an implicit Approximate-Factorization method (AF) or SymmetricGauss-Seidel (SGS) scheme.

2.1. Low Mach Number Modifications

Numerical algorithms for the simulation of compressible flows become inefficient and inaccurate a t very low Mach numbers. The difficulties are due to the formulation of the governing equations in their discretized form. The problems have been addressed by many previous investigators, including Shuen et al. [SHU 931, Pletcher et al. [CHE 931 and Edwards et al. [ROY 981. The main problem is the stiffness of the governing equations a t low Mach numbers. The condition number of the Jacobian matrices (ratio of the maximal to the minimal eigenvalues) increases to infinity as the Mach number approaches zero. The time marching step therefore is restricted because of the large disparity of the eigenvalues of the Jacobian, representing the convective and acoustic signal speeds, respectively. The second problem results from the pressure singularity in the momentum equations a t very low Mach numbers. The ratio of the magnitudes among the pressure term p and the convective terms p~~is inversely proportional to the Mach number squared M2. The large difference in magnitude will yield a large roundoff error. To enable efficient numerical solutions of the equation system a t low Mach number, a pseudo-time term is added to the time-dependent compressible Navier-Stokes equations. The primitive variables are employed as unknowns, rather than the traditional conservative variables. A preconditioning matrix I? is used in order to eliminate the time-step difference between the convective and acoustic characteristic speeds at low Mach number. The preconditioned

432

Finite volumes for complex applications

equations can be discretized in the delta form:

where the Jacobian matrices are defined as,

Here Q is the vector of the conservative variables and Q is the vector of the primitive variables (p, u, v ,w, T)T.A more detailed description of the matrix r can be found in [YAO 971. To circumvent the problem of large difference in magnitude of the convective and pressure terms in the momentum equations, a gradient splitting of the Euler flux into convective terms and a pressure term can be considered: v ( p u 2 p) = v ( p u 2 ) v p . (6)

+

+

Both split gradients are of the same order of magnitude, independent of the Mach number. The Liou7s Advection Upwind Splitting Method is applied to treat the convective and pressure terms separately. The convective terms are upstream-biased using an appropriately defined advection Mach number a t the cell interface, while the pressure term is strictly dealt with by using acoustic waves. One of the advantages of the method is that the upwind effect can be easily reached with few modifications of the programs.

2.2. Verification

The present numerical algorithm was subjected to verification of it's temporal and spatial accuracy and consistancy. The scheme is formally second order accurate in space, since the viscous terms are obtained from second order central differences. The scheme was first verified using the usual grid refinement study for the case of viscous flat plate flow a t a free stream Mach number of M , = 0.5. Defining the global error as the L2 - norm of the deviation of the present solution from the Blasius solution, second order accuracy was verified (von Lavante [LAV 901). Next, the combination of the present solution scheme with the computational grid was investigated for the case of the "new design" bluff body (see below) using three different grids. Table 1 summarizes the results. It should be noted that the finest grid very closely approaches the Strouhal number of the corresponding experiment.

Fields of application

Meshpoints

At [s] rel. CPU-time

f [Hz] Strouhal number

5000 8.8. 1 342 0.267

12300 5.88. 10 339 0.265

41600 2.06. 35 335 0.262

433

exp.

0.263

Table 1. Comparison of different grids

The temporal accuracy was tested by simulating the 2-D flow about a cylinder, with free stream Mach number Mm = 0.1 and a Reynolds number of Re = 200. The resulting Strouhal number of the vortex separation is compared with known experimental and numerical results in Table 2. The agreement is also very good.

I

1

Method

I Strouhal num. /

Table 2. Strouhal number in cylinder wake flow

3. R e s u l t s

In the present research project, several shapes of the bluff body were investigated experimentally and numerically. The experimental results have been given in detail by Hans et. al. [HAN 981. Main emphasis was put on linear dependency of the vortex-shedding frequency as a function of the mean flow velocity. In the present paper, only the three most interesting bluff body shapes, shown in Figure 2, will be discussed. Two of the bodies, a "T"-bar and a rectangle, were fairly simple, the third one was optimized by the present authors for the cleanest signal and was therefore called "new design".

Figure 2. Design of the bluff body shapes The horizontal part of the T-bar was facing upstream, which was in contrast t o other applications of this shape [MIA 931. The Strouhal number (Figure 3) was constant for the whole velocity range corresponding to Reynolds numbers

434

Finite volumes for complex applications

from 10,000 t o 300,000. Experimental and numerical results showed a very good agreement. The standard deviation was within 2%.

0.30

:

1- New

Design

-

,

0.00 0

,

, ,

numerical

:, - experimental , , ,

,

,

,

, ,

,

,

, , ,

,

,

,

,

,

,

, ,

,

,

,

,

,

50000 100000 150000 200000 250000 300000

Re

Figure 3. Strouhal number versus Reynolds number Although the two-dimensional simulations predicted vortex-shedding frequencies that agreed very well with the experimental data, it was decided to investigate the flow about the promissing "new design" using three dimensional approach. No symmetry was assumed, allowing the specification of asymmetric inflow velocity profiles. The dominant vortex structure makes this case interesting for large eddy simulation (LES) turbulence treatment. Therefore, no turbulence model1 was used, although the inflow would correspond to a fully developed turbulent pipe flow. The computational grid consisted of 22 blocks. Two grid resolutions were used: coarse with 300,000 cells and fine with 1.07. l o 6 cells. The coarse grid was implemented in order t o study the process of the main vortex generation. No attempt was made in this case to properly resolve the boundary layer a t the pipe wall. There were only a few grid points in this boundary layer. The second, finer grid consisted of 1.07. lo6 points, making the full simulation of all the viscous layers possible. The minimum grid spacing a t the wall was less than yf = 1.0, and of the same order of magnitude as the Kolmogorov length. There were a t least 20-40 grid points in each of the shear layers. The grid was exponentially stretched toward regions with smaller gradients of the flow variables. The resulting flow displayed highly three-dimensional nature, with main vortices curved toward the walls, and with a complex system of secondary flow features. After separating from the bluff body, the primary vortices, seen in Figure 4, were convected downstream. They dissipated toward the pipe walls; their central part became straight, assuming almost two-dimensional form. In order to visualize some of the lower order features, such as secondary vortex structure, the enstrophy, defined as w = (V x U ) 2was computed and displayed in contour plots. The enstrophg contours on a cylindrical surface close t o the pipe wall are shown in Figure 5. Here, the so called "horse shoe vortex" is clearly visible in front of the bluff

Fields of application

435

body. Behind it, the highly vortical, unsteady flow associated with the primary vortices can be seen. Several secondary and tertiary vortices, some aligned with direction normal to the main vortices, could be observed, indicating that at least some of the turbulent structure was simulated.

Figure 4. Density contours over the pipe cross-section

Figure 5. Enstrophy contours at the wall PIP0

dimless. time

Figure 6. Pressure evolution for pulsation frequency of lOOHz

436

Finite volumes for complex applications

Finally, t h e effects of pulsating inflow on the function of the vortex-shedding flowmeter are currently investigated. As can be seen in Figure 6, pulsations in the flow with frequencies close t o the natural vortex-shedding frequency had a n pronounced influence on the operation of the meter. Between two pulsation peaks the former shedding frequency recovers.

4. Conclusions

A verified and validated Navier-Stokes solver for two-dimensional and threedimensional simulations of compressible, viscous and unsteady flow over a wide range of Mach numbers is presented. The solutions achieved by this program show good agreement with experimental data. The effects of disturbed inflow conditions are currently investigated. 5. Acknowledgement This project is supported by the Deutsche Forschungsgemeinschaft (DFG).

6. References [ROY 981 [HAN 981

[MIA 931

[CHE 931 [SHU 931 [VAT 871

[LAV 901 [LAV 931

[YAO 971

J. R. EDWARDS A N D C. J . ROY,AIAA Journal, Vol. 36, No. 2, pp. 185-192, 1998. E. v., PERPEET,S.: VortexHANS,V., POPPEN,G . , LAVANTE, shedding flowmeters and ultrasound detection: signal processing and bluff body geometry. Flow Measurement and Instrumentation 9 (1998), 79-82. MIAU,J . J . , YANG,C.C., CHOU,J.H. A N D LEE, K.R., A T-shaped vortex shedder for a vortex flowmeter, Flow Meas. and Instr., Vol. 4, NO. 4, 259 - 267, 1993. A N D K.-H. CHEN,AIAA-93-3368-CP. R. H. PLETCHER J.-S. SHUEN, K.-H. CHEN,A N D Y. CHOI,J. of Comp. Phy. 106, pp. 306-318, 1993. VATSA,V.N.,THOMAS,J.L.,WEDAN,B.W., Navier-Stokes computations of prolate spheroids at angle of attack, AIAA Paper, (87-2627), 1987. VON LAVANTE, E., The Accuracy of Upwind Schemes Applied to the Navier-Stokes Equations, AIAA Journal, Vol. 28, No. 7, 1990. VON LAVANTE, E., YAO, J . , 1993. Simulation of flow in exhaust manifold of an reciprocating engine. AIAA 24th Fluid Dynamics Conference. Proceedings of 7th International J . YAO A N D E. VON LAVANTE, Symposium on CFD, pp. 689-694, Sept. 15-19, 1997, Beijing, P R China.

A Finite Volume Scheme for the Two-Scale Mat hematical Modelling of Tic Ignition Process

A . Aouji LIMHP- C N R S Avenue JB Clement 93430 Villetaneuse

V. Rosenband Faculty of Aerospace Engineering Technion Haifa-32000

W e consider the problem of the ignition of a cylindrical sample of infinite length composed of T i particles coated with a thin T i C layer and placed into a furnace at T. W e describe an implicit finite-volume scheme, for the two spatial scales modelling, on a moving mesh at the particle level, applied to the numerical study of the ignition process.

ABSTRACT

Iiey Words: Reaction-Diffusion Equation, Adaptive Finite- Volume, Stefan Problem, Moving-Mesh, Ignition Process, Solid-Solid Combustion.

1. Introduction

In this paper we present a two-spatial scales mathematical modelling for -+ TiC ignition process [AOU 98, LAP 96, SEP 951. At the particle level, we have a Stefan problem which is discretized by a finite-volume scheme on a moving mesh. The paper will be organized as follows, in section 2 we describe the mathematical modelling, in section 3 we present the numerical scheme we have applied t o the treatment of this two-scale , and describe the finite volume scheme. Section 4 presents some numerical simulations.

Ti + C

438

Finite volumes for complex applications

2. Mathematical Modelling

The reader is referred to [AOU 981 for more explanations about the physical analysis of the process.

Titanium Particle

Figure 1: Scheme of the reaction modellang

2.1 Mass Balance We consider the mass-diffusion balance expressing Fick's law for a spherical particule of Titanium of radius Rp. We denote by R = R2 ( t ) u R 1( t ) = [0, R ( t ) ] U [R ( t ), Rp] = [0, R,,] .The mass-diffusion coefficients follow an Arrhenius-type law.

2.2.1 Dissolution Domain , Zone II

V Z E [O1Rc], V r E [0, R ( t ) ]= Rz ( t )

2.2.2 Carbidization Domain , Zone I

V z E [O, Rc] , V r E [R( t ), RP]= Rl ( t )

Fields of application

dC' r2 dr

at

439

d ~ ' dr

r2 D' (T (z. t ) ) -(r,r, t))

c1(2, R ( t ),t ) = C; ( T )

(5)

2.2 Stefan Problem

We express the condition at the interface which indicates that due to concentration gradient in both domains I and 11, the interface moves with the velocity dR(t) d, .

-

(T ( >1

ac'

) 7( 2 ,R (t) , 1) + - D"

(T ( 2 , t ) )

ac" Y&-

(T'

= (C; ( T )- C:) .TdR (t)

R (4 t) 1

(7)

2.3 Energy Balance

We evaluate the overall energy balance at the sample macroscopic level, for which we take into account the heat released at the mesoscopic scale - i.e. particle scale- by the exothermic mass diffusion processes described below.

2.3.1 Carbidization Domain , Zone I We denote by the rate of heat released during carbon formation - by diffusion process in zone I-, defined by

2.3.2 Dissolution Domain , Zone II the rate of heat released during carbon dissolution We denote by by diffusion process in zone 11-, defined by

:[lR"'

dQ" (t) = 4aNTi dt

A H ~ C " ( r , r, t ) r2 dr

I

440

Finite volumes for complex applications

2.3.3 Global Heat Balance

ac

We consider the cylinder domain = [0,Rc] and write for V z E [0,Rc] The heat balance takes into account the heat released locally at the particle level for the two diffusion processes by:

aT P - C P( z ,~t ) = div(X grad T ( z , t ) )

+

dt

dt

2.3.4 Boundary Conditions Symmetry Condition at the center of the cylinder

Heat supplied by the furnace maintained a t temperature T ,

3. Numerical Scheme

We integrate from t , to t,+l, the three different balances, assuming that a non uniform temporal mesh is given. We use a variable stepsize backward difference formula of order 1, which is well suited for the numerical treatment of stiff equations. We focuss on the numerical methodology used to discretise the mass balances, written in conservation form, on a variable domain. In order to track efficiently the variation of the computational domains Ql ( t ) and Q2 ( t ), we have considered the integration of the mass balance on a cell ICi ( t ) = [ri ( t ), ri+l ( t ) ] which evolves with time and denotes its boundary 8Ii.i ( t ) = {ri ( t ), ri+l ( t ) ). We adopt a cell-centered formulation. 3.1 Integration on a moving mesh

dC

- ( z ,r , t )

dCl =

L"" J,.,,,

div ( D (T ( z ,t ) ) grad C ( 2 , r, t ) ) dQ (13)

The transport theorem of Reynolds [N'KO 941 affirms that: d

C ( r ,r , t ) r2dr

ac

- ( z ,r , t )

r2dr+

J

aK.(t)

C ( z ,r , t ) r 2dr -dt dt

Fields of application

441

+c( 2 , ri+1 ( t ), t ) ri+l (t)' Vi+l ( t ) dt - C ( z , ~i( t ), t ) ~i (t)' vi ( t ) dt where vi+l ( t ) = and vi ( t ) = are the velocities of the extremities of the interval1 on which the mass balance is integrated. We outline the procedure, and omit the details of the algebraic computations. We integrate the ( z , r, t ) from its definition previous relation from t , t o t,+l and substitute into the mass balance. We apply Green's Formula for the discretization of the divergence of the mass fluxes. We apply the first order accurate backward Euler scheme for the discretization of the right hand side integrals. We denote A t , = - t,. and my = mes (I{:) = ~ ' " + l ( ' " ) r2dr = ri+l (t,) 3 - ri (tn13],the r,(tn) volume of the cell and end up with,

&

We point out that the following term, which takes into account the transport of the concentration field, due to the variation of volume of the integration cell appears :

(

(

-

1

i

(

n

1

) 1

I

)

)

ri+i ( t , + ~ Vi+l ) ~ (tn+1) A t ,

ri (tn+ll2~i (tn+i) Atn

(16)

Since C is constant in cell Ii'; (t,) , we may use "local information" i.e. related to this cell, and therefore write the following continuity relation ( j

i

t n +t n + )

C

(

We now have the following relationship

i

(

+

)I

)

=c

1

(17)

442

Finite volumes for complex applications

After some algebra and leaving the convected mesh terms in the right hand side, we are led to a three diagonally dominant matrix system which can be efficiently inverted by the classical -double sweep- Thomas Algorithm. At this stage we have made no assumption on the nodes velocities. 3.2 Nodes Velocity

We have to prescribe the speed of the mesh; this is realized through a linear speed formula, according to [MUR 591, taking into account the fact that the velocity at the interface between domain I and I1 is known from the stefan problem equation. 3.3 Boundary Conditions

An important issue is the discretization of the boundary conditions on a moving mesh, because they will lead to a different set of equations that will influence significantly the behaviour of the numerical solution, if a wrong formula is supplied, for example consider:

C ( 2 , R ( t ) , t ) = C i =Constant We integrate the equation on cell [rI (t,) , r ~ + (t,)] l assuming that CIKn= C i , therefore (z, r , t )(

K?

( t n )= = 0 and

the equation is,

Using the previous arguments on the continuity of the concentration field we get

Fields of application

443

In fact the variation of the computational volume can be understood as a source term.

3.4 Time Step Adaptation

The stiffness of the problem means that once the heat released by the chemical kinetics is enough, a sharp rise of the temperature occurs inside the sample from the exterior surface to the interior of the cylinder. A time-stepping adaption procedure t o dynamically reduce the time-step with respect t o the temperature field behaviour has been considered.

5. Numerical simulations Ignition temperature is reached, when a sharp increase in temperature field occurs. This ignition process depends on particle size, initial thickness of T i c interfacial layer, heating rate, and value of T,. We present below a sample of such temperature profile.

Temperature evolution of different positions in the cylinder Tfurnace = 1200 K 2000.0

Figure 2: Temperature profile T ( R c ,t)

444

Finite volumes for complex applications

V . , Mathematical modelling and numer[AOU 981 . AOUFIA . , ROSENBAND ical simulation of Titanium/Carbon Ignition , p d International High-Energy Materials Conference and Exhibit, Dec 8-10 (1998). [LAP 961

. LAPSHINO.V. A N D OVCHARENKO A.E. , A mathematicalmodel of high-temperature synthesis of the intermetallic compound Ni3Al during ignition , Combustion, Explosion and Shick Waves , Vol. 32, No 2,(1996),158-164.

[MUR 591 MURRAYW.D. A N D LANDISF . , Numerical and Machine Solutions of Transient Heat-Conduction Problems Involving Melting or Freezing , Transactions of the ASME, (May 1959), 106-112.

[N'KO 941 N'KONGAB . AND GUILLARD H . , Godounov type methods on nonstructured meshes for three-dimensional moving boundary problems , Comput. Meths. Appl. Mech. Engrg. 113 (1994). [SEP 951

SEPLYARSKII B.S. AND GORDOPOLOVA I.S., Ignition of condensed systems interacting through a layer of high-melting products , Combustion, Explosion and Shock- Waves, Vol. 31, N04, (1995),405-410.

Two Perturbation Methods to Upwind the Jacobian Matrix of Two-Fluid Flow Models.

K u m b a r o A., T o u m i I. CEA Saclay, DRN/DMT/SYSCO F-91191 Gif-sur-Yvette Cedex, France [email protected] C o r t e s J. CEA Cadarache, DRN/DTP/SMET F-13108 Saint-Paul lez Durance, France

W e examine the eigenstructure of a two-fluid model. Because of complex interphase interactions, perturbation methods are used to get approximations of the eigenelements. W e compare two significant perturbation methods. Mathematical and numerical results are provided.

ABSTRACT

Key Words: two-phase flows, eigenstructure, upwinding, perturbation methods.

1. I n t r o d u c t i o n

In this paper, we examine the eigenstructure of a two-fluid model, derived from [TOU 961, when regularising terms are added t o it. We think that such a study is of relevance to construct upwind schemes. Unfortunatly, it turns out t o be a difficult task in two-fluid systems. Hence, several approaches are proposed t o get practical approximations of the eigenelements : A numerical algorithm. However, it may be expensive in CPU time for 3D finite volume calculations, although improvments are in progress ([ALO 981). Also intensive use of symbolic calculation packages may help obtain closed form related expressions.

446

Finite volumes for complex applications

A perturbation method using a small parameter. We distinguish -

-

:

the density perturbation method to analyse the eigenstructure when the density ratio is small ( E = << I), see [COR& 98],[COR 981. Taylor expansion of the eigenvalues when the relative velocity is small compared to the speed of sound ( ( = $ << I), see [KUM 961, [TOU 961, [TIS 971.

Here, our aim is to study the numerical relevance of the two perturbation methods, as applied t o the Roe scheme [ROE 811, [TOU 961. The results obtained using these different perturbation methods will be examined. Moreover the numerical influence of the mathematical regularisation on such non-conservative models will be underlined. 2. Regularisation of the two-fluid model

Two-fluid models are regularised by the addition of interface pressure correction terms [POK 971. Indeed, in the basic model, surface tension effects are neglected, and it is assumed that pressure equilibrium exists between the two phases : p = pl = p, # pi, with pi the interface pressure. Here, we give the standard homogeneous form of the physical set of conservation equations [TOU 961 :

where the subscript k refer to the gas phase ( k = g ) or to the liquid phase ( k = 1 ). ak is the volume fraction ( a, +al = 1 ), pk the density, v k the velocity, ek the energy and hk the enthalpy ( hk = ek + ). Moreover, we note by Hk and Ek respectively the phasic total enthalpy and energy. For simplicity, we will consider a perfect gas. Several interface pressure correction models exist in the litterature. They must vanish when the velocities of the two phases become identical (v, = 0 ). We will apply the two perturbation methods to : the pressure correction proposed in [LAH 921

Fields of application a

447

the pressure correction proposed in [TOU 961

Actually, setting 2 p - p; = agplbv,2 and p - pi = CpplvT

we see that Cp = a g b leads t o correction (3) whereas 6 = 0 and C p = C p ( a S ) leads to correction (2). We can compute the Jacobian matrix A of the two-fluid system :

As mentioned in the introduction, the interphase interactions yield a complex eigenstructure. Hence, perturbation methods may be a convenient way to get practical approximations of the eigenvalues and eigenvectors. 3. The two perturbation methods

3.1. The Taylor expansion of the eigenvalues

In order t o determine the eigenvalues of the system, we must find the roots of a polynomial of degree six P(X) given by : P(X) = d e t ( A - X I d )

(4)

A straightforward computation of this determinant leads to the following polynomial :

To get approximations of the roots of PA, one can use a perturbation method by introducing the ratio [ = -V T am

448

Finite volumes for complex applications

&(T

and the new variable z = - A), with the 'characteristic' speed of sound in the two-phase mixture given by a i = yP(-

1

Pg

+ -).

ffl

Plffu

We look for a first order approximation of the eigenvalues and the eigenvectors (see [KUM 961). Let the polynomial PA(X)be writen as

with l q ( ~ , J )5 l (1+ I z I ) ~ ~ (;J ) E N and 4(E) + 0 as

J + 0.

We look for the roots of P ( z ; J) in a neighbourhood of a root zo of the polynomial po(z). We will carefully distinguish the single roots and the double we can find first roots of po(z). Hence, introducing the notation 60 = y*, related approximations of the two-fluid eigenvalues :

where lc = 1 corresponds t o the pressure correction (2) and lc = ag correspond t o the correction (3). Following the same method, the first order approximation of the eigenvectors can be derived (see [TOU 961). 3.2. T h e d e n s i t y perturbation m e t h o d

In [COR& 981, it is shown that a scaling of the densities in the two-fluid system yields a splitting of the Jacobian matrix :

Fields of application c is the average density ratio

E

449

= pi/pP and

+

+A-1 A. would represent the Jacobian matrix of a two-fluid system with no pressure gradient in the liquid phase. Hence, it is expected that its eigenelements are easy to compute. For instance, we get 1 - A - ~ + A~ = P D P - I

(12)

€

(k e) 0

with D =

.

0

The computations yield :

A6

and p =

/I ! I

I

I

I

I

A) I

(13)

450

Finite volumes for complex applications

+yp(vg- X 2 , 5 ) -

-

$ 6 ~ ~ v ? ( y v g X Z , ~ )

+9 ~ + ;

$6agv;) X 2 , 5 ( 3 - ?)vg In particular, when A-1 = 0 , the perturbation theory for linear operators and

02,s

=

&(~2,5(p

-

provides a convenient way to obtain higher approximations of the eigenelements of the original matrix ([HIN 911). For instance the correction (2) yields A-1 = 0 and in these circumstances, we can expand the eigenelements into a first order approximation in 6 ( [COR& 981).

4. Numerical results The presence of non-conservative products yield significant difficulties. Here, we mainly follow [KUM 961. Hence we construct the Roe type numerical scheme as follows :

with the numerical flux given by

Fields of application

45 1

The positive and negative part of the Roe-averaged matrix are obtained thanks to first order approximations of the eigenstructure above. Some two-phase shock tube tests are computed using the two perturbation methods described in 3 and the two pressure corrections mentioned in 2. Moreover, we take y = 1.093 and pl = 720. The initial conditions for these shock tube problems are :

TEST 1 Left state Right state

ag 0.25 0.10

TEST 2

a,

Left state Right state

0.8 0.8

vg (m/s) 0 0 vg (m/s) 150 100

vl (m/s) 1 1 vl (m/s) 5 1

P (MPa) 20 10 P (MPa) 0.15 0.1

h, (kJ/kg) 3092.7 3099.9 h, (kJ/kg) 12 12

hl (kJ/kg) 1338.2 1343.4 hl (kJ/kg) 3 3

The CFL number is 0.9 and the computations have been done with 400 cells. Results are provided after 200 iterations. The theorical results obtained in the previous section and the numerical results obtained in this one will be fully discussed in the presentation.

Une me'thode ite'rative pour de'centrer sans diagonaliser, pri[ALO 981 F. ALOUGES, vate communication

[COR& 981 J . CORTES,A . DEBUSSCHE A N D I. TOUMI, A Density Perturbation Method to study the Eigenstructure of Two-Phase Flow Equation Systems, Journal of Comp. Physics, Vo1.147, No.2, 1998, pp. 463-484

[COR 981 J . CORTES,A n Asymptotic Two-Fluid Model for Roe-Scheme Computation, Computational Fluid Dynamics ( ECCOMAS'98 proceedings ), John Wiley & Sons, Ltd., Vol.11, 1998, pp. 416-422 [EYM 991 R .EYMARD, T . GALLOUET, R. HERBIN, Finite Volumes Methods, to appear in the Handbook for Num. Anal., eds C. Lions, North Holland

[GOD 911 E. GODLEWSKI, P . A. RAVIART, Hyperbolic systems of conservation laws, Mathbmatiques & Applications, Ellipses, 1991

[HIN 911 E. J . HINCH,Perturbation Methods, Cambridge Texts in App. Math., 1991 [KUM 961 I. TOUMI,A. KUMBARO, A n approximate Linearized Raemann Solver for a Two-Fluid Model, Journal of Comp. Physics, Vo1.124, 1996, pp. 286-300 JR., The prediction of phase distribution and separation [LAH 921 R. T. LAHEY, phenomena using two-fluid models, Boiling Heat Transfert, Elsevier Science Publishers, 1992, pp. 85-121

452

Finite volumes for complex applications

M. MORIA N D V. H. RANSOM, Regularization of Two[POK 971 H. POKHARNA, Phase Flow Models : A comparison of Numerical and Differential Approaches, Journal of Comp. Physics, Vol. 134, 1997, pp.282-295 [ROE 811 P. L. ROE,Approximate Riemann solvers, parameter vectors and difference schemes, Journal of Comp. Physics, Vo1.43, 1981, pp. 357-372

A n Upwind Numerical Method for TWO-FluidTwo-Phase Flow [TOU 961 I. TOUMI, Models, Nuclear Science and Engineering, Vo1.123, 1996, pp. 147-168 [TIS 971

I. TISELJ, S. PETELIN, Modelling of Two-Phase Flow with Second-Order Accurate Scheme, Journal of Comp. Physics, Vo1.136, 1997, pp. 503-521

Figure 1: Test 1. Eigenstructure obtained by an exact calculation and the two perturbation methods

.

w

.

.

.

.

.

T.""-a.T.WlSnh

.\.

.

.-

',., 0 7 s

-

:.. . ,, . / .

ii :\

07,s.

. :.

.

i, !

g .,,,,

..

,

Olr.

0 7B75

-

0-7

-

. ,.

,, "

D M I

Y

0 )

" 02

"

" LU

'L; ' 0.4

0.5

Xlml

Be

0,

08

0.

(

o

0,

01

02

a4

(15 X IM

08

a,

oa

oe

I

Figure 2: Test 2. Eigenstructure obtained by the two perturbation methods

Finite volumes simulations in magnetohydrodynamics

M. Hughes, L. Leboucher, V. Bojareuics, I<. Pericleous, M. Cross Centre for Numerical Modelling and Process Analysis University of Greenwich London SE18 6PF - UIir

Magnetohydrodynamics involves the interaction of magnetic fields with fluid flows, so that simulations in this area of physics and its applications couple the resolution of Navier-Stokes equations and Maxwell equations. While most simulations in fluid dynamics are performed with finite volumes, most of the electromagnetic computations are traditionally performed with finite elements. The straight forward temptation is to solve each problem separately using one of the available finite volumes and finite element software tools, and feed back the results of one code into the other. This method can work reasonably well when the flow does not influence the magnetic field, that is to say when the flow only is influenced by the magnetic field. Then the amount of data exchange between the two codes is affordable. However, if both fluid flow and magnetic field have an influence on one another, then the data exchange and iterative computation between the two systems becomes too large. In such a case it is preferable to solve both problem within the same code and, for simplicity, with the same method, i.e. finite volumes or finite elements. While electrical engineering finite element software often gives approximate conservation of electrical quantities without serious inconvenience, fluid dynamics codes are very sensitive to the conservation of different physical quantities, especially the conservation of mass. Therefore, a good option is to solve both fluid dynamics and electromagnetic equations with the same conservative method, i.e. the finite volume method. Examples of conservative discretisations and simulations of electromagnetic systems will be given at the conference. ABSTRACT

Key Words: MHD, magnetohydrodynamics, finite-volumes.

454

Finite volumes for complex applications

1. I n t r o d u c t i o n

An electrically conducting fluid of conductivity o and velocity u interacts with a magnetic field B when induced or imposed electric currents of density j create a Lorentz force j x B within the fluid. This force may either be used to improve some casting processes or have unwanted stabilizing or destabilizing effects on liquid metal flows. The Ohm law in a moving fluid submitted to an electric potential @ and a time-dependent vector potential A defines the current

The electric potential may be solved from the divergence of this law, assuming Coulomb's gauge V . A = 0 and the conservation of electric charges V . j = 0

The magnetic field can be solved from the curl of Ohm's law ( I ) ,from Maxwell's equation V x B = poj and from the identity V x V x B = -V2B which holds for any conservative field - V . B = 0.

This is the induction equation. Finally, the Navier-Stokes equation with the Lorentz force is

where p and v are the density and the kinematic viscosity of the fluid and where the pressure p is obtained from the divergence of this equation, taking into account the conservation of mass V . u = 0.

+

V Z p= V . ( p v - 1 ~ 2 uj x B - pu . V U )

(5)

2. I m p o s e d m a g n e t i c fields

Rewriting the induction equation in dimensionless form gives exactly the same equation with the exception that the magnetic diffusivity (pan)-' is replaced with the inverse of the magnetic Reynolds number Rm = pouuL. In case where this number is small, corresponding either to a low electrical conductivity ( p o o ) - l ,to a small velocity or to a system of small length scale L, the magnetic field is unaffected by the motion of the fluid. Then the induction B may be computed by any mean, e.g with an electrical engineering simulation software. The same software may be used to compute the electric potential

Fields of application

455

Figure 1: magnetohydrodynamic velocity field of the moten aluminium in an electrolysis cell. Stirring of the liquid metal by the electromagnetic forces. and the resulting currents and Lorentz force. This force may then be added as a source term in any other computational fluid dynamics code. This is done easily with the finite-volume simulation software PHYSICA [PHY 991 where the electric currents and magnetic fields were taken from the finite-element code CADEMA [CHI 981 in order t o solve the magnetohydrodynamic flow of a molten aluminium pad lying on an electrolysis cell of an aluminium production plant. Alternatively, once the magnetic and velocity fields are known, the electric potential may be solved from eq. (2) within the fluid dynamics code using the same finite volume method as for the pressure equation (5). This method is used to model another phenomenon of the same aluminium production process: the instability of the position of the interface between the aluminium pad and the electrolyte layer lying upon it. This instability is very similar t o the waves a t the surface of oceans and rivers apart from the fact that it is submitted to an electromagnetic perturbation coupled to the variations of the interface position. Here only the perturbation to the electric potential is computed within the fluid dynamics code [CHI 981. The finite volume mesh is shown on figure 2. Note that the electric potential @ is shifted from the other scalar variable 7 denoting the height of the interface, in order to obtain the value of its derivatives d@/dx and d@/dy at the positions where the electric currents and the resulting Lorentz force need to be known, i.e. at the positions of the momentum variables U and V. An example where all electromagnetic quantities are part of the fluid dynamics finite-volume code is the flow of a liquid metal in a rectangular

456

Finite volumes for complex applications

Figure 2: finite volume mesh for magnetohydrodynamic system formulated in terms of the fluid variables U , V, q and the electric potential a. Note the position of @ at the corners of the cell and its control volume shown as the light dotted rectangle. pipe [HUG 941 [LEB 991 submitted t o a constant magnetic field. Figure 3 shows the electric currents induced by the magnetic field in the fluid moving perpendicularly t o the plane of the figure.

3. Induced magnetic fields

When the magnetic Reynolds number is not small compared t o 1, then the magnetic field is affected by the velocity field according t o eq. (3). This

Figure 3: electric currents in the cross section of an electrically insulating duct with increasing magnetic field strength from left t o right.

Fields of application

457

Figure 4: iterative solution of Navier-Stokes and Maxwell's equations.

equation may be rewritten as a convection-diffusion equation

It appears then that the induction equation may be discretised and solved with the same finite difference scheme as the Navier-Stokes equation (4). The induction equation could however be solved by any electrical engineering finite element code. But exchanging the velocity and magnetic fields between the fluid dynamics and the electrical codes until both fields have converged may require too much time, especially if the data need to be read and written on the disk between each call to one of the two codes. Another reason for solving the induction equation with finite volumes is the conservative properties of this method. Some finite elements codes don't satisfy V . B = 0 . This may not be a problem for steady electrical engineering simulations, but it becomes a serious drawback when time dependent simulations or iterative computation of fluid velocity and magnetic fields are performed. Indeed the error in V . B can amplify with time or with the iteration procedure, leading to completely wrong results or even to numerical instabilities. An example of magnetohydrodynamic channel flow where the magnetic field is convected by a large value of the fluid velocity is shown on figure 4. The Navier-Stokes and induction equations are solved iteratively with the same finite volume method.

4. Conclusion

The nature of magnetohydrodynamics where all vector fields u, B and j are divergence free together with the dynamical properties of fluid flows makes the finite volume method suit particularly well to coupled electromagnetic and fluid mechanical problems. The same finite difference scheme can be applied to

458

Finite volumes for complex applications

both of them, making magnetohydrodynamic codes shorter, simpler and more accurate than classical finite elements.

5. Bibliography

[PHY 991

http://physica.gre.ac.uk

[CHI 981

CHIAMPI,M., REPETTO, M., CHECHURIN, V., KALIMOV, A., LEBOUCHER, L., 8th International I G T E Symposium on Numerical Field Calculation in Electrical Engineering, Institut fiir Grundlagen und Theorie der Elektrotechnik, Graz, Austria, September 1998.

[HUG 941

HUGHES,M ., Computational Magnetohydrodynamics, thesis, University of Greenwich, 1994.

[LEB 991

LEBOUCHER,L., Monotone Scheme and Boundary Conditions for Finite Volume Simulation of Magnetohydrodynamic Internal Flows at High Hartmann Number, J. Comput. Phys., Vol. 150, pp.181-198, 1999.

Finite Volume Method for Large Deformation with Linear Hypoelastic Materials

K. Maneeratana and A. Ivankovic Department of Mechanical Engineering Imperial College of Science, Technology and Medicine, London,

UK

ABSTRACT This paper describes a development of the finite volume method for modelling of structural problems involving geometrical non-linearities. The rate of Green strain tensor and second Piola-Kirchhoff stress rensor are used as the work conjugate pair in total and updated Lagrangian descriptions whilst the deformation rate tensor and Cauchy stress tensor with Jaumann objective stress rate are employed in the deformed Lagrangian (Eulerian) description. The law of conservation of linear momentum i s discretised and resulting systems of algebraic equations are solved, using an iterative segregated procedure. The method is applied to a number of test cases and the results are shown to be in good agreement with analytical results. The accuracy, simplicity and adaptability of the method for non-linear structural problems are clearly demonstrated. Key Words: Finite Volume method, large deformation, hypoelastic materials

1. Introduction This work concentrates on the development of the cell-centred FV method to geometrically non-linear problems involving linear hypoelastic materials. Since the method is inherently iterative, the computational effort for solving non-linearity is not expected to be vastly difficult. The law of conservation of linear momentum in the integral form is employed as the governing equation with linear hypoelastic materials as the constitutive relationships or equations of state. In the total strain or deformation theory (DT) scheme, the Green strain and second Piola-Kirchhoff stress are used. As hypoelastic constitutive equations have to express relationships between stress and strain rates,

460

Finite volumes for complex applications

solids described by total strain formulation are, therefore, not hypoelastic materials. However, the St. Venant-Kirchhoff hyperelastic model, which has similar constitutive relationships to hypoelastic materials, are employed in this study [BON 971. In the incremental approach, however, there are several possible schemes. The total Lagrangian (TL) and updated Lagrangian (UL) schemes are based on the initial and updated initial configurations, respectively. In both schemes, the rates of Green strain and second Piola-Kirchhoff stress are used as the work conjugate pairs. In the deformed Lagrangian (DL) scheme, the rate of deformation and Cauchy stress with Jaumann objective stress rate are employed. This scheme is also commonly called Eulerian [MCM 751 even though the formulations are not truly Eulerian as conventionally defined. In the incremental schemes, two methods of obtaining incremental governing and constitutive equations are considered. The derivation approach, denoted by 'deri' in the rest of the paper, simply differentiates the relationships with respect to time, while the total difference approach, denoted 'diff', considers the difference of total forces and deformations in the control volume. The governing equations are discretised by a cell-centred finite volume technique in line with earlier studies [DEM 941, [WEL 981. The presented method is, then, employed to solve simple problems and numerical results are compared with analytical solutions.

2. Governing and Constitutive Equations In this section, the governing and constitutive equations are briefly outlined. Details can be found in most continuum mechanics textbooks, such as [CRI 911 and [BAT 961. Position of a particle within a body at a reference time to is defined by position vector x,. A motion, caused by applied forces, brings the particle to a new location x at a subsequent time t . In a TL formulation, all static and kinematic variables are referred to the initial configuration at time to. Even though the formulation for DT scheme can be categorised in this approach, the abbreviation TL will be specifically referred to the incremental formulation hereafter for clarity. The UL formulation is based on the same procedure as TL scheme but variables are referred to the most recently calculated (updated) configuration. For the pure UL scheme, employed in this study, a new reference configuration x, is calculated for every increment of time or load. This updating is, in effect, resetting the displacement vectors to zero at the beginning of every increment and the TL formulations can be, therefore, simplified as will be shown in the next section. On the other hand, the deformed Lagrangian (DL) equations are based on the current deformed configurations x .

2.1. Governing Equations The basic law of conservation of momentum for the deformed body of volume v, bounded by deformed surface with normal vector a pointing outwards, is:

Fields of application

461

where u is the displacement vector, 0 is the Cauchy or true stress tensor, b is the body force, p is the current density and a dot above a quantity represents the time derivative of that quantity. As the Cauchy stress is based on the actual force, acting on deformed volumes, the second Piola-Kirchhoff stress tensor, S , is introduced as the mathematical (F-I)' mapping of the Cauchy stress to the undeformed volume as S = JF-' .o. where F = I + G is the deformation gradient tensor, G = &/ax , I is the identity tensor and the Jacobian J = det(F). Therefore, equation [l] can be rearranged for DT approach for an undeformed body with initial surface vector a,, volume v, and density p , as:

where deformed surface vector a and volume v relate to a, and v, by da = J(F-I)' .duo and dv = Jdv,. The equation [3] can be expressed in TL scheme using 'derivation' and 'difference' schemes as:

Equations [3] and [4] can be simplified for UL formulations to:

d dli -Ip,,dt dt "

dv,,=~(~,,+~,,~~)da~+~~,,bdv,,,

where the second Piola-Kirchhoff stress S,, rate of deformation gradient F,, density p,, surface vector a, and volume v, are referred to the last updated configuration. The reference position vector xu is updated every increment by setting it equal to x of the previous increment, giving F , = I since the tensor F , is now based on the updated configuration. The rate form of equation [I] can be written for the DL formulation, using simple time 'derivation' and 'differencing' schemes as:

462

Finite volumes for complex applications

~ ah and velocity gradient L = &I&. where chi = ( t r ( ~-) (L)T). If the force difference approach is used, the rate form of DL formation can be expressed by:

2.2. Constitutive

Equations

In order to obtain a well-posed system, the relationship between stress and strain measures for a given material is required. For the DT scheme, the Green strain E can be expressed using the deformation gradient F as:

The linear relationship between second Piola-Kirchhoff stress S and Green strain E can be expressed by:

where p and A are Lam6 constants. For the TL scheme, the rate form of equation [lo] can be expressed as:

where the rate of Green strain obtained by simple 'differentiation' and 'different' schemes are:

For the UL scheme, the formulations of strain rate in equations 5121 and [13] can be simplified to the Green strain rate in the updated configuration, E,, as:

As UL and TL schemes are based on different configurations, constitutive relationship [ I 11 must be corrected in order to obtain the equivalence between the TL and UL results [BAT 961. The corrections is based on relationships between the TL and UL stress and strain rates as:

Fields of application 463 This correction converts UL stress and strain tensors to the TL formulations but it also heavily complicates the expression. As such, there is no clear advantage of the UL formulations over the TL ones. However, if the strain is small, the equation [ l l ] can be employed as the relationship between the rate of stress and strain measures for large deformation modelling. Therefore, this study employs this formulations for UL scheme with small strain restriction as:

Due to the change in reference configuration in UL scheme, the stress S , at the start of an increment is equal to (T in the last increment. Therefore, the total value of S, is calculated from:

where n is the time step counter. In the DL formulation, the rate of a is frame different or not objective. As stress and strain measures in a constitutive relationship must be frame indifferent, an objective stress rate has to be employed. The rate of Cauchy stress 6 is related to the Jaumann objective stress rate 6, by:

where W = + ( L - LT) is the rotation rate tensor and D = + ( L + LT) is the deformation rate. 2.3. Mathematical

Model

By introducing constitutive relations into the governing equations, a general form of the transport equation is obtained in the form:

temporal

diffusion

sources

which can be regarded as an equation for the dependent variable 6,where 6 stands for displacement vector or displacement rate vector for total strain and incremental approaches, respectively, I +,+ is the temporal coefficient, D ,++ is the diffusion coefficient, Q$, is the surface source term, Q , is the volume source term, r is the surface vectors and R is the volume.

3. Discretisation Procedures The general governing equation [20] is discretised by employing a finite volume discretisation which has been successfully developd for CFD applications by

464

Finite volumes for complex applications

[DEM 941 and [WEL 981. The presented method takes the full advantage of control volumes of arbitrary shapes. It employs the fully implicit time discretisation and assumes piece-wise linear distribution of dependent variables and material properties in space. The temporal term is approximated by the mean value theorem. Surface and volume integrals are discretised using the mid-point rule. The non-orthogonality in the diffusion term is approximated by a simple orthogonal correction approach. The spatial discretisation is second order accurate while the temporal discretisation is only first order accurate. The resulting sets of coupled non-linear algebraic equations are segregated and the resulting algebraic systems of linear, de-coupled equations are iteratively solved by a pre-conditioning conjugate gradient method.

4. Test Case

In order to demonstrate the method's accuracy and efficiency, few examples are modelled and numerical results are compared with analytical solutions in order to validify the code [MAN 991. A problem presented here is a cube subjected to simple tension. The cube, with sides of length 15, is discretised by 27 cubic Cvs. It is subjected to displacement Vapplied uniformly over one side while the opposite side is assumed to be symmetry plane. Other sides are free surfaces. The force F, required to deform the cube, is calculated from the resulting stresses on the loaded boundary. Material properties used in the calculation are E = 200 GPa and v = 0.3. Figure 1. shows the comparison between the numerical and analytical results. The analytical load - displacement solutions are based on the total Green strain and natural strain definitions [CRI 9 11. For the solutions based on second PiolaKirchhoff stress and Green strain, the DT and TL-diff results coincide exactly while TL-den results agree well with the analytical analysis. Results obtained by the U L schemes are shown to correlate with small strain solution. This is not surprising, as it is said in section 2.2., that the UL approaches in this study are restricted to large deformation problems under small strain conditions.

0.0

0.1

0.2 UIL

0.3

Figure 1. Displacement -force relationships for different schemes.

0.4

Fields of application

465

error (%) 6.0

Figure 2. Errors of F for different schemes. Similarly, DL results agree well with log strain solutions. Overall, the total difference schemes are more accurate than the simple derivation schemes (Figure 2.) because there are less problems of finite increment size used in the numerical procedures, which is assumed to approach zero in the rate equations. Comparisons of CPU times are shown in Figure 3. Times needed for these cases under small deformation in both total strain and incremental approachs are given as a guideline to the extra amount of time needed for large deformation problems.

total CPU

time (s)

60

0.0

0.1

0.2 UIL

0.3

0.4

Figure 3. Comparison of total CPU time required for different schemes.

466

Finite volumes for complex applications

5. Conclusions A numerical method for the analyses of a hypoelastic material is presented. The method is based on the FV discretisation of the law of conservation of linear momentum. Various work conjugate pairs of stress and strain measures are used in constitutive relationships. The developed technique is shown to be simple, easy to understand and manipulate since the underlying concept is simple and straightforward: the balance of forces in the solution domain. This procedure ensures both local and global conservativeness, resulting in meaningful solutions even for coarse numerical meshes. The non-linearity is handled with relatively small additional cost as the method is inherently iterative. This is achieved by using the segregated solution procedure which also offers very efficient memory management. It is clear that the DT approach is the most accurate but it is also limited to materials with single valued stress - strain curves (loading - unloading follow the same path) and is therefore not readily applicable to inelastic problems. The applicability of UL scheme is restricted to small strain cases while the DL approach with Jaumann stress rate is theoretically limited by since the Jaumann rate produces cyclic stresses under shear [KHA 951. Thus, the TL scheme with 'difference' approach is considered to be the most flexible method; it is accurate and it can be further developed for inelasticity applications. Furthermore, this approach does not require expensive remeshing of the solution domain as the calculation progresses. S o far, the method has been successfully employed for simple geometrically non-linear elastic problems. 6. References [BON 971 BONET J. et al., Nonlinear Continuum Mechanics for the Finite Element Analysis, Cambridge University Press, 1997. [MCM 753 McMEEKING R.M. et al., Finite-element formulations for problems of large elastic-plastic deformation, International Journal of Solids and Structures, 1 1 , 1975, p. 601-616. [DEM 941 DEMIRDZIC I. et al., Finite volume method for stress analysis in complex domains, International Journal for Numerical Methods in Engineering, 37, No 2 , 1994, p. 3751-3766. [WEL98] WELLER H.G.et al., A tensorial approach to computational continuum mechanics using object oriented techniques, Computers in Physics, 12, 1998, p. 620-631. [CRI 911 CRISFIELD M.A., Non-linear Finite Element Analysis of Solids and Structures volume I , John Wiley & Sons, 1991. [BAT 961 BATHE K.-J., Finite Element Procedures, Prentice-Hall, 1996. [MAN 991 MANEERATANA K. et al.. Finite volume method for geometrically non-linear stress analysis applications, The Seventh Annual Conference of the Association for Computational Mechanics in Engineering ACME'99 (1999). p. 117-120. [KHA 951 KHAN A.S. et al., Theory of Plasticity, John Wiley and Sons, 1995.

A finite volume formulation for fluid-structure interaction. C.J. Greenshields, H.G. Weller and A. Ivankovic Department of Mechanical Engineering Imperzal College of Science, Technology & Medicine London S W 7 2BX, UK

The finite volume method has dominated computational fluid dynamics for many years and has recently emerged as a viable numerical method for stress analysis i n solid structures. In this work we have adopted the technique for solving problems involving interaction between fluids and structures. The use of a single code enables simple transfer of information at the fluidstructure interface and an implicit solution procedure for the whole system within each time step. The method is used t o study wave propagation i n liquid filled, flexible pipes where, since the pressure wave is influenced by the flexibility of the pipe wall, the solution algorithm must be strongly coupled to obtain convergence. ABSTRACT

Iiey Words: finite volume; fluid-structure interaction; coupling.

1. Introduction

Recent decades have witnessed the development of computational methods to solve problems involving interaction between solid bodies and fluids. Fluidstructure interaction (FSI) covers a wide range of engineering disciplines but a large proportion of all studies fall into the categories of flow-induced vibration, noise and hydrodynamics. Given such a limited scope of application, there is surprisingly little conformity in the numerical methods and coupling procedures adopted throughout research and industry [HAM 951. The required level of fluid-structure solution coupling depends on the nature of the problem we want to solve. If the deformation of the structure is significant, we would preferably like to solve both components implicitly within each time step. At present, implicit coupling appears hindered by limited availability of a general purpose method capable of treating a variety of problems and by the complexity of

468

Finite volumes for complex applications

the methods currently practised. However, the finite volume (FV) method has recently emerged from the realms of computational fluid dynamics (CFD) as a viable alternative for structural analysis [DEM 93;DEM 941. This presents a new possibility for FSI in which both fluids and structures are solved using the same approach [DEM 951. The aim of this paper is to illustrate the simplicity and effectiveness of the FV method for FSI modelling through the analysis of wave propagation in liquid filled, flexible pipes.

2. Mathematical Formulation

This work is concerned with a simple mathematical description of the fluid and solid in which energy contributions are ignored. The continuum is governed by the following: mass balance, or continuity equation

momentum balance (neglecting body forces)

where p is the density, V is the velocity and a is the stress tensor. In the case of a linear elastic (Hookean) solid, the continuity equation need not be considered and since deformations are sufficiently small, the convection term in the momentum equation can be ignored and V becomes dU/dt, where U is displacement. The momentum equation then becomes:

where ,u and X are Lamd's coefficients and I is the identity tensor. For a Newtonian fluid, the continuity equation must be rigidly obeyed and the momentum equation becomes:

where r] is the fluid dynamic viscosity and p is the pressure. In this work, the system of fluid equations is closed by relating pressure and density for

Fields of application

469

compressible flow in liquids by a linearised barotropic relationship

where the subscript '0' represents reference values and Iil is the bulk modulus. The specification of a problem is completed with the definition of a solution domain and initial and boundary conditions. At the initial instant of time, values of all dependent variables must be specified throughout the solution domain and boundary conditions must be specified at all times either by the value of the dependent variable or the gradient of the dependent variable.

3. Finite volume discretisation

The FV method is based on numerical integration of the system of equations over spatial and temporal domains and can be found in most CFD textbooks [FER 961. Notable details of the method adopted here are: face variables in the convection term are calculated using the Gamma differencing scheme [JAS 981 which locally blends second-order accurate central differencing with unconditionally bounded upwind differencing to maintain boundedness; the P I S 0 algorithm [ISS 961 is adopted to ensure that the velocity field in the momentum equation satisfies the continuity equation. The described discretisation procedure reduces both fluid and solid motion t o a set of linear vector equations which are solved in a segregated manner, where each component of the dependent variable is solved separately, treating inter-component coupling terms explicitly. The structure displacements are solved using the Incomplete Cholesky Conjugate Gradient method and the fluid velocity and pressure are solved using the Biconjugate Gradient method [HAG 811.

4. Fluid-structure coupling

The discretisation method described above has been implemented into the Field Operation and Manipulation C++ library [WEL 981. The solid and liquid models are combined within a single code to model the transient behaviour of a flexible pipe and a contained compressible liquid. The fluid and structural parts of the solution domain form separate meshes but the interface boundary share

470 Finite volumes for complex applications

START TIME STEP LOOP Solve fluid Momentum (V) Pressure (p ) Velocity correction (v)

3PISO

Solve structure Transfer wall fluid pressure to structure Momentum (U) converge

2

Displace fluid mesh boundary with structure boundaryu Recalculate volumes and surface areas for new fluid mesh Equate fluid boundary velocity (V) to structure boundary velocity @ J / d t ) I

END OF TIME STEP

A

Figure 1: Implicit solution scheme for fluid-structure interaction the same location in space. The respective systems of equations are solved for both meshes and boundary conditions are passed as shown in Figure 1. 5. Fluid transients in water filled pipes

The test case concerns wave propagation in a water filled unplasticised PVC (PVCu) pipe of external diameter, D = 50 mm and wall thickness, t = 3.8 mm. The dynamic modulus, E = 4.0 GPa and Poissons ratio, v = 0.32 were measured at 20 "C for the PVCu material used in experimental work and its density was 1420 kg/m3. The properties of water used in the calculations were 11' = 2.2 GPa and = 998.2 kg/m3. The first simulation looks at the wave speed through water in a straight 200 mm length of pipe. The problem is axi-symmetric and the solution domain is 500 cells in the axial direction by 20 across the fluid and 8 across the wall section; the fluid is given an initial absolute pressure of 2 bar and zero velocity, and is contained at both ends of the pipe. The simulation begins by applying a fluid outlet condition of fixed 1 bar pressure at one end of the pipe. The first 50 ps of wave propagation is presented in figure 2 as pressure along t,he

Fields of application

0

10

20

30

40

50

60

70

471

80

Distance [mm] Figure 2: Wave propagation along a flexible pipe pipe axis. At 10 ps the fluid does not see the pipe wall deformation and, as expected, the initial wave travels at the speed of sound in unconfined fluids, co = = 1485 m/s. By 20 ,us, the fluid pressure behind the unconfined wave front begins to rise in response to the pipe wall contraction. The initial unconfined wave is quickly suppressed by the surrounding regions of high pressure, as observed in experimentally [THO 691, and a true water hammer wave begins to propagate at an axial speed is 505 m/s. The classical analytical solution for fluid wave speed in an unconstrained, thick-walled pipe [WYL 931 gives a comparable speed of 546 m/s. The analysis only accounts for the change in pipe area which leads to an overprediction in wave speed of 7.5% compared to our numerical solution. 6. Solution stability and Convergence

The efficacy of any iterative method is limited by the rate of convergence and worse still, by numerical instability. The coupling procedure adopted here

472

Finite volumes for complex applications m.na in pressure

AP

stafi

a) stable (I&i\pl+O) -structure cell more rigid than fluid cell

pressure change in

tI

Ap structure

b) unstable ( p p l - t m ) - fluid cell more rigid than structure cell

Figure 3: Stability of the fluid-structure coupling procedure uses an outer iterative loop over the fluid and structure solver routines whose stability is examined by following the principal steps within a single iterative loop. Initially, the fluid equations are solved on the fluid mesh which has been deformed by the motion of the structure boundary within the previous outer iteration, provoking a change in pressure at the boundary, Ap. The structure equations are then solved and the cells adjacent to the boundary respond to Ap by a displacement Ub. The fluid mesh is then deformed once again and the process is repeated. A condition for stability is that JApj must tend t o zero with successive iterations, which is illustrated schematically in figure 3. The pressure change at the boundary between successive iterations is depicted as a linear function of the change in displacement for both the fluid and structure. It is clear that if the structure cells are more rigid than the fluid cells a t the boundary, the solution stabilises with successive iterations; if the fluid cells are more rigid, the procedure is unstable. In practice, the nlethod is unstable for very flexible ~truct~ures such as thin-walled plastic containers, hoses and arteries. In such cases, the instability can be eliminated by introducing a form of under-relaxation where, following the structure analysis, the fluid mesh is only deformed by a fraction o of the structural displacement. However, the approach is somewhat contrived since a must be carefully selected for the specific simulation under consideration to ensure reasonable convergence. Rapid convergence is only possible if there is little variation in cell size along the interface and therefore the method is only ideally suited to problems involving simple geometries and small deformations.

Fields of application

473

7. Conclusions

Both fluid and structural analyses can be solved in a coupled, fully implicit manner using the FV method within a single code. The method has been validated using the analysis of wave propagation in liquid filled flexible pipes. In our 2 dimensional solution, there are only 5 unknowns and the coupling procedure adopted is easy to understand with the data transfer made simple by the use of one code and the same discretisation procedure for fluid and structure. The numerical prediction of wave speed and peak pressure give good agreement with analytical solutions and experiment and will improve with application of more accurate boundary conditions. Some under-relaxation is necessary to ensure stability of the coupling procedure in systems where the structure is comparatively more compliant than the fluid. The method can be applied to 3 dimensions with minimal additional effort and used to study problems such as water hammer in pipes, blood flow in arteries and impact of fluid filled containers.

Acknowledgements The authors wish to thank Mr Tom van der Laan for the data from his pipe impact tests. CJG is currently funded by the EPSRC and HGW is funded by Computational Dynamics Ltd.

References

[DEM 931

DEMIRDZIC, I. & MARTINOVIC, D., Finite volume method for thermo-elastic-plastic stress analysis, Computer Methods in Applied Mechanics & Engineering (1993) 109 331-349.

[DEM 941

DEMIRDZIC, I. & MUZAFERIJA, S., Finite volume method for stress analysis in complex domains. International Journal for Numerical Methods in Engineering (1994) 37 3751-3766.

[DEM 951

DEMIRDZIC, I. & MUZAFERIJA, S., Numerical method for coupled fluid flow, heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology. Computer Methods in Applied Mechanics & Engineering (1995) 125 235-255.

474 Finite volumes for complex applications

[FER 961

FERZIGER,J . H . & PERIC,M., Computational methods for fluid dynamics, Berlin: Springer-Verlag (1996).

[HAG 811

HAGEMAN, L.A. & YOUNG,D.M., Applied iterative methods. New York: Academic Press (1981).

[HAM 951

HAMDAN, F . H . & DOWLING, P.J ., Fluid-structure interaction: application to structures in an acoustic fluid medium, part1 : an introduction to numerical treatment, Engineering Computations (1995) 12 749-758.

[ISS 861

ISSA, R.I., Solution of the implicitly discretised fluid flow equations by operator-splitting, Journal of Computational Physics 62 40-65.

[JAS 981

JASAK,H. et al, High resolution N V D differencing scheme for arbitrarily unstructured meshes, International Journal for Numerical Methods in Fluids in press.

[THO 691

THORLEY, A . R . D . , Pressure transients in hydraulic pipelines ASME Journal of Basic Engineering (1969) 91 453-461.

[WEL 981

WELLER, H.G. et all A tensorial approach to continuum mechanics using object-oriented techniques, Computers in Physics (1998) 12 620-631.

[WYL 931

WYLIE, E.B. & STREETER,V.L., Fluid transients in systems, Englewood Cliffs, New Jersey: Prentice Hall (1993).

BOUNDARY CONDITIONS FOR SUSPENDED SEDIMENT Vittorio Bovolin, Luca Taglialatela University of Salerno Department of Civil Engineering Via Ponte Don Melillo

I84084 FISCIANO (SA) ITALY Ph. +39 089964087 Fax +39 089964045 Email bovo@ bridgediima. unisa. it

ABSTRACT This paper deals with the problem of setting appropriate boundary conditionfor sediment concentration at the channel bed. Sediment transporr is a common and very complex phenomenon. in many real situations suspended sediment represents the tmst important component of the total sediment load. In order to predict sediment load and morphological evolution numerical models are a very useful tool, application of these models requires the setting of boundary condition for the sediment transport equation. In this paper the boundalv condition for sediment concentration is set according to a semiempirical relation that links the sediment flux to the local shear stress value. Calculations are carried out using a numerical model solving the averaged Navier-Stokes equations. the standard k-E model is used as turbulence model. Numerical results are compared with experimental data obtained in strong non unifonn transport conditions such as: sediment pick-up by afZow initially free of sediment and sediment distribution in a steep sided trench. Key Worii.~:srdirnenr rmnsporr, rurbulence, boundary condirion

1. Introduction

Sediment transport by turbulent flows is commonly encountered in practical applications such as rivers, navigation canals, estuaries and reservoirs. Sediments can be transported both as bed load and suspended load, the predominant mode of transport depends on the size, shape and density of the particles in respect to the velocity and turbulent field of the water body. The prediction of sediment transport with the aid of laboratory experiments may be very time-consuming and costly and, for many real life problems, impossible, hence there is a great need for powerful mathematical models to predict it. With the advance of computing methods in fluids,

476

Finite volumes for complex applications

it is possible to obtain, at a reasonable computing time, numerical solutions for complex geometry. The boundary condition criteria for sediment concentration at the first grid node of the computational domain close to the bed are not as well established as are for the hydrodynamic counterparts. This paper tests a semi empirical condition proposed by [NAO 971, comparison of numerical result with experimental data shows good agreement. 2. Suspended sediment equation Sediments are subjected to two contradicting actions: velocity fluctuations lift up and keep them in suspension, gravity induce them to settle down. The upward flux of sediment due to the turbulent motion is E, while the downward flux due to gravity is w,,C, where v and c are the fluctuating components, respectively. of the velocity in the transverse direction and concentration, w,is the fall velocity and C is the local volumetric sediment concentration. Assuming an uniform sediment size and a single particle fall velocity w,.the differential transport equation for suspended sediment for steady 2-D turbulent open channel flow is given by:

where V and U are the mean velocity components in the orthogonal and in the flow direction. We consider only cases where the sediment concentration is so low that it does not interfere with the hydrodynamic. In these cases there is no coupling between flow and sediment equations and therefore equation [I] can be solved separately once the mean hydrodynamic field and the settling velocity are known, and when a model for the turbulent sediment diffusion has been adopted. In the present paper distribution of velocity, eddy diffusivity as well as friction velocity are calculate by numerical integration of the Reynolds equations with the standard k-E model as closure model. The computational procedure is based on the finite volume method using the SIMPLER algorithm for the pressure coupling [PAT 721, [LAU 741, [PAT 801, [ROD 801 and [VER 951. Solution of the hydrodynamic model is used as a "frozen" in equation [I]. 3. Boundary condition

In computational fluid dynamics it is customary to bridge the layer close to the wall by employing empirical formulae, called wall functions, these formulae provide near-wall boundary conditions for the mean-flow and turbulence transport equations [ROD 801, [VER 951. These formulae therefore connect the wall conditions to the

Fields of application

477

dependent variables at the near-wall grid node. The near-wall grid node is presumed to lie in the fully-turbulent part of the flow. The advantage of this approach is that it escapes the need to extend the computation down to the wall avoiding in this way the need to account for viscous effects in the turbulence model. The wall function approach also includes an option to permit wall roughness effects to be simulated via the specification of an equivalent "sand-grain" roughness height k , [JAY 691. Once the hydrodynamic is known in order to solve equation [I] it is essential to set the appropriate boundary conditions at the bed. Generally speaking boundary condition can be of the fixed value type (Dirichlet), fixed flux type (Neumann) or mixed one. Fixing sediment concentration or flux at their equilibrium values may be an unrealistic option because, under non equilibrium condition, near-bed concentration and flux change drastically in the downstream direction before the equilibrium value is finally reached. It seems therefore essential to set mixed type boundary condition that may mimic sediment flux variation following changes in entrainment and deposition. Near the bed equation [1] reduces to [CEL 881 and [RAU 901:

Integrating once yields:

left hand side of equation [3] represents the net flux J of sediment across an horizontal plane. The net flux J is the difference between the entrainment rate E and the deposition rate D. While the latter is equal to wsC the former is not immediately known and therefore it is the main problem in modelling suspended sediment transport in non-equilibrium condition. Starting from equation [3], applying the Reynolds analogy and mixing length concept, [NAO 971 developed a semi empirical boundary condition for uniform and non-uniform situations. Applying Reynolds analogy, [SHI 951 and [ROD 801 yelds:

where I,,, and I, are, respectively, the mixing lengths of momentum and particle concentration, being the velocity gradient multiplied by the mixing length proportional to the shear velocity, equation [4] can be rewritten as:

478

Finite volumes for complex applications

where the proportionality coefficient o,,fixed by the authors, as suggested in literature, equal to 0.50 is known as the Schimdt number, substituting equation [5]in equation [3] a differential equation for C i s obtained.

Integration of [6] in the range b
where z = ( o , w , ) / ( ~ ~ *is) the Rouse number and Cl the concentration at the first grid node. To overcome the need to evaluate Cb, it can be assess [NOA 971 that part of the turbulent eddies produced at the upper sublayer penetrates periodically the lower one and picks up sediments at a rate proportional to Cb,,vlq , where v' is the root mean square of the vertical velocity fluctuation, replacing v' by U*, the net flux is expressed as:

where q is the picking up efficiency assessed on theoretical basis: q = b-4.52/d+)/(4.52/ d')

[91

and limited to d+ = U,dlv > 5 . Evaluating Cb from equation [8] and substituting it in equation [7], the net flux J finally becomes:

Equation [ l o ]is the general form for non uniform condition. The expression for uniform condition is obtained setting J = O in [lo], this is equivalent to fix the sediment concentration at the fist grid node. Equation [lo]can be used once the reference level b, and the concentration Cb,m,,, are specified. To evaluate the bed concentration we follow [RIJ 841 which gave the following expression:

Fields of application

479

[Ill in which the mobility parameter T is:

where o'is the standard deviation of effective bed-shear stress, T ~ ,and , ~Tb,C;cr,~ are the instantaneous critical bed shear stress, respectively, along and counter local flow direction, J, and Jz are integrals representing the pick up action of the bed shear stress. The non dimensional grain is:

in which D5,is the particle size, s is specific density (pJp), g the acceleration of gravity and v the kinematic viscosity coefficient (Np). 4. Results

In order to test the proposed boundary condition we compare numerical results with experimental data obtained by [ASH 821 (as reported by [CEL 84]), by [RIJ 811 and [RIJ 851. The first two data set ([ASH 821 and [RIJ 811) refer to experiments in which initially clear water flow, passing over a sand layer, entrains sediment into suspension until the full transport capacity is reached. The second data set ([RIJ 851) refers to measurements of sediment concentration profiles in steep sided trenches. Table 1 contains all the relevant information for the simulated cases.

Table 1. Relevant dutu for simulated cases

480

Finite volumes for complex applications

For sake of simplicity following [RIJ 841 the reference level b has been set equal to 0.01d where d is the flow depth. Inlet boundary (x=O) are of fully developed channel flow. Comparison of calculated and measured sediment concentration profiles allows the assessment of the boundary condition capability to simulate localised effects, furthermore from a practical point of view it is also interesting to verify the capability of the model as a whole to reproduce the total suspended sediment discharge. Figure 1 depicts comparison of calculated sediment concentration profiles with experimental data [RIJ 811, [ASH 821, the general agreement is fairly good. Figure 2a. for the previous data, shows comparison of bottom sediment concentration, it gives an idea of the boundary condition capability to reproduce the pick up action by the flow, particularly noteworthy is the agreement between numerical results and experimental data in the fist three sections where stronger is the flow non uniformity. In order to assess the model capability figure 2b reports comparison of calculated and measured total suspended load, also in this case the agreement is certainly encouraging on the model effectiveness to reproduce real life cases. The capability of the model to reproduce sediment concentration adjustment in case of non constant depth is another interesting test on the way to simulate morphological evolution. Figure 3 and 4 depict sediment concenuation profiles and total suspended load for a steep sided trenches. This case is particularly demanding because the model has to simulate. in a short stretch, both net deposition, in the enlargement zone, where the flow decelerates and net entrainment in the zone where the flow is under a strong acceleration.

5. Conclusions This paper reports preliminary results of a study on boundary condition for sediment concentration. Numerical results shown fairly good agreement with experimental data. Further research is needed in order to assess the capability of the model to reproduce more complex situation as such as three dimensional case and morphological changes. 6. Acknowledgements

The financial support for the research presented in this paper has been provided by a grant of the Italian Ministry for University and Scientific and Technological Research (M.U.R.S.T.) in the framework of the 1997 National Research Project (P.R.I.N.) "Swirling, turbulent and chaotic processes - Water works and Environmental applications".

Fields of application

48 1

7. Bibliography [CEL 881 Celik I. and Rodi W., Modelling suspended sediment transport in non equilibrium situations, Journal of Hyclraulic Engineering ASCE Vol. 114, No. 10 (1988) [JAY 691 Jayatilleke C. L. V., 'Ilne influence of Prand~lnumber and surface roughness on rlze resi.rtance of the Irr~ninursublayer to momentum and heath transfer. Prog. Heat Mass Transfer, Vol. 1, p.193 (1969) [LAU 741 Launder B.E. and Spalding D.B., 7ize numerical compula~ionof turbulent flow, Comp. Meth. in Appl. Mech. & Eng., Vo1.3, p269, (1974). [NAO 971 Naot D. and Nezu I., Wall funclions for the calculation of turbulent 3 0 sediment transport in open channels X X W IAHR Congress Vol. 2 San Francisco USA pp. 1268-1273 (1997) [PAT 721 Patankar S. V. and Spalding D. B., A calculation procedure for heat, mass and momentum transfer in three dimensional parabolic flow.^, Int. J. Heat Mass Transfer Vol. 15 pp. 1787 (1972) [PAT 801 Patankar S. V., Numerical heal transfer and fluid flow, Hemisphere publising corporation, Taylor & Francis Group, New York (1980) [RAU 901 Raudkivi A.J., Loose Boundaiy hydraulic.^, Pergamon Press, 3th edition (1990) [ROD 801 Rodi W., Turbulence models and their application in kydrcutlics, state-of-the-artpaper. International Assoc. for Hydr. Res., Delft, The Netherlands (1980) [SHI 951 Shiono K., Falconer R. A,, Berlamont J., Elzier M. and Karelse M., A note on statified flow in compound channel, Hydra 2000 XXVIth IAHR Congress, 3B8 pp. 134-139 (1995) [RU 841 Van Rijn L.C., Sediment transport. part II: Suspended load transport Journal of Hydraulic Engineering ASCE, 110 (1 1). pp. 1613-1641 (1984) [RU 861 Van Rijn L.C., Mathernutical modelling of suspended sediment in non-unifol-m flows Journal of Hydraulic Engineering ASCE, 112, pp. 433-455 (1986) [VER 951 Versteeg H. K. and Malalsekera W., An introduction to Computa~ionalFluid Dynamics,thefinite volume method, Longman Scientific &Technical (1995)

Figure I . Sediment concentrution profiles

a ) Ashidu [ASH 821 6) van Rijn [RIJ 811

482

Finite volumes for complex applications

........................................................................... Near-Bed Concentration 0

<

I: :

Okabe (1982, Run 5 ) Calculated

I

Okabe (1982, Run 5) Calculated

I

< $

: j.

$ $ $

i

Figure 2. a) sediment concentration at the bottom b) suspended load [ASH 821 .......................................................................... Concentration Profiles

Concentration Profiles

Figure 3. Sediment concentration profiles in trenches, test2 and test3 [RIJ 851

Suspended Sediment Load

Suspended Sediment Load

Trench Teal-2 (Qsrr/=30)

0

Trench Test-3 (Qsrq=30)

Qs-exp (Rijn, 1985) Calculated

0

Qs-exp (Rijn. 1985)

Calculated

1

Figure 4. Suspended sediment load, respectively, in test2 and test3 [RIJ 851

Second order corrections to the finite volume upwind scheme for the 2D Maxwell equations

B. Bid6garay

J.-M. Ghidaglia

MIP - CNRS UMR 5640 Universitk Paul Sabatier 118 route de Narbonne 31 062 Toulouse Cedex France

CMLA - CNRS UMR 8695 ENS de Cachan 61 avenue du president Wilson 94235 Cachan Cedex France

ABSTRACT W h e n computing solutions t o Maxwell equations wath finite volumes m e -

thods one often faces mesh dependent structures. W e describe a way to add second order corrections to the finite volume upwind scheme for the 2D Maxwell equations that are designed t o overcome this dificulty. This is done by using exact solutions t o the wave equation for each component of the electromagnetic field. W e illustrate the method with numerical results on simple test cases.

Key Words : 2D Maxwell equations, finite volumes, upwind scheme.

1. Introduction

When computing solutions to Maxwell equations with finite volumes methods one often faces mesh dependent structures. Indeed, fluxes are computed across edges that may have privileged directions in some parts of the computational domain. Our goal is to write a modification of the classical upwind finite volume method that handles this drawback.

484

Finite volumes for complex applications

Here we more precisely address the 2D Maxwell equations in the T E Z polarization. Setting Q = (Q1, Q2, Q3) = ( E z ,E y ,H z ) they read

In an homogeneous domain ( i t . E and p constant) each component of Q satisfies the wave equation

a

where

C

~

-u-c2Au =O at2 =E 1. ~Both formulations are used t o derive our scheme.

(2)

2. Leading o r d e r t e r m s

The computational domain is decomposed into triangles. Integrating (2) on one of them, K, and on the time interval [0,t], one gets

The computation of IK = -

/x

udxl

is performed through the usual upt =O

wind scheme (see e.g. [GHI 961) for hyperbolic systems using formulation (1). More precisely equation (1) is written as Q t P

IK = -

+ div(AQ) = 0 and IK

reads

A(n)Qdg where n is the external unit normal vector on the boun-

dary of the triangle K , d K . The eigenvalues of A(n) are 0 and fc. For the computation of IK this integral may be split in three contributions considering separately each edge of K . Then we decompose A ( n ) Q over the eigenvector basis of A(n) and associate Q = Q ( K ) = to eigenvalue c and Q = Q(K)t o eigenvalue -c where K is K ' s neighbor across the considered edge of K . This is the usual upwind scheme. This is a particular case of our scheme and we will use it to evaluate its performances.

Fields of application

485

3. Second order corrections The second part, UK = c2

lK (lt

(o, S)dS) d", which happens to be

a second order correction, is computed using the exact solution of the wave equation (2) with initial condition 1 on triangle K and 0 elsewhere. This approach has already been used by Abgrall [ABG 941, Gilquin et a1 [GIL 91, GIL 941 and Cha'ira [CHA 951 for solving the Riemann problem for gas dynamics and writing Euler equations as a wave equation, but in our case we have no conformal invariance. t

For the computation of UK we set W(X,t ) =

( u ( x , t ) - u o ( K ) ) d r . The J

0

insertion of uo which is formally bound t o disappear in the next differentiation ensures that we preserve a constant solution over space (and time) if the initial data is constant, which is the first step towards flux conservation. We notice that this function w is solution to the wave equation

with initial data tuo

-

0, initial time derivative wl = uo - uo(K) and right hand

side f = ul = -div(A(uo-uo(K)). With this notation, UK =

LKz

c2

-((a,

t)du.

An exact solution to this wave equation is given by Kirchoff formulae

1 H(C"t2 - x2 - Y2) where G(x, y; t) = and H denotes the Heaviside func27rc ,/&2 - x2 - V2 " tion. Hence the computation of UK leads t o multiple integrals on the edges of the triangles that read ( C and C' being equal or adjacent edges) Y

LxC. 10 10 lxc.

W -

-

and

W' = -

c H ( c % ~- lo

?;; Jc2t2

-

ct H(c2t" 27r

10 -

-

-

-

u'I2) dado' u'12

- u')~)(u 0') . nc dudu'. o'l2Jc2tZ - lo - 01/2

that may be expressed after tedious calculations by means of classical functions of the variables (lengths and angles in the triangle). We make an approximation here by computing only integrals over edges of K or its neighbors but not

486

Finite volumes for complex applications

triangles that only have one vertex in common with K, otherwise the computation of UK would be exact. This part of the derivation of our scheme is specific to the 2D case since the fundamental solution G to the wave equation has different expressions in different dimensions. The 1D case is easy to compute but is of no interest since there is no point in correcting direction dependent structures. After this modification the scheme is no longer a finite volume scheme : there is no exact balance of fluxes. The formula for U K shows some discontinuity a t time t = 0. For C' = C , 2c2t - c17r

and Wl = 0, and if we denote by a the angle between edges Wo= 27r C and C' when are distinct, a acute

a obtuse c t - a1

IT

ct7r-llrl

27r sincv

w1

- 3cosa-

sin a

-

~COSQ~---

sin a

These formula? are continuous for a = 5 and are a puzzle for computer algebra systems. To obtain explicit formulae and avoid numerical integration is a major advantage for numerical computations. This computation is performed with a condition on the size of triangles and the time step (that implies, in particular that no length appears in the final result). More general conditions may be taken into account but then the integration should be performed on more triangle edges. To take account of the discontinuity a t time t = 0, we do not use the full correction but only a fraction of it given by a factor 8 E [O, 11. This means that we make a balance between time t and time 0 where UK = 0. Since we deal with a second order correction this does not affect the consistence of the scheme, whatever the choice of 8 is. The CFL ratio and this parameter 8 are the two parameters to tune in order to obtain a "good" scheme. They are strongly linked in 1D but may be chosen independently in 2D. Numerical results show that 8 = 0 is not the best choice with respect t o norm conservations for example. But the best 6' also depends on the mesh and is therefore difficult to choose.

Fields of application

487

4. Numerical results This work is still in progress and up to now only simple test cases have been performed. Different boundary conditions have been implemented (following [ENG 77, ENG 79, JOL 891, namely absorbing boundary conditions, perfect conducting surfaces and perfect reflecting boundaries. Different initial conditions and incident fields have also been tested. The main drawback of the second order correction is that it induces some extra dissipation. This is very easy to see on a 1D equivalent of our correction. This dissipation is larger for a large 0. This leads us to choose 0 E [O, .3] for numerical simulations. The best conservation of norms is usually obtained in this interval. The following figure shows that for a test where the L2 and Lm norm is supposed to be conserved, the choice the upwind scheme without correction (0 = 0) is not the better scheme for norm conservation. On Fig. 1 only the Lm norm of E, is represented. The different curves correspond to different meshes for the same test case. Curves are similar for other components of the field or the L b o r m . The fact that curves are decreasing is not generic but specific to this field in the computed case. Our correction seem to benefit finer meshes but criteria to adjust 0 t o a particular mesh are not obvious t o find. Since we only add a small perturbation to the original scheme we can not expect any real improvement of its main defaults, like phase shift. The initial purpose of the introduction of our UK was the correction of mesh dependent structures and we have tested the propagation of a wave front with different angle of incidence. For small angles (otherwise our boundary conditions have to be improved) we show a real improvement of the straightness of the front. This shown on Fig. 2 and 3.

5. Perspectives

Perspectives of this work may be found in different directions. First we may go towards more realistic test cases and try to model real physical structures. A complete study of the case of an inhomogeneous media has to be performed. This would also contribute to more physical test. The 3D case is also of interest and formal calculations of the correction have to be derived in this context.

488

Finite volumes for complex applications

6. Bibliography [ABG 941

ABGRALL R., <<Approximation du problkme de Riemann vraiment multidimensionnel des Cquations d'Euler par une mCthode de type Roe (I) and (II)>>, C.R. Acad. Sci. Paris 319, 1994, p. 499-504 and 625-629.

[CHA 951

CHAIRAS., Sur la rksolution numkrique des kquations d'Euler de la dynamique des gaz par des schCmas multidimensionnels, PhD Thesis, Ecole Normale SupCrieure de Cachan, 1995.

[CIO 931

CIONIJ.-P., FEZOUI L. A N D STEVEH., <>, Rapport INRIA 1867, 1993. ENGQUIST B. A N D MAJDAA., <>, Math. Comput, 31, 1977, p. 629-651.

[ENG 771

[ENG 791

[GHI 961

[GIL 911

ENGQUIST B. A N D MAJDAA., <>, Commun. Pure Appl. Math., 32, 1979, p. 313-257. GHIDAGLIA J.-M., KUMBARO A. A N D LE COQG., <>, C. R. Acad. Sci., se'rie I, 332, 1996, p. 981-988. GILQUIN H. A N D LAURENS ' J . , < >, 1991.

[GIL 941

GILQUINH., LAURENS J. A N D ROSIERC., <>, ENS de Lyon, UMPA 133, 1994.

[JOL 891

JOLYP . A N D MERCIER B., <>, Rapport INRIA, 1047, 1989.

Fields of application

489

F I G . 1: Norm conservation The Lm norm is represented as a function of 0 and for different meshes, the exact L2 norm being the horizontal line. The 'cross'-curve correspond to the finer irregular mesh (others are 'star' and 'circle'-curves) and the 'diamond' curve is a regular mesh that is finer than the 'squareJ-curve.

490

Finite volumes for complex applications Ex

FIG. 2: Front straightness :8 = 0 Ex

1

0.8 0.6 0.4 0.2 0

F I G . 3: Front straightness :8 = 0.1

A MHD-Simulation in Solar Physics

t

Institut fur Angewandte Mathematik, Universitat Freiburg, Germany Centre de Mathe'matiques Applique'es, Ecole Polytechnique, Palaiseau, France

ABSTRACT A Riemann-solver-based FV-scheme for the numerical solution of the equations of magnetohydrodynamics(MHD) i n two and three space dimensions is proposed. Using recently developed techniques for the efficient combination of high-order resolution, adaptive mesh refinement and parallel computing the method is designed for complex flow problems i n solar physics. Here we consider the rise of patterns of concentrated magnetic fields i n the sun's convection zone. While rising to the solar surface these so-called flux tubes are fragmented by instabilities of Kelvin-Helmholtz- and Rayleigh-Taylor-type. Key Words: MHD, Riemann solver, high-order FV-schemes, adaptive mesh refinement, magnetohydrodynamical instabilities.

1. Introduction

In the last decade the numerical solution of hyperbolic and hyperbolicparabolic conservation laws made considerable progress. In particular this applies to the case of several space dimensions. Features like upwind techniques, shock capturing, and formally high order resolution are standard now, even on unstructured grids. Furthermore the concepts of adaptive mesh refinement using grid indicators respectively error estimators and grid alignment became widely used. On the level of model problems a lot of corresponding analytic results on convergence and rigorous error estimates are available. Concerning physically relevant problems it is noteworthy that these techniques -despite their general applicability- have been utilized merely for the system of gas dynamics. Up to now, less has been done for more complex situations. Motivated by problems of solar physics this paper adresses two issues. First we present a numerical scheme that extends the methods mentioned above to the equations of ideal magnetohydrodynamics (MHD) in Section 2. Specific difficulties caused by the structure of the MHD-system are discussed, for in-

492

Finite volumes for complex applications

stance the divergence free condition for the magnetic field. The second issue of this paper is the presentation of numerical results for physically relevant two and threedimensional MHD-simulations. We start with the discussion of two basic problems: the Kelvin-Helmholtz instability and the Rayleigh-Taylor instability (Sections 3.1 and 3.2). Finally we present results of ongoing numerical studies for a problem in solar physics which is strongly influenced by these instabilities (Section 3.3). To be specific we study the question how far a so-called flux tube, that is a pattern of concentrated magnetic field, can rise to the solar surface. If a flux tube can rise far enough this would support the conjecture that flux tubes cause the sun spots which can be observed from earth on the solar surface. 2. N u m e r i c a l Solution of t h e MHD-System in 2 D / 3 D 2.1. T h e M H D - S y s t e m

The MHD-system describes the interaction of the motion of a compressible, electrically conducting fluid with a magnetic field. For a spatial domain R C R3, it is given by[Ca65, Ch611:

0x

d -(pE) at

+

(2x ii)

V . (pZE + PC)

= pgu,,

Here p(2, t ) stands for density, G(P,t) for velocity with components u,, u,, u,, B(Z, t) for the magnetic field with components B,, B y ,B,, and E for the total energy. For the pressure p(3, t ) , the matrix 'P(Z, t ) = (pij(Z, t)) E RsX3is defined as

To close the system, we add the thermodynamical relations

where T(Z, t) is the temperature and E = &(p,T) the internal energy. Note that, for g(Z, t) E R, the effects of gravity are included in the system above. We recall that the MHD-system above can be written as a system of hyperbolic

Fields of application

conservation laws with the vector of unknowns 3 := ( p , pii, 8,pE) E U fluxes g,G,i? : u -+ R8 and source 2: R x IW - >x ~ U -+ R8: ct

493

c

R8 ,

+ +(C), + G(O), + Z ( C ) , = $2, t, C ) .

Initial and boundary conditions will be specified when needed.

2.2. The Numerical Scheme

In this section we will present our scheme to solve the MHD-equations numerically. For sake of simplicity we will restrict the presentation to a polygonal domain R C R2 that is covered by a grid of triangles. A part of the grid is displayed in Figure 1 to introduce notations.

Figure 1. Triangular grid and Notations. For initial data

Go : R -+ U ,the FV-scheme

takes the form

Here At, denotes the local time step constrained by a suitable CFL-condition and ITj 1 the area of T j . As usual, the numerical flux is thought to be an approximation of t"+'

3

2 ( C ( 2 ,t ) )

in,

( ( 2 , )

(3

dz d t ,

where fii = ( f i l l , . . ,6il)E R8, i = 1 , 2 . The term results from integrating the conservation law with respect to the time interval and the triangle T,. Using the rotational invariance of the MHD-system the integrand can be expressed solely in terms of the flux g:

2(6) ( G ~ ~ ) )

( )

=

~

j

i ( z j l )rotation. ,

494

Finite volumes for complex applications

This reduction to a onedimensional setting allows us to use (approximate) Riemann solvers. Based on an extensive study of different solvers in the MHDcase we favor the one suggested by Dai and Woodward, since it offers a good compromise between computational effort and accuracy [We98]. For details on this scheme we refer to [DW95]. By the linear reconstruction technique of Durlowsky, Engquist and Osher and the use of a second order Runge-Kutta method the basic FV-scheme described above has been extended (formally) to second order. To allow local refinement and coarsening of the grid we have to introduce an error estimator or a grid indicator. Error estimators for nonlinear hyperbolic systems in multiple space dimensions seem to be out of reach for the moment. For the Euler equations density-based indicators are well-established. Note that, in addition t o the density, the magnetic field has to be taken into account for MHD. Even in one space dimension there exist discontinuous solutions of the MHD-system (Alvkn waves, entropy wave) which do not undergo a jump in density and the magnetic field. Extending a grid-indicator for the Eulerequations we propose the following grid indicator for refinement respectively coarsening.

The triangle Tj has to be refined, if ref-ind(Tj) > ref-limit, and may be coarsened, if crsind(Tj) < crslimit, where ref-limit, crslimit > 0 are some threshold values. Another MHD-specific problem is caused by the condition that ~ . fhas i to vanish. While this property is conserved by exact solutions for initially divergence free magnetic fields, it is not automatically satisfied in numerical schemes. A disregard of this problem can lead to severe numerical problems [BB80]. As a remedy we add an artificial source term, which is proportional t o V . l? [As96, Po94]. In our experience this approach works well in many situations but seems to have merely a stabilizing effect and does not necessarily reduce the divergence error. In particular problems with stagnation points remain a challenge. Even the computational advantages, that we achieve by local adaptation and higher order schemes, do not suffice to resolve small scale structures using a one-processor machine. Therefore our 2D-Code has been implemented on a parallel computer with a shared memory architecture. This gave good speedup results for large grids. Unfortunately computers with shared memory architecture and a large number of processors are not easily available. In 3D we use a code which can be run on machines with distributed memory by using MPI to communicate. This code is based on a grid concept which was developed by

Fields of application

495

Schupp [Sc99]. To gain still from local grid refinement a concept for dynamic load balancing is integrated. 3. Numerical Results

As outlined in the introduction of this paper our code is designed for the simulation of problems in solar physics. In particular we are interested in the rise of flux tubes in the sun's atmosphere. These strong magnetic fields are assumed to develop in the deeper convection zone. When they rise they are subject to instabilities which break them up and slow down their ascend. One type of instability is caused by the lower density in the interior of the tube compared to the density of the atmosphere through which it rises. This results in Rayleigh-Taylor instabilities which increase the boundary of the tube and decrease its rise t o the solar surface. At the same time the sides of the tube are fragmented by Kelvin-Helmholtz instabilities due to the shear flow between the atmosphere and the rising tube.

Figure 2. The time evolution of a KH instability for t = 0.4,0.6,0.8,1.0using 256218,475374,571044,727138 elements respectively. Starting with [Ch61] both types of instabilities have been studied thoroughly in the literature for their own sake. The results show that the amplification of small scale disturbances can be reduced or even suppressed by a magnetic field tangential t o the separating interface of the fluids. The magnetic field can be

496

Finite volumes for complex applications

used to increase the stability of the rising flux tube, which leads to a rise into higher regions of the atmosphere without being too strongly fragmented. This stabilization is obtained by twisting the magnetic field in the flux tube so that it is tangential at the boundary. For physical background we refer to [Sch79]. In our calculations we show the performance of the code for pure KelvinHelmholtz and Rayleigh-Taylor instabilities. Then we present two results for the flux-tube problem: with and without stabilization by twisting. For a quantitative study consult [DRW99]. 3.2. Kelvin-Helmholtz Instability (KH)

In the first problem we consider a KH instability in a rectangular domain R that is separated in two parts by a horizontal line. For the initial conditions in 0, we take

The velocity in x-direction is u,, = f5 in the upper/lower part of 0. To start the instability we perturb uy periodically. As boundary conditions we choose periodic conditions for the vertical boundaries and outflow conditions for the horizontal boundaries. The results for the transport of the density with the calculated velocity field at different time levels are displayed in Figure 2. For visualization we use a special tool to treat vector fields[BR98]. Note that gravitation is not included.

Figure 3. RT instability at time t = 4.5 for B,/&

= 0.00,0.01,0.05,0.1.

3.3. Rayleigh-Taylor Instability (RT)

The second test case shows a perturbed horizontal layer of lower density in a stratified atmosphere. Under the influence of gravitation this layer rises. Because of a perturbation of the initial values it produces the typical "fingervstructure of the RT-type instability. We show the announced stabilizing influence of a growing tangential magnetic field in Figure 3 (Displayed quantity:

Fields of application

497

density). As boundary conditions we have periodic conditions for the vertical conditions. For the the horizontal boundaries and the construction of the atmosphere we refer t o [DRW99]. 3.4. Rise of flux tubes to the solar surface

We turn to calculations of a flux tube in 2D. In the top row of Figure 4 we show B, for a flux tube with a magnetic concentration normal to the plane of calculation. This magnetic field cannot reduce the fragmentation through the instabilities so that the flux tube is strongly fragmented and hardly rises.

Figure 4. Flux tube in 2D.

In the lower row of Figure 4 the calculation is repeated with a twisted magnetic field. This prevents the fragmentation and allows the flux tube t o rise faster and higher than in the first calculation. Figure 5 shows a first calculation for a Aux tube in three space dimensions.

Figure 5. Flux-tube in 3D for a fixed time t = 0.6. Top: yz-plane for x = 0, bottom: xz-plane for y = 1, y = 3.3, y = 5.6.

498

Finite volumes for complex applications

ACKNOWLEDGEMENTS The authors were partially supported by the DFGSchwerpunktprogramm "Analysis und Numerik von Erhaltungsgleichungen" and the EU-TMR research network for Hyperbolic Conservation Laws. Bibliography [As961

A ~ L A N., N Two-Dimensional Solutions of MHD Equations with an Adapted Roe Method, J.Num.Meth.Fluids 23, 1996, p. 1211-1222.

[BR98]

Becker J., Rumpf M.,, Visualization ~f Timedependent Vector Fields by Texture Transport Methods, In Proceedings of the Eurographics Scientific Visualization Workshop '98, 1998.

[BB80]

BRACKBILL J.U., BARNESD.C. The effect of nonzero V . B on the numerical solution of the magnetohydrodynamic equations, J . Comput. Phys. 35, 1980, p. 426-430.

[Ca65]

CABANNES H., Magnetodynamique des fluides, Paris (1965).

[Ch61]

CHANDRASEKHAR S., Hydrodynamic and hydromagnetic stability, Oxford (1961).

[DW95]

DAI W . , WOODWARD P . R . , A Simple Riemann Solver and High-Order Godunov Schemes for Hyperbolic Systems of Conservation Laws, J.Comp.Phys. 121 ( I ) , 1995, p. 51-65.

[DRW99]

DEDNERA., ROHDEC., WESENBERG M., A numerical study on magnetohydrodynamic instabilities i n three space dimensions, in preparation.

[Po941

POWELL K.G., A n Approximate Riemann Solver for Magnetohydrodynamics (That Works i n More than One Dimension), ICASE-Report 94-24.

[Sc99]

SCHUPPB., Entwicklung eines efizienten Verfahrens ziir Simulation kompressibler Stromungen i n 3 0 auf Parallelrechn e m , PhD-thesis, Freiburg (1999).

[Sch79]

SCHUSSLER M., Magnetic Buoyancy Revisited - Analytical and Numerical Results for Rising Flux Tubes, Astronomy and Astrophysics 71, No. 1-2, 1979, p.79-91.

[We981

WESENBERG, M., A Note on MHD-Riemann-Solvers, print (1999).

Pre-

A Zooming Technique for Wind Transport of Air Pollution* P.J.F. Berkvens, M.A. Botchev, W.M. Lioen, J.G. Verwer CWI, f? 0. Box 94079 1090 GB Amsterdam, The Netherlands http://www. cwi.nU e-mail: [berkvens, botchev, waltel; janv]@cwi.nl ABSTRACT In air pollution dispersion models, typically systems of millions of equations that describe wind transport, chemistry and vertical mixing have to be integrated in time. To have more accurate results over specijic jixed areas of interest-usually highly polluted areas with intensive emissions-a local grid refinement or zoom is often required. For the wind transport part of the models, i.e. for finite volume discretization~of the transport equation, we propose a zoom technique that is positive, mass-conservative and allows to use smaller time steps as enforced by the CFL restriction in the zoom regions only. K E Y WORDS: finite volumes, advection schemes, local refinement, air pollution, high performance computations

1 Introduction Mathematical problems often encountered in air pollution modelling are transportreaction problems of the form

where c, are the concentrations of m chemical species in the atmosphere. The species under consideration are not only pollutants, but all the main chemical substances present in the atmosphere. In real applications m lies between 25 and 100. The first term in (1) denotes the time rate of change of c,, the term V . (gc,) describes the transport of the species by a given wind field g. The term V ( g ,cs) on the right-hand side appears as a result of parameterization of transport processes not resolved on the 'This work has been done within the program "Wiskunde Toegepast" ("Mathematics Applied) of NWO, the Netherlands Organization for Scientific Research, project no. 613-302-040. This work was supported by NCF, the National Computing Facilities Foundation, under Grant NRG 98.02.

500

Finite volumes for complex applications

grid. It will be referred to as vertical mixing. Vertical mixing is often modelled by means of turbulent diffusion parameterization. The stiff reaction term R describes the chemical reactions among the species c,. Usually, (1) is discretized in space with a finite-volume technique. For typical grid resolutions and numbers of chemical species m, (1) yields a system of millions of equations that has to be integrated in time. More background information on air pollution modelling can be found in the recent books [4, 111 and the survey [7] giving an overview of numerical techniques used in the field.

1.1 Model In air pollution models problem (1) is typically discretized in space with a finitevolume technique and then integrated in time with operator splitting. This means that the whole advance in time consists of separate advection, vertical mixing and chemistry advances. Chemistry and vertical mixing are integrated in time implicitly, to avoid a severe restriction on the time step At imposed by the chemistry-mixing stiffness. Advection is usually integrated in time explicitly. In a particular atmospheric dispersion model we are working with, a successor of [5], several modern advection schemes are used. The two basic schemes are the Slopes scheme [9] and the Split scheme, a third order flux-limited upwind scheme [6, 81. Both schemes are onedimensional and applied with directional splitting. For the atmospheric application it is important that mass is conserved and concentrations remain positive (c,(t) 2 0, t 2 0). Both advection schemes mentioned have these properties on uniform grids, even for divergent flows, which occur in the atmosphere. Positivity of species concentrations is guaranteed for time-step size restrictions such that no negative air masses can occur during any one of the split advection steps, which is a natural constraint. 1.2 Why zooming? To capture local phenomena-as occurring over highly polluted areas-without increasing the cost too much, it is often desirable to have a nonuniform grid with local refinement (zooming) over the areas of interest. The local grid refinement or zoom methods have been subject of active research (see e.g. relatively early works [ l , 31). Most of the effort has been to develop powerful adaptive grid refinement. In our case, however, adaptivity is not needed and not even welcome. First of all, areas of interest are known in advance (that can be Europe, for example). Second, meteorological data as e.g. wind fields are often not available on a fine grid in the whole domain (see Figure 1). Third, the position of the zoom regions is often determined by the chemistry part of the model: for example, one has to refine in the regions with high emission activity. Thus, our goal is an efficient robust zoom algorithm for@ed zoom regions and we propose a simple strategy for this.

2 Zooming technique 2.1 Requirements for zooming Since mass conservation and positivity are important for the atmospheric application, we wish our zooming technique to preserve both these properties (something we have

Fields of application

501

Figure 1: An example of zooming. Zoom regions are numbered. not managed to find in the literature). Due to the smaller size of the finite volumes, the CFL restriction is more severe in zoom regions and a smaller time step At needs to be taken. Positivity and mass conservation are more difficult to preserve if we want to have smaller step sizes in the zoom regions only, so that e.g. in a zoom region with refinement factor two, two time steps are to be done within each global advance in time. Our zooming technique gives a simple and reliable way to have smaller timestep sizes in the zoom regions only while preserving positivity and mass conservation across zoom interfaces.

2.2 How to zoom in l D We explain first how to perform zooming for the one-dimensional (ID) transport equation ct + (uc), = 0 and then how to extend this approach to the three-dimensional (3D) case. We use advection schemes in the mass conservative flux formulation. Each cell i contains a mass pi being the integral of c over the ith cell volume. In ID fluxes F,+,,, are calculated giving the amount of chemical species transported per time step between cells i and i 1. They depend on the given air-mass fluxes determined from the velocity field and on the masses pi. The time advance has the form

+

where F Z l l 2 are the fluxes at time level n. On the boundaries of the zoom region its advection scheme is adjusted to the scheme on the coarse level. As an example, consider the situation with refinement factor ref = 3 (see Figure 2). This adjustment consists of two elements. First, the three outermost cells of the zoom region are lumped together and considered to be one single cell (the interface cell). The tracer mass in the interface cell is assumed to be distributed uniformly (or in accordance with its slope for the slope scheme) over its fine grid subcells, and the fluxes between these subcells are never computed. Second, on the boundary of the zoom region we calculate the flux FC that is coarse in both space and time. For each coarse time step, there are three corresponding time substeps that have to be done in the zoom region. The coarse boundary flux F~ is applied at

502

Finite volumes for complex applications

u w

tn+,

grid cells w

c

e cell V

zoom region

Figure 2: 1D zooming: one time step t, + tn+l corresponds to three time substeps in the zoom region. The "coarse" flux F~ is applied on the first substep. once at the first fine time substep, for the second and for the third substep zero flux is taken at the boundary of the zoom region. It can be shown that this simple strategy guarantees mass conservation and positivity of the chemical species for many modern advection schemes including the Slopes and Split schemes (for more details see [2]). The time-step restriction for positivity is basically the same as in the uniform case, namely that the coarse air mass flux on the one wall plus the three fine air mass fluxes on the other wall do not take out more mass than there was in the cell. The strategy is also easily generalized to the multidimensional case, with a similar time-step restriction per direction. Another choice would be to equidivide F' in three parts and apply them consecutively at the same times as the fine fluxes on the other side of the interface cell. In 1D as well as in multidimensional cases this gives no longer a positive scheme, unless a more severe time-step restriction is enforced. The problem is that parts of the coarse flux, which was calculated with the old concentration field, are applied to a new concentration field. The latter may have changed in such a way that positivity is lost.

2.3 How to zoom in 3D We now give more details about our zooming algorithm in 3D. For simplicity reasons we restrict ourselves to block-shaped zoom regions in 3D. For our applications this is sufficiently general. A zoom region has interface cells all along its edges forming the boundaries with its parent region. The 1D algorithm described above can then be applied in a split manner. The only complication as compared with the 1D case is caused by the fact that the advection scheme is directionally split. We use a Strang (symmetric) splitting scheme consisting of six split steps as follows: X - Y - 2 - 2 - Y - X where , x denotes the application of the advection operator in the x-direction during half a time step, At/2, etc. For one triplet of advection steps-one in each direction-on the coarse level, e.g. X-Y-2,in the zoom region we perform ref triplets of advection steps in the same and the opposite order, thus 2-6-2-5-94?in case ref = 2. Here 2 denotes the application of the advection operator in the x-direction during a quarter time step, At/4, etc. For example, in Figure 3 we show the sequence of the directional substeps in a zoom region with refinement factor 2. Clearly, Strang splitting order is preserved in the zoom region, which is important for accuracy reasons. The coarse fluxes computed by the parent at the boundary with the child region

Fields of application

coarse region:

zoom region:

X YZ XY ZZ

YX

Z YX ZYXXY

503

Z

Figure 3: The sequence of advection steps in a coarse and a fine region with refinement factor 2. are used as boundary conditions for the child region and are applied at once at the first time substep per direction. After the zoom or child region has carried out its ref triplets, the concentration field in the child region is copied back to its parent after suitable coarsening. This then allows the parent region to advance another three directional substeps and to provide new boundary conditions for the child region. This 3D zooming algorithm can also be shown to be mass-conservative and positive for the advection schemes we are interested in. The complete description of our algorithm can be found in [ 2 ] .

3 Experiments With our tests we want to demonstrate that (i) our zooming technique gives an accuracy comparable with the accuracy one would get on the overall fine grid provided that important phenomena remain in the zoom regions; (ii) this accuracy is achieved with significantly less computational effort; (iii) even when important phenomena occur outside the zoom regions, our algorithm still gives a solution which is at least as accurate as one would get on the overall coarse grid. The situation (i) is quite common for atmospheric modelling since one of the typical needs for the local refinement is concentration of the emission sources in highly polluted areas. We note in passing that zooming also leads to a more economical memory use, compared to the overall fine grid. We present 2D numerical results obtained with our zoom implementation for the Slopes advection scheme. As a model problem for our experiments we take the solidbody rotation test commonly known as the Molenkamp-Crowley test [lo]. In this test the advection equation on the (unit) sphere

is solved for the velocity field u = 27r cos x sin y , v = -27r sin x. Here x E [O; 27~1 and y E (-r/2; 7r/2) are the longitude and the latitude coordinates respectively. The chosen velocity field provides a solid-body nature of the air rotation over the sphere, so that after one full rotation (at time t = 1) the solution will coincide with the initial distribution c(x,y, 0). For the initial distribution in our tests we take a cone of height 1 centered at the point (x, y ) = (37r/2,0). We stress that the cone provides a severe test for any advection scheme. The radius of the cone base is taken to be 21 grid cells with respect to the finest zoom region used. In Figure 4, we show the velocity field and the position of the zoom regions used in the tests. Overall, we have a nonuniform grid consisting of three regions: the coarsest,

504

Finite volumes for complex applications

4)

90

45

135

225

180

270

315

360

longitude

Figure 4: The velocity field and the position of the zoom regions. with resolution A x = Ay = 4.5O and two zoom regions refined with factors 2 and 6 (resolutions A x = Ay = 2.25' and A x = A y = 0.75' respectively). First of all we have compared the zoom grid solution with the solution obtained on the overall fine grid (with the same resolution as in the finest zoom region). As desired, it turned out that the difference between them is negligible as long as the moving cone remains in the zoom region. The difference in the solutions is explained by the fact that the order of the directional substeps in the zoom algorithm alternates (as in Figure 3) whereas it remains the same for the uniform grid algorithm. Performing one full rotation (0 t 1)on the nonuniform zoom grid means that the cone has to travel over the whole sphere, leaving and then entering again the zoom regions. Of course, with the local zoom regions fixed in space, i.e. not moving with the cone, globally one can not expect much better accuracy than on the overall coarse grid. This is confirmed in our tests. We have performed one full rotation on the overall coarse ( A x = Ay = 4.5'), the zoom, and the overall fine (Ax = Ay = 0.75') grids. In each case, we have measured accuracy by comparing the solution cn after one full rotation with the initial distribution c0 (the exact solution would give no difference with cO).We have computed the following errors (representing minimum, maximum, scaled 1 2 , mean and variance errors respectively):

< <

emin =

err0 =

min c t - min c:, max,:c

JC ~j

,

(c;j - cP,j )' max cy,

emax =

,

max c[

-

max c:,

max cy,

C ~ j c ? jerr1 = C Yi4,j

,

errl=

C ~i( ~ l l2f ~- 1, C~j(c:,j)~

where yj is the (i, j)-cell air mass divided by the total mass of the air (cells near the poles have less mass), and the sums and the minimalmaxima are taken for all the grid cells within the finest zoom region. To be able to compare errors of different space resolutions, on the overall fine and on the zoom grids we computed the errors on the data coarsened up to the coarse grid resolution. We have summarized our accuracy observations in Table 1. In Figure 5 (left picture), we plot the zoom solution after the full rotation. As we

Fields of application

505

Table 1: Errors. "Local error" means with the cone cone still being within the finest zoom region, "global error" means after one full rotation Errors fine zoom coarse

local:
emin -5.9e-3 -1.8e-2 -1.9e-2

emax -3.le-2 -0.12 -0.21

global: err0 2.5e-3 1.7e-2 3.3e-2

err 1 1.4e-3 1.8e-3 2.6e-3

err2 -3.5e-5 -0.12 -0.15

Figure 5: The cone after the full rotation see, the cone shape is strongly deformed. This is not only because it has travelled through the grid that is 6 times coarser (we should emphasize that the resolution 4.5O is quite coarse for this test) but also because of the accuracy losses near the poles. For comparison reasons, at the right plot we present the solution for the same test but with half the grid size in the coarse region (in other words, cone is to travel through the grid that is only factor 3 coarser, with basic resolution Ax = Ay = 2.25'). As expected, deformation is significantly decreased. Finally, we comment briefly on the computational expenses. Rough estimates taking into account total number of grid cells and the CFL restriction on the time step size show that the computations on the overall fine grid would be 5.2 times more expensive than on the zoom grid. This is however a too optimistic speed-up estimate which does not take into account communication overhead for the zoom algorithm. With the current implementation (SGI workstation) we observed that our zoom algorithm is approximately 2.6 times faster than the uniform algorithm on the fine grid. However, code optimization has to be performed yet.

4 Conclusions We have presented a positive and mass-conservative local grid refinement (zoom) algorithm for advective transport. The algorithm can be applied to many modern advection schemes with directional splitting in space and explicit advance in time. With our approach, a smaller time step (due to the stricter CFL condition) is taken within the zoom regions only.

506

Finite volumes for complex applications

The described zoom algorithm has been implemented as a code that allows to use an arbitrary number of zoom regions that can lie inside each other in an arbitrary way, provided they are of block shape and strictly embedded. To have high performance of the uniform grid code preserved as much as possible, the work of the zoom code is organized in a uniform region-wise manner, i.e. the zoom grid is split into a cascade of uniform grid regions. For more details on the implementation we again refer to [2]. Our zoom code will be incorporated in the air pollution model TM3, a recent successor of [5]. The TM3 model is operational at the Institute of Marine and Atmospheric Research (IMAU, Utrecht University), the Dutch Royal Meteorological Center (KNMI) and the Dutch National Institute of Public Health and the Environment (RIVM). References

[ l ] M. J. Berger. Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations. PhD thesis, Department of Computer Science, Stanford University, Stanford, CA 94305, Aug. 1982. [2] P. J. F. Berkvens and M. A. Botchev, W. M. Lioen and J. G. Verwer. A zooming technique for wind transport of air pollution. Report MAS-R99xx, CWI, 1999. [3] J. H. Flahery, P. J. Paslow, M. S. Shephard, and J. D. Vasilakis, editors. Adaptive methods for Partial Differential Equations. SIAM, Philadelphia, PA, 1989. [4] T. E. Graedel and P. J. Crutzen. Atmosphere, Climate and Change. Scientific American Library. Freeman and Company, New York, 1995. [5] M. Heimann. The global atmospheric tracer model TM2. Technical Report 10, Deutches Klimarechenzentrum (DKRZ), Hamburg, 1995. [6] W. Hundsdorfer and B. Koren, M. van Loon and J. G. Verwer. A positive finitedifference advection scheme. Journal of Computational Physics, 117:35-46, 1995. [7] J. G. Verwer, W. Hundsdorfer and J. G. Blom. Numerical time integration for air pollution models. Report MAS-R9825, CWI, 1998. [8] A. C . Petersen, E. J. Spee, H. van Dop, and W. Hundsdorfer. An evaluation and intercomparison of four new advection schemes for use in global chemistry models. Journal of GeophysicalResearch, 103(D15):19,253-19,269, Aug. 1998. [9] G. L. Russell and J. A. Lerner. A new finite-differencing scheme for the tracer transport equation. J. Appl. Meteor, 20: 1483-1498, 1981. [lo] D. L. Williamson and P. J. Rasch. Two-dimensional semi-Lagrangian transport with shape-preserving interpolation. Monthly Weather Review, 117:102-129, 1989. [ l 11 Z. Zlatev. Computer treatment of large airpollution models. Kluwer Academic Publishers, 1995.

Computational Solid Mechanics using a Vertex-based Finite Volume Met hod

G. A. Taylor, C. Bailey and M. Cross Centre for Numerical Modelling and Process Analysis University of Greenwich, Woolwich, London SE18 6PF, UIi' E-mail: [email protected]

p~

ABSTRACT '4 number of research groups are n o w developing and u s i n g f i n i t e volume

( F V ) methods for computational solid mechanics ( C S M ) . T h e s e methods are proving t o be equivalent and i n s o m e cases superior t o their finite element ( F E ) counterparts. In this paper we will describe a vertex-based FV method with arbitrarily structured meshes, for modelling the elasto-plastic deformation of solid materials undergoing small strains in complex geometries. Comparisons with tradite'onal F E methods will be given.

Key Words: Vertex-based, Finite Volume, Solid Mechanics, Elasto-plastic.

Over the last three decades the FE method has firmly established itself as the pioneering approach for problems in CSM, especially with regard to deformation problems involving non-linear material analysis [OH80, ZT891. As a contemporary, the FV method has similarly established itself within the field of computational fluid dynamics (CFD) [Pat80, Hir881. Both classes of methods integrate governing equations over pre-defined control volumes [Pat80, Zie951, which are associated with the elements making up the domain of interest. Additionally, both approaches can be classified as weighted residual methods where they differ with respect to the weighting functions that are adopted [OCZ94]. Over the last decade a number of researchers have applied FV methods to problems in CSM [Tay96]. It is possible to classify these methods into two approaches, cell-centred [DM92, HH95, Whe96, Whe991 and vertex-based [FBCLSl, OCZ94, BC95, Tay961. The first approach is based on traditional FV methods [Pat801 as applied to problems in CFD and suffers from the same difficulties when applied to complex geometries involving arbitrarily structured meshes [DM92, HH95]. The second approach is based on traditional

508

Finite volumes for complex applications

FE methods [ZT89] and employs shape functions to describe the variation of a variable over an element, and is therefore well suited to complex geometries [FBCLgl, OCZ941. Both approaches apply strict conservation over a control volume and have demonstrated superiority over traditional FE methods with regard to accuracy [Whe96, Tay961, some researchers have attributed this to the local conservation of a variable as enforced by the control volumes employed [FBCLSl, BC95] and others have attributed it to the enforced continuity of the derivatives of variables across cell boundaries [Whe96]. The objective of this paper is to describe the application of a vertex-based FV method to problems involving elasto-plastic deformation and provide a detailed comparison with a standard Galerkin FE method. 2. E q u i l i b r i u m E q u a t i o n s and B o u n d a r y C o n d i t i o n s

In matrix form, the incremental equilibrium equations are

[ L ] ~ { A+ ~ ){ b ) = (0)

in i2,

(1)

where [L] is the differential operator, {Au) is the Cauchy stress, { b ) is the body force and R is is the domain. The boundary conditions on the surface = rt U Fu of the domain Cl can be defined as [ZT89, OCZ941 [ R ] ~ { A ~= ) {Au)

{tp)

= {up)

on Ft and

(2)

ru,

(3)

on

where i t p ) are the prescribed tractions on the boundary Ft, {up) are the prescribed displacements on the boundary r, and [R] is the outward normal operator [OCZ94, Tay961.

3. C o n s t i t u t i v e R e l a t i o n s h i p In matrix form, the stress is related to the elastic strain incrementally as follows; { A a ) = [D]{Ate), where [Dl is the elasticity matrix. For the deformation of metals, the von-Mises yield criterion is employed and the elastic strain is given by {Ac,) = {A€) - {AtuP),where {Ac) and {Acup) are the total and visco-plastic strain, respectively. The visco-plastic strain rate is given by the Perzyna model [Per661

where a,,, oy,y , N and s are the equivalent stress, yield stress, fluidity, strain rate sensitivity parameter and deviatoric stress, respectively. The < x > operator is defined as follows;

<x>=

0 x

when when

x x

<0

> 0.

and

Fields of application

509

The total infinitesimal strain is {A€) = [L]{Au), where {Au) is the incremental displacement.

4. Vertex-based Discretisation Erllploying the constitutive relationship of the previous section in equations (1) and ( 2 ) , and assuming the boundary conditions as described by equation (3) are directly satisfied by the vector {Au), . - the method of weighted residuals can be applied to the equations to obtain the following weak form of the equilibrium equation [ZT89];

where [W] is a diagonal matrix of arbitrary weighting functions. At this point the unknown displacement can be approximated as [ZT89]

{Au)

2

{Ah) =

E[N]~{AB)~ =E[I]N~{AB)~,

(6)

j=1

j=1

where { A u ) ~is the unknown displacement at the vertex j , N j is the shape function associated with the unknown displacement and [I]is the identity matrix. The displacenlent approximation can be introduced into equation (5) if the arbitrary weighting functions [W] are replaced by a finite set of prescribed for each vertex i [ZT89, OCZ941, functions [kV] = Cy=l[W];,

i=l,n.

for

(7)

Equation (7) can be expressed as an incremental linear system of equations of the form [Ii]{Aii)-{ f ) = {O}, where [Ii']is the global stiffness matrix, {Aii) is the global displacement approximation and { f ) is the global equivalent force vector and can be formed from the summation of the following contributions;

[Ii]ij =

l, L,,

Lut lu,

[L w ] ~ [ D ] [ L N dQ ]~ -

+

[W]'tP dl. -

[RW]T[D][LN]~ dl.

[RW]:[D]{~~UP) d r j

and

(8)

5 10

Finite volumes for complex applications

Figure I: 2D control volumes, (a) overlapping F E and (b) non-overlapping FV. where Ri is the control volume associated with the vertex i and is the boundary of the control volume.

ri = I?,

U

rt,

4.1. Standard Galerkin FE Method In the standard Galerkin FE method the weighting function associated with a vertex is equal to the shape function of the unknown associated with that vertex [ZT89, Hir88, OCZ941, [WIi = [NIi. The shape functions describe the variation of an unknown over an element and there can be a number of elements associated with each vertex. Hence, it is apparent that control volumes described by weighting functions of this form will always overlap. This is illustrat,ed in Figure l ( a ) for a simple two dimensional case of two adjacent nodes i and j , where the control volumes Ri and R j have contributions from all the elements associated with their respective vertices i and j . Hence, for the standard Galerkin FE method the contributions as described by equations (8) and (9) are

where [BIi = [ L N ] ; . It is important to note that if the boundary of the control volume, such as that described by ri in Figure l ( a ) , coincides with the external boundary of the domain, the shape functions are not necessarily zero along that part of the boundary. Thus, if a flux is prescribed such as a traction this will not necessarily disappear and may contribute to the equivalent force vector as described in equation (11). Additionally, the symmetrical nature of the stiffness matrix as indicated by equation (10) should be noted. The Galerkin approach is

Fields of application

511

accepted as the optimum technique for treating physical situations described by self-adjoint differential equations, particularly those in solid mechanics, as the inherent symmetrical nature is preserved by the choice of weighting functions [ZT89, OCZ941. 4.2. Vertex-based FV Method In the vertex-based FV method the weighting functions associated with a vertex are equal to unity within the control volume, [WIi = [I],and zero elsewhere. This definition is equivalent to that for the subdomain collocation method as defined in the standard texts [Hir88, ZT891. Though it is important to note that weighting functions defined in this manner permit a variety of possibilities with regard to the control volume definition [OCZ94]. This is because the weighting functions are not restricted to to a direct association with the cell or element as in the Galerkin case. This is an important consideration and requires the recognition of the vertex-based FV method as a discretisation technique in its own right [Hir88]. For the vertex-based FV method the contributions as described by equations (8) and (9) are

It is important to note that the traction boundary conditions can be applied directly as another surface integral, but in the previous Galerkin approach an additional surface element is generally included on the domain boundary. A non-overlapping control volume definition suitable for a vertex-based FV method is illustrated in two and three dimensions in Figures l ( b ) and 2(a), respectively. The Figures illustrate the assembling of vertex-based control volumes from their required sub-control volumes [Tay96]. Additionally, the asymmetric nature of the contributions to the overall stiffness matrix as described by equation (12) does not ensure that symmetry will always be preserved. For this reason FV methods were initially argued as being inferior, but in the light of recent research where different control volume definitions have been proposed, the extent of this inferiority has come into question [OCZ94, Zie95, BC951. 5. Results and Conclusions In this section the vertex-based FV method is applied to a three dimensional validation problem and compared with the standard Galerkin FE method. The non-linear solution procedure adopted in for both these methods is based upon

Finite volumes for complex applications

512

Conuol volurnc

Yf

Sub-conuol

L I

1

--_

_

,'

, /

Vertex

X

InlegraUm pa1nI.i 3

4

Figure 2: (a) 3D assembly of FV sub-control volumes and (b) spherical vessel. that of Zienkiewicz and Cormeau [ZC74, Tay961. Both methods utilised an explicit technique with regard to time stepping of the Perzyna equation (4). It is important to note that the FV solutioil procedure only differs from that of the FE in contributions to the global equivalent force vector and the global stiffness matrix. Hence, allowing an accurate comparison of the two methods [Tay96]. The methods are compared with regard t o accuracy and computational cost. They are also analysed for a variety of meshes with different element assemblies.

5.1. Test case: Internally pressurised spherical vessel For this validation problem a thick walled spherical vessel, consisting of an elastic-perfectly plastic material, undergoes an instantaneously applied internal pressure load. The pressure load is 30 dNmm-', the Youngs modulus and Poisson ratio required to define the elasticity matrix are 21,000 dNmm-' and 0.3, respectively, and the yield stress is 24 dNmm-'. This problem is rate independent and the final solution is equivalent to that of an elasto-plastic analysis [ZC74]. A closed form radial solution is available [Hi150]. Numerically the problem can be modelled in three dimensional Cartesian coordinates, with the displacement componeilts fixed to zero in the relative symmetry planes. The spherical vessel is then reduced to an octant as illustrated in Figure 2(b)'. Examples of meshes consisting of linear tetrahedral (LT),bilinear pentahedral (BLP) and trilinear hexahedral (TLH) elements are illustrated in Figures 2(b)', 2(b)3 and 2(b)4, respectively. Firstly, the problem was analysed with a series of meshes consisting of TLH elements. The hoop stress profiles, along the radii, as obtained from one of the numerical analyses are plotted and compared against the reference solution in Figure 3(a). The profiles illustrate the stress in the plastic and elastic regions, and the radial extent of the plastic region according to the analytical solution. The close agreement of the two methods is illustrated. However, it

Fields of application

IS Oee 10

10 Hoop 8-1

e

Hoop s(rcs.

5 (Wmm-=)

e

5 13

rfi

0

I

150 Radial dialrow (m)

Radial dirlrocc lmm)

Figure 3: (a) 950 TLH and (b) 4,800 LT elements. is important to note the closer agreement between the reference solution and the FV method when a coarse mesh is employed. These observations may be associated with the higher order, trilinear nature of the elements employed in the three dimensional analysis at this stage. With regard to the FV method, the implementation of pressure loads (tractions) will involve bilinear face elements for T L H elements. Hence, when considering the application of pressure loads for the two methods as described in equations (11) and (13), the contributions are different due to the individual weighting technique associated with each method. Furthermore, the weighting technique employed for the FV method may be more complementary, when applied generally, as all the terms are integrated conservatively at a local level. Conversely, for the FE method the weighting is not locally conservative which may introduce errors when pressure loads are employed. These conclusions are tentative and rely on the interpretation of the present observations, but they agree with the findings of other researchers [Whe96] and strongly suggest that further research of the FV method is worthwhile. Secondly, the problem was analysed with a series of meshes consisting of B L P elements and there was much closer agreement between the methods [Tay96]. This is attributable to the lower order, bilinear nature of the element concerned and the linear nature of the triangular faces over which the pressure loads were applied. As illustrated in Figure 2(b)3 the BLP elements are orientated so the pressure load was prescribed over a triangular face. This was an outcome of the automatic mesh generator employed [Fem] and it is possible to further study the element when pressures are applied to the bilinear, quadrilateral faces, though it was not studied in that research. Thirdly, the problem was analysed with a series of meshes consisting of LT elements. The hoop stress profiles from one of the analyses are plotted in Figure 3(b). There is complete agreement between the methods with regard to LT elements as the global stiffness matrices and global force vectors constructed by the two methods are identical. This is a consequence of the linear nature of

5 14

Finite volumes for complex applications

Figure 4: (a) CPU times on a SPARC 4, 11OMHz. both the element concerned and the triangular faces over which the pressure is applied. It is possible to demonstrate this equivalence analytically [Tay96] by extending to three dimensions, a two dimensional analysis which has been applied to elastic problems involving linear triangular elements [OCZ94]. Finally, the methods were compared with regard to computational cost. Considering LT elements, as the matrices are identical and symmetric a conjugate gradient method (CGM) is applicable in both cases. As illustrated in Figure 4(b), the FV method (FV-CGM) requires more CPU time than the FE method ( F E C G M ) even when the same linear solver is employed. This is expected as the FV method visits six integration points, while the FE method visits a single Gauss point when adding contributions to the linear system of equations [Tay96]. Considering TLH elements, the geometrical nature of this validation problem prohibits an orthogonally assembled mesh. Hence, for the FV method a bi-conjugate gradient method (Bi-CGM) is required due to the asymmetric nature of the coefficient matrix obtained [Tay96]. Conversely, for the FE method a CGM is sufficient as the matrix obtained is symmetric. These requirements agree with the discussions in the previous section. As illustrated in Figure 4(a), the FV method (FV-BiCGM) requires approximately twice the CPU time as the FE method (FE-CGM). This is also expected due to the computational requirements of the two different linear solvers employed. Also for TLH elements, the FV method visits twelve integration points per element, while the FE method visits eight Gauss points per element. Hence, it can finally be concluded that any improvement in accuracy obtained by employing the vertex-based FV method must be offset against the extra computational cost required. Bibliography [BC95]

C. Bailey and M. Cross. A finite volume procedure to solve elastic solid

Fields of application

5 15

mechanics problems in three dimensions on an unstructured mesh. Int. Journal for N u m . Methods i n Engg., 38:1757-1776, 1995. [DM921

I. Demirdzic and D. Martinovic. Finite volume method for thermo-elastoplastic stress analysis. Computer Methods i n Applied Mechanics and Engzneering, 109:331-349, 1992.

[FBCLgl] Y.D. Fryer, C. Bailey, M. Cross, and C.-H. Lai. A control volume procedure for solving the elastic stress-strain equations on an unstructured mesh. Appl. Math. Modelling, 15:639-645, 1991. [Fem] [HH95]

Femview Ltd., Leicester, UK. F E M G E N / F E M V I E W .

J.H. Hattel and P.N. Hansen. A control volume-based finite difference method for solving the equilibrium equations in terms of displacements. Appl. Math. Modelling, 19:210-243, 1995.

[Hi1501

R. Hill. The Mathematical Theory of Plasticity. Clarendon Press, Oxford, UK, 1950.

[Hir88]

C. Hirsch. Numerical Computation of Internal and External Flows: Fundamentals of Numerical Discretisation, volume 1. John Wiley and Sons, 1988.

[OCZ94]

E. Onate, M. Cervera, and O.C. Zienkiewicz. A finite volume format for structural mechanics. Int. Journal for Num. Methods i n Engg., 37:181-201, 1994.

[OH801

D.R.J. Owen and E. Hinton. Finite Elements i n Plasticity: Theory and Practice. Pineridge Press Ltd., Swansea, UK, 1980.

[Pat801

S.V. Patanker. Numerical Heat Transfer and Fluid Flow. Hemisphere,

[Per661

P. Perzyna. Fundamental problems in visco-plasticity. Aduan. Appl. Mech.,

Washington DC, 1980. 9:243-377, 1966. [Tay96]

G.A. Taylor. A Vertex Based Discretisation Scheme Applied to Material Non-linearity within a Multi-physics Finite Volume Framework. PhD thesis, The University of Greenwich, 1996.

[Whe96]

M.A. Wheel. A geometrically versatile finite volume formulation for plane

[Whe99]

M.A. Wheel. A mixed finite volume formulation for determining the small strain deformation of incompressible materials. Int. Journal for N u m . Methods i n Engg., 44:1843-1861, 1999.

[ZC74]

O.C. Zienkiewicz and I.C. Cormeau. Visco-plasticity-plasticity and creep in elastic solids-a unified numerical solution approach. Int. Journal for Num. Methods in Engg., 8:821-845, 1974.

[Zie95]

O.C. Zienkiewicz. Origins, milestones and directions of the finite element method - a personal view. Archives of computational methods in Engg., 2:l-48, 1995.

[ZT89]

O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method: Volume I : Basic Formulation and Linear Problems. Magraw-Hill, Maidenhead, Berkshire, UK, 1989.

elastostatic stress analysis. Journal of strain analysis, 31(2):111-116, 1996.

Control volumes technique applied to gas dynamical problems in underground mines

Elena Vlasseva Associated Professor Department of Mine Ventilation and Labour Safety University of Mining&Geology "St. Ivan Rilski" Sofia 1100, Bulgaria

ABSTRACT. Paper presents an application of Control Volumes' Method in the field of underground mining. Modeling deals with noxious gas distribution (concentration) it1 time and space along mine roadway or network. Models take into account variable velocity, density and outside gas inflows in mass balance of airflow. Different source functions, which due to technological or accidental reasons, provoke occurrence of transient process, are presented in the paper. Numerical and computer models are verifed with real data, obtained unfortunately from great disaster, which took place in one Bulgarian underground coal mine in 1997. Models serve as a tool used to reveal the circumstances lead to that disaster. Modeling results might be also applied to help introduction into practice of engineering solution, which can be analyzed due to their practical implementation. Key words: control volumes; gas distribution; underground mining

1.

Introduction

Air flows in underground mines are of great importance for human lives underground. Normally flows transfer not only fresh air but different impurities, liberated from rocks or as a result of mining operations. Keeping of these impurities into safety limits is one of the main task for underground staff. Once liberated into underground air, gas is distributed along ventilation paths in difficult to predict ways and time duration. Gas might be observed on unlikely for common understanding places. By that reason preliminary knowledge about gas dynamics is extremely valuable. One possible approach might be real mine experiments. This approach however faced great difficulties due to problems with simulation of realistic gas liberation characteristics. Problems with control of such simulation cannot be neglected as well. All the above points show that mathematical and computer

5 18

Finite volumes for complex applications

modeling can be of favor in prediction of gas distribution along ventilation paths. They can reveal specific relationship between gas liberation and its influence on underground air flows.difficu1ties with control and safety of the experiment. In that connection mathematical and computer modeling in close with reality decsription is very important and liable technique. It can reveal specific relationships between gas source and its influence on ventilation system and its important parameters. The process of gas liberation and distribution in a single path and its mixing with fresh air is presented by convection-diffusion equation with source function, varaible velocity and density of air gas mixture. It is solved simultaneously with continuity equation. Control volumes method with exponential scheme is applied for numerical modeling.. Errors due to discterization and limits of method for this class of problems are outlined. Solutions for one ventilation path are addapted to a network modeling where interrelation between air flows with different characteristics are of a great importance to the whole process of safety working conditions in mine. Results of modeling are validated with practice in the following ways: > it was used as investigation tool for one methane explosion which took place two years ago in Bulgaria. Revealing of most likely reasons for its occurrence became possible; P it was also applied to in one case study for inertization with nitrogen of a typical mine configuration. This technical activity is performed to supress mine fire development.

1.

Governing equations and source functions

Let us assume mine roadway with characteristics shown on figure 1. Air flow is well developed in direction of coordinate s. Length of roadway (L) is times greater compared to its width or cross sectional area (f) and this give a ground to present mathematical model into one dimensional way. Independent variables are: s - along the length of mine roadway, m; 7 - time for tracing the process, s.

rr;i0 ulrl -

c . ~ )

I

-

.-

woo

c,ir)

C/O.

N

.. .

2

x,

-P

&'

-L

1

U

Figure 1. Physical model.

I

I

I _r 1

L

(-

5

r:

Figure 3. Discretization of calculation area

Known functions are: u ( s , ~ -) velocity of air gas mixture, rnls;

Fields of application

5 19

q(s,z) - gas liberation, m3/s; p(s,2) - density of air gas mixture, kg/m3. Unknown function is C(S,T) - concentration of gas impurity into air gas mixture. Gas distribution in thus describe physical model obeys mass conservation for gas impurity and air and equation of state for air gas mixture:

a~ +-~ ( P u )-- P, q -

as

a7

(1)

f~

a ( ~ C ) a ( ~ u C ) - D, 37 as P +

a2

(puc)

as2

-

P, (4 f~

(2)

(3) =In the above written equations index g referred to gas impurity, index m - to air gas mixture. Initial and boundary conditions are given with the terms: C(s,O) = Co(s) (4) Pm

ac I.;r =O

C(0, z) = CH( T ) ;

(5)

ax

qk qL=2 e

a

Source functions q(s,z) which present gas liberation or provocation of transient process are presented on figure 2. Their characteristics may change either in time or in space or combinations of both. They are chosen on typical behavior of gas sources: a - gas liberation from walls or during coal transportation; b, c - gas liberation from already mined areas; d - point gas source; e - gas liberation due to repeatable operations (diesel power, blast work); f - gas liberation from mined areas with increase of its surface; g - gas desorption from newly opened surfaces.

qk* qk qLrqL q l ~qL > b

f

L

C

d

L

g

h

Figure 2. functions

Typical

source

520

Finite volumes for complex applications

Combinations of above presented source functions are also possible. For istance: a + d - point gas source with constant emission rate in time; c + f - distributed along the length source with variable in time emission rate; b + g - linearly distributed source along the length with pick in time and then gradually decreasing emission rate; d + e - point source with periodically changeable emission rate. Presented on figure 2 source functions can be easily approximated with polinom or with sum of functions. Numerical treatment of above presented mathematical model - expressions (1-3) plus (4-5) makes possible taking into account all written into model variations.

2. Numerical and computer modeling 2.1. Schemes, mesh-type, discretization

Mathematical model (1-5) is solved by application of Control Volumes Method [PAT84]. Presented on fig. 1 physical model is transformed into regular mesh of control volumes (fig. 2) with size As. They are defined by mesh points and control volumes boundaries. Number of points are Concentration

C(S~,T~)is

defined

I:,;

N, = - + 1 . in

mesh

points,

air-gas

flows

?

Exponential profile [PAT84) for concentration variations is assumed. Following the above points numerical analogue, binding three adjacent points (W,P,E) can be written in the way:

where:

e D

e D

Upper index 0 referred to a previous time step. Boundary conditions (4-5) are transformed into numerical schemes in the same way by assuming first order boundary condition at left boundary (s=O) and second order - at the right boundary.

Fields of application

521

Full procedure on transformation of differential problem into numerical schemes is given in [VLA93]. Having expressions (7) for each three adjacent points linear system with three diagonal matrix for all points i ~ ( t N,) 1 is obtained. Its solution under TDMA gives concentration in each mesh point and in any time moment C(si,rj).

2.2. Approximation, stability, convergence, errors, application limits Numerical schemes (7) approximate differential problem (1-5). This statement was proved by application of Taylor's series. Approximation error is from first order in regard of T and from second order in regard of s. Numerical scheme (7) is absolutely stable. It was proved by applying Matrix criterion [SHISS] by presenting numerical scheme in way from one time layer to the next one and examining eigen vectors and their eigen values [SHI88]. Convergence is a concequence from approximation and stability, following Lax theorem [SHI88]. I I Applicability limits to gas dynamical problems are investigated. For most common parameters - velocities, lengths etc. error analysis was performed. On figure 4 relative errors (numerical/exact solution) in regard of

u AT As

Cu number ( CU = -).

lc-A

10.'

1

!i

Figure 4. Relative errors due to discretization

3.

umber of

mesh points, even limited to the minimal range (3) lead to 8% errors, which for purpose of gas dynamical problems is agreeable. Physically unlikely results has not been obtained.

Modeling validation with practice

Presented herewith mathematical model and its numerical interpretation reach their computer realization in both aspects: for a single roadway or set of sequential ones [VLA93]; for a complex network, where solutions for a single roadway were harmonized at places where they cross each other [STE87]. Modeling was applied in numerous engineering problems from practice. Two of them are presented in this paper: investigation of one methane explosion; technological solutions for inertization with nitrogen of fire zone. 3.1. Methane explosion

Finite volumes for complex applications

522

Severe methane explosion took place on the 1-st of September 1997 in "Ivan Russev" coal mine. Operations in one section were canceled due to annual holidays from IS' August to 1'' September. As a result great amounts of methane were accumulated. Degazation (transfer of this gas out of mine) was to be performed. Mine does not have remote control system and no data (besides miners1 evidences) was available. As a result of incorrect activities of miners, performing degazation, explosion took place causing death for more than 10 workers.

/if

to Ill section

Q3

+

Figure 5. Path in mine for distribution of methane during degasing Investigation about circumstances [MIC98] was performed by application of

U

428

o

t

o

2

J

~

@

Y

11mrm

3

m

m

m

w

1

l I

10

'I

10

75

93

l l m e mln

Figure 7. Concetration at the entrance of Figure 6. Air volumes at the entrance of observed path observed path Changes in cross sections can be seen there. Transient process was initiated by variable air flow (figure 6) and methane concentration (figure 7) at starting point 0. They reflect workers actions, such as: variable cross-sections along the route (fig. 11); assembling of ventilation curtain to direct higher air flow towards the gassed section; switching on/off of booster's fan operation, causing changes in ventilation conditions (air volumes

Q, ,Q2 and Q,

Fields of application

523

variable methane concentration (C2) of in-flowing air into 503 crosscut (Q2)

70 6~

so h

40

:, ,'I I

t

20

10

Or,

ZO

1)

U1

-

a

TR.

KO

lcx,

120

Explosion took place between points 5 and 6 and modeling results (figure 8) show the same - high methane concentration with more than 7 minutes duration stay at the place of explosion. Unfortunately ignition source was also available.

Figure 8. Concetration in time of observation in some points of path

3.2. Inerting with nitrogen of fire zone In some cases during mine operations evidence for [MIC97] development of mine fire can be observed. Then one of anti-fire measures is to inertize atmosphere at a danger zone so that oxygen does not be available to support burning. The model presented herewith can deal with one impurity into air-gas mixture. Inerting of air however presumes more than one ipurity, namely: methane inflow from mined zone (points 5-6 on figure 9) and from mine workings (points 1-2-3-4 on figure 9); oxygen from ventilation flow and from injected technical nitrogen; nitrogen from air and from injected technical nitrogen. In order to evaluate concentration of the above mentioned three gas components via a model constructed for a single component, the author has applied consecutive diffusion mixing. It pressumes appropriate definition of transporting medium and impurity as well as suitable presentation of gas sources ~ ( s , z.) Calculational passes three stages (methane release and distribution, nitrogen outflow from gob area, nitrogen injection at a given place in the panel and its further distribution in the area which must be inert). For any of these stages computer modulus were developed - METHANE, INERT-GOB and NITRO. Common input data for the three computer programs are geometric characteristics of mine workings (fig.9). The three programs interact and their incorporating in the total inverting strategy makes it possible the composition of general program procedure INERTIZATION r & 12 4 3 . 2 Numerous solutions were performed, 4 corresponding to specific fire situations. rtant parameters were needed to inert the observed 4

4".

34

3

Figure 9. Object for inertization

concentration of flammable gas in all points of observed area (this

Finite volumes for complex applications

524

information is very important in order to keep the atmosphere out of explosion); effectiveness of operations performed on fire supression. On figure 10 is shown methane concentration along the path with length 1500 m. Figure 11 shows time delay in inertization from point 2 and 3.

--

0

U E. u

I2

h OU

-

--

Elapsed T i m e , m l n

i)

Figure 10. Time delay in inertization

h)

L e n g t h (I-12-2-3-34-4), m

Figure 10. Methane concentration

4. Bibliography

[VLA93]

[MIC98]

[MIC97]

[PAT841 [SHI88] [STE87}

Vlasseva E.D., Mathematical Modeling of Convection-diffusion Processes in Underground Mines, Ph.D. Thesis, UMG, Sofia, 1993, pp. 200 Michaylov M.A., E.D.Vlasseva, Simulation Analysis of Methane Explosion, Second International Symposium on Mine Environmental Engineering, July 29-31 1998, Brunel University, UK,p. 1-16. Michaylov M.A., E.D.Vlasseva, Modeling of Preventive Nitrogen Inertization in Underground Mines, 1.5'~Mining Congress of Turkey, 6-9 May 1997, Ankara, pp. 203-210. Patankar S., Numerical Methods in Heat Transfer, Sovremennoe Mashinostroene, Moskow, 1984, p. 150 (Russian translation) Shi D., Numerical methods in Het Transfer Problems, Moskow, Mir, 1988, p. 544 (Russian translation) Stefanov,T.P., E.D.Vlasseva, E.E.Arsenyan, Unsteady Gas Flows in Mine Ventilation Networks, 22 International Conference of Safety in Mine Research Institutes, Beijing, China, oct.1987,pp. 115-124.

Simulation of salt-fresh water interface in coastal aquifers using a finite volume scheme on unstructured meshes B. Bouzouf, D. Ouazar LASH, EMI 14 Av. Ibnsina Rabat, Maroc I. Elmahi IVG, University of Duisburg, Germany

ABSTRACT This paper is devoted to the numerical study of seawater intrusion into coastal aquifers. The cell-centered finite volume method is adopted here to solve the set of simultaneous partial differential equations describing the motion of saltwater and freshwater separated by a sharp interface. These equations are based o n the Dupuit approximation and are obtained from integration over the vertical dimension. I n order to have fEexibility upon complex configurations domain, non structured grid meshing is utilized. To approximate the diffusion fluxes, GreenGauss type reconstruction, based on Diamond cell and least square interpolation, is performed. The model is first validated using academic test case studies with known close from solutions. A real case study concerning the Gharb aquifer in North West of Morocco is carried out to show the overall trend of saltwater fronts.

Key Words: Coastal aquifers, seawater intrusion, finite volumes, unstructured meshes, Green-Gauss reconstruction. 1. Introduction The modelling of groundwater in coastal aquifers is an important and difficult issue in water resources. The primary difficulty resides in efficient and accurate simulation of the movement of the saltwater front. Freshwater and saltwater are miscible fluids and therefore the zone separating them takes the form of a transition zone caused by hydrodynamic dispersion. For certain problems where the transition zone is relatively small compared to the aquifer extent and thickness, the simulation can be simplified by assuming that two fluids are immiscible and separated by a sharp interface (sharp interface model). This last assumption, together with the Dupuit approximation, permits the integration of the equations in the vertical direction [BEA 791. The objective of this paper is to present a cell-centered finite volume based approximation to calculate the position of the sharp interface. This class of methods

526

Finite volumes for complex applications

is becoming one of the commonly used techniques for partial differential equations in engineering calculations and computational physics. Their popularity is due mainly to their ability to faithful to the physics conservation and the possibility of solving the problems on complex geometries. The diffusion contributions here are approximated by using Green-Gauss type interpolation. This technique is found to be very robust, it can be used on general mesh, not satisfying necessarily Delaunay condition on the triangulation. Time integration is performed by an explicit Euler scheme in order to keep the memory requirement reasonable. 2. Mathematical Model We assume here that the saltwater and freshwater are separated by a sharp interface, thus two domains are considered. For each flow domain the equation of continuity may be integrated over the vertical dimension reducing the determination of the position of the interface to a 2D problem. The system of equations can be written as follow ([ESS 901):

o f and R,

represent here the fresh and salt water flow domain respectively, K f z and K S z (respectively K f y and K S y )are the hydraulic permeabilities in the fresh and salt water in x-direction (respectively in y-direction), h f and h, are the heads, B f and B, are the thickness of fresh and salt water zone and n is the porosity. Pf , where p f and p, are the specific weights in fresh We note also by b = Ps - P f and salt water, and by 0 for confined aquifer 1 for unconfined aquifer Invoking continuity of the pressure at the interfaces, the interface elevation can be calculated from the freshwater and saltwater heads by

The system (1) represents two coupled parabolic partial differential equations which should be solved simultaneously for the freshwater head ( h f ) saltwater head (h,). Once these values are known, the interface elevation ( C ) can be obtained from (2). The set of the equations (1) can be written in the vectorial form:

Fields of application

with W = ((Sf Bf

+ n(a. + 6))hf - n ( l + 6)h,,

(S,B,

+ n ( l + 6))h,

-

527

nbhf)t

3. Finite volume discretization To solve the system of equations (3) we have considered a triangular cellcentered finite volume formulation ([EBVGPH 9911, where the state variables Wp are the average values for the cells a t time level n:

Integrating eq. (3) on a control volume Ci yields in explicit formulation:

w;+'

=

w:+

At

[R,(Wn) n,

+ R, ( W n )n,] d a + Area(Ci) S(W:)

(4) where A t is the time step. The discretization of equation (4) requires the approximation of terms such as

l,,

( B ~ K L , ,,do ~ )

where 1 = f , s and

and

l,,(& 2) K'

n~ do

rij is the interface separating two cells Ci and C j .

Y. Coudiere et al. [CVV 961, have studied an elliptic problem -div(AVu) = f ulr = 9

in St C lR2 over I' = ail

(5)

Where A is a symmetric definite positive matrix with coefficients a,j in C1 (01, f E CO(R) and g E C2(I'). They have used a Green-Gauss type interpolation to construct the gradients at the interfaces of the mesh. The gradient on each edge is approached by the Green theorem and then a first order Gauss quadrature formula, for which requisite values at the vertices P are interpolated from the states on the neighbourhood of P. The weak consistency of this scheme was

528

Finite volumes for complex applications

proved under some assumption on the weights of the interpolation. We took inspiration from this scheme for devising our numerical procedure and discretize the diffusive contributions. We begin firstly by writing

One constructs the co-volume Cdec centered at the interface F i j and connecting the barycenters Gi and G j of the triangles that share this edge and the two endpoints N and S of rij (see figure 1).

Figure 1: Diamond shaped co-volume

ah1 To calculate - - I r i j , the divergence theorem is applied to the co-volume ax surrounding rij,which gives

Cdec

represents an edge of the co-volume Cdec and n,, is the axial component of the outward unit normal to E .

E

If we note by e = [ N 1 N2], , One can write also

Where h l N , and hlN, are respectively the values of hl on the nodes Nl and N2 of the edge E . The data a t the centers Giand G j are known exactly while the data at the vertices N and S must be determined by some interpolation procedure. For one node P

Fields of application

529

of the mesh, one utilizes a linear approximation v of hl on the set of cells which share the vertex P.

Where V ( P )is the set of triangles K surrounding P, h l K the head a t the center of triangle K and Q K ( P )are the weights of the interpolation corresponding to the node P. In order to ensure the consistency of the scheme, the weights of the interpolation are calculated using a least square procedure (see [CVV 961 for details). 4. Model validation To verify and validate the numerical solution obtained from the finite volume model, numerical simulations have been compared to existing analytical solutions. 4.1 Steady state

Two cases have been checked: confined and unconfined aquifer. For both of them the initial values of hf and ( are arbitrarily fixed. The analytical solutions are as follows: Unconfined aquifer ([VER 681, [VN 751):

with qo = 1 0 m 2 / d a y ,

P

= 0.741 and K = 6 0 m l d a y .

Confined aquifer ([GLO 591, [RH 621):

with go

=

5 m 2 / d a y and

zt = - l o r n .

4.2 Unsteady state Keulegan [KEU 541 presented an analytical solution for the interface in a confined aquifer of uniform thickness:

with D = 1 0 m

,

n = 0.3 and K = 39.024mlday.

530 Finite volumes for complex applications

Figure 2: Comparison of analytical and numerical solutions for unconfined aquifer.

Figure 3: Comparison of analytical and numerical solutions for confined aquifer.

Figure 4: Comparison between analytical solution and finite volume method.

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53 1

The numerical solutions are in good agreement with the analytical solutions as depicted in figures (2), (3) and (4). 5. Application to the Gharb basin, Morocco

We have applied the finite volume based model to the Gharb aquifer which is located in North West of Morocco. The surface area of the coastal Gharb is estimated to about 4000 km2. It belongs to the structural domain of the Morocco Atlantic plain.

Location of the Gharb basin In figure ( 5 ) , the areas of pumping in the Gharb aquifer are depicted. Figure (6) shows the actual front saltwater corresponding to pumping schemes of Figure (5). 6. Conclusion

Characterization of certain coastal aquifer systems may be accomplished by assuming that saltwater and freshwater are separated by a sharp interface. Invoking the Dupuit assumption and performing a vertical integration results in quasi-three-dimensional, the equations may be solved to give freshwater head, saltwater head and interface elevation. Cell-centered finite volume scheme on a unstructured mesh is used to approximate the partial differential equations. Comparisons of the finite volume approach adopted in this paper, with known analytical solutions have shown close agreement. The model was also applied to a real case concerning the Gharb aquifer in North West of Morocco.

BIBLIOGRAPHY [BEA 791

BEARJ., Hydraulics of groundwater, McGraws-Hill, New York, 569 pages, 1979.

[CVV 961

COUDIERE Y ., VILAJ . P. A N D VILLEDIEU P ., Convergence of a finite volume scheme for a diffusion problem, F . Benkhaldoun

532

Finite volumes for complex applications

3

4

.

8 4.2 4.4 4 . 6

1

3.8

4

4.2 4.4 4.6

I o5 I o5 x(m> Fig.5 Areas of pumping in GHARB Fig.6 Simulated freshwater and saltwater xcm)

aquifer

interface

and R. Vilsmeier eds, Finite volume for complex applications (Hermes, Paris), pp. 161-168, 1996. [EBVGPH 991

ELMAHII., BENKHALDOUN I?., VILSMEIERR., GLOTHO., PATSCHULL A. A N D HANELD., Finite volume simulation of a droplet pame ignition on unstructured meshes, J . of Comput. and Appl. Math., Vol 103, 1, pp. 187-205, 1999.

[ESS 901

ESSAIDH. I., A quasi-three-dimensional finite difference model to simulate freshwater and saltwater flow i n layered coastal aquifer systems, U.S. Geological survey Water-Resources Investigations, Report 90-4130. Menlo Park, California, 1990.

[GLO 591

GLOVERR. E . , The pattern of freshwater pow i n a coastal aquifer, J . of Ground Water Resour., 64, pp. 439-475, 1954.

[KEU 541

KEULEGAN H. G., A n example report on model laws for density current, U.S. Natl. Bur. of Stand., Gaitherburg, Md, 1954.

[RH 621

RUMMERR. R. AND HARLEMAN D. R., Intruded saltwater wedge i n porous media, U.S. Geol. Surf. Prof., Paper 450-B, 1962.

[VER 681

VERUIJTA., A note o n the Ghiben-Herzberg formula, IASH bull. 13, pp. 43-45, 1968.

[VN 751

VAPPICHA V. N. AND NAGARAJA S. H., Steady state interface i n coastal aquifer with a vertical outfEow face, National Symposium on Hydrology, Rurkee, India, 1975.

Progress in the flow simulation of high voltage circuit breakers X. Ye, L. Miiller, K. Kaltenegger and J. Stechbarth ABB High Voltage Technologies Ltd., 5401 Baden, Swilzerland ABSTRACT In this paper progresses in thephysical and numerical modelling, which lead to inzprovement in the accuracy and capability of simulation for capacitive switching design oj circuit breakers, are introduced Numerical results and measurement results are conzpared and discussed One inzportant progress lies in the successful treatment of the artificial viscosity. To maintain the numerical stability but at the same time to keep the artrJicial viscosity so small that the physical viscosity is not distorted, the upwind biasing essentially local extremum diminishing (ELEDJ scheme has been adopted and improved. Another progress is associated the moving grids technique, where additional terms have been added to the governing equations for the moving grid without deformation and new grid lines are added or renzoved in the deformed moving grid by the solver in the progress of the calculation. The Chimera boundary interpolation method is used to enable the communication between two blocks with relative motion. Key Words: circuit breaker, moving grids, upwind biasing scheme, Chimera boundary

1. Introduction

In a high voltage circuit breaker a gas with good dielectric and thermodynamic properties such as SF6 is used to extinguish the electric arcing which occurs as electric contacts move apart. The current interruption can only then be realised. The capacitive switching, i.e., the current interruption at high voltage but with low current, represents one important case among various cases of tests and design. The ability to perform capacitive switching without electric breakdown is one of the defining parameters for the speed of the circuit breaker and therefore strongly cost relevant. Furthermore, only the density of gas @) and its electric field strength (E) play an important role in gas breakdown between the contacts of the circuit breaker (s. Fig. 4). As the criterion of the ratio of E/p for gas breakdown is well known, CFD can be employed to simulate the flow field and to subsequently produce the distribution of gas density in the circuit breaker during the design. It is obvious that for such simulations, a high degree of accuracy of flow calculation is required.

534 Finite volumes for complex applications

Since later 1980's researchers and engineers have begun to apply CFD tools in the development of circuit breakers. Most of them, e.g. [CLA 971, concentrated their efforts on using the flow simulation with implemented arcing models to investigate the ground flow effects in circuit breakers, e.g., pressure build-up in pressure chamber, plasma jets, and to identify the limit and capability of CFD tools. Their efforts to verify CFD tools were, however, constrained mainly in the comparison of the simulated pressure build-up in the pressure chamber with experiments. There are only few works (s. [TRE 911) done for enhancing the accuracy of CFD tools for predicting dielectric strength in capacitive switching and for verification of CFD tools in the regions of nozzles and electrode contacts. As a result, the ability and accuracy to predict the dielectric strength of a circuit breaker with CFD tools have been not satisfactory. The following two points remain unclear. 1) How can the moving parts be treated correctly and which numerical methods are to be introduce to obtain a sufficient high accuracy? 2) How can a CFD tool be verified for predicting dielectric strength in capacitive switching? In this paper, these questions are addressed, with concentration on the simulation of cold SF6 gas with moving electric contacts and further on the prediction of the dielectric strength of a circuit breaker during the capacitive switching. The cold gas simulation is not as trivial as to be anticipated, because the low energy level of cold gas forces a CFD code to consider, with sufficient accuracy, all macroscopic and microscopic flow effects, examples of which are: flow separation, influence of wall and viscose layer and their transient evolution, turbulence transport and suck effect caused by moving contacts. Further, to guarantee the numerical stability, artificial viscosity must be introduced, and hence a sophisticated scheme must be used to keep the artificial viscosity so small that it can maintain the numerical stability effectively but does not confuse the physical viscosity on a viscose layer. A sophisticated moving grids technique has to be introduced both for obtaining a high accuracy and for performing an efficient computation. In section 2, the mathematical models and numerical methods, such as ELED scheme, moving grids technique and Chimera boundary, are introduced. In section 3, examples are presented and discussed for flow simulation where the code verified through the comparison with experiment. Subsequently, the method and example are introduced for coupling of the flow field and the electric-static field in circuit breakers with the consideration of the influence of roughness. Finally, in section 4, our results are concluded with further improvement suggested. 2. Mathematical models and numerical methods

2.1 Mailtematical models

For the fluid flow in a circuit breaker it is necessary to use the complete NavierStokes equations in their time averaged form, i.e. the so called Reynolds equations

Fields of application

535

with an adequate turbulence model. The fluid flow will be then governed by the Navier-Stokes conservation equations for mass, momentum and energy. These equations have the differential form as shown in eq. [I].

with the variables

with the shear stress T and heat flux J a s follows:

where p,, p, are molecular and turbulent viscosity, A heat conduction coefficient, O temperature, Pr Prandtl number, V Nabla operator, Re Reynolds number, W velocity vector. These equations can be transformed to curvilinear coordinate system with the transformation {={(x,y) and v=q(x,y). To close the equation system, the standard k-E turbulence model (s. [LAU 741) was used to obtain the turbulent viscosity p,.

2.2 Numerical methods

The basic equations are solved with a multi-block finite volume Runge-Kutta multisteps time-marching method. The code (HT206) was previously applied for fluid flow in turbo systems (s. [SCH 911 and [SCH 981) and has been extended for the development of circuit breaker. The numerical methods of the code, which are relevant to the flow simulation of circuit breaker, are described as follows: 2.2.1 ELED scheme

The spatial discretisation of the present code is based on a high order non-oscillatory scheme, which consisting of central discretisation and artificial viscosity in the following form (s. [JAM 8 11):

536

Finite volumes for complex applications E(4)

where q denotes the variables in the equation [I] and E and I+: are the coefficients of the scheme. If the formulation of these coefficients is taken from [JAM 811, it is then the classic Jameson-Schmidt-Turkel (JST) scheme. A more advanced formulation of these coefficients is introduced in the present code, that is the essentially local extremum diminishing (ELED) scheme of [JAM 941: ,+;

with

Q I + - , =R(Aq I + :,,Aq 1- , ) '

C

R(% v)

a

2

1

The valuation of the wave speed I + , in 4 and q direction here will be treated generally for variable q overall in the flow field as

where f and g are convection terms in 5 and q direction, U and V are defined in eq. [3] and [4]. Based on our experience the numerical constants r, E, C, and C2 can be , C2=1.5 - 8.0. This formulation is defined to be selected as ~ 1 . 5~, ; 1 0 " ~CI=2.0, scheme which satisfies the condition that in the limit as the mesh width Ax + 0, local maxima are non-increasing, and local minima are non-decreasing.

2.2.2 Moving grids technique To simulate the flow field containing electrical contacts moving during the current separation, two types of moving grids are generally used. The type I moving grid moves only its form and the number of grid will be not changed as shown in Fig. la. In contrast with the type 1, the type I1 moving grid will be expanded or compressed by adding new grid lines to or removing existing ones from left orland right side as shown in Fig. lb. Some methods have to introduced to treat both types of moving grids correctly. additional grid ITS inserted Chimera

a. Type I: whole grid moves

b. Type II: only the grid line on grid side moves

Figure 1. Two types of moving grids

Fields of application

537

2.2.2.1 Treatment of moving grid type I

In the type I moving grid, the variables will be kept in the same cells during the movement at a time step. This will produce error if no additional measures will be taken, because the positions of these values in the flow field have been changed through the moving of the grid cells. To avoid such errors, additional terms in eq. [ l ] must be introduced. In the case that a block of grid moves with a speed of x, and y, as its x and y components, the velocities U and V in the curvilinear coordinate system with 5 = t(x,y,t) and q = q(x,y,t) will then be (s. [STE 781):

cr

tr

where =-xkx - yJy and =-xJX - y J y , then the convection term in eq. [l] after the transformation to the curvilinear coordinate system is

2.2.2.2 Treatment of moving grid type II

For the type I1 moving grid, the movement only involves the two side grid lines, i.e., almost all the cells of this grid are not in motion, therefore, there is no need to apply the additional terms. However, because of the change in the cell size at every time step, there would be implicit unphysical energy and mass loss as grid compresses and unphysical energy and mass increase as the grid expands, if the values of energy and mass after the change of cell size were taken simply from the old before the change of cell size. Therefore, sophisticated treatment of the value of energy and mass is very important to get the accuracy of calculation. As grid expands or compresses, the value of the density and the total energy in the changing cell after the change of cell size will be then

where x, ,-xi is the cell size, new and old denote the values after and before the change of cell size.

538

Finite volumes for complex applications

2.2.3 Chimera boundary

Because of the movement relative to neighbouring grids, the moving grids will have discontinuity of grid line at interfaces with their neighbouring grids. Hence, the Chimera interpolation method (s. [STE 871) will be applied to enable the information exchange between two grids with discontinuity of grid lines. The basic concept is that two layers of ghost cell of a block will overlap with the neighbouring block. To obtain values in the ghost cell, the values of the nearest cell centres of the neighbouring block are interpolated to the cell centre of the ghost cell (s. Fig. 2). The values in the ghost cells can then be used for flux building. Same as the central discretisation, the Chimera interpolation is non-conservative, however, the upwind character is obtained through the ELED scheme.

Figure 2. Overlapping blocks for Chimera interpolation

3. Results and Discussion

3.1 Improvement of numerical scheme

To investigate the improvement of the numerical scheme especially in its capability to resolve the flow discontinuity and to keep a small disturb in shear layer, two test cases are considered.

a) Shock tube problem

b) Fully developed turbulent pipe flow

Figure 3. Two test cases for numerical schemes

Fields of application

539

The first test case is the shock tube problem as described in [SCH 981. In Fig. 3a, the pressure distributions calculated with the JST and ELED schemes are compared with the exact solution of [HIR 901 at the time t=6.1 ms. As can be seen, the scheme of JST oscillates strongly just before the shock front whereas the ELED scheme resolves the discontinuity without oscillation. The use of ELED scheme resulted in only 7 percent increase in the computing time. The other test case is the fully developed turbulent pipe flow as shown in Fig. 3b, where the calculated fully developed radial velocity distributions, which is normalised by the velocity in the middle of pipe Urn,are compared with the experiment results of [NIK 321. It can be seen that the near-wall shear layer is strongly disturbed by the JST scheme, while the result of ELED scheme agrees very well with the experiment.

3.2 Verification with experiments of a circuit breaker

Fig. 4 shows a schematic diagram of the core part of a circuit breaker. To interrupt the current, the electrical contact "finger" will be moved together with isolating and auxiliary nozzles toward left, while the electrical contact "plug" will stay still. The gas in the pressure chamber (not illustrated) will be then compressed through this motion and flows from left into the isolating nozzle, there will be a highly transient and transonic flow. from pressure chamber

-------

\ moving elec. contact (finger)

\ fixed elec. contact (plug)

Figure 4. Schematic diagram of the simulated circuit breaker To verify the code for the capacitive switching design, measurement of static pressure on the 6 points showed in the Fig. 4 was carried out for cold. These 6 measured points are located in different flow regions. O and O are in the diffuser region with flow separations; O is in the geometric throat; 0 and O are in the channel flow region with boundary layer character; 8 is behind the shock front and presents the pressure lost over the shock. Therefore, the values at these 6 positions reflect all flow details. For the calculation, an inlet was defmed as shown in Fig. 4 and the pressure measured in the pressure chamber was used as the inlet boundary condition, so that the leakage in the pressure chamber which is difficultly to be estimated can be ignored. In Fig. 5 the simulated pressure distributions at O to 8 are

540

Finite volumes for complex applications

time in ms

0.0

12.0

24.0

36.0

time in ms

48.0

time in ms

0.0

12.0

24.0

36.0

time in ms

time in ms

48.0

0.0

12.0

24.0

Figure 5. Comparison ofpressure in circuit breaker: simulation, 0 measurement

-

36.0

time in ms

49.0

Fields of application

541

compared with the measurement. The simulation results agree very well with the measurements. The discrepancy of P5 is caused by the selection of the inlet where the boundary layer begins downwards from the start point in the reality.

3.3 Coupling of the electrical andflow fields Based on the test results on a circuit breaker where breakdown voltages were measured in dependence of travel positions, the present code can be verified through comparison of the ratio E/(pkJ with the critical value of 1480 kvm2/kg for 6 bar absolute filling pressure of SF6 from the streamer theory (s. [BEY 861). In this case E/(pk) must be higher than the critical value, where k, is a roughness factor which accounts for the microscopic effects which intensify the local field strength and is obtained from an semi-empirical function of local density and roughness. The electrical field strength E was calculated with the ABB electrical field program ACE. The gas density p was obtained from the flow simulation with the present code. Fig. 6 shows the distribution of E/(pkr) in the circuit breaker. The maximum of this ratio lies on the surface of the plug contact and it overruns the critical value slightly. The simulation results correspond very well with measured breakdownlhold values of the applied voltage observed in measurements.

Figure 6. E/(pkJ distribution and its maximum at trave1=115 mm

4. Conclusion

The progresses in the numerical methods, including numerical scheme and moving grid technique, have been made, leading to the development of the code presented in this paper which is able to fulfil the requirements of capacitive design of high voltage circuit breaker. The effectiveness of the code is confirmed through the following facts: (1) The adoption of the more advanced ELED scheme resolves flow discontinuity efficiently; (2) the physical viscosity is not confused; (3) the calculation results agree very well with the measured results; (4) the calculated

542

Finite volumes for complex applications

density w a s coupled with electrical field strength, and the resulted ration o f W(p kJ predicted the gas breakdown correctly. T h e code will b e further improved and developed, in particular, arcing model will b e implemented and verified, so that it can b e used t o predict the dielectric strength under high temperature a n d pressure.

5. References [BEY 861 Beyer. M., Boeck, W., Moller, K., Zaengl, W.: Hochspannungstechnik, SpringerVerlag, Berlin Heidelberg New York, 1986 [CLA 971 Claessens, M., Moller, K.. Thiel, H.G.: A computational fluid dynamics simulation of high- and low-current arcs in self-blast circuit breakers, J. Phys. D: Appl. Phys. 30, p. 2899-2907, 1997 [HIR 901 Hirsch, C.: Numerical computation of internal and external flows, Vol. 2, John Wiley & Sons, 1990 [JAM 811 Jameson, A,, Schmidt, W., Turkel, E.: Numerical solutions of the Euler equations by finite volume methods with Runge-Kutta time stepping schemes, AIAA paper 8 1-1 259. January, 198 1 [JAM 941 Jameson. A,: Analysis and design of numerical schemes for gas dynamics 1: Artificial diffusion. upwind biasing, limiters and their effect on accuracy and multigrid convergence, Int. J. of Computational Fluid Dynamics, August, 1994 [LAU 741 Launder, B.E., Spalding, D.B.: The numerical computation of turbulent flows, Computer Methods in Applied Mechanics and Engineering, Vol. 3, p. 269-289, 1974 V I K 321 Nikuradse. J.: GesetzmilPigkeit der turbulenten Stromung in glatten Rohren. Forsch. Arb. 1ng.-Wes. Heft 356, 1932 [SCH 911 Schafer, 0 . : Application of a Navier Stokes Analysis to turbomachinery bladecascade flows, 1 9 ' ~International Congress on Combustion Engines, CIMAC, Florence, 1991 [SCH 981 Schafer, 0. et al: Last advances in numerical simulation of aerodynamic forces on turbine blades of turbochargers for pulse charged engines, 22nd CIMAC , International Congress on Combustion Engines, 19-21 May , Kopenhagen , 1998 [S'FE 781 Steger, J.: Implicit finite-Difference simulation of flow about arbitrary twodimensional geometries, AIAA Journal, Vol. 16, No. 7; July. 1978 [STE 871

Steger, J., Benek, J.: On the use of composite grid scheme in computational aerodynamics, Computational Methods in Applied Mechanics and Engineering, Vol. 64, NO.1-3, 1987

[TRE 911 TrCpanier, J.Y. et al: Analysis of the dielectric strength of an SF6 circuit breaker, IEEE Transaction on Power Delivery, Vol. 6, No. 2, April, 1991

River valley flooding simulation

Francisco Alcrudo Area de Meccinica de Fluidos Maria de Luna, 3 C P S - Universidad de Zaragoza 50015 Zaragoza, S P A IN

--

ABSTRACT

D a m break flood wave propagation along a reach of a river valley located i n the Italian Alps is mathematically modeled with package S W Z D that solves the nonlinear Shallow Water equations. Simulation results are compared with data obtained from a physical model of the river valley operated by ENEL (Italy). The dificulties encountered during the modelisation process and the solutions adopted are explained in this paper. K e y Words: D a m break, Flood, Shallow Water, Physical Model

1. Introduction

Considerable efforts are being presently devoted to the validation of numerical models describing dam break flows, mainly due to the need for modern risk assesment and mitigation tools. Real life experimental data concerning actual dam break or severe flooding are very difficult t o obtain because of the unpredictable nature of the phenomenon. However, measurements obtained from reduced scale physical models can provide excellent validation information because the experimental conditions can be more precisely defined. The work reported in this paper colicerns the comparison of the simulation results obtained with SW2D program [ALC92] with measurements of the flooding experiments carried out by ENEL (Italian Utility Company) in a reduced scale physical model of the Toce river. The physical model is some 50m long by l l m wide and is built mainly in concrete (see figure 1). It represents a 5km long reach of Toce river which is located in the italian Alps.

544

Finite volumes for complex applications

*-

4b

TOCE RIVER VALLEY "

,-

--,

(Physical Model 1:100)

9nr'

Figure 1: Digital Terrain image of Toce river valley physical model (ENEL) The model reproduces many details of the actual valley geometry including the river bed and some villages, hydraulic structures and a reservoir located in the middle of the reach that depending on the intensity of the flood is overtopped and eventually filled with water across its embankments. The upstream end of the river reach model is connected to a small water tank fed by a hydraulic pump. Flooding is initiated by starting the pump that rapidly fills up the tank, overtops the entrance to the reach and rushes downstream. The pump capacity is such that the process takes place very rapidly thus simulating an abrupt irruption of water into the valley model. ENEL personnel located water stage probes at 32 different positions in the model valley. Among them one in the river bed at the entrance section and another one in the middle of the feeding tank that can be used to impose the boundary conditions together with the pump discharge versus time that was also recorded. Experiments were carried out for two flooding intensities: The first one such that no overtopping of the reservoir takes place (peak discharge of 0.21 m3/s)and the second one with reservoir overtopping (peak discharge of 0.36 m3/s). Measured water stage readings at several probe locations were compared to water levels obtained with SW2D model for different friction coefficients and flood intensities and overall satisfactory agreement was found. The physical model geometry was distributed by ENEL as a Digital Terrain Model (DTM) covering the model area at regular intervals of 5cm, therefore specifying the bottom elevation function z ~ ( x9), in some two hundred thousand points. The simulations reported here were run on a platform of comparable computing power to that of a Pentium processor. In order to have reasonable run times (a few hours) the size of the DTM grid had to be coarsened by a factor of three.

Fields of application

545

2. Mathematical model It is commonly accepted that bulk flood flow can reasonably be well described by the non linear Shallow Water equations that simply express conservation of mass and momentum in the plane of movement of water. Since they are obtained after and integral mass and momentum balance in the horizontal directions (or by averaging the Navier-Stokes equations across the vertical) no information regarding vertical velocities is obtained. Usually no flow shear forces are taken into account when the problem is convection dominated as it is the case in severe flooding. Fi-iction forces with the bottom are accounted for by empirical formulae such as Manning's or Chezy's. The Shallow Water equations can be written in integral conservation vector form as follows:

1

~ at / U ~ V + / [ F . ~ , + G . ~ ~ HdV ] ~ S = Here t represents time, dV an elementary volume and n, and n, the cartesian components of the normal vector to the elementary surface area dS enclosing the considered volume. Think that in 2-D a volume means in fact an area and an area is actually a line. U is the vector of conserved variables and F and G are the cartesian vector fluxes of mass and momentum.

Here h, u and v represent water depth and the two cartesian velocity components respectively and g is the acceleration of gravity. Source term H accounts for bed friction and bottom slope:

where n is Manning's friction coefficient. The mathematical model SW2D solves the Shallow Water equations in two dimensions by means of a finite volume spatial discretization in multiblock structured meshes coupled to an explicit two step time integration scheme. This is done by applying equation (1) to every cell of the computational domain in the usual Finite Volume approach. Cells can be quadrangles of arbitrary shape but sound judgement has to be exercised so as to avoid very distorted or stretched control volumes that may degrade the overall accuracy. Numerical fluxes are evaluated at cell faces through MUSCL variable extrapolation with limiting to enforce monotonicity. After variable extrapolation, Roe Riemann solver is applied a t each cell interface. Bottom slope and bed friction represented by Manning's formula are spatially integrated pointwise.

546

Finite volumes for complex applications

Figure 2: Cartesian g r i d used i n the computations totalling 22000 points Once the spatial discretisation has been done the solution is advanced in time by a predictor-corrector sequence. Source terms are implicitly time integrated with no extra cost because the operator remains diagonal if they are pointwise spatially discretised. Details of the algorithm can be better found in [ALC98]. 3. Boundary conditions

Proper computation of the flow variables a t the upstream end in order to reproduce the correct flood characteristics at the inflow of the model valley is crucial if good agreement with downstream located probes is sought. Downstream boundary conditions do not exert such a strong influence on the global flowfield mainly because water leaves the reach in critical or supercritical conditions. Available initial data from ENEL were the inflow rate, Q ,the reading of the water level probe located in the inflow tank, named S1, and the reading of the water level probe located at the inlet section, named S2. After considerable efforts it was determined that good agreement with experimental data at the entrance could be found only by imposing a subcritical inflow condition based on the available total head at the feeding tank. Despite the advise given by ENEL that flow conditions are critical at the entrance section, the model could not be run under this assumption. Numerical experiments showed that failure to accept critical flow at inlet was due to a slight adverse slope in this area that led to flow reversals, because critical flow can only be reached at the top of an upslope. Since subcritical inlet conditions require that two flow variables be imposed, they can be implemented by either imposing flowrate, Q, water level at the inflow section (S2 probe reading) or far upstream on the reservoir (S1 probe reading) together with the inlet angle, a.

Fields of application

547

Imposing flowrate is not adequate in a two dimensional computation involving an irregular inlet section because it is difficult to obtain an appropriate criterion t o distribute the available discharge among the inlet section cells. Using the water level a t the inflow section is an interesting option but it is better to use the water level a t the feeding tank (S1 probe reading) as the Total Head available in order to have water level a t the inlet section (S2 probe) as an accuracy check. It must be borne in mind that the inflow rate, Q, can also be used as a check. Due to the size of the tank with respect to the entrance to the reach, the velocity in the former can be considered negligible, and the reading given by probe S1 is considered as the total head h~ that is available at the inflow section. This can be written as follows: (h~)inflow= (h +

T)+ u2

v2

= (h),, inflow

being (h),, the reading of probe S1 a t the considered time and the other variables with subindex inflow are evaluated a t every cell of the inlet section. Also from the outgoing bicharacteristic (see [ALC92] one has:

where u is the velocity vector and n the locally outward pointing unit vector. Subindex bich corresponds to the expression transported by the outgoing bicharacteristic from the inside of the computational domain. Once water depth and modulus of the water velocity at every inlet section cell are determined from the above equations the two cartesian components can be computed if an inflow angle, a, is imposed. In the tests run ct was varied from zero to a few degrees with no significant changes in the computed results.

4. Testcases

Besides the two inflow hydrographs (of different intensities) tested by ENEL, several simulations were performed varying Manning's friction coefficient above and below the value of n=0.016 suggested by ENEL. Also and more importantly, runs were made both with and without the buildings composing the valley villages. The DTM geometry did not contain buildings, but these could be included by modifying the bottom surface function z B ( x ,y) appropriately. However, due t o the low resolution of the grid used, buildings are represented very roughly as figure 1 shows: Villages can be seen as groups of mushroom like sprouts. Their influence on the solution is nevertheless very substantial. For every run made, great attention was paid to matching the inflow rate supplied by the pump (Q) and the water level measured by the probe located in the river bed a t the inlet section (S2 probe). This guaranteed that at least the inflow flood wave was close to the actual one. Figures 3 and 4 show the

548

Finite volumes for complex applications

INFLOW RATE S W D EXPERIMENTS x BUILDINGS n4.015 + BUlWlNGS nd1.02 0 NO BUILDINGSn=O.02

'

.

, 0.WO

1,

,

25

, ,

,,

,

, ,

,

, , ,

,

:,, 0

50

75

-

Bul:T

SOLID EXPERIMENTS BUILDINGS n=O.O15

,

1W

,

NO BUILDINGS n.0.02

125

150

175

Figure 3: Inflow rate and S2 probe reading for the moderate flood event

0 350

0 300 i

r 0

0 250

BUILDINGS n.0015 BUILDINGS n=0 02 NO BUILDINGS n.0 02

0 ZW

0 1%

0 tm

SOU0

0 050

r + 0

25

50

75

1W

125

150

175

25

50

75

1W

EXPERIMENTS

BUILDINGS n;0 015 BUILOINGS rr002 NO BUILDlffiSD002

125

1%

175

Figure 4: Inflow rate and $2 probe reading for the severe flood event comparison between computed and measured inflow discharge (left) and water depths in centimeters (right) at probe S2 (located just at the inflow section) versus time for the two tested flood events. Computational results (crosses, plusses and circles) correspond to different model options as shown. Although agreement in both flow and water depth is quite good, runs with buildings follow better the experimental values. Figures 5 and 6 show the comparison between calculated and measured water levels at probes S4 and P8 for both flooding events. Both probes are located around the central part of the valley, S4 being about 5m and P8 some 16m downstream of the inlet section. Although runs without buildings show larger errors than those including buildings (especially at probe S4) the situation is reversed at other probes not shown here for lack of space. Finally figures 7 and 8 show the same comparison at probes named P13 and P21. Both lie close to the river bed. P13 is located in front of the central reservoir about 21m downstream of the inlet section while P21 is located some 7m further downstream. Overall agreement at the considered locations can be judged acceptable.

Fields of application 0075

.. ,

.

.,

,

.,.

.

. , .., ., . .. ,. 54

,

,

,

,

o

,

'

549

m

OlM

0 I00

0 050

SOLID EXPERIMENTS x BUILDINGS n=O.O15 +

0 25

50

75

100

BUILDINGS n=0.02 NO BUILDINGSn.O.02 125

150

SOLID EXPERIMENTS BUILDINGS 0=0.015

x +

-

0 25

175

30

75

100

BUILDINGS nd02 NO BUILDINGSn.0 02 125

150

175

Figure 5: Comparison at S4 and P 8 probes for moderate flood

SOLID

EXPERIMENTS

SOLID EXPERIMENTS x BUILDINGS n-0015 r BUILDINGS n-002 0 NO BUILDINGSn-0 02

0 WO

I 25

50

75

1W

125

150

115

25

50

75

lm

I r25

r50

175

Figure 6: Comparison at S4 and P 8 probes for severe flood

0 I50 0 150

o1m 0

tw

0 050

0OM

S X l O EXPERIMENTS x BUILDINGS m 0 1 5 + BUILDINGS h 4 02 0 NOBUILDINGS h = M 0 MO

SOLID EXPERIMENTS x BUILDINGS n-0015 BUILDINGS n=OM 0 NOBUILDINGS n-0 02 OWO

25

Y)

75

101

I25

lso

175

I

Y)

75

rm

125

150

Figure 7: Comparison at P I 3 and P21 probes for moderate flood

175

550

Finite volumes for complex applications

0 150

0 IM

0 050

SOLID EXPERIMENTS BUILDINGS n=0015

SOLID EXPERIMENTS BUILDINGS ~ 0 0 1 5 r BUILDINGS -0 02 0 NOBUILDINGS n;O 832 i

0000

50

25

75

100

125

1%

175

25

50

15

>W

125

t50

175

Figure 8: Comparison at PI3 and P21 probes for severe flood 5. Concluding remarks

Although discrepancies between measured and computed water levels can be important a t certain probe locations, the mathematical model used provides a reasonably accurate description of the two flooding events considered. Due to the very valuable assistance that this kind of tool can provide in tasks such as land use and emergency planning or risk assesment studies, it seems worthwhile to carry out further validation and improvement work. 6. Acknowledgements

The author would like t o thank ENEL and especially Dr. G. Testa for providing the experimental data and clarifying many technical questions. Finantial support provided by the European Union under CADAM concerted action is also gratefully acknowledged.

7. References [ALC 921

[NUJ 951

[ALC 981

ALCRUDO F., Esquemas de alta resoluci6n para el estudio de flujos discontinuos de superficie libre, Ph.D. Thesis, Universidad de Zaragoza, 1992 NU JIC M . , Efficient Implementation of non-oscillatory schemes for the computation of free surface flows, Journal of Hydraulic Research, 33, No. I , 1995, p. 101-111 ALCRUDOF . , Dambreak flood simulation with structured grid algorithms , Proceedings of the 1st C A D A M (Concerted Action on D a m Break Modelling) Meeting, (1998), Published b y the EU, in press.

Modelling vehicular traffic flow on networks using macroscopic models. J.P. Lebacquel, M.M. Khoshyaran2

' CERMICS-ENPC. FRANCE. email: [email protected]. TASC. USA.

ABSTRACT: In this paper, we describe a macroscopic model for vehicular traffic flow, with several extensions, resulting in a flow model on a network. These extensions require the introduction of link boundary conditions, partial flow dynamics and intersection models. Some numerical schemes based on the Godunov scheme are proposed for the discretization of the model. Key Words: Godunov scheme, Traffic flow, LWR model, partial flows

1

Introduction

Macroscopic modelling of vehicular traffic flow goes back to the pioneering work of Lighthill and Whitham [LW 551 and Richards [Ri 561, which introduced the celebrated LWR (Lighthill Whitham Richards) model of traffic on an infinite track. This model relies on the continuum hypothesis, i.e. the asumption that vehicular traffic can be described by macroscopic variables, the density K ( x ,t ) , the flow Q ( x ,t ) , and the speed V ( x ,t ) , as functions of the position x and the time t. These variables are related by the following equations: aQ - 0 -+= dt= KV aK

-

V = V, ( K , x)

conservation equation definition of v behavioral equation.

(1)

or simpler:

Qe and V, represent the equilibrium flow-density resp. speed-density relationships ( Q ,( K ,x) dgfKV, ( K ,2 ) ) . Their aspect is the following:

552

_

Finite volumes for complex applications

Qmax (x)

...---

k i t (x)

-

Km(x)

vm v,,,, (XI - - - - -

-

,

K krit(x)

Km&)

Of course, considering the continuum hypothesis, the above LWR model (1) should be considered as a phenomenological model, but it is usually accepted that it provides a reasonably good description of the dynamics of traffic flow a t a space scale of a hundred meters and a time scale of 10 seconds. Actually, the LWR model (I),also refered t o as the first order macroscopic traffic flow model, constitutes but one among several competing approaches to macroscopic traffic flow modelling (see [LL 991 for a general discussion). Other notable modelling approaches include the second order macroscopic traffic flow models (see [Le 951 for an overview and [Sc 881 for relations between second and first order models) and the kinetic traffic flow models [PH 711, [Ph 791, [He 971. Neither experimentation nor theory has provided arguments strong enough to support one model unambiguously. There is also no real consensus concerning " the exact functional form of the equilibrium relationships, but the shapes suggested in the above illustration are generally accepted, up t o a few variations. In the sequel, we shall concentrate on the LWR model, which is simple, enjoys obvious physical meaning, and provides results generally in good agreement with measurements.

2

The Godunov scheme for the classical LWR model on the line

The entropy solution of (1) is the only solution considered usually in the literature on traffic flow modelling. In entropy solutions, the decelaration of trafic generates shockwaves, whereas the acceleration of trafic induces rarefaction waves. Entropy solutions are also characterized by the fundamental fact that they maximize locally the flow [Le 961. The Godunov scheme [GR 911, [Kr 971 provides a numerical solution of the classical LWR model, as shown in [Le 961, [Da 951. This solution is satisfactory for applications: it approximates the entropy solution. Let us introduce the equilibrium supply and demand functions:

(the symbols + and - represent right- and left-hand limits). The following illustration describes these functions, that represent respectively the greatest possible inflow (supply) and the greatest possible outflow (demand) at point x.

Fields of application

553

With these notations, the expression of the Godunov scheme is straightforward:

[

3 3.1

=

~f

+

y

- Q:)

(conservation equation) (flow equation), QI = M i n [A, (K:, i ) , C , (Kf,, ,i 1) (4) ef with as usual (i) d= [xiPl, xi] the cell i of length li, t the index of the time-step, K t the average density in cell i a t time t At, Qi the average flow a t point xi during time-step t. The flow equation expresses that the flow is the minimum between the downstream supply and the upstream demand. The flow equation in (4) is the expression of the analytical solution of the Riemann problem, which can be obtained even if the upstream and downstream equilibrium relationships differ [Le 961. The flow maximizing property of the entropy solution is crucial for this result. The practical necessity for considering space-dependent and even space-discontinuous equilibrium relationships is obvious: lane drops, intersections provide contexts for such discontinuities. ~ ; + l

+ 1

Extensions of the basic LWR model Link boundary conditions

For applications, extensions of this basic LWR model are indispensable. A first and obvious generalization concerns the extension of the model to networks, which implies two steps: the definition of proper boundary conditions for links, and the description of intersections. The equilibrium supply and demand concepts provide the proper framework for the definition of link boundary conditions. Considering now a link such as the following:

the boundary conditions are the upstream demand A,(t) and the downstream supply C d ( t ) [Le 961, [LK 981. The link inflow Q(a, t ) at any time is the minimum between the link supply C, ( K ( a + , t ) , a ) and the upstream demand Au(t). Similarly, the link outflow Q(b, t) a t any time is the minimum between the link demand A, (K(b-, t ) , b) and the downstream supply Cd(t). Thus: Q(b,t)

= M i n [C, ( K ( a + , t ) , a ) , A u ( ~1 ) =Min[A,(K(b-,t),b)

(5)

554

3.2

Finite volumes for complex applications

Partial flows on links

Before considering intersections, it is necessary to consider partial flows on links. Indeed, the fashion in which the traffic flow separates (according to preselection lanes) or does not separate (so-called FIFO flow) in the incoming links of an intersection determines the way in which the intersection works. Further, in many advanced applications, various categories of users must be considered: users would be distinguished according to destination, information availability, path, etc. The macroscopic variables are disaggregated according to some assignment attribute d:

K = C Kand~

Q

=

~

Q

~

The partial flows and densities are related by the trivial relationships: -0 x+x aKd

aQd

Qd = K d V d

conservation equation for Qd, K d

definition of speed Vd of the partial flow .

(6)

These equations must be completed by a phenomenological model. The simplest possible model is the so-called FIFO model:

(vehicle speed independent of attribute d). This model results in a straightforward advection equation relative to the composition coefficients x d def - Kd/K of the flow:

in which the global velocity of the flow V results from the resolution of (2)). A more realistic model is the lane assignment model [LK 98). In this model, vehicles may have restricted access to lanes according to the assignment attribute d. Let I be the set of lanes, Idthe set of lanes accessible to vehicles d, yiKmaxthe maximum density of lanes i, K4 the density of vehicles d in lanes i. Then the K4 are the unknowns of the lane-assignment problem and are subjet to the following constraints:

The K~ constraints express the split of K d into the K!, and the yiKmax constraints express that the total density in lane i cannot exceed the maximum density yiKmax of this lane. The Kd constitute the dynamic data and the

Fields of application

555

constitute the geometric data of the lane assignment problem. The unknowns K,d can be determined by solving either

yi and

(maximizing locally the total flow), or

(Wardrop optimum), subject to constraints (9) in both instances. The meaning of (11) is to assign users t o lanes in such a way that all users having the same attribute d have the same speed on all the lanes they use effectively (otherwise, users would switch lanes in order t o drive faster: this is an individual optimum). These lane assignment models result in systems of conservation laws for which approximate Riemann solvers are under study. A simple case (2 user types, 2 lanes) was analyzed in [Da 971. Supply-Demand models for partial flows can also be defined. The principle is to calculate partial supplies and demands for all superscripts d and to determine the corresponding partial flows by comparing partial demands to partial flows. Partial demands are defined as:

which is a FIFO-like model. The partial supply model really defines the user behavioral model. Let us first define coefficients ,Bd which determine the maximum density (i.e. pdKma,) of the lanes available to vehicles d. If we refer to the notations of the preceding subsection,

We propose the following two models for the partial supplies: Model 1: Cd(x,t) = PdC(x,t) (linear model), Model 2: Ed((z, t) = p d E e , x) (homogeneous section model) Model 1 is extremely simple but allows K d to exceed Pd~,,,. Model 2 does not have this drawback but still does not take partial flow overlapping into account as precisely as the lane-assignment models, since the data formed by the Idsets and the coefficients yi has been simplified and only coefficients pd are left. It would also be natural to define the partial flows Qd as:

(w

Nevertheless, since

EDdis usually >

1 (because of the partial flow overlap-

d

ping), it is possible that partial flows calculated according to (13) satisfy:

556

Finite volumes for complex applications

thus implying a model inconsistency. This inconsistency can be resolved by using the following expressions: Qd = Mas

M i n ( C d , a d )C M i n (z6,n6)]

[c,C s

Q

= M i n [ C , C M i n ( ~ " A ~ 1. 6

The partial flows Qd can be viewed as solutions of the following program: d

0

5 Qd 5 Min [Ed,Ad]

def with y~ some concave increasing functions such as: qd(q) = q - M i n $ d , A d j . The above functions p d are not intrinsic (they depend on the local partial supplies and demands instead of the local geometric attributes such as the pds or the maximum flow). Other functions (entropic functions for instance) would yield the same expressions while being intrinsic.

3.3

Modelling intersections

Modelling intersections is more difficult, since intersection models are phenomenological by nature. They describe for instance, in the case of a merge, bhe local priority rules, or the gap acceptance process. Two modelling schemes can be considered. 1. Modelling intersection as objects of finite extension, by trying t o remodel produce the movement dynamics. This was the idea of the STRADA [BLLM 961, in which exchange zones generalize cells inasmuch as they behave similarly but are endowed with several entry- and exit- points and provide upstream demands respectively downstream supplies for downstream respectively upstream cells or exchange zones. Exchange zone models are discrete by essence. 2. Pointwise intersection models. These were considered in [Le 961 [LK 981 and derived from zone models by letting the zone extension become vanishingly small. The study of these models is the subject of ongoing research. Let us still give one example and consider the node depicted hereafter. Let us denote Cj(t) the supply of exit link (j) of the intersection a t the node point, and Ai(t) the demand of the entry link (i) of the intersection at the node point. A proportion yij of users about to exit link (i) chooses link (j) (the coefficients yij are called assignment coefficients and must be considered exogeneous to the flow model). Thus the partial demand of traffic from link (i) to link (j) def is given by: Aij(t) = yijAi(t). Split supply coefficients pij (depending on the link geometry) can be introduced in order to disaggregate the link supplies Ci(t). We can deduce partial supplies Cij(t) by applying for instance model def 1 (the simplest): Xij (t) = PijCj(t). Since it is possible and even likely that

Fields of application

Pij > 1, a formula similar to

557

(14) should apply, with the same rationale, to

i

yield the partial flows Qzj(t) : (i) + ( j ) :

4

Discretization of partial flows in links and intersections

Let us consider the Supply-Demand model for partial flows (notably equations (12) and (14). If we consider two consecutive cells (i), (i l), the following relationships result (discretized model):

+

Cf+, = C, [K:+,, i + 11 (expressing the traffic supply of cell (i function of the cell mean density K;+, ,

+ 1) as a

+

' Ct+l,d = Pi+l,d Ci+l

or Cf+,,, = Pi+l,d Ce [Kl+l/Pi+l,d,i I] (expressing the partial supplies, according to partial supply models 1 or 2, A: = A, [Kt,i] (expressing the traffic demand of cell (i) as a function of the cell mean density K:, = ( K & / K f ) At = X;,~A;,expressing the partial demands according to the FIFO-like model, -

, expressing the partial flows between ,I ( c : + ~,A:,&)] ,&

Min (Ef+,,,&

Qt'd - M a r

[xi+,

,)

c + 1

in

.c

cells (i) and (i + l" according ) t o (14),

Kt+' z,d = Kt, equation, Kf = 6

+

I,

[Qt

K:,&and Q: =

-

Q;,,], expressing the discretized conservation yielding the total cell density and flow as

6

the sums of partial densities and flows. The similitude between the intersection model (16) and the Supply-Demand partial flow model (12), (14) is evident, thus the discretization of the intersection model follows the same lines as the discretized Supply-Demand partial flow model and need not be described in detail here.

558

Finite volumes for complex applications

5

Conclusion

The development of extensions of the basic LWR is still an ongoing process. The only intersection model for which there exists any kind of experimental support is the Supply-Demand model [LK 981, and link partial flow models are still tentative. Nevertheless, suitable discretized models should be developed, in order to be able to choose between alternative modelling schemes. The finite volume method, combined with the search for analytical solutions, seems to be the best approach to the investigation of numerical solutions of the LWR model and its extensions.

References [BLLM 961 C. Buisson, J.P. Lebacque, J.B. Lesort, H. Mongeot. The STRADA model for dynamic assignment. Proc. of the 1996 I T S Conference. Orlando, USA. [Da 951 C.F. Daganzo. A finite difference aproximation of the kinematic wave model. Transportation Research 29B. 261-276. 1995. [Da 971 C.F. Daganzo. A continuum theory of traffic dynamics for freeways with special lanes. Transportation Research 31 B. 83-102. 1997. [GR 911 E. Godlewski, P.A. Raviart. Hyperbolic systems of conservation laws. SMAI. Ellipses (Paris). 1991. [Kr 971 D. Kroner. Numerical schemes for conservation laws. Wiley Teubner. 1997. [He 971 D. Helbing. Verkehrsdynamik. Springer Verlag. 1997. [Le 951 J.P. Lebacque. L'Cchelle des modeles de trafic: du microscopique au macroscopique. Annales des Ponts. 1st trim., 74: 48-68. 1995. [Le 961 J.P. Lebacque. The Godunov scheme and what it means for first order traffic flow models. Proc. of the 1996 ISTTT (J.B. Lesort ed.). 647-677. 1996. [LK 98) J.P. Lebacque, M.M. Khoshyaran. First order macroscopic traffic flow models for networks in the context of dynamic assignment. E U R O Work Group on Transportation 1998, Goteborg (Sweden). CERMICS Report. To be Published. [LL 991 J.P. Lebacque, J.B. Lesort. Macroscopic traffic flow models order. 14th ISTTT. Accepted for publication. 1999.

:

a question of

[LW 551 M.H. Lighthill, G.B. Whitham. On kinematic waves 11: A theory of traffic flow on long crowded roads. Proc. Royal Soc. (Lond.) A 229: 317-345. 1955. [Ph 791 W. F. Phillips. A kinetic model for traffic flow with continuum implications. Transportation Planning and Technology, Vol. 5, 3, pp 131-138. 1979. [PH 711 I. Prigogine and R. Herman. Kinetic theory of vehicular traffic. American Elsevier, New York. 1971. [Ri 561 P.I. Richards. Shock-waves on the highway. Op. Res. 4: 42-51. 1956 [Sc 881 S. Schochet. The instant response limit in Witham's non linear traffic model: uniform well-posedness and global existence. Asymptotzc Analyszs. 1, pp 263-282. 1988.

Finite volume method applied to a solid/liquid phase change problem

El Ganaoui M., Bontoux P. IRPHE- Universite' d'Aix-Marseille I I I M T , 38 Joliot Curie 13451, Marseille Mazhorova 0. Iceldysh institute Moscow

A second order accuracy method of time and space based on finite volume approxzmatzon i n a fixed m,esh zs developped for Navier-Stokes and energy equations extended to solid/liquid phase change problems. This fixed grid method validated with respect to an interface tracking method i s able to describe the interaction of steady and oscillatory melts with the interface during Bridgman crystal growth.

ABSTRACT

Iiey Words: Finite volume, phase change, interface.

1. Introduction Free and moving boundary problem requires the simultaneous solution of unknown field variable and the boundaries of the domains on which these variables are defined. Phase change during directional solidification of semiconductor crystals by the Bridgman technique is a typical example of such a complex process. Each method of solution must solve the appropriate heat, mass and momenturn transfer equations and deternliiles the melt solid inter-

560

Finite volumes for complex applications

face. It was still necessary to satisfy the Stefan or similar derivative condition on that boundary. Furthermore it was sometimes be difficult or even impossible t o track the moving boundary directly [CRA 841. The possibility, therefore, of reformulating the problem in such a way that the transmission conditions at the interface are implicitly bound up in a new form of the equation, which applies over the whole of a fixed domain. The moving boundary appears, a posteriori as one feature of the solution. One possibility of reform~lat~ing the problem is to introduce an enthalpy function in the energy equation and a porous model in the momentum equation. The enthalpy function is the sum of the specific heat and the latent heat required for the phase change. In the momentum equation, we assume that the liquid turn to solid in an intermediate region t o be a porous medium. In this way on prescribing a Darcy source term the velocity value arising from the solution of the momentum equation are inhibited, reaching values close to zero on complete solid formation [VOL 801. The coupled enthalpy porosity model gives a single set of homogenous NavierStokes and energy equations adapted to the problem of phase change during directional solidification [MOR 991. The finite volume method is validated with respect to an interface tracking method [ELG 961. It uses a fixed grid and the interface position is given from the thermal field (solidification isotherm). The resulting interface shape is also studied and some insight on cristal constitution are given.

2. Formulation For directional solidification, A cylindrical ampoule with radius R and length L contains melt and crystal. The ampoule must be moved relative to a prescribed external temperature gradient. This motion of the ampoule is acounted for by supplying a melt to the top of the computational space at a uniform velocity Ut and with drawing cristal from the bottom with the same velocity. The heat transport between the furnace and the ampoule is modelled with a re scribed furnace temperature profile with three zones, cold (T = T,), adiabatic (aT/dn = 0 ) and hot one (T = T h ) . The length L , the velocity a / L and the thermal difference Th - T, are used as reference scales to give dimensionless form of the variables ( x ,u , 0 ) . x ( r , z ) represents the courant point with radial r and the axial z components, u = (u,, u,) represents velocity with radial and axial coordinates u, and u,, respectively. Liquid, solid and intermediate medium are distinguished by the suffixes 1, s and sl. For the energy equation a continuous enthalpy function is introduced :

Fields of application

561

where E is a prescribed small regularisation of the temperature, fl a mesure of liquid fraction and Ste = Lf/c(Th- T,) is the Stefan number. T h e corresponding enthalpy is continuous and piecewise linear. In this way the energy equation takes the following form in all the dimensionless domain 0 < r < 1 et 0 <2 < L/R

If Ste-' goes t o 0, the equation (2) goes to the classical one phase energy equation. For the momentum equations a permeability term is introduced:

T h e momentum equations takes the following form

where Ra = g/?(Th- T,)/vcu, is the Rayleigh number, Pr = vlcr is the prandtl number characteristic of the material and Da = L2/l
562

Finite volumes for complex applications

3. Solution method

3.1 Finlte oolume method

Equations ( 2 ) , (4) and (5) are discretized by the finite volume method [EYM 971. To explain the approach we consider the two-dimensional convection diffusion equation for a general variable cp with the velocity field in cartesian coordinates u :

F(p) = up - y, grad cp is the advection-diffusion tensor of cp with the convective part F c = u cp and the diffusive part F d = -y, g r a d p . Equation (6) gives the expression of the conservation of cp in an infinitesimal domain, it is equivalent to write in any subdomain V and for all time t and t ' :

where nr(x) is the normal vector t o the boundary d V at point x , outward t.o

v.

In order to define a finite volume scheme, the time derivative is approximated by a finite difference scheme on an increasing sequence on time ( t n ) n E I N with t o = 0. A mesh M of the physical dom ain R of E2 (R = UV, V E M ) is introduced. The discrete unknowns at time t , = n6t (St is the time step), are expected to be an approximation of cp on the cell V around the point Mi,j and noted 4:?. Equation (6) is integrated over each cell V using the Gauss divergence theorem :

where ( d p l d t ) " is given b y the time scheme at the time step t, = n St in the control volume V. The next step is the approximation of the convective part F C . v and the diffusive part Fd.rn of the projected flux 3 . 7 ~over the boundary dV of each control volume.

Fields of application

563

3.2. Ti,me discretization

The time integration is performed implicitly using a three levels, given a second order truncation error in time.

3.3 Flux approximations

The diffusive part of the flux is discretized with a second order truncation error in space:

Different discretizations for the convection are possibles, central schemes apply a symmetric interpolation for cpi+l/z , Upwind schemes apply a one side interpolation. Leonard [LEO 901 has introduced a QUICK and other scheme as a mixtures between the two kinds of interpolation. We have used QUICK scheme with a second order truncature in space. we write - ~ i + ~ l ~ , j c p ~pi+l/2,j + ~ / ~ ,is j ,an approximation of cp at the east side of the volume V

3t+112,j

the subscript m is equal -1 for

> 0 and 0 for

ui+l~z,j

u;+1/2,j

< 0.

4. R e s u l t s We consider a two-dimensional model in cartesian coordinates corresponding to a diameter section of the growth ampoule. This model permits the investigation of asymmetric flow. We consider unstable configuration (IVB), which foster the amplification of hydrodynamic instabilities. In this case the computation exhibits steady symmetrical solution up t o Ra = 8 x l o 4 . This solution is characterized by the presence of two contra-rotating cells with equal sizes and a flat interface. Transition to asymmetrical solutions occurs

564

Finite volumes for complex applications

at Ra = 1.6 x l o 5 . The asymmetrical instability is shown to become tirnedependent at 4.2 x l o 5 5 Ra 6.4 x l o 5 and interacts with the interface shape. A periodic solution is obtained for Ra = 8 x l o 5 . A description of the unsteady regime effect on the interface shape during the growth process is given in figure 1. In a space of time corresponding to one period of the periodic regime, the two contra-rotating cells which compose the melt exchange their sizes in time: at t = t o T / 6 (T is the period) the flow is dominated by the cell of the right which brings about the interface, with the presence of a second cell in the left. This situation changes at t = t o T / 3 in favour of the left cell which becomes dominated at t = t o T / 2 and configuration is still the same until t = t o 5 T / 6 . The interface interacts with the previous hydrodynamic regime. The crystal characterised by successive solidification interfaces is represented in the figure 2 on the diameter crystal. This result shows a significant structure which appeared t o be relevant of some patterns exhibited in the experiment.

<

+

+

+

+

5. C o n c l u s i o n

The finite volume method is able to simulate accurately complex physical phase change problem. It represents the flow pattern and the heat transfer during directional solidification. The method enables to determine the various symmetric, asymmetric, steady and time-dependent flow solutions and to analyze the transitions between regimes. A recent extension concerns the effects of inequal thermal conductivity in solid and liquid phases on the interface shape and hydrodynamic transitions. Acknowledgment We would like t o thank Dr J. Ouazzani from Arcofluid and Dr D. Morvan from IRPHE for helpful and comments. We thank also the CNRS for support on CRAY C98 of IDRIS-CNRS Center in Orsay. Support from CNES are also acknowledged.

Fields of application

565

Figure 1: Description of various hydrodynamic configurations during one period of the periodic regime Ra = 8 x lo5. The signal of the temperature T(lo,lo)(t) is represented in the point (10,lO) in the melt function of time (t - t i n i t ) / 6 t ,6t =

-__. -'

I

--

__.-

- -_.__ ._

-._. ....

-.

.,--

Figure 2: Resulting crystal by considering successive interfaces during one period of the periodic regime Ra = 8 x lo5.

566

Finite volumes for complex applications

References [CRA 841

CRANKI . J., Free and Moving B o u n d a r y Problems. Press, Oxford, U. K., 1984.

[VOL 801

V. R. VOLLERAND M. CROSS.Accurate solutions of moving boundary problems using the enthalpy method. Int. J. Heat Mass Transfer, 24,P. 545-556, 1980.

[ELG 961

EL GANAOUI, M. et al. Stiff problems in Thermofluid dynamic using finite volume method. In Finite volume for complex applications. Eds. F . Benkhaldoun and R. Vilsmeier, Hermhs, 1996, p. 717-724.

[MOR 991

D. MORVAN,M. EL GANAOUI, A N D P . BONTOUX. Numerical simulation of 2D crystal growth problems in a vertical BridgmanStockbarger furnace, latent heat effects and crystal-melt in interface morphology. In1.J. Heat Mass Transfer, 4 2 , p. 573-579, 1999.

[ELG 991

EL GANAOUI M., BONTOUX P . ET MORVAN D. Capture d'un front de solidification en interaction avec un bain fondu instationnaire. C. R. Acad. Sciences Paris, t,. 327, SCrie I1 b, p. 41-48, 1999.

[EYM 971

EYMARD R., GALLOUET TH.,AND HERBINR. Finite Volume Methods, in Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions (Eds), North Holland. Preprint 1997 (to appear).

[LEO 901

LEONARD B. P. AND MOKHTARI S. Beyond first-order upwinding the ultra-sharp alternative for non oscillatory steady state simulation of convection. Int. J. Numerical Methods Engineering, 30: 729-766. 1990.

Clarendon

Integrating finite volume based structural analysis procedures with CFD software to analyse fluid structure interactions Marcus A Wheel, Andrew Oldroyd, Thomas J Scanlon and Pan Wenke Department of Mechanical Engineering, University of Strathclyde, Glasgow, GI IXJ, U K

ABSTRACT A finite volume based procedure for predicting deformations in statically and dynamically loaded two dimensional structures is briefly presented in this paper. The paper goes on to describe the integration of this procedure within an existing finite volume based CFD code to provide a facility for analysing fluid structure interaction phenomena. The integrated facility has been used to analyse a flow induced vibration problem in which a moving fluid induces oscillatioizs in a bluff flexible structure around which it is flowing.. Preliminary results obtained from this analysis are included in the paper. Key Words: finite volume method, structural analysis, fluid structure interaction.

1. Introduction Numerical structural analysis is dominated by the finite element (FE) method because of its geometric versatility that allows it to represent the complex geometries of loaded engineering structures. In computational fluid dynamics (CFD) an alternative technique the finite volume (FV) method is popular because it automatically satisfies the conservation principles of fluid mechanics on a local scale. Conventional FV procedures have however lacked the geometric versatility associated with the FE method. Although FE and FV methods have been successfully employed in structural analysis and CFD applications respectively there are many engineering analysis problems in which complex mechanical systems interact with a contained or surrounding fluid. Typical examples include flow induced vibration, wave loading of marine structures and sloshing of liquids in containers. Forming processes such as casting also involve interactions between solid and fluid phases. In these typical interaction problems the deformation of the structure is intimately coupled to the motion of the fluid. In order to analyse these problems in detail a facility with numerical structural analysis and CFD capabilities is required. Existing FE and CFD software packages could be used concurrently to provide such a facility but the two packages would have to exchange information through data files. For a typical transient interaction problem the effort involved in continuously acquiring and storing data in these files will be computationally intensive and the whole analysis

568

Finite volumes for complex applications

procedure will therefore be inefficient. One alternative approach to producing a facility for analysing fluid structure interaction problems would be to develop entirely new software with both structural analysis and CFD capabilities. A second alternative would be to develop procedures that could be integrated with existing software in the form of user defined functions to provide the additional features required for the analysis of interaction problems. The recent development of geometrically versatile FV based numerical structural analysis procedures [DEM 94, BAI 95, WHE 961 has enabled both of these alternative approaches to be pursued. The FV based structural analysis procedures exploit features employed in FV based CFD software. Most importantly, the unknown displacement variables are located at the centres of the cell that are used to discretize the structure. Individual cells can be multifaceted and cell geometry is not restricted to a number of predefined shapes as in the conventional FE method. Unstructured cell meshes can be assembled to represent the complex geometries of loaded engineering structures. Although these procedures were originally developed to predict small strain deformations in linear elastic structures there capabilities have subsequently been extended to analyse problems involving nonlinear materials [TAY 951 and large deformations [PAN 991. 2. A finite volume procedure for predicting deformations in loaded structures

To predict deformations in two dimensional loaded structures the FV procedure divides or discretizes the structure into a series of interconnected elements or cells. The geometry of individual cells is unrestricted, each cell can have an arbitrary number of edges and traction forces act on these edges. A typical cell is shown in figure 1. The resultant force found by integrating the traction forces acting on the edges of a particular cell is either equal to zero, in which case the cell is in equilibrium, or equals the rate of change of momentum of the cell. Mathematically this condition can be expressed as

a

au,

at,

at

I o v . r z j d=-Sp-dv ~ S

Here, S and V are the surface area and volume of the cell respectively, o,,, n, and ui correspond to the surface stress, surface normal and cell centre displacement components and p is the material mass density. The finite difference approximation

a

au.

-jp-dV at,

=-

at

is used for the temporal discretization of the momentum term on the right hand side of equation 1. The superscripts m-1, m, and m+l refer to previous, current and future time steps while At is the time step size. To minimize time damping in the numerical procedure the temporal weighting

Fields of application

569

is applied to the resultant force term on the left of equation 1. The weighting parameter 8 can be set within the range 0 I 0 I1. By using a suitable constitutive relationship the stress component terms within equation 3 can be replaced by terms involving the strain components and constitutive properties. It is assumed that the strain components are constant on each edge of the cell so these can be approximated by linear displacement variations involving the unknown cell centre displacements [WHE 961. On substituting these approximations for the strain components into the resultant force term and equating the resulting expressions to the discrete momentum term a system of algebraic equations containing the unknown cell centre displacements is produced. When 8 is set to zero the unknown displacements at the future time step are given explicitly in terms of the displacements evaluated at the current and previous time steps. The time step size will however be subject to an appropriate restriction. If 8 is non zero then the algebraic equations are implicit and the future displacements must be obtained by solving the equations simultaneously. T o validate the FV based structural analysis procedure briefly summarized here a test problem consisting of a unfilled 0.5 m square box supported internally by a 0.0179 m thick cantilever beam was selected. The thickness of the box wall was twice that of the cantilever. The Young's modulus and density of the box material were 5.0 MPa and 500 kgm-3 respectively. Although the majority of the mesh representing the box and cantilever consisted of regularly arranged quadrilateral cells, as shown in figure 2. the intersection between the box and the cantilever was represented by an unstructured mesh of triangular cells, as illustrated in figure 3. A tensile stress of 1 ~ mwas- applied ~ to the lower half of the right outer face of the box and a corresponding compressive stress was applied to the upper part of this face. This induced a state of pure bending in the internal support beam. The computed vertical static deflection of a point located midway up the intersection between the right end of the cantilever and the inner surface of the box was 2.26 mm. The theoretical value for the end deflection of a cantilever of the same length loaded by the same moment is 2.32 mrn. The static loads were then removed and the structure allowed to oscillate. The subsequent time dependent deformation of the block and cantilever were computed using the FV procedure with the temporal weighting parameter 0 set to 0.25. The time step size was set to 0.04 s. The variation in the vertical displacement of the point midway up the intersection with time is shown in figure 4. It can be seen from this figure that the numerical procedure does not suffer from time damping because the amplitude of the computed oscillations does not decay with time. The average peak to peak elapsed time is 4.18 s implying that the structure is oscillating at a frequency of 0.2392 Hz. A prediction of 0.2396 Hz was obtained for the first natural frequency of the structure from a modal analysis performed using the commercial FE code ANSYS with the same mesh geometry and material properties. The accuracy of the

570

Finite volumes for complex applications

deflection and frequency predictions demonstrate the suitability of the FV procedure for calculating static and dynamic deformations in loaded structures.

3. A CFD analysis of laminar vortex shedding The commercial CFD code PHOENICS was used to simulate laminar vortex shedding around a square bluff body within a confined channel flow. The geometry of the flow domain is shown in figure 4. The aspect ratio of the block, A/B, was fixed at 1.0 and the blockage ratio, B/H, was set at 0.25. All dimensions were normalized with respect to B which was set to 1.0. Dimensionless density and inlet velocity were both set to 1.0 and pressures, velocities and time were all normalized with respect to these specified values. A Reynolds number of 600 was specified to ensure that the flow was predominantly laminar. The computational domain was constructed using a multi block fine grid embedding technique in which the cell mesh was progressively refined in the region around and immediately downstream of the block. This aimed to ensure that the details of the vortex shedding process would be simulated accurately. The fine grid embedding necessitated the use of a collocated grid with pressure and velocity variables stored at the cell centres. An upwind convection scheme was also used. To initiate vortex shedding an initially asymmetric flow field was required. This was generated by performing a steady state analysis with the wall velocities of the block fixed at 1.0 in a clockwise sense. Previous studies demonstrated that the vortex shedding behaviour observed in the subsequent transient analysis was insensitive to the magnitude or sense of the wall velocity prescribed in the steady state analysis. A time step size of 0.04 was selected to perform the transient analysis. The computation was performed using 1875 time steps on a Pentium 200 MMX personal computer with 64 MB of memory. Total run time was approximately 5 days. Pressure was monitored at a point located close to the upper rear corner of the block. Figure 5 shows how the predicted pressure at this location varied with time. The Strouhal number obtained from measuring the average peak to peak elapsed time was 0.1852 for this case. This compares extremely well with an experimentally obtained value of 0.182 for these conditions [DAV 841 and represents an improvement on a previously obtained computational result of 0.16 obtained using an upwind convection scheme on a staggered grid [SCA 991.

4. Response of the block to fluid pressure loading

At each step in the transient CFD analysis the static gauge pressure field around the block was recorded. The complete time history of the pressure field was then applied to the FV based model of the box and cantilever test structure discussed previously. The response of the structure to the pressure loading is shown in figure 5 in which the variation in the vertical displacement midway up the intersection between the cantilever and the box with time is depicted. This response displays a

Fields of application

57 1

characteristic 'beating' behaviour with the amplitude of oscillation increasing and decaying at regular intervals. A Fast Fourier Transform revealed that the two frequency components present in this response correspond to the frequency of the oscillating pressure field and the first natural frequency of the structure. The beating behaviour displayed by the response can therefore be attributed the interaction between these two frequency components.

5. Coupling the CFD and structural analysis The uncoupled analysis discussed in the previous section acknowledges that the motion of the fluid influences the structural response but ignores any effect that the movement of the structure may impose on the fluid. This can only be determined by performing a coupled analysis incorporating the interactions between the fluid and the structure. The FV based structural analysis code was incorporated into the PHOENICS CFD code as a user defined function. Initially, only the flow field is computed in each time step to ensure that periodic vortex shedding is properly established. In each subsequent time step the flow field is computed first. The calculated pressure distribution at the interface between the fluid and the box is then supplied to the structural analysis function and the deformation of the structure determined. The structural deformation is then used to calculate the change in the flow field geometry. The flow field mesh is adjusted to account for any change in geometry. The computation is advanced to the next time step and the adjusted mesh is used to compute the new flow field. The most important feature of this cycle is that all required data is maintained within the physical memory of the PC and continuous access to disk storage is avoided. Currently the cycle does not supply the boundary velocity of the structure to the flow field computation and consequently the transfer of momentum from the structure back to the fluid is neglected. The addition of the structural analysis procedure within each time step added two days to the total solution time. Figure 6 shows the response of the structure predicted by the fully coupled analysis. This response is very similar to that observed when the oscillating pressure field was applied to the structural analysis procedure independently. The beating behaviour exhibited by the earlier response is also observed in the fully coupled response. However, a detailed comparison of the two responses reveals that the maximum amplitude of oscillation is slightly increased in the fully coupled case. When the fluid pressure variation close to the upper rear corner of the box was examined it was found that there was a 2.25% increase in Strouhal number from 0.1852 to 0.1894. This change in Strouhal number represents an increase in the frequency of the forcing function being applied to the structure and, since it is now closer to the natural frequency of the structure, it produces a corresponding increase in amplitude of the structural oscillations. The increase in Strouhal number and amplitude of oscillation also demonstrate that the fluid flow and structural response influence one and other in an intimately coupled way, the details of which can only be modelled by a fully coupled analysis procedure such as the one presented here.

572

Finite volumes for complex applications

6. Conclusions A finite volume based procedure capable of predicting the deformation in statically and dynamically loaded structures has been tested on a benchmark structural analysis problem. Comparisons with analytical and numerical solutions have demonstrated that the procedure can accurately predict the static deformation and transient dynamic response of the test structure. The procedure has subsequently been integrated with an existing FV based CFD package. The integrated facility has been used to model the laminar fluid flow around a bluff flexible structure. Vortex shedding from the rear of the structure creates pressure fluctuations within the flow which induce vibrations in the structure. The structural vibrations increase the frequency of the vortex shedding above that observed when the structure is rigid. The increase in vortex shedding frequency is accompanied by an increase in the amplitude of the structural displacement. Future work will focus on improving the analysis by incorporating damping into the structural analysis procedure and including the momentum transfer to the fluid arising from the structural motion. An investigation to examine how the temporal resolution influences the accuracy of the solution will also be conducted.

7. Bibliography [BAI 951

Bailey, C. and Cross, M., 'A Finite Volume Procedure to Solve Elastic Solid Mechanics Problems in Three Dimensions on an Unstructured Mesh', Int. J. Num. Meth. in Eng. 38, 1757-1776, 1995.

[DAV 841

Davis, R.W., Moore, E.F. and Purtell, L., A Numerical-Experimental Study of Confined Flow Around Rectangular Cylinders. Physics of Fluids, 27,46-59, 1984.

[DEM 941

Demirdzic, I. and Muzaferija, S., 'Finite Volume Methods for Stress Analysis in Complex Domains', Int. J. Num. Meth. in Eng., 37,3751-3766, 1994.

[PAN 991

Pan Wenke and M.A. Wheel, M.A., A Finite Volume Method for Predicting Finite Strain Deformations in Incompressible Materials, Proc. ECCM 1999, 3 1 August 3 September 1999, Munich, Germany, 1999.

[SCA 991

Scanlon, T.J., Stickland, M.T. and Oldroyd, A., 'A Numerical Analysis of Vortex Shedding Within a Confined Channel Flow', to be published in Proc. IMechE Part C, J. of Mech. Eng. Sci., 1999

[TAY 951

Taylor, G.A., Bailey, C, and Cross, M., 'Solution of the elastic visco plastic constitutive equations: a finite volume approach', Appl. Math. Modelling 19, 747-760, 1995.

[WHE 961

Wheel, M.A., 'A Geometrically Versatile Finite Volume Formulation for Plane Elastostatic Stress Analysis', J. of Strain Analysis, 31, 111-116, 1996.

Fields of application

Cell 2

Node Nd3

P

/

573

n Side 2

Node Ndn

4 1

\

Figure 1. A typical multifaceted cell

--

.-

Figure 2. Cell mesh used to represent box and cantilever structure

Figure 3. Detail of cell mesh at intersection of cantilever and box

1

574

Finite volumes for complex applications

Figure 4. Geometry offlow domain used to model bluff body vortex shedding

'Iime (Dimensionless) -

-

Figure 5. Fluid pressure variation downstream of block

-0.003

I

I

1

4 0

10

I

20

40

30

50 I

Time (s) -

-

-

-

-- -

-

- --

1

-

Figure 6. Response of structure tofluid pressure loading (uncoupled analysis)

--

0 003 E 0002 E 0001

E

0

"f

f

0 -0.001 -0 002 -0.003 0

10

20

30

40

50

'Iime (Dimensionless)

Figure 7. Response of structure tofluid pressure loading (coupled analysis)

A generalized parcel method for the spray dispersion computation

B. Nkonga Mathe'matiques Applique'es de Bordeaux, Univ. Bordeaux I 351, Cours de la Liberation, 33405 Talence cedex, F R A N C E [email protected]

W e propose a generalized formulation of the parcel method for t h e computation of a dispersed spray in a turbulent flow. T h i s approach take i n t o account a n exponential distribution of the radius and the temperature in a parcel of droplets. A s y s t e m of ordinary differential equations is obtained for t h e conservative quantities transported by t h e parcel. T h e analytical solution of t h e parcel method s y s t e m (of O D E ) is achieved i n a t i m e step with t h e classical assumption of fixed characteristics of the surrounding flow and for a simplzjied model f o r droplets rate of change. I n order t o improve t h e methodology o n i n dustrial geometries, we have defined reflecting, outgoing and incoming boundary conditions for parcels. T h e numerical scheme for t h e gas phase is developed in a finite volume/finite element context for unstructured moving meshes. A generalized Roe solver for multi-species flows and moving meshes i s used for the convective flux and a finite element discretization is used for diffusive t e r m s [ N K O 971. T h e t e r m s coupling the Eulerian and t h e Lagrangian formulations, as in t h e classical P I C methods, are obtain by conservative interpolations. Finally, the methodology is performed o n the spray injection in a simplified piston engine.

ABSTRACT

K e y Words: Multi-phases flow, Mixed Lagrangian/Eulerian Formulation, parcel Method.

Introduction

We focus on the behavior of the spray after the complete atomization: the T h i n Spray, the Very T h i n Spray regimes and the transition to gas phase by

576

Finite volumes for complex applications

evaporation. Then we assume that collisions effects and liquid breakup are negligible. Moreover we consider that the spray is composed of spherical drops with uniform temperature. Let us denote f ((, t) the spray probability density function (pdf). In the following analysis, we shall use the phase variable ,$ = (x, v , r, 6') where x is the position, v the velocity, r the radii and 6' the temperature of the droplet. According to the kinetic model for spray introduced by Williams and the formulations derived in [ORO 81](24-34) and [CLO 971, we shall use the following Fokker-Planck type equation for the spray:

where v is the particle acceleration and DEgis the diffusion coefficient associated with the deterministic model of the surrounding gas turbulence effects. Generalized parcel method

<

Let us assume that the gas flow is not a function of microscopic variables and decompose the function f as a sum of numerical particles fe solutions of the kinetic equation (1). We also assume that there is no interaction between numerical particles and no correlation between the radius distribution, the temperature distribution and the other distributions. The numerical particle is then defined by:

Y cpe (r)dr and (Y, CPe) =

where (Y, cpe)T =

1+

Y de(6')d6'.

The behavior of a numerical particle (1) is then defined by its number of droplets Np = (1,fl)<, its total mass Me(t), its mean radii re(t), its mean temperature Te(t), its mean position Xe, its mean velocity Ve, its total energy Ee(t) = Mt(Ke + CUTe)and its characteristic size Se = Jae - Xe . Xe: Me MeVe MeKe Mere

(mer3, fe) , (mer3v, fe\[ , = ( m e r 3 y , fe)<, = (mer4,f i ) <, = =

MeXe Meue Meue MeTe

(mer3x1fe)<, (mer3x - v , f e ) < , (mer3x . x , f e ) < , = (mer301fe)S , = = =

(3)

where me is the mass of a droplet with a radius one, C, is the specific heat at fixed volume and, for a given function Y, we have:

Fields of application

577

Proposition 1 Let us assume that droplets cannot collapsed or break-up, then for a given parcel k', the number Ne of droplets is constant and the behavior is defined by the following differential equation (ODE):

9

=

dXe dt

= Ve,

dv' dt

=

{+}!,

doe dt

=

ve,

-

-

dre =

Me[%],

e

dt

[ ddr4 t]t-re[%]

5 dt

=

!!.!!

=

2Ke+{x.G)(,

=

{b}

dt

"'

{V.G}~+~D~~, (4)

dt

where, for any function Y: { Y l e =

7

e

dt

2 ( r 3 y , fe)<

e

a n d [yIe= (yl Ve)r / (r39~ e ) ,

We denote by G the differential operator defined by equation (1). This proposition is obtained by computing (Y (x, v , r) , Gfe) t for the values of Y ( x , v , r ) = 1, r 3 , r 3 x , r 3 v , r 3 v . v , r 3 v . X, r 3 x . x, r30. Let us denote W the molecular weight, p the density, p the pressure, Y the mass fraction, T the temperature, p the gas shear viscosity, X the thermal conductivity , Re the Reynolds number and Pr = 0.75 the Prantl number. We assume that the diffusion process and the energy balance control the exchanges rates. Then the exchange rates of the kinetic equation are defined as functions of the transfer number 8 , the Nusselt number M u , and the averaged temperature on the drop surface T,. The three parameters (B,n/u, T,) are defined by the following non linear system [OR0 81](94):

where the subscript (d) is used for the drops, (g) for the gas, (v) for the vapor species (produced by drop evaporation), cr is the void fraction ( volume occupied by the gas), R is the perfect gas constant, we chose N u d = 1, p; is the normalized equilibrium vapor pressure defined by the Clausius-Clapeyron equation, and Y,* is the averaged vapor species mass fraction over the drop's surface, p: (T,) =

Pd

exp

(-

IC(T$ Ww

(& $) ) , -

YJ ( T ~ )=

P: wv

wi- p; ( w i wt,). -

578

Finite volumes for complex applications

where the subscript (i) is used for the inert species parameters. The Reynolds number and the latent heat of vaporization ( L ( T ) )are given by:

Re =

2rllu

+ u' - vllpg , P9

L ( T ) = L(To) + ( ~ P V- c d ) ( T - To)

u is the gas velocity and u' his fluctuation. Then, the rate of drop radius change (.i-),the rate of temperature change (8) and the mean aerodynamic drag force ( v d r a gare ) , given by:

The fluctuations of gas velocity produced a external force for the droplets. This force is defined by: Vturb

=-

C ( u + u',V ) - C ( U ,V ) (V - U ) r2

+ C ( u +r u',v) u'

where u' is the gas velocity fluctuation. Let us denote a a vector of random numbers, a = ( a l ,..aj, ..ad) with -1 5 aj 5 1. We assume that the fluid turbulence is isotropic and Gaussian distributed. Then, for a given vector a the turbulence force in a time step is approximated as follows [DUK 80](240-241):

wliere k is the turbulent kinetic energy and T the turbulence time scale. In practice T is of order At. Then the rate of change v, taking into account the jet propulsion effects, the generalized Archimedes, the gravity, the mean drag and the turbulence forces, is defined as follows:

Fields of application

579

where p is the coefficient of the jet propulsion effect, u is the mean flow velocity around the particle, and

In order to develop a numerical method for parcel of drops, we assume that during a time step ( t - s ) the parameters K , P, p, C and g are constant. Then we can compute the analytical solution of the generalized parcel method system.

Proposition 2 Let us consider an exponential distribution for the radii:

1

00

rrn+pexp(-in)di and h = ( 1 m , n ( 3 ) ) re. w e suppose 2C+3 K + ,u C and i = - +V + g , where the that = - O), i = -JL 2r coeficients C (drag), K (evaporation), p (jet propulsion), g (external forces) and the surrounding gas velocity, are constant. Moreover, the status of the numerical particle is supposed known at the time s. Then [rPIe= cupr:-3 and the behavior of the numerical particle is defined

when I,,,(p)

=

g(~,

0

by:

re(t) = J r ; ( s ) - ~ ( (-t S ) = r e ( s ) E ( t s, ) 3 Me(t) = M e ( s ) R c V e ( t ) = V e ( s )- V e ( s ) + ( t - ~ ) G + E & ~ e ( s ) p -R + + ~ x ~ ( ~ X e ( t ) = x e ( s ) + Xe(s>+ ( t - s ) ( V e ( s )- Ve(s))+ y Ke(t) = l o ( s ) + 72'12 ( s ) - E4&4( s ) ( t - s)Et ( s ) ( t - s)R2Et.(s) + R % & ( s ) - ~ s + ~ & ~ . (R s%)E ~ ~ ( S ) Te(t) = T,+(T~-T,)R?

+

< = 4a2 - 301, = V d s )- y + *g,

where Vt(.)

2r2(s)

y, Et.(s) =

& t ( ~ )=

(8)

+3 p ) ,

* g8G, y = al(2C

+

G = *g,

U = a1 C U ,

2rZ'" V!(S), &o(s)= U . ( V , ( s ) - V e ( s ) ) X(s) = Y &4(s) = -&t+(s), EY(s) = fw,Y

-

r2(s) ( g . (Ve( s ) - Ve( s ) )+ 3 *g.Ve(s), &2(s)= '""lt( s )and E2-, ( s ) = Ke(s) - &o ( s ) - 82 ( s ) &4 ( s ) - fY ( s ) EY*(3). (3) =

+

Proof: We have:

$1

ca

J ( P )=

rP*(r)dr =

(L,. (3)) . &T

+

Then [rp]

7

~,

a , r , ~ - ~where

~

~

580

Finite volumes for complex applications

a, =

#;

furtheremore, we can note that mean radlus. Therefore,

= 1, so that rt is the Sauter

-

--

and

{$It

We finally obtain the following simplified system:

(9) Then, the proposition is obtained by the integration of the previous system of equations, when using the relations:

and

Note that lim 72% = e - 7 , so that at the limit when K --+ 0 we recover n-0 the solution of the non vaporing spray. We can also obtain, by an analytical integration, the behavior of the size St of the numerical particle. Previous investigations [DOM 971 have focus on the case of a Dirac radii distribution. In this new context, the Dirac radii distribution can be viewed as a particular case where a p = 1 for all p. Some aspects of this analytical solution was taken into account in the numerical approach proposed by P. O'Rourke [OR0 811. st

Numerical model We assume that the motion of the Newtonian fluid phase is governed by the Navier-Stokes equations for time dependent spatial domain. Let us denote W =

[ p,

pG, E

] T the conservative variables in the gas phase, Pt =

Fields of application

[ Me,

M

e &

IT

581

the conservative variables for a numerical parcel and

l3(2e, Se) the approximated ball in which live a computational parcel. The equations for vaporizing liquid spray, with the parcel method formulation writes as:

(10) where is the convective flux, 6 and d are the functions describing the parcel motion. Note that if we add a diffusive terms in the gas equation or use a multi-species formulation, there are minor change in what will be developed. For a parcel Y setting in the mesh element r;-' a t the time tn-*, the characteristics of the surrounding flow are defined as an average W; (at the time tn) of the values on the vertices (cells) of T;-' . Assuming that those characteristics and the physical parameters (drag coefficient, evaporation rate, ...) are constants and for a given initial condition, the ordinary differential describing the behavior of the computational parcel is integrated exactly. We rewrite the numerical scheme for spray as follows:

As we assume constant flow characteristics, the stability and the accuracy of the coupled system (gas-spray) is insured only under a CFL type condition [OR0 811.

Parcels localization and boundary conditions 'n-

l

associated t o the Let us consider a parcel with a previous location X, -n+' is computed we define mesh element r;-;. When the new position X, .-$,.+a approximating the domain recovering the parcel a set of elements 7,

582

Finite volumes for complex applications

trajectory during the time step: ,,-n-f,n+i -

e

/

-{T

3tE[tn-i,tnti],

n- l

where Te,l = rt inequality:

2 t ( t ) ~ T } = { ~ Te.2, ,~,

...,

Tt,N)

(13) and for strait lines r t , j and rt,j+l are related by the max

max

( t ).

(14)

2 t ( t ) ~ ~ t , j

,(g-

n+: = l ,$;ti ) = T ~ , NThe construction of 7 ,n-' "n+' ' is Therefore, re achieved by a recursive algorithm. For a given element Te,j the space domain is decomposed in four unrecovered zones defined by: the element T, the domains +n+ 4 f l l , flz and 03. If X, E Te,j then re-N= ~ t , and j the sequence is achieved.

If 2rii t fli and there exist an neigbhor element ri of r t , j such that ri E fli then ~ e , j + l= ri. Otherwise, the trajectory is crossing the physical boundary. j ni is on the boundary of the physical The segment sli associated to ~ t , and domain associated to a boundary condition that can be a reflection ( elastic or non elastic) or an open boundary. For reflecting conditions, the trajectory is modified, a new position is defined and then the algorithm is performed with this new position. For an open boundary, the parcel is just deleted from the actual list. This construction set out a natural way to applied the boundary conditions for parcels "crossing" the physical boundary. According to the CFL n-+,n+i is cost-less. Indeed, it is in general condition the construction of 7, included in the set of the adjacent elements to the nodes of r ~ - - ~ . A mixed finite volume/finite element formulation is used for the discretization of the gas equation. The representation is "vertex centered" and therefore cell-averages are stored a t the vertices of the mesh and the control volumes are the dual cells. Let us denote (Ci(t))i=l,,.,n,a set of non-intersecting cells (dual cells). The integration of the gas equation on the space-time volume gives the following numerical scheme:

where a: is the volume of the cell Ci, a"'" a conservative approximation of the convective flux with the accuracy of order rn = 1 or m = 2, the source term produced by the gas-parcel interaction, W: are components of components of Pn-3 and aPn = pn+f-pn-f.

w", P;-+

are

Fields of application

583

Let us denote V ( i ) the set of the neighboring cells and T ( i )the set of the @? are defined as follows for neighboring elements of the cell Ci.Then @"nd the first order finite volume scheme:

The accurate computation of B:e is costly. When i is a vertex of the triangle r and the parcel is defined by an element r at the time

-

we can use the approximation

(elt

2

tn+t (d;++ E r) then

-n+

8+:),

where Bi,!

4

is the barycentric

coordinate of 2,"" associated t o the vertex i in the triangle. Otherwise, we can assume that !?:,( 0. The fluid step is resumed by the following system:

This system is linear (diagonal) for explicit schemes (p = n) and non linear for implicit scheme (p = n 1). In practice we use, in this last case, a linearized implicit scheme obtain as follows: +',m

(wn+l)

-

+

+r,m ( f i n ) + j ( l )(wn+l

- wn where 9(')=

)

-

(17) . . The linearized implicit scheme is then reduced t o the resolution of a non symmetric linear system. The resolution of this scheme is in general achieved by Gauss-Seidel relaxation method. It is the costly part of the implicit algorithm.

Application: Injection in a piston engine Let us focus now to the piston engine application. In this case, the jet(s) position(s) and characteristic are assumed known. The fluid initially at rest is spray compressed by the piston motion in the Ricardo prechamber. Then the + is injected at the position defined by 2 i n j ( t ) , with a mean velocity K n j ( t ) 115 m/s and a mean drop radius rin, = 4 * 10-6 m. The injection starts after 27ms and ends a t 28ms with a total mass of 6.8g injected. A total number of 6060 numerical parcels are injected. The numerical behaviour of the spray and the boundary conditions are validated in this context (Fig 1 ). Computations has been performed with a PVM parallel code and the figures are ploted in the case of 4 sub-domains partition.

-

584

Finite volumes for complex applications

[CLO 971

CLOUET,J.F. AND DOMELEVO, K . Solution of a kinetic stochastic equation modeling a spray in a turbulent gas flow. Math. Models Methods Appl. Sci., 7(2):239-263, 1997.

[DOM 971

DOMELEVO, K. AND SAINSAULIEU, L. A numerical method for the computation of the dispersion of a cloud of particles by a turbulent gas flow field. J. Comput. Phys, 133(2):256-278, 1997.

[DUK 801

A particle-fluid numerical model for JOHN K. DUKOWICZ. liquid spray. Journal of Computational Phisics, 35:229-253, 1980.

[NKO 971

B. NKONGA.On the conservative and accurate cfd approximations for moving meshes and moving boundaries. To be published in Computer Methods in Applied Mechanics and Engineering.

[OR0 811

PETER J . O'ROURKE.Collective drop effects on vaporizing liquid spray. PhD Thesis, Los Alamos National Laboratory, 83, November 1981.

Figure 1: Spray density evolution and Velocity field at 23.06ms Particles, Implicit scheme, moving mesh of 4202 vertices.

:

6060 total

Finite Volume Methods for Multiphysics Problems

C.Bailey, M.Cross, li.Pericleous, G.A.Taylor, N.Croft, D. Wheeler, H. Lu Centre for Numerical Modelling and Process Analysis Unzverszty of Greenwich Woolwich, London SE18 6PF, UK

There is a growing need for accurate and robust multi-physics based simulations tools that have the ability to model fluid flow; heat transfer, possibly with solidification or melting; and solid deformation in an tntegrated manner. Engineering Processes are now benefiting from the use of such tools where potential defects are being identified at an early stage in the desrgn process. This paper details a finite volume formulation and solution strategy for predicting the physics associated with solidification based processes. ABSTRACT

Key Words: Multzphysics Modelling; Solidification; Casting; Electronic Packaging

1. Introduction A number of real industrial processes are governed by interacting physical laws, where the materials involved in the process may be both solid and/or fluid. An example of this is the casting of metal components where the metal, originally above its melting point, is placed into a mould cavity then allowed to cool and solidify into its required shape. T h e challenge for the modelling community, provided by this type of problem, is to capture all the relevant physics taking place and ensure that there interactions are properly siliiulated using a single modelling framework. Although finite volume methods are very well established for fluid flow phenomena, the dominant discretisation technique for solid ~llechanics(hence stress) is the finite element method. Lately, finite volume methods have been used to analyse solid mechanics problems on unstructured meshes [DEM 951,

586

Finite volumes for complex applications

[BAI 951, [TAY 96].These methods have proved to be as accurate as classical finite elements for both linear and non-linear materials. Integration of this type of stress analysis within a finite volume CFD framework provides the capability of solving fluid-structure interation problems using a single computational framework.

2. Governing Equations

2.1 Fluid Flow The general equations for the conservation of momentum and mass are:

where g is the velocity vector and ,u, p , and p are the dynamic viscosity, density and pressure a t time t . The source term S contains the buoyancy and Darcy terms due to temperature changes and solidification.

The conservation of energy is given by

where h , k and T are the enthalpy, thermal conductivity and temperature. L is the latent heat, which is the energy released due to solidification, and fi represents the liquid fraction (1 = liquid; 0 = solid).

2.3 Solid Mechanics

T h e incremental equilibrium equations are :

Auij,j = 0

i, j = x,y, z

Using Hookes Law the incremental stress, Au, is given by:

AU = [Dl ( [ L ] A d- Att - Acplp)

Fields of application

587

where [Dl is t,he elasticity matrix and A € , Act and AcUpare t,he total, thermal and viscoplastic incremental strains respectively. [L] is the differential operator matrix and Ad are the displacement increments.

3. Discretisation

Figure 1 details two types of control volume that can be used in the finite volume discretisation procedure. Cell-centred control volumes are the same as the mesh elements used to represent the domain. Vertex based control volumes Vertex based control volumes

Cell centred control volume

Figure 1: Control Volumes are built up around each node where the boundary for each control volume is the face that joins the element centre to the centre of the element faces.

3.1 Cell centred For the cell centred approach, velocity components, temperature, pressure, and liquid fractions are stored at the centres of each mesh element making up the control volume. The general form of the conservation equations for mass, momentum, and energy can be written as:

where, from left to right, the terms represent the transient, convective, diffusive and source contributions respectively. The dependent variable is represented by

588

Finite volumes for complex applications

q5 which is unity for mass, the velocity component for momentum, and enthalpy for energy. Integrating the above equation over each control volume and using the divergence theorem gives:

Linear approximations are used for the transient term and source terms over the control volume. The diffusive and convective terms are then approximated at the faces of the control volumes. This gives:

mn,

Af and dap are the where the summation is over the number of faces. dependent variable at neighboring control volume, face area, and distance between centres of neighboring cells. To obtain accurate values of the diffusion coefficient, r f , at the face the harmonic mean is used. For pf , in the convection term, upwinding is used and to evaluate the normal component of the velocity at the face, ( g . ~ )the , Rhie-Chow interpolation method is used. The value of a t the face is calculated using arithmetic averaging.

3.2 Vertex based IJnlike the flow and heat equations, above, the stress equations are integrated over vertex based control volumes. Integrating the equilibrium equations over a vertex-centred control volume and using the divergence theorem gives:

1

AoijSjdv=

C Anij.nAf

(9)

f where the incremental stress components, Auij, are approximated at the faces of each vertex based control volume. Using shape functions, the displacement at any point within a mesh element can be written as a combination of the values stored a t the nodes of that element, therefore:

Ad = [ N ]Av

(lo)

where [N] is the matrix of shape functions and Av is the vector of vertex stored displacements. The equilibrium equations, integrated over each control volume,

Fields of application

589

can now be written in terms of nodal incremental displacements [TAY 961:

4. Solution Procedure The above discretised equations for dependent variables at both cell-centred and vertex-centred control volumes can be assembled to provide systems of equations for the dependent variable 4 representing velocities, temperature. fluid pressure, and nodal displacements. The form of each systerll of equations is: [A]$ = b (12) where [A] is the system matrix and b contains source terms. These integrated systems of equations are solved using the following solution procedure:

(1) Solve flow equations using the SIMPLE procedure, [PAT $01 (2) Solve Energy equation for temperatures and liquid fractions [VOL 911 (3) Evaluate other variable quantities, eg. physical properties, etc (4) Repeat steps 1-3 until convergence (5) Solve incremental solid displacements (6) Update total stress variables. a ,, = cold + A a (7)Recalculate geometrical quantities (8) Repeat steps 1-7 for time-step advancement Note, in step 7, the geometrical quantities are recalculated due to the changes in displacement from the computational solid mechanics analysis, and the equations in steps l and 2 are rediscretised over the updated mesh.

5. Results

Figure 2 shows the computational domain of a silicon chip that is bonded to a printed circuit board using very small solder bumps. This is achieved by passing this assembly through a furnace, heated to 230C, where the solder, in the form of a paste, melts then, as the assembly leaves the furnace, solidifies t o form the required joint. Under the solder bumps, copper connections are used to carry d a t a into the printed circuit board. The PHYSICA modelling

590

Finite volumes for complex applications

framework [PHY 981 is used to simulate this fusion process, where the physics of heat transfer (solidification) are coupled with stress to predict the resulting stress profiles in the assembly. Figure 3 shows both the solidification

Figure 2: Computational domain for electronic package. profiles for the solder bumps and the stress profiles throughout the assembly (dark contours representing high stress). Clearly we can see that the thermal miss-match between the materials in the assembly is giving rise to stress concentrations around the copper via connections. The next example illustrates coupled flow, solidification, and stress modelling for the casting of lead ingots. Here molten lead is poured into an iron mould and then allowed to cool and solidify. Figure 3 shows the computational domain for this analysis. A quarter of the domain is shown with symmetry planes at the centre. It is assumed that the mould is instantly filled and that the cast and mould temperatures start at 730C and 300C respectively. As the molten lead cools, thermal gradients will evolve resulting in thermal convection as lead flows from hotter to cooler regions. Figure 4 shows the magnitude of thermal convectioil after 30 minutes of cooling. Figure 4 also shows both PHYSICA and thermocouple temperature profiles at the centre of the ingot. The affect of latent heat release during the phase change from liquid to solid can clearly be seen. The PHYSICA predictions compare very favourably with the gathered data. Figure 6 (left) shows the solidification profile (dark area being liquid) after 100 minutes of cooling. As expected the lead solidifies inwards from the

Fields of application

Figure 3: Solidification profiles in solder bumps and stress in package.

Figure 4: Computational domain for mould and lead ingot.

591

592

Finite volumes for complex applications

mould walls towards the centre of the casting.

Temperature 800

---4

Corn loom Plant datai6Dmml

100 2 0 0 300 400 500 600 700

Time'mins'

Figure 5: Thermal convection and temperature comparison.

Figure 6: Solidification front and stress in mould. As the lead cools it heats the surrounding mould in a non-uniform manner. The thermal gradients occurring in the mould material results in thermal stress. Figure 6 (right) shows the magnitude of stress throughout the mould (dark area being high stress) after 100 minutes of cooling. These type of predictions provide useful information for mould desiginers to minimise stress concentrations and hence cracks in the moulds.

Fields of application

593

6. Conclusions

The rationale behind the PHYSICA framework is to supply modellers with a tool which enables the development of computational mechanics based models where interactions between the participating physics can be accomplished in a robust and efficient manner. This paper has summarised the discretisation and solution procedures used within the PHYSICA framework. The capability of this modelling approach for real world processes is illustrated for electronic packaging and lead casting.

Bibliography [BAI 951

BAILEYC . CROSSM . , A Finite Volume Procedure to Solve Elastic Solid Mechanics Problems in Three Dimensions on an Unstructured Mesh, Int. J . Num. Meth. In Eng., 38, 17571776.

[DEM 951

DEMIRDZIC I. MUZAFERIJA S., Finite Volume Methods for Stress Analysis in Complex Domain s, Int. J. Num. Meth. In Eng, 37, 3751-3766

[PAT 801

PATANKAR S., Numerical Heat Transfer and Fluid Flow, Hemisphere, 1980.

[PHY 981

PHYSICA, A Softwa.re Environment for the Modelling of Multiphysics Phenomena: http://phys ica.gre.ac.uk/

[TAY 961

TAYLORG . A . BAILEYC. CROSS M., Solution of Elastic/Viscoplastic Constitutive Equations : A Finite Volume Approach, Appl. Ma.th Modelling, 19, 746-760 (1996).

[VOL 911

VOLLERV. SWAMINATHAN C., General source-based Method for Solidification Phase Change, Num Heat Transf - Part B, 19, 175-189.

A Finite Volume discretization and multigrid solver for steady viscoelastic fluid flows

H. A1 Moatassime S. Raghay and A . Hakim Faculte' des Sciences et Techniques, Dkpartement de Mathematiques et Informatique, B.P. 618, Marrakech, Maroc

ABSTRACT A finite volume technique has been introduced i n an attempt to simulate a viscoelastic flow. The steady flow of a Oldroyd-B fluid through a 4:1 abrupt contraction has been chosen as a prototype example due to the existence of previous simulations i n the literature. The finite volume method ( F V M ) is used to discretise the conservation and constitutive equations and a multigrid technique is used. The F V M is proven to be quite capable to handle numerically, viscoelastic models with low computational cost. Its use is recommended as a viable alternative to the solution of viscoelastic problems using a variety of constitutive models. Key Words: Numerical simulation; finite volume; planar contraction; viscoelastic flows; Oldroyd-B model; non-uniform staggered grid; multi.qrid technique. 1. Introduction For the past several years, numerical simulation of viscoelastic flows has been a powerful tool for understanding the fluid behaviour in a variety of processes of both industrial and scientific interest. Polymeric fluids, owing to their viscoelastic character, are of particular interest in the numerical simulation because of their wide applications in material processing and their behaviour different from that of Newtonian fluids in ways which are often complex and striking. Although there have been many successful numerical complexities that arise due to the change of type, i.e., hyperbolic-elliptic or hyperbolicparabolic. The earlier numerical schemes such as central finite differences [Per 771, Galerkin finite elements [Cro 841 and spectral finites elements [Ber 871 to solve the equations are unstable when the elasticity of the flow becomes

596

Finite volumes for complex applications

significant. While each of them has its advantage and disadvantages, the search for even better and/or faster methods still continues. In that respect, it was inevitable that the finite volume method should also be tried within the viscoelastic context. However, the viscoelastic simulations with FVM are very limited [Sas 951. Contraction flows of viscoelastic fluids are of importance in fundamental flow property measurements as well as in many industrial applications involving molding and extrusion of polymer melts and solutions. They are also a challenging class of problems for analytical and numerical work. Numerous theoretical studies have been devoted to the study of viscoelastic fluids in contraction flow, and there is a large literature on this subject; see for example ([Bog 871, [Mit 901). However, the theoretical prediction of entry-flow for nonNewtonian fluids remains a different task. The constitutive equation used to describe the rheology of the polymer has fading memory effects, and may contain nonlinear terms that add to the complexity of the problem. The presence of a geometrical singularity poses another major challenge to the numerical simulation of contraction flows. In this paper, we conduct numerical simulation of the planar contraction flow of viscoelastic fluid. The FAS ( f i l l Approximation Storage) multigrid algorithm is used to simulate large Weisenberg number.

2. Governing equations

For the two-dimensional flow of an incompressible viscoelastic fluid, the governing equations are continuity, the x and y component momentum equations, and three component stress equations from the Oldroyd-B constitutive equation. The solution of these equations, with appropriate boundary conditions, gives u, v, T,,, rVV,rXy and the pressure p as functions of x, y. The equations are expressed below in terms of dimensionless variables.

+

+

Re(u.v)u v p = (1 - w,) A u v.7 div u = 0 T + We[(u.V)r P(T, VU)] = 2w,D

+

Boundary conditions The boundary conditions employed for the velocity are given below:

Fields of application

597

Figure 1: Computational domain for contraction pow In inlet and outlet section : Fully developed Poiseuille flow is imposed

where Uu = $ and Urn = The extra-stress components are fixed along the inlet section. They correspond to a fully developed flow between parallel plates.

at the centerline we impose symmetry conditions:

and adhesion at the wall:

3. Numerical method

To solve the above coupled non-linear equations at moderate cost, the use of an iterative numerical method has to be made. It has to be noticed that all of the governing equations can be written in the form of the general transport equation as follows:

where 9 is the working variable which can be a component of a vector or tensor and even a constant. The coefficients A and I? have different meanings

598

Finite volumes for complex applications

Figure 2: The control volume AV for grid point P for different equations, and S+ is called the source term which includes all the terms that cannot be t,aken into account in the convective and diffusion terms, and has different contents for different equations.

A simple finite volume formulation is used for the spatial discretisation. The flow domain is divided into a set of control volumes A V around P with bounding surface area A as shown in Figure 2. Integrating equation (1) over the control volume A V and using the divergence theorem, we have

where Sa is the integral of the source term S*. This can be linearized as

Fields of application

599

in whichx, is the part of the SQ that does not explicitly depend on G and Sp is the coefficient of Gp,which is made negative t o enhance the numerical stability of the discretized equation system (2). where, an overbar means that the applied values are evaluated using the known fields for iteration level n. By using a proper spatial variation approximation scheme, the final discretized equations relating the Gp t o its neighbouring grid point values can be expressed symbolically in a general form in every control volume:

where

where Dnb and P, are respectively the diffusion coefficient and the Peclet number. sign(nb) = when nb E {w, s) and sign(nb) = -when nb E {n, e). The summation is to be taken over all the neighbouring grid points nb of the central point P, and the coefficients anb are the functions of the working variable, and their structures depend both on the approximation scheme used and the form of the cell. These coefficients determine the spatial accuracy of the final solution. In our calculations, the Hybrid scheme proposed by Patankar [Pat 801 is employed t o solve momentum equations,

+

f ([Pel) = m a x ( 0 , l - 0.5 [Pel) while Upwind scheme is used in the constitutive equations for the stresses.

f (IPel) = 1. In viscoelastic flows the convective terms are too small in comparison with the source terms and special treatment of the source term is needed. The computation of these terms requires the first gradient of T,,, T,, and T,, . The term d ~ , , / d x , as like as the other terms, is assuming quadratic variation of r,, along the x-direction. Thus , d ~ , , / d x is written as 2ax b.

+

Solution m e t h o d To obtain the kinematic fields, an equation for the pressure is obviously necessary because it is also an unknown. The strategy of pressure correction is utilized to produce the pressure equation, in which the continuity of the field is enforced via a pressure correction so that the resulting pressure relation, which couples the pressure and the velocities, replaces the continuity relation while the momentum equations retain their role for determining the velocity field. To avoid physically unrealistic field, such as the checker-board velocity and pressure distribution, staggered control volumes have to be used. As shown

600

Finite volumes for complex applications

in Figure.3, for the P-centered control volume for scalar fields, the velocities uk (k = x , y)are discretized using their values on the faces normal to the k direction; thus, the location of the control volume for uk in the momentum equations is staggered only in k direction relative to the control volume for the fields (cf. Figure 3). The steps of present algorithm are as follows :

* Computations of velocities.

* Computation of pressure and correction of velocities

* Computation of stresses. * Convergence control and return to first step if necessary. Multigrid Procedure After discretization we get a non linear finite dimensional operator Ln and we have to solve L,(u,p, 7) = bn In order to solve this problem we use a FAS (Full Approximation Storage) multigrid algorithm. Let (Qk)llkln a set of n grids of domain Q, Rn is the finest grid and R1 is the coarsest grid. For k = 1,.- . ,n let Lk a non linear finite dimensional operator approximation of L on the grid nk. We denote by (uk,pk, rk) an denotes the extension operator approximation of (u,p, 7) on the grid Rk. from grid flk t o grid flk+l and 7Zt-l is the projection operator from grid Qk to grid flk-1. On each grid we solve

EL+'

by a relaxation procedure Mk and we denote by R ~ ( u ~ ( ~ ) , ~ ~ ( ~the ),T~(') vector obtained after m iterations with the initial vector (uk(l),pk(l), ~~(l)). The object of the FAS algorithm is to solve

on the finest grid 0, by solving intermediate problems (4) on coarser grids in order to reduce the number of iterations on the finest grid.

Fields o f application

601

Figure 3: Two-dimensional staggered mesh and the control volumes for v, v,p well as I-

FAS algorithm (0) (u,, p,, 7,) initial approximation of solution of (5) (1) k := n , rk) = M 2 k(uk1pk,T ~ (mk ) iterations of relaxation on grid flk) (2) ( ~ kpk, (31 Compute bk-1 = ~ ; - l ( b k - Lk((vk,pk,~k)) + L ~ - ~ ( R : - ~ ( U ~r ,kP) )~ , (~k-i,pk-1,rk-1) = 72k-l ( z L ~7 k, )~ (initializing ~ , of solution on grid RkPl) k := k - 1 (crossing to the coarser grid) (4) If k > 1 go to (2) Else (~1,pl,.rl)=M~'(~1,pl,~1) (ml iterations of relaxation on the coarsest grid R1) ( 5 ) Correct the approximation obtained on flk+l with the extension of

k := k + 1 (crossing to the finer grid) ( 6 ) (vk,pk,T ~ =) MEk( u k , p k r, k ) (pk iterations of relaxation on grid Rk) ( 7 ) If k < n go to (5) Else convergence test If convergence done, good, end Else go to (2) In practice for the FAS procedure we take ml = 10, p=3 and mk = 2(k = 2 , . . . , n) for small Weissenberg numbers and ml = 50, p=20 and mk = 20(k = 2, . . . , n) for large Weissenberg numbers.

4. Numerical Results

The flow along the centreline has an elongational nature, and the elongational properties of the rheological models play an important role in determining the way it responds to the accelerating forces when it approaches the corner. Velocity profiles for Weissenberg number We = 1 are plotted in Figure

602

Finite volumes for complex applications

4. This figure shows an important velocity overshoot near the coner. Marchal and Crochet [Mar 871 have also predicted similar velocity overshoot and high stress peak near the singular region in their high De calculations with Oldroyd-B fluid. First normal stress difference profiles for different values of the Weissenberg number are reproduced in Figure 5. The maximum of (T,, - T,,) is reached just upstream of the entry section. As We increases, the value of this maximum increases sharply. Profiles of the normal extra-stress r,, as well as T,, are presented in Figure 6. This figure shows a high T,, and T, peak near the corner. In Figure 7 we present distribution of T,, and T,, for (Re,w,, We) = (0.01,0.8,1). It is clear that the re-entrant corner represents a singularity, many lines pass a cross this point.

Conclusion The planar 4 : 1 contraction flow of Oldroyd-B fluid has been simulated by utilizing a finite volume method with multigrid technique on a non-uniform staggered grid system. With FVM the velocity and pressure fields are made t o satisfy one the same momentum equation a t the end of each step, and the pressure correctors are used to correct the velocity field only. To make the method suitable for viscoelastic flow computations, decoupled techniques are employed, and artificial diffusion terms are introduced on both sides of the discretized constitutive equations to stabilize the numerical calculation. With the method, thanks to the modest demand on memory, it becames possible t o solve large problems on small computer systems. As a results, the present numerical simulation have allowed us to reproduce such of the experimental results.

Bibliography [Per 771

M.G.N. Perera, K. Walters, Long-range memory effects in flows involving abrupt changes in geometry. Part I. Flows associated with L-shaped and T-shaped geometries, J. Non-Newton. Fluid Mech. 2 (1977) 49-81.

[Cro 841 M.J. Crochet, A.R. Davies, K. Walters, Numerical Simulation of Non-Newtonian Flow,Elsevier, Amsterdam, 1984. [Ber 871 A.N. Beris, R.C. Armstrong, R.A. Brown, Spectral/finite element calculations of the flow of a Maxwell fluid between eccentric rotating cylinders,J. Non-Newton. Fluid Mech. 22 (1987) 129-167. [Bog 871 D.V. Boger,Ann. Rev. Fluid Mech., 19 pp. 157-182, 1987.

Fields of application

603

Figure 4: U and V on y = 0 and y = 1 respectively

Figure 5: (rxx- T,,) on y = 1 for We = 0 . 8 , l and 1.2 [Mit 901 E. Mitsoulis, Numerical simulation of viscoelastic fluids, in : N.P. Cheremisinoff (Ed.),Encyclopedia of Fluid Mechanics, vol. 9, Polymer Flow Engineering, Gulf Publishing Company, Dallas, Texas, USA, 1990, pp. 649-704. [Pat 801 S.V. Patankar, Numerical Heat Transfer and Fluid Flow,McGrawHill, New York, 1980. [Sas 951

G.P. Sasmal, A finite volume approach for calculation of viscoelastic flow through an abrupt axisymmetric contraction, J. Non-Newton. Fluid Mech. 47 (1995) 15-47.

[Mar 871 J.M. Marchal and M.J. Crochet,J. Non-Newton. Fluid Mech. 26 (1987), 77-114.

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Finite volumes for complex applications

Figure 6:

T,,

and

T,,

on y = 1 respectively

Figure 7: asolines of r,, and

T,,

respectively

Complexity, Performance and Informatics

Various CG-type methods applied to finite volume scheme

'

0. Schmid DLR,Cologne, Germany A . Buflmann, E. von Lavante and M. Moczala University of Essen, Essen, Germany

This paper compares comprehensively preconditioned classical, Lanczos and Arnoldi methods for a subsonic turbine cascade. The well known symmetric Gauss-Seidel (SGS) is selected, representing a classical method. General minimal residual method ( G M R E S ) is chosen as a member among the Arnoldi methods and Bi conjugate gradient stabilized ( B C G S T A B ) represents a truncated non-symmetric Lanzcos method. Two preconditioners are tested. The first one is the prolific diagonal incomplete lower upper (DILU) preconditioner with zero fill-in. The second one is the lower upper symmetric Gauss-Seidel (SGS) preconditioner. ABSTRACT

Key Words: Computational Fluid Dynamics, Navier-Stokes Equations, Finite Volume Method, Implicit Method, Upwind Shemes, MUSCL-Extrapolation, CG-type Methods, Classical Methods, Steady Flow, Convergence, Viscous Flow, Iterative Solution.

1. Introduction The numerical simulation of typical turbomachinery components, such as compressor and turbine rows or stages, requires excessive amounts of computational resources. This is not only due to the highly complex physics of the flow, which makes very fine computational grids necessary, but also because of the need for high timewise accuracy.

'presently University of Essen, Essen, Germany

608 Finite volumes for complex applications The flow in these components is almost always unsteady; however, at times, it might be admissible t o filter out the time dependent part of the flow. Under this assumption, the flow can be considered steady, and the computation might converge to a steady state result. In this case, one is interested in accelerating the rate of convergence in a way that is not necessarily time accurate, marching in a properly defined pseudo-time. The problem of acceleration of a numerical scheme for solving the compressible Navier-Stokes equations in complex components of turbomachines can be approached by implementing an appropriate implicit procedure. The implicit operator should provide fast convergence to steady state results for the steady computations. Numerous different implicit algorithms for solving the Navier-Stokes equations have been published to date. These include, for example, the work of Barth [BAR87], Jameson [JAMSl], Pan and Lomax [PAN861 and Wright et a1 [WRI96], to name just a few. What was missing, in our opinion, is a comprehensive comparison of the different implementations of implicit solvers with regard to their ability t o predict complex flows in modern turbomachinery, such as highly loaded turbines or compressors. In the present work, several implicit methods were obtained by using different approximate inversion procedures of the same descrete delta form of the governing Navier-Stokes equations. These were applied to numerical simulations of several simple test cases as well as the experimentally tested and documented TI06 turbine cascade. The computed data were compared with available theoretical or experimental results.

2. Basic Algorithm The governing equations are the two dimensional Navier-Stokes equations in divergence form written as

where Q is a vector of dependent conservative flow variables and F and G represent the flux vectors. The well known details of this model equation can be found in [JAM911 and [PU85].

Complexity, performance and informatics

609

Discretization of equation (1) using first order accuracy in time yields:

where the superscripts denote the time level and the subscripts denote the spatial location. The fluxes F, G, F, and G, are the time properly defined numerical fluxes, obtained from a particular spatial discretization. 6< and 6, are the difference operators acting on the numerical fluxes. In the present work, the spatial discretization of the governing equations was carried out by Roe's Flux-Difference Splitting procedure [ROE84]. The interpolation slope of the primitive variables was limited by the modified Albada limiter, according to the previously published MUSCL type procedure (see, for example, [EVL94]). The implicit operator is defined in the delta form of equation (2), obtained by a proper linearization of the numerical fluxes F, G, F, and G,. It can be and B = written, after splitting the Jacobians of the flux vectors A = into a matrix with non-negative eigenvalues A+ and non-positive eigenvalues A-, as

In the above equation, M and N are the Jacobian matrices obtained from the linearization of the viscous fluxes F, and G,, respectively. They were cenb f trally differenced and denote backward and forward differencing respectively.

&, &

The convective fluxes in the explicit part were linearized by assuming that Roe's numerical damping term was locally constant. Taking, for example, the numerical flux 1 2

Pi++= [ F i+ Fi+l - l A l i + + ~ ~ i + + ] , this results in +

( A

+A

)

and

;4 ;-

$(Ai+l - IAli+;),

with the rest of the Jacobian matrices evaluated correspondingly. 2.1 Inversion of the Implicit Operator

Equation (3) can be expressed in a simple form as

MAQ:j = RHS where M denotes the implicit operator.

(4)

610 Finite volumes for complex applications

The solution of equation (5) involves the inversion of the implicit operator matrix M , which is rather large, sparse and linear. A non-iterative direct method t o invert it would be, among several possibilities, the Gad-elimination. However, this method of inversion of M would result in a dense matrix and would therefore require large amounts of computer memory. Worse yet, many fill-ins would have to be computed, making the algorithm slow. Another possible method of inverting M would be one of the approximate factorization (AF) schemes, such as the AD1 scheme using the Thomas algorithm. This method is simple and cheap t o implement, but introduces a factorization error that effectively limits the CFL numbers to rather small values. It has yet another significant disadvantage: in the case of zonal grid decomposition it requires special implicit treatment of the interzonal communication. With a simple interzonal boundary condition, the AD1 scheme induces large errors on the zonal boundaries delaying the convergence significantly. Therefore, the present authors decide to use and test several iterative methods that, in our opinion, offer an acceptable compromise between computational efficiency and complexity. The iterative inversion of M allows updating the interzonal boundary values after each subiteration, providing a simple implicit-like boundary condition. 2.1.1. Relaxation Methods Using equation (5) as a starting point, relaxation methods split the implicit operator in two parts. After some trivial manipulations, one gets the following equation, which describes every relaxation method:

where k denotes the subiteration level. For the Symmetric Gad-Seidel scheme (SGS), the matrix P is

Here D is the diagonal part of M , L is the lower part of M and U is the upper part of M. The solution will be reached by a forward and a backward sweep. 2.1.2 Conjugate Gradient Methods

CG-type methods use the non-symmetric or the symmetric Lanzcos process t o build up the Krylov subspace. After that the residuum is minimized over the given subspace by a Galerkin or a least-square condition. Subiterations are introduced in the following way

Complexity, performance and informatics

The minimum vector

61 1

ck is calculated through a least-square procedure,

+

llrkll~

= rninll60 - MVk&;;1I2 = 0,

(9)

or by implementing a Galerkin condition

-

s-T r, =

s ; r ~-~ zTlI1fiy',

= 0.

(10)

Without preconditioning, the tested CG-type methods behave very poorly. For this reason, two preconditioners, the distributed incomplete lower upper preconditioner (DILU), and the symmetric Gad-Seidel preconditioner (SGS) were implemented. Both preconditioners are well suited for block matrices. All the tested solvers are using the same termination condition for the maximum number of subiterations. The subiteration process is stopped when either the term is smaller than lo-', or the maximum of 500 iterations is reached.

3. Results The above schemes were tested on several two-dimensional configurations possessing increasing degree of complexity. The initial simple test cases were a bump in a channel, a flat plate and a subsonic diffusor. There are well known test cases treated in detail previously and will not be, therefore, discussed here.

3.1 TI 06 Turbine Cascade

The axial TI06 turbine cascade has been designed and extensively experimentally tested by Fottner et al. [FOT78] and is therefore well suited for further validation of the present results. The flow in the T106 turbine cascade was computed for an inflow Mach-number of 0.29 using the implicit methods mentioned above. The computational grid, consisting of 5 blocks of grids (zones) is shown in Fig. 1. Here, the body-fitted 0-grid is split into two parts, connected by three H-grids. The convergence informations for the implicit schemes discussed in the previous chapter are summarized in the tables below. The resulting parameters shown are: P2, the exit flow angle avaraged over the cascade width; <, the total pressure loss parameter, defined as C = ( p t l - pt2)/(ptl-pa); maximum available CFL-number; the maximum computer RAM requirement and the total CPU time on a SGI-Power Challenge given in hours. Where applicable, the computed values are compared with experimentally measured data. All the simulations were carried out using the k-w turbulence model.

612

Finite volumes for complex applications

In this case, the fastest convergence t o 6 degrees of magnitude was obtained using the SGS relaxation scheme. The most effective CG-type procedure was the BiCGSTAB method, combined with the SGS preconditioner, directly followed by the method of steepest descent using the SGS preconditioner. CG met,hods with DILU preconditioning were not really competitive.

Figure 1. Computational grid for the TI 06 turbine 11

measured

11

JAC

I

GS

I

BGS

I

SGS

]

Table 1. Relaxation-type schemes I 8 2 lo]

C [%I

measured

SD+SGS

BiCGSTAB+SGS

FOM(S)+SGS

GMRES(S)+SGS

26.80 2.08

27.42 8.73 40.0 1.81 1861

27 70 8.72 40.0 1.77 1920

27.39 8.73 40.0 2.46 2008

27.39 8.73 40.0 2.51 2023

CFL C P U - t i m e [h] memory [KB]

Table 2. CG-type schemes with SGS preconditioning 0, [Yl C [%I

CFL C P U - t i m e [h] memory [ K B ]

measured

SD+DILU

BiCGSTAB+DILU

26.80 2.08

27.30 8.97 35.0 3.20 1920

27.40 8.73 40.0 2.47 1979

I

FOM(S)+DILU 27.39 8.73 40.0 3 15 2067

GMRES(5)fDILU 27.39 8.73 40.0 3 14 2082

Table 3. CG-type schemes with DILU preconditioning The density contours, shown in Fig. 2, are a clear indication of the relatively simple flow in this configuration. Evident is the viscous wake behind the turbine blade and the boundary layer, thickening on the suction surface towards the trailing edge due to the adverse pressure gradient.

-

Complexity, performance and informatics

613

Figure 2. Tl06 density contours 4. Conclusions Different iterative schemes for implicit solutions of the Navier-Stokes equations were tested on several configurations from the area of turbomachinery. Several iteration strategies were implemented. The subiterations resulted in implicit-like interzonal boundary conditions. The fully implicit procedures were effective in accelerating the timewise computation, allowing much larger time steps than the explicit formulation resulting in up to six-times faster convergence. For the present configurations, the most effective implicit procedure was the SGS, followed closely by BiCGSTAB with SGS as preconditioner. Overrelaxation methods using a relaxation parameter between one and two burst after a few time steps, due t o the high relaxation parameter. Every preconditioned CG-type method converges faster than the Gaufi-Seidel method. Preconditioned CG-type methods depending on the truncated nonsymmetric Lanzcos process are faster than those restarting the symmetric Lanzcos process t o construct the Krylov-subspace. Nevertheless all of them can not reach the convergence history of the symmetric Gaufi-Seidel method, bringing up t o mind that the use of the SGS scheme is really an advisable choise t o calculate fluid dynamic problems.

614

Finite volumes for complex applications

5. Acknowlegements and Bibliography A major part of the present work was done during the lead authors stay at the DLR, Institute for Propulsion Technology. Many calculations were done with the TRACE-S code developed by the lead author's project leader D.T. Vogel(DLR) who helped the lead author in several tasks.

[BAR8 71

BARTH,T. J . , Analysis of Implicit Local Linearization Techniques for Upwind and TVD Algorithms, AIAA paper 870595.

[JAM911

JAMESON, A., Time Dependent Calculation Using Multigrid with Applications to Unsteady Flows Past Airfoils and Wings, AIAA paper 91-1596.

[PAN861

PAN,D. A N D LOMAX, H., A New Approximate LU Factorization Scheme for the Reynolds-Averaged Navier-Stokes Equations, AIAA paper 86-0337.

[WRI96]

WRIGHT, M. J . , CANDLER,G . V., A N D PRAMPOLINI, M., Data-Parallel Lower-Upper Relaxation Method for the Navier-Stokes Equations, AIAA Journal, Vo1.34, No. 7, July 1996.

[ROE841

ROE, P . L. AND PIKE,J., Efficient Construction and Utilisation of Approximate Riemann Solutions, Computing Methods in Applied Sciences and Engineering, VI, pp. 499-516, INRIA, 1984.

[EVL94]

LAVANTE,E., HILGENSTOCK, M. A N D GROENNER, J., Simple Numerical Method for Simulating Supersonic Combustion, AIAA paper 94-3179.

[PUL93]

Pulliam, T. H., Time Accuracy and the Use of Implicit Methods, AIAA paper 93-3360-CP.

[PU85]

PULLIAM, T. H. AND STEGER,J . L., Recent Improvements in Efficiency, Accuracy and Convergence for Implicit Approximate Factorization Algorithms, AIAA paper 85-0360.

[FOT78]

FOTTNER,L., HAPPEL.H.W. ET AL, Anwendung neuer Entwurfskonzepte auf Profile fiir axiale Turbomaschinen, Teil 11, AbschluBbericht Zukunftstechnik Luftfahrt 1978 MTU 4.14-2.

[STE96]

STEINERT, W., SCHREIBER, H. A., WEBER,A., Experimente am transsonischen Verdichtergitter, DLR-TSG-89-5, DLR Kijln 1996.

VON

A Newton-Relaxation Finite Volume Scheme for Simulation of Dynamic Motion Bruce A. Jolly, Magdi Rizk Sverdrup Technology, Inc. Eglin Air Force Base, Florida

A computational fluid dynamic method is used to simulate relative motion between moving bodies. A CFD scheme called Beggar, is a finite volume scheme designed for the specific purpose of simulating flow about stores and store separations from aircraft. This paper describes the dzfierent aspects of the method. First, the automated grid assembly scheme, which uses polygonal mapping tree data structures is discussed. Second, the flow solver based o n a Newton relaxation scheme for time-accurate flow solutions is described. A MK84 general-purpose bomb trajectory from the inboard wing station of a fighter aircraft is predicted and compared t o experimental data. Also, a triple release of three generic stores from a wing is demonstrated. ABSTRACT

K e y Words: Upwind, Newton, Polygonal mapping, Trajectory.

1. Introduction

The USAF is continuing the development of a CFD method called Beggar. This code originated in the Wright Laboratory [BEL 951 and was transferred to the Air Force Seek Eagle Office (AFSEO) in 1993 for application. Currently, the use of CFD coupled with a dynamic rigid body solver or 6 degree-of-freedom (6-DOF), provides separation data that can be used in certification recommendations [BRK 961. The cost per event is a fraction of a single flight test and can be repeated within 24-48 hours on multi-processing computers. Beggar also offers the advantage of easy and simplified input. All fluid boundaries such as blocked, interpolated, singularities, and freestream conditions are automatically detected. User specified boundary conditions are easily described using common English terms. Such boundaries include no-slip, tangent, and mass flow conditions among others. The solution for the coupled, time accurate, CFD and 6-DOF simulation consists of four operations: grid assembly, flow solution, force and moment calculations, and integration of the equations of motion. Each operation is executed a t every time step. The flow

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Finite volumes for complex applications

calculations are run across multiple processors and executed concurrently with the other three operations [PRW 991.

2. Grid Communication

To accommodate moving bodies, an overlapping grid system is used. Such a grid is limited only by its ability to interpolate with surrounding grids and is labeled a superblock. For relative simple geometries and those containing fixed fins, a blocked grid system (by point-to-point matching) is often desired to form a superblock. A blocked grid system can be used t o construct a single superblock but no two superblocks, by definition, can construct a blocked system. Superblocks combine to form a d y n a m i c group. Each moving body and the global static system is represented as a separate dynamic group. The core of the Automated Grid Assembly Scheme (AGAS) is the polygonal mapping (PM) tree, which is a combination of an octree and binary space partitioning (BSP) tree data structures [SAM 901. To maximize efficiency, point classification relative to other superblocks is performed by a three-tier search. At the first level, the point is compared to a superblock bounding box to rapidly determine the general location of the point relative to a superblock. Once determined to be near a superblock, the octree data structure quickly determines the point t o be in, out, or near a boundary of the superblock. Finally, if the point lies within an octant containing superblock boundary point, the BSP tree is used to determine whether the point is inside or outside the grid. 2.1 D a t a structure

The octree data structure is one in which a region of space is recursively subdivided into octants. At n levels, a spatial domain can be divided into 8" sub-domains. A grid point can easily be located within a child octant by comparing its coordinates to that of the parent octant. The BSP data structure is one in which each node in a binary tree is represented by a plane definition. The two children of each node represent the in and out sides of a facet boundary. If a facet is clipped by an existing plane in the BSP tree, the clipped pieces of the same plane definition will be placed in different branches of the tree. The PM tree data structure makes use of both the octree and BSP data structure to increase efficiency and reduce the BSP levels. The BSP tree is used to determine whether a point is either inside or outside a grid boundary. Since it is not guaranteed to be well balanced, it is limited t o the leaf octants that contains a boundary facet. The PM tree data structure is constructed by refining the octree until a single octant contains no more than one grid point from the same dynamic group. The boundary faces of the cells attached to this

Complexity, performance and informatics

617

point are used to define facets inserted into the BSP tree. All superblocks in a dynamic group share a single octree data structure but the BSP tree is unique to each superblock. Once the PM tree data structure is complete, it never has to be determined again. During dynamic simulations, coordinate transformations are used to examine points that have not moved relative to the original data structure of the moving grids. 2.2. Grid assembly

When a solid surface from one superblock is embedded in another superblock, a hole must be cut into the latter. AGAS uses the facets of the superblock boundary representing the solid surfaces as hole-cutters. Cells to be cut by the facets are determined from the PM tree and the nodes of the cell are labeled to be either inside or outside the cutting grid. If the facet vertices do not lie in neighboring cells then the facet is subdivided until they do. A superblock that is cut by a surface will require flow variables to be interpolated from other superblocks just outside the resulting hole. An interpolated point is one that requires its flow variables to be updated via an interpolation donor and represents a fringe or boundary point for the flow solver calculations. The interpolation donor is provide by valid fluid points from other superblocks and is determined by the P M tree. Once the P M tree identifies which superblocks the grid point that needs interpolation lies in, stencil jumping using a Newton method determines the final position of the point relative to potential donor cells. If no valid donor is found for an interpolated point, the point is labeled as out and its neighboring cells are marked as needing interpolation. The resulting hole propagates until either a valid donor is found or failure is declared. 3. Flow Solver Formulations and Schemes

The Navier-Stokes equations may be written relative to a general coordinate system, <(x, y, a, t), ~ ( xy,, Z , t ) and <(x,y, a, t), in the form

where Q = Jq = [Jp, Jpu, J p v , Jpzli, ~

+

Ki, = J ( q ~ t finz

K,, = J(f,n,

e

]

~

+ gin, + hi&*)

+ guny+ h v n z )

w.

J is the Jacobian of the inverse transformation, given by J = Ki, and K,, are respectively the inviscid and viscous flux vectors associated with

618

Finite volumes for complex applications

the K curvilinear coordinate. They are equal to Fi, G i , Hi and F v , G,, H v if K. equals to (, q or C, respectively. Similarly, fir gi, hi and f,, g,, h, are respectively the Cartesian inviscid and viscous flux vectors per unit area in the x, y and z directions. (u, v, w) is the Cartesian velocity vector, p is the density and e is the total energy per unit volume. The inviscid flux vector is given by

+

+

+

where 0, = nt UK, VK, wn, The viscous terms are discretized using central differences. The thin-layer approximation which neglects cross derivative terms is used. The eigen system in the n coordinate direction is determined by the flux Jacobian A, = dKi,/dq. This may be expressed as A, = R,A,Ril where R, is the matrix of right eigen vectors and A, is the diagonal matrix of eigen values. In the Steger-warming scheme, the matrix of eigen values is split into a matrix with only negative eigen values and another with only positive eigen values. This allows the splitting of the flux vector Ki,, which may be expressed as Ki, = A, q, into two flux vectors associated with disturbances propagating in the negative and positive K, directions. Upwinding can then be applied separately to the two split flux vectors. In the Roe scheme, a Reimann problem (shock tube problem) is solved approximately at each time step to determine the flux vector at each of the cell faces. The initial conditions for the Reimann problem at both of the cell face sides are obtained by MUSCL extrapolation using the Van Albada limiter. The difference between the flux vector at the cell face and the flux vector associated with either of the initial conditions is expressed in terms of the summation of the flux differences across the characteristics curves propagating from the cell face. The Euler equations in characteristic form [WHI 841 along the n direction may be linearized and expressed in terms of a set of Riemann invariants. Combining the appropriate set of these equations with the far-field conditions results in the characteristic far-field boundary conditions. Noncharacteristic boundary conditions are used at solid boundaries, where a zero pressure gradient boundary condition is used in the case of inviscid flow and the no-slip boundary condition is applied in the case of viscous flow. Numerical solutions of the flow governing equations require discretized approximations of the equations. Time is discretized into finite time intervals, while the computational space is discretized into finite volumes. Discretized flow variables are assigned to each time interval and finite volume. Let i, j , k be the cell indices along the 5,q ,5 coordinate directions, respectively. The discretized equations at the n+l time step, using the first order Euler implicit time discretization, are given by

Complexity, performance and informatics

619

where

provided the computational mesh is non deforming. Here, F, G and H are the numerical fluxes associated with the three curvilinear coordinates, while 6<, 6, and dC are central difference operators. The nl is the index for the n+l step. For a time accurate problem, Eq. ( I ) is satisfied a t each time step, iteratively, by using Newton's method [WHI 911. The application of this method leads to

where superscript m and ml are the index for the current and next Newton iteration respectively. While Eq. (1) is the discretization of the flow governing equations a t a single cell, Eq. (2) combines the equations for all cells within an individual grid into a single vector equation. Here, AG, 4 and R are vectors which combine the five-component vectors Aqi,j,k, qi,j , k and Ri,j , k l respectively, for all the cells within a grid. A q and q are vectors which combine the five-component vectors Aqi,j,k and qi,j,k, respectively, for all the phantom cells extending beyond the grid boundaries for the purpose of applying the boundary conditions. j is a diagonal matrix with the cell volumes along its diagonal. A T is a diagonal matrix which is introduced so that the fixed time step may be replaced by a variable time step in the iterative procedure. While the right hand side of Eq. (2) is derived based on either second order Roe fluxes or second order Steger-Warming fluxes, the flux Jacobians in the left hand side are approximations derived using fluxes from Steger-Warming flux vector splitting and first order differences. The incremental correction to the flow variables in the m l iterative step within the n l time step is calculated by solving Eq. (2) for A q n l ~ m literatively, , using the symmetric Gauss-Seidel (SGS) relaxation scheme defined by

where, superscript 1 is the index for the SGS iterations. Each of the iterations includes a forward pass ( p = 0) and a backward pass (/I = 1). The coefficient a block strictly lower triangular matrix, matrix for Aqnl~mlis split into Dn1>" a block diagonal matrix, and unl,m a block strictly upper triangular matrix. For a given inner iteration the phantom cell incremental cell solutions, Aq"l>ml, are held fixed while the interior cell incremental solutions are

620

Finite volumes for complex applications

updated. The phantom cell solutions are then updated so that the boundary conditions are satisfied. While the boundary conditions are applied explicitly for each inner iteration, at convergence of the inner iterations, an implicit updating of the boundary conditions will have been achieved [VAN 931. 4. Dynamic Solutions

4.1. MK-84 separating f r o m aircraft

A MK-84 general-purpose bomb trajectory from the inboard wing station of an F-15E aircraft is predicted [COL 961. The store is released in level flight at a Mach number of 0.90. The computed store trajectory is compared to experimental data obtained using the captive trajectory system (CTS) at the Arnold Engineering and Development Center's 4 foot transonic wind tunnel. The CFD solution is computed assuming inviscid flow. The total grid system consisted of 2.2 million points in 33 grids, making up 23 superblocks. Figure 1 shows the trajectory graphically through composite snapshots in time. The physical time step during the dynamic calculations was .5 milliseconds and the code was executed using a single processor on a CRAY C90. The total computer time required to complete 0.2 seconds in physical time was 55 hours. Today, this same solution can be repeated on an workstation in 24 hours. As seen in Figure 2, the trajectory compares well with experiment.

'

XB-wr

.

YB.%T

YB-CFD ZB-W 5

U--

$-

I

B

I

0

Figure 1: MK-84 Trajectory

005

0 1

011

'rim(SoF)

02

Figure 2: MK-84 Trajectory Comparisons, Position

4.2. Ripple release demonstration

For demonstrations purposes, the release of three generic finned-stores was

Complexity, performance and informatics

621

computed [PRW 961. This shows the versatility of the dynamic method and Figure 3 illustrates the mesh interaction between moving grids. The three stores, wing, and interface grids consisted of 2.2 million points in 16 grids (for more efficient parallelization the grids have been split to as many as 67), making up 10 superblocks, and four dynamic groups (the wing is considered a dynamic group though frozen). Figure 4 shows the trajectories graphically through composite snapshots representing 0.3 seconds in physical time from release. This configuration has been calculated on many platforms including CRAY C90 and SGI ORIGIN 2000. It is being used as a data set for research being conducted in the efficient parallelization of AGAS [PRW 991.

Figure 3: Trzple Release Grid Systern

Figure 4: Triple Release Trajectories

5 . Summary

The USAF is developing and using a unique computational method for the prediction of trajectories. This method consist of an integrated set of tools that enable the classification and assemble of grids into a composite grid using novel data structures. With all grid boundaries and interpolation stencils defined, an unwind finite volume flow solver is employed to calculate fluid motion. Coupled with a 6-DOF trajectory simulation, and an efficient parallelization scheme, the capability exists to accurately and rapidly simulate aircraft and store separation events. Future development plans include multi-thread parallelization, finer grain parallelization of the AGAS, and optimization of flow solver convergence.

Acknowledgements This work was performed under the direction of the SEEK EAGLE Office of the United States Air Force. The authors would like to acknowledge the

622

Finite volumes for complex applications

contributing work of the entire team including Dr. Ralph Noack, Steven Standley, Shawn Westmoreland, Larry Coleman, and Joseph Keen. Special thanks to Jim Brock who is the Air Force lead for this team, Ray Maple and Nathan Prewitt for their creation and advancement of AGAS, and Dr. Davy Belk for his visionary ideas from which Beggar was created. Bibliography [BEL 951

BELK,D.M., MAPLE,R.C., "Automated Assembly of Structured Grids For Moving Body Problems,'' AIAA Paper 951680.

[BRK 961

BROCK, JAMES, Computational Fluid Dynamics Inviscid Store Separations Verification, AFTDC T R 96-01, March 1996.

[COL 961

COLEMAN,L.A., JOLLY, B.A., CHESSER,B.L., BROCK, J .M ., "Numerical Simulation of a Store Separation Event From an F-15E Aircraft," AIAA Paper 96-3385.

[PRW 961

PREWITT, N.C, BELK, D.M., MAPLE, R.C., "Multiple Body Trajectory Calculations Using the Beggar Code," AIAA Paper 96-3384

[PRW 991

PREWITT,N.C., BELK,D.M., SHYY,W . , "Parallel Implementations of Time-Accurate CFD Applications With Moving Chimera Grids," Proceeds of the 9th SIAM Conference on Parallel Processing for Scientific Computing, March 1999.

[SAM 901

SAMET, H. ,Applications of Spatial Data Structures: Computer Graphics, Image Processing, and GIs, Addison-Wesley, 1990.

[VAN 931

VANDEN,K . J . , WHITFIELD,D.L., "Direct and Iterative Algorithms for the Three-Dimensional Euler Equations," AIAA Paper 93-3378, July 1993.

[WHI 841

WHITFIELD,D.L., JANUS,J.M., "Three-Dimensional Unsteady Euler Equations Solution Using Flux Vector Splitting," AIAA Paper 84-1552, June 1984.

[WHI 911

WHITFIELD,D.L., TAYLOR,L.K., "Discretized NewtonRelaxation Solution of High Resolution Flux-difference Split Schemes," AIAA Pa,per 91-1539, June 1991.

An Attempt to Develop a Multi Purpose FAS Multigrid Algorithm Luc Fournier*,Oliver Gloth** *INRIA, Projet Sinus, Sophia Antipolis (France) **Institutefor Combustion and Gas Dynamics, University of Duisburg (Germany)

One of the most recent challenges in scient$c computing is creating a Pexible and open development platform for numerical simulations. From the implementation point of view, modern programming languages offer poweifiul tools forpexibility, such as the inheritance of object oriented programming or C++'s template mechanism. The numerical approach however should be flexible as well. The present paper mainly treats the numerical aspects of an FAS multigrid algorithm built in a such context to improve the numerical eficiency. To keep such a platform open for new developments, time explicit schemes together with ajnite volume discretization are used. This discretization allows the usage of a volume agglomeration technique to construct a sequence of meshes. This is close to be a black box multigrid algorithm, accelerating the convergence of various problems. The main difficulties as well as approaches to overcome those are presented here. For all this parts of the FAS algorithm, flexibility in terms of the equations to be solved, as well as the discretization used, was a major guideline, although not fully achieved by now.

ABSTRACT

KEY WORDS :FAS

multigrid, explicit schemes, object oriented programming

1. Introduction The resolution of non linear PDEs can be done using time explicit as well as time implicit formulations. By saying explicit, it is meant that the variables on a new time level only depend on the state of the previous time level. Implicit schemes are generally based on solving a linear system of equations for every time step. They are known to be numerically efficient. Unfortunately, new physical problems or new discretizations require more implementation work. Consequently, explicit methods are suitable for an open development platform if their inefficiency for steady state problems can be overcome. Since a few decades, multigrid methods are used to accelerate the convergence properties of numerical methods. Due to the non linearity of explicit schemes, a Full Approximation Scheme (FAS) multigrid method (described in [BRA82]) has to be used for those. Explicit schemes provide an easy way of implementing different equations in an existing unstructured development platform. It seems to be natural to also find a sufficient way of generating mesh sequences for the FAS algorithm, without

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demanding too much user interaction. Please note, that user does not only stand for end users, but also for members of a research team mainly working on discretizations for physical problems. The volume agglomeration technique as presented in [LSD921 is considered as a black box method for generating finite volume coarse levels. Some more recent works have proved the efficiency of this in the field of computational fluid dynamics [MV96]. The first part of this article presents the implementation of an FAS algorithm as a basis for efficient iteration schemes for various problems. All developments presented in this paper have been done using the MOUSE C++ library [GVH97]. The development of this library has been initiated approximately two years ago. Its major intention is to be a flexible framework for finite volume computations. The second main part is devoted to some numerical experience, made with the solution of the Euler equations for compressible gas flow.

2. A multigrid approach 2.1. The FAS method

The FAS multigrid method is used to solve the following non linear equation:

where u denotes the field of unknowns, @ the non linear operator and S a source term. h and H, used as subscripts, refer to the fine and the coarse grid. As in every multigrid method we need inter-grid operators. I: and :f are the operators to restrict the unknowns and the residual from the fine to the coarse grid. The prolongation of the correction from the coarse to the fine grid is written as ~ hThe . ideal two grid FAS algorithm can be described using four steps:

1. pre-srhoothing: 1f u f ) " = u f ) is the initial value of the unknowns, we find a new estimate u p ) ' 1 of this field using several iterations of an explicit smoother (simply an explicit iteration scheme of the equation). r?)" = S h - @ h ( u p ) " ) now represents the residual of this operation. 2. Solution of the coarse grid equation: The following equation in U H is solved. H (k),l -H ( k ) , l @ H ( U H ) = S H = @ H ( I ~ U ~ ) -I h r h

[21

For the ideal two grid algorithm this has to be done until a sufficient level of convergence is reached. 3. correction: Using the converged solution of [2], it is now possible to correct the variables on the fine grid. u(k)82 - u h( k 1 3 1+ I & ( U H - I hH uh ( k ) , l) 131

4. post-smoothing: A new estimation u!") of the field is found, using several iterations of the explicit smoother, with u f ) " as initial value.

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2.2. Volume agglomeration technique

The agglomeration technique is used to build a set of coarse finite volume grids. A coarser level can be obtained by clustering neighbouring volumes. Each cluster of volumes will form a control volume of the coarsened mesh. Used recursively, this technique produces a sequence of grids with nested control volumes. Figures 1 and 2 show the principle of this agglomeration algorithm applied to a mixed element sample mesh. The most used technique to agglomerate control volumes is a greedylike algorithm.

Figure 1. elements and associated control volumes

Figure 2. elements and agglomerated control volumes

But for 3D problems, Mavriplis[MV96] proposes a priority list to initialize a pure greedy technique. In [ON0981 a coloring edge-based method is found. These variants show an interest on the quality of the agglomerated cells. The basic agglomeration of the implementations presented in this paper is improved by using a local optimization criterion. It can be observed that the regularity of the agglomerated levels influences the convergence rate.

2.3. The Object Oriented Library As mentioned above the FAS method, as it is described here, is used to extend an object oriented development platform. One of it's main design principles is the encapsulation of physical, mathematical and computer related parts of the code as much as possible. The main features of this library are: - dynamic data storage classes - interpreted language for high level numerical implementations - template based operator mechanism - meshing and adaptation classes - basic visualization tool.

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The multigrid operators easily fit in this framework. Once implemented in C++ they form an extension to the control language and can thus be used to implement different iteration techniques. One major goal should be the independence of the equation to be solved. This could not be completely reached by now, since the development within this framework is fairly young. The authors however, are optimistic to be able to finish the work towards a flexible toolkit for finite volume applications using FAS for convergence acceleration. Up to date information on the developments can be obtained from http://www,vug.uni-duisburg.de/MOUSE.

2.4. Implementation of multigrid operators 2.4.1. Spatial discretization using the defect correction technique The second order flux computation is performed on the fine grid using the MUSCL interpolation [VL79]. This technique is based on a reconstruction of the state on the cell interfaces, using linear extrapolation. The extrapolation is performed using the gradients of the variables, which are computed using Green's formula for field nodes and a least square approach for boundary nodes. On the coarse levels, created by the above described agglomeration technique, the position of the nodes and the control volume borders becomes irregular. This makes a second order discretization using MUSCL extrapolation very difficult. Experiences show that such a discretization is very critical in terms of stability for the explicit time stepping. Therefore it does not seem to be reasonable to apply a higher order approach on the coarse grids. The defect correction method [DH95] proposes to solve the equation [I] iteratively by building a convergent series, based on a problem which is simple to inverse. At each iteration the following problem has to be solved:

sP)

@h,l(?hp+l)) =

with

s?) = Sh -k @ h , ~ ( ~ -p )@) h , 2 ( u p ) )

) respectively denote a first and a second order discretization where ( ) and on the fine level. Each system [4] is partially solved using one cycle of the FAS algorithm. When converged (i.e. GhVl( u( 7,1 + 1 ) ) = @ h , l ( u p ) uh ) ) is the solution of [l].

2.4.2. Runge-Kutta for smoothing The FAS algorithm allows to treat non linear equations. On every level the exact equation, including all boundary conditions, has to be modeled. In the presented FAS implementation an explicit Runge-Kutta is used for smoothing, as well as for solving the equation on the coarsest level. For linear multigrid methods a Jacobi or GaussSeidel relaxation is often used for smoothing. In [WES91] a study of this technique for a set of linear model equations can be found. Unfortunately, this requires a point implicit formulation of the discretization. To allow the method to be easily used with different discretizations, a fully explicit Runge-Kutta method has been chosen. The

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main problem when using Runge-Kutta as a smoother, is that the smoothing properties are dependent on the number of sub-steps, the coefficients used, and the CFL number. This is shown by a von Neumann analysis of the one dimensional convection equation [5]. It is discretized using an adjustable first order upwind scheme [6].P = 1 represents a full upwind and P = 0 a central scheme. Time integration is done by a three step Runge Kutta scheme with the coefficients a1 = a2 = $, a3 = 1.

a,

Figure 3. amplification factor

(P = $)

Figure 4. amplification factor (P = 1 )

The von Neumann analysis gives an amplification factor of this scheme as a function of the time step At and an angle O E [O..II]. O = II represents the shortest and O = 0 the longest resolvable wavelength. As it is noted in [WES91], the smoother does not have to be convergent. It has to be efficient for high frequency disturbances. Unfortunately, within the scope of this work, it has not been possible to apply a theoretically unstable Runge Kutta scheme with good smoothing properties for high frequencies. Figures 3 and 4 show the amplification factor as a function of At and 0 . Dark colour represents a high amplification. Isolines are drawn for amplification values of (0.1,0.2,0.3, ..., 1.0). Please note that values > 1 indicate an unstable scheme. 2.4.3. Boundary conditions

The FAS multigrid algorithm needs a correct expression of the boundary conditions on every grid level. As explained in 2.1 a partially converged solution is obtained on each grid. On every level the same set of equations is modeled. The only difference can be found in the position of the boundary nodes. Only on the finest mesh the nodes are on the boundary itself (half cells). This makes the implementation of strong (nodal) formulations for boundary conditions easy. For the agglomerated levels this is not the case anymore. The node position can no longer be assumed to be on the boundary. Figures 5 and 6 illustrate this.

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Figure 5. Fine grid control volumes

Figure 6. 1st agglomerated level

Furthermore, geometric properties (e.g. surface normal vectors) are difficult, if not impossible, to reconstruct on the coarse levels. To still imply the boundary conditions on the coarsened levels, they are included in the source term of the defect correction scheme. The second order discretization @h,2(U(hn1) includes a nodal formulation of the boundary conditions (see equation [4]). This formulation works well in case of the compressible Euler equations. Here a nodal boundary condition forces the velocity vectors to be parallel to surfaces. For Navier Stokes computations, where a pure Dirichlet boundary condition is demanded (non slip condition on surfaces) it is not working this way. This contradicts with the demand of a flexible algorithm. So the boundary condition problem is only partially solved by now. However, one explanation is that for the inviscid computations a weak formulation of the boundary conditions (pressure flux on surfaces) is always present. This is not the case for viscous computations. To compute a friction-caused flux on non-slip surfaces, velocity gradients are needed. Unfortunately, these gradients are very hard to compute on an agglomerated mesh. 2.4.4. Inter-grid operators Transfer operators are usually considered to be the weak part of the volume agglomeration technique. The restriction for the residual is simply a summation of all fine grid cells which contribute to one agglomerated cell. For the variable vector an averaging over all contributing cells is employed. Please note that this is done, using the control volume sizes as weighting factors. There are two different kinds of prolongation (interpolation) operators implemented. One is a canonical injection as in [LSD92], meaning that the fine grid values are simply set to the corresponding coarse grid values. Another operator is a linear interpolation using the gradients on the coarse nodes . These gradients are computed using the same combination of a Green integral and a least square algorithm as described earlier.

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3. Numerical experiments The test which is presented in this paper is a fully subsonic ( M , = 0.3), inviscid and symmetric flow around a NACA0012 airfoil. Two different grids (3 114 nodes and 12284 nodes) have been employed. In figure 7 the larger mesh can be seen. Figure 8 shows isolines of the Mach number for a converged solution. The inner and outer operator (a2and @ I in equation [4]) are dicretized using an adjustable upwind scheme (see also equation [ 6 ] ) .For the inner iteration a half upwind (,L? = has been used for all runs.

i)

Figure 7. NACA0012 (12284 nodes)

Figure 8. Isolines of the Mach number

Convergence histories can be found in the figures 9 and 10. Each plot shows three curves (full upwind explicit, full upwind FAS, half upwind FAS). The x-axis represents the number of FAS V-cycles and the y-axis the residual. Fastest convergence could be obtained by a full upwind discretization of @ z for the defect correction scheme. The oscillatory curve in both figures represents a mono grid solution obtained by an explicit three step Runge Kutta time integration. The x-axis is scaled to represent approximately the same CPU time per unit as for the FAS solutions.

Figure 9. convergence (31 14 nodes)

Figure 10. convergence (12284 nodes)

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4. Conclusions The present paper has depicted the numerical and technical aspects of a flexible (black-box like) multigrid implementation. Numerical flexibility, as it is demanded, introduces difficulties. The agglomeration technique, as well as the inter-grid operators, are not be dependent on the physical problem. However, the boundary conditions are. Thus different strategies for coarse grid boundary conditions, for different physical problems seem to be necessary. One future approach could be to classify the boundary conditions and to offer flexible and reusable solutions for every class. Dirichlet boundary conditions introduce most problems, since there are no pure boundary nodes on an agglomerated grid. If a general way of prescribing boundary values (e.g. least square extrapolation) can be formulated, the treatment of boundary conditions becomes problem independent. First numerical experiments show strong interactions between different parts of the algorithm. Therefore a better theoretical understanding of the different components (e.g. defect correction scheme, explicit smoother, inter-grid operators ...) is desired. Another challenge for this multigrid approach is the computation of time-dependent problems. A possible approach to be able to do this is a dual time stepping method [BH93].

References [BH93] M. BREUERand D. HANEL. A dual time-stepping method for 3-D, viscous, incompressible vortex flow. Comput. Fluids, No 22, p. 467-484, 1993. [BRA821 A. BRANDT.Guide to multigrid development. Multigrid methods, in Lecture notes in Mathematics, No 960, p. 220-312, 1982. [DH95] J.-A. DBSIDBRIand P. W. HEMKER. Convergence analysis of the defect-correction iteration for hyperbolic problems. SIAM J. Sci. comput., No 16, p. 88-1 18, 1995. [GVH97] 0 . GLOTH,R. VILSMEIER, and D. HANEL.Object oriented programming for computational fluid dynamics, HiPer'97, Krakow, Poland, 1997. [LSD921 M.-H. LALLEMAND, H. STEVE,and A. DERVIEUX. Unstructured multigridding by volume agglomeration: current status. Comput. and Fluids, N021, p.397-433, 1992. [MV96] D. J. MAVRIPLIS and V. VENKATAKRISHNAN. A 3D agglomeration multigrid solver for the reynolds-average navier-stokes equations on unstructure meshes. Int. J. for Num. Meth. in Fluids, N023, p. 527-544, 1996. [ON0981 N. OKAMOTO, K. NAKAHASHI, and S. OBAYASHI. A coarse grid generation algorithm for agglomeration multigrid method on unstructure grids, 36th Aerospace Sciences Meeting and Exhibit, N098-0615 in AIAA, Reno, 1998. [VL79] B. VANLEER.Towards the ultimate conservative difference scheme V: a second-order sequel to Godunov's method. J. of Comput. Phys., N032, p. 361-370, 1979. [WES91] P. WESSELING.An introduction to multigrid methods. John Wiley & Sons, 1991.

On Higher Order Accurate Implicit Time Advancing for Stiff Flow Problems

Ce'cile VIOZAT(**), Eric SCHALL(*), Alain DERVIEUX(*), David LESERVOISIER(*)(***)

(*) I N R I A , BP 93, 06902 Sophia-Antipolis Cedex (France) (**)CEA-SA C L A Y , 91 191 Gif-sur- Yvette ceder (France), (***)SNECMA, Centre de Villaroche, 77550 Moissy Cramayel (France)

It is proposed a study of the application of a third-order accurate implicit time advancing for compressible flows at large time steps. A two-stage Runge-Kutla scheme is applied to low Mach number flows. The possibility of using an adaptive time step is addressed.

ABSTRACT

Iiey Words: High-order time accuracy, adaptive time step, implicit RungeIiutta method, unsteady %ow, low Mach number, natural convection.

1. Introduction The accuracy of numerical approximation is before all determined by the amount of information devoted to the description of a given phenomenon. In advection phenomena, not only advected quantities have to be well described by the spatial mesh, but also time step size has to be fine accordingly (except for characteristic methods, for which part of the discretisation is avoided) ; the limitation on the time-step size for accuracy is expressed by the CourantFriedrichs-Lewy (CFL) condition (that is also a stability condition for explicit time advancing schemes). If we apply an implicit time advancing, using a larger time step then, either the numerical scheme is highly dissipative and dissipation errors will be dominant, or Gibbs oscillatior~swill appear.

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Now, not only such dissipative schemes are often first order accurate ones, but also, even with high order accurate schemes, the practical accuracy cannot be better than first-order accuracy when details are not captured. By "practical accuracy order", we mean the apparent order of accuracy for a given size of time step, by opposition t o the formal order of accuracy, that is an asymptotic one i.e. applying only for "small enough" time steps. This remark is in fact no true for all kind of flows. A special category is the case of low Mach flows. In the case where asymptotic models are very accurate, then their combination with implicit schemes (that are mandatory for these models) allowing highorder accuracy is a very efficient strategy (large time steps, good accuracy). In the case where a complete compressible Navier-Stokes model is applied, the question of the efficiency of a strategy relying on large time steps and high-order accurate time advancing is open. The purpose of this paper is to examine the use of a third order accurate implicit scheme in a complete compressible Navier-Stokes model with some emphasis on low Mach flows. Third-order accuracy is rarely implemented in conlplex CFD codes. Yet, third-order accuracy is accompanied with small phase errors for advection phenomena ; then spurious oscillations are also reduced as compared with second order accuracy. We first describe the CFD code and discusse the motivation for choosing the Norsett time advancing, then we examine the calculation of a natural convection phenomenon in which acoustics are initially predominant. The question of finding adapted time steps is finally addressed.

2. Implicit Time Advancing in CFD We concentrate on implicit advancing for compressible flows, and more precisely on Godunov-type upwind methods.

2.1 Upwind Method

In this section we describe in short the upwind spatial discretisation that we consider in the sequel. The numerical scheme is described and discussed in [DER 921. Starting from a given triangulations, equipped with nodes attached t o vertices, we build a dual control volume tesselation by dividing each triangle into three parts. A first-order accurate Godunov method for the Euler equations is written in terms of the conservative variables W = ( p , pu, pv, E) by expressing the following flux equality:

where a, is the cell area, V ( i )holds for the direct neighbors of vertex i, qij the

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integral of normal vector along the common boundary between cell i and cell j , @(Wi,Wj, q;j) corresponds to the integration of the Euler fluxes through the common boundary with the application of a Riemann solver. Symbol "B.F." holds for boundary fluxes for which we do not want to give details here. In practice, we apply the Roe approximate Riemann solver. The right and left input of the Riemann solver are cell values Wi and Wj, which results in a standard manner into a spatially first-order (at best) scheme. In this work we use the Roe approximate Riemann solver. The necessary last feature is the (spatially) second-order accurate extension. We follow the lines of van Leer's MUSCL method ([VL 771) and write the second order accurate fluxes in terms of cell wise reconstructions at cell interface:

In the sequel, the semi-discretised system will be written in short as follows:

where "B.T." holds for boundary terms. The above approximation is enough accurate for transonic flows, but degrades for low Mach number ones. During the last ten years, many authors have shown that the direct application of the Roe Riemann solver results in very bad approximations. In [VIO 981, some insight is given about the problems that arise in this context, and it is confirmed that the introduction of for example the Turkel preconditioning (see e.g. [VIO 981) permits to recover accuracy for flows with low, but rather uniform Mach number. 2.2 Implicit Tame Advancing : BDFl

The Backward Euler or first-order (in time) backward differencing (BDFl) scheme applied to (3) writes: A~w"" = -Q2

(wntl)

(4)

where 6Wn+l = Wn+l - W ' q s the increment between Wn and Wntl corresponding to time levels n and n + 1, and in which A is the diagonal mass matrix, the input of which are:

where a; and At: hold respectively for the area of cell i and the related time step length. When the flux function XP2 is differentiable, it can be linearised through Taylor's formula:

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then introducing the development (5) in expression (4)) we get a linearised version written under the 6-form :

From a truncation analysis of scheme ( 6 ) for a 1D model we derive the error order is 0 (at,ax2). In practice, the flux Q z is not differentiable, or difficult to differentiate, the memory requirement of the resulting sparse matrix is large and it has been found interesting to introduce linear operators that are approximations of the jacobian aQ2/aW.

Linearised Scheme with First Order in Space This hybrid order onelorder two scheme is a simplified version of ( 6 ) where a linear operator relying on order one spatial approximation is applied t o the "implicit part" : ( A P n ) 6wn+l = -*2 ( W n ) (7)

+

where the linear operator Pn is the sum of a genuine jacobian for viscous fluxes and a first-order (in space) approximate jacobian for Euler fluxes:

Further, in the t.urbulent case, mean flow and closure variables are treated by separate matrices t.hanks to the freezing of production terms and turbulent viscosity. The advantages of this matrix is that its storage concerns only direct neighbors and it is diagonally dominant for simplified models ; this second feature allows applying rustical algorithms such as Jacobi or Gauss-Seidel. At each iteration, the linear system is actually solved by a multigrid scheme described in [FRA 981. For unsteady laminar flows, the accuracy of the above scheme turns to be of 0 (At, A t A x , Ax2), which in practice is not much accurate and limits the size of time steps usuable for a given accuracy. For turbulent flows, the accuracy may be degraded by the decoupling between mean variables system and closure variables system.

2.3 An Implicit Runge-Kutta Scheme

We can predict that passing to second-order accuracy in time will reduce the enormous amount of dissipation that is introduced by B D F l since in the usual truncation analysis, it is a second derivative in time that will disappear. From this point of view, stability will be notably degraded. Passing to third

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order will only "kill" a dispersion term (third order derivative), which means that the dissipation situation will essentially not change, but we reduce the phase error, an important feature for unsteady calculations. This work concentrates on passing to third-order accuracy. While second order accuracy in time is easily obtained with the extension BDF2 relying on the second-order difference in time:

that will produce an unconditionally stable scheme for both advection and diffusion, on the contrary, BDF3 is unconditionally stable only for diffusion operators. We have chosen a nonlinear scheme with two implicit systems to solve a t each iteration, the Singly Diagonal two-stage Implicit Runge-Kutta (SDIRIc2) scheme of Norsett [HA1 871:

Third-order accuracy can be related by a truncation analysis to the following necessarv conditions:

+

with -y = (3 &)/6. It is known that the above scheme is third-order accurate in time, uncondition77ally stable for any complex gain coefficient of nonnegative real part (Asta-

)(: >

bility), which ensures stability for both advection and diffusion At each stage of the SDIRK scheme, a nonlinear system has to be solved. We solve it by two nested loops. First a Newton-like or Defect Correction iteration is preconditioned by the above first-order (in space) simplified linearised system, we perform a number (I. of Defect Correction iterations. Second, the linearised system is solved by a multi-grid cycling.

3. A Numerical Experiment

In many transient flows of industrial interest, the Mach number is not so large, but the actual transient phenomena cover the different time scale of that kind of flow: the acoustic and convection scales. For representing this kind of

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1.8 1.6

2z

-

SDIRK2, dl=O.l --

1.4

1.2-

Figure 1. Various time length Figure 3. Nusse,lt number obtained for calculations started from time with digerent strategies, t = 0.1. a =5, Ra = lo3.

flow, we consider a transient version of the test case of De Vahl Davis [DVD 831 by specifying an initial condition : the fluid in the square cavity is a t rest with uniform temperature equal t o the (right horizontal) cold wall. Evolution will be forced by putting suddenly the temperature of the left vertical wall to a larger level and maintaining it at this level, the warm wall temperature. After fast acoustic phenomena, convection will install and progressively get steady. The Mach number in this case is of the order of l o p 5 .

3.1 A Global Comptrtation

Since time scale varies a lot, a constant time step is not adapted t o the whole calculation. After some trial and errors we have succeeded in computing the flow with some representation of each phase by imposing the time step t o increase linearly with the physical time. For result evaluation we concentrated on the Nusselt output, a number indicating the rate of calories through the cold wall. For presenting its evolution as a function of time we have t o use a logarithmic scale for time (Figure 3 ) . We concentrate on the final phase of the flow. In Figure 1, we observe that for a much larger time step, the third-order time advancing is still much more accurate; in fact, for a same level of accuracy, SDIRK is 16 times less expensive.

3.2 Adaptive Time Stepping

Since both time advancing schemes are a priori unconditionally stable, time steps as large as we want can be used stably. But accuracy can be a mess. Indeed, for BDFl applications, waves which are smaller than time discretisation may be completely damped, and for SDIRK applications, too large time steps may produce results much less good than BDF1. This is why we have to

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+n-

1

1

0.1

0.1

637

--

-

dt adapt. Order 3 dV16 Order 1 et 3 (Old) ..

0.01 -

0.01

+

, 0.001 -

0.001

-

~0.0001

0.0001 1e-05

Figure 2. Tame step adaptation : time step length for a sensor based on cl(t) (left) and for a sensor based on cAUg(t)(right).

consider the automatic adaptation of the time step length. Such adaptation is not so easy due to a particular characteristic of the flow: acoustic waves, that are quasi periodic, and that present amplitudes decreasing progressively of several order of magnitude. Many classical choice in measuring the time truncation will be faced t o the resulting signal, with large oscillations, leading to too large and too small time steps. For example, a sensor relying on maximum (in domain) of the pressure time derivative:

will produce the "thick" signal presented in Figure 2 (on the left). We consider q(t) = dpldt and its average ~ ~ u ~=( tCici(t)/nt, ) where nt is the number of nodes. The actual sensor that we propose is:

The ability of this sensor in evaluating the heterogeneity of the pressure field is much more stable, since, taking the inverse as a time step produces a rather regular time step as illustrated in Figure 2 (on the right); in fact this adaptive time step turns t o be very similar to the one obtained by trial and errors. In Figure 3, an adaptive computation with a reinforced iterative convergence at each st.age is shown t o produce a better result (with larger time steps) than a previous version especially for the final phase of the flow.

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4. Conclusion

We have shown that third order time accuracy can be observed on nontrivial problems computed with implicit time advancing. The explanation is that pressure fluctuations, that are too fast for a good representation with a large time step are in fact of too small amplitude for interfering with the overall accuracy measure. From this point of view, full compressible approximations enjoy the same advantage as asymptotical models. For too large time steps, however, we emphasize that the third order scheme can be less accurate than the first order BDF1. This is why such schemes have to be applied in combination with adapted or adaptive time step length. We have addressed the problem of choosing such an adaptive time step for a rather particular class of flows in which pronounced periodicity was a serious obstacle to a standard timetruncation evaluation. Although still a prospective choice, it proved to be an efficient solution for the problem under investigation. Acknowledgements The authors wish to thank Roger Peyret and Jerome Francescatto for their kind help in this study.

Bibliography [DER 921

A. DERVIEUX,J . A . D ~ S I D E RCompressible I, Flow Solvers using Unstructured Grid, Rapport INRIA, No 1732, 1992.

[HA1 871

E. HAIRER,S.P. NORSTETT,G . WANNER.,Solving Ordinary Differential Equations I, Springer-Verlag, 1987.

[VL 771

B. VAN LEER,Upstream Centered Finite Difference Schemes for Ideal Compressible Flow, J.Comp.Phys., Vol. 23, pp. 263275, 1977.

[VIO 981

C. VIOZAT,"Ecoulements stationnaires et instationnaires h petit nombre de Mach et maillages dtirds", T h b e d 1'universite' de Nice, 1998.

[FRA 981

J . FRANCESCATTO, A. DERVIEUX,"A semi-coarsening strategy for unstructured MG based on agglomeration", Int. J. Num. Meth. Fluids, 26, pp. 927-957, 1998.

[DVD 831

G. DE VAHL D ~ v ~ s , " N a t u r aconvection l of air in a square cavity : a benchmark numerical solution", Int. J. Numer. Methods Fluids, Vol. 3, pp. 227-248, 1983.

Numerical Solution of Steady 2D and 3D Impinging Jet Flows

Karel Kozel, Petr Louda, Jaromir Pfihoda

Department of Technical Mathematics C T U Prague Karlovo na'mbta' 1 3 CZ-121 35 Praha 2 Institute of Thermomechanics AS CR Dolejikova 5 CZ-182 00 Praha 8

ABSTRACT This paper deals with the numerical solution of 2D and 3D incompressible turbulent impinging jet flows. Numerical method is based on an artificial compressibility method, discretisation is done using cell-centered finite volume method with quadrilateral and hexahedral cells i n 2D and 3 0 case respectively, applied to structured multi-block grid. W e used several explicit and implicit schemes and tested their eficiency. The performance of eddy viscosity low-Re two-equation turbulence models is tested. Modification of Chien k-c model is proposed, which excludes the direct dependence on friction velocity. In 2D case, an algebraic Reynolds stress model is also used.

Key Words: artificial compressibility method, explicit and implicit schemes, incompressible impinging jet, turbulence models

1. Introduction

The subject of this study is to compare predictions of the averaged flow field in the steady incompressible impinging jet provided by the different turbulence

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models within the framework of artificial compressibility finite volume method. The turbulence models considered are low Reynolds number eddy-viscosity two-equation models by Launder-Sharma, Chien (see [PAT 85]), and YangShih k-6, and Menter's SST model ([MEN 941). The Chien's version, however, is modified in order not t o depend explicitly on friction velocity. For 2D mean flow, a non-linear model of Reynolds stress ([PAT 99]), recently developed at the Dept. of Mechanical and Thermal Engineering, RUG Gent, is being tested. This model implicitly includes the influence of streamline curvature. Since it requires the kinetic energy and time scale of turbulence t o be specified, it is used in conjunction with linear eddy viscosity models solving for these quantities. 2. Mathematical model

The turbulent flow of incompressible newtonian fluid in the volume V is supposed to satisfy Reynolds-averaged Navier-Stokes equations in the form

where vi denotes (time) averaged velocity vector, t time, ni unit outer normal vector of the surface d V , uij averaged stress tensor given by where p is averaged static pressure divided by constant density of fluid, dij Kronecker's delta, u kinematic viscosity, Sij = + strain rate tensor, and rij is Reynolds stress tensor.

1(2

2)

2.1 Turbulence model

To solve equation (1)for unknown velocity and pressure, the Reynolds stress tensor must be specified. The linear model follows Bousinesq's hypothesis, the non-linear one contains terms of second and third order in terms of power of mean velocity derivatives; the third order term captures the effect of rotation. The constitutive relation for deviatoric part of Reynolds stresses in linear model is given by 2 7 . . - -kd.. ~~2Sij.

"

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641

For 2D mean flow, the general representation of Reynolds stresses is as follows

where strain rate and rotation tensors are non-dimensionalised by the time scale of turbulence T and their invariants are

and C's are functions of invariants 3 and fi. When using constitutive relations for Reynolds stress mentioned above, one has to specify k , v ~ and , T . In this study, the low-Reynolds number twoequation models are used. The generic form of modelled equations for turbulence kinetic energy and dissipation rate is then

where q stands for k or

E,

and eddy viscosity is given by

Considering the modified Chien k-E model, the source terms are given by

where (with y being the distance to the solid wall)

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Finite volumes for complex applications

3. Numerical solution In order to obtain the stationary solution W = col(p, u , v, w, k, 6) of the system of equations (1) and (2)

with steady boundary conditions, we use the artificial compressibility method, i.e. we seek solution of the above system now with Q = diag(1, 1,1,1,1,1) and the same boundary conditions for t + cm. 3.1 Discretization in space

To discretize equation (3) in space, the cell-centered finite-volume method on a structured grid consisting of quadrilateral or hexahedral cells in 2D or 3D case respectively is used. Approximating convective transport of the quantity q, we use AUSM type scheme ([VIE 98]),which for hexahedral cell i in structured grid reads J 1=1 av where the unit flux fi = vlnl through the cell face 1 between cells i and i f i = (vi vi+l)n1/2 and value ql is interpolated by

+

+ 1 is

with e.g. K = 1 for central scheme and K = 113 for 3-rd order upwind scheme. Discretization of non-viscous stresses is central, as well as for viscous ones. Derivatives in viscous stresses are approximated by their mean values in the dual octahedral cell constructed over each face 1with vertices located on vertices of face 1 and in centres of the cells i and i 1. For stability to be maintained, the second order differences based artificial damping of pressure is added to the right hand side of continuity equation.

+

3.1 Discretazation in time

After dicretization in space is done, the system to solve becomes

Complexity, performance and informatics

643

with W being the cell average value. We use m-stage Runge-Kutta scheme, which has the form

For e.g. 3-stage scheme is a1 = a 2 = 112, a3 = 1, C F L = 2, and for Pstage scheme a1 = 114, a2 = 113, a s = 112, a 4 = 1, C F L = 2 f i . Following scheme is the implicit one (a # 0):

with Newton linearization of non-linear terms in the cell i:

(pi is set of cells in the stencil) we obtain linear algebraic system with 13 or 33-block-diagonal matrix in 2D or 3D case respectively. The system is solved iteratively using block tri-diagonal systems in one direction and relaxation in other directions. Only the negative part of source terms in turbulence models is treated implicitly, with approximate, empirical, Jacobi matrices. The first and third order terms of Reynolds stress is treated implicitly as long as the effective viscosity is positive (invariants 3, fi are kept constant during time step), otherwise explicitly as well as the second order term. 4. Numerical results

The grid for 2D cases is orthogonal. Fig.1 compares the efficiency (in terms of number of time steps required) of the central 3-stage Runge-Kutta scheme and implicit scheme ( a = 112, C F L = 50) with other 2 schemes not described here (see [KOZ 971) using log La norm of steady residuals. Fig.3 shows multiblock grid in (x, 9)-plane for 3D jet with axisymetric geometry. In z-direction we consider both confined and semi-confined geometry (ERCOFTAC's testcase) - there is only small difference in static pressure on the impingement wall between both cases for the distance of the walls equal 2 diameters of inlet pipe (fig.2). Isolines of k in 3D case, by Launder-Sharma k-E model and implicit scheme ( a = 1, C F L = 400), are shown on fig.6. Figures 4 and 5 show velocity and turbulent kinetic energy k on the jet axis of 2D impinging jet. Although

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Finite volumes for complex applications

Figure 1: 2D laminar impinging jet (Re = 50) - convergence histories non-linear model of Reynolds stress gives higher k around stagnation point, the mean velocity is almost the same as achieved by linear model. Figure 8 shows velocity vectors of 3D confined impinging jet.

conlined geomelIy + ERCOFTAC geomety -+--

1.5 1.4 n

1.3 1.2 1.1

1 0

0.2 0.4 0.6 0.8

1

rlD

1.2 1.4 1.6 1.8

2

Fig. 2: 3D turbulent impinging jet (Re = 28000) - static pressure along impingement wall

Fig. 3: Grid in the (2,y ) -plane for 3D jet

Complexity, performance and informatics

Fig. 4: 2D turbulent impinging jet (Re = 20000) - velocity on the jet axis

Fig. 6: 3 0 turbulent impinging jet (Re = 28000) - isolines of k and grid used

645

Fig. 5: 2D turbulent impinging jet (Re = 20000) - turbulent kinetic energy on the jet axis

Fig. 7: Isolines of turbulent kinetic energy in 2D impinging jet (non-linear model, Re = 12000)

Figure 8: Velocity vectors for confined 3D impinging jet (Re = 20000).

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Finite volumes for complex applications

Bibliography [PAT 851

PATEL V. C ., RODIW., SCHEUERER G .: Turbulence Models for Near-Wall and Low Reynolds Number Flows: A Review, AIAA Journal, Vo1.23, 1985, pp. 1308-1319

[MEN 941

MENTERF. R.: Two-Equations Eddy-Viscosity Turbulence Models for Engineering Applications, AIAA Journal, Vo1.32, No.8, 1994, pp. 1598-1605

[KOZ 971

KOZELK . , LOUDAP.: Numerical Solution of 2D and 3D Free and Impinging Jet, Proceedings of Conference "Numerical Modelling in Continuum Mechanics", Part 11, Prague 1997, Ed. M.Feistauer, R.Rannacher and K.Koze1, pp. 367-374

[VIE 981

VIERENDEELS J . , RIEMSLAGH K., DICK E.: A Multigrid Semi-Implicit Line-Method fo Viscous Incompressible Flows, Proceedings of Seminar "Euler and Navier-Stokes Equations", Prague 1998, Ed. K.Koze1, J.Pfihoda, M.Feistauer, pp. 89-92

[PAT 991

PATTIJN S. : Non-linear, Low-Reynolds, Two-Equation Turbulence Models, Ph.D. Thesis, RUG Gent, 1999 (http://allserv.rug.ac.be/-edick/tw03v/saskia/ index.html)

Acknowledgement This work was partially supported by grants No.l01/98/KO01 of GA CR, COST OC F1.70197 and Research Project of MSMT No.304/98/2lOOOOOlO.

Triangular, Dual and Barycentric Finite Volumes in Fluid Dynamics* JiFi Felcman and Miloslav Feistauer Institute o f Numerical Mathematics Faculty o f Mathematics and Physics Charles University, Prague Malostranske' n h . 25 118 00 Praha 1 Czech Republic

'The present research has been supported under the grant No. 201/99/0267 of the Czech Grant Agency and the Grant No. CEZ J13/98,113200007.

The robust adaptive higher order method is devised for the numerical simulation of the compressible gas flow. Since the viscosity and heat conduction coeficients of gases are small, the V ~ S C O U Sdissipative terms could be considered as perturbations of the inviscid Euler system. Therefore the method is based on general class of flux vector splitting schemes for the finite volume solution of the inviscid Euler equations. The structure of the viscous terms offers the application of the finite element method. The choice of the shape of finite volumes with respect to the use of conforming/nonconfonniag finite elements is demonstrated on practical examples.

ABSTRACT

Key Words: Navier-Stokes equations, transonic flow, finite volume method.

1. Formulation of the problem

W e consider gas flow in a space-time cylinder QT = fl x (0, T ) , where R C R2is a bounded polygonal domain representing the region occupied by the fluid and T > 0. The complete system o f viscous compressible flow consisting o f the continuity equation, Navier - Stokes equations and energy equation can be written in the form

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Finite volumes for complex applications

for the unknown vector-valued function of conserved quantities w. The functions f, and R, are called inviscid (Euler) and viscous fluxes, respectively. The above system is equipped with initial and boundary conditions. For their definition including the periodicity condition in the solution of a cascade flow see [FFL 951.

2. Discretization

We split (1) into the inviscid system and the purely viscous system

and discretize them separately. The time discretization of (2) and (3) is carried out with the use of a partition 0 = to < tl < t2 < . . . of the time interval (0, T) by the forward Euler method. The inviscid system (2) is discretized by the cell-centred finite volume (FV) method on a mesh D h = {DiIiEI ( I is a suitable index set), where the finite volumes Di are closed polygons. The boundary dDi can be expressed in is either the intersection 3Di n d D j or the form dDi = UjEs(i) where rij C 3 0 . S ( i ) is a suitable index set. We set lDil = area of Di, n i j = unit outer normal to dDi on hij = length of F i j . The averages of the sought solution w on Di a t time tk are approximated by values wt. The purely viscous system (3) is discretized by the finite element (FE) method on a mesh Th,that generally differs from D hused for the finite volume method. Using the above ideas, we discretize the complete system (1) via operator inviscid-viscous splitting. One time step tk + tk+l is divided into two fractional steps.

rij,

rij

rij,

Step I (inviscid FV step on the mesh Vh):Assume that the values w f, i E I, approximating the solution on the finite volumes Di at time tk are known. Compute the values w f t t , i E I , from the FV formula

equipped with inviscid boundary conditions. In (4), H is a suitable numerical flux. For its construction see e.g. [FEI 931. From the point of the adaptive mesh refinement the Osher - Solomon numerical flux seems to be the most

Complexity, performance and informatics

649

robust (for the numerical tests see [FDF 94, FFD 961). The higher order extension of the scheme (4) was proposed in [DFE 981. For a 3D version see [FSO 981.

Step I1 (viscous FE step on the mesh Th): define the finite element function - k+ 3 from the values wtitcomputed in the FV step. Apply the viscous wh as Dirichlet boundary conditions. Compute the finite element function the solution of the following problem

&+'

(i) G;+' satisfies the viscous Dirichlet conditions, -k+

,

4

for all test functions ph E V h . The definition of V h corresponds to the choice of the used finite volume mesh Dh and the type of Dirichlet boundary conditions in the viscous problem. w p dx and In ( 5 ) , ( w , p) and ah(w , p) denote the approximation of dx, respectively, obtained with the aid of the numeriR,(w, p) cal quadrature. The above system (4) - (5) is purely explicit. The stability condition from [FFD 961 is used.

c:=~

3. Combined FV - FE method

Often the structured quadrilateral FV grids are used in the numerical solution of the Euler equations. Despite their good resolution of channel flow (e. g. GAMM channel) and flow arround an airfoil (e. g. NACA 0012) the results for cascade flow (due to the complicated domain and periodic boundary conditions) are rather poor (cf. [FDF 941). Therefore we abandoned them and concentrate us on triangular generally unstructed meshes. We denote the primary triangular mesh in the combined FV-FE method by Sh = {TjIjEJ (see Fig. 1). From this mesh the unstructured FV grid Dh and F E grid Th are derived. .3.1 Dual F V and triangular FE

Denoting by a h = {Pi)iElthe set of all vertices of all triangles T E S h , we associate each Pi E a h with the dual finite volume Di constructed in the following way: Join the centre of gravity of every triangle T E S h , containing the vertex Pi, with the centre of every side of T containing Pi. If PiE crh fla R , then we complete the obtained contour by the straight segments joining Pi with the centres of boundary sides that contain Pi.In this way we get the boundary a D i of the finite volume Di. (See Fig. 2.) We set Dh := {Di)iEI and 7' -h := Sh. We use conforming piecewise linear finite elements for the discretization of purely viscous system (3). The components of the vector w are approximated

650

Finite volumes for complex applications

by functions from the finite dimensional space

Xh = { q ~E C(2); cplT is linear for each T

E

G )4 .

(6)

We set in (5) Vh

=

{ ~ ~ = ( ~ 1 , ~ ~ , ~ . ~ ~ ) E X h . ~ = O o n t h e p a r t o f

where wi satisfies the Dirichlet condition -k+i

1

(7)

-k+$

and define w h E Xh such that w h (Pi) = wZtt, i E I. The numerical quadrature, called mass lumping, is used in the evaluation of (., .)h and ah(., .) in (5) (see [FFL 951). 3.2 Triangular FV and triangular FE

The mesh Vh = Sh is used in the discrete scheme (4) (see Fig. 3). The conforming linear finite elements on a triangulation 7) of are used in the is compatible with Vh in the following sense: scheme (5). The triangulation The set of all vertices of the triangles T E 7 h consists of the barycentres Bi, i E I, of all Di E Vh and the vertices of Di E Dh lying on 8 0 . In this case the spaces Xh and V h are defined in the same way as in (6) and (7).

rh

3.3 Burycentric FV and triangular FE

By {Si)iEI we denote the set of all sides of all T E S h . We construct the barycenter finite volume mesh Vh = {Di)iEI over the gridTh: Each side Si $ dR is associated with the quadrilateral Di for which the boundary consists of segments connecting the endpoints of Si with the barycentres of the triangles adjacent to the side Si. For Si C 80, Di is defined as the triangle with sides formed by the above segments and Si. (See Fig.4.) The finite volume approximation of the exact solution w at time tk is sought as a piecewise constant function in R with constant values on D,,i E I. The scheme (4) is applied on the mesh Vh . The finite element approximation (5) uses the Crouzeix- Raviart noncon:= Sh forming finite elements (cf. [FEI 93, Par. 8.9]), linear on each T E and continuous at midpoints of Si, i E I. The components of the vector w are approximated by functions from the space

xXR =

{cp E L'(R), piT is linear for each T E G , cp is continuous a t midpoint of Si for each i E I

vtR =

{Vh = (91,92.93, p4) E xfR1 9, = 0 at the midpoint of

(8)

Complexity, performance and informatics

St E a R , where wi satisfies the Dirichlet condition.)

651

(9)

-k+$ ,k+i We define w, E x:~ such that w, (Qi) = w f t 4 , where Qi is the midpoint of the side Si € {Sj)jEI.The numerical quadrature using the midpoints as integration points is used in (5). The scheme (5) with and the test function space vFRis applied.

i%Yidefined above

4. Computational results The combined FV - F E method was applied to several test problems. The goal was to obtain steady viscous transonic flow via time stabilization for tk + m. Here we present the application of this method to a technically relevant flow past a cascade of profiles. The results were obtained in cooperation with V. DolejSi. The computational results are compared with a wind tunnel experiment (by courtesy of the Institute of Thermomechanics of the Czech Academy of Sciences in Prague). The experiment and the computations were performed for the following data: angle of attack = 19" 18', inlet Mach number = 0.32, outlet Mach number = 1.18, y = 1.4, Reynolds number zz 1.5 . lo6. Two cases, namely triangular FV/conforming F E and barycentric FV/nonconforming FE were computed. Fig. 5 and 6 show the mesh Dh and Thfor b0t.h cases. The initial coarse triangular mesh was refined adaptively according to the algorithm described in [DOL 981. Fig. 5 and Fig. 6, where the pressure distribution along the profile is plotted, indicate a very good agreement of computational (-) and experimental (0) results for the case from Fig. 9 and Fig. 10, respectively. In order to support theoretically the computational results, the combined FV - FE schemes were analyzed in the case of a model scalar nonlinear convectiondiffusion problem. In [FFL 971 and [FSS 991 the convergence of the method combining dual FV and conforming F E is proven for semiimplicit and purely explicit schemes, respectively. The work [FLW 991 is concerned with error estimates of the semiimplicit method. In a similar, but more sophisticated way the combined barycentric FV/triangular F E method was analysed in [ADF 981 and [DFK 991.

References [ADF 981

Angot Ph., DolejSi V . , Feistauer M., Felcman J., Analysis o f a combined barycentric finite volume - finite element method for nonlinear convection-diffusion problems, Appl. Math. 43, No. 4, (1998), 263-310.

[DOL 981

DolejSi V . , Sur des me'thodes combinant des volumes finis et des 6Je'ments finis pour le calcul d'6coulements compressibles sur des

652

Finite volumes for complex applications maillages non structurgs, PhD disertation, Charles University & Universitb Mbditerranbe, 1998.

[DFK 991

Dolejsi V., Feistauer M., Klikova A., Error estimates for barycentric finite volumes combined with nonconforming finite elements applied to nonlinear convection-diffusion problems. Preprint. Faculty of Mathematics and Physics, Praha, 1999.

[DFE 981

DolejSi V., Felcman J . , Investigation of order of accuracy for higher order finite volume schemes, Engineering Mechanics, Vol. 5, No. 5, (1998), 327-337.

[FEI 931

Feistauer M., Mathematical Methods in Fluid Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 67 (Longman Scientific & Technical, Harlow, 1993).

[FFD 961

Feistauer M., Felcman J., DolejBi V., Numerical simulation of compressible viscous flow through cascades of profiles, ZAMM 76 (1996) S4, 297-300.

[FFL 951

Feistauer M., Felcman J., LukLEovA-Medvid'ova M., Combined finite element - finite volume solution of compressible flow, J . Comput. Appl. Math. 63 (1995), 179-199.

[FFL 971

Feistauer M., Felcman J., LukLEovCMedvid'ova M., On the convergence of a combined finite-volume - finite element method for nonlinear convection-diffusion problems, Numer. Methods Partial Differential Eq. 13 (1997), 163-190.

[FLW 991

Feistauer M., Felcman J., LukLEovbMedviclovB M., Warnecke G., Error estimates of a combined finite volume -finite element method for nonlinear convection-diffusion problems, accepted in SIAM J. Num. Anal.

[FSS 991

Feistauer M., Slavik J., Stupka P., On the convergence of a combined finite volume - finite element method for nonlinear convectiondiffusion problems. Explicit schemes, Numer. Methods Partial Different,ial Eq., 15 (1999), 215-235.

[FDO 961

Felcman J., DolejSi V., Adaptive methods for the solution of the Euler equations in elements of blade machines, ZAMM 76 (1996), S4, 301-304.

[FDF 941

Felcman J., Dolejli V., Feistauer M., Adaptive finite volume method for the numerical solution of the compressible Euler equations, Proceedingsof the Second European Computational Fluid Dynamics Conference, ECCOMAS 94, 5-8 September 1994, Stuttgart, Germany, John Wiley and Sons, 894-901, 1994.

[FSO 981

Felcman J., Solin P., On the construction of the Osher-Solomon scheme for the 3D Euler equations, East-West J . Numer. Math., Vol. 6, No. 1 (1998).

Complexity, performance and informatics

Figure 1. 23-iangular mesh Sh

Figure 2. Dual FV mesh

D h

Figure 3. niangular FV mesh

and triangular FE mesh

D h

Figure 4. Barycentric FV mesh

D h

Th

and triangular FE mesh

Th

and triangular FE mesh

Th

653

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Finite volumes for complex applications

Figure 5. Triangular FV and conforming FE - pressure distribution

Figure 6 . Barycentric FV and nonconforming FE - pressure distribution

Concepts for parallel numerical solution of PDEs

Guntram Berti Institut fur Mathematik, B T U Cottbus UniversitMsplatz 3-4 03044 Cottbus, G e r m a n y bertiQmath.tu-cottbus.de

ABSTRACT The parallelization of numerical codes on unstructured grids still is a demanding programming task, which is often performed i n an application-specific manner. However, many of the algorithms used i n this field exhibit strong structural similarities. A concept exploiting the latter is developed i n this paper. Furthermore, it will be shown how the abstract concepts can be implemented i n a reusable generic way, thus easing the actual work to be done for building parallel PDE software. Keywords: parallel scientific computing, distributed grid data structures

1. Introduction

For applications in the field of numerical PDE solution, the increased computational power of parallel hardware offers a promising way of surpassing limitations of resources. However, parallelization of numerical code can be a demanding task, especially when performed from scratch. Taking advantage of existing tools for parallel computing is therefore a sensible option. For applications operating on regularly structured data (e. g. Cartesian grids) a good deal of tools are available. For unstructured grids, on the other hand, there is still a lack of sufficient tool support, see [Bir98] for a different approach and an overview. We propose a generic, data-structure based solution, which makes only minor modifications t o existing code necessary. This is no automatic parallelization concept; the act of data exchange remains under the control of the application program.

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Finite volumes for complex applications

Our approach emphasizes logical grid distribution; it is applicable for distributed as well as for shared memory, and for multi-block type grids. The organization of the paper is as follows: First, a short overview over the strategy of geometric partitioning is given. Then the concepts of distributed overlapping grids are introduced, and some aspects of their implementation are discussed. Finally, we review some implementation-related issues.

2. Geometric Partitioning for parallel PDE solution

Algorithms for grid-based numerical PDE solution typically work locally on data with a predictable pattern of access, with work roughly proportional to grid size. Many of these algorithms do not depend on the order grid elements are processed (this is not true e. g. for some linear solvers used for convectiondominated problems). Therefore, geometric partitioning, or domain decomposition, is an adequate parallelization strategy [Zom96]: Equal-sized parts of the computational grid are assigned to different processes. These parts have an overlap which acts like a cache in order to avoid direct access to remote data; its size depends on the client algorithms used by the application. Computations in parallel (ownercomputes rule) alternate with synchronizing the locally copied information in the overlap with the neighbor processes. In order to implement domain decomposition, several concrete problems have to be solved. A given grid has to be partitioned into equal parts; an appropriate overlap has to be derived from the algorithms and to be added to these parts. We must maintain some correspondence relation between grid regions duplicated in the overlap, and provide a means to ensure consistency of duplicated data on the grid. These question have to be tackled in the basic case of static distribution. If the grid distribution is dynamic, there are the additional difficulties of incremental load re-balancing and migration of grid-related data structures. We concentrate here on the static case, because it is required for every application. Also, it covers most of the parallel application-level code as well as the actual communications load, so it is the key to both parallel performance and ease-of-use. Migrating a distributed grid data structure definitely is not a trivial task, but can be made largely transparent to the application-level code. This problem will be discussed in [Ber99]. We do not treat partitioning of grids in this paper. It is a well-investigated problem ([Els97]) and is considered a preprocessing step here. 3. Concepts for distributed grids Manageable data structures for distributed grids are the key to parallel implementations of PDE solvers. They should both hide any detail of no interest

Complexity, performance and informatics 657 to the application programmer and at the same time provide the high-level constructs needed to express the logical nature of data distribution. In particular, a simple yet expressive way of dealing with the logical parts of grid overlap according to their ownership semantics proves essential. The following three sections are concerned respectively with the inner structure of the overlap, maintaining consistency of data on the grid (distributed grid functions), and finally the construction of the overlap itself, driven by the stencils of client algorithms. 3.1. Distributed overlapping grids - DOGS Our point of departure is a simple 'sequential' grid, henceforth called local grid. For parallelization of existing code, it is crucial that the representation of this grid does not need to be changed. On top of this local grid sits an overlapping grid, which manages overlap-related information. It also has access to a coarse partition grid, the cells of which correspond to the local grids, and establishes the link to neighboring local grids.

..-

distributed overlapping grid (DOG) partition grid

global grid local grid

(Gi)

(6) local overlapping grid

Figure 1: components of a distributed grid Thus far, the grid entities have a local meaning only. Now, we add as an additional layer a distributed grid, which stands for the ensemble of all local grids. If there is single global memory, there needs to be only one such entity that contains the array of local overlapping grids. In the case of a physically distributed environment, this layer is represented by a distributed grid for each physical memory. The distinction between overlapping and distributed grid is essential for the meaning of global reduction operations. It also leads to a better factorization of functionality, such that an overlapping grid implementation can be reused for many distribution scenarios.

658 Finite volumes for complex applications

The concepts of overlap structure will be introduced here in an informal manner; the reader interested in a formal treatment is referred to [Ber99]. As mentioned above, each local grid Gi contains a certain overlap Oi which can be thought of as a 'halo' surrounding it. The local grid parts are tied together by means of identification mappings between their overlaps, defining the global grid 6 as the union of the Gi with the proper identifications. On each local overlap, we impose ownership relations by distinguishing between parts Eij that belong to Gi (and are copied in Gj), ranges Sij that are shared between Gi and Gj, and parts Cij that are copied from Gj, cf. fig. 2. These bilateral ranges support the efficient data-exchange among neighbors. They are complemented by total ranges Ei,Si and Ci given by union over all neighbors j. These are central for guiding the parallel execution of algorithms. For completeness sake, we mention also private ranges Pi not duplicated elsewhere, imported ranges (Si U Ci) and exported ranges Ei U Si. 3.2. Distributed grid functions - DGFs

Static communication patterns exchange only data on the grid, a notion captured by the concept of distributed grid functions, As before, we distinguish between local ('sequential'), overlapping, distributed and global grid functions, corresponding to the type of grid they are thought to 'live on'. For a proper definition of the notions of consistency, an overlapping grid function F has to be considered jointly with an algorithm 5 responsible for calculating its values, a write-range W where F is locally computed, and a read-range R, where F is locally accessed. What about the class of algorithms 5 we consider here? First, according to the SPMD paradigm, 5 is thought to be a function on the global grid computing a global grid function F . This can only be achieved by computing locally the Fi if 5 is data-parallel: It does not depend on the order of computation, i. e. not on the values of F itself. If this not the case (e. g. Gauss-Seidel iteration), which is a block-version of the original 5 . This we obtain another algorithm may or may not be tolerable, and must be decided by the numerical scientist. Assuming now 5 is 'sufficiently data-parallel', we call a tuple (F,5,W, R) locally up-to-date if F(e) = 5(e) for all e E W; it is locally consistent if also F(e) = 5(E) for all e E R with remotely owned or shared counterparts E. A distributed grid function (Fi,5, Wi, R i ) 1 5 i l ~ is said to be globally consistent if each (Fi,5, Wi, R,) is locally consistent. Then the global grid function F is well defined and (ideally) identical to the outcome on a global sequential grid. The act called synchronization of establishing global consistency is a central service offered by a distributed grid function. If DGFs are used for a parallel program, we have first to decide about appropriate ranges W and R. In most cases, W = Pi U Ei USi is adequate; the range R depends on the set of algorithms using F , so taking R equal to the whole local grid is always safe but may be wasteful.

zP

Complexity, performance and informatics

659

Second, we have to control when a synchronization should occur. Again, this is safe when all local computations are done; we can overlap computation and communication (compute-and-send-ahead) by first calculating on exported ranges, initializing data exchange, calculating the rest, and finishing data exchange. The task performed by DGFs may be a bit more involved. So it is sometimes useful to calculate results only partially on the shared parts of W, e. g. the assemblation of FEM stiffness matrices. Then synchronization means reduction (typically addition) on shared ranges. It may also be useful to work with data that is only locally up-to-date, and it depends on the sequence of algebraic operations, when a consistent state has to be established, see [Bas94]. 3.3. Incidence-based stencils and overlap generation

The efficiency of the approach strongly depends on the size of the local support or stencil of $, which is for each grid element e the minimal neighborhood U(e) such that the evaluation of g(e) uses only information defined on U(e). Two questions now arise: how can the stencils of algorithms on unstructured grids be described, and how can one produce the correct overlap from that and a partitioned grid? Attacking the first problem, we confine ourselves to stencils expressible in terms of the grids incidence relations. To formalize this idea, we introduce incidence sequences I = (do,. . . ,dn) with 0 di 5 d = dim(G), meaning "go from elements of dim. do to incident elements of dim. dl to . . . ". Given a set A (or germ) of elements of dimension do, we may define the hull X I ( A ) generated by I on A by taking the union of all elements visited by following the path given by I , starting from the elements of A.

<

-,

exposed shared

e copied

-

-

private exposed rl copied (a) atencil ( d ,d - 1, d )

(b) corresponding hulls

Figure 2: Simple stencil and overlap generated by MARK-HULL

660

Finite volumes for complex applications

o a

0

n (a) stencil (d, 1, d)

exposed shared copied private exposed copied

(b) corresponding hulls

Figure 3: Another stencil and overlap generated for one grid part

Consider the following simple flux calculation example: for all Cells c E G do for all Cells n adjacent to c do fluxsum(c) tfluxsurn(c) flux(U(c),U(n)) The stencil of this algorithm is given by (d, d - 1,d), which means that for a given cell, each neighbor cell is accessed, via the corresponding facet, as depicted by figure 2(a). If some kind of second-order recovery is added, involving interpolation on vertices, a more complicated stencil results (figure 3). Knowing this local stencil, it is now possible to derive an appropriate overlap, given a partitioned grid. In essence, this reduces to the calculation of the hull generated by the stencil, taking the boundary of the grid partitions as germs. An algorithm MARK-HULL using space and time proportional to the size of the generated hull has been developed and implemented, but cannot be reproduced here due to space limitations, see [Berg91 for a discussion. For the flux-algorithm stencil above, this results in an overlap as in figure 2(b). In a typical application, there are several stencils involved, and thus different hulls may need to be computed. The total overlap of the grid is then composed by the union of partial overlaps imposed by different algorithms.

+

4. Generic implementation and practical experience

The concepts presented so far are useful in their own right. However, we would like to offer an implementation which is usable without change for a large variety of concrete grid data structures. In order to achieve such a grid-independent realization, it proves necessary to adopt a more abstract approach to grid data structures. In [BB99] is shown how an abstract grid interface enables a generic programming style, producing implementations that do not rely on a particular

Complexity, performance and informatics 661 grid representation. See [Berg91 for a detailed description of these concepts. We just remark that our approach makes a clear separation between combinatorial grid, geometric embedding of the grid and grid functions that allow to store data on grid elements. This corresponds well to the requirements of the distributed grid data structures, as it mainly depends on the combinatorial properties, and in the case of a static distribution, grid functions are the main actors. Most importantly, incidence information is captured by iterators, which essentially answer queries like "give me all vertices incident to this cell, one after another", making them a suitable basis for the MARK-HULL algorithm mentioned above. On top of this grid abstraction, data structures for representing distributed grids, overlaps and grid ranges, as well as the algorithms for generating them from incidence-type stencils, have been developed in a fully generic fashion. Thus, the implementation may be directly used with any grid that satisfies the abstract interface. In practice, it proves rather straightforward to adapt existing grid implementations to this interface, or more precisely, to provide a separate implementation for it. According to our experiences, this requires only one or two days of work, which could probably still further reduced. Once this has adapted been done, the generic implementation of distributed grids can be used 'off the shelf?, fig. 4. ~t remains to I distributed I I generlc I I generlc I II----------' grid (generic) I I component 2 1 I component 3 1 I----------' I----------# insert the appropriate calls to synchronization routines and global reduction operations. Most of the code Figure 4: using generic components with exmay continue to use the 'native' grid isting codes data structures. It should be noted, that as a result of this adaptation, not only the distributed grid components may be used, but all generic algorithms and data components developed on top of the interface are immediately available. The techniques discussed so far have been successfully used to implement distributed solvers for the 2D Euler equations for one- and two-component systems. These solvers use a fluctuation-splitting algorithm on unstructured grids with a second-order recovery step, and an explicit time integration. The mathematics are described in [SBB97]. The implementation of this solver is based on a C++ -realization of the generic grid interface mentioned above. As such, the parallelization of the code essentially involved substitution of grid functions with their distributed counterparts, and addition of synchronization calls. Also, the maximal stencil of the algorithms used has been determined by simple pencil-and-paper inspection. Due to a parameterization of the solver over the actual grid type, the same numerical code can be used both for the parallel and the sequential version.

662

Finite volumes for complex applications

Currently, t h e parallelization of a 2D Navier-Stokes solver with implicit time integration and a two-level grid is underway. This code has been developed using its own grid d a t a structure; thus a n adaption t o the generic interface had t o be done.

5 . Conclusion We have presented a novel approach t o the development of parallel applications involving grid-based algorithms of data-parallel type, such as used in solvers for partial differential equations. In particular, the concepts allow the programmer t o concentrate on the abstract aspects of grid distribution. It offers automatic generation of grid overlap based on the stencil of the client algorithms, and is therefore directly applicable t o a broad class of applications. Furthermore, a generic implementation of these concepts make it easily usable.

6. Bibliography

[Bas941

Bastian, P. Parallele adaptive Mehrgitterverfahren. PhD thesis, Universitat Heidelberg, 1994.

[BB99]

Bader, G. and Berti, G. Design principles of reusable software components for the numerical solution of PDE problems. In Hackbusch, W. and Wittum, G., editors, Concepts of Numerical Software. Vieweg Verlag, 1999 (to appear).

[Berg91

Berti, G. Generic software components for Scientific Computing. PhD thesis, BTU Cottbus, in preparation.

[Bir98]

Birken, K. Ein Model1 zur efizienten Parallelisierung von Algorithmen auf komplexen, dynamischen Datenstrukturen. PhD thesis, Universitat Stuttgart, October 1998.

[Els97]

Elsner, U. Graph partitioning - a survey. Technical Report SFB393/97-27, Technische Universitat Chemnitz, December 1997.

[SBB97]

Schenk, K., Bader, G., and Berti, G. Analysis and approximation of multicomponent gas mixtures. In Feistauer, M., Kozel, K., and Rannacher, R., editors, Proceedings of the 3rd Summer Conference Numerical Modelling in Continuum Mechanics, Prague, 1997.

[Zom96]

Zomaya, A. Y., editor. Parallel and distributed computing handbook, chapter Scientific Computation. McGraw Hill, 1996.

Performing parallel direct numerical simulation of two dimensional heated jets. S.Benazzouz, V.G.Chapin & P.Chassaing

Laboratoire de Mkcanique des Fluides E N S I C A , Toulouse F R A N C E

ABSTRACT Parallel direct numerical simpllations of a two dimensional heated jet is perjormed. The relevance of a decomposition domain strategy is shown. A performances comparison between a parallel vectorial computer (FUJITSU VPPSOO) and a parallel superscalar one (IBM SP2) is done. Key Words: Parallelization, acceleration factor, communications, processus, vectorial architecture.

1. Introduction

Our work consists in studies about the self excited global mode which develop in two-dimensional heated jet when the density ratio between the jet and the ambiant fluid is sufficiently low. We need many long and hard simulations t o analyse the physics of the problem. Simulations done on single processor become too long. Our objectives by using parallel computations is to reduce the return time (CPU time) of our simulations with attention to limit the increase of the total computing time (CPU time * processors number). The second objective is to increase the size of the tractable problem in doing a memory partition. The paper will be organized as flollow. The next part shows the numerical methodology, the third part of our work explicit the optimization strategies. Part number four shows our results. The last part derives some conclusions from our results.

664

Finite volumes for complex applications

Figure 1: Computing domain

2. Numerical methodology Our code resolves on Euler equations with added fourth order numerical dissipation. T h e temporal scheme is a 5-stage second order explicit RungeK u t t a scheme [Jameson, Schmidt et Turkel (1981)] with a CFL number of 3. The spatial discretization is based on second order central differences. The boundary conditions based on a monodimensional characteristic method are following ones : reservoir conditions at the inlet, slip conditions a t the lateral boundary and s non-reflecting conditions at the outlet [Hedstrom (1979)l. T h e initial jet field has been choosen following [Yu et Monkewitz (1990)l paper. It consists in an hyperbolic tangent profil applied on the density and the longitudinal velocity. The characteristic non-dimensional parameters are the velocity ratio A = u:+u, = 1 and the density ratio which is choosen in

' the range0.25 < S = 2 < 0.55 for all simulations presented in this paper.

Our grid has two zones: a first one named the uniform zone, with a constant space between gridpoints and a second one, named the geometric zone, with a geometric progression between each gridpoints. The region of physical interest is the uniform one. We made this in order to make our simulations cheaper in computation time and to smooth unphysical reflection from the outlet boudary to the region of interest. Figl. resumes boundary conditions and details of our grid structure.

3. Optimization strategies The main goal of any parallelization method is to reduce the CPU time without a too high degradation of the global performance which will be analysed late on through the acceleration factor. Because the code is explicit, this made relatively easy its parallelization. Every processus needs only d a t a located in

Complexity, performance and informatics

665

Figure 2: Computing domain decomposition his subdornain except some frontiers data. Differents methods are actually used. We have choosen a multidomain decomposition strategy. This method consists in affecting a part of the global domain for each processus. This method can be also called SIMD (Single Instruction, Multiple Data) with distributed memory. Each processor computes into his own subdornain and communicate only frontiers data with neighbor processors. Different solutions exist to decompose the global domain. The decomposition can be based on longitudinal or transversal subdomains or a rnixing of both methods. In order t o optimize our parallel solver, we have to reduce the communication time. Our grid contains more gridpoints in longitudinal direction ( N I > N J ) . For thoses reasons our calculation domain is decomposed through the longitudinal direction. Due to the nature of our grid, subdomains have the same number of gridpoints if the processors frequencies are the same. If the processors frequencies are not the same a simple load balancing is used through an adaption of the gridpoints number for each subdomains. Note that -in the case of same frequencies- due to our grid, each processus compute the same gridpoints but not necessarly the same field area. Fig.2 show a schematical decomposition of the global domain. To analyze the performances of our parallel code we use a parameter noted M P called the acceleration factor:

where Tm is the multiple processus running time (for each processus) , T, the single processus running time and N p is the processus number. In a parallel computation the total time (Tm) is in general composed by communication time Tcommand computation time Tcom,. Following this it becomes :

666

Finite volumes for complex applications Mshpomtr n e c M bypmasrm P m SCHEMA t

i

Ma~hpavlbnctdsdbypmns~P lnDlSSlP

I -I

Figure 3: Frontiers treatment Where Tcomp can be modelized -in a domain decomposition method- by

Equation (1) become

3.1 Subdomain communication

Due to the nature of our performing strategy, each processus needs t o communicate with at least another processus. All processus, except the processus which compute the inlet and the outlet subdomain send and receive updated data t o and from the previous and the next subdomain processus. As shown in fig.3 only subroutines SCHEMA and DISSIP need data located outside the subdomain limits t o compute derivates of the physical quantities. Almost all the communication time (T,,,,) consists in those data communications. We have choosen to apply non-blocking sending communications and blocking receving communications in order to reduce the communications time which is capital in optimizing the acceleration factor.

3.2 Vectorization method

The parallel vectorial version of our code is based on vectorization of the largest DO loops. The main difficulty in vectorizing a code is to make all

Complexity, performance and informatics Number of processus Time (minutes)

I 1

1 642

14 ( 1 173 1

8 87

1 1

16 44

32 22

667

64 12

Table 1: Return times

DO loops explicit. In our solver, the main computing time is consumed into subroutine SCHEMA and DISSIP, where the DO loops are effectively explicit and hence ready for vectorization

4. Performance evaluation All results will be shown wit,h the same number of gridpoints in each subdomains because the processor frequencies are the same.

4.1 Acceleration factor versus processors number

All experiments presented in this section are made with a grid defined by

N I = 285, N J = 213 and where uniform zone dimensions are 0 < X / D < 5 and -1 < Y / D < 1 In a first part we will show the dependence of the acceleration factor with the processor number. Our experiments show two important results. T h e acceleration factor is relatively high M p > 0.9 and it decrease versus ilunlber of processors. Table 1 shows return time results and Fig. 4 shows the evolution of the acceleration factor with the processus number.

From equation (4) the acceleration factor can be written

We can see clearly t h a t the increasing of the processus nulilber has some negative effect on the acceleration factor , this phenoinena can be largely explained by the previous equation. In fact, the conlmunication time are widely independent on the processors number so the ratio LT? is constant. I t is just depending on transversal number of gridpoints ( N J ) , because almost all comnlunications are vectors of length NJ or 2*NJ.

668

Finite volumes for complex applications Dependence of acceleration factor with the processors number

0.50

20

40 nuv~bcrOJprocrrrorr

60

80

Figure 4: Acceleration factor 4.2 Grid size efect

In this part all results are based on experiments made on 16 processors. A second part will the influence of the grid size on the acceleration factor. We can note that the length of each subgrid (NIINP) has more important influence than grid width (NJ). In a first approximation we can modelize the cornmuilication time by :

And the single processus time by

Equation (5) become

The direct dependence of the acceleration factor with NI is verified in fig.5 with a constant value for N p ( N p = 16) This result is perfectly in agreement with the global domain decomposition strategy which is a longitudinal decomposition.

Complexity, performance and informatics

669

Evolutron of the ucceleratron factor 100

I

on grtddrnm~tom

-

r

1

-'

0.M

1W

200 3W Long;tud#nrrl gridpoinls nurnbrr 1Nli

I

403

Figure 5: Grid size effect Computer Time (minutes)

I S P 2 I VPPSOO 1

173

1 45

VPP3OO with vectorial performin 16

Table 2: Return times

4.3 Vectorial performances

A last part compare performances between a parallel vectoriel computer and a parallel scalar one. We notice that for the same number of processors, it is more profitable to compute on a vectoriel architecture. This comes primarily owing to the fact t,hat the most computation time is consumed in the largest DO loops which are easily and high performed vectorisable loops. The power of vectorial architecture come from his hardware SIMD, unfortunately all loops which want to be vectorizable have to be explicit. As we see in Fig.6 the acceleration factor of the VPPSOO (this computer has just 6 processors) is near the acceleration factor of the SP2. But, we obtained a return time divided by three by using vectorial instructions 5 . Conclusion

We saw the performances dependence on many factors. With return time reducing, another effect of the parallelization is the memory profit, in this case large and hard simulations have to be computed on a parallel computer with attention to adapt it to the grid size.

670

Finite volumes for complex applications SPZNPP300 Cumparison

number ojprocruor~

Figure 6: Comparison SP2/VPP It is more profitable to decompose a computation domain through only one direction, in order to minimize the communication time and made more profit in return time. It is best to decompose through the direction which have the greater gridpoints number. However, for computations which are not requiring large memory, but are expensive in computation time, our interest is to used a weakly parallel vectorial architecture. This numerical study shows the best way to optimize our parallel code, how to choose a decomposition strategy and what kind of architecture to use. Bibliography

[HED 791

HEDSTROM,G. W. Nonreflecting boundary conditions for nonlinear hyperbolic systems, J. C.P. 30, p222-237 (1979) .

[JAS 811

JAMESON, A . , SCHMIDT,W. ET TURKEL,E. Numerical solutions of the euler equations by finite volume methods using runge-kutta time-stepping schemes, A. I.A.A. paper 81-1259 (1981).

[YUM 901

Yu,M.H. ET MONKEWITZ, P.A. The effect of nonuniform density on the absolute instability of two-dimensional inertial jets and wakes, Plays. Fluids A 2(7), July 1990, pll75-1181 (1990).

Two-Dimensional Riemann Problems Assessment Tests for Upwind Methods for Multi-Dimensional Supersonic Flow Problems

J. van Keuk, J. Ballmann Lehr- und Forschungsgebiet fur Mechanik der R W T H Aachen Templergraben 64, 52062 Aachen, Germany

Numerical simulations of two-dimensional R i e m a n n problems using different upwind methods are presented i n this paper. These problems are well suited to analyze the different methods with regard t o robustness and accuracy, since they include m a n y complex interaction phenomena as they can also appear i n applied problems like super- and hypersonic intake flows. S o m e positive and negative features of the different methods are shown and, finally, first results of a three-dimensional intake flow simulation using one of the favorite schemes are presented.

ABSTRACT

K e y Words: two-dimensional R i e m a n n problems, upwind-methods, robustness, accuracy, limiters, grid refinement, multi-dimensional supersonic flow problems.

1. Introduction The purpose of the authors's research project is to analyze three-dimensional supersonic and hypersonic flows in intakes of airbreathing propulsion systems, which are characterized by complex interaction phenomena between shocks, contact discont.inuities and expansions. These all may occur simultaneously e.g. in the intake's corners. To resolve these phenomena in a physically reasonable way the use of upwind methods is necessary, but it is not sufficient to concentrate the choice of the method only on one or the other phenomenon or interaction. In a paper by P. D. Lax and X.-D. Liu [LAX 981 two-dimensional Riemann

672

Finite volumes for complex applications

problems with 19 genuinely different solutions were analyzed. These problems are well suited t o investigate upwind methods regarding their positive and negative features in connection with robustness and accuracy. To pick out the best methods available for the simulation of the flow problems mentioned above is the purpose of this paper.

2. Physical Model The Navier-Stokes Equations for three-dimensional flow form the basis for the numerical simulations planned in the authors's research project. For a simply connected, three-dimensional domain V with boundary dV these equations can be written as

PI where U = (p, pu, pv, pw, pE)T is the vector of conservative variables. F,, G , and H, represent the convective flux functions and F , , G,, H, are the corresponding diffusive ones. The vector of heat conduction is modeled by Fourier's law and Sutherland's formula is adapted for the laminar viscosity as a function of the static temperature. The description of the heat conductivity is carried out via a constant Prandtl number and the gas is assumed to be thermally and calorically perfect.

3. Numerical Method

3.1 Basic Flow Solver

The DLR Navier-Stokes Solver FLOWer (Version 114.1) forms the basis for the simulation of the flow problems considered in this paper. This code is formulated as a cell vertex centered finite volume scheme in a block structure and it has been extended by different upwind methods (e.g. van Leer / Hanel, AUSM, AUSM+, AUSMDV, AUSMPW, LDFSS, HLLE, Roe (Harten / Yee)) t o better capture the directed propagation of information inherent in the inviscid part of the equations [VKE 981. Some of them will be described more precisely in the next section. Central discretization is applied t o the diffusive part according to its parabolic character in space and time. Time integration is performed by an explicit &Stage Runge Kutta scheme. For the simulation of flow problems with asymptotically steady state solutions, convergence acceleration techniques like local time-stepping and the well known

Complexity, performance and informatics

673

multigrid method in FMG-FAS formulation are available in the FLOWer-Code.

3.2. Upwind Methods

A large variety of different upwind methods has been presented in the last 20 years. Basically, one distinguishes between flux vector splitting (FVS) and flux difference splitting (FDS) techniques. The simpler variant is t o consider only the Eigenvalues of the flux' Jacobian of the inviscid flux vector and t o split the flux terms according t o the sign of the associated propagation speeds. This is done in FVS schemes. Sophisticated methods t o introduce more physical properties into the numerical scheme are the FDS methods, where local Riemann problems at the cell interfaces are solved either exactly or approximately. The upwind methods described below are first order accurate in space. This does not suffice for the numerical solution of the Navier-Stokes Equations for reasons of consistency, because of the second order derivatives appearing in the diffusive terms; it is therefore necessary to increase the order of accuracy in space concerning the discretization of the convective terms. One possibility to formally achieve second order accuracy in space is to use MUSCL-Extrapolation (Monotonic Upstream Scheme for Conservation Laws). The idea is t o consider a piecewise linear distribution of the variables ( conservative or primitive ones) within each control volume, see van Leer [VLE 791 This leads to the following expressions for left and right extrapolated values of any scalar flow variable U :

where

and a ( * ) are the so-called "limiter functions" to guarantee the TVD-property of the scheme. Widely used limiters were proposed by van Leer [VLE 791 and Roe [ROE 811 ("minmod , "superbee"):

amin(T) =

{

;in(r, 1) if r > 0 otherwise @sup(r) = max [O, min (2r, 1) , min (r, 2)]

avL(r) =

l+r

,

PI

The interface values can be considered as resulting from a combination of forward and backward extrapolations. The parameter K switches between these possibilities.

3.2.1. van Leer Flux- Vector Splitting Van Leer imposed continuous flux functions for the whole range of Mach numbers and eigenvalues of F:?, which are either positive/negative or zero.

674

Finite volumes for complex applications

In the subsonic regime this leads t o the following expression for the split fluxes as functions of the Mach number with a second order polynomial in a slightly modified formulation by Schwane and Hanel [HAE 891. 1

+

I

1

f" = *;pa*

where

:( * 1)

2

[5]

a* = J ~ ( K- l ) / ( ~ 1)Ht is the critical speed of sound. There is no splitting

in supersonic flow since all eigenvalues have the same sign:

Fi+; = FL; F ~ + =F,; +

, F;+; = 0

, F;;

=o

for for

M M

>1 < -1

[GI

3.2.2. AUSM Flux-Vector Splitting (LRouiSteffen) The Advection Upstream &litting Method was originally proposed by Liou and Steffen in the early 90's [LIO 931. First, the inviscid flux vector is split into an advective and a pressure term. These two terms are then split separately, leading to the following expression for the flux a t the cell interface

where Ai+; (0) = {e}i+l - {*Ii and Mi+; = M: + Mt;, . The split Mach numbers M & are definded according t o van Leer's splitting as described above. The pressure flux terms are assumed to be governed by acoustic wave speeds. An expression using second-order polynomials of the Mach number is proposed for the pressure splitting:

3.2.3. A USMDV Flux-Vector Splitting (Wada/Liou) Yet another approach for the splitting of the inviscid flux vector as a mixture of a more FDS based scheme and a more FVS based one was proposed by Wada and Liou [WAD 941 and is called AUSMDV.

Complexity, performance and informatics

675

where s is a switch as a function of the local pressure gradient.

)

where

K = 10

[lo]

It is clear t o see that the AUSMD and AUSMV differ only in the treatment of the term pu2 in the x-momentum flux. The corresponding expressions are:

The velocity splitting within the AUSMDV is similar to the original proposal of van Leer extended by terms designed t o capture stationary and moving contact discontinuities. UL;IUL~]

+

51

if Cm

PLL+/~LI

otherwise

-7

[I31

where CYL

=

~ ( P I PL ) + (P/P)R

r

=

C ~ R

2 ( ~ / ~ ) ~

(PIP)L + ( P / P )R

[I5] and c,, = m a x ( c L , c R ) The pressure splitting is the same as in the original formulation and finally, the mass flux for the AUSMDV is (P/P)L

3.2.4. HLLE Flux-Difference Splitting The most simple flux difference splitting scheme was proposed by Harten, Lax, van Leer and modified by Einfeldt [EIN 911. The solution of the whole Riemann problem is replaced by a model consisting of three constant states separated by two shocks yielding for the flux a t the cell interface

676

Finite volumes for complex applications

where the Af denote the speeds of the fastest and slowest wave in this model. 3.2.5. Roe Flux-Difference Splitting

A more sophisticated flux difference splitting scheme was developed by Roe [ROE 811. Roe replaces the approximation of the solution of the nonlinear Riemann problem by the exact solution of the linearized problem that has to be extended by the approximation of discontinuous solutions.

A,: is the propagation speed of the k-th wave of the linearized Riemann problem and lk, r,: are the corresponding left- and right-eigenvectors. The so constructed formulation of the flux function consists of a central part supplemented by an upwind term, which has to be computed using average values of the conservative variables. One possibility for the construction of these average values is Roe-averaging, see Grotowsky [GRO 941. A problem appears if one of the eigenvalues changes its sign. For centered expansion fans with sonic point the scheme then leads t o an expansion shock and generates non-physical solutions such as the "carbuncle phenomenon'' when calculating hypersonic blunt body flows. To circumvent this difficulty Harten proposed a modification of the modulus function in Eq.(18)

I&(

=

+G) lXk

1

for / A k / < 6 else

where 6 is a small number often referred as "entropy fix". For this method there exists an alternative way of increasing the formally order of accuracy in space, that was originally proposed by Harten and Yee. This Ansatz comprises an approximation of the truncation error of the first order scheme and is known as "modified flux approach" [GRO 941.

Complexity, performance and informatics

677

4. Results First, two-dimensional Riemann problems as they have been investigated by P. D. Lax and X.-D. Liu [LAX 981 are considered. In these problems a quadratic solution domain is divided up into four quadrants and the initial data are constant in each of them. The initial conditions are restricted such that only one elementary wave, a shock, an expansion, or a contact discontinuity appears a t each interface. Depending on the initial conditions complex interactions between the different waves are evolving in time. Fig.2 shows the results obtained with the different upwind methods for the problem number 13. The initial data for this problem consist of two stationary contact surfaces and two moving shocks. The computational domain has been discretized with 300 grid points in each direction in an equidistant manner. The density distribution is displayed. Obviously that the overall best results were obtained with the Harten / Yee FDS and the AUSMDV in connection with van Leer's flux limiter. The moving shocks are sharp, the stationary contact discontinuities correctly captured. Also the secondary contact surfaces and the small vortex structure in the center are fairly resolved. Use of the original AUSM scheme leads to spurious oscillations behind shocks and contacts due to the non-monotonous behaviour of this scheme across discontinuities. Finally, the van Leer / Hanel FVS and the HLLE are characterized by the unappropriate capturing of stationary contact surfaces and, in addition, as a consequence of the higher amount of artificial viscosity inherent to both schemes the resolution of the secondary contacts and the small vortex structure is worse, too. Fig.3 shows an investigation of the influence of three different flux limiters (Roe's "minmod" and "superbee" limiter and the one of van Leer) on the results for two different problems, what is often neglected, and a surprisingly strong influence is obvious that was not expected. In dependence of the limiter location within the TVD region a dramatic improvement of the resolution of the moving contacts in the Lax 5 problem and a better resolution of the triple points in the Lax 12 was detected. In addition, in both cases the use of the "superbee" limiter leads to small oscillations as a consequence of the vicinity t o instability of this limiter, so that perhaps the van Leer limiter should be the best choice. As some kind of a grid sensitivity analysis Fig.4 shows solutions for the Lax problem number 5 on a refined grid (700 grid points in each direction) and in nice agreement with the work of Lax and Liu small vortex structures appear that could not be resolved with any method on the coarser grid. The fact, that the Harten / Yee FDS resolves only one vortex structure may be due to some kind of a direction dependency in the way of increasing the formal accuracy in space. In conclusion again the influence of the flux limiter is clear to see when looking at the pressure distributions obtained with the van Leer / Hanel FVS. Use of van Leer's limiter yields a vortex resolution almost comparable to the AUSMDV solution whereas the "minmod" limiter nearly prevents any vortex to appear.

678

Finite volumes for complex applications

Finally, Fig.1 shows first results of a threedimensional intake flow simulation using the Harten / Yee method. On the upper left hand side of this figure the result for the density distribution for the two-dimensional case is plotted. For the three-dimensional case the interior part of the intake is assumed t o have sidewalls beginning a t the intake lip. On the lower left hand side of the figure the density distribution in the symmetry plane of the inlet is showed. The phenomena in front of the intake lip are obviously exactly the same as for the two-dimensional case, but also the shock positions and angles as well as the separation region on the upper side of the interior part are a t least similar, because the three-dimensionality does only less influence the effects in the symmetry plane. Finally, on the right hand side of the figure two cross sections in the interior part are plotted, which show the three-dimensional effects in the lower corners produced by the two shocks originating from the intake lip and the sidewalls. 5. Conclusions

Numerical solutions of 19 two-dimensional Riemann problems using different upwind methods have been realized. Three of them are presented. The overall best results were obtained using the Roe FDS in the Harten / Yee formulation and the AUSMDV. The original AUSM scheme showed a nonmonotonous behaviour across discontinuities resulting in pressure and density oscillations behind them. Furthermore, a surprisingly high influence of the flux limiter was detected. So deficiencies of the schemes exhibiting more numerical viscosity could partly be cured by an appropriate limiter. Finally, first promising results of a three-dimensional intake flow simulation using the Harten / Yee method were presented. Unfortunately a considerable high value for the entropy-fix had to be chosen for reasons of stability, so that subject of future work in this project will be simulations using the AUSMDV, since it proved t o be comparable accurate in the resolution of even complex interaction phenomena a t a lower computational cost and higher stability. 6. Bibliography

[LIO 931

LIOUM.-S., STEFFENC. J., <, J. Comp. Phys. 107, pp. 23-39, 1993.

[EIN 911

EINFELDT B., M U N ZC. D., ROE P. L., SJOGREEN B., >, J. Comp. Phys. 9 2 , pp. 273-295, 1991.

[GRO 941

GROTOWSKY I. M . G . , ~ E i nnumerischer Algorithmus zur Lijsung der Navier-Stokes-Gleichungen bei ~ b e r -und Hyperschallmachzahlen, PhD Thesis, RWTH Aachen, 1994.

Complexity, performance and informatics

679

[HAE 891

SCHWANE R . , HANEL,D., <>, AIAA Paper 89-0274, 1989.

[LAX 981

LAXP. D., L I U X.-D., <<Solutionof Two-Dimensional Riemann Problems of Gas Dynamics by Positive Schemes >>, SIAM J. Sci. Comput. 19, No. 2, pp. 319-340, 1998.

[ROE 811

ROE P. L., <Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes >>, J. Comp. Phys. 43, pp. 357372, 1981.

[VKE 981

VAN

[VLE 791

VAN LEER B., <>, J. Comp. Phys. 32, pp. 101-136, 1979.

[WAD 941

WADA Y., LIOU M.-S., <
0.061

-

0.05 0.05\

KEUKJ . , B A L L M A NJ N . , SCHNEIDER A . , KOSCHELW., <>, AIAA Paper 98-1526, 1998.

3D-Simulation (Symmetly Plane) ensity Di.tribullon. (Density

Harteflea H a r t e f l i FDS)

0.04

1 >0.03 0.02 0.01 0

Figure 1: 2D/3D Intake Flow Simulations

1

0.02

--

0.015

E a

0.01

0.005 0

0.01

0.0.

680

Finite volumes for complex applications Harten I Yee

AUSM (minmod)

AUSMDV (van Leer)

AUSMDV (minmod)

X

van Leer (minmod)

X

HLLE (minmod)

Figure 2: 2D Riemann Problem according t o Lax Problem 13 (Two Shocks/Two Contact Surfaces)

Complexity, performance and informatics AUSMDV (rninrnod)

X

AUSMDV (van Leer)

X

AUSMDV (superbee)

68 1

AUSMDV (rninrnod)

X

AUSMDV (van Leer)

X

AUSMDV (superbee)

Figure 3: Influence of the Flux Limiter for Lax Problems 12 (Two Shocks/Two Contact Surfaces) and 5 (Four Contact Surfaces)

Finite volumes for complex applications

682

Harten IYee (Density)

Harten IYee (Pressure)

AUSMDV (Density,vL)

AUSMDV (Pressure,vL)

X

X

van Leer IHaenel (Pressure, minmod)

van Leer1 Haenel (Pressure,vL)

1

1

09

09

08

08

07

07

06

06

>05

>05

04

04

03

03

02

02

0 1 0

01 "

"

1

'

'

"

05

x

1

0

"

'

'

l

" 05

'

'

L

x

Figure 4: Effects of Grid Refinement and Flux Limiters

Robustness and accuracy on unstructured grids Numerical experinients on finite volume schemes

Ernst A. Meese

Skjalg E. Haaland

C F D normay as Tcknostallen P r o f e s s o r Brochsgate 6 N- 7030 T r o n d h e z m , N o r w a y

Departinent of Applzecl hlechanzcs, Ther-mo a n d Fluzd D y n a m z c s NTNU N - 7 4 9 1 Trondhezm, N o r w a y

Wzth the turo-dzmenszonnl zncompresszble h'auzer-Stokes equatzons zn mznd. the focus of the present u~orkwas on finzte zlolume schemes on trzangulnr and qundrtlateral unstructured grrds. Nzamtrzcnl cornparntzve studzes u~ereperformed o n selected schemes t o determane the schemes' robustness of accuracy wzth respect t o grzd qunlzty. The tests uiere performed o n large serzes of randomly dtstorted yrzds wzth Z ~ yzng I degree of dzstortion. Cell centred arid vertex centred dzscrete dzffzrszon operators roere consrdered. '4 vertex centred scheme had the louiest complexzty of constructtor2 for second order ncctrrocy. The convectzon operators were currently lzlnzted to vertex centred schemes. Varzo~astnterpolatzons nnd locatrons of quadrature poznfs uvere sttadzed S o m e sttpcrconvergent behnczozrr on strongly dzstorted grzds mas observed for both dzffuszon nnd concectzon operators. Lastly, an 2rnstrtrctzrred vzon-lrnear pztz lzrnzter was developecl from esfablzshed structuied methods. It tocis tested and found strccessful ABSTRACT

h e y Cliords: finzte volume, trnstructured grzd, exper~nientson accuracy, stipercorluergence, cell centred, ucrtex centred, quadrzlnternl cell, frzangtilar cell, jlttx ltmzter

1. Introduction It is conlnlon t o judge the accuracy of a finite volume scheme by an analysis of the order of it's truncation error. This is, however, not sufficient on general grids. 'The truncation error may be very poor due to grid skewness, but good solutions may still be obtained. Hard analysis of the global error is usually out of reach. Hence, numerical comparative studies are necessary t o determine the relative performance of the schemes, particularly because such studies see111 t o

684

Finite volumes for complex applications

be rather rare in the lit,erat,ure. It is cornmon t o assess the quality of finite volume schemes by nurllerical experirnents on a set of well established benchmark problems. These berlchnlark comparisons are often formulated on quite smooth grids. However, unstruct,ured grids are often generated automatically. and must be expected t,o cont,ain quite distorted cells in some regions. Distorted grid cells may great,ly reduce accuracy, and it is our opinion that tests on smooth grids are insufficient for the general verification of a finite volume scheme. Therefore, the present comparative studies were performed on randomly distorted grids. hilethods performing well on such est.reme grids may be used with confidence under more regular conditions. Two strategies for locating the unknowns were considered in the present. work. In the cell centred (CC) (primary grid) strategy the unknowns are located a t the centre of each grid cell. In the vertex centred (VC) (secondary or dual grid) strategy, the unknowns are located a t the vertices of the grid cells, and new control volumes for the equations are formed around each vertes. Which of the two strategies t o be considered best for general purposes are under dispute in the literature [Mav 971. The schemes considered here all have truncation error worse than second d but they have truncation error t o second order on order on d i ~ t o r t ~ egrids, uniform grids. Arguments have been made that such schemes really are second order [TMa 85, T D B 961. However, these arguments assume the existence of a grid distribution function x = x(<) = g ( [ / N ) t h a t is independent on N , in which case the local stretching parameter r = hi+l/hi will go uniformly t o 1 as iV goes t o infinit,y, and the truncation error will eventually show second order behaviour. In real application, r and skewness are seldom very small, and low order behaviour should be expected. However, it is not truncation error, but global error t h a t we want t,o reduce. Standard theory states that the global error is never worse than the truncation error, but there is room for it t o be better, and this is in fact the case for several simple schemes, however, not for all. In the following, some finite volume schemes relevant for the convection and diffusion operat.ors of the inconlpressible Navier-Stokes equations and the test result,s on A non-linear flux limiter is also tested. Due t o the space them are pre~ent~ed. limitations the presentation is quite brief, but a thorough presentation may be found in [EAM 981.

2. Discretisation of the diffusion operator T h e diffusion operator of the incompressible Navier-Stokes equations may be stated as

Complexity, performance and informatics

685

where the sum is over the edges of a polygonal shaped control volun~eR and Gj is the outward unit normal vect.or a t edge E j . T h e line int,egra.ls along each edge may be approxiirlat,ed by any preferred one-dimensional quadrature. For the CC schemes we use t,he nlidyoi~itrule along each e_dge. Assunning a c o ~ l s t ~ a ~ v ,l t we need an estiinate of Gq5 t,o compiete a finit,e volunie sche~nefor the operator (1). The Naive Cell Centred ( N C C ) met~hod is sin~plyto use the difference between the values of 4 in two neighbouring cells divided by the distance b e h e e n the cell centres. On non-uniform grids, this eslilllate both niisses the direction of the gradient and the location of the quadrature point. To improve the gradient estimate in the spirit of the finite volunie method (see e.g. [GMe 96]), the integral mean value theorem is applied as

The control volunle R is selected as the polygon formed by the six nodes at the centres of the cells in the immediate vicinity of the edge e I11 compass notation on regular quadrilateral grids this amounts to the nodes P, E , N, NE, S, and SE. The extension to triangular cells is trivial. To form the rightmost integral of Eq. ( 2 ) , the trapezoidal rule is used along each line segment of R . A drawback of this Cell Centred scheme wzth Integral euuluatzon of the gradzent ( C C I ) is that it becomes quite complicated for the grid edges sharing a vertex with a boundary edge. In this situations one or more of the six neighbouring nodes are missing and some other R has t o be formed. The last cell centred scheme we considered was based on a leust sqzrares appror~rnatzonof the gradient (CCLS). Our formulation is motivated by a need for the formation of the matrix in an iinplicit method. Other methods are suggested for explicit methods [Bar 891. The estimate is formed by the linear combination

(am)

=Ccimi, i

where the sum is over t,he nine neighbouring nodes P, W , E, SW, S, SE, NW, N , and NE using compass notation. Only regular quadrilateral grids were considered, and these nine nodes are used for all four edges of the cell under consideration to get a compact stencil. This makes the gradient estimate different for a given edge when viewed from its two adjacent cells, but this has posed no problem. Using Taylor series analysis we may find six equations for the nine coefficients of Eq. (3) to satisfy for second order accuracy of the gradient estimate. To make the solution unique, we aim t o minimise the expression

using the six equations mentioned above as side constraints.

686

Finite volumes for complex applications

xmary grid cells into secondary cells. The shaded area as the part of the secondary volume around vertex 1 contazned an C.

Finally, the vertex centred scheme (VCI) based on Eq. (2) was considered. A control volume is formed around each vertex by drawing lin_es from the centre of each edge to the cell centre (see Fig. 1). A gradient V4; for each primary cell Ci is formed by Eq. (2) using the trapezoid rule along each edge. This gradient estimate is used along the line segments of R in Eq. (2) residing within C;. It is possible to show that this reduces to a particularly simple formulation [EAM 981.

3. Discretisation of the convection operator

The part of the integral of the convection operator around the vertex 1 in Fig. 1 residing within the primary cell C is

iip,

where u'?, q5P, and 4p are representative values at line segments 1 and 4 respectively. These values are interpolated from values at the vertices of the primary grid cell C as

where for a primary cell Ci with mi vertices. Four locations for the quadrature points were considered. Referring to Fig. 1, on uniform grid these points amounts to x the midpoint scheme over each secondary edge, A the midpoint scheme on each secondary edge segment within the primary cell, in the fourth order GauBian quadrature over the secondary edge, and 0 in the trapezoidal rule over each secondary grid edge. Two means of selecting the coefficients cjk of Eq. (6) was used. Firstly, the coefficients obtained on a uniform primary grid can be used also on deformed cells. We call this the constant coeficient schemes. Secondly, it is also possible

+

Complexity, performance and informatics

687

t o obtain lznenl- reconsti-uctzon within each primary cell. Snlall equation systems (3 x 3 for triangular and 4 x 4 for qtradrllateral cells) are obtained for the coefficients c J k by Taylor analysis, demanding a second order truncation error of the interpolation These systems are solved numerically for each primary cell. The flux limiter was implemented on the scheme with i? located at A and a t x . However, instead of finding q5r by the arithmetic average, an interpolation 4p = (1 - C Y ) ~Q.? E , [O, 11, was used assuming the flow along the edge & to be from vertex 1 to vertex 2. For cr > 112 the scheme becomes upstrean1 weighed. On structured grids, it has become c-ornrnonplace to adopt the normalised variable $c = ( d c - C$[i)/(d~ - &i) to govern the amount of upstream bias as suggested by Leonard [Leo 881 and C;askell and Lau [GLa 881. For a given cell face, the nodes are the first downstream ( D ) , the first upstream ( C ) , and the second upstream (TJ) of the face. For our case, C and D correspond to nodes 1 and 2 respectively. If we have a gradient e+s t ~ m a t eava~lablea t 1, we may rewrite the indicator as $c = 1 - (& - qhu)/2Vd1 . & . The gradient was estimated by an integral on the form of Eq. (2) over the secondary grid cell around vertex 1. We may now use established flux limiters like e.g. QUICK on unstructured grids. T h e limiter ACUTER

#

+

was developed for this purpose [EAM 981.

4. Numerical Experiments The random grids were generated by disturbing the uniform grid mesh length h into hi = 1 - drnd#i for edge i, where rnd#i E [O,l] is some random number. When d = O there is no disturbance and for d = 1, any grid length might be the result. T h e uniform mesh length is here normalised to 1. A uniform triangular grid consists of only equilat,eral triangles. Some smoothing is necessary on the most distorted grids t o ensure valid cells. In all results presented here, the maximzim norm of the global error versus the largest edge length in the grid is used.

4.2 Experiments on the diffusion operators T o study the isolated properties of the discretisations for the diffusive terms of the incompressible Navier-Stokes equations, Poisson equations are feasible,

688 Finite volumes for complex applications

Figure 4. CCI on Eq. ( 8 ) on quad. Figure 5. CCLS on Ey. ( 8 ) on quad. cells

cells ,,

..

..

i,

..:

,

:.

,,

:,

..

,

.

.: x, ; , : . : 2

Figure 6. VCI on Ey. (8) on quad. Figure 7. 1fCI on Eq. ( 8 ) on trzancells

yrilar cells

Complexity, performance and informatics

689

A t,est case looking somewhat like a top hat, was proposed. The solution is

and the right hand side function f of (8) is selected to fit this solution. The domain of definition is ( x , y) E [0, 11 x [0, 11. The "Top Hat" Poisson equation is designed to study discretisations a t the interior domain, as the flat solution at the boundaries gives little influence on the global error from boundary discretisation errors. Furthermore, the rather sharp gradient simulates situations met in fluid flow problems. All methods tested showed good second order accuracy on rectangular grids. However, the NCC scheme is seen t o perform very poorly on skewed grids (Figs. 2 and 3 ) . The CCI scheme is somewhat better (Fig. 4), but we have to go to t,he CCLS scheme go get second order accuracy (Fig. 5 ) for a cell centred scheme. More surprising is the seco~lclorder of the VCI method (Figs. 6 and 7). The low order truncation error is hence of a kind that d o not manifest itself into the global error. The VCI scheme is considerably less complex than CCLS, but yields comparable results.

4.3 Experzments on the convection operators

The velocity field for the convection was the solid body rotation ,u(x,y ) = y and v ( x , y ) = -z over the donlain ( x , y) E [-I, 11 x [O,1] with inlet and outlet 0 and 0 < x < 1 respectively. The on t.he boundaries y = 0 and -1 < x equation solved was ( r =

dm)

with solutioli e ( ~ =)

<

exp(-2 r ) sin4 .rr I. 0

<

r 1 else

'

Some results are presented in Fig. 8. Runs were also made on rectangular grids, in which case all schemes were second order accurate even for very large aspect ratios. The best of the constant coefficient schemes for convection are not superconvergelit t o second order, however, some superconvergence is in effect. Really, all t.he schemes has zero'th order truncation terms, but the average global error lies between first and second order for most cases. Looking at the results for the variable coefficient schemes, it seems that sonie of this superconvergerice was lost on quadrilaterals. The only scheme that is linearity preserving is the one with both u and 4 represented at A. This scheme has the best order estirliate of all tested schemes, however, the average error level is 2-3 times higher than the best constant coefficient schemes.

690

Finite volumes for complex applications

Triangular Grids

Figure 8. The plots show curve fits t o the scatter maxlmum global error data for each combznatzon of reconstructzon of u and 4 . The vertzcal spread of these scatter data zs quzte large, zadzcatzng szgnzficant dependence on grid skewness The two top plots are for constant coeficzent schemes, the two bottom for lznear reconstructzon schemes, the two left most on quadrzlateral grzds, and the two rightmost on trzangular grzds.

Complexity, performance and informatics

Figure 9. VCI for Eq. (12) on qtlads Figure 10. 1'CI for Eq. (12) on wzthout flux lzmztzng angles wzthout flux lzlnitzng

691

tri-

Figure 11. VCI for Eq. (12) on Figure 12. VCl for Eq. (12) on quads with ACUTER qrtuds with AC1TTER

4.4 Experiments on a convection-difftzlsion equation

To study the effect of combining the collvection and diffusion operators, and also to study the flux limiting, the equation

over the unit square was selected. The result,s presented in Figs. 9 through 12 are for v = 1/20. In this case, the solution has a "boundary layer" along the lines x = 0 and y = 1 and flux limiting is essential on coarse grids, and we actually see that ACUTER, in addition to giving monotone solutions, yields somewhat better results than central differences even on fine grid. The distor1096, o 15%, x 20%l, tion level is labelled as; @ - Uniform, o - 5%, - 35%, A - 55%, V 90%. The dotted lines indicate the second order slope. -

-

-

-

+

692

Finite volumes for complex applications

The data show a clear second order accuracy trend, and the vertical spread is small, indicating small dependence of the accuracy on the grid cell skewness. This is opposed to the pure convection case.

5 . Conclusions

Experiments on large sets of random grids have revealed properties of the global error of the schemes under study. Superconvergence to second order was demonstrated for the CCLS and VCI schemes on the diffusion operator, and also for the VC diffusion-convection scheme both with and without flux limiting. The flux limiter ACUTER provided both monotone solutions on coarse grids and some enhanced accuracy. The VC convection operators are less robust with respect to grid skewness, but it is interesting to note that constant coefficients proved t o be just as good, or even better, as linear reconstruction. It seems clear that neither primary nor secondary grid methods, as such, could be claimed to be the more accurate. A more relevant question, although still not precise, is with what simplicity or ease a prescribed accuracy may be achieved on either primary or secondary grids. In this respect the VCI scheme comes out a winner in the present tests. [Bar 891

BARTH, T . J., Recent developments in high order k-exact reconstruction on unstructured meshes, A I A A paper 93-0668,1989.

[GLa 881

GASKELL,P. H . AND LAU, A. K . C . , Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm, Int. J . Numer. Meth. Fluids, 8 (1988), pp. 617-641.

[GMe 961

GRASSO,F. AND MEOLA,C., Euler and Navier-Stokes equation for compressible flows: finite-volume methods, Handbook of computational fluid mechanics, Peyret, R., ed., Academic Press, 1996, pp. 160-277.

[Leo 881

LEONARD,B . P., Simple high-accuracy resolution program for convective modelling of discontinuities, Int. J. Numer. Meth. Fluids, 8 (1988), pp. 1291-1318.

[Mav 971

MAVRIPLIS,D. J . , Unstructured grid techniques, Annual Review of fluid Mechanics, Annual Reviews inc., 1997, pp. 473-514.

[EAM 981

MEESE, E. A., Finite volume methods for the incompressible Navier-Stokes Equations on unstructured grids, dr.ing. thesis, Norwegian University of Science and Technology, 1998.

[TMa 851

THOMPSON, J. F.

AND MASTIN,C. W., Order of difference expressions in curvilinear coordinate systems, J . Fluid Eng., 107 (1985), pp. 241-250.

[TDB 961

TREGUIER,A. M . , DUKOWICZ, J. K.,

AND BRYAN, K., Properties of non-uniform grids used in ocean general circulation models, Journal of Geophysical Research, 101 (1996), pp. 631-644.

A validation of an efficient numerical method for 3-D complex flows

E.A. ~ a d l u n 'S., ~eonardi',R. verzicco2 & I? Orlandil Universita di Roma, "La Sapienza," Dipartimento di Meccanica e Aeronautica via Eudossiana 18, 00184 Roma, Italy. 2~olitecnicodi Bari, Istituto di Macchine ed Energetics, Via Re David 200, 70125, Bari, Italy.

'

A second-order accurate, highly eficient method is developed for simulating unsteadjl three-dimensional incompressiblejows in complex geometries. This is achieved by using boundary body forces that allow the imposition of the boundary conditions on a given suqace not coinciding with the computational grid. The governing equations, therefore, can be discretized and solved on a regular mesh thus retaining the advantages and the eficiency of the standard solution procedures. ABSTRACT

Key words: 3 0 complex$ows, Immersed boundaries, Finite-diflerences

1. Introduction The continuously growing power of computers is strongly encouraging the engineers to rely on computational fluid dynamics for the design and testing of new technological solutions. Numerical simulations allow the analysis of the phenomena without resorting to expensive prototypes and difficult experimental measurements. Flows in simple geometries use regular grids and hence very efficient numerical methods. Flows in complex geometry requiring body-fitted curvilinear or unstructured meshes are less efficient and more difficult to code. In addition, in most of industrial applications, geometrical complexity is combined with moving boundaries which increase the computational difficulties since they require, regeneration or deformation of the grid.

694

Finite volumes for complex applications

In view of these difficulties it is clear that a numerical procedure that can cope with the flow complexity but at the same time retains the accuracy and high efficiency of the simulations performed on fixed regular grids would represent a significant advance in the study of industrial flows. One possibility for the solution of this problem is the introduction of a body-force field f such that a desired velocity distribution V can be assigned over a boundary S . In principle there are no restrictions for the velocity distribution V and for the shape and motion of S; therefore a wide variety of boundary conditions can be imposed. The main advantage of this approach is that f can be prescribed on a regular mesh so that the accuracy and efficiency of the solution procedure on simple grids are maintained. Indeed this idea is not new since it has already been pursued in the past by methods slightly different each other but in principle almost equivalent (Peskin ( I ) , Goldstein, Handler & Sirovich (2)). Recently Mohd-Yusof (3) has derived an alternative formulation of the forcing that does not affect the stability of the discrete-time equations. In this case the formulation o f f is flow independent. The expression for the forcing f derived by Mohd-Yusof (3) is:

where RHS'+'/~contains convective, viscous terms and the pressure gradient. If now we ask which value for fIf 'I2 will yield ul+' = v'+'on the immersed boundary this is simply given from the above equation

An adapted version of the forcing of Mohd-Yusof (3) has been developed to combine body-forces with finite-differences.

2. Results

2.1 Vortex ring formation The generation of a vortex ring is generally obtained by pushing a finite amount of fluid through a curvilinear nozzle. The ring was generated in water and injected fluid was colored by di-sodium fluorescein illuminated by a laser sheet in a plane crossing the axisymmetric nozzle through a diameter. This provided flow visualizations that could be compared with the numerical simulations. As the fluid flows along the inner walls of the nozzle some vorticity is generated and this separates at the corner in the exhaust region. This vortex sheet is very unstable and starts rolling up forming a compact toroidal structure. During the roll up, the primary vorticity induces at the external wall of the nozzle a secondary vorticity that

Complexity, performance and informatics

695

Figure 1: Comparison between numerical and experimental vortex trajectories at Re = 500 ( a ) ,Re = 1500 ( b ) ,Re = 2500 ( c ) .filled A numerical experimental primary ring. filled V numerical filled o experimental secondary ring. H numerical x experimental tertiary ring. also separates at the corner and forms a secondary vortex ring. This structure, has a circulation oppositely signed with respect to the primary ring, thus propagating backward inside the nozzle. Depending on the Reynolds number, i.e. on the strength of the primary vortex ring, the secondary vortex can be strong enough to generate a tertiary vortex ring that propagates in the positive axial direction. This complex behaviors is described by the trajectories of the vortex patches in 1. The Reynolds number flow dependence observed in the laboratory has been confirmed by the numerical simulations and figure 1 shows the good agreement between numerical and experimental trajectories. Before concluding this section we wish to stress an important points: the adequate description of the different flow regimes implies that the strength of the vortex structures is correctly predicted. This means that not only the velocities at the walls but also the vorticity generation is well described by the present approach.

2.2 Flow around a sphere In this section we consider the flow around a sphere where the separation is determined solely by the viscous processes at the wall. This should evidence eventual inaccuracies in the treatment of the boundary. In this flow a uniform velocity U is imposed at a certain distance from a sphere of diameter D.The Reynolds number is Re = U D / v and, depending on its value, the flow exhibits different regimes ranging from steady to unsteady up to fully turbulent. As a first test we have reproduced the Fornberg's (4) simulations; he used boundary fitted coordinates to simulate the axisymmetric flows ranging from Re = 100 up to

696

Finite volumes for complex applications

Figure 2: (a): length of the separation bubble vs Re; present results, rn results by (4). linear fit. (b): pressure profiles along the sphere surface at Re = 100,200,500 and 1000 (for increasing pressure values); -present results, .....-.. data from (4) Re

CD (Fornherg (4))

CD (Present results)

% error

100 200 500 1000

1.0852 0.7683 0.48 18 0.3 187

1.0794 0.7567 0.4758 0.3209

0.53 1.50 1.24 0.69

Table 1: Comparison of the drag coefficient for the flow around the sphere at different Reynolds numbers.

Re = 5000. We are aware that for Re 2 300 the hypothesis of axial symmetry is not physical because the flow develops three dimensionalities in the wake, however, the high-Re axisymmetric simulations were done to validate the numerical procedure. The flow around a sphere does not separate up to Re 2. 24 and for increasing Reynolds the axial length of the separation bubble grows linearly up to Re 2 100. This result has been found by the present numerical simulations as it is shown in figure 2. For Re 100 the pressure distribution over the surface of the sphere and the force coefficients were calculated by Fornberg (4); figure 2b shows that also for these quantities the present numerical methods is very accurate. From the pressure and velocity field around the sphere we have computed also the drag coefficient Cd that, as shown in Table 1, agrees very well with the results of Fornberg (4) and the error never exceeds the 2%. In order to test the capability of the code to handle fully three-dimensional flows and to capture flow transitions we have performed simulations in which the hypothesis of axial symmetry was removed. Similar simulations were performed by Mittal (5) using boundary fitted grids and he found that the flow preserves the symmetry for Re 270. For 270 Re 300 the wake becomes wavy but it still preserves the symmetry about a plane crossing the sphere in the centre. Finally for Re 2 450 the flow becomes fully three-dimensional and any degree of symmetry is lost. This is confirmed by the vorticity contour plots of figure 3 showing the oscillation of the wake only in one plane. Mittal ( 5 )obtained essentially the same results removing any symmetry by giving a random perturbation. The last simulation, at Re = 500, which

>

<

> <

Complexity, performance and informatics

697

Figure 3: Contour plots of azimuthal vorticity at Re = 300 ((a)-(c)) and Re = 500 ((d)-(fl) through orthogonal sections. (Ame = $2). Grid 97 x 97 x 193 gridpoints in the azimuthal, radial and axial directions. is above the second critical Reynolds number, by the flow visualizations of figure 3 shows the loss of the previous symmetry. In fact in this case the waviness of the wake is in two orthogonal planes.

2.3 Flow stability in a channel with cavities

The previous cases were simulated by plugging the body force method in a code in cylindrical coordinates. To study the flow past a cavity in a channel with periodicity in the streamwise direction the body force method has been inserted in a code that was previously used to perform the DNS of the turbulent channel flow. As a first simple case the critical Reynolds number for the plane channel has been evaluated; the value of 5850 agrees satisfactory well with the theoretical value of 5772. Ghaddar et al. (9) by a spectral element method investigated the effects of a cavity to reduce the transitional Reynolds number. They evaluated the critical Re by linear theory and by DNS; in the two cases they respectively found 900 and 1000, which agree well with the present value of 1050. To excite the unstable modes, a random disturbance at each point has been added to the steady state solution; after a transient, the velocity oscillates as V = ~ o e ~ ( ' - ~ ~ ) ) s in(o(t to)). From the time history of one of these signals o and o can be calculated. Table 2 shows that there is a good agreement between the present values and those given in Ref. (9). By our simulations it has been investigated the variation of transitional Reynolds number as the ratio height (h)l width ( I ) of the cavity was varied. For h / l = 1/4,1/3 and 112, respectively, the critical Re is 2758,2138 and 1120 showing that the deeper is the cavity the stronger is the effect induced on the flow. The reason is that a deep cavity produces a thin vorticity layer in the region between the cavity and the channel. This layer rolls up and modifies the Tollmein Schlichting waves that are the typical vortical structures of an unstable flow in a channel. Figure 4 shows that in the case of a very shallow cavity the waves are very similar to those in a channel while these have

698

Finite volumes for complex applications Re 5 25 800 900

Ghaddar et al. (9) theory linear -.043 -.0075

Present -.0419885851 -.0145530291 -.0075892 1262

Table 2: Comparison between the exponential decay of o of the perturbation velocity 93

Figure 4: Contour lines of the fluctuating component of the stream function 7 = the flow moves from left to right a completely different wave number when the cavity is deeper. In this case quantitative comparisons were not possible; the study was performed because the aim is to carried out the DNS of the flow in similar geometries to investigate how the shape and the spacing between the cavities affect the near wall vortical structures. The simulations are currently running and the results will be presented elsewhere.

2.4 Flow inside a IC piston In this last example we show the capability of the method to simulate high-Re turbulent flows in complex geometries for which a turbulence model is needed. In addition this example includes a moving boundary and this feature was missing in the previous flows. The configuration chosen is a simplified axisymmetric piston-cylinder assembly with a fixed central valve. For this configuration experimental measurements are available for the validation of the numerical results (Morse et al. (6)). The profiles are available at lOmm increments starting from the cylinder head for crank angles 36' and 1440 after top dead center. In the experiment the piston was externally motored and the valve was fixed and a tiny annular gap was left open between the valve and the cylinder head, no compression phase was included in the flow dynamics. The piston was driven by a simple harmonic motion at a speed of 200rpm cz 2 l r a d s which for the present geometry yields a mean piston speed of = 0.4m.s (when averaged over half cycle). The Reynolds number of the flow based on and on the piston radius is Re = 2000 in air. Details of the subgrid-scale model used in this computation are given in Verzicco et al. (7). In Verzicco et al. (7) extensive analysis of the flow inside the piston has been performed showing the changes in the dynamics when the axial symmetry was enforced,

6

v,

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699

Figure 5: Flow inside the IC piston: Vector plots at t = n/2 of the oscillation cycle ( a ) and (0). The r - 8 section (b) is taken lOmm below the ceiling of the cylinder. Grid 65 x 65 x 151.

Figure 6: Comparison between numerical and experimental averaged vertical velocity profiles for the crank angle 36" respectively 10 (a), 20 (b) and 30mm ( c ) below the cylinder ceiling. -present results, o experiments by (6). when threedimensionalities developed and when the turbulence set in. In this paper we only show the turbulent case for which the comparison with the experiments was made. In figure 5 snapshots during one instant of the oscillating cycle are given and the the high three-dimensionality of the flow can be appreciated form the vector plots in orthogonal sections. Radial profiles of axial velocity were obtained by phase averaging the fields over four piston cycles and then averaging in the azimuthal direction. These profiles are shown in figure 6 for several distances from the ceiling of the cylinder at two different crank angles and the comparison with the experimental data shows that the agreement is always very good. In Verzicco et a[. (7) profiles for additional sections and rms profiles of axial velocity are also given showing that the quality of the agreement is always very good. Finally in Haworth (8) the present results are compared also with the data obtained by a code using unstructured boundary fitted deformable meshes. In that paper it is shown that the quality of the results is the same even if the immersed boundary technique is incomparably less expensive.

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3. Conclusions In this paper we have summarized some of the results obtained by a numerical method based on Cartesian or cylindrical coordinates in combination with boundary body-forces. We would like to point out that the coding is very simple and in our group has been found that undergraduate students can learn to use the code in few weeks. In addition we found that it is very important to have an efficient and friendly software of interface between a CAD package and the flow solver and we are working on it. We are really confident that in a short while our group is going to produce a package to solve laminar flows past any kind of two and three-dimensional bodies. On the other hand to have a reliable package to solve realistic flows at high Reynolds numbers, it is important to spend a lot of time to test different kind of subgrid models. This is a difficult task that can not be achieved by a small group as that at the University of Roma. More collaboration is necessary and also on this point we are working together with some scholar at the Center of Turbulence Research at Stanford. [ l ] Peskin, C.S., 1972, Flow patterns around heart valves: a numerical method. J. of Comp. Phys., 10,252-271. [2] Goldstein, D., Handler, R. & Sirovich, L., 1993, Modeling a no-slip flow boundary with an external force field. J. of Comp. Phys., 105, 354-366. [3] Mohd-Yusof, J. 1997, Combined Immersed boundariesIB-splines methods for simulations of flows in complex geometries. CTRAnnual Research Briefs 1997, NASA AmesIStanford Univ., 3 17-327. [4] Fornberg, B., 1988, Steady viscous flow past a sphere a high Reynolds number. J. Fluid Mech., 190,47 1-489. [5] Mittal, R., 1999, Planar symmetry in the unsteady wake of a sphere. AIAA J., 37, 388-390. [6] Morse, A.P., Whitelaw, J.H. & Yanneskis, M., 1978, Turbulent flow measurement by Laser Doppler Anemometry in a motored reciprocating engine. Report F S n 8 n 4 , Imperial College Dept. Mech. Eng. 171 Verzicco, R., Mohd-Yusof, J., Orlandi, P. & Haworth, D.C., 1999, LES in complex geometries using boundary body forces. Proc. of the 1998 CTR Summer Program. VII, 17 1-1 86. [8] Haworth, D.C., 1998, Large-Eddy-Simulation of in-cylinder flows. in Multidimensional Simulation of Engine Internal Flows, IFP, RueilMalmaison, France. [9] N.K. Ghaddar, K.Z. Korczak, B.B. Mikic & A.T. Patera, 1986 Haworth, D.C. & Jansen, K., 1997, Numerical investigation of incompressible flow in grooved channels. JFM, 163,99-127.

Comparison of Two Finite Volume Methods for 3D Transonic Flows through Axial Cascades J.FoEt, J.Fiirst, J.Halama, K.Koze1 Department of Technical Mathematics C T U Prague Karlovo namdsti 13 CZ-121 35 Praha 2

This paper deals with the numerical solution of 3 0 inviscid transonic flow through axial turbine cascades, mathematically modeled by the system of Euler equations. Numerical solution is computed on structured grids using two different finite volume methods: simplified cell-centered T V D MacCormack scheme and cell-vertex Ni's based scheme. Two 3D axial cascades, one stator and one rotor, from last LP turbine stage of ~ k o d aPilsen-Turbines factory have teen solved. A comparison of achieved numerical results for the stator and the rotor cascades is presented. Influence of body forces in the case of rotor flow is discussed.

ABSTRACT

K e y Words: Euler equations, transonic flow, turbomachinery

1. I n t r o d u c t i o n

Numerical simulation of transonic flows in turbomachinery branch is subject of our long time cooperation with industry. Several numerical methods for the computation of 2D flow through cascades have been developed. Results have been thoroughly verified by measurements taken in real turbine and compressor cascades. 3D results of two independent numerical methods have been compared each other, because there are no proper experimental data available in 3D case. This enables validation of numerical results and improvement of numerical methods.

2. 2D p r o b l e m s and methods Inviscid flow through a cascade has been modeled by Euler equations in

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Cartesian or curvilinear coordinates. Modified system of equations describing streamtube thickness variations has also been involved. Laminar viscous flow has been modeled by Navier-Stokes equations. Following numerical methods have been employed: (a) finite volume MacCormack scheme (TVD and the classical form) (b) Ron-Ho-Ni finite volume scheme (c) multistage Runge-Kutta finite volume scheme (d) E N 0 and W E N 0 schemes (e) implicit Hwang-Liu TVD scheme (f) explicit and implicit forms of Osher scheme All schemes are of cell-centered type and only (b) is of cell-vertex type. Schemes (a), (b), (c) and (d) have been used on H-type quadrilateral grid, the scheme (e) on unstructured triangular one and the scheme (f) on unstructured hybrid grid. Schemes (a), (b) and (c) are central schemes and schemes (d), (e) and (f) are upwind schemes. Schemes (a), (b) and (c) have been extended to 3D. Mentioned numerical methods have been tested and validated with experimental data obtained for different geometrical configurations. For example, a Pilsen, measured in transotypical turbine cascade SE 1050 of ~ k o d Turbines, nic wind tunnel of I T CAS, see [FOJ 941 and [FOR 941 and VKI-LS82 turbine cascade, see [HUL 921. 2D inviscid methods combined with computation of a boundary layer have been used in turbomachinery design. Evaluated choke characteristics and loss predictions have been successfully compared with experimental data gathered by ART1 Prague. A comparison of results of unsteady flow through SE 1050 cascade, with time dependent outlet pressure, achieved by methods (a) and (b) was published in [FOJ 941. Two techniques for the solution of modified system of Euler equations have been developed. The first one simulates the influence of AVDR (Axial Velocity Density Ratio) by modified continuity equation [FOJ 941 and the second one by using S1 surface [FOJ 941 (quasi 3D approach of Wu). The second technique has been applied for the computation of flow through radial cascades. 3. 3D flow through axial blade rows

3.1. Governing equations

3D inviscid transonic flow in relative frame of reference is described by the

Complexity, performance and informatics

703

system of Euler equations in non-dimensional form:

where p denotes density, wi are the Cartesian components of relative velocity vector, e is total energy per unit volume, p is static pressure, 0 is angular velocity, K is ratio of specific heats and R is the radius. The system of equations is closed by the equation of state for ideal gas:

The right hand side of (1) is equal to zero for the flow in absolute frame of reference, i.e. with R = 0. 3.2. Boundary conditions

The non-permeability conditions d . 6 = 0 (6 denotes normal vector) are prescribed along walls. Stagnation values of density pa and speed of sound a0 and spatial distribution of velocity angles ci are imposed on the inlet section. Distribution of outlet static pressure p = p(R) is given. 3.3. Numerical solution Numerical solution has been computed on structured hexahedral grid by the methods (a) and (b). Equations (3) represents 1D version of method (a) in absolute frame of reference. The third equation is a TVD correction.

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Finite volumes for complex applications

where

v

=

At At -PA=-(IuI+a), Ax Ax

2

P P

a =y-

= (Awi-$, nwi+4)/\lAWi*g(12 @(r) = minmod(2r, 1) = max[O, min(2r, 1)] 1.'

The modification, where v is computed using the smallest absolute value of eigenvalue instead of spectral radius p ~has , lower amount of artificial viscosity (see [ANG 961). The second method (b) approximates relation (4)

+ 2At2 40)

a~

(Fwdiv(F, G , H),Gwdiv(F, G , H ) , Hwdiv(F, G , H ) ) n d S

where the first two terms are first and second order convective terms and last two terms are first and second order terms including body forces. D is control volume consisting of eight cells and D is the cell of dual mesh. Artificial viscosity is implemented in the form of additional redistribution terms in each grid cell.

3.4. Numerical results

Two axial cascades of the last LP steam turbine stage of ~ k o d aTurbines, Pilsen are presented within this subsection. The first one is the stator cascade with strong divergence of the tip part. The inlet flow is of axial direction and outlet static pressure depends on radius. The results are plotted in the form of Mach number isolines on the blade surfaces (Fig.1 and Fig.2). The second one is the rotor cascade with highly twisted blades, typical for the LP steam rotor cascades. Following design procedure of a rotor blade which places usually several profiles (verified in linear cascade configuration) a t

Complexity, performance and informatics

Fig.1.a: pressure side, method (a)

705

Fig.1.b: pressure side, method (b)

v'

Fig.2.a: suction side, method (a)

Fig2.b: suction side, method (b)

a different radial positions, it has been proposed to consider the rotor cascade as a non-moving cascade in relative velocity field. Rotor inlet flow field parameters have been computed from the solution at stator outlet and the outlet static pressure is again a given function of radius. To fulfill correct nondimensionalization, these two ratios: PO and a t the rotor inlet have to

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Finite volumes for complex applications

be prescribed, i.e.

Relative Mach number a t points on the inlet is evaluated each step of marching procedure and W components are then computed using common isentropic relations. The results plotted in the same form like for the stator are in Fig.3 and Fig.4. This model does not take into account the influence of body forces, i.e. Coriolis and centrifugal forces. The next approach keeps the above mentioned formulation and adds body forces to show how much do they influence the flow field structure. Fig.5 shows the results, achieved by method (b), of the rotor flow with body forces in the form of inlet and outlet relative Mach number distribution (xaxis corresponds to radius and y-axis to relative Mach number). Considerable changes close to hub can be seen. The flow in hub section accelerates strongly in case there are no body forces (see also relative Mach number isolines in Fig.G.a), whereas in the case there are body forces the acceleration rate almost disappears (see isolines in Fig.6.b). The presence of body forces has only minor

Fig.3.a: pressure side, method (a)

Fig.3.b: pressure side, method (b)

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influence at tip part of the cascade.

~-4.

~-4,

Fig.4.a: suction side, method (a)

Fig.4.b: suction side, method (b) M-4" c M.wI +~

16 14 12 1 I8

*.....+ -* ~+ * ".
.,"....--.

b:

1;: 0 08

1

1.2

14

16

i.8

Fig.5.a: without body forces

I

2

__~". .A--

*..*..

04

02

-

0 08

1

12

11

18

18

2

Fig.5.b: with body forces

\ Fig.6.a: without body forces

-

Fig.6.b: with body forces

708 Finite volumes for complex applications Bibliography

[ANG 961

ANGOTP., F ~ ~ R S J .T, KOZELK.: TVD and E N 0 Schemes for Multidimensional Steady an Unsteady Flows - A Comparative Analysis, Proc. Finite Volumes for Complex Applications - Problems and Perspectives, editors: Benkhaldoun F., Vilsmeier R., Rouen 1996, Hermes, pp 283-290

[CAU 891

CAUSOND.M.: High Resolution Finite Volume Schemes and Computational Aerodynamics, Notes on Numerical Fluid Mechanics Vo1.24, Braunschweig 1989, Vieweg, pp 63-74

[FOR 941

FORT J . , HUNEKM., LAIN J . , KOZELK . , SEJNA M., VAVRINCOV M.: ~ Numerical Simulation of Steady and Unsteady Flows Through Plane Cascades, Proc. 14th ICNMFD, editors: Desphane S.M., Desai S.S., Narasimha R., Bangalore 1994, Lecture Notes in Physics, Springer, pp 461-465

[FOJ 941

FORT J . , HULEKT., KOZELK . , VAVRINCOVA M.: Numerical Simulation of 2D Steady and Unsteady Transonic Flows, Transport Theory and Statistical Physics, 23(1-3), 1994, pp 385-407 editors: Desphane S.M., Desai S.S., Narasimha R., Bangalore 1994, Lecture Notes in Physics, Springer, pp 461465

[HUL 921

HULEKT., HUNEKM., KOZZLK.: Numerical Solutions of Euler and Navier-Stokes Equations, Proceedings of the European CFD Conference, Vol.1, ed. Ch. Hirsch, J . Periaux, W. Kordula, Elsevier Science Publisher D.V., Amsterdam 1992, pp 61-68

iNI 811

NI R. H.: A Multiple Grid Scheme for Solving Euler Equations, AIAA Journal, Vo1.20, No.1, 1981

Acknowledgement

This work is supported by the grants No. 101/98/1229, No. 201/99/0267 and 101/98/K001 of Grant Agency of Czech Republic and by the grant COST OC F1.70/1997.

An efficient and universal numerical treatment of source terms in turbulence modelling. B. Merci, J. Steelant and E. Dick Department of Flow, Heat and Combustion Mechanics Universiteit Gent, Sint Pietersnieuwstraat 41, B-9000 Gent, Belgium. Tel. +/32/9/264.33.14 Fax +/32/9/264.35.86 e-mail: Bart.Merci @ rug.ac. be A B S T R A C T A careful treatment is requiredfor the source terms in turbulence models in order to obtain a stable discretization. Implicit or explicit treatment is possible. The analysis shows that the choice depends on the source terms' Jacobian, which can be constructed either exactly or approximately (on the basis of individual terms). Numerical results prove that the rigourous method is generally applicable and superior to the approximate method, concerning convergence speed. The reason is that the multigrid technique can be applied, which is not at all evidentfor turbulent$ow. The analysis indicates the needfor a time step restriction, too, for n ~ i merical robustness. Practical results show this is not always necessary. Drfferent two-equation turbulence models are investigated conjirming the generality of the approach.

Key Words:

low-Reynolds turbulence models ( k - ~ ; k - w )source , terms, multigrid

1. Introduction Nowadays, low-Reynolds two-equations turbulence models are often used in CFD codes. They describe the physics of turbulence more universally than algebraic models and are less complex than Reynolds' stress models. When using these models, the source terms should be appropriately handled in order to obtain a stable discretization. A numerical analysis shows that the choice between implicit and explicit treatment depends on the Jacobian of the source terms. This Jacobian can be constructed either exactly or approximately on the basis of individual terms. Independently of the turbulence model, the latter method is mostly used. In this work, a more fundamental study is carried out to analyse in a general way differential equations with (non-linear) source terms. A comparison of convergence results with both methods is carried out. Different turbulence models are investigated in order to check the generality of the approach. Finally, convergence acceleration using the multigrid technique is studied. Previous studies [STE 94, STE 971 showed that the application of the multigrid technique on turbulence equations is not evident. Gerlinger et al. [GER 971 observed the same difficulties and proposed to freeze the non-linear parts in the source term. In this study, the MG-technique is applicable without damping or freezing any parts of the source terms, if the proposed discretization method is used. Convergence is much faster than can be obtained by solving the turbulence equations only on the finest grid.

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2. Treatment of source terms 2.1. Scalar equation 2.1.1. Linear source term

In order to investigate the influence of discretization schemes for source terms in differential equations, a simplified equation is considered first with a linear source term:

where A is a real or complex constant. It is well known that A > 0 requires an explicit treatment, whereas an implicit treatment is best suited when A < 0.If A is complex, the same conclusions are valid, but now the sign of Re(A)is decisive.

2.1.2. Nan-linear source term The clear distinction of discretization in the previous paragraph is a consequence of the fact that the source term S(4)and always have the same sign. This does not hold any longer for a non-linear source term, making the choice of discretization less evident. To investigate this, eq. [I] is generalized to:

$$

with S a non-linear function in 4.Depending on the source term S($),a non-zero steady solution can be obtained if a value 4, # 0 exists for which S(4,) = 0. The steady solution is assumed to be stable. In figure 1 a possible evolution of the source term S(4)is given in function of 4.Analytically, 4,is reached in a monotonic way. Therefore it seems a reasonable requirement that the discretization always results in an update 64in accordance with the source term sign:

S(4)2 0 * 64 2 0, S(4)5 0 3 645 0.

[31

There are two other requirements, more specifically related to turbulence modelling. The first demand is that 4cannot become infinitely large, and the second one is that 4 cannot become negative. An appropriate discretization that fulfills these three requirements has to be looked for. Implicit discretization of eq. [2] results in:

Complexity, performance and informatics

71 1

Figure 1 : Source term S(q5)in function of q5 Explicit discretization gives:

The amplification factors G E and G I are respectively:

For a prescribed amplification rate G, the necessary time step to obtain this rate is determined as:

Four different situations are possible (points A,B,C and D). Two characteristic examples are considered here. First point A ( S > 0 and > 0) is considered. The explicit treatment [5]guarantees lG( > 1 independent of the time step. However, the solution 4n+1 + cc for A T -+ rn which is not desirable, since a finite, stable steady solution is assumed. The best - time step should result in a solution close to the exact one. If an approximation q5 can be estimated for the steady solution I$,, the corresponding explicit time step can then be calculated from eq. [7]with G = 4/dn.

$$

The implicit treatment [4] demands

&2

I

in order to assure IGI 6"

> 1. For

the upper limit of the timestep, the solution will also' + co. Explicit treatment is the best choice for point A, with the time step calculated from e q [71. < 0 ) is studied. The As second characteristic example, point B ( S > 0 and situation does not differ much from point A, except that within a linear approximation, a 'natural' additional upper time limit can be proposed. With Taylor's expansion, the (qYt1 - 4"). As we source term S is approximated as ~ ( 4 " ~3' )S(4")

+ $$Imn

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Finite volumes for complex applications

S(4"11)=

0, this results in

mntl

=

1

mn - w 47%, which corresponds for

&=

80

. The corresponding dn+' I + . coincides with point B' in figure 1. If by a non-linear theory a better estimation 4 can an explicit treatment [6] to a time step

-

a a+

be found, the needed time step should be given by eq. [7]. In most cases, however, some hypotheses have to be made to obtain this non-linear approximation. To ensure a stable iteration process, the maximum time step is taken as the smallest time step from the linear and the non-linear analysis:

An implicit treatment [4] always fulfills the condition IGI > 1, and moreover guarantees IGI < LYI for all AtI. Therefore implicit treatment is preferable for point B. An infinite time step, however, corresponds to a solution 4n+1= 4B',which can be far away from 4,. The time step to be taken for a non-linear approximation is given by eq. [7]. For the same reasons as in the explicit discretization, the most conservative time step is taken:

4

Point C (S < 0 and < 0) is very similar to point B. The only difference is that the source term is negative, so that the desired amplification 0 < G < 1. The conclusions and time step restrictions are the same as in the previous case. For the same reason, the situation of point D corresponds to point A. Again, the difference is that, as S(4)< 0 the amplification should be 0 < G < 1. Thus it is seen that neither of the two possibilities (explicit or implicit treatment) can fulfill all the three mentioned requirements on itself in any of the four situations. The introduction of a time step restriction seems therefore necessary. For an explicit treatment this is obvious, since an infinite time step leads to an infinite (positive or negative) value of 4.For an implicit treatment, however, it can be necessary, too. In < 0, implicit treatment is practice, however, it is difficult to approximate G. When more robust than explicit treatment. In order not to counteract the implicit treatment, equations [8] and [9] are slightly altered. In [8], the sum is taken instead of the maximum. A closer look at eq.[7] reveals that this corresponds to dropping the term 22 Id,"

in [9]. Based on this discussion, the time step to be taken in any case is given by:

$$

> 0, and an implicit combined with an explicit treatment of the source term S(4)if one if a < 0. It is clear that the practical use of equation [lo] requires a reasonable + approximation of G.

a

Complexity, performance and informatics

713

2.2. Coupled system The source terms in two-equation turbulence models are in general strongly nonlinear and coupled. Two major methods exist to handle the source terms. In the first method [VAN 86, WIL 931, the positive parts of the source terms are not linearized and treated explicitly. The negative parts are linearized in such a way that negative real eigenvalues are obtained. They are treated implicitly. The second method [MER 931 starts with the construction of the exact Jacobian of the source terms: a Depending on its eigenvalues, this matrix can be split into a pos84'

$ %.

itive and a negative part: = + According to the previous analysis, the distinction between positive and negative here is made on the basis of the real part of the eigenvalues. The negative part is then treated implicitly, the positive part explicitly. An additional time step restriction is in principle necessary for robustness. For k-E or k-w based turbulence models, this time step can be determined from a simplification of the turbulence equations ([MER 991). Numerical results, however, show that for realistic flow situations this time step restriction is less stringent than time step restrictions coming from the convective or the diffusive terms. These restrictions were not encountered in the previous analysis because there the equation was simplified towards one containing a source term, but no convective or diffusive terms.

3. Numerical results Both source term discretization methods are numerically investigated. The lowReynolds k - E model by Yang-Shih and the low-Reynolds k - w model by Wilcox are studied for incompressible flows. A multistage time stepping scheme is used to reach the steady state solution [VIE 981. From now on, the method with the approximated Jacobian will be called 'approximate method', whereas the other method will be called 'rigourous method'. Fig. 2 shows the convergence evolution for a fully developed channel flow. The convective terms and the diffusive terms are set to zero, so that the source system's behaviour can be studied. Within the approximate method (curves 1 and 4), there is always a time step restriction, resulting from the implicit treatment of the negative parts of the source terms. For the rigourous method, a calculated time step restriction has to be introduced (curves 3 and 6), or strong underrelaxation is necessary, resulting in a worse convergence (curves 2 and 5). This shows the (theoretical) necessity of a time step restriction in order to retain a robust method. Fig. 3 shows the convergence history for a flat plate flow on a stretched grid (145x89 points). An alternating line solver is used. The approximate method performs equally well as the rigourous method (curves 1 and 2, and curves 4 and 5). The reason is that the convective and diffusive time steps are more restrictive than the source term time step so that they cover the influence of the source term discretization. For the same reason, the introduction of a calculated time step restriction is not necessary here (curves 3 and 6). Fig. 4 shows convergence results for the k - E model for a flat plate flow (193x97 gridpoints) using multigrid. The cpu times for the Navier-Stokes equations and for the turbulence equations have to be added to obtain the global cpu time. A substantial im-

714

Finite volumes for complex applications

provement is seen with the rigourous method when both the Navier-Stokes equations and the turbulence equations are solved on the coarse grids (curves 5 and 6). Curves 3 and 4 correspond to the Navier-Stokes equations solved with MG and the turbulence equations only on the finest grid. Curves 1 and 2 show the result for all the equations only solved on the finest grid. However, no convergence is obtained when the first method is used in combination with full multigrid (curves 7 and 8). This indicates the superiority of the second method. For the k - w model, the difference between the two methods is much smaller. Both methods converge with multigrid, again leading to a substantial improvement over single grid calculations (results not shown). Fig. 5 shows convergence results for the k - E model for a backward-facing step flow. Similar conclusions as for the flat plate can be drawn. However, now the first method also converges with multigrid. Similarly as mentioned in [GER 971, the flat plate flow seems more demanding for the multigrid technique. Similar results are obtained for the k - w model (fig. 6).

Figure 2: Convergence history for channel flow.(l: k - E, approximated Jacobian; 2: k - E , exact Jac.; 3: k - E , exact Jac., with time step restriction; 4: k - w, appr: Jac.; 5: k - w, exact Jac.; 6: k - w, exact Jac., with time step restriction)

Figure 3: Convergence history forflat plateflow with an alternating line solver:(l-6: see above)

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715

Figure 4: Convergence acceleration with the multigrid technique for jlat plate flow.(l,2: residual of NS, resp. turbulence, eqs., all solved single grid; 3,4: res. of NS, solved with MG (4 grids), and turb. eqs., solved single grid; 5,6: res. of NS with MG ( 4 grids), and turb. eqs. with MG (4 grids), rigourous method; 7,8: res. of NS with MG ( 4 grids), and turb. eqs. with MG (4 grids), approx. method)

Figure 5: Convergence acceleration with the multigrid technique for BFS jlow.(l-8: see above)

4. Conclusion A method of source term discretisation is presented that is robust and, most importantly, allows the use of the multigrid technique for convergence acceleration. The method is independent of the turbulence model.

5. Acknowledgements The first author is aspirant at the Flemish Science Foundation (F.W.O.).

6. References

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Finite volumes for complex applications

Figure 6: Convergence acceleration with the multigrid technique for BFSflow. (1-8: see above) [GER 971

Gerlinger P. and Briiggemann D.", Multigrid Convergence Acceleration for Turbulent Supersonic Flows, Int. J. for Numerical Methods in Fluids, volume 24,p. 1019-1035, 1997.

[MER 931

Merkle C.L., Weiss J. and Venkateswaran S., Efficient Implementation of Turbulence Modeling in Computational Schemes, Proc. Second U.S. National Congress on Computational Mechanics Washington, D.C., August 1993, 1993.

[MER 991

Merci B., Steelant J., Vierendeels J., Riemslagh K. and Dick E., Treatment of Source Terms and High Aspect Ratio Meshes in Turbulence Modelling, Proc. 14th AIAA CFD Conference, Norfolk City, June 1999, in press, 1999.

[STE 941

Steelant J. and Dick E., A Multigrid Method for the Compressible Navier-Stokes Equations Coupled to the k - E Turbulence Equations, Int. Journal of Numerical Methods in Heat and Fluid Flow, volume 4 (2), p.99-113, 1994.

[STE 971

"Steelant J., Dick E. and Pattijn S., Analysis of Robust Multigrid Methods for Steady Viscous Low Mach Number Flows, Journal of Computational Physics, volume 136, p. 603-628, 1997.

[VAN 861

Vandromme D. and Ha Minh H., About the Coupling of Turbulence Closure Models with Averaged Navier-Stokes Equations, Journal of Computational Physics,volume 65, p. 386-409,1986.

[VIE 981

Vierendeels J., Riemslagh K, and Dick E., A MultigridSemi-Implicit Line-Method for Viscous Incompressible and Low Mach Number Compressible Flows, Proc. of the 4th ECCOMAS Computational Fluid Dynamics Conference, Athens, John Wiley, p. 1220-1225, 1998.

[WIL 931

Wilcox D.C., Turbulence Modeling for CFD, Griffin Printing, Glendale, California, 1993.

Comparison of numerical solvers for a multicomponent , turbulent flow

E m m a n u e l l e X e u x e t C.E. M. I. F. Evry A l a i n Forestier C.E.A. Saclay J e a n - M a r c H Q r a r d E. D.F. Chatou

This contribution's topic is the resolution by different numerical solvers of a multicomponent, compressive, turbulent pow. The unique associated Riemann Problem's solution is identified thanks to an entropy characterization. An exact Riemann solver is implemented and called by Godunov scheme. Some numerical simulations are introduced to exhibit a comparison between Godunov scheme, Vfroe-nc and Rusanou scheme.

ABSTRACT

Key Words: Turbulent flow - Godunov scheme - Linearized solvers

1. I n t r o d u c t i o n

In this contribution is exhibited the resolution of the hyperbolic system which describes a compressive multicomponent turbulent flow. The model is written for a polytropic isentropic gas. With compressive flow, Favre's average is used to select a mean flow and a turbulent one. In this work, we are interested by the one order closure model and particularly the coupling between turbulence and pressure. Reynold's tensor is described through the turbulent kinetic energy It' of the mixture. The system is closed thanks to the li' evolution equation. 2. A t u r b u l e n c e m o d e l t o describe m u l t i c o m p o n e n t flows

The average variables describing the flow are : (p, p a , pu,li')

71 8

Finite volumes for complex applications

The density of the mixture is noted by p. pa stands for the density of one of the fluid, with a the mass fraction of one component of the mixture. U stands for the velocity and K for the turbulence of the mixture. Setting W = (C, I<),this system is conservative in C(p, p a , pU) variable, but not in K variable :

With all these assumptions we obtain the following first-order system in 2D :

+ ax(PU) + a, (PU) = o at(pa) + ax( P ~ u+) ay(pav) = o atp

(3)

at(PU) + a x ( p ~+2 -32K + P) + ay(puv)= o at(pv)

+

2 + ay(pv2+ -A+P)=o 3

2.1 From a 3D problem to the 1D Riemann Problem

It is well known that finite volume upwinding schemes are efficient methods to solve such no linear system. The most natural finite volume method is the Godunov's method. It requires the exact solution W = (C,K j on the boundary dRj. The Riemann solution of the multidimensional (S) system is unknown. So we have to come down to the solution of the associated one-dimensional problem written in the normal direction of the interface. x We set by (PR) the associated Riemann Problem, and by W * ( TWl, , W,.) its solution :

Come back to the Godunov scheme, we introduce the notations V ( i )the neighboring cells of cell "i" (not including cell i) lij the interface measure : lij = (8Qi n Qj 1 nij the unit normal vector from the i cell to the j cell.

Complexity, performance and informatics

719

We set by 3jk(C*, njk)the normal numerical flux at the interface between Wj and W k: 3jk(C*, njk) = ~ ( C * ( ~ Oj~, k ),) . + j k (4)

And, for the nonconservative equation

Ii'j

is an average of the Riemann solution on

aQj

2.2. Mathematical analysis of the hyperbolic system

The (Sn) system is hyperbolic, nonstrictly. E' is the turbulent celerity in a two phase flow. 10 K C" = aE2+ -- with E2 = P'(Pcr) (8) 9 P The eigenvalues are the following ones

And, the associated five eigenvectors are linearly independent in R5. The first and fiveth characteristic fields are genuinely nonlinear a t the sufficient condition 1 that pressure is a convex function of the - specific volume :

P

a2P

dP

>O 2a(Pa) PaPa(Pa)

+ VXl.rE < O and

vX5.r; > 0

(10)

+

The treble characteristic field is linearly degenerated : t t vXl.rtl = VX2.r2 = VX3.r3t = VXq.r4 =O

(11)

The solution consists in at most six constant states separated by shock waves, rarefaction waves or contact discontinuities. The rarefaction curves are :

IGpS

(a, p,u,,u,, I<), a = ag,p> 0, u, = u , ~ ,li = 7 ,

P;

720

Finite volumes for complex applications

( a , p , u nu,, , K ) ,a = ad, u, = U r d , p

> 0, I< = 7 , ~ r 3

Shock curves are :

4 ~- P i (a,p,un,UT,IC)= , a a g , u , = u r g , p >O,Iil= 4 ~ 1- P un = ung -

(a, p , ~ n U ,T , IC),a = ad, U , = u7d,p

Un

= Und

4 ~- Pr I(r, > 0, I< = 4 ~r P

-

+

I{,)

+(P

-

P,)]

PPr

The contact discontinuities verify [un]= 0

and

2 11'

+PI = 0

(16)

2.3. Entropy characterization and uniqueness of the solution

The generic form of the mathematical entropy is :

pU2 p = C1(T

+

1

=dp p2

+ IC) + C2p + C3pa + C4pU + C sPC,I + C6

(17)

Our system ( S ) admits two supplementary conservative variables :

In keeping with the second thermodynamic principle, a p convex entropy is growing on a physical shock.

The equivalence between Lax inequalities and compressive shock is shown. But the equivalence between entropy shock and compressive shock is demonstrated for only the physical entropy E . 3 has no physical sense, because its growing on

Complexity, performance and informatics

721

1-shock curve implies incompressive shock. The computation of the Riemann solution on the entropy shock curves ensure the uniqueness of the result. 3. Other numerical methods and preferences

The advantages of the exact solver are well known : positivity respect, entropy solution. But we have to balance these advantages by the fact that the method uses more CPU than linearized solver which doesn't require analytical computations. Then we introduce different schemes to analyze where a method is more or less efficient than an other. 3.1 A linearized solver : Vfroe-nc Vfroe scheme was introduced by Faille, Gallouet, Masella in 1996. It is based on a local resolution of a linearized Riemann problem. The numeric flux is defined, like Godunov scheme, by the physical flux computed at the interface solution of the linearized problem. An extension of this scheme was introduced by Buffard, Gallouet and Hkrard [BUF 981. Vfroe-nc uses the nonconservative variables to preserve Riemann invariants through contact discontinuities. Thus we prefer the variables (P(pcr),U ) to (pa, pU). 1 - Yl Y,. With Y = ( a ,u,, u t , I<,P), T = - and the linearized variable Y = 2 ' 0 we have to solve the linear systek :

+

10 With the turbulent celerity '2 = (+yP+ -I<)+ we obtain the eigenvalues : 9 Defining the intermediate states by a combination of the eigenvectors r; :

Y,. = Y j

+ C airi(Y)

The linearized Riemann solution Y * (y, Y, , 0) is given by :

722

Finite volumes for complex applications

The extension of Vfroe-nc scheme t o nonconservative systems is given by :

The linearized solver generates intermediate states that may not respect physical positivity. For example in the mirror states for a double rarefaction wave Uni< 0, we have to set :

It is also possible that the linearized velocity u, implies a bad choice of a . A big jump of cr may generate, in the next time step, such a pressure that the deduced mass fraction would be out of the ( 0 , l ) definition set.

3.2. Rusanov scheme

Rusanov scheme was introduced in 1961, it doesn't need any exact or approximative Riemann problem's resolution. In a multidimensional framework, we have the nonconservative extension of the scheme :

We introduce

Sitjdepending on the A eigenvalues :

With lij = 180; n Rj), and Wij =

wi

+

2

W j , we

have the following scheme :

(31) Rusanov scheme has been chosen for its respect of p, pa and K positivity, and because it preserves the mass fraction in the good definition set : 0 < a < 1.

Complexity, performance and informatics

723

3.3. Numerical results

We present, in the following page, a 2D test case of a onecomponent flow through a tube with obstacles. The inlet and outlet are free entrance, the others walls are reflecting sides. The air is coming from the left to the right side of the tube, with an initial velocity of 600 m . s - l . The initial conditions are : ( p , u , v ,a , Ir', P) = (1.1,600, 0,1,1000,100000) This case of a tube with obstacles reveals the advantages and deficiencies of the different methods. Vfroe-nc generates some negative turbulence in spite of the boundary positivity. With Rusanov method, positivity problems do not appears, but the I( evaluation is very approximate.

4. Conclusion

We conclude that in some extreme tests the cost of an exact solver is justified by the robustness required. With Godunov scheme the entropy solution is safely captured, without any positivity problem. It is not the case with the other linearized solvers which have shown their limits. We advocate to use successively the two schemes : the exact one for initialization and to deal with the boundary conditions, and a linearized one to go on with a reduced cost.

5. Bibliography [BUF 981

BUFFARDT . , GALLOUET T . , H ~ R A R JD. M., A sequel to a rough Godunov scheme : application to real gases, ~ c o l e s CEA-EDF-INRIA pp.363-382, 1998.

[GOD 961

E. GODLEWSKI, P . A . RAVIART,"Numerical Approxzrnatzorz of Hyperbolic Systems of Conservation Laws", Applied Mathematical Sciences 118, ED. Springer, 1996.

[HER 941

J.M.HERARD,A. FORESTIER, X. LOUIS," A Non Stricly Hyperbolic System To Describe Compressible Turbulence", Rapport E.D.F/D.E.R. HE-41/94/11A.

[LOU 951

X. LOUIS,"Mode'lisation Nume'rique de la turbulence compressible", PhD Thesis, univ. Paris VI, 1995.

[RAN 921

RANDALL J . LEVEQUE,"Numerical Methods for Conseruation Laws 'I, Birkhauser, 1992.

[SMO 831

J . SMOLLER, "Shock waves and reaction-diffusion equations", Springer-Verlag, 1983.

724

Finite volumes for complex applications

Husanov scheme

Figure 1: Comparison of turbulence at the critical time where It' < 0 by Vfroe

Parallel Overlapping Mesh Technique for Compressible Flows

Jacek Rokicki Zentrum fur Hochleistungsrechnen, T U Dresden, 0-01062 Dresden, Germany Warsaw University of Technology, Nowowiejska 24, 00-665 Warsaw, Poland { jack I jmajewsk I jzoltak ) @meil.pw.edu.pl

Dimitris Drikakis Queen Mary and Westfield College, University of London, Mile End Road, London E l 4NS, United Kingdom

Jerzy Majewski Warsaw University of Technology, Nowowiejska 24, 00-665 Warsaw, Poland

Jerzy ~ 6 l t a k Aviation Institute, Al. Krakowska 110/114, Warsaw, Poland

I n the paper w e present the Chimera overlapping m e s h technique applied t o the solution of compressible flow problems. T h e presented algorithm is particularly suitable for cases with large number of meshes overlapping in almost arbitrary m a n n e r (including multiple overlaps).

ABSTRACT

Key Words: Compressible Bows, Overlapping meshes, Parallel computation

1. Introduction The Chimera overlapping mesh technique (e.g. [HEN 94, ROF 991) is employed here as an efficient tool for grid generation in complex geometries and as a method for paralelisation. This technique is based on the subdivision of the physical domain into overlapping subdomains. Subsequently, the system of flow equations is solved on each subdomain separately (see Fig. 1). The global solution is obtained by iteratively adjusting the boundary conditions on each subdomain.

726

Finite volumes for complex applications

Figure 1: Grid system B1 around the SKF-1.1 airfoil. 2. Implicit unfactored solver

On each mesh the compressible flow is simulated by solving the Euler equations. Implicit time discretization is used and the nonlinear system of equations is solved by an unfactored Newton-type method. The Riemann solver [DRD 941 calculates the inviscid fluxes at the cell faces using the characteristic values of the conservative variables. A Godunov-type upwind scheme up to third-order accuracy has been employed for the calculation of the characteristic cell face values. 3. The overlapping-mesh algorithm

We consider here the general case in which the flow domain fl is fully covered by several overlapping subdomains R1,. . . ,OK, O = fll U . . . U flK. If I? and rj denote the boundaries of 0 and Rj subdomains, respectively, then yj = rj \ r will denote the interface between the subdomains in which boundary information takes place. In order to obtain convergence of the global solution, each interfacial point must be well covered by at least one of other subdomains. Each subdomain R j is covered by a suitable regular grid. However, the grid points in the overlap region do not necessarily coincide (Fig.1). As a result

Complexity, performance and informatics

727

each interfacial grid point has to be localised in the reference frame of each grid in the system. Subsequently biquadratic interpolation procedure is used in order to prolongate the local solution onto the whole subdomain. The second difficulty is related to the fact that the solution, obtained on various subdomains, remains different in the multiply overlapping regions. To deal with it weight functions xP are defined (henceforth called blending functions), which are used t o construct the global solution (see [ROF 991 for details). 4 Global solution and full algorithm

Let us assume that UIplis the solution vector defined on the subdomain Rp and extended to the whole 0 by assigning U[,](C) 0 for C # R,. The global solution U ( C ) is defined then as K

U(C) =

C xp(C)U[p](C)

p= 1

1

c t fl .

(1)

The full algorithm encompasses the following steps: 1. Initialise boundary conditions a t the interfaces yp (p = 1.. . . , K ) . 2. Perform a few iterations using the Euler solver on each local grid to obtain new local solutions U[pl, (p = 1,.. . , K ) .

3. Interpolate each local solution on the prescribed interfaces and calculate global solution.

4. Check the convergence on all interfaces and if there is no convergence return to step 2. The step 3 cannot be performed in parallel since it requires interprocessor communication, if different subdomains are served by different processors. Nevertheless, the computational cost associated with the step 3 is negligible compared t o the cost of step 2.

5. Numerical results The numerical quality of the method was assessed using the case of the transonic flow around the supercritical SKF-1.1 aerofoil with the deflected maneuver-flap [STT 791. The free-stream Mach number was Mm = 0.60 with the angle of attack cx = 2".

728

Finite volumes for complex applications

Figure 2: Mach number distribution around the SKF-1.1 airfoil for M , = 0.60, a = 2" (case B3). Three grid systems B1, B2, and B3 with increasing degrees of refinement, were chosen to demonstrate the convergence of the numerical solution. The systems consisted of 20 grids (25 for B3) as shown in Fig. 1. The number of grid points was 18410 for B1, 42183 for B2 and 55304 for B3. The Mach-number field corresponding to the grid system B3 is shown in Fig. 2. The shock wave is well visible in the middle part of the main airfoil, while a second, much weaker shock appears over the flap. Figure 3 presents the comparison of the computed C, distributions on the main airfoil and on the flap for all grid systems used in the computations. 6. Parallel implementat ion Parallelisation of the method is based on the assignment of a single grid or a group of grids to each processor. Therefore, the decomposition of the flowfield into subdomains determines, at the grid generation stage, the granularity of the parallel problem. As a result, the number of processors L cannot exceed the total number of grids K. Moreover, since grids are of different size, the number of processors L must be significantly lower than K, otherwise an uneven

Complexity, performance and informatics -2 5

-20

CP

L/

1

, :1

-

gnd system

.....

gnd system

729

B1 BZ

gnd system B3

Figure 3: Grid convergence study for the SKF-1.1 airfoil for M , = 0.60, a = 2". load distribution will severely affect the parallel performance. The parallel implementation scheme is shown in Fig. 4. The overall computational effort and total number of iterations did not depend on the number of processors. They were, however, slightly dependent on the number of grids in the system, overlap size, etc. (several results from numerical experiments can be found in [RDM 991). To estimate the parallel speed-up we have measured the execution time TL necessary for performing a fixed number of iterations (in our case 100 iterations) on an L-processor system. The speed-up factor c r ~and the corresponding efficiency q~ were defined as a~ = rl/rL, q~ = a L / L . In order to quantify the load-balancing effects the coefficient /3 was defined L by: pp = w,/;C,=,w, , P = rnax,,l ,...,L ,Clp , where w, stands for the workload assigned to the p t h processor (w, is proportional to the number of grid points assigned t o the p - t h processor). In order to minimise the coefficient ,8, for a given system of grids, a simple grouping algorithm was used according to which grids (in descending load order) were assigned to processors with the lightest load. Various parallel tests were performed for the flow around the SKF 1.1aerofoil using the 64-processor CRAY T3E. Two different grid systems were used. The first consisted of 25 grids with the total workload of 53344 (grid B3 from previous Section), with the largest and the smallest grids having workload of

730

Finite volumes for complex applications

Communication buffor - redistribution of boundary information --

Performs few iterations of the local solver on the group of grids

I

II

-

-

Performs few iterations of the local solver on the group of grids

(. . .)

Performs few iterations of the local solver on the group of grids

Communication buffor - collects solutions from a I processors Generation of the new global solution

NOCONVERGENCE

I

I

rz&$zxq Figure 4: Parallel implementation scheme.

10000 and 35, respectively (allowing the use of up to 6 processors). Table 1: The acceleration factor Q I L total efficiency QL and load balancing coefficient @ for a 64-processor CRAY T3E (grid system B3). L T L [see] QIL QL P

Table 1 collects obtained the values of c r ~ 7~ , , P. The case L = 6 is already poorly balanced and, therefore, no further acceleration is possible. For L = 5 the communication-to-computation performance ratio is visualised in Fig. 5, as obtained with the VAMPIR performance analysis tool [NAG 961. The second grid system contained 57 grids allowing the use of a larger number of processors. The total workload was equal to 36747, and the workload of the largest grid was 1800. The parralel communication was organised either by PVM reduce or by PVM send/receive. The performance is presented in Fig. 6.

Complexity, performance and informatics

73 1

Figure 5: Single iteration of the solver for CRAY-T3E using 5 processors communication to computation performance ratio. 6. Conclusions

The present results reveal that the overlapping-mesh technique can be efficiently used on a multi-processor system in conjunction with implicit methods and characteristics-based schemes. Grid-independent solutions were achieved for the case considered here. The efficiency of such an approach will increase with the complexity of the problem, provided that the workload can be distributed between the processors as evenly as possible. This can be achieved by further subdivision of existing grids or by re-grouping grids assigned to processors. Acknowledgments

This work has been supported by the KBN 7 T07A 022 14 and KBN 9 T12C 069 14 Grants. Performance tests were run on the CRAY T3E a t ZHR, TU-Dresden. Bibliography

[DRD 941

DRIKAKIS, D., DURST,F . , Investigation of Flux Formulae in Tran-

732

Finite volumes for complex applications 1

ILI

LOAD I M B M N C E REOUCE

'2

SEHDlRECElVE

Figure 6: T h e acceleration factor all total efficiency 7 ) ~a n d load balancing coefficient p for a 64-processor CRAY T3E (57 overlapping grids system). sonic Shock-Wave/Turbulent Boundary Layer Interaction, Int. Journal for Numerical Methods in Fluids, 18, p. 385-413, 1994. [HEN 941

[NAG 961

HENSHAW, W.D., A Fourth-Order Accurate Method for the Incompressible Navier-Stokes Equations on Overlapping Grids, J. Comput. Phys., 113, 1994. NAGEL,W.E., ARNOLD A., WEBERM., HOPPEH-C., SOLCHENK., VAMPIR: Visualization and Analysis of MPI Resources, SUPERCOMPUTER 63, 12, No. 1, p.69-80., 1996.

BACH

[RDM 991

ROKICKI,J., DRIKAKIS, D., MAJEWSKI, J . , ZOLTAK,J., Numerical and Parallel Performance of the Overlapping Mesh Technique Coupled with Implicit Riemann Solver, ZHR Internal Report, ZHR-IR-9901, 1999.

[ R O F 991

ROKICKI,J., FLORYAN, J.M., (1999) Unstructured Domain Decomposition Method for the Navier-Stokes Equations, Computers and Fluids, 28, p.87-120, 1999.

[STT 791

STANEWSKY, E., THIBERT,J . J . , Airfoil SKF-1.1 with Maneuver Flap, AGARD Report AR-138, 1979.

A comparison of Finite Volume and High-Order Finite Difference Schemes for the Solution of the Navier-Stokes and Euler Equations

M. Meinke, Th. Rister, R. Ewert Aerodynamisches Institut W u l l n e r s t r a f l e zw. 5 u n d 7, 52062 A a c h e n , G e r m a n y Tel.: +49-241-804821, E-mail: [email protected]

--

--

-~

~

p

-

p

~

ABSTRACT T h e properties of finite volume and high-order discretization schemes are

compared first for the large-eddy simulation of turbulent flows and secondly for computational aeroacoustzcs. For the LES of turbulent flows a finite volume method of second-order accuracy based o n a slightly modified A USM formulation, jLS93, MSR98] zs compared wzth a compact finzte digerence method followzng Lele [Lel92]. It is found that n o advantage is obtained from the high-order finite digerence scheme for a plane turbulent jet flow due to the fact that filtering of the solution is required to avoid high frequency oscillations. I n the field of aeroacoustics the application of high-order methods is, however, indispensable. This is demonstrated wath solutaon of the Euler equations for acoustic benchmark problems, which involve the simulation of the propagation and interaction of acoustic and entropy waves. For diflerent cases the accuracy of a second-order finite-volume and high-order discretization methods are compared, a compact finite-dzfference scheme derived by Lele and the dispersion-relation-preserving ;/DRP) scheme of T a m [TW93a], which minimizes errors by usir~goptzmized approxirr~utiorisof first derzvatives irr, the wave number and frequency space.

Key W o r d s : High-order methods, Large-Eddy S i m u l a t i o n , Aeroacoustics.

1. Introduction

The dirc,c.t computation of sound generated by a turbulent flow field is difficult, since the fluid mechanical pressure variations are several orders of magnitude larger than the acoustic signals. One way to solve this problem is t o extract

734

Finite volumes for complex applications

information about the acoustic field from a flow simulation and to use this information as source terms in computational aeroacoustic (CAA) methods. The LES of turbulent flow field seems t o be well suited for the determination of the source terms, see e.g. [I. For the simulation of the sound propagation in a non-uniform flow field the Euler equations have to be solved. Discretization schemes for these two problems are considered in this paper. Schemes of higher than second-order are usually more difficult to formulate and less robust, but promise a more accurate solution with the same amount of grid points. This can be important in the field of LES, where the truncation error of the numerical scheme can have a substantial influence on the solution and interferes with the contribution of the subgrid scale model. High-order discretization schemes and a finite volume method of second-order are compared for the solution of the Navier-Stokes equation for the LES of a turbulent plane jet and for the solution of the Euler equation for the simulation of wave propagation. 2. Method of Solution

The LES of the turbulent flows was carried out with different solution schemes for the Navier-Stokes equations, a second-order finite volume method based on a AUSM method and a compact finite difference scheme of 6th-order accuracy in space. The discretization of the viscous fluxes is performed with central differences of second-order accuracy. Test simulations with a sixth-order scheme for the viscous terms in combination with the compact finite difference scheme did not provide more accurate results compared to those obtained with the second-order scheme. For the simulation of sound propagation the Euler equations are solved additionally with a fourth-order Dispersion Relation Preserving (DRP) scheme. In the following the discretization methods for the convective terms are described briefly. In all cases the Navier-Stokes or Euler equations are discretized in general curvilinear coordinates for collocated structured grids.

A modified AUSM-scheme is used for the discretization of the convective fluxes with a second-order accuracy. The formulation for the convective flux F in direction B is as follows:

Complexity, performance and informatics

735

Herein, p is the density, u is the velocity, p the pressure and e the internal energy. Superscript and denote right and left interpolated variables which are obtained with a quadratic MUSCL interpolation of the primitive flow variables. M D = 3 is the average of the left and right interpolated Mach number. The speed of sound a is also computed as an average of the left and right interpolated values. The parameter x is used to control the amount of numerical dissipation within this scheme. Its influence on the solution is demonstrated in [MSR98, Sch97] for the LES of a turbulent channel flow. In case x is set to zero, a central discretization of the pressure derivative is obtained, which exhibits only a very little numerical dissipation and was therefore found t o be best suited for the LES of the different flow problems investigated. For strongly curvilinear or stretched grids a value of x=0 may result in an unstable scheme, so that slightly higher values must be used. For the CAA applications only values of ~ = were 0 used.

2.2 High- Order Compact Finite Diflerence Scheme The high-order compact finite difference scheme proposed by Lele is implemented for the skew-symmetric formulation of the governing equations. The inner scheme of sixth-order accuracy is given by FL+1 + 3FL

+ FA-,

1

7 + ?(FP-1 - Fp+l)

= z(Fp-2 - Fp+z)

.

FL denotes the first derivative of the convective fluxes Fp a t the grid point defined by the subscript p. Approaching the boundaries with non-periodic conditions the order of the scheme is reduced in steps from 6th to 4th and then t o 3rd order. The reduction of accuracy near boundaries leads locally t o larger errors and causes the generation of high-frequency oscillations. These oscillations could be avoided with a higher grid resolution in those regions. A substantial reduction of the spatial steps may be acceptable for direct numerical simulations, but not in a context of a LES. Therefore an explicit filtering procedure is applied every 5th time step with an one-parameter family of fourth-order accuracy:

-

Q , denotes the filtered variable at the node p. The parameter a allows a control of the filtering. Values only slightly smaller than a = 0.5 result in a cutoff filtering with a filter width equal to the grid spacing. Values of a in the range of [0.49-0.4991 were used to damp high-frequency oscillations. All details of the present implementation are presented in [Ris98].

736

Finite volumes for complex applications

2.3 Dispersion Relation Preserving Scheme

The DRP-scheme following Tam and Webb [TW93a] is used for the solution of the Euler equations. With the help of Fourier transformation in space and Laplace transformation in time the Euler equations can be recast into the dispersion relation. If the same procedure is applied to the discretized Euler equations and the dispersion error is minimized for a certain wave number range [TW93b], the coefficients aj of a finite difference approximation

can be determined with the restriction that a formal accuracy of 4th order is obtained. The coefficients then read: a0 = 0, al= -a-1 = 0.770882380518, a2 = -a-2 = -0.166705904415, as = -a-3 = 0.0208431427703.

2.4 Integration in Time The integration in time of the differential equations is carried out with an explicit Runge-Kutta method for both discretization schemes. For the secondorder scheme a low storage method, [Jam83], with second-order accuracy is used. ,4 conlparison of a classical Runge-Kutta method of 4th order and the low storage version for the high-order method showed that the same accuracy is obtained for the cases considered here, since the time step is relatively small due to stability restrictions. The computational effort for the integration of the same physical time is about four times higher for the high-order scheme than for the second-order scheme. This is due t o the fact, that the computation of the space derivatives is about two times more expensive for the compact scheme. Another factor of two results from the smaller time step in the 4th order Runge-Kutta method, which can only be chosen half the size of that for the low storage version. Therefore, the low storage version was used in most of all cases with the compact scheme.

3. Results

In the following results are presented from solutions of the Navier-Stokes equations for the LES of turbulent flows and from solutions of the Euler equations for C A A applications.

3.1 LES of a Turbulent Plane Jet It is known that jet inflow conditions have a large influence on the downstream development of the jet, [RS93]. In order to provide a physically mean-

Complexity, performance and informatics

737

ingful inflow condition for the mass flow, a turbulent inflow condition is used, which is generated by the LES of a fully developed turbulent channel flow. Both, the jet and channel simulation, are conducted simultaneously, so that at each time level an instantaneous solution from a cross section of the channel flow can be inserted in the inflow plane of the turbulent plane jet. Plane Jet case 1 case 2

grid points 81 x 149 x 65 65 x 129 x 33

L, x L, x L, axl 1 0 0 x 8 0 x 2.50 0.05D 12.5D x 8D x 2.5D 0.15D

ayl

0.005D 0.005D

Table 1: Grid parameters for the LES of the turbulent plane jet at a Reynolds number of 7600 and a Mach number of 0.1. x denotes the streamwise, z the spanwise direction. A x l and A y l are the smallest spatial steps i n x - and ydirection, respectively. The fully developed turbulent channel flow is simulated for a Mach number of Ma=O.l. Quasi periodic boundary conditions are used in streamwise direction, in which pressure and temperature fluctuations and the mass flow rate are considered to be periodic. Periodic boundary conditions are used in spanwise direction. A detailed description of the results obtained for the fully developed channel and pipe flows with the applied algorithm can be found in [Sch97, MSR981. The other boundary conditions correspond to the case of a turbulent jet exhausting from an opening in a wall of infinite extension into a fluid a t rest. On the lateral boundaries the mass flux was assumed to be the same as for the first inner surface and the pressure was held constant. In the exit plane nonreflecting boundary conditions, proposed by Thompson, [Tho87], and Poinsot & Lele, [PL92], formulated for general curvilinear coordinates, [Ris98], are applied. This boundary condition allows t o use a small domain of integration, thus leading to a reduced computational effort. In spanwise direction periodic conditions are applied.

case 1, AUSM, no SGS-model case 1, Pade-31416 no SGS-model case 2.AUSM. no SGS-model case 2, AUSM, dynam~cSGS-model

dD

a

x/D

Figure 1: Simulation of a plane turbulent jet at a Reynolds number of 7600

with different solution schemes. Left: decay of centerline velocity u,~/u,~(x=O), Right: Half width B of the jet. The turbulent plane jet flow is computed with both the second-order AUSM

738

Finite volumes for complex applications

and the compact finite difference scheme. The corresponding grid and flow parameters are summarized in Table 1. The grid points of the channel flow mesh are continued identically into the mesh for the jet flow. The Reynolds number is 7600 based on the width D of the jet which corresponds t o Re,=200 of the channel flow calculation. In Fig. 1 solutions of the second-order scheme are compared with those from the higher-order Pad& 31416 scheme. For the compact finite difference scheme an explicit filtering with a=0.499 is used to remove high frequency oscillations in the flow field, which are generated by the reduced accuracy near boundaries. It can be seen that the same length of the potential core is predicted with case 1 and case 2 grids and also with the different discretization schemes. The decay of the centerline velocity is predicted with the same accuracy until the end of the domain of integration. The spreading rate is in good agreement with the theoretical value of 0.097 (Fig. 1).

?L D

r D

a D

Figure 2: Comparison of the turbulence intensities i n streamwise direction ~ ( u " ) . Results of the AUSM scheme with diflerent SGS-models. +: case 1, without model, X : case $ without model, 0 : case 2, dynamic model. In Fig. 2 the turbulence intensities a(ul') a t different locations near the inlet sections obtained with the AUSM scheme are presented. Only negligible differences between the calculation with and without SGS-model on the coarse grid (case 2) are visible. The turbulence intensity computed on the finer grid (case 1) differs from that on the coarser grid (case 2) especially in the core region. This is partly caused by the different solution of the channel flow, i.e. the inflow condition of the jet, and partly by the different resolution in the jet domain. The second moments in Fig. 3 of the Pad63/4/6 and the AUSM scheme for case 1 and case 2 calculations again show only small differences for the cases with and without SGS-model. The turbulence intensities for the compact scheme on the coarse grid (case 2) are higher in the vicinity of the inlet section, because of small high-frequency oscillations generated by the decreased order of accuracy near the physical boundaries. The usual way to avoid such oscillations is by grid refinement near these boundaries. On the finer grid, which has a three times smaller spatial step in streamwise direction, these oscillations vanish totally.

Complexity, performance and informatics

739

Figure 3: Comparison of the turbulence intensity i n streamwise a(ul')and spanwise direction a(wl')at diflerent locations near the inlet section: : case 1, without model, P A D E - ~ /with ~ / ~a = 0.499; - - -. . case 2, dynamic ~ / ~ a = 0.499; . . - . .. . : case 2, without model, PADEmodel, P A D E - ~ / with 3/4/6 with CY = 0.499; -. . case 1, without model, A USM; ---: case 2, without model, A USM Further variations of numerical parameters were carried out, but are not presented here for brevity. For a higher Reynolds number of 22000, the same tendency as described above is observed. A detailed description of the numerical methods and all results can be found in [Ris98].

3.1 Simulation of C A A Benchmark Problems To examine the properties of the different solution schemes for CAA applications, the Euler equations are solved for the test cases of category 3, problem 1 and 2, which were defined on the first CAA-workshop [HRT95]. In this case the propagation of density, pressure and velocity pulses with Gaussian shapes are simulated in a base flow with Mach number 0.5. The initial position of the entropy and vorticity pulse are selected in a way that they reach the outflow boundary simultaneously. In Fig. 4 the density distribution is shown for a dimensionless time of 60. The boundary conditions applied are the non-reflecting boundary condition

740 Finite volumes for complex applications

0.1

t

:\

anaiyc sdution AVSM scheme. Thompson BC -------. Pade schema. Thompson BC DRP scheme, Tam BC - - - - -

1;

f':

'-l2

0.04 0 02

-0.02 -0.04 -100

-50

0

50

100

X

Figure 4: Left: Density distribution a dimensionless time T = 60. DRP scheme, Tam's boundary condition. Right: Comparison of the density profile on the symmetry line for the different schemes. following Thompson [Tho901 and the asymptotic conditions of Tam [TW93a]. In Fig. 4 the density profile is compared for the three different solution schemes a t a dimensionless time of 60. The second-order method clearly shows the largest deviation from the analytic solution, which confirms that it has the largest dispersive and dissipative errors. The entropy and vorticity pulses also show a deviation from the circular shape, which indicates a stronger anisotropic error distribution compared to the high-order schemes. Both high-order methods show good agreement with the analytic solution, also for later time levels. For this case the computing time for the DRP scheme is about three times smaller than for the compact finite difference scheme. , O5 -

;

0.03 -

,

,

,

. .

0.02 0.01 -

"

x

,

analytic solullon -Pade scheme. Thompson BC Pade scheme, Tam BC - - - - -

004

*

,

40

-20

0

"

20

-

40

60

80

100

x

Figure 5: Left: Density distribution for a dimensionless time T = 100. Pade scheme, Thompson's boundary condition. Right: Comparison of the density profile on the symmetry line for the dierent boundary conditions. The performance of the different boundary conditions is illustrated in Fig. 5 for the compact finite difference scheme. Although high frequency oscillation of small amplitude are visible with Tam's boundary condition the deviation of the solution near the outflow boundary are much smaller. These results are in

Complexity, performance and informatics

741

agreement with comparisons by other authors, see e.g. [HSM95].

4. Conclusion High-order schemes and a second-order finite volume schemes have been compared for two different applications, the LES of turbulent flows and computational aeroacoustics. In the case of the LES of a plane turbulent jet the applied compact finite difference scheme showed no advantage in terms of accuracy over a second-order method. The computational costs for the compact scheme are about a factor of four higher than for the finite-volume method. In both cases the solution was found t o be more or less independent of the application of subgrid scale models. In solutions of the Euler equations for CAA applications, it is known that high-order schemes are advantageous. The simulation of two benchmark problems on equidistant grids showed that the DRP scheme of Tam and the compact finite difference scheme produce results of similar accuracy, where the DRP scheme is about three time less expensive than the compact scheme. Currently, these comparisons are extended also t o stretched and curvilinear grids. First test simulations have shown that filtering techniques or damping terms have t o be used t o avoid high frequency oscillations in the solution.

References [HRT95]

[HSM95]

[Jam831

[Lel92] [LS93] [MSR98]

[PL92]

HARDINJ. C., RISTORCELL~ J. R., A N D TAM C. K . W., ICASEILaRC workshop on benchmark problems in computational aeroacoustics (CAA). Technical Report CP-3300, NASA, Hampton, VA, May 1995. HIXONR., SHIHS.-H., A N D MANKBADI R. R., Evaluation of boundary conditions for computational aeroacoustics. Technical Report 95-0160, AIAA, 1995. JAMESON A,, Solution of the Euler equations for two-dimensional transonic flow by a multigrid method. Applied Math. and Comp., 13:327-355, 1983. LELES. K . , Compact finite difference schemes with spectral-like resolution. J. Comput. Phys., 103:16-42, 1992. LIOUM. S. A N D STEFFEN JR. CH. J . , A new flux splitting scheme. J. Gomput. Phys., 107:23-39, 1993. MEINKEM., SCHULZ C., A N D RISTERTH., LES of Spatially Developing Jets. In R. Friedrich and P. Bontoux, editors, Computation and Visualization of Three-Dimensional Vortical and Zbrbulent Flows. NNFM 64, pages 116-131. Vieweg Verlag, 1998. POINSOT T. J . AND LELES. K., Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys., 101:104129, 1992.

742

Finite volumes for complex applications POPE D . S., A ~ ~ S C O U S / ~ C O Usplitting S~~C technique for aeolian tone prediction. In Second Computational Aeroacoustics (CAA) Workshop on Benchmark Problems. NASA C P 3352, 1997. RISTERTH., Grobstruktursimulation schwach kompressibler turbulenter Freistrahlen - ein Vergleich zweier Losungsansiitze. Diss., Aerodyn. Inst. RWTH Aachen, 1998. Russ S. A N D STYKOWSKI P . J . , Turbulent structure and entrainment in heated jets: The effect of initial conditions. Phys. Fluids, A 5(12):3216-3225, Dec. 1993.

C., Grobstruktu~~zmu~ation turbulenter Freistrahlen. Diss., SCHULZ Aerodyn. Inst. RWTH Aachen, 1997. K. W., Time dependent boundary conditions for hyperTHOMPSON bolic systems. J. Comput. Phys., 68(1):1-24, 1987.

THOMPSON K . W . , Time dependent boundary conditions for hyperbolic systems ii. J. Comput. Phys., 89:439-461, 1990. TAM C. A N D WEBB J., Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys., 107:262-281, 1993.

TAMC.K.W.

A N D WEBB J . C . , Direct computation of nonlinear acoustic pulses using high-order finite difference schemes. Technical Report 93-4325, AIAA, 1993.

Simulation of 3D turbulent flow through steam-turbine control valves Agaphonov B. N.", Goryachev V. D.~',Kolyvanov V. G.'), Ris V. v.~',Smirnov E. M.~',Zaitsev D. K . ~ ' " Energotech Ltd., P.S. Malii ave. 52-26, St. Petersburg, 197110 Russia " State

Technical University, Tver, 170024 Russia State Technical University, St. Petersburg, 195251 Russia, E-mail: [email protected] -

-

ABSTRACT The paper deals with three-dimensional and axisymmetric conzputatrons of turbulent steady-state gas flow through steam turbine control valves of complex geometry. Using a finite-volume second-order multi-block Navier-Stokes solver, parametric computations have been carried out for two valve models. The flow essential features are discussed, and pressure losses calculated are compared with experimental air-test data.

Key Words:finite-volume merhod, 3 0 turbulent flow, steam turbine, control valve.

1. Introduction

Control valve is one of the crucial parts of a steam turbine. Efficiency and reliability of the valve depend, to a great extent, on flow behavior in its flow passage, which has a complicated geometry. A typical valve flow passage includes an inlet tube, a partially blocked steam chest, an annular contractor, a conical diffuser and an outlet tube. As a rule, the center-line of the inlet tube is perpendicular to the axis of the main part of the valve body that produces 3D flow inside the valve. The main parameter affecting the flow and pressure losses is the valve lift. Today's achievements of CFD allow routine 3D computations to be performed for development of more efficient and cost-effective valve design. An example of application of 3D numerical simulation for getting more insights into the behavior of flow through a steam turbine valve is reported in [COF 961. The present work describes the first experience of use of an advanced Navier-Stokes finite-volume multi-block solver (named as SINF code), for studying turbulent compressible gas flow in steam turbine valves designed within the Russian industry.

744

Finite volumes for complex applications

2. Test cases and mathematical models Systematic computations of steady-state flow for different valve lifts have been performed to compare a control valve designed following [KOL 981 with a prototype manufactured for many years for high-power steam turbines. Computational examples are formulated accounting for the conditions adopted during the air test of two control valve models [KOL 981. For both models, the seat diameter, D , is 70 mm. Data for three cases are presented below. The main geometrical and operating parameters adopted at the calculations are given in Table 1 (G' is the critical mass bulk).

Table 1. I

I1

I11

0.3

0.21

0.3

Mass bulk, GIG*

0.485

0.344

0.485

Inlet-tube Mach number

0.127

0.090

0.127

Reference velocity, m/s

95.7

67.9

95.7

Case Lift, hlD

Cases I, I1 correspond to the valve model with an improved passage (Figure 1 ) . The valve lift, h, is 0.30 for Case I, that is the maximum designed value. For Case 11, the lift is reduced to 0.210. Despite Figure 1 covers only the major part of the valve passage, the flow through the valve has been calculated simultaneously with the flow in the inlet and outlet tubes dropped in the plot. Note also, that the grid shown on the valve boundary surfaces consists of lines connecting centers of finitevolume boundary faces, and of inherent boundaries of computational blocks.

Figure 1. Example of control valve passage geometry

Complexity, performance and informatics

745

Case 111 is associated with the valve of an old construction, the passage of which is far from a perfect aerodynamic form. The shutoff element has a plate-like form that does not fix the position of 3D-flow separation, the seat has a hollow, and there are two backward steps in the diverging part of the valve passage. The valve lift is taken the same as for Case I. For description of 3D turbulent flow within the valve body, the Reynoldsaveraged Navier-Stokes equations are applied in combination with the highReynolds-number k - E model of turbulence and the wall-function technique [LAU74]. Since the standard k - E model produces an excessive turbulence generation in the contractor region; a simple procedure of eddy viscosity limitation is used up to the downstream edge of the shutoff element. The limit is taken to be equal to the maximum eddy viscosity within the inlet tube. The latter is defined using relations typical for fully developed round tube flow. As well, calculations based on axisymmetric approximation have been carried out to analyze the contribution of 3D effects on pressure losses in the valve design adopted. To get an axisymmetric formulation of the problem, the inlet tube of the real valve is replaced with an upstream annulus.

3. Computational aspects The SINF (Subsonic and INcompressible Flows) code is based on the second-order finite-volume spatial discretization using the cell-centered variable arrangement and body-fitted grids. Cartesian velocity components are used as primary variables. Upwind second-order schemes are applied to compute convective parts of fluxes (the results presented have been obtained with the QUICK scheme [LE079]). The artificial compressibility approach and implicit schemes incorporating the approximatefactorization and convergence-acceleration techniques are employed to solve both incompressible and subsonic compressible flows [SMI 931. In the recent multi-block version, exchange of data at block interfaces is carried out at each artificial time step via creating an auxiliary virtual interface block, that involves geometrical and flow-filed information from near-interface layers of connecting grids. Computations for the virtual-block cells are performed with the same rules as for real blocks. This technique provides for full conservation of convective and diffusion fluxes even on non-uniform and strongly skewed grids. The 3D computations have been performed assuming the flow to be symmetrical with respect to the middle plane. For each valve model, a multi-block grid of 80,000 cells was generated. For that, as well for data post-processing, possibilities of the SELIGER system [GOR 971 are used. As an important part, this system includes an advanced grid-generation and geometry-modeling tool denoted ORIGGIN (Operative and Rational Interactive Grid Generation INstrument). 2Dl3D block-structured grid generation possibilities are provided by implementation of various algebraic techniques and an elliptic equation solver. Another part of the SELIGER system is the graphic postprocessor LEONARDO.

746

Finite volumes for complex applications

4. Results and discussion 4.1. Axisymmetric computations Figure 2 shows results of axisyrnmetric computations conducted for Case I. A distribution of velocity vector normalized with the reference value is given together with isolines of non-dimensional total pressure. One can see that a relatively small separation region is formed past the shutoff element, and the velocity defect is retained until the outlet section of the downstream diffuser. Note that the maximum value of the Mach number achieves 0.6. No flow separation is observed on the outer wall of the passage, but a wake-type layer developing along the outer wall gives a very significant contribution to the integrated pressure loss. The development of this, relatively thick, layer starts after a steep turn of the flow at the beginning of the diffuser. As a whole, the results of simulation point that the "eddy" part of pressure losses is dominant. That, in particular, promotes a successful use of the standard k - E turbulence model that works well at predictions of free shear layers. For the flow region shown in Figure 2, the pressure loss calculated is 1.52%. However, available experimental data were obtained with a rig where a tube five times diameter had been placed downstream the diffuser and a small backward step had been arranged in the connection plane. Simulation of axisymmetric flow developing in the region downstream the diffuser has resulted in a pressure loss of 0.54%. The overall rate of pressure loss is given in Table 2.

Figure 2. Velocity vectors (a) and total pressure contours (b) computed with axisymmetric formulation for Case I A reduction of the lift (Case 11) leads to an increase of the separation zone past the shutoff element. Moreover, a higher non-uniformity of velocity field within the diffuser is observed. That would result in an increase of pressure losses. However, the reduction of the Mach number adopted for this case compensates the effects of

Complexity, performance and informatics

747

flow field transformation, and the overall pressure losses have retained at the same level as in Case I (Table 2). Table 2. Case

I

I1

I11

Pressure loss, AP/P,,,,,, % - air test data - 3D computation - axisymmetric approximation

2.0 2.3 2.1

2.0 2.3 2.1

3.8

4.0

Flow configuration computed for Case 111 (Figure 3) differs considerably, as compared with that for Case I. The geometry of the diffuser including two backward steps may come under obvious criticism. The overall rate of pressure losses computed has risen to 4%.

Figure 3. Velocity vectors (a) and total pressure contours (b) conlp~lted with axisymmetric formulation for Case Ill

4.2. Three-dimensional computations Flow structure computed on the base of 3D formulation is illustrated below for Case I only. Figure 4 shows distributions of the Mach number and total pressure over the symmetry plane. Note, that the gas inflows into the steam chest from the right. Comparing Figure 2 and Figure 4, one can extract the 3D effects. First, the adopted size of the steam chest is not enough to create a flow field that would be uniform in the circumferential direction. The velocity of gas in the gap between the seat and the shutoff element is higher on the inlet tube side than that on the opposite side. A significant circumferential non-uniformity of the flow field is

748

Finite volumes for complex applications

observed even in the narrowest part of the flow passage Consequently, the shutoff element is acted upon by a non-zero lateral force. The separation zone is deformed, as compared with the axisyrnmetric data, and the low-speed wake is shifted to the right, i.e. in the direction opposite to the intake velocity.

Figure 4. Mach number (a) and total pressure (b) contours over the symmetry plane. 3 0 computations, Case I To get more insights into the 3D-flow structure, let us consider velocity distributions over several representative cross sections identified in Figure 1. Data for two sections (A, B) given in Figure 5 illustrate the development of flow inside the steam chest. Over both the sections, the velocity field non-uniformity is retained almost same, despite the velocity level is considerably increased in the second section. The cross flow patterns around the central body look similar for the most part of its length. The main flow passes a small flute of the central body without a massive separation, so the influence of this flute on pressure losses is not significant. The flow swirl generated within the steam chest due to the lateral intake, in combination with the separation past the shutoff element, results in a formation of longitudinal vortex structure in the lower valve body. It is illustrated in Figure 6 covering velocity distributions over other two sections (C, D). The cross flow developing inside the diffuser is of rather high intensity. Remarkably, that the

Complexity, performance and informatics

749

direction of cross circulation is opposite to that typical for flow in curved tubes. It is reasonable to suggest that the real vortex cores developing in both halves of the valve body to be in an interaction. That produces their periodical displacement in the cross direction, and leads to development of low-frequency oscillations being a possible reason of the valve vibration. To study numerically these effects one needs to conduct 3D unsteady simulation.

Figure 5. Cross velocity vectors and whole velocity isolines, (a, b) and (c, d ) correspond to Sections A and B as given in Figure 1

Figure 6. Cross velocity vectors and whole velocity isolines, (a, b) and (c, d ) correspond to Sections C and D as given in Figure I

750

Finite volumes for complex applications

Despite accounting for 3D effects changes the valve flow structure considerably; pressure loss data obtained with the 3D and axisyrnmetric formulations are practically the same. Table 2 shows also that the results of computations are in a good agreement with the air-test measurements.

Acknowledgments The development of the SELIGER system possibilities is partially supported by the Russian Foundation of Basic Research, grant 99-07-90103.

Bibliography [COF 961

COFERJ. I. et al., Advances in steam path technology, General Electric Power Generation Paper, GER-3713D, 1996.

[KOL 981

KOLYVANOV V. G., AGAPHONOV B. N., Control valve for steam turbine, Russian patent, RU 21091 143, C1, 1998.

[LAU 741

LAUNDER B. E., SPALDING D. B., The numerical computation of turbulent flows, Comput. Methodr Appl. Mech. Eng., 1974,Vol. 3, N 1, p. 269-289.

[LEO 791

LEONARD B. P., A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comput. Methods Appl. Mech. Eng., 1979, Vol. 19, p. 59-98.

[SMI 931

SMIRNOV E. M., Numerical simulation of turbulent flow and energy loss in passages with strong curvature and rotation using a three-dimensional Navier-Stokes solver, Report on the "Research in Brussels'92" grant, VUB, 1993.

[GOR 971

GORYACHEV V. D., SMIRNOV E. M., A computer and information system for computer fluid dynamics: SELIGER, Proc. of the 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, Berlin, Vol. 3, 1997, p. 47-53.

Adaptivity, Tracking and Fitting

An Adaptive Hybrid Object-Oriented Code for ~ ~ ~ - ~ ~ ~ l i c- a t i o Adhoc3D

U. Tremelj., H. Bleecket, G. Brenners, G. Greinert $ NEC-CCRLE research laboratories, 5 Institute of Fluid Mechanics, University of Erlangen, t Institute of Computer Science, University of Erlangen

This paper presents an object-oriented ( 0 0 ) approach for CFDapplications with attention being paid to conserve high computational performance. This is demonstrated by implementing an 00-simulation environment for flow simulations on hybrid unstructured adaptive meshes using 00 programming in C++ combined with a lean flow solver kernel in FORTRAN 77/90. ABSTRACT

Key Words: Object-Oriented Programming, High-Performance Computing, CFD.

1. Introduction In the field of numerical simulation object-oriented programming (OOP) is used only to a limited extent although the benefits like greater productivity, flexibility, maintainability and reusability ([GAM 951, [STR 981) of this modern type of software engineering compared t o subroutine-based procedural programming are generally accepted. However it is usually not used in the high-performance community since it is suspected of being the reason for severe performance losses (due to runtime- and memory-requirements), in particular on vector computers. In the present paper, an 00-approach is described for numerical simulations particularly in the field of CFD which allows to retain high computational performance using the above mentioned features of OOP advantageously. The basic idea is t o use the 00 paradigm wherever useful and t o define clear and

754

Finite volumes for complex applications

simple interfaces in order to encapsulate highly optimized modules. The possibilities of such an approach are demonstrated through the implementation of an 00-numerical flow simulation environment on hybrid unstructured adaptive meshes in C + + . The solver object encapsulates a numerical method for the integration of the Euler and Navier-Stokes equations which has been realized in FORTRAN 77/90, interacting through well defined interfaces. This wrapping technique enables the use of modular, encapsulated and highly optimizable subroutines for numerical computations and improves maintainance and exchangeability. Parallel aspects, considered in the design phase, strongly influence the simulation environment components. A parallel implementation based on message passing will be realized in the near future. The main focus of the implemented system, described in the next section, is on complex meshing management, a solution based adaptation process and, closely linked to that, a surface reconstruction process. The solver object, including the encapsulated lean flow solver kernel interface, and the basic principles of the underlying flow physics are described in section 4. First results and performance measurements are presented in section 5, showing that a high computational performance could be achieved. 2. 0 0 - S i m u l a t i o n Environment for CFD

The basic components of a CFD simulation environment are well suited to be encapsulated into three high level objects:

A control-object (Adhoc3D1 see figure 1) managing the actions during runtime, i.e. controlling 10,solution and adaptation process. A mesh-object describing the discretized computational domain, in this case a hybrid unstructured adaptive mesh, internally subdivided into other objects. A solver-object solving the governing PDEs and computing the numerical solution depending on the given initial and boundary conditions. The methods used internally for the numerical computations are encapsulated within this object.

3. Mesh-Management The complex administration of the adaptive hybrid mesh is encapsulated in different objects. The underlying principle is a decomposition of the domain in piecewise disjoint and non-empty subdomains containing sets1 of elements l ~ o t that e here the use of a set implies that every item occurs not more than once in this set.

Adaptivity, tracking and fitting

755

Figure 1: Adhoc3D-Shell of different type whereas the union of all subdomains is the whole mesh. Each element is uniquely contained in one subdomain. Each subdomain-boundary is completely covered through a set of boundary patches belonging to this subdomain. These boundary patches contain a set of faces. All faces in a patch have the same boundary type which indicates, e.g., a special boundary condition. Neighboring subdomains share a common boundary patch exactly, i.e. all faces in this patch must be shared between the subdomains. Therefore the mesh itself contains only a set of all subdomain objects and a set of all boundary patches. This decomposition is necessary to support the message passing parallel programming model. Such a general description of the mesh and its subdomains can be expressed in OOP quite well using class hierarchies with abstract and derived concrete types. A subdomain is defined as an abstract type, i.e. that a subdomain class only defines a common interface and common properties of subdomains in general. For example, each subdomain must have a unique ID and should read itself from or write itself to a file. The mesh, using these methods, does not take care about the concrete type of subdomain behind this object. If the method is common to all subdomains (like asking for the unique ID) it will be implemented in the subdomain class itself. Otherwise the method is defined as a virtual function which must be implemented in derived classes defining concrete types. The correct implementation is then looked up at runtime. A concrete type of a subdomain is, for example, an unstructured region containing elements of different type or a structured block containing only one type of elements connected in a structured manner. In figure 1 these kinds of concrete types are shown. The use of common boundary patch objects enables the transfer of information like marker values, flags etc. over subdomain boundaries and also between different subdomain types. New subdomain types

756

Finite volumes for complex applications

(Cartesian subgrids, phism.

...)

can be added in an elegant way, using this polymor-

3.1. Unstructured Regions The first concrete subdomain type currently implemented is an unstructured region. This object encapsulates sets of elements of different type connected in an unstructured manner. Supported elements are tetrahedra, pyramids, prisms and hexahedra. These elements have common faces like triangles and quadrilaterals which are the face types supported. The main design aspect was to realize compact and efficient objects for the management of all elements of one type in a certain subdomain. To achieve this the classes are designed like pools containing arrays of elements and not as arrays of fine grained objects. This means that these classes act like wrappers for internal, partly low level structures (one dimensional arrays). These arrays are used without duplication or adjustment in the solver low level kernel routines. This reuse of memory and data is an important aspect if only limited memory resources are available. It also helps to increase performance because no additional work is done for adjusting or translating these data. The internal arrays are totally encapsulated within these classes, other objects have only access to the internal data through well defined higher level methods. The elements and faces are not considered as independent objects. All data are managed in a pool. Each unstructured region contains a pool object for each element and face type. Through the use of generic programming these different element and face types are easily implemented. On top of the hierarchy there is a class containing common data used from all these pools. From this class a template is derived which contains all properties common to all elements and faces used here. These properties are, for example, the number of nodes, temporary marker data, links to parent and children elements. More special templates, namely the element-pool and face-pool templates are derived from this template. These pool templates contain the common properties of elements and of faces respectively. Concrete classes for the different element and face types are derived from these pool templates.

3.2. Adaptivity

A local tetrahedral mesh adaptation facility has been implemented in order t o accurately resolve flow field features. This facility is based on a combination of red (regular) and green (irregular) refinement, first proposed by R.E. Bank et al. in [BANK 831. It is based on the idea that all elements marked for refinement are refined in a regular way first and then to correct the hanging nodes by applying local irregular refinement rules t o the elements concerned. Irregular refined elements will first be regularly refined if further refinement is

Adaptivity, tracking and fitting

757

Figure 2: Refinement patterns necessary. The algorithm used for the local refinement of tetrahedra is based on the approach of Bey in [BEY 951 and extended t o create a stable and consistent triangulation2 for decomposed meshes. This corresponds to a synchronized parallel closure of the mesh, currently sequentially simulated. Note that the local refinement could also be reverted which will be done automatically if elements are marked for coarsening. The possible refinement patterns, currently implemented for a tetrahedron and the corresponding refinement patterns for a triangle are shown in figure 2. If a triangle is regularly refined, the edge midpoints are connected resulting into four child triangles of equal size congruent to the parent triangle. An edge midpoint is connected with the opposite node resulting into two triangles, in case of an irregular triangle refinement. For a tetrahedron the regular refinement consists of cutting away four congruent child tetrahedra at the corners. The remaining octahedron is divided along the shortest diagonal into four tetrahedra. This regular refinement leads t o eight child tetrahedra of equal volume. The irregular refinement patterns of a tetrahedron describe a split a t one edge, a t two opposite edges and a t a face. Results of the adaptation process are shown in figure 4.

3.3. Surface Reconstruction Surface reconstruction is necessary in order to correctly represent the surface contour for newly inserted mesh nodes during the refinement process. The implemented procedure corrects the interpolated boundary nodes so that the original boundary is reconstructed up to a certain tolerance. Each boundary face is supplemented with a set of original surface points (so called progressive mesh information) which is used t o reconstruct the surface if this boundary face is refined. Details of the geometric operations for this adaptation post'1n the sense defined in [BEY 951.

758

Finite volumes for complex applications

processing step can be found in [TRE 991. Some pictures to illustrate the procedure are given here (see figure 3).

Figure 3: Surface reconstruction through progressive mesh information

4. Solver

Currently a solver object is available for inviscid and laminar flow simulations on adaptive hybrid meshes consisting of tetrahedra, pyramids, prisms and hexahedra using the dual-mesh idea combined with an edge-based data structure. This object takes care of extracting the necessary information needed in the kernel routines and sets up the structures used in the interface, without producing redundant data, i.e. setting the coordinates, variables, connectivity lists and adaptation marker arrays for the tetrahedra (no copy, only a start address is given), creating arrays containing the boundary nodes with the corresponding boundary type and computing the boundary normals, creating an array containing all edges and computing the normals and control volumes belonging to each edge and node respective. After setting up these structures, the FORTRAN 77/90 flow solver kernel is called which performs the main computational work. 4.1. Flow Solver Kernel

The fluid motion is described by the time-dependent laminar Navier-Stokes or the Euler equations in conservative form for compressible perfect gases. A box-type approach is used by constructing a secondary or dual-mesh defined by the centroids of the primary grid cells and grid edges as vertices. This secondary mesh defines the vertex-centered control volume^.^ The edge-based contribution t o the cell residual at a given node is obtained using a modification of the advection upstream splitting method of Liou and Steffen [LIOU 93, WADA 971 3This approach may be interpreted as a Finite-Element Galerkin method with mass lumping

Adaptivity, tracking and fitting

759

for the inviscid terms and central differences for the viscous terms. Second order accuracy of the original first order upwind scheme is achieved using the MUSCL approach with flux limitation. The gradients of primitive variables associated to the vertices of the primary grid are obtained employing the Green-Gauss formula. The resulting algebraic system of equations is integrated using a 4th other Runge-Kutta scheme. Convergence acceleration is achieved by local time-stepping. Presently, boundary conditions may be imposed for supersonic or subsonic in- and outflows based on the theory of characteristics. No slip wall boundary conditions axe prescribed for viscous flows. Symmetry or a weak formulation for the "impermeable wall" condition is prescribed for the case of inviscid flows.

4.2. Adaptation Sensor

The adaptation process is invoked depending on user settings and starts with a scan of the flow field based on a simple pressure sensor. The absolute pressure difference for each edge is scattered to the nodes. The sensor returns a list of markers for the tetrahedral elements, depending on the sensor values at the nodes and user settings.

5. Results and Performance Measurements First results are calculated on a HP-V-Class architecture, using the supersonic flow past a wedge ( M , = 3.0) with and without mesh adaptation. A fully converged steady-state solution was achieved in 800 iterations, including 5 refinement levels. The starting mesh has 2,599 nodes and about 12,230 tetrahedral elements. After five refinement levels, the mesh has 277,949 nodes and 1,665,685 elements. The overal computation time requires 12,410 secs (wall clock; without 10),of which the flow solver kernel takes 11,979 secs (96.5%), the flow solver wrapper 113 secs (0.9%) and the adaptation process 318 secs (2.6%). On the finest refinement level, the total memory requirement is 739 MB, of which the flow solver kernel takes 254 MB and the wrapper 60 MB.

Figure 4: Supersonic flow past a wedge without and with mesh adaptation (for each case mesh and contour is given)

760

Finite volumes for complex applications

Portability of this method across various platforms is proven through different tests on machines with RISC architecture like SGI, SUN and HP. A port for both vector-parallel machines (the NEC-SX4) and massive parallel machines (the NEC-Cenju3- and NEC-CenjuCsystem) is in progress. After finishing the parallelization of the current system, the integration of an industrial structured solver is planned in combination with dynamic load balancing through the use of the DRAMA-library [DRAMA 981. The latter is necessary to avoid load-imbalance of the processors through the adaptation of the mesh. 6. Bibliography

[STR 981

STROUSTRUP B., The C++ Programming Language, 3rd Edition, Addison Wesley 1998.

[GAM 951

GAMMA,E. et al., Design Patterns, Addison Wesley 1995

[TRE 991

TREMEL, U., Simulation kompressibler nichtviskoser. 3D-Stromungen auf unstrukturierten adaptiven hybriden Gittern. Diplomarbeit am IMMD9, Universitat Erlangen-Niirnberg, 1999.

[BANK 831

BANK,R.E. et al., Refinement algorithms and data structures for regular local mesh refinement Scientific computing (Stepleman, R., ed.), pp.3-17, IMACS / North Holland, Amsterdam, 1983.

[BEY 951

BEY, J., Tetrahedral Grid Refinement. Computing 55, pp.355-378, Springer-Verlag, 1995.

[DRAMA 981

LONSDALE, G. et al., Dynamic Re-Allocation of Meshes for parallel FE-Appl. 4th ECCOMAS CFD Conference, Athens 7-11 Septem., 1998.

[LIOU 931

LIOUM.S., STEFFENC. J. et al., A new Flux Splitting Scheme, Journal of Computational Physics, 107, 23-39, 1993.

[WADA 971

WADA,Y., LIOUM.S., An Accurate and Robust Flux Splitting Scheme for Shock and Contact Discontinuities, SIAM J. Sci. Comput., Vol. 18, No. 3, pp. 633-657, 1997.

Adaptive mesh refinement for single and two phase flow problems in porous media

Mario Ohlberger Institut fur Angewandte Mathematik Herrnann-Herderstr. 10 79104 Freiburg (Germany)

This paper is devoted to the study of a posteriori error estimates and adaptive methods for a mixed finite element - finite volume approximation for single and two phase flow problems i n porous media. In a first step we analyze a scalar nonlinear convection diffusion equation and give a rigorous a posteriori error estimate for the Cauchy problem. I n a second step these results are used to implement an adaptive method for the systems of density driven flow and two phase flow i n porous media.

ABSTRACT

Key Words: A posteriori error estimates, convection dzffusion equation, porous media, two phase Bow, finite volume schemes, unstructured grids

1. Introduction

As a generalized mathematical model problem for single and two phase flow problems in porous media let us look a t the following nonlinear system of partial differential equations. Let 0 c IRd, d 3 be a bounded domain and J = ( 0 , T )n lR a time interval. Let OT = 0 x J denote the space-time domain. We now look for a Darcy velocity u, a pressure p and a concentration or saturation c, satisfying in flT the system

<

Here 0 < a0 < a , y,b and f denote given functions, where D may be degenerate.

762

Finite volumes for complex applications

The system is closed by imposing suitable initial and boundary conditions. Discretising the system we use an operator splitting technique, where the equations ( 1 ) and (2) are discretised via mixed finite elements, using RaviartThomas elements, and the transport equation (3) is discretised by a cell centered upwind finite volume scheme. This choice of the numerical scheme is adapted to physical situations, where the flow is convection dominated, which is for instant the case in density driven ground-water flow or in an imiscible two phase flow regime. For a simplified model problem of the system, given above, the convergence of the scheme was proved in [OHL 971. In this work we are interested in a posteriori error estimates and adaptive methods. Due to the difficulties of the system, the first efforts in this direction are the analysis of mixed finite element methods and finite volume methods separately for the corresponding scalar partial differential equations. In Section 2 we define a finite volume scheme for scalar convection-diffusion problems and give rigorous a posteriori error estimates. These results are obtained by using Kruzkov or Kuznetsov type techniques, such that no regularity of the solution is proposed. In Section 3 we define a mixed finite element method for a scalar saddle point problem corresponding to the equations (1) and (2) and state an a posteriori result of Wohlmuth and Hoppe [WOH 991. Combining the mixed finite element and finite volume method we finally design an heuristic adaption strategy for the full system ( 1 ) - (3) in Section 4. Finally in Section 5 adaptive numerical experiments are given, which underline the theoretical results. 2. A posteriori estimate for finite volume schemes

Let us first look at the Cauchy problem for the scalar nonlinear convection diffusion equation ct + V - (ub(c))- EAC= 0, E 2 0 and let c E Lioc((O,T ) x p) be an entropy solution of this equation if for all n E R:

for all cp E C?(JRd x R+)with cp 2 0. For the data of problem (4) we assume the following conditions. co E u E

L ~ ( R n~B )V ( R ~ ) , b E c1( R ~ ) , C 1(JRd x R+)n L"(JRd x R') with V . u = 0.

<

(5)

Furthermore let A, B E IR be defined such that A co 5 B. Let J := {to, ...,t N ) be a partition of [O,T] and Atn := tn+' - t n the step size of J. Furthermore for each n E (0, ..., N ) let T n = {Tjlj E I n ) be a regular triangulation of Rd. The joint edge of Tjand Tt will be denoted by

Adaptivity, tracking and fitting

763

Sjl. The oriented set of edges En is defined as

En

:= {(j, l)(Sjl is an edge of

Tn and cj > cl),

where cj, q denote piecewise constant approximations of the exact solution c on the elements T j , G . Let hki, := minjern diam (Tj) and let x j be the circumcenter of triangle T j . Define the distance between the circumcenters of two neighbouring cells T j , Z E 7" as djl := Ixj - xll and let hjl denote the maximum of the diameters of those cells. We assume that there exists an a > 0 such that we have for all hj := diam (Tj) aha

5 ITjl,

a ldTjl

5 hy-'

and

ahj

5 djl

(6)

for all j , l E I . For any j, 1 E I n and t , E JRf let gjn, : R2 3 R be a C1 numerical flux, satisfying the following conditions for all w, v,w', v' E [A,B].

where L, is a given constant and njl denotes the outer unit normal to Sj1 with respect to T j . Now the upwind finite volume scheme for computing the approximate solutions to (4) is defined by

for all n E (0, ...,N ) and j, 1 E In. Here N ( j ) denotes the indices of the neighbouring cells of Tj. Given the discrete values c? let us denote the approximate solution ch : JRd x [O,T]+ W. by ch(z,t) := c7,if z E Tj,tn t < t n f l . For the time step Atn we assume the following CFL-condition

<

<

where E ( 0 , l ) and L, is the Lipschitz constant from (8). With this definitions let us now state the main theoretical result of this paper.

764 Finite volumes for complex applications Theorem 1 (A posteriori error estimate) Let ch be the discrete solution defined i n (10) and c the entropy solution of (4). Furthermore let the Assumptions (5)-(9) and (11) be fulfilled. Then we have

Here q0, qt , qc and qd are defined as

where

+

KO, r l ~ , ~V ,E , ~ are defined as KO = l/T(clev 2K:L,Lblcl~v + I c o l ~ v ( 1+ llullrnLbK:), qE,o= K z l c o l ~ v & , %=, ~~ K Z I C I B V and E KI, and Ka are constants independent of the data. L, and L b denote the Lipschitz constants of u , b respectively.

Proof: The proof of this estimate is based on the Kruzkov or Kuznetsov 0 technique and can be found in [OHL 991. It can be shown, that the dependence of the right hand side of this estimate (cf. [OHL 991). on h, At and E is of order ( E + fi+ 3. A posteriori result for mixed finite elements With the notations of Section 2 let us now define a mixed finite element method for a scalar saddle point problem corresponding to (I), (2). Therefore ) ) ~u E ~ ' ( 0 ) )and W = ~ ' ( 0 ) let us define the spaces V = {u E ( L ~ ( S ~1 div Furthermore for v, w E V, cp E W and s E Lw(S1) let the bilinear forms A and B be defined as:

Adaptivity, tracking and fitting

765

Then we look, for a given function s E L m ( 0 ) ,a t solutions ( u , p ) E V x W of the weak saddle point problem with homogeneous Dirichlet boundary conditions:

To discretise these equations let us introduce the Raviart-Thomas subspaces Vh,Wh of V , W:

Wh = {wh E WI w h l ~= const V T E f i ) , Vh = {vhl vhlT E ( P I ) ~ VET f i , vh has continuous normal components over the edges of Th). Then the mixed finite element solution ( u h , p h ) E Vh x Wh is defined as

In the case where the right hand side y = 0 and d = 2 Wohlmuth and Hoppe obtained the following a posteriori result [WOH 991. T h e o r e m 2 Let ( u , p ) , ( u h ,ph) be defined as in (13)-(16). Then there exist hmax > 0 and c, C , independent of the mesh size, such that for all h 5 h m a z u

where

2

2 5 I l ~ - ~ h 1 1 & ( n+ ) 1 1 1 ~ - ~h111&, 5 C u ~ u ,

"

(l7)

quij - 119 - n o g l l i z ( ~ , )+ h;11a-1(s)uh11i2(~,)

+

wjlajllSjl)l)[a-'(s)uh ' t j l ] ~ ) ) $ ( ~ , )

LGN(3)

Here I1, denotes the L~ projektion onto the space of piecewise constants, tp denotes a tangent vector to S j l , ['IJ denotes the jump across this edge and 1 1 lu1 [ I 2 . = Jn a V 1 ( s ) u. u + Jn a - l ( s ) d i v ( u ) d i v ( u ) . The weighting factor wjl is given y 0.5, if S j l is an interior edge and 1 elsewhere. c t j l denotes the mean value of the extrema of a ( s ) on T j and 3 (cf. [WOH 991 for details).

da"

4. A d a p t i v e m e t h o d for single a n d two phase flow

Combining the a posteriori results of the Sections 2 and 3 we now want to develop a heuristic adaptive scheme for the system (1)-(3). Therefore let us define the error indicator fjoij, qn;j on elements T j E Z as

766

Finite volumes for complex applications

ST,

@/lTjI and K t , K c ,Kd are where Q; is defined as in Theorem 1, G j := given constants. Furthermore let vu;, be defined as in Theorem 2. Then the adaptive algorithm for solving (1)-(3)is: 0. Do { cy := JT, co/lTj1, V j t In } (while a d a p t o ( P ,T O L O c:) , ) For n = 0 to N do:

1. For given s := c: let ( u h ( . , t n ) , p h ( . , t n ) E) Vh x W h be defined as the solution of the mixed finite element scheme (15),(16).

3. For given uh(.,t n ) ) ,ch (., t n ) calculate ch (., tn+l) with the finite volume scheme:

Here adapto adapts the initial grid using the error indicators f j o i j and returns false if it has done, whereas adapt calculates the grid Tn+' from the given data:

a d a p t ( T n ,TOL,, TOL,, c:, uh(.,t n ) ) { for all T j E Tn { calculate ( v n ; j )(vuij) , from the given data; coarse T j if (vnij) BTOL, or (vUij) 5 OTOLu; refine T j , until (r],,j) 5 T O L , and ( q u i j )5 TOL,; , return the resulting grid as Tn+l;

<

1 Here 0 E ( 0 , l ) is a given value, which depends on the refinement rules of the mesh. The tolerances TOLo,TOL, and T O L , depend on TI the number of elements of Tn and a fixed prescribed tolerance T O L (cf. [KRO 991 for a possible choice).

5. Numerical experiments As a first example we look at the following scalar nonlinear and degenerate parabolic equation for the saturation s on R = ( 0 , l ) x (0,0.5) C IR', which models imiscible two phase flow in porous media

Adaptivity, tracking and fitting

767

where u = (1,0), b ( s ) is the fractional flow rate and d ( s ) the capillary diffusion given by b ( s ) = X l ( s ) / ( X l( s ) A,(s)), D ( s ) = A2 (s)b(s)p',(s). The mobilities X I , A2 of the phases and the capillary pressure p, are given by X I ( s ) = s 3 / ( 2 p 1 ) ,X Z ( S ) = ( 1 - s ) ~ / and ~ z p,(s) = Furthermore in this example we choose pl = 1,112 = 3, @ = 0.2 and K = 0.2. As initial condition we choose s = 0 in fl and as boundary conditions we impose s = 1 on ( 0 ) x (0,0.5) and homogeneous Neumann boundary conditions on all other boundaries. Figure 2 shows the result of the adaptive numerical scheme, whereas Figure 1 shows the numerical result in the case, where we choose p, = 0 (hyperbolic case). In this case the equation reduces to the so called "Buckley-Leverett" equation. Both calculations where done with the same threshold value for the grid refinement.

+

- d m .

Figure 1. Adaptive grid and saturation for the hyperbolic problem at t=O.l (right) and comparison of saturation profiles on a horizontal slice (left).

Figure 2. Adaptive grid and saturation for the degenerate parabolic problem (right) and comparison of saturation profiles on a horizontal slice (left).

768

Finite volumes for complex applications

As a second example let us look at the Elder problem, which is an example of density driven flow in porous media, described by the full system (1)-(3). Here a closed aquifer, which is initially filled with fresh water, is bounded by a salt dome at the middle part of the upper boundary. Thus salt water enters the domain and is distributed in the aquifer. Thereby the density differences between salt and fresh water initiate a convection field, which dominates the flow (cf. [SEG 941 for details). Figure 3 shows the adaptive grid, the concentration distribution of the salt water and the velocity field after 4 and 8 years.

Figure 3. Adaptive grid, concentration distribution and velocity field of the Elder problem after 4 and 8 years. 6. Bibliography

[WOH 991 B.1. WOHLMUTH AND R.H.W. HOPPE, A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements. Math. Comp., posted on May 19, 1999, PII: S 0025-5718(99)01125-4 (to appear in print). [KRO 991 D. KRONERAND M. OHLBERGER,A-posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions. To appear in Math. Comp., 1999. [OHL 971 M. OHLBERGER,Convergence of a Mixed Finite Element - Finite Volume Method for the Two Phase Flow in Porous Media. EastWest J. Numer. Math., 5:183-210, 1997. [OHL 991 M. OHLBERGER,A posteriori error estimates for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations. Preprint 3, Mathematische Fakultat, Freiburg, 1999. [SEG 941

G. SEGOL, Classical Groundwater Simulations: Proving and Improving Numerical Models. PTR Prentice Hall, Englewood Cliffs, New Jersey, 1994.

Parallel solution of hyperbolic PDEs with space-time adaptivity

Per Lotstedt and Stefan Soderberg I n f o r m a t i o n technology Dept. of Scientific Computing Uppsala University SE-75104 Uppsala, Sweden [email protected] and [email protected]

A n algorithm i s developed for adaptive solution of hyperbolic partial differentzal equations o n structured grids partitioned i n t o blocks. T h e space steps and the t i m e steps are chosen s o that estimates of the local errors satisfy given tolerances. T h e equations fulfilled by the global error are integrated o n a coarse grid. T h e method is implemented o n a parallel computer with d y n a m i c load balancing. T h e numerical examples are a scalar equation and a s y s t e m of equations i n t w o dimensions.

ABSTRACT

K e y Words: adaptivity, hyperbolic partial differential equations, numerical solution, parallel computation

1. Introduction Adaptive methods for numerical solution of hyperbolic partial differential equations (PDEs) have the advantage of saving memory and computation time by spending the efforts in areas of the computational domain where it is necessary for the accuracy of the solution. In other areas, fewer grid cells are needed and larger time steps can be taken. Furthermore, if the refinement and coarsening of space and time steps are based on estimates of the numerical discretization errors, then the global error in the solution can be estimated. An error tolerance given by the user controls the acceptable error level. We describe an adaptive method for structured grids and finite volume discretization~in space. The grid is refined and coarsened by halving or doubling

770

Finite volumes for complex applications

the step size in a multiblock partitioning of the computational domain. All cells in a block are refined or coarsened. This simplifies the administration of the grids. At the block boundaries there may be jumps in the cell size. The time derivatives are approximated by a predictor-corrector method and an implicit scheme for problems with large discrepancies in wave speeds. The errors in the time discretization are estimated in the same manner as in solvers of ordinary differential equations. The chosen techniques are suitable for parallel computation using message passing. A load balancing algorithm dynamically redistributes the blocks on the processors in an implementation of the method. The numerical tests are made in two space dimensions with a scalar equation and a system of equations. Grid refinements in patches of structured grids with application to flow problems are found e.g. in [BEC 891 [FEL 991 [PAK 961 [WRO 971. In these papers either the steady state problem is solved or there is no adaptivity in time. A parallel implementation of an adaptive solver is described in [DLR 941. Our work differs from the above cited references in that the local errors in space and time are controlled and integrated to obtain an estimate of the global error in the solution.

2. Space adaptivity

The computational domain in space is partitioned into a number of blocks of a fixed physical size. Each block is a topological rectangle in 2D and has a structured grid of cells. At the boundaries of a block there is one extra column of ghost cells to facilitate the coupling to the neighboring block. There are jumps in the step size of a factor 2 along the common boundary of certain blocks where there is a refinement of the grid. The average of the solution in a cell is updated by computing the fluxes through the cell faces in the usual finite volume manner. This discretization will be second order accurate at least on Cartesian grids. The flux through a cell face at a block boundary is calculated using the variables in the adjacent ghost cell. If the cell in the other block has a different size then the values in the ghost cells are determined by averaging for a coarse ghost cell and bilinear interpolation for fine ghost cells. The interpolation in the fine ghost cells only give first order accuracy in the approximation of the derivative. Still, second order accuracy is obtained in the interior of a block when the characteristics of the solution cross the boundary [GUS 751. The truncation error in the space discretization is estimated in the following way. In every block, a coarse grid is created by doubling the grid size in both directions. Suppose that the approximation of the space derivatives is Rh(uh), where u h is the solution on the fine grid. Then the leading term in the error on the fine grid, assuming second order accuracy, is given by

Adaptivity, tracking and fitting

771

where p u h is the prolongation or the volume weighted average of u h in the four cells corresponding to one coarse cell. The estimate TR is computed for groups of four cells in the fine grid. If max IrRI exceeds a tolerance E R , then the grid is refined in a block. If max ( T R1 5 0 . 1 . for ~ ~all groups of cells in a block, then the grid is coarsened in that block. Since we do not allow the jumps in the grid size to be greater than 2 at block boundaries, neighboring blocks may have to be refined after a change. The check of the spatial error is made in every time step. If TR is too large, then a new grid is generated and the new time level is computed with the new grid spacing. If the estimate is sufficiently small, then the grid in the block is coarsened in the next time step.

3. Time adaptivity

The time derivatives are discretized by linear multistep methods. The advantage with a multistep method compared to a Runge-Kutta method is that there are no internal stages where data have to be sent between the blocks in a parallel implementation. Two second order schemes are considered. One method is the predictorcorrector pair Adams-Bashforth (AB) and Adams-Moulton (AM) and the other is the backward differentiation method (BDF), see [HNW 931. The coefficients in the methods are adjusted to a possible change in the global time step Ati between two time levels i and i + 1 to retain second order. Both AM and BDF are implicit and A-stable [HNW 931 and the imaginary axis belongs to the stability region. This is a desirable property for solution of hyperbolic problems. In the first method, the solution at level n 1 is predicted by the explicit AB. Then the solution to the AM scheme is obtained by fixed point iteration. With a sufficiently small time step, only a few iterative steps are necessary for convergence. Convergence of the iteration sometimes introduces a more severe restriction on the time step than the accuracy. The local truncation error due to the time discretization is estimated by comparing the two solutions at t,+l computed by AB and AM using the same old data. The local errors in the solutions are proportional to At;

+

where u(t)is the analytical solution and CAB and CAM are weakly dependent on At,. Hence, the local error in is

u~+M'

The error-per-step r t is computed for every cell in the blocks. Then At, is calculated such that A& = J-ht,

772

Finite volumes for complex applications

If ] A t , - Atnl/At, 5 0.1, then Atn+' = At,. Otherwise, =Ac. In the second method, the explicit second order BDF discretization is used as predictor. Then un+' is computed with the implicit BDF method. A system of linear equations is solved by GMRES in each step as described in [HLR 991. There is no restriction on At, for stability in time or convergence of GMRES. Only accuracy requirements introduce bounds on the time steps. This discretization and solution method is suitable for systems of equations with a large span of wave speeds. The local truncation error is estimated by comparing the explicit and implicit BDF schemes as in (2) and(3). For a linear system of hyperbolic equations

the error e = u(tn) - un in the solution approximately satisfies a similar equation et Ae, Be, = rt T R , (6)

+

+

+

where e ( x ,y, t ) interpolates the errors a t the discrete points in the space-time grid and TR is defined in (1). This equation tells us how the truncation errors are spread t o the errors in the solution. We control only the right hand side of (6), but we integrate (6) to obtain an estimate of the global error e at every time level. This equation is integrated simultaneously on the coarse grid with twice the time step. The cost is only 1/8 of the cost of integrating ( 5 ) on the fine grid. 4. Parallelization

The partitioning of the grid into blocks simplifies the parallelizat,ion of the adaptive algorithm and is suitable for processors with limited cache memory. The neighbors of a block are the same from the beginning, but the grid size across the boundary may change. Each processor is the host of one or more blocks. Solution variables a t the boundaries are needed in an adjacent block to update the ghost cells. If a block and its neighbor belong to the same processor, then data are moved in the memory of the processor. If they are located on separate processors, then data are sent over the internal network with MPI. For parallel efficiency, the load balance must change dynamically as the work per time step in a block increases with refinement or decreases with coarsening of the grid. In order to keep the cost for load balancing at a reasonable level, only entire blocks are distributed not individual cells. This reduces the size of the data set passed to the load balancing algorithm to the number of blocks. The algorithm used in the numerical experiments is the recursive spectral bisection method (RSB).

Adaptivity, tracking and fitting

773

5. Numerical results The algorithms are tested on two simple hyperbolic equations (5) on a square, where A = B = 1 in the scalar case and the linearized and symmetrized Euler equations

for the system of equations. Periodic boundary conditions are assumed at the outer boundary of the computational domain.

5.1. Scalar problem The equation (5) is solved with A = B = 1. Initially, u(x, y , 0) is a Gaussian pulse. It is advected along the diagonal. After leaving the domain in the upper right corner it returns in the lower left corner. The error tolerances are EL = ER = 5 . and there are 8 blocks in both the x and the y directions. In Fig. 1, the initial grid and the grid after one period are displayed. They are almost identical.

Figure 1: The computational grid initially (left) and after one period (right).

Comparison along the diagonal of the square between estimated (dashed) and analytical (solid) local errors TR (left), rt for the BDF method (center) and rt for the A M method (right).

Figure 2:

774

Finite volumes for complex applications

In Fig. 2 the estimated local errors in time and space are compared to the leading terms in T R and TL computed analytically. The errors overlap almost everywhere. The estimated global error from (6) is compared to the exact global error in Fig. 3. They agree very well. The time step of the predictor-corrector scheme is here restricted by the convergence of the fixed point iteration.

Figure 3: Integrated global error along the diagonal of the square from (6) (dashed) is compared to the exact error (solid) for the BDF method (lefi) and the A M method (right).

5.2. System of equations

The equation (5) with the coefficients in (7) is solved with c = 1 and the initial data u1 = 0,u2 = O,u3 = 0.3exp(-a((x - 0.5)2 ( y - 0.5)2)),cw= 300. The error tolerances are ~t = E R = 5 . and the domain is partitioned into 8 x 8 blocks. There is no analytical solution t o compare with in this case. The solution in u g a t t = 1 is found to the left in Fig. 4. The estimated error in u3 at t = 1 is plotted to the right. It is of the order of the tolerances as could be expected.

+

-0 0-

--I0

0.1

.

0

r - 7 , m

-0.0-

--

0.-

A

7 0 I

Figure 4: Solution of (5) (left) and computed error from (6) (right).

Adaptivity, tracking and fitting

775

5.3. Parallel execution

The adaptive algorithm is implemented in Fortran 90 using MPI for the message passing. The experiments are executed on a set of three Digital Alpha Servers with four EV5/300MHz processors each. The load balance is updated in every 25th time step. For the scalar problem, the distribution of the blocks on 4 processors is depicted in Fig. 5. Here, a redistribution of the blocks is necessary since the refinement moves from the center to the corner of the computational domain.

. . .:.z...:..~. o,e. : : . ;:; ............... :.:...>,:z :I .::-.,-.* ,

,

..-- .........

, . --.------. . . . . . .. .-: ~.~ .. -<' --.

0

L1

oz

01

01

01

01

!:.

or

(I*

.

,

: '

0.

1

Figure 5: The distribution of blocks at t = 0 (left) and t = 0.25 (right) for the scalar equation. Let Tp be the clock time for solving the system of equations (7) between t = 0 and t = 1 on p processors. In Fig. 6 the fixed size speedup 2Tz/Tp and efficiency 2T2/pTp are shown as a function of p. The comparison is made for 2 processors since the code does not run in single processor mode.

ow-I

3

.

,

I

I

I

.

?O

11

12

Figure 6: Fixed size speedup (left) and efficiency (right) as a function of the number of processors.

776

Finite volumes for complex applications

6 . Bibliography

[BEC 891

M. BERGER,P . COLELLA,
[DLR 941

J. DE KEYSER,K . LUST, D. ROOSE,->, Parallel Comput., 20 (1994), p. 1069-1088.

[FEL 991

L. FERM, P. LOTSTEDT,Blockwise adaptive grids with multigrid acceleration for compressible flow, AIAA J, 37 (1999), p. 121-123.

[GUS 751

B. GUSTAFSSON, The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comput., 29 (1975), p. 396-406.

[HNW 931

E. HAIRER,S. P . NBRSETT,G . WANNER,Solving ordinary differential equations, 2nd ed., Springer-Verlag, Berlin, 1993.

[HLR 991

L. HEMMINGSSON-FRANDEN, P. LOTSTEDT,A. RAMAGE, S. SODERBERG,>,in preparation.

[PAK 961

N. G . PANTELELIS, A. E. KANARACHOS, <>,Int. J. Numer. Meth. Fluids, 22 (1996), p. 411-428.

[WRO 971

J . W U , H. RITZDORF,K. OOSTERLEE,B. S T E C I ~ E LA., SCHULLER, ->,Int. J. Num. Meth. Fluids,24 (1997), p. 875-892.

Dynamic mesh generation with grid quality preserving met hods

Andreas Wick and Frank Thiele Hermann-Fottinger-Institut fur Stromungsmechanilc

Technische Uniuersitat Berlin 10623 Berlin, Germany

A B S T R A C T T h e various methods employed so far t o accomodate computational grids

t o a moving boundary often fail t o preserve grid quality. In this paper, a detailed analysis o f thzs issue is conducted. Based o n the findings a new method i s presented that performs substantially better. T h e theoretical results are confirmed by numerical studies. I i e y Words: m o v i n g boundaries, deforming grids, d y n a m i c meshes, spring analogy

1. Introduction

The incorporation of moving boundaries and deforming domains has become an important task in numerical flow simulation. There are many applications in which this issue must be faced. To give a few examples, we can mention aerodynamic design optimization, fluid-structure interaction, free-surface flows and multi-element airfoils with moving flaps. In order to take into account the effects of time varying domains in an adequate and straightforward manner numerical algorithms are commonly based on the Arbitrary Lagrangian-Eulerian (ALE) formulation of the conservation laws (see e.g. [ANJ97]). There is no need for special search and interpolation algorithms as is the case with some other methods (e.g. [KA095]). Compared to a standard finite-volume method that operates on a fixed grid (pure Eulerian approach) the numerical solution of the ALE equations additionally requires at. each time step new grid point coordinates and the corresponding velocities with which the cell surfaces move.

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While the consistent and efficient computation of the grid velocities has been investigated thoroughly [ZHA93], the adjustment of the mesh to moving boundaries is still an open problem, at least for Navier-Stokes computations which require grids with very high aspect ratios. Specifically, for nontrivial geometries with boundaries exhibiting strong curvature or sharp bends, overlapping of mesh cells is likely to appear. In this case, a local or global remeshing becomes necessary and may induce unwanted interpolation errors. Therefore, of the methods proposed in literature, in section 2 we revisit shortly those having the potential to preserve the quality of the initial grid for the whole mesh evolution. Besides the well-known spring analogy model [BATSO], these are methods based on differential equations [ILI94, CRU96, LOE961. The numerical implementation of the equations is addressed in section 3. In section 4 the methods are subjected to a deformation analysis which reveals serious weaknesses. The observed deficits also become apparent in numerical test cases. In section 5 the conclusions that can be drawn are summarized. 2. Grid generation methods

2.I Problem formulation

We assume that we start with a high quality grid at the first time level. Our aim is to find grids for the proceeding time levels that fit the prescribed boundary and that possess a quality comparable to the initial grid. Specifically, cross-sections of coordinate lines must not occur. To accomplish this, we have to inspect a very basic geometrical configuration, typical both for unstructured and structured grids. It is a line segment, connecting two neighbor vertices, which is subjected to a deformation. The difference between the coordinates at the new time level and the preceding time level defines the displacement vector.

We hope to find a displacement vector w that leaves the length of all line segments unaltered while transforming the grid of a former time level to a new one. Of course this is only possible if the prescribed boundary displacements correspond to a rigid body motion. If the boundary moves in another way, causing the interior domain to deform, the resulting stretches have to be equidistributed in a certain way. Otherwise a high local concentration of stretches will lead to a quick degradation of the grid quality.

Adaptivity, tracking and fitting

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2.2 Methods A very well-known method is the spring analogy model. The lines connecting neighbor vertices are seen as tension and/or torsion springs. The imposed boundary displacement generates forces in the springs, leading to a new equilibrium state. By minimizing the energy of the spring system, the grid is optimally redistributed. A particular simple model that gave good results in many practical applications ([BATSO, FAR941) has been proposed by Batina [BATSO].

V is the set of vertices and N ( i ) the set of neighbors of i. In a structured 2D grid the set of neighbors N(i) consists of the east, west, south and north nodes. Despite some desirable properties of a numerical scheme derived from this equation, it has some flaws that gave rise to the development of alternative methods. Ilinca [ILI94] proposed t o solve a Laplace equation for the grid velocity i. By an integration over the time step the displacement field is obtained. This numerical integration necessitates an assumption regarding the time dependence of the grid velocity (e.g. j. = const, for tn < t < tn At). The error introduced thereby leads to a time step restriction. It is therefore a better choice to use a Laplace equation for the displacement itself.

+

The introduction of a diffusion coefficient in this formula allows for more control over the grid evolution.

Crumpton and Giles [CRU96] choose the diffusion coefficient as a function of the cell volume. They argue that small cells are most endangered with regard to crossing of grid lines. Lohner and Yang [LOE96] prefer the diffusion coefficient to be connected to the wall distance, since large stretches are supposed to be close to the wall. While the cell volume is an intrinsic quantity of the discretisation, the wall distance is not, and as a consequence it might not be available. To keep the formulas valid for cases as general as possible we set

for the diffusion coefficient. The function Vol(xi) returns the volume of the dual grid cell which is associated with vertex xi. The Laplace equation [3] can also be interpreted as a special case (E = const., divergence free displacement field) of the following equation:

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Finite volumes for complex applications

This formula is also closely related to equation [2]. Similar to the derivation of the spring analogy model, the fact that the minimization of an appropriate energy function leads to optimally redistributed grids is exploited. But while the spring analogy is based an a discrete model now we apply a continous model. That is to say we consider the domain 0 as a Hookean medium with zero Poisson's ratio. Then equation [6] represents the well-known Navier's equation of the classical linear theory of elasticity (see e.g. [ERI62]). Following the reasoning of Crumpton and Giles the Hook medium should be stiff for small cells. The most simple way to do this is to take the reciprocal value of the cell volume as elastic modulus E.

The coefficients of equations [4] and [6] are now strongly dependent on the chosen discretisation which might be considered as a disadvantage. A better choice should be to link the elastic modulus to the stretches whereby equation [6] becomes nonlinear. However, due to limited space we restrict ourselves to linear methods here.

3. Solution Procedure The discretisation of the differential grid generation equations are based on the Finite Volume Method (FVM). To allow complex geometries to be accounted for, a block-structured approach with general non-orthogonal coordinates is adopted here. The strong conservation form of the equations is retained by expressing the vectors and tensors in Cartesian components. The displacement vector is assigned to the vertices ("Cell vertex scheme"). First derivatives, appearing at cell faces, are approximated by central differences leading to a second order scheme. The numerical stability is enhanced by using the deferred correction approach which makes the scheme partly implicit. The system of linear equations obtained from the discretisation is solved by an iterative ILU method [ST068]. 4. Theoretical and numerical analysis

With regard to the grid quality the methods presented in section 2 have to meet two requirements. The main task is the equidistribution of stretches in a manner that avoids high local concentrations of them, since they are responsible for the overlapping of grid cells. Even more fundamental is the preservation of rigid-body displacements. It is evident that a method should not create stretches for boundary displacements that allow for a rigid displacement of the whole domain. It is important to note that these two basic requirements are sufficient to maintain a high grid quality, provided that the deflection of the

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boundary is small. This is normally the case as the movement of the boundary is subdivided in successively applied steps in order to resolve the time scales of the fluid flow. For small deformations there is a fundamental theorem which states that a general deformation can be considered as resulting from a linear combination of a rigid body translation, a rigid body rotation and stretches along the principal axes of strain ([ERI62]). If a method behaves well for these three types of deformations it is also able to deal with a general deformation since the latter is nothing but a superposition of the elementary ones. In the following section the equidistribution of stretches is further investigated. 4 . 1 Disc with a hole subjected t o a rotated boundary

The undeformed domain is covered by orthogonal grid lines. For the ratio of inner radius ri, and outer radius rout a value of 0.4 is chosen. The outer boundary is rotated in clockwise direction for an angle of a = 40'. Discretisations of the deformed disc with 20 x 10 grids are displayed in figure 1. For this

Figure 1. Deformation of a grid with constant spacing in radial direction by rotation of the outer boundary in clockwise direction; (a) Laplace equation, (b) Navier's equation with E = V o l ( x i ) - l , (c) spring model problem equations [3], [4] and [6] (provided that the diffusion coefficient and the elastic modulus are constant) have an analytical solution. It can be obtained by first transforming the equations from the cartesian coordinate system ( x , y) to a polar coordinate system (9,r ) and then exploiting the rotation symmetry of the probiem. Inserting the analytical solution in equation [l]yields the grid coordinates for the deformed domain: (;)n=(;)n-l+(cosp

where

F=-

(

sinp

-sinp cosy

--

--

r

4 rout 25 r

)(

F(cosa-1) Fsina

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Finite volumes for complex applications

As a measure of grid quality we use the angle between the grid lines after the deformation. It is obtained from the updated grid point coordinates. The angle is depicted in figure 2a and 2b as solid lines. The exact solution of equation [3] is very well represented by the numerical solution (circels in figure 2). We can therefore assume the discretisation to be fine enough. For the first test series a discretisation with constant cell volume V ( a i ) is selected. As can be seen in figure 2a, if the coefficient in the equations [4] and [6] are determined from the cell volume, the solutions coincidence with the analytical reference solution and hence lead to bad grid quality. While the discretisation with constant cell volume is rather academic, it is more natural to use constant spacing in radial direction. The same equations then give almost satisfactory results. While the Laplace equation leads to a worst angle of 16O, it is greater than 30' for Navier's as well as the diffusion equations, whereby the latter performs slightly better. It is important to note here that the distribution of stretches due to the coefficients [5] and [7] is further improved when the mesh is more refined towards the walls. For the two discretisations investigated, the spring model also preserves the grid quality in a satisfying manner, as can be seen in the figures l c and 2.

Figure 2. Angle over normalized radius; (a) constant cell volume, (b) constant spacing in radial direction

4.2 Solid Body Rotation of a disc

As already mentioned, it is important to examine the performance of the methods introduced in section 2 for the case of rigid-body deformations. To do so, the displacement vector that corresponds to a rigid-body deformation of the entire domain is inserted in the equations. In the case of rigid-body translation, the displacement vector is constant, which is indeed a solution of the equations [2], [3], [4] and [6]. Let us now take a closer look at the rigid-body rotation. The domain S2 shall rotate by an angle of a . The position vector X

Adaptivity, tracking and fitting

783

of the deformed domain is obtained by letting a rotation tensor a act on the position vector x of the undeformed domain.

With zn s X and

sn-'

r x we find for the displacement vector:

Here 6 denotes the identity tensor. Taking into account that gz = 6, it is clear that the displacement vector as given by [lo] is the solution of the Laplace equation [3]. This is not the case for the diffusion equation. If equation [lo] is inserted it remains:

This equation can be considered as a linear, homogeneous system of equations which determines the components of the gradient However, these components are already determined by equation [5]. In general this components will neither be zero nor will they have other values that fullfil equation [ll]. To render equation [ll]valid, the expression in brackets therefore has to vanish. For a Cartesian coordinate system this condition reads expanded:

vk.

cosa sin o

-sina cosa

) ( i !) -

=0

This equation is not fulfilled unless the rotation angle is zero. Hence we conclude that equation [4] is not able to represent rigid-body rotation. Note that it makes no difference here if the coefficient k is determined from the deformed grid, as proposed by Crumpton and Giles. A similar analysis for the Navier's equation leads to: cosa = 1, which is approximately fulfilled for small rotation angles. We conclude that the Navier's equation is able to represent rigid-body rotation, provided that the rotation angle is small enough. It remains to examine the spring model. Inserting equation [lo] into equation [2] we obtain:

The sum vanishes only if the whole domain is covered by straight coordinate lines, because in this case the contributions from opposite neighbor nodes cancel each other out. This is, of course, an unacceptable constraint on the spatial discretisation of complex geometries. Consequently, the first bracket is required to vanish which leads to the same condition as for the diffusion equation. Hence also the spring model cannot correctly represent a rigid-body rotation. The foregoing theoretical investigations are supported by numerical computations that are performed on a grid with constant spacing in radial direction,

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Figure 3. Grids generated for revolved boundaries; (a) diffusion equation with k = V o l ( x i ) - ' , (b) Navier's equation with E = V o l ( x i ) - ' , (c) spring model

which was also used in the previous section. Both the inner and the outer radius are rotated in clockwise direction by an angle of 40'. The Laplace equation and also diffusion and Navier's equations reproduce the desired rigid-body rotation, provided the coefficients k and E are constant respectively. A slight deviation from orthogonality can be observed in figure 3b. This grid was generated with the Navier's equation with a variable coefficient k. For smaller angles this deviation disappeared. The use of the diffusion equation with variable coefficient led to a significant deviation from orthogonality (fig. 3a). Also the spring model fails to reproduce the rigid-body rotation, as can be seen in figure 3c. This weakness is probably the origin of the problems that were encountered by many authors (e.g. [RAU93]) when using the spring model. The degradation of the grid quality is expected to be larger the more the mesh lines deviate from straight lines in other words for strongly curved coordinate lines. 4 . 3 A complex test case

After we investigated how the different grid generation methods perform for elementary deformations, we turn now to a general deformation. The geometry of this test case consists of a channel that contains a triangular shaped obstacle which fills one third of the channel high. In figure 4, magnifications of the mesh around the upper corner of the triangle are shown. They are generated by different methods in the following manner. First the triangle is rotated around its center in clockwise direction by an angle of 15'. Subsequently, the interior grid points of the initial grid are adjusted to the new boundary, which yields a new grid. Then the triangle is moved back to its original position again. From the grid produced in the latter stage and the known deflection of the triangle, a new grid is generated that fits the boundary of the initial state. It is clear that after this succession, the initial grid should be retrieved, or else the method employed to generate the grids is not quality preserving.

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785

The spring model fails completely, as it leads to overlapping mesh cells (figure 4c). In contrast, both diffusion equation and Navier's equation generate grids that can be used for fluid flow calculations. However, only the Navier's equation leads to a full recovery of the initial grid. When the diffusion equation is used the near wall orthogonality gets lost, as can be seen in figure 4a. This confirms the observations of the previous section.

Figure 4. Detail of the grid around the upper corner of the triangle after a succession of boundary displacements; (a) diffusion equation with k = V o l ( z i ) - l , (b) Nauier's equation with E = V o l ( x i ) - ' , (c) spring model

5. Conclusion

Many authors observed that the methods employed so far for generating grids for complex time varying domains fail to preserve grid qualtity. By means of a deformation analysis, we were able to give an explanation for this. The spring model and the diffusion equation lack the ability to reproduce rigid body rotations. Problems will occur in applications where the boundary or a part of it rotates, e.g. with an oscillating airfoil. The Laplace equation, on the other hand, doesn't give satisfactory results in the presence of moving boundaries that exhibit strong curvature or sharp bends, as high concentrations of stretches are not equidistributed in a sufficient manner. On the basis of our findings! we proposed a new method that resolved these problems. This new method can be seen as a link between the other methods. It is important to note that the analysis conducted in this paper is valid in a very general setting. It does not rely on a particular coordiante system nor on the structured spacial discretisation. Hence, the drawn conclusions hold also for the threedimensional case and for unstructured grids. There are no significant differences with regard to the computational expense required by the different methods. However, in comparison with the cornputational expense of the fluid flow calculations, the overhead incurred by the grid generation is still too high. The implementation of more efficient solution procedures will be subject of further research.

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Acknowledgment T h e first author wishes t o express his appreciation for the financial support t o this work provided partly by the Dr. Fritz Walter Fischer-Stiftung a n d t h e DFG (grant No. A F 3112-2).

Bibliography [ANJ97]

ANJUA. et al., 2-d fluid-structure interaction problems by an Arbitrary Lagrangian-Eulerian finite element method, Int. J . Comput. Fluid Dyn., 8(1), p. 1-9, 1997.

[KA095]

KAO K.-H. and LIOU M.-S., Advance in overset grid schemes: From Chimera to DRAGON grids, A I A A Journal, 33(10), p. 1809-1815, 1995.

[ZHA93]

ZHANGH. et al., Discrete form of the GCL for moving meshes and its implementation in CFD schemes, Computers Fluids, 22(1), p. 9-23, 1993.

[BATSO]

BATINAJ.T., Unsteady euler airfoil solutions using unstructured dynamic meshes, A I A A Journal, 28(8), p. 1381-1388, 1990.

[IL194]

ILINCAA., Calcul des Ccoulements compressibles tridimensionnels sur des maillages en mouvement et adaptifs, thhse de doct,orat, Ecole Polytechnique de Montreal, MontrCal Quebec, Canada, 1994.

[CRU96]

CRUMPTON P.I. and GILESM.G., Implicit time accurate solutions on unstructured dynamic grids, A IAA-95-1671- CP, p. 285-293, 1995.

[LOE96]

LOHNERR. and YANGC., Improved ALE mesh velocities for moving bodies, Communications i n Numerical Methods in Engineering, (12), p. 599-608, 1996.

[FAR941

FARHATC. and LANTERIS., Simulation of compressible viscous flows on a variety of MPPs: computational algorithms for unstructured dynamic meshes and performance results, Comput. Methods Appl. Mech. Engrg., 119, p. 35-60, 1994.

[ERI62]

ERINGENA.C., Nonlinear Theory of Continuous Media, McGraw-Hill Book Company, 1962.

[ST0681

STONEH.L., Iterative solution of implicit approximations of multidimensional partial differential equations, S I A M J. Numer. Analysis, (5), p. 530-558, 1968.

[RAU93]

LEE-RAUSCH E.M. and BATINAJ.T., Calculation of AGARD wing 445.6 flutter using Navier-Stokes aerodynamics, A IAA-93-34 76, 1993.

A Finite Volume Method for Steady Hyperbolic Equations M. J. Baines, S. J. Leary, M. E. Hubbard Department of Mathematics The University of Reading Reading, RG6 AX, UK

1 Abstract A finite volume method is presented for steady conservation laws on unstructured meshes which incorporates mesh movement. The method can substantially improve the resolution of sharp features (contacts, shocks) by solving the problem on an optimal mesh. Results are presented for a number of steady state test problems, including scalar advection and the shallow water equations in 2-D. For scalar equations there is a close relationship with the method of characteristics [5]. For shocked flows we describe a discontinuous least squares method which uses a nonlinear shock jump residual to adjust the mesh.

2

Introduction

Finite volume methods for these equations include the cell-based multi-dimensional upwinding schemes. These schemes have been very successful in producing good approximate solutions to the above problems [I] but there is still the possibility of even greater resolution of sharp features by a careful deployment of the mesh [2]. A great deal of effort has been put into mesh refinement near shocks using mesh subdivision, but similar improvements in resolution can also be obtained much more cheaply by minor adjustments to the mesh (see e.g.[3]). In this paper we consider a descent approach using a least squares measure of the residual.

3

Fluctuations

We are interested in systems of steady conservation laws such as the Euler Equations or the homogeneous Shallow Water Equations, of the form

where

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Finite volumes for complex applications

In integral form this becomes

where dS is measured into the arbitrary surface S. In determining approximate solutions U of these equations we may associate with the differential form (1) the residual R = divF(U) while with the integral form (2) we associate the fluctuation

where ST is a triangle. In fluctuation/distribution schemes such as the Multidimensional Upwind schemes a weighted amount of qh in each triangle is added to the values of the solution a t its vertices. The weights may be chosen so that the schemes are conservative, positive and linearity preserving. In particular, conservation is assured if the weights in each triangle sum to unity. In the Least Squares method, which minimises

the use of descent techniques t o achieve the optimisation also adds weighted amounts of qh in each triangle to the values of the solution at its vertices. However, this time the sum of the weights is zero and the method is not conservative in the usual sense. The procedure has the effect of redistributing solution values.

4

Counting Issues

On a computational mesh the number of equations given by (2) is equal to the number of cells in the mesh while the number of unknowns depends on the number of nodes in the mesh. In general these are different. When the number of equations exceeds the number of .unknowns, as for example on an unstructured triangular mesh, it is impossible t o choose the unknowns to satisfy all the equations. Moreover in any iteration of distribution type in which fluctuations are added to the vertices of the mesh with weights, convergence does not imply that the fluctuations vanish because there is a null space. Likewise, in the least squares approach the norm (4) cannot be driven down t o zero because of the existence of the null space. We are interested in allowing the coordinates of the vertices t o be additional unknowns of the problem. Typically this will make the number of unknowns exceed the number of equations. In that case, setting all the equations to zero does not determine the unknowns and there are then many solutions which make the fluctuations zero. A fluctuation distribution type of iteration will yield one of the many solutions at convergence.

Adaptivity, tracking and fitting

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A unique solution may be obtained if the number of unknowns is equal to t.he number of equations. For a scalar equation and a smooth solution this may be achieved on a triangular (or tetrahedral) mesh by including a sufficient number of coordinates per node in the list of unknowns. The fluctuation may then be driven to zero by a fluctuation distribution scheme and the accuracy of the approximate solution depends on the validity of relying on the fluctuation as a measure of the error and the coarseness and/or connectivity of the mesh. We shall discuss the use of Least Squares as a fluctuation distribution scheme in this context. Recall that such a scheme has the advantage of a norm to minimise but is not conservative in the usual sense.

5

A Scalar Problem

We first consider the scalar two-dimensional equation

where g is divergence-free. In the least squares method we minimise (4) over both the nodal values Uj and the nodal coordinate N j in the direction perpendicular to a. We use a steepest descent method of the form

to achieve the optimisation, where Y may be either Uj or N j and T is a relaxation factor. Fastest convegence occurs when the sweeping takes into account the hyperbolic nature of the original equation. We take scalar examples in which is a function of x or a function of U . (i) g = (y, -2) in a rectangle -1 5 z 5 1,O 5 y 5 1 with initial data

u = { 01

-0.6 5 x 5 -0.5 otherwise

on one inflow side and zero on all others. Results are shown in Figs.1 and 2. As expected the norm (4) has been driven down to machine accuracy and the solution shows the characteristics. The redistribution effected by the Least Squares optimisation is attempting to equidistribute q5 amongst the triangles [I], driven by overwriting the inflow values. (ii) Consider now the system in which A is a function. of u, corresponding to the Shallow Water Equations. We discuss two regimes corresponding to subcritical and transcritical flow. We consider flow in a channel with a bump. In the subcritical case we obtain good results by the Least Squares method which are comparable with Multidimensional Upwinding. The mesh does not alter much when nodes are allowed to move. In the transcritical case however the results are poor and we turn our attention to a variation of the method which uses degenerate triangles.

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Finite volumes for complex applications

6

Use of Degenerate Triangles

In the presence of shocks or contact discontinuities least squares methods tend t o give oscillatory solutions. A possible elaboration is to introduce degenerate triangles at the shock and to use the Least Squares method with moving nodes to adjust the position of the shock. In effect it is a form of shock fitting. Note that the fluctuation is square-integrable because 4 is always bounded, even at shocks where U is discontinuous. On the other hand, the residual is not square-integrable as a result of divF(U) being unbounded at shocks. T h a t is, on a non-degenerate mesh the l2 norm of the residual is well-defined but on a degenerate mesh it is not. However, the norm (4) is always well-defined. We show results from three examples. x 5 0 , 0 5 y 1 with initial data (iii) g = (y, -2) in a square -1

<

on the inflow side. (iv) a = ( u , 1) in a rectangle 0

<

< x 5 1,0 < y <

with initial data

on y = 0. Notice the double value of U a t inflow, introducing a contact discontinuity. The degenerate triangles are inserted vertically t o connect the triangles on either side of the discontinuity. Results are shown in Figs.3, 4, 5 and 6 (v) The final example is (ii) again in the transcritical case for the flow in a channel, which exhibits a shock in the constriction. An approximate solution is first obtained on a fixed mesh by the method of multidimensional upwinding. This is t o approximately locate the shock. Degenerate nodes are then chosen in the approximate position of the shock and the Least Squares minimisation procedure used to position the shock by finding the best position of the shock nodes. Results are shown in Fig.7.

Adaptivity, tracking and fitting

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Numerical results

Figure 1: Initial grid and solution for example(i).

Figure 2: Final grid and solution for example(;).

Figure 3: Initial grid and solution for example(iii).

791

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Finite volumes for complex applications

Figure 4: Resulting grid and solution for example(iii).

Figure 5: Initial grid and solution for example(iv).

0

(Is

Figure 6: Resulting grid and solution for example(iv).

>

Adaptivity, tracking and fitting

Figure 7: Results for example(v).

793

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Finite volumes for complex applications

8

Conclusion

There exist good methods for the approximate solution of first order PDEs using the fluctuation/distribution technique of Multidimensional Upwinding, achieving conservation, positivity and linearity preservation. However, by the nature of the technique, on unstructured meshes the fluctuations are not driven to zero because of the null space. If we are to rely on the fluctuation to generate a better solution, we need to introduce more degrees of freedom which may be done by adding node locations to the list of unknowns. In this way we can ensure that the fluctuation is reduced to machine accuracy. A convenient way of iterating towards the solution for all variables is to minimise the Least Squares norm of the fluctuation. Although this is not a conservative procedure it is a redistribution procedure and we may think of it as redistributing both the solution and the node locations. The effect is to equidistribute the fluctuation amongst the triangles and so proceed down to the limit in a uniform way. It is useful to regard the procedure as sharpening up a conservative solution. References

[I] H.Deconinck, P.L.Roe and R. Struijs (1993). A Multidimensional Generalisation of Roe's Flux Difference Splitter for the Euler Equations. Computers and Fluids, 22, 215. [2] P.L.Roe (1996). Compounded of Many Simples, in Proceedings of Workshop on Barriers and Challenges in CFD, ICASE, NASA Langley, August 1996, (Ventakrishnan, Salas and Chakravarthy (eds.)), p241, Kluwer, 1998.

[3] M.J.Baines. Least Squares and Equidistribution in Multidimensions. Numerical Methods in Partial Differential Equations (1999), to appear. [4] M.J. Baines and S.J.Leary. Fluctuation and Signals for Scalar Hyperbolic Equations on Adjustable Meshes. IJNME (submitted).

[5] M.J.Baines and M.E Hubbard. Multidimensional Upwinding with Grid Adaptation. In Numerical Methods for Wave Propagation (E.F.Toro and J.F.Clarke (eds.)), Kluwer (1998).

Moving grid technology for finite volume methods in gas dynamics Boris N. Azarenok and Sergey A. Ivanenko Computing Center of RAS, Vavilov str. 40, GSP-1, Moscow, 117967, Russia e-mail: [email protected]

, [email protected]

A B S T R A C T Adaptive grid generation procedure is coupled with the Godunov-type solver. The problem to construct the harmonic coordinates on the surface of the graph of control function is formulated. The projection of these coordinates onto a physical region produces an adaptive-harmonic grid. R e ~ u l t sof computations for the nonstationary pow in the wind tunnel containing a step are presented. Iieywords: Adaptive grid, harmonic mapping, numerical methods in gas dynamics

Recent development of robust adaptive grid generation techniques which can give locally refined information in areas of interest as well as fitting boundaries precisely is of great importance for the practical problems, particularly for simulating the moving shocks in the nonstationary proble~nsof gas dynamics. In the present study the r-refinement adaptation is used with the Godunovtype finite-volume solver of the second order accuracy t o model a gas flow. The approach is based on the grid cells convexity concept [2] and the theory of harmonic maps between surfaces. The harmonic functional (Dirichlet's functional) is approximated in such a way that its minimum ensures all grid cells to be convex quadrilaterals. The lnain property of this approximation is that the discrete functional has an infinite barrier on the boundary of the set of grids with all convex cells and it guarantees nondegenerate (unfolded) grid generation a t each time step. This folding-resistant property is very important in the problems with interior lines following thin and strongly bent layers of high gradients. Another important property is that the grid generator is solver-independent . 1. Governi~lgEquations and Finite-Difference scheme We use the governing conservation laws expressed in integral form to describe the planar flow of the ideal gas. The volume integrals in the space x-y-t may be transformed to the surface integrals by virtue of Gauss's theorem as it is shown below for the law of conservation of mass

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Finite volumes for complex applications

Thus, a ~rariat~ional formulation of the probleill is written as following [4] :

/L

Udrdy + F(U)dydf + G(U)dtdr = 0 ,

where the vectors of conserved variables U , F(U) and G(U) have the following conlpo~lents

here r~ and u are t , l ~ velocity e components, p and p are the pressure and density. Let us int,roduce t#hemoving grid in space x-y-t. Integrating the system over t,lle oriented surface being the boundary of t.he con~put,illgcell we get a cell-centered finite-volume discret,ization of the governing equat,ions

-AIL 1+112,j+1/3are the areas of t8he upper and lower faces n+1/3 n+l/? of the cell a t t"+' and t n . Each of four vector values Q i + l l ? , j , Qi+l/?,j+lt n+l/? Q"+'/' L , J + l l l ? ,Qi+, j + l l ? is an average flux of mass, impulse and energy through tlie where --lzL',j+ll,.

corresponding intercell surfa.ce t,owards t.he out,sicle normal vector. To calculat,e tlie values a t t,he nest t,ilne level n + l we use t,rvo-step procedure "predictorcorrect,or" applying a monotonicit.y algorit,hm and s o l ~ t ~ i oofn the Rieman~i problenl [I]. The sc-heme is of t.he second-order accuracy in time alld space. 2. Dynamic solution-adaptive lnesll algorithm Suppose we are given a sinlply connected donlain R with a snlootll boundary in the .r, y plane. Consicler t,he surface z = f ( x , y ) of t,he graph of the c-ont.rol function f(.t.. y). It is required t.o filid a mapping of bile parametric square 0nt.o the domain Q under a given ]napping between boulldaries such that the mapping of t.he surface onto the parametric square be harmonic. Thus, t,he problelll is t80nlinimize t,he harmonic f~rict~ioilal which car1 be writ.t,en in the forin

Adaptivity, tracking and fitting

797

This functional defines an adaptive-harmonic grid clustered in the regions of high gradients of the function f ( e , y). Detailed account of tlie minimization algorithlil is presented in [2]. One time step t o solve the 2-D equations of gas dynamics with automatic grid adaptation contains the following stages: 1. Generate the mesh a t the nest time step. As a n initial approach we can either use the mesh from the previous time level or, that is most effective, define the node coordinates by their velocity from the previous time level. 2. Comput,e the gas dynamics values a t t,he next time level using the secondorder Godunov-type scheme described in sec.1. 3. Update tlie gas dynamics value chosen as a control function from the cell centers t o nodes. The result is a control function value fij a t each mesh node. 4. Eva1uat.e the derivatives ( f,)ij and ( f,)ij at each mesh node. 5. Make one iteration step and compute the new values of x i j and yij. 6. Repeat starting wit,h step 2 t,o convergence. Note that this approach can be extended t o the general structured multiblock grid configurations.

3. Results of t,est computations As a test example the planar nonstationary flow in the wind tunnel containing a step [3] has been considered. T h e wind tunnel is 1 unit wide and 3 units length, step is 0.2 units h e ~ g h tand begins a t x=0.6. Initial conditions for ( u , z ~ , pp) , are ( 3 , 0 , 1 ,y) t h a t corresponds t o the flow-111 boundary condition a t .r=O as well Here y is the ratio of specific heats and equal 1.4. The exit boundary has transmissive conditions since a t x=3 the flow is supersonic. Along the walls of the tunnel reflecting boundary conditions are applied. T h e corner of the step is the center of a rarefaction fan and hence is a singular point of the flow. Just above the step there is a thin boundary layer. Shock wave interacts with this layer and qualitative nature of the flow near the step is altered. When c o n ~ p u t i n gon the rectangular mesh we apply the additional boundary condition t o the density and velocity near the corner of the step [3] to minimize numerical errors generated by the corner. Results of computations on the uniform and rectangular mesh with steps h,=h,=1/160 are presented in Fig.1. T h e mesh contains 480x160 cells. Isolines of the density are shown in Figs.la-d corresponding t o time moments t=O.5,1,2,4. T h e results of Fig.1 are the same that in 131. At time t=O 5 (Fig.la) the bow shock is formed, then extending by t = l it reaches the top wall and reflects from it (Fig.lb). In some time, a t t=2 (Fig.lc), the shock reflects from the step and near the top wall the Mach stem emerges and the contact discontinuity emanates from the triple point. By time t=4 (Fig.ld) the shock wave has already reflected triply, the contact discontinuity twice intersects the shock. In Fig.2 the results of computations on the adaptive moving grid 150x60 with applying the procedure t o capture the bow shock are presented a t t=4.

798

Finite volumes for complex applications

Fig.%a,bcorresponds to adaptation with the density as the control function. In case, presented in Fig.2c,d, we perform adaptation to IVI. The density contours are shown in both cases. Note that the Mach sten] is described rather precisely by the moving boundary but the shock, emanating from the triple point, is not resolved well by nodes clustering. In the first case of adaptation to the density (Fig.2a,b) the procedure does not provide grid clustering t o the contact discontinuity. At the same time adaptation to IT/'[ produces weak grid clustering to the contact discontinuity (Fig.2c,d). In the case of global adaptation without shock capturing t,he method produces excellent results shown in Fig.3,4 at the same time moments as in Fig.1. T h e mesh contains 180x60 cells that seven times less t hall for the rectangular grid (Fig.1) and here at the same accuracy it is required less computer time by factor of 1.5. Here, like for the uniform grid, the shock wave is smeared within 3 or 4 cells and the mesh with strong clustering looks like a set of the blocks with the boundaries defined automatically as the lines of grid nodes clustering. Inside every block the mesh is quasiuniform and the solution is smooth, see F i g . 4 ~ . This approach may be considered as further developrnent of the moving block technology presented in the monograph [4]. Advantage of the moving adaptive grid technology is "the automatic generation of moving block boundaries" that can not be done by hand in many cases. This problem is essential difficult in three-dimensional cases. 4. Conclusions Modeling of the norlstationary flow in the wind tunnel containing a step has shown that the accuracy of the solution obtained on the adaptive grid increases greatly in comparison with the rectangular grid. Numerical experiments confirm the high reliability and accuracy of the algorithms which employ adaptive-harmonic method t o construct the grid. It is shown that for the problems with moving shocks adaptive moving grids appear to be more economical, a t the same resolution as on the rectangular grid, and in our case reduce the memory requirements by the factor of 10. Presented solver-independent solution-dependent grid generation technique is stable and robust. In its application we can avoid human intervention into the solution process. It can be extended to the three-dimensional case and can be used in other problems of computational fluid dynamics. References [l] Azarenok B.N. On a realization of high-order Godunov's scheme, Computing Center of RAS, (Moscow, 1997). [2] Ivanenko S.A. Harmonic mappings, Chapt. 8 in: Handbook of Grid Generation. J.F. Thompson at al, (CRC Press, Boca Raton, F1, 1999). [3] Colella P. and Woodward P.R. J.Comput.Phys., 54 (1984), pp. 115-173. [4] Godunov S.K. at al, Numerical solution of multi-dimensional problems in gas dynamics (Nauka Press, Moscow, 1976).

Adaptivity, tracking and fitting

Fig. 1

799

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Finite volumes for complex applications

Fig. 2

Adaptivity, tracking and fitting

801

--

(dl

05

10

15

Fig. 3

20

25

30

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Finite volumes for complex applications

0.5

1.O

2.0

1.5

Fig. 4

2.5

3.0

Numerical Simulation of Lifted Turbulent MethaneAir Diffusion Flames

Ming Chen, Nobert Peters Institut fur Technische Mechanik, RWTH-Aachen, Templergraben 64, 0-52056 Aachen, Germany

This paper presents the numerical simulations of the lifted turbulent methane-air diffusion flames using a flamelet concept. To describe turbulent flame and stabilization i n inhomogeneous mixtures, combined flamelet formulations for partially premixed flames have been applied, which inelude transport equations of the mean mixture fraction 2 and its variance and an equation for a smooth scalar field G , determining the mean location of the flame front. The turbulent burning velocity for partially premixed combustion is described. The level set approach is applied to solve the G-equation. The simulations suggest that the mechanisms of stabilization can be explained by the partially premixed flame propagation and flamelet quenching. The stabilization point is determined by the local flow velocity, the local turbulent burning velocity, and the local scalar dissipation rate. The stabilization points are found not to be located at the isoline of the stoichiometric mixture, but at the lean side. ABSTRACT

z"~,

Key Words: flame lift-off , turbulent combustion, numerical simulation.

1. Introduction

Despite a lot of research efforts, the physical mechanisms of the flame stability are still not clearly explained. Pitts [PIT 881 has reviewed the theories and experimental findings of the lifted diffusion flames. The mechanisms responsible for the flame stability may be divided into three groups: 1) premixedness model; 2) flamelet quenching model; and 3) flame extinction due t o large and small scale turbulent structures. The premixedness model [VAN 661 assumed that fuel and air along the

804 Finite volumes for complex applications

stoichiometric mixture are fully premixed and flame stabilization occurs at the position where the axial mean velocity equals the turbulent mean burning velocity. However, the premixedness model has been challenged by several new theories [PET 83, BYG 85, MUL 941, which argued that the stabilization of the lifted diffusion flames can be explained by flame extinction processes. Peters and Williams [PET 831 proposed that non-premixed combustion occurs and flame propagation proceeds along instantaneous surfaces of stoichiometric mixtures up to the point where the flame is quenched. Byggsoyl and Magnussen [BYG 851 proposed that the extinction occurs in the smallest vortices of the turbulent flow. Broadwell et al. [BRO 851 suggested a large scale mixing model t o explain the lift-off and blowout behaviors. A numerical calculation of lifted diffusion flames has been done by Miiller et al. [MUL 941, they used the combined flamelet formulations for premixed and non-premixed combustion t o calculate flame propagation and lift-off heights in jet diffusion flames. In this work, the numerical calculation of the lifted turbulent methane-air diffusion flames have been carried out with the purpose t o gain more understanding of the lift-off behavior from the point of view of numerical simulation. The nieasurements by Tacke [TAC 981 showed that the stabilization point of a lifted methane diffusion flame is not located a t the isoline of the stoichiometric mixture, but at the lean side, which is contrary t o the expression of the turbulent burning velocity used by Miiller et al.. To investigate the position of the stabilization point, the formulation of the turbulent burning velocity applied in the present study is more general. Following a describtion of the combined flamelet formulations of partially premixed combustion, numerical methods will be introduced. Finally the results of the calculation are presented and discussed. 2. Mathematical Model

The flamelet model [PET 871 has been proven to be very useful to describe non-equilibrium chemistry in combustion systems in the flamelet regime, when chemistry is fast compared t o the mixing time scales of the flow. The concept allows to decouple the effort to describe combustion chemistry from the calculation of the turbulent flow field. Originally, flamelet models have been developed to treat non-premixed combustion problem [PET 871, and later premixed flames [WIR 921. In order t o describe turbulent flame propagation and stabilization in inhomogeneous mixtures, both approaches have been combined by solving scalar equations of the mean mixture fraction 2,its variance z"', and a mean G-field equation [MUL 941.

-

2.1. The flamelet model for non-premzxed combustion

Additionally to the basic conservation equations for mass, momentum, tur-

Adaptivity, tracking and fitting

805

bulent kinetic energy & and the dissipation rate of turbulent kinetic energy 2, the transport equations of the mean mixture fraction and its variance should be solved, which read

where the Schmidt numbers Sc2 and Sc5 tion rate 2 is modeled as

are constants. The scalar dissipa-

The energy conservation equation is solved in the form of the total enthalpy:

Here PTT is the turbulent Prandtl number, and qk denotes a radiation heat loss term. 2.2. The G-Equation for partially premixed combustion

To describe the location of the pemixed flame front, the scalar function G is applied in the flow field. The scalar G is so defined that G = Go describes the location and geometry of the flame front. A mean turbulent G-equation may be derived from the kinematic consideration [PET 971:

where the mean flow velocity V u is defined as the conditional velocity in the unburnt mixture ahead of the flame front and v; is its fluctuation. Sg is the laminar burning velocity of a unstretched planar flame, K is the flame curvature and D L = S i C is the Markstein diffusivity. The absolute value of the gradient of the scalar G is represented by a = IVGI, and the normal vector n is defined as n = -VG/IVG/. It has been shown that Equation (5) can be modeled as [PET 971

806

Finite volumes for complex applications -

where the diffusivity DT,E is , ~ T / S CThe ~ ,term ~ S i u can be expressed as

where P ( Z , X , a ) is a joint pdf of Z, x and u. It is assumed here that 2,x and a are statistically independent, and the fluctuation of the scalar dissipation rate can be neglected, then one obtains

= Letting P ( a ) u do and using an ad-hoc expression that modifies the laminar buring velocity S;(Z) by a quenching term, in terms of the scalar dissipation rate X [MUL 941

SL(Z,X)= S;(Z) max

(9)

one obtains,

The scalar dissipation rate a t quenching X, is 6.4 s-' for methane-air diffusion flamelet. With the relation following Wirth M. et al. [WIR 921

s;m

= s;loGI,

(11)

Equation (10) can be written as

Here SF denotes the turbulent burning velocity for premixed flame. With the expression [WIR 921

one obtains

Adaptivity, tracking and fitting

807

Here the constants bl = 1.5,b2 = 0.8, b3 = 1.4 are obtained from experiments and direct simulations. Instead of using Equation (3), the scalar dissipation rate jj in Equation (14) is modeled following [lo]:

where A Z F is the flame width in mixture fraction space, which is assumed as AZF = 2 Z S t .

2.3. Treatment of the partially premixed combustion The present method treats the discontinuous variation of all states as continuous crossing from unburnt to burnt states in a region sealed with turbulent flame brush. The weighted value of the mass fraction of species i is calculated by: = f b x , b + (1- fb)yTu. (I6) The fraction of burnt flame in each cell f b is obtained by assuming that G fluctuations satisfy a Gaussian distribution:

-

-

where the variance of G is assumed proportional t o the squre of the integral length scale with a proportionality factor of 114. The local mass fraction of unburnt mixture Y,,, is obtained by a quenched diffusion flamelet while the is obtained from the burnt diffusion flamelet. mass fraction of burnt mixture Yi,b The mean mass fraction of species of burnt state is obtained by weighting the laminar values with an assumed pdf, i.e. a beta-pdf.

The code structure for partially premixed combustion is shown in Fig. 1.

3. N u m e r i c a l m e t h o d s for solving G-equation

In this work, the level set approach of Sussman et al. [SUS 941 is used to solve the G-equation. Instead of explicitly tracking the interface, the level set approach captures the interface implicitly, in which the interface is identified as the zero level set of a smooth function. To guarantee that. G is a smooth function in space, it is useful to keep G a distance function. Therefore, when G # Go, the condition lVGl = 1 has t o be satisfied. The smoothing algorithm of Sussman et al. is used solving the following equation to a stea.dy state: - = Sgn(G(x,t o ) - Go)(l -

at

IVgl);

g(x, t = to) = G(x, to)

(19)

The application of a conservative grid adaptation technique to 1D unsteady problems

Manuel J. Castro-Diaz Numerical Analysis. University of Ma'laga 29080 Ma'laga Pilar Garcia-Navarro Fluids Mechanics University of Zaragoza

I n this contribution, we present an eficient conservative mesh adaptation algorithm appl~edto 1-D shallow water equations. This algorithm is suitable for unsteadv situations and discontinuities of the solutions are well captured. Numerical results are presented.

ABSTRACT:

Key Words: Grid adaptivity, Q-Schemes of Roe and Van Leer, Implicit and explicit T V D methods, unsteady shallow water flows.

1. I n t r o d u c t i o n

Numerical methods for predicting the water profile and discharge in steady as well as unsteady situations of hydraulic systems have become a common tool. In particular, finite difference applications of numerical schemes have been widely reported. One of the basic problems in unsteady hydraulic systems is the location of solution discontinuities and shocks. In order to solve this problem, an efficient conservative grid adaptation algorithm applied to the resolution of the shallow water equations is presented. First, the equations t o be solved are presented. They are essentially the well known shallow water equations written in conservative form. The discretization of the system is done using the numerical method proposed by BermudezVBzquez in [BER 941. A high order method as the TVD-McCormack scheme (see [GAR 92-11 and [GAR 92-21) has also been used to compare numerical solutions.

810

Finite volumes for complex applications

A posteriori error estimator to control the error of the numerical solution is constructed using a metric tensor M , being M the solution of a minimization problem. Once M is computed, a simple version of an anisotropic Delaunay algorithm for one-dimensional domains is used to adapt the mesh. A local conservative interpolation algorithm is used to guarantee conservation of the variables during mesh adaptation. Finally numerical results are presented and comparisons with non conservative interpolation algorithms and other numerical schemes are given. 2. Shallow water equations Shallow water equations represent mass and momentum conservation along the direction of the main flow. They constitute an adequate description for most of the problems associated with open channel flow modelling and can be written as the following system of equations:

where A is the wetted cross sectional area, Q is the discharge, g is the acceleration due to gravity and Il represents an hydrostatic pressure force term:

and 12 accounts for the pressure forces due to longitudinal width variations,

beeing B ( x ) the breadth of the channel that it is supposed to be locally rectangular. The right hand side of equation (1) also contains the sources and sinks of momentum arising from the bed slope and friction losses. 3. Discretization of the equations The discretization of the system is done using the numerical method proposed by Bermudez-VBzquez in [BER 941, that is, an explicit extension of the

Adaptivity, tracking and fitting

81 1

Q-scheme of Roe with upwinding in flux and source terms. For more details see [ROE 811, [ROE 861 and [BER 941. An high order method as the TVDMcCormack scheme (see [GAR 92-11 and [GAR 92-21) has also been used t o compare numerical solutions.

4. Error estimator: metric computation

A posteriori error estimator to control the error of the numerical solution is constructed using a metric tensor M , being M the solution of the following minimization problem (see [CAS 971 and [CAS 961 for more details): Find a metric tensor M , so that, the adapted Delaunay mesh constructed from M minimizes the interpolation error

where, W n = (An, Qn)T is the solution of the shallow water equation at time t = tn and IIh[Wn] is a continuous piecewise linear interpolation of Wn over the mesh Th. In general, this metric tensor is given in terms of Hessian matrices of the variables. In this particular case, as An and Qn: IR -+IR, the metric tensor can be computed as:

where D2An and D2Qn are, respectively, the second derivatives of An and Qn, eo is a positive control parameter and T,61 a truncated function that avoids metric degeneration

and l,in, the maximal where, usually, €0 = 1/12,, and €1 = l/lgin, being I, and minimal allowed length for mesh edges, respectively. Note that An and Qn are unknowns as they are the solution of the problem a t time t = t,, there fore, the metric tensor M is approximated using the numerical solution a t time t = t,, Ax, Q;. For more details and the extension t o bidimensional domains see [CAS 971 and [CAS 961.

81 2

Finite volumes for complex applications

5. Conservative mesh adaptation algorithm

Once the metric tensor M is computed, the mesh is adapted using a anisotropic Delaunay algorithm. For one-dimensional meshes the algorithm is simple (see [CAS 971 and [CAS 961): let di the length of the edge ai with respect to the metric tensor M. Three possible cases can be distinguished: If d, > dm,, (dm,, x 1.4) then ai is cut in two edges. The length of the new edges is computed and they are split until the length of all the new edges is smaller than d,,,. If di < dmin (dmi, x 0.6) then the edge ai is suppressed. As this process implies that neighbour edges have changed, we must check if their lengths are larger than dm,,. In that case, the previous step is applied to the corresponding edges. If dmin 5 di 5 dm,,, ai is kept. One of the most difficult problems on mesh adaptation is the interpolation of the numerical solution onto the adapted mesh. This is critical if the studied phenomena are unsteady. A deficient interpolation could spoil the good properties of the numerical scheme, as conservation and monotonicity. The usual interpolation operator in mesh adaptation is the linear one, but this operator, in general, is not conservative, that is, given the numerical solution of the shallow water equations over the mesh Th a t time step n,(A:, Q:),

being IIhf [A:], nh'[Q:] a continuous piecewise linear interpolation of (A:, Q:) over the adapted mesh, TL, a t time t = t,. In order to guarantee the conservation of variables during the mesh adaptation process we propose the following interpolation operator: II;, [A:] is the continuous piecewise linear function over the mesh Thj such as

where, vj is a vertex of Th,, Cj is the cell associated to v(iand ICiI is the length of cell Cj. Note that A: could be difficult and expensive to calculate if Th # Thj. We can avoid this problem if we perform this computation during the mesh adaptation loop since Cj n Th is easy to determine. Using conservative interpolation together with mesh adaptation, the discretization error and CPU time can be reduced substantially for unsteady problems (see table 1).

Adaptivity, tracking and fitting

813

6 . Numerical results

6.1. D a m break problem

This is an interesting problem to test the efficiency of conservative mesh adaptation algorithm for non-steady flows with shocks since it has an analytical solution. This problem is generated by the homogeneous one-dimensional shallow water equations with the initial conditions hL if x

<$

h(x, 0) =

Q(x, 0) = 0. hR i f x > $

In this case, hL = 2, hR = 1 and L = 60. If the calculation times used are so as to avoid interaction with the extremities of the channel, the boundary conditions are trivial. Table 1 summarizes the results obtained for the Q-scheme of Roe with an uniform mesh, non-conservative adaptation, conservative adaptation and for the TVD-McCormack scheme (TVD-MC) with uniform mesh at time t = 2.5 seg. As can be observed, the Q-scheme of Roe+conservative adaptation only needs 332 nodes to obtain an error of 0.053 units. If an uniform mesh is used, the number of nodes must be about 3000 for a similar error. For a higher order scheme as the TVD-MC the number of nodes is about 1000. Note that, if a non-conservative mesh adaptation algorithm is used, the approximation error increases up to 0.0583 units. The reduction of CPU time for a similar tolerance error is significant, if mesh adaptation is used and note that the computational cost for conservative and non-conservative adaptation is practically the same. Figure 1 shows a comparison between the numerical solution for the dam break problem with conservative mesh adaptation and the exact solution at time t = 2.5 seg.

conservative

TVD-MC Roe

Table 1: Table 1: Dam break problem. 6.2. Surge propagation through converging-diverging channel

In this example, the geometrical domain for the flow is an interval of L = 500 m with flat bed and a sinusoidal width variation given by B(x) =

5 - 0.7065 (1

+ cos (2n (-)))

if 1x - 2501 5 150 otherwise

814

Finite volumes for complex applications 2

h approx. b",', ..-~-.. h exact. --

1.5 -

1

-

0.5

-

0

0

10

20

30

40

50

60

Figure 1: Dam break problem with conservative mesh adaptation. Comparison with the exact solution a t time t=2.5 seg. hL = 2, hR = 1. In this case the exact solution does not exist and only comparisons with other schemes can be performed. The time evolution of a surge is considered. A bore 9.79 m deep of 1000 cumls propagates downstream over still water 1 m deep. A 2 m weir is supposed t o be placed downstream. At time t = 150 s the downstream end is reached by a front similar t o the initial one so that it is partially reflected and partially transmitted over the weir (see figure 2). Only 142 nodes are needed to obtain a good approximation if conservative mesh adaptation is used, while a uniform mesh of 600 nodes is needed if TVD-McCormack is used. The reflected surge starts travelling upstream and it propagates until it becomes a stationary hydraulic jump in the contracting region. This final steady st,ate is shown in figure 3. The total CPU cost when using mesh adaptation is 186 seconds while the total CPU time when using TVD-McCormack is 360 seconds in a PENTIUM I1 (333 Mhz.)

7. Conclusions Conservative mesh adaptation has been applied with success t o 1D unsteady problems. The idea of the method is simple and it can be easily applied by other users. The application to 2D and 3D configuration is straightforward. Conservative interpolation guarantees the conservation of all the variables during mesh adaptation and, as the results show, the numerical error is reduced when it is used. The global CPU requirement is significantly reduced compared wit,h a direct computation on a uniform fine mesh. Acknowledgements: The author is indebted to J. Macias and E. VAzquez

Adaptivity, tracking and fitting

81.5

Figure 2: Surge propagation in a converging-diverging channel. Comparison between conservative adaptation and TVD-MC scheme a t time t=150 s.

Figure 3: Surge propagation in a converging-diverging channel. Comparison between conservative adaptation and TVD-MC scheme a t time t=600 s.

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Finite volumes for complex applications

for many valuable discussions. This work has been supported by the Comision Interministerial de Ciencia y Tecnologia (C.I.C.Y.T.), project MAR97-1055CO2-01. 8 . Bibliography

[BER 941

BERMUDEZ, A., V ~ Z Q U E Z M.E. , Upwind methods for hyperbolic conservation laws with source terms. Computers and Fluids,23(8),1994, p. 1049-1071.

[CAS 971

CASTRO.M . J , F. HECHT,B. MOHAMMADI A N D 0. PIRONNEAU.Anisotropic Unstructured Mesh Adaptation for Flow Simulations. I.J. for Numer. Meth. in Fluids, 1997, 25, p. 475-491.

[CAS 961

CASTRO-DIAZ,M. Generacio'n y Adaptacidn Anisdtropa de Mallados de Elementos Finitos para la Resolucidn Nume'rica de E.D.P. Aplicaciones. Ph.D. Universidad de Malaga. 1996.

AR 92-11

GARCIA-NAVARRO, P. A N D ALCRUDO,F. Implicit and Explicit TVD Methods for Discontinuous Open Channel Flows. Proc. of the 2nd Int. Conf. on Hydraulic and Environmental Modelling of Coastal, Estuarine and River Waters, Vol. 2, Edited by R.A. Falconer, K. Shiono and R.G.S. Matthew, 1992, Ashgate.

[GAR 92-21

GARCIA-NAVARRO, P . , A N D ALCRUDO, F. 1D Open Channel Flow Simulation Using TVD McCormack Scheme. J. of Hydraulic Engineering, ASCE, 118, 1992, p. 1359-1373.

[MCC 711

MCCORMACK, R.W. Numerical Solution of the Interaction of a Shock Wave with a Laminar Boundary Layer. Proceedings of the 2nd International Conference on Numerical Methods in Fluid Dynamics, 1971, p. 151-163.

[ROE 811

ROE, P.L. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics, 43, 1981, p. 357-371.

[ROE 861

ROE,P .L .Upwinding differenced schemes for hyperbolic conservation laws with source terms. In Carasso, Raviart, and Serre, editors, Proceedings of the Conference on Hyperbolic problems, Springer, 1986, p. 41-51.

Application of mesh adaptive techniques to mesh convergence in complex CFD

David Leservoisier(*)(**), Alain Dervieux(**), Paul-Louis George(***), Olivier Penanhoat(*)

(*) SNECMA, Centre de Villaroche, 77550 Moissy Cramayel, FRANCE (**) INRIA Sophia-Antipolis BP 92, 06902 Sophia Antipolis Cedex, FRANCE (***) INRIA Rocquencourt BP 105,78153 Le Chenay Cedex, FRANCE

This paper is devoted to the illustration of the fact that mesh adaptation is a natural path for mesh convergence for complex problems. We apply an anisotropic reconstruction mesh adaptor. The field of application considered is that of turbulent compressible flows.

ABSTRACT

1

Introduction

It is a remarkable fact that up to now nearly only second-order methods have been efficiently and routinely applied to complex CFD such as RANS (ReynoldsAveraged Navier-Stokes) finite volume simulations of compressible flows. Secondorder accurate schemes are evidently much better than first-order accurate ones, but pioneering experiments in applying higher order schemes to practical configurations are often disappointing. In fact, the actual order of convergence of both - second and higher order- finite volume methods on practical problems is questionable; indeed, transonic flows involve singularities for which nurnerical schemes produce only first-order accurate results. Even nonlinear limited schemes (TVD, etc.) have permitted second-order accuracy on most part of the domain but they do not capture a shock with an approximation error better than first-order in the usual L2 (root mean square) functional norm. Then, making the spatial mesh finer will make the error on the discontinuity grow bigger that the rest of the error and decide the size of the asymptotic error of the global calculation, that is, O ( O x ) . These discontinuities are in fact the prototype of any detail (smooth or not) of the flow that is not accurately represented by the mesh. Think of wakes, slip lines or surfaces, boundary layers, ... In most complex CFD calculations, the mesh is not fine enough for capturing well these details; moreover, numerical dissipation makes this situation worse; as a result, second-order convergence to continuous solutions is generally not observed. The purpose of this paper is t o

8 18

Finite volumes for complex applications

study some conditions that would allow actual second-order convergence for 2D RANS CFD calculations. The key point on which we shall rely is the use of m e s h adaptation. It is clear that for a lot of complex calculations in which small details arise in very localised areas of the computational domain, the rnesh is in fact one of the main unknowns of the numerical problem. This explains why in many new adaptive approaches, the mesh is globally reconstructed instead of only modified in order to be better adapted to the solution. In (31, the authors describe a method in which a local matrix called "metric" is computed from an estimate of the hessian derivative of the unknown; then a new anisotropic mesh is rebuilt by a Delaunay technique taking account of the adaptation metric. In this study, we propose, as a complement of this method, a strategy for researching mesh convergence. For a given number of unknown (degrees of freedom) we make correspond a "discrete answer" defined as the association of a adapted mesh and a corresponding EDP discrete solution. We suggest that this discrete answer allows second order mesh convergence in many cases where the usual mesh refinement produce only first-order convergence. Further we claim that this strategy allows mesh convergence for complex problems such as industrial CFD flows. This assertion will be illustrated by several turbulent flow calculations.

2

Mesh adapted solution

In the recent bibliography on mesh adaptation, it is recognized that mesh adaptation is a mean for compensating lack of accuracy due to singularities and steep gradients in the solutions. Quantitative assessment of this property is possible by considering the number of nodes instead of mesh size: for singularities, second-accuracy is observed, while for regular gradient stiffness, second-order accurate behavior arises with a considerably much smaller number of nodes than for uniform meshes. For a rather simple presentation of these facts, see [I], in which the distribution of a given number of nodes in an interval. Let us define an adaptive algorithm which associates to a given n u m b e r of nodes N the couple ( M N , U N ) of a adapted m e s h and a numerical solution. The solution UN is computed on the mesh M N ,and the mesh M N is adapted to the solution U N . M e s h adaptation: The principle of mesh adaptation is an already rather popular combination of Hessian evalutation and anisotropic mesh recorlstruction with total mesh size control [3].

2.1

Application to adaptive interpolation

The second-order convergence to continuous solution depends of both the adaptive process and the approximation of PDE.

Adaptivity, tracking and fitting

819

Mesh coupling: In order to obtain the above described couple, we parametrise

Figure 1: Adaptation t o an analytic "eight-shaped" function: isotropic and anisotropic algorithm. the remesher in order t o build a new adapted rnesh with (in fact approximatively) the prescribed number of nodes. Then, a new solution of the PDE is computed and produces a new metric. The process is repeated uritil convergence ; a sufficient iterative convergence can be recognised if we observe a rather srnall, but not so srnall, variation of the PDE solution between two meshes. The rest

il

(is1

iis

it31

I<<<*

i~
IL<<,CU

Mesh vertres nu&

Figure 2: Mesh convergence for the i~lterpolatior~ of the eight shape function on a sequence of uniform meshes: multiplying the number of nodes by four does not produce a solution four tirnes more accurate. of the paper will concentrate on the verification that the proposed algorithm allows second-order accurate convergence towards the contir~uoussolution, that is, for a norm of integrable furictions (typically : L"), llUN - UII = O ( N 0 . 5 ) .

820

Finite volumes for complex applications

For a first study that be independant of the approxinlation of the PDE, we consider first replacing the solution of the PDE by a simple PI interpolation of an analytical function. Examples of isotropic adapted mesh and anisotropic adapted mesh are depicted in Figure 2. The given fonction has steep gradients along an eight-shaped curve. For each element of a sample of values of N, number of nodes, we have iteratively converged the sequence described in Fig.1 (in which "solution of EDP" is replaced by "interpolation of analytic function"). This process yields for each N a couple ( M N ,U N ) and an error e,v = ~ I U N - UIIL?. Convergence curves of this error to zero for the uniform, isotropic adaptive and anisotropic adaptive strategies are depicted in Figures 2, 3 and 4. Second-order convergence

Figure 3: Mesh convergence for the interpolation of eight-shaped analytic furiction with a sequence of adapted isotropic meshes (both coarse and finer meshes are perfectly adapted by iterating mesh-solution coupling): second-order accuracy appears only if more than 10000 nodes are used. to continuous solution appears already for a small number of nodes when the anisotropic adapted algorithm is applied, see Fig. 5. With the isotropic algorithm, second-order convergence appears only for number of nodes larger than 5,000 (Fig. 4). Second-order convergence with the uniform algorithm is not obtained, even with 30,000 nodes Fig. 3.

3

Application to turbulent flows

We now concentrate on assessing the described strategy on a set of typical flows of industrial interest. These flows will be computed with a 2D compressible flow solver with k - r turbulence modelling and wall law, relying on a MUSCL approximation on unstructured meshes with triangles, see [2] for details. The hessian sensor is computed with the six conservative variables, including

Adaptivity, tracking and fitting

821

Figure 4: Mesh convergence for the interpolation of the eight-shaped analytic function with a sequence at adapted anisotropic meshes: Assuming you have a t least 100 nodes, four times more nodes brings four time more accuracy. density, moments, energy, turbulent kinetic energy, turbulent dissipation rate.

3.1

Flat plate flow

We start with the capture of -a part of- a boundary layer by computing a supersonic flow past a flat plate (with the wall law, only the fully turbulent part of the boundary layer is captured). We consider the friction velocity on a point after the stagnation point. Two strategies are compared: the isotropic one and the anisotropic one. We observe that the isotropic strategy indeed allows second-order convergence, but only for meshes finer than 15,000 nodes. The anisotropic strategy give second-order convergence already for 6,000 nodes.

3.2

Back-step flow

An interesting prototype of such flow is the famous back-step flaw (41. We concentrate on an important but difficult output, the abscissa of the reattachment point. Results are sketched in Fig. 7. We observe that the results are no so good since the convergence order is only slightly more than one. In fact such an output as the position of a vanishing point is rarely approximated with the accracy of other more standard output.

4

Concluding remarks

This paper discusses the research of mesh convergence for a class of industrialtype flow calculations. We claim that mesh adaptation is the key for secondorder convergence (or further) for such calculations. This is illustrated by a

822

Finite volumes for complex applications

Figure 5: Mesh convergence of friction velocity for the flat plate case with isotropic adaptation

Figure 6: Mesh convergence of friction velocity for the flat plate case with anisotropic adaptation (dots for second-order, dashes for first order slopes)

Adaptivity. tracking and fitting

823

Figure 7: Mesh convergence reattachment abscissa for the back step flow with anisotropic adaptation few examples. For the plate flow, we illustrate that already fine meshes are necessary for observing second-order convergence. With ar~isotropicadaptation, the necessary fineness is less (by a factor two) than with the isotropic one. For the prediction of a reattachment point, even with mesh adaptation, second-order convergence is not obtained. We errlphasize that the model choosen is a n o n stiff one (from the approximation point of view) in accordance to the common opinion, and also as shown Carrau and Dervieux in 171. In despite of this, the size of necessary meshes for secondorder convergence is large. We can expect an amplification of this property for low-Reynolds modeling and of course 3D calculation. In both cases, the application of anisotropic mesh adaptation seems to us mandatory.

References [ I ] Palmerio, B. and Dervieux, A . Multimesh and multiresolution analysis for mesh adaptive interpolation. Applied Numerical Mathematics, 22 (1996). [2] Carre', G. An implicit multigrid method by agglomeration applied to turbulent flows. Conlputers and fluids, 26, No. 3, pp 229-320, (1997).

131 Castro-Diaz, M.J., Borouchaki, H., George, P.L., Hecht, F. and Mohammadz, B . Anisotropic adaptative mesh generation in two dimensions for CFD. Computational Fluids Dynamics '96, pp 181-192, (1996). [4] Look, P . H., McDonald, M . A. and Firmin, M . C.P. Pressure distributons and boundary layer and wake Measurements. AGARD AR 138, A6-1 A6-77, (1979).

824

Finite volumes for complex applications

[ 5 ] Laug,P., Borouch,aki, H. and George, P.-L. Maillage de courbes gouverne

par une carte de metriques. INRIA Rocquencourt, Rapport de recherche no 2818 (1996). [GI Dhatt, G. and Touzot, G. Une presentation de la methode des elements finis. Collection Universite de Compiegne. Maloirle S.A. Editeur (71 Carrau, A. and Dervieux, A. Application of a mesh adaptive capture strategy for solvirig 1D turbulent layers", Proc. of the Fourth Internatioonal Conference on Numerical Grid Generation in Computational Fluid Dynamics arid related topics, Swansea, UK, 6-8 april., 1994, N.-P. Weatherhill, P.-R. Eisernan, .J. Hauser, J.-F. Thompsori (eds.), Pineridge Press Ltd., 513-526, Swansea, Wales (1994)

Figure 8: Obtaining the adapted couple (mesh, solution): Better adapted meshes, with (about) the same number of nodes and new flow solutiori on these meshes are iteratively rebuilt until mesh and solution do not vary much.

Multidimensional Fully Adaptive Finite Volume Schemes for the Numerical Simulation of Stiff Combustion Front Propagation in Condensed Phase

A. Aouji L IMHP- C N R S Avenue JB Clement 93430 Villetaneuse France

A mathematical modelling of multidimensional solid-solid combustion propagation based on a nonlinear reaction-difjusion equation is presented. A fully adaptive, finite-volume scheme on a structured non uniform mesh is detailed. The computational efficiency of this numerical scheme is discussed in the case of several multidimensional simulations. ABSTRACT

Key Words: Reaction-Di~usionEquation, Adaptive Finite- Volume, Adaptive Time-Stepping, Phase-Change Modelling, Solid-Solid Combustion.

1. Introduction

We consider the mathematical modelling of the ignition and the propagation of stiff combustion fronts in SHS ( Self-propagating High Temperature Synthesis) process of Ti + C -+ TiC in various geometries - slab, cylinder and parallelepipedic volume-, by a reaction-diffusion equation coupled with a stiff differential equation for the kinetics . This solid-solid reaction propagates inside a sample (made of a stoechiometric mixture of micronic titanium and carbon particles) if sufficiently enough heat is supplied on a significant part of its exterior surface. We will focuss in this paper on the description of the numerical scheme we have designed and the presentation of some results concerning the propagation of a solid-solid combustion front. In section two, we present the mathematical modelling. Section three is devoted to the description of the temporal discretisation method. Section four outlines the main features of the finite volume scheme applied to the enthalpy balance. Section five details

826

Finite volumes for complex applications

the conservative, spatial adaption procedure. Section six compares the computational cost of the finite-volume schemes in one, two and three dimensional numerical simulations of the propagation of a stiff combustion front. 2. Mathematical Modelling

2.1 Chemical kinetics

We denote by 1(T - T a p )the Heaviside function, and consider the one-step exothermic kinetics whose rate follows an Arrhenius law for T i C + T i c , with an ignition temperature Tap.The conversion rate Y behaves like:

+

2.2 Enthalpy balance and boundary conditions The total enthalpy of the system, takes into account two isothermal phase changes. The first one occurs at T=TaB = 11661i' and the second one at T=Tsl = 194311' with the respective latent heat of fusion Lapand L S I .Denoting by To, a fixed reference temperature, the enthalpy of the system exhibits a discontinuity at T = Tapand T = TSl,[BON 731, and writes

H ( T )=

L

Cp( 0 )dO + Lap.1 (T - T a p )+ L s l . l (T - T,,)

(2)

The governing enthalpy balance equation ca.n be cast in the form

. A H f dY a (dtp ' H ) = div ( k ( T ,Y ) grad T )+ pM T ~at ~ '

In cartesian coordinates, we have the following boundary condition

- k ( T , Y ) grad T.n = h ( T - T,)

+E.CT.

[T~ - T ,

(t)l]

(4)

The stiffness of the problem, both from spatial and temporal point of vue, requires an adaptive procedure to compute accurately and at a low cost the shape and the velocity of the front.

Adaptivity, tracking and fitting

827

3.0 Temporal scheme

The temporal discretization is based upon a Backward Difference Formula of order 2 with a non-uniform timestep, which handles efficiently such stiff systems. This is a three-level scheme , second order accurate finite-difference approximation.

.4n adaptive procedure based upon cite9 is implemented to dynamically increase or decrease the time-step. The reaction is only active after the first phase transition a t 1166K, therefore during the first part of the process, below that critical temperature, the time step is around 0.1 s. In the second part of the process, when the reaction is active, it is decreased down to approximatively 0.001 s. We remark that the chemical kinetics equation can be written in a more general form as

dY = U (T) - Y.V(T) dt

(-&)

with U (T) = V (T) = koenp .l( T - T a p ) . We notice that an > O,Pn < 0, yn > 0. We have the "explicit" equation of the conversion ratio at the new timestep, which is iteratively solved by a fixed point Picard method.

4.0 Spatial discretization scheme

We emphasize below a three-dimensional finite volume discreti~at~ion principles. The reader is referred to [AOU 99B],(resp. [AOU 99C]), (resp.[AOU 99D]) for the detailed description of the scheme in one , (resp. two), (resp. three) dimensions and extensive discussion about numerical simulations.

4.1 Computational domain and numerical problem

We describe in this section the numerical methodology used in the threedimensional case with the computational domain fl = [0, L] x [O,1]x [0, H] plit i=I j = J k = K

into a finite number of parallelepipedic volumessuch that R =

U U U

;=I j = 1 k = l

fii,j,k,

on which the enthalpy balance is integrated. We assume having cell-centered

828

Finite volumes for complex applications

formulation, where a constant unknown is located at the center ci,j,k of the ] lengths parallelepipedic cell Ri,j,k = [xi, xi+l] x [yj, yj+l] x [zk, z ~ +, ~with ui = xi+l - xi , b = Yj+1 - Y j l C k = zk+l - zk,and volume mi,j = Jn,,,,, d a = ~ i . .ck 6 ~. The finite volume method is well suited for the discretization of such parabolic equations expressed in conservation form, with Neumann boundary conditions [GAL 921. We must smooth the Heaviside function used to express the phase change, and introduce the phase fraction of phase a@, denoted by fap E [0, 11, and the phase fraction of phase sl, denoted by fsl E [O,1]. We have rT H (T)= Cp (t)) dt) fa@ .Lap + fsl. Lsl (8)

ITo

+

The discontinuity of the enthalpy a t the phase transition t,emperatures Tapand Tsl is replaced by a small intervall over which the phase change occurs. This is achieved by defining T& = Tap- t , and T:~= Tap t , where t = 0.01 x Tmp is the phase change intervall. In fact, this method introduces a "nlushy zone", which is &-dependent. Assuming that the density is constant throughout the process, and omitting the details, we are led to the following formulation

+

aT &AHj dY p.Cp(T, Y ).- = diu ( k (T, Y ) g r a d T)+--p.Lap MTiC dt

dt

'

afap

dt

afsl

p.Ls1.-

at

(9) We are now able to formulate, the numerical problemby integrating the enthalpy balance over the time-space finite volume [t, , t,+l] x

8T

A ( T , Y )- dR dt at

=

diu ( B(T, Y ) g r a d T) di2 dt

A11 iterative update procedure for the liquid phase fractions f,p defined identically by:

and

fsl

is

It has been shown in [SHY 941, that an efficient stable update procedure can be devised by substituting the previous equations directly into the enthalpy balance to get an another relationship namely the T-based method update.

Adaptivity, tracking and fitting

829

4.3 Laplacian discretization

We apply Green's Formula, an denoting t ain

div (B"+' grad T . " )

Bn+l

= B (Tn+',Y n + l ) ,, we ob-

di2

nt,,,k

We must now, discretize the integrals and evaluate the thermal conductivity a t the interface between cell i2i,j,k and i 2 i + l t j , k . We apply the mean formula, and evaluate the integrand at the enter of the cell R i j,k.We compute the diffusion fluxes at the interface between the cells fli and R i + l , j , k by a centered approximation,first order(resp. second order ) accurate on a nonuniform (resp. uniform) mesh. This requires the computation of the distance between the centers of two adjacent cells. An important issue, is the computation of the thermal conductivity at the interface between the cells i 2 i I j , k and a , + l , j , k . We use the continuity of the diffusion flux at the interface to write an harmonic mean relationship for the thermal conductivity a t the interface. This ensures the conservativity of the diffusion flux across the interface.

4.4 Matrix formulation

After some simple, but rather tedious algebraic calculations, we are led to matrix formulation m i , j , i .A:::. -

-

[

an

.T?'' + Pn . q y j , , + y n ~,yj,,!]

+

(I3)

m i , j , k . c Z ~ ~m l i,j,k.O:L;

+E:::. +~ ~ y j t ; .

- q;;,!) ,k

- T;;;)

+u::;. (..:;it- T"l , l , kl )

w;::. (q;'; - q:;;j,k - s;;; . - T?;':~

-

- D;;:.

&;; k; -;:7~ -

,k

1 1

We get an heptadiagonal , nonsymmetric, strictly dominant matrix which is inverted by the simple straightforward Jacobi iterative method.

830

Finite volumes for complex applications

5.0 Spatial adaption s c h e m e

In order to maintain the structure of the mesh, we consider an extremely simple, straightforward, easy to implement adaptive procedure based on a finite volume discretization of the reaction-diffusion. In multi-dimensional simulations, it is applied in the same way, successively to the x,y,z or r,z directions. It has the advantage, of using the same matrix solver throughout the numerical simulation. We describe here the main features of this method along the x-axis in the three-dimensional case. We assume given a parallelepipedic cell Qi,j,k, and compute an adaption criteria to decide whether t o refine or unrefine the finite-volume mesh.

5.1 Adaption criteria We assume given cP, (value of the criteria for unrefinment)and cPM (value of the criteria for refinment). We compute the adaption criteria, based on the curvature of the temperature solution, along the x axis, where a and ,B are given positive weights, and decide wether to refine or coarse the mesh.

5.2 Refinmerat procedure If @;is greater than Q M ,then we split Slij,k U Slitlf,kinto three equal volumes, along the x-axis, compute the new value 6 of the field U ( for example TI Y ), at the second cell such that the field U is conserved by:

We have a linear, barycentric combination of Uiand Ui+i. The sum of the weights affected t o UiandUi+~isequal to 1 and we have the following property. 2ai - ai+l 2 0 2ai+i - ai 0

>

u -ai+i 5 ai 5 2.ai+l 2

-

U U, E [min (Ui, U i + l ) , max(Ui, Ui+l)]

(16) 5.3 Unrefinment procedure

If Qiis smaller than @,

, then we fuse !2i,j,k U Qi+l ,jIkintoone single volume,

Adaptivity, tracking and fitting

83 1

A

along the x-axis, and compute the new value Y )by:

U of the field U ( for example T,

A

We have a linear convex combinaison and therefore

U;E [min (U,, Ui+l), max (Uill J ; + l ) ]

6.0 N u m e r i c a l s i m u l a t i o n s a n d discussion

All the numerical experiments presented in this paper were conducted on a Power Challenge and on the Origin 2K of Crihan. The various programs were coded in C programming language and were run in 64-bit precision. We present a table of the computational gain obtained by the use of adaptive methods. Id

N=100 nodes

N=200 nodes

N=400 nodes

N=100 nodes,At adapt

N=100 nodes,At adapt

N=100 nodes,At adapt

N=100 nodes

N=200 nodes

1053 s

9033 s

K=100 nodes,At adapt

N=200 nodes,At adapt

N=400 nodes 13779 s N=400 nodes,At adapt

269 s At adapt,,Ax adapt 240 s N=100 nodes 10530 s N=100 nodes,At adapt 2955 s At adapt,,Ax adapt 2500 s

1060 s

6510 s

N=200 nodes

N=400 nodes

32000 s N=400 nodes,At adapt 9140 s

K=400 nodes,At adapt

4" At adapt,Ax adapt 2d

3d

150 000 s 42 000 s

F i g u r e 1. Temperature profile along X axis F i g u r e 2 . Te,mperature profile along X , Y- axis

7.0 C o n c l u s i o n a n d p e r s p e c t i v e s

A variety of simula tion codes in (ID, 2D and 3D) has been written to investigate numerically the relationship between heat supply - surface of application and combustion front propagation[AOU 99E]. Work is in progress to determine the most efficient matrix solver in two and three climensions. Acknowledgment Extensive numerical computations were performed on Origin 2K1 through grant 199009 from CRIHAN.

832

Finite volumes for complex applications

[AOU 99A]

AOUFI A., VREL. D., PETITET J . P . , "Modeling of SelfPropagating Synthesis of T i c " , Intern. Journal of SelfPropagating Hagh-Temperature Synthesis, 6, 1, 41-53, (1997).

[AOU 99B]

A.AOUFI," Self-Adaptive Finite-Volume Scheme for reactiondiffusion equations with Phase Change. Application to the numerical simulation of one-dimensional self-propagating fronts", submitted to Journal of Computational and Applied Mathematics. A.AOUFI, "Fully Adaptive Finite-Volume Scheme for reaction-diffusion equations with Phase Change. Applicatioil to the numerical simulation of bidimensional self-propagating fronts", submitted to Int. J . Num. Meth. Eng.

[AOU 99D]

A. AOUFI,"Three Dimensional Fully Adaptive Finite-Volume Scheme for Reaction-Diffusion Equations with Phase Change. Application to the numerical simulation of self-propagating fronts", submitted to Applied Numerical Mathematics. A. AOUFI,D. VRELA N D J - P PETITET," Bidimensional Numerical Analysis of Critical Parameters for Stable Combustion Synthesis Study of T i c " , Gordon Research Conference on Oscillations and Dynamic instabilities in Chemical Systems, June 6-11, 1999, I1 Ciocco, Barga, Italy. Accepted for Presentation (Poster).

[BON 731

BONACINA C., COMINI G . , FASANO A . A N D M . PRIMICIERO ,"Numerical Solution of Phase-Change Problems", Int. J . Heat Mass Transfer., 16, 1825-1832, (1973).

[GAL 921

T . GALLOUET, "M6thodes des Volumes Finis", Collection Problkmes No11 Linkaires Appliquks, CEA-EDF-INRIA, 2830 Octobre 1992.

[SHY 941

W. SHYYAND M . M . RAO," Enthalpy based formulations for phase change problems with application to g-gitter", AIAA paper No. 93-2831.

[VER 941

VERWER J . G . "Gauss-Seidel iteration for stiff odes from chemical kinetics", Siam J . Sci. Computing. Vol 15. No. 5, pp. 1243-1250, September 1994.

[VRE 951

VREL D., LIHRMANN J . M . , PETITETJ . P . , "Synthesis of titanium carbide by self-propagating powder reactions: part 1, enthalpy of formation of TIC", J. of Chem. and Eng. Data, 40, 280-282, 1995.

Mathematical and numerical modeling of a two-phase flow by a Level Set method. Sandra ROUY1 and Philippe HELLUY2 DCN INGENIERIE C E N T R E SUD S / D LSM B.P. 30 83800 Toulon Naval F R A N C E I S I T V - Laboratoire Mode'lisation Nume'rique et Couplages B.P. 56 83162 La Valette du Var CEDEX F R A N C E

ABSTRACT This study is devoted to the numerical simulation of a two-phase pow. Because of its formal simplicity, a Level Set method has been chosen. The interface is thus located by the zero level set of a smooth function which is convected at the speed of the fluid. A conservative system can also be written which is solved by the Finite Volume method. Special care is needed for the pressure law at the interface between the two fluids. Some numerical results are presented. Key Words: two-phase Bow, finite volume, level set, conservation laws, Roe's method

1 Introduction The purpose of this study is t o describe the separated and time-dependent gasliquid flow which appears in the cooling chamber of a gas generator where the temperature can reach 2500 K and the pressure 180 bar. The time scale is about one microsecond. We have two objectives. The first one is t o follow the evolution of density, velocity, pressure and temperature in both fluids. The second one is to accurately capture the shape of the interface separating the two fluids. The aim of this research is also to implement a simple method in order to achieve these goals. The most common model for describing two-phase flows is the two-fluid flow model obtained via a space and/or a time average of the local instantaneous equations of conservation laws for each phase (ref [SAIN 931). These equations have then to be coupled with some transfer terms between the phases. This model is especially concerned with dispersed two-fluid flows because the geometry of this type of flows allows a formulation of these transfer terms. In this way a system is obtained which can be easily solved.

834

Finite volumes for complex applications

On the contrary, when dealing with separated two-phase flows, it is more judicious to use an interface tracking method. Two different approaches are possible: the lagrangian approach and the eulerian approach. In the lagrangian methods the interface is located by marker particles (ref [UNV 921). These methods are accurate and close to physical phenomena. However they require a lot of particles when the interface is complex. Moreover regridding algorithms must be employed in order t o properly track the interface. The eulerian methods are divided into two categories: the VOF methods and other related methods (ref [HIR 811) and the LEVEL SET methods (ref [MUL 921). The Volume-Of-Fluid methods introduce a new variable: the volume fraction of one of the fluids defined in each cell of the grid. This function is equal to 1 if the cell is filled with fluid and equal to 0 if it is empty. For a mixed cell which contains both fluids: the VOF function lies between 0 and 1. A specific numerical process is used to compute the transport of this quantity and to ensure the conservation of both liquid and gas phase mass. This process requires to reconstruct the interface a t each time step. Different techniques of reconstruction, more or less complicated, have been proposed. These methods are especially used for incompressible fluids with free surfaces. In the Level Set methods the interface is represented as the zero level set of a smooth function 4 defined on the entire physical domain and satisfying a transport equation. This function is initialized as the signed distance from the free surface. In this formulation the interface can merge or break up with no special treatment. It presents several advantages: there is no need to reconstruct the interface or to regrid the neighbourhood of the discontinuity. Moreover it can be easily generalized to three dimensions. For these reasons, we have chosen a Level Set method. This type of method seems to be the simplest way t o treat the interface. In all the methods described above, it is very difficult and sometimes impossible to deal with two-phase flow with a large density ratio. Our main concern is to overcome this difficulty.

2 2.1

Governing equations Equations of motion

We consider all phases as a single fluid which is supposed t o be compressible and inviscid ; we neglect gravity, phase changes and surface tension. The fluid is then governed by Euler equations:

where p is the density of the fluid, d the velocity, E the specific total energy, P the pressure and e the internal energy with the relation E = e $u2.

+

Adaptivity, tracking and fitting

2.2

835

The level set formulation

The interface r is then located by the zero level set of 4 (ref [MUL 921). I? is defined by r = {x/4 (x, t ) = 0). The level set function is positive in the gas and negative in the liquid. Hence we have:

> 0, ifx in gas < 0, ifx in liquid Since the interface moves with the fluid, q5 satisfies the transport equation 4t + + d . V 4 = 0. This equation is not conservative. In order to have a conservative equation, we consider quantity p4 which satisfies the conservative equation:

2.3

Closure of the system

The system is thus the following:

In order to close this conservative system, an equation of state is needed which must be relevant to the different states of the fluid. This is the difficult point of the modeling. We use the ideal gas equation of state for the gas and the Stiffened-gas equation for the liquid (ref [COC 961). Therefore the pressure depends on the density, the internal energy and 4. The equation of state is then:

P = P (P, e l 4) = a (4)Pe + 0(4) P = (y - 1) pe with y = 1.4 if 4 > 0 P = ( N - 1) pe - NP, with N = 5.5 and P, = 4.9211538 Pa if

4 <0

We can already notice when internal energy is small, the Stiffened-gas equation provides negative pressure. We shall see later that this is one of the major difficulties of our model.

3

Resolution of the system

We solve system (1) in two dimensions by the Finite Volume method which is well-adapted t o the equations of conservation. A first order Godunov-type method which consists in solving Riemann problems between cells was chosen as a starting point for investigations (ref [GOD 961).

836

Finite volumes for complex applications

At the interface between two cells, we shall solve a problem of type:

+

Ut div f (U) = 0 U(x < 0) = UL U(X > 0) =

uR

T .

with f (U) = (fl (U) , f 2 ( u ) l T , U = (p, p u ~pu2, , PE, p41T , u = ( u l , u2) 1s the velocity of the fluid, f l (U) = ( p u l , ~+ ~ P, ? P U I U Z , ( P E P ) u l , ~ 4 ~ and 1 ) ~

+ + P, ( P E + P ) u2, ~

(U) = (PW, pu1u2, pu; 4 ~ 2 ) ~ Because of its simplicity we decided to use Roe's method. This method consists in a linearization of the above system that leads to solve (v is the outward normal to the interface): f2

Let us recall the main features of this method. A system of type (2) admits a Roe-type linearization, if for any (UL,UR) in E x E , there exists two matrices A1 (UL,UR) , A2 (UL,UR) , such that the three following points are verified: (i)

fi (UR) - f~(UL) = AI (UL,UR) (UR - UL)

(UR) - f 2 (UL) = A2 (UL,UR) (UR - UL) (property of conservativity) the Jacobian matrix of f l (ii) A1 (U, U ) = ,U (property of consistency) the Jacobian matrix of f 2 A2 (U, U) = (iii) for any direction u = (ul, u2)T E R2 such that Iul = 1, the Roe matrix A, (UL, UR) "in the direction of v" defined by vl A1 (UL, U~)+v2A2(UL, UR) has real eigenvalues with a complete family of eigenvectors (property of hyperbolicity). f2

In practice, the Roe matrix is determined by computing an averaged state between UL and UR. We apply a change of variables W + U(W) such that U ( W ) and f ( U ( W ) )are homogeneous quadratic functions of W and so we can write the following relation:

A, (UL,UR) = A,

(U) with

= U (W*) and W* =

(WL

+ WR).

In our case, we can prove the existence of the Roe matrix defined by (iii) where JZ, fi$)T with f i = the average state is given by: U = (fi,

m,

fiz,

$ (fi + 6) and T = (flzi+6in) f i + & for r = U I , u2, H, 4.

Adaptivity, tracking and fitting

837

Then Al and A2 write:

-

0 1 1--2 -uI2tZaIuI K(2-5) --u1~2 '112 ~(-~l++alfil~)

0

-

Al

(U) =

-

-(YUZ

-

z-m2

u1 -au1u2

0 (a+l)E

4

0

0

--

-

-4G

-

0 a

-

0 0 0

o

-

u1 -

The pressure law coefficient E is equal t o y - 1 in gas and to N - 1 in liquid. The main difficulty here is to properly calculate a in such a way that it produces a correct pressure at the interface between the two fluids. Several choices will be discussed in section 4. The eigenvalues and the corresponding eigenvectors of Roe's matrix are given by: X1 = u.u - c, A2 = A3 = X4 = 7i.u , X.5 = 7i.v C , 7-1

= 1

-

u

-

H

-

-

E

l

)

-

+

T

,

7-2

= (l,Ei,%,;

(24-2),o)

T

,

T

rg = (0, -u2, vl, u1% - VZG, 0) , rq = (0,0,0,0, l)Tand r5 = ( ~ , K + F U ~ , G + ~ ~ ~ , B + E U . U ~ ~ ) ~ . that is to say in

E is the sound speed and is equal to

+ P,) +f .

and in liquid i? = J N ( P gas c = enthalpy which satisfies the relation H = E

/p

, H is the

total specific

Roe's scheme associated to the linearization is then:

x*

represents the summation over all edges e of the control volume E,, n, where is the normal direction t o the edge and U," is the approximation of the average value of U (., t ) on the control volume. The numerical flux is given by: 4 (Up, Ujn, n e ) = A, (Up, Ujn) Ujn + A;= (UF, Ujn) U p A+ and A- are defined by the relation Af = TA*T-~ where A* = diag Xf = max(0, A) and A- = min(0, A).

(At),

For stability, a CFL condition has to be respected: E m a x IXkI

la&] 5 1

838

Finite volumes for complex applications

4

Results

In order to validate Roe's solver, we compute the well-known test case of the shock tube which reduces to a one dimensional Riemann problem:

There are three characteristic families corresponding to the distinct eigenvalues. We can explicitly compute the one-parameter families of shocks, simple waves and contact discontinuities. The family of shocks is computed thanks to the Rankine-Hugoniot jump conditions which are the following: ff

[PI = [PUI

a [pu] =

+

[P pu2]

a [PEI = [ ( P+ PE) ul 0

[ P ~ =I [ P 4 4

The families of simple waves and contact discontinuities are computed thanks to the Riemann invariants which are u and P for the contact discontinuities and s (the entropy ), u f a c and C$ for the simple waves (ref [SMO 941). We obtain via this resolution an interesting information: the level set function is equal to zero at the contact discontinuity. An example of the solution is shown on the following figure.

' PI

contact discontinuity

x As a first validation, we consider a two gases flow where the pressure laws are different. Density, velocity and 4 are represented on the following figures.

Adaptivity, tracking and fitting

density

velocity

pressure

level set function

839

We can see that the capture of the contact discontinuity is not satisfying. Moreover, this leads to oscillations (ref [COC 961). Another point is that we have to introduce an intermediate equation of state when # L and # R have different signs. We have considered two different equations: an interpolation between the two laws and an "upwind" law. This intermediate law must provide relevant pressures. This point becomes especially critical when we consider the gas-liquid flow. Indeed computations in this case are not successful. Moreover as we said previously, the water pressure may become negative when the internal energy is too small. This adds another difficulty.

840

Finite volumes for complex applications

5

Conclusions

The level set method is a simple method to track the interface. Indeed we obtain a conservative system which can be easily solved by the Finite Volume method. Moreover, no special treatment is needed for tracking the interface. However in the case of a gas-liquid flow, we have t o overcome the fact that we have two different types of equation of state. An intermediate law has to be introduced, which keeps physical sense. Moreover, the Stiffened gas equation has to be modified t o avoid negative pressure. Finally, in order t o properly capture the contact discontinuity, a second order scheme a t least must be employed. Some results with a gas-liquid flow will be presented during the symposium.

6

References

[SAI 931 SAINSAULIEU, L.-An Euler System Modelling Vaporising Spray-Dynamics of Heterogeneous Combustion and Reacting Systems,1993,Vol.l52,pp.280-305. [UNV 921 UNVERDI, S.O. and TRYGGVASON, G.-A Front-Tracking Method for Viscous, Incompressible, Multi-fluid Flows-Journal of Computational Physics- 1992,Vo1.100,pp.25 37. [HIR 811 HIRT, C. W. and NICHOLLS, B.D.-VOF Method for the Dynamics of Free Boundaries-Journal of Computational physics,l981,Vol.39, Ni.l,pp.201-225. [MUL 921 MULDER, W., OSHER, S, and SETHIAN, J.A. -Computing Interface Motion in Compressible Gas Dynamics-Journal of Computational Physics,l992, Vo1.100,pp. 209-228. [COC 961 COCCHI, J.P., SAUREL, R. and LOR AUD, J.C.-Treatment of Interface Problems with Godunov-Type Schemes-Shock waves,l996,Vol.5,pp. 347-357. [GOD 961 GODLEWSKI, E. and RAVIART, P.A. -Numerical Approximation of Hyperbolic Systems of Conservation laws-Applied Mathematical Sciences,l996- 509p. [SMO 94) SMOLLER, 3. -Shock Waves and Reaction-Diffusion equations- SpringerVerlag,1994-632p.

Multiresolution analysis on triangles: tion t o conservation laws

applica-

A. Cohen, S.M. Kaber, M. Poster Laboratoire d'analyse nume'rique Universite' Pierre et Marie Curie (Paris 6), France http://www. ann.jussieu.fr

A multiresolution algorithm is coupled with a finite volume scheme t o solve scalar bidimensional conservation laws. The originality resides i n the adaptivity of the multiresolution decomposition, which takes into account the possible appearance of discontinuities and their displacement. Numerical simulations on triangular meshes point out the advantages of the method i n terms of CPU and memory costs. ABSTRACT

Key Words: Multiresolution - finite volumes - adaptive scheme.

1. Introduction

We are interested in this work in solving conservation laws on polygonal domains. It is well known that such equations can develop localized discontinuities in finite time. In such areas where the solution is not smooth a fine resolution is necessary and furthermore, higher order schemes for flux computations are necessarily nonlinear - including for instance E N 0 reconstruction. These are very costly techniques and it seems reasonable to use all available information on the local smoothness of the solution to decide whether they should be used or not. This can be done within multiresolution framework. The original idea of combining the advantages of this method - data compression, smoothness indicators - into conservation laws solvers in order to reduce the number of flux computations is due to Harten [Har94]. Initially implemented in one dimension, it was then extended to bidimensional cartesian grids, and smooth deformations of rectangular grids, (see references in [CDKP99]). Preliminary approach for unstructured grid is explained in [Abg97]. In the case of triangular meshes, which are more flexible in modeling complex geometries, a complet,e implementation with a detailed analysis of the encoding/decoding algorithm can be found in [KP99, CDKP991. In this former approach, the solution is represented everywhere on a uniform fine grid and the multiscale analysis is used to speed up the flux computations in the smooth areas. What we propose here is a fully adaptive scheme which

842

Finite volumes for complex applications

make use of the compressed solution - encoded at each time step by its most significant coefficients. This approach gives way to new difficulties, for instance in the analysis of the stability and precision. These points have been studied in detail in the one dimension case in [CKMP99] and are currently under investigation in the triangular meshes case. The outline of the paper is as follows: we recall in section 2 the multiresolution and the finite volume algorithms for triangular meshes which are then combined to produce the adaptive scheme presented in section 3. Numerical simulations and analysis of performances on tests cases follows in section 4.

2. Multiresolution analysis and finite volumes

We briefly describe a multiscale transformation of a function described by its mean - and not point wise as usual - values on a triangular mesh. The equation is defined on a polygonal domain fl which is discretized by a coarse grid R0 made of No triangles. A hierarchy of nested grids Re (0 5 C 5 L) is built. The grid Re+' is obtained by dividing each triangle of Re into four smaller triangles by joining the three edges midpoints. Triangles from the level C are denoted by Tf for 1 lc 5 Ne = 4 e ~ o We . denote by iif the average of a function u on the triangle Tf and iie = (iif):~~. The mean values tie+l of a function u being given on the fie+' grid, the mean values on the coarser grid fle can be computed by

<

where T$' are the four triangles of Ref' obtained by subdividing the triangle T!. The center triangle is denoted by T$'. The relation (1) defines a projection operator ?:+' from the resolution level e 1 on the level 4 by p;+'iie+' = iie. Multiscale decomposition relies on the existence of this operator as well as a prediction operator Q$+l from the resolution level e to the level e 1. They should be compatible in the sense that P;+'&:+~= I d . The predicted values iie+l = Qe,+,tie are approximations of the exact mean values iie+' obtained by local linear combinations of values at level e. Details on every triangle are defined as the differences between exact and predicted values df,j = G::' - iifS1, for j = 1 , . . . 3 . The multiscale representation of tiL is denoted by f i M R = (iio,8 , . . ,dL-l). Going from iiL to iiMR and reverse is achieved through wavelet-type algorithms described in [CDKPSS]. The two types of data representation have the same memory requirement. However, in the multiscale representation case, the amount of data can be reduced by thresholding the sufficiently small details. More precisely this consists in setting df,j = 0 wherever I d i j 1 is smaller than a level dependent tolerance E l . We refer to [CDKP99] for justification. In the same reference, the stability and

+

+

Adaptivity, tracking and fitting

843

of reconstruction operators are studied. This leads t o

We now recall how the numerical treatment of the 2 0 scalar conservation law dtu div f(u) = 0, (3) can be improved within the multiresolution framework. The unknown u(x, y, t ) is defined on a polygonal set R , for t > 0. An initial condition u(x, y, 0) = uo(x,y) is imposed as well as boundary conditions on the edges of R. We first consider that equation (3) is solved by a finite volumes scheme on the fine grid R L . The costly step of this scheme is the flux evaluation 2 ) f ' n x 1 div f(R)dxdy. Since the solution can present discontinuities - even

+

with s&oth initial data - the computations may have to be locally very precise. On the other hand, the point wise evaluation of a function known by its mean values requires a reconstruction step (using the operator R in the flux evaluation, not t o be mistaken with the prediction Q:+, defined by ( 2 ) ) . This step can also be very costly if E N 0 techniques are used for instance. It is therefore crucial to limit the number of these flux computations and this is where multiscale analysis comes into play. In Harten's framework, the details provided by the multiscale decomposition of the solution are used as criteria to choose the flux computation method as follow. Starting from the coarsest level where fluxes on all triangle edges are precisely computed. The levels are explored in turn: wherever details are large, fluxes are computed using pre1

cise and costly numerical flux approximations D: = p lTkl

1

,f

j

lr:,ilfL,, where

1rf;Llf;t;with fh denoting the numerical flux across the

:

rkcr;,, edge rk, (of length Irk 1) between two triangles belonging to the finest grid R L . Elsewhere, the mean values of flux divergences are computed by interpolating the same quantities on the immediately coarser level, using the prediction operator This algorithm was successfully implemented in [KP99, CDKP991. It leads to a CPU reduction by a factor 2 to 3 depending on the test case. Its precision is governed by the precision of the underlying finite volume scheme on RL since at each time step the time evolution of the solution is performed on this grid. Therefore the gain in CPU time remains linked to this discretization,

844

Finite volumes for complex applications

and there is no memory saving. A way to overcome these two limitations is provided by a fully adaptive scheme in which the solution is computed on a nonuniform time dependent grid. In the next section we modified the initial scheme in order to compute the solution on such a hybrid grid, fine where large fluctuations are foreseen and coarse in the smooth areas. We refer t o [CKMP99] for a detailed analysis of the algorithm in the one dimensional case. We are principally interested here in showing the potential of the method in the case of multidimensional grids - and more precisely triangular ones. 3. Fully adaptive schemes We introduce a hybrid grid 3t: c ueRe defined at time step t , by selecting subsets out of each Re. We will show how this grid is built in such a way that it reflects the smoothness of the solution. Since this regularity is time dependent, the hybrid grid will have to evolve in time, this will also be described. Eventually we will explain how the finite volume scheme is performed on such a grid, so that both the solution mean values and the details in the multiresolution analysis evolve correctly in time. We first describe how 3t: is built using the initial data uo. It is initially set equal to 51'. Until the maximum level of resolution if reached, each triangle of ?i is:tested to decide whether its subdivisions should be added to 3t:. The criterion is the comparison of the mean value of the solution on the four subdivisions with the values predicted by interpolation A smoothness indicator R:, associated t o a triangle T:, is activated (R: = 1) if one of the four details di,, is greater in absolute value than the threshold tolerance E I . The triangles whose indicators are activated are actually divided into four triangles who are themselves included into the computing grid 3t:. Another requirement is t o preserve the grid nestedness. As a corollary we must always be able to interpolate a t the finer level. This has two consequences: If a triangle T: is divided because = 1, all its neighbors on the same level C are included in 'HZ, even if their smoothness indicators are not activated. An auxiliary grid G: containing RfI, is defined: If a triangle belongs to the grid 31: for one of the two previous reasons, then all its neighbors a t level C belong to G,O. On triangles belonging to G,0\3t: no actual computation takes place and the mean values of the solution will be estimated by Q:+,. When the initial data is given by its dual multiresolution representation, pathological cases with non zero details only on the fine level can occur. To handle such cases, all the triangles should be actually divided up t o the finest level, even if the details are negligible a t a coarser level, and the thresholding should be done iteratively starting from the finest level. This is in practice very difficult t o realize without giving up all the advantage of the multiresolution in terms of memory savings. In the test cases we have studied so far, the initial solution is given by an explicit expression, which ensures that the tree structure of ?i: is achieved by the previous algorithm which explores and refines the levels

Qi+,.

RE

Adaptivity, tracking and fitting

845

starting from the coarsest. The hybrid grid may have to be modified a t each time step in order to follow the discontinuities propagation and capture shocks formation. To this effect, we define a larger grid '??:+I, containing 'Id:, whose time evolution can be predicted. We use here the hyperbolicity of the problem ensuring that if the CFL condition is respected the discontinuities do not propagate further than one space step in one time step. To foresee possible propagation in any direction the term of neighbors of a triangle is extended to all triangles sharing a vertices with it, and not only the three triangles having a common edge and participating in the interpolation scheme Q:+,. A corresponding grid containing ??:+I as well as all its neighbors a t each level is also defined. At each time step the mean values of the solution on the coarsest grid and the details corresponding t o triangles in 3-12 are modified (as detailed in algorithm 1). Whenever one of the four details of a triangle of 3-1: is higher than the level tolerance, its smoothness indicator and that of all its neighbors is activated. If a detail is higher than a still larger tolerance, the corresponding indicators a t the finer level are also activated. These operation can make triangles previously in &\%: part of A:+' and even create new subdivisions a t the finer level. The nestedness of the new grid '@+l is ensured as in the initial step - which Eventually some triangles of can induce the creation of subdivisions in G:+l. @\??: may now be useless and therefore removed. An important difference with Harten's algorithm is that the computing grid Ifi: is incomplete. In other words, the depth of resolution is spatially variable and defined for a triangle T: of level e by a local depth i. This has direct consequences on the precision of scheme. In both approaches the precise numerical flux evaluations are performed only on the interfaces of the hybrid grid. In Harten's case the solution is everywhere available on the finest grid so the maximum precision can be achieved. In the adaptive case, the solution could theoretically be computed down to the finer level by applying the &:+, as many times as necessary starting from i where it is actually known. This would be completely inefficient in practice since it would destroy all the benefits of the multiresolution. On the other hand, brutal use of the available mean values on the hybrid grid t o compute the fluxes leads t o an order of approximation governed by the coarser level. This problem is explained in details in the one dimension case in [CKMP99] and we simply summarize here the main conclusions. We have so far two ways t o tackle this difficulty: In one dimension it is possible to reconstruct directly - that is linearly mean values a t the finest level from values at a not immediately coarser level as soon as the details are null on a sufficient number of consecutive intervals on this coarse level (four intervals in the case of linear reconstruction). Taking into account the areas of high gradient where actual decoding must be performed leads to a complexity in Nlog(N) where N is the cardinal of A:. The second method consists in using a higher order E N 0 type scheme to compute the fluxes at points where the solution is known by its mean values on

846

Finite volumes for complex applications

a coarse or intermediate grid level. In that case the complexity is that of the hybrid grid multiplied by a fixed coefficient which depends on the complexity of the E N 0 reconstruction. The precision is that of the coarse grid to a power related to the E N 0 reconstruction order. From the parameter study performed in [CKMP99], it seems that the first method is the most efficient one if we compare the performances for a given accuracy. However its extension to the triangular grid case seems more complicated. In this preliminary implementation we have used the same EN0 type reconstruction as in [CDKP99]. We summarize in the following adaptive algorithm ( I ) , how the finite volumes scheme is used to compute the time evolution of the solution and the details of the multiresolution on the hybrid computing grid fi;.

Algorithm 1 Finite volumes + adaptative multiscale Initialization of 3-12 and uLR Loop on time step n = 0,1, ... '

Determination of '@+l Partial decoding. Evolution o n the coarse grid: VT; v:'" = $ E J & IrP,jl& with ,$3,n+l k

f1°

E

n fi:+l

lri,jlqj:= Cracrr,, Irilfi

- iiO,n A~v~J"'

Loop o n levels 4! = 0 p L - 1 : v TL E Re n / ~ k,.il + -e+i,n

-

&C

j ve+i,n k,j dl,n+l - de,n k,j - k,j

I

e+i

l &:+I

e+i

I f ,

with

e+i

Irk,

-!+I

..-

Er~crk>l 1~L1.f;

(e:+lae>n)k,j at ( 8 ; ; l . n

- fje+l,n k,j

4. Numerical experiments

To illustrate the performances of the algorithm we first present the test case of a linear flux on an initial discontinuous solution. We choose the function uo(s,y) = 1 if < 0.1 and zero elsewhere on the unit square with periodic boundary conditions. Performances in term of CPU time, memory requirements and precision are evaluated for the adaptive multiresolution scheme and the reference finite volumes scheme on the finest level. The numerical fluxes are computed with the second order E N 0 method already implemented in [KP99, CDKP991. It requires a small CFL number chosen here equal to 0.2 on the finest grid. There are 50 triangles on the coarsest level. We first show results corresponding to a multiscale decomposition on a maximum of 5 levels. In this case the finest grid on which the finite volumes reference solution is computed counts 12800 triangles. The direction of propagation is parallel to

Jw

Adaptivity, tracking and fitting

847

the Ox axis and the solution is computed over one period of time. Figure 1. shows the initial solution and the solution after one period computed with the adaptive scheme on a maximum of five levels with a total of 1209 triangles. The shape of the solution is well preserved, and the finer levels are used only in the vicinity of the discontinuity. On the next group of figures 2, the performances of the adaptive scheme are displayed for different maximum number of levels. For each value of this parameter L, the standard finite volume solution on the finest grid - level L - is computed. CPU time and memory requirements along with the error with the exact solution in the L1 norm can be compared. The precision of the adaptive scheme in this case is very satisfying since the two error curves are always close to each other. Actually, in this test case, the source of possible errors is the discontinuity and in its vicinity both schemes are similarly discretized. As far as the memory requirement are concerned the advantages of the adaptive case are indisputable. We are actually restricted t o a few number of levels because the finite volume scheme will not run on our work station for a grid finer than the 5th level ( L = 4). We can predict that for 204800 triangles it would require about 13 hours of CPU. In this limit case L = 6, the adaptive scheme actually discretizes the solution on 2198 triangles, among which only 1228 belong to the finest grid and takes only 1.5 hour to run. One remarks that the CPU time does not depend linearly upon the number of triangles because some expensive overhead computing must be done which is inherent to the multiscale decomposition like the creation and the deletion of triangles. The adaptive scheme is nevertheless much faster than the standard finite volume scheme, all the more so as the number of levels in the multiresolution increases.

5. Conclusions A fully adaptive scheme based on the multiresolution decomposition of the solution is proposed to improve the performances of a standard finite volume algorithm. The mean values of the solution are computed on a time varying grid made of cells belonging to different levels of decomposition according to the local smoothness. The preliminary numerical tests which are presented exhibit very good performances in term of CPU and memory savings. The immediate future works consists in developing the vector case algorithm necessary t o handle gas dynamics. From a more theoretical point of view, we are currently studying the convergence of the scheme.

Bibliography

R. Abgrall.

Multiresolution analysis on unstructured meshes: application t o CFD. In Experimentation, modelling and conabustion. John Wiley & Sons, 1997.

848

Finite volumes for complex applications

Figure 1: Multiresolution solution on five levels at time t=O and t = l (one period)

Figure 2: Comparison of CPU times, memory occupation and precision for finite volumes and multiresolution algorithms [CDKP99]

A. Cohen, N. Dyn, S.M. Kaber, and M. Postel. Multiresolution schemes on triangles for scalar conservation laws. Submitted 1999.

[CKMP99]

A. Cohen, S.M. Kaber, S. Muller, and M. Postel. Adaptive multiresolution finite volume schemes. I n preparation, 1999.

[Har94]

A. Harten. Adaptive multiresolution schemes for shock computations. J. C o m p . Phys., 115:319-338, 1994.

[KP99]

S.M. Kaber and M. Postel. Finite volume schemes on triangles coupled with multiresolution analysis. CRAS 328, s6rie 1:817,822, 1999.

A local level set method for the treatment of discontinuities on unstructured grids L. Tran, R. Vilsmeier, D. Hanel University of Duisburg, FB7 / IVG, Germany E-mail: {linhbao, hjOOOvi, hj454ha)Qivg.uni-duisburg.de The paper is concerned with the treatment of discontinuous prob1e.m~ i n field simulations on arbitrary grids. The location and movement of the discontinuities are described by a level-set function with restricted, dynamic definition range. The physical transport of the discontinuities is decoupledfrom normalizing the level set function within its definition range and an error correction mechanism is employed t o minimize unphysical effects. The back influence from discontinuities t o the continuous part of the field simulation is provided by flux separation, where no sub-cell resolution is required. Object oriented programming based on MOUSE allows an arbitrary number of discontinuities for arbitrary physical effects within the same computation.

ABSTRACT

I i e y Words: discontinuities, jitting/tracking, local level set, 00-programming

1. Introduction

In the past decades remarkable progress was made concerning the efficiency and accuracy of capturing methods for the field simulation of problems with embedded discontinuities. Corresponding work was mainly motivated by computations of compressible flows where discontinuities together with continuous solutions are resolved by the same conservative method. In all capturing inethods the discontinuities are not represented as sharp jumps but they are smeared over a certain number of nodes, which leads t o artificial, intermediate states. In many problems, in particular where large scales have t o be resolved, these effects may be acceptable. However, if strong interaction between discontinuities and the surrounding field are present, these artificial states can remarkably falsify the solution. Typical examples for that are reacting flows, like detonations or flames. Front tracking enables numerically a representation of discontinuities as jumps and tracks them in space and time. Although the mathematical and physical principles of weak solutions of conservation laws are well known, the numerical realization is complex and has lead to different approaches. Among the different ideas for tracking a front the level-set approach [KER 88, MUL 92, SUS 941 was found to be well suited to develop a tracking concept on arbitrary grids.

850

Finite volumes for complex applications

2. G o v e r n i n g e q u a t i o n s and solution concepts

Consider the following general system of equations modelling a physical problem of interest:

"

a t J Q ~ v + ~ ~ L Z ~sdv A = V

A

1

(11

V

where Q, H and S represent the variables, fluxes and source terms respectively. The system of equations is solved on arbitrary grids employing a nodal finite volume method, figure 1. A discrete formulation of the above system reads as follows:

AQ -Ivd At

+ Resn,vd = 0

with:

Figure 1: Control volume

1 Resa,vd = Hi n'i AAi Vvd i Z l

(2)

where Vd is a control volume with the size VVdand the sum is carried out over all bounding segments. As underlying software, the MOUSE package [GLO 971, currently in development at the cite of the authors, is used. The following methods for front-tracking are, however, not yet available in the present public release. 3. M e t h o d f o r f r o n t p r o p a g a t i o n

An embedded discontinuity is a feature, that is one dimension below the actual problem considered. It is a point in a 1-D problem and a line or a surface in a 2-D or 3-D problem respectively. An obvious trial is to transport the discontinuity by a Lagrangian approach through the mesh. This approach works well in 1-D, but yields some problems in 2-D and 3-D, since the topology of the line or surface must be respected, which is not evident. To overcome this drawback, an Eulerian formulation can be introduced, yielding the well known level-set methods, 0. Before proceeding, we may classify the discontinuities in two distinct types: Passive discontinuities, which are simply transported by a given carrier speed. Referring to fluid dynamics, this is the speed of the fluid and examples are material interfaces, contact discontinuities or free surfaces. Active discontinuities, which are able to move relative to a carrier. Examples in fluid dynamics are shocks or combustion fronts (if described discontinuously). Comparing both types, the remarkable difference for a computational approach is, that the propagation speed of active discontinuities is only defined on the discontinuity, while the carrier speed governing the motion of a passive discontinuity is available everywhere in the field. 3.1 D e s c r i p t i o n of t h e location a n d m o t i o n of discontinuities

In the present work a level-set formulation is used on a fixed computational grid. The method introduces a continuous function G ( x ,[y, z ] ,t ) ,while the actual position of the discontinuity is given by a discrete iso-value, typically 0.

Adaptivity, tracking and fitting

85 1

T h e motion of the discontinuity can thus be described by a scalar convection equation:

where c'is the propagation speed of the discontinuity. Figure 2 sketches such a discontinuity on a grid, consisting of arbitrary elements. For this and the following subchapters some sets S are defined here. Points are referred t o by the letter P , edges by I< and levels by L. For example, a set of edges around a node i is referred t o as SIi'-P(i), a set of nodes in level 1 by S P - L ( l ) and the set of nodes neighbouring a node j by Sf'-P(j). Furtheron the level 1 of a node i or an edge k is defined as L P ( i ) or LIi'(k) respectively.

Figure 2: Discontinuity on a 2-D mesh and membership of nodes and edges to corresponding levels, P, s p q r n )and K , E S I q A ( m )

E

For the present paper, discrete values for G are stored a t the nodes of the mesh. It is however not required to define the function G in the whole computational domain, instead it can be described locally in two neighbourship levels only. The definition of a level for an edge k E SI<-L(m) reads:

V m*

< m-

1 ; SI<-K(k)fl SIi'-L(m*) = 0 ) and SIi'-I<(k)nSIi'-L(m - 1 ) #

0

and the edges hitting the front are of level 0. For a node i E S P - L ( m ) we write:

V ?n* < m

-

1 ; SIC-P(i) n SIC-L(m*) = 0 and

SIi'-P(i) n S K - L ( m - 1 ) # 0

?'he restriction to two neighbourship levels imposes, that for each step of motion, only nodes i E S P - L ( l ) can be overrun, if any. In this case, the function G can still describe the position between the first and the second level. After each time step, the definition range is adjusted accordingly. T h e above equation ( 3 ) is discretized in space and integrated in time. For a space discretization employing a finite volume method, propagation speeds are needed everywhere on the surface of the control volumes. Unfortunately, these are not available in the case of active discontinuities, thus requiring a corresponding extrapolation already a t the nodes i E SP-L(1). Due to this drawback, a differencing variant is preferred. The term Z.VG can be directly evaluated for all edges k E SIi'-L(0). T h e gradient V G can easily be computed on all nodes i E S P - L ( l ) employing a Gauss-type integration on the finite volumes or a least squares method as shown below. A simple directional upwinding is applied in the direction of the edges, combined with a central formulation for the transverse component due to lack of information. Since the discrete values for G are stored a t the nodes of the mesh computed a t the edges k E S K - L ( 0 ) a redistribution of the time derivative onto the appending nodes has to be performed. As the propagation speed is generally not available a t edges k 6 SK-L,(O), a discretization on these is per-

g,

852

Finite volumes for complex applications

formed with extrapolated speeds. This residual is however only required, if the corresponding node in the first neighbourship gets overrun. 3.2 Normalization of the level set function The precision of the method is directly related to the curvature of the Gfunction normal to the discontinuities. Therefore it is useful to provide a constant slope as far as possible. This is the aim of normalization. It is desired, that the function G satisfies the normalizing condition lVGl = C , where C is a constant # 0, typically chosen C = 1. This equation corresponds to the description of a sand hill, where the position of the discontinuity can be thought to be the outer cut line at constant altitude. To achieve normalization, several methods can be used. Modifying the basic equation (3) by adding a normalizing contribution controlled by a switching function S, with S = 0 for G = C , yields:

This formulation has shown to he less useful here. Although the original equation is restored for S = 0, the physical transport is falsified, since the nodes and edges for which S is applied are not located right at the discontinuities. Upon the experiences with the present work it is therefore preferable to separate the physical transport and the normalization completely. Assume that equation (3) has been solved for a time step. A normalization of the G-function can now be achieved with a pointwise iteration process for the nodes in the definition range around the discontinuity. The iteration process starts at the nodes of the first neighbourship level SPL(l),iterating up to convergence, and then proceeds from level to level in increasing order. This method corresponds to parabolic space marching although the problem is in fact hyperbolic. Within a level, the pointwise iteration is carried out by a two stage method, based on least squares. In a first step, the direction of the gradient VGn is computed, in the second step the functional value of Gn+l is set to satisfy the slope condition. Step 1: Consider the following least squares ansatz:

where

E,,, (i) = S P - P ( i )

n

(SP_L(LP(i))

u S P L ( L P ( i ) - 1))

(5)

>

with G i j = T j - Ti and a distance weighting exponent q 0. The formulation is one-sided due to the restricted consideration of neighbours in Epa,(i). Differentiation and minimization yields the following system (exemplary for 2-D):

C

1 lrsiij/r

j€EpOr(~)

(

( ~ x i j ) ' ~ x iAjy i j ) (Gi,z) A ~ i j A ~ i j( ~ ~ i j Gi,y ) ~

( G j - Gi) Aai,

C

=i€Epor(i)

16ij/9

(flyij)

allowing the computation of a one-sided gradient VG? at iteration level n.

Adaptivity, tracking and fitting

853

S t e p 2: Again with least squares, but now projected in the direction of the gradient V G r , the following ansatz is obtained:

where

EhYp(i) = S P P ( i )

n

u

SP-Ii(k)

k € S K L ( L P ( i ) - 1j)

Considering a @ h y p / d ~ ~ n=t 0 l )for minimization and setting C = 1 yields:

(8)

For the nodes i E S P - L ( l ) this approach must be modified, since the position G = 0 on the edges k E (SIq-P(i)nSI<-L(0)) crossing the discontinuity must be preserved. For these edges, * the actual neighbour node j on the opposite side of the discontinuity is replaced by a phantom node j* at the crossing position with G ( j * )= 0, see figure aside. Since the distance j 7. f j 1 IGZ2* 1 can approach 0 as the discontinuity moves across the mesh, it is useful to expand the fraction with the minimal distance /G,,,(4 found, thus replacing the terms by -, and these terms hold in the

i*;

I ;m,lq

xjIEhyPiz) >

IGnlanlq l i t 6 ,+ 0 Also 1 , since the addend mmln is contained. A d d i t i o n a l e r r o r correction: The above described normalization can still alter the position of a discontinuity. Since depending on the curvature, a shift of the front line appears, an additional correction method is required. For this, the exact position of the front before normalizing the nodes i E S P L ( 1 ) is stored and compared t o the position after the normalizing step. Considering a linear representation of G over the edge, an error correction is done, computing an increment AGk for the edges, that satisfies:

where Al,,,,, is the shift after the iteration in the first level is com leted. ) Since several edges k E SIC-L(0) may be appended to a node i E S P - ~ 6 the corresponding correction is taken as average. Although this additional correction is simple, its effect is very remarkable as shown in figure 3. 3.3 P r o p a g a t i o n s p e e d , back influence a n d o v e r r u n n i n g n o d e s In the previous chapter a propagation speed Zwas used, without stating how this is obtained. Considering a passive transport, the carrier speed is easy to

854 Finite volumes for complex applications

Figure 3: Passive convection of a circular line on a field with constant carrier speed. Cornparison with and without additional error correction. Computed solutions.

access. For active discontinuities, the propagation speed depends on the corresponding values at both sides of the discontinuity. Consider as example a shock, whose propagation speed can be computed upon the pressures at both sides. For the edges k SIC-L(0) the left and right states can be computed upon projected variables from the ending nodes. Since the position and motion of discontinuities can be treated as described, it is now interesting, how the discontinuity affects the surrounding fields. In the present work, a flux separation scheme was used, not requirering a sub-cell resolution. In this method the original control volumes of the mesh are preserved. Referring back to figure 1, each edge of the mesh carries a segment of two adjacent control volumes. In smooth regions, a projection from the nodes storing the variables to the cell interface is performed and a flux comof the fluxputed in a central or upwind manner. The flux is Figure 4: then used for both the adjacent control volumes. For ail edges k E SIC-L(0) this is no longer the case. The flux for the "left" side is computed using the "left" side projection, and the the flux for the "right" side with the right side projection accordingly, see figure 4. This method is very simple and can be employed on any grid. As a node of the mesh gets overrun by a front, the variables stored at the corresponding nodes switch to the other side. At present these new values are obtained by extrapolation from the the surrounding neighbours at the new side. Note, that this method does not ensure conservation. However, a fully conservative variant of the method is currently in development and expected to be available soon. -

4. C o m p u t a t i o n a l e x a m p l e s The above presented methods have been developped without any preference for a specific application. However, the examples below are related to fluid dynamics. The figures 5 and 6 show computed results for a shock tube problem where both, the front shock and the shear layer are tracked. The computation

Adaptivity, tracking and fitting

855

was performed on a 2-D unstructured mesh with triangular elements. As a second example, the inviscid, transient computation to steady state for a cylinder at Ma, = 2 is shown. As initial condition a straight shock is located in front of the cylinder. The computation is performed on the grid shown in figure 7. Figures 8, 9 and 10 show the tracked position of the bow shock, the corresponding subset grids and isolines of density for the initial condition, an intermediate and the converged solution.

expansion ..

...--.. i....

~ s o l ~ nof es

contact

shock

..

Figure 5: Shock-tube example: Isolines of density, subset meshes and location of shock and contact discontinuit~es1n1t ~ a condition l ( t o p ) ,after 300 time steps (below) and zoom of the contact discontinuity after 300 time steps (left). These figures are computed results.

- tntt~alcond!l~on

-lnldal condlbon loo timesleps

100 limesteps 200 timesteps 300 tlmssleps

- rI - y :-

2.0.

Figure 6:

-:>

4.0

.

2.0

.

Shock-tube example: cut through a 2-D solution at different time levels.

References

[KER 881 A. KERSTEIN,W. ASHURST,F . WILLIAMS:Field Equation for Interface Propagating in an Unsteady Homogeneous Flow Field. Phys. Rev. A , uol. 37, pp2728-2731, (1988).

[MUL 921 MULDER,OSHER,SETHIAN:Computing Interface Motion in Compressible Gas Dynamics. J. Cornp. Phys. , vol. 100, pp 209-228, (1992). P. SMEREKA, S. OSHER:A Level Set Approach for Comput[SUS 941 M. SUSSMAN, ing Solutions to Incompressible Two-Phase Flow. J. of Comp. Physics, vol. 114, pp 146-159, (1994).

[GLO 971 0 . GLOTH,R.VILSMEIER,D. HANEL:Object oriented programming for com~ u t a t i o n a lfluid dynamics, HiPer' 97, Krakow, Poland, (1997).

856

Finite volumes for complex applications

Figure 7:

Top: computa-

tional mesh

Figure 8: Series at top, right: Tracked position of the bow shock, initial (left), intermediate (middle), and final state (right). Figure 9: Series at right: Subset meshes, initial (left), intermediate (middle), and final state (right).

Figure 10: Series at right: Density isolines, initial (left), intermediate(middle), and final state (right).

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A Stabilized Version of the Wang's Partitioning Algorithm for Banded Linear Systems * Velisar Pavlov Center of Applied Mathematics and Informatics, University of Rousse, 7017 Rousse, Bulgaria

Abstract The parallel partitioning algorithm of Wang for arbitrary nonsingular banded systems is stabilized. Some numerical experiments (including random matrices) are presented.

Key words and phrases. Iterative refinement, parallel partitioning method, banded systems, perturbation. AMS(M0S) subject classification. 65G05,65F05,65Y05

1

Introduction

Banded systems of linear equations appear in many problems and are the computing time consuming kernels of various applications. The systems arise either directly, as in the difference approximations of ordinary differential equations, or after suitable rearrangement of equations and unknowns, as in finite element methods for elliptic problems. Such systems we can solve in parallel by the so called partition methods. A typical member of these methods in the case of tridiagonal systems is the method of Wang [8]. This method gives an efficient parallel algorithm for solving such systems. Full roundoff error analysis for the whole algorithm in the case of nonsingular tridiagonal matrices is presented in [lo]. Generalized versions of the partitioning algorithm of Wang for banded linear systems are presented in [2, 61. Full roundoff error analysis in this case can be found in [ll]. In this work it is shown that the algorithm is numericaly stable for some special classes of matrices, i.e. diagonally dominant (row or column), 'This work was supported by Grants 1-702197 and MM-707197 from the National Scientific Research f i n d of the Bulgarian Ministry of Education and Science.

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Finite volumes for complex applications

symmetric positive definite, and M-matrices. Unfortunately when the matrix (even though well conditioned) of the system does not belong to the above mention classes, the algorithm can breack down or behave poorly. In our paper we present a stabilized version of the generalized Wang's algorithm for arbitrary nonsingular banded linear systems. Let the linear system under consideration be denoted by

+

where A E Rnxn,which bandwith is 2 j 1. For simplicity we assume that the number of superdiagonals j is equal to the number of subdiagonals. The partitioning algorithm for solving (1) can break down when it is necessary to divide by numbers which are less than a certain limit 6. In such cases we improve the algorithm perturbing the inputs or intermediate data. But the result which we get is perturbed. In order to make the solution more accurate we use iterative refinement (see [3]). Hence, it is necessary to solve (1) several times with different right hand sides. A similar perturbation approach is used in [l]for a Strassen-type matrix inversion algorithm, and in [4] for a fast Toeplitz solver. The convergence of the iterative refinement is analysed in [7,91.

The outline of the paper is as follows. Section 2 presents the partitioning algorithm. In the next section we consider perturbations and iterative improvement of the solution. Finally, in Section 4 we present some numerical experiments (including random matrices) in MATLAB.

2

The partitioning algorithm

For simplicity we assume also that n = ks - j for some integer k, if s is the number of the parallel processors we want to use. Let us note that our assumptions are not essential for the consideration. We partition matrix A and the right hand side d of the system (1) as follows:

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861

where Bi E ~ ( " - j ) ~ ( ~ - j i) ,= 1 , 2 , . . . ,s , are band matrices with the same bandwith as matrix A , &,Ti are matrices of the following kind

whoseelementsa(i-l)k+l,~~~l ~ R j and finally

~

aj i k, , b i k , ~ ke R j X j , i =1 , 2,..., 3 - 1 ,

Now we define the following permutation

[ l : k - j , . . ., ( i - 1 ) k i - 1 : i k - j , . . . , ( s - l ) k + l : s k - j , k - j + l : k , . . . ,i k - j + l : i k , . . ., ( s - 1 ) k - j + l : ( s - l ) k ] , of the numbers [ I , . . . ,sk- j ] , and denote the corresponding permutation matrix by P . By applying this permutation to the rows and columns of matrix A we obtain the system

A=

A P x = Pd,

PAP^

=

All

A12

where All = diag{B1, B 2 , . . . ,B,) E ~ " ( " - j ) ~ " ( " - j ) ,

ak

Ck

0

."

0

. . . a2k

C2k

"'

0

A21 = 0

... a s - 1

C(s-l)k

.

"

here Aal E ~ j ( ~ - l ) ~ ~ ( " jA22 ) , = diag(bk, bak,. . . , b(,-l)k), E The algorithm can be presented as follows. Stage 1. Obtain the block LU-factorization

A=

( All

-421

A12 A22

by the following steps:

)

=

LU =

( All A21

0 Ij(s--1)

)(

Is(;-j)

R S

)

0

862

Finite volumes for complex applications

1. Obtain the LU-factorization of All = PILIUl (with partial pivoting,

if necessary). Here P1 is a permutation matrix, L1 is unit lower triangular, and Ul is an upper triangular matrix with diagonal elements u,(1), u2(1), . . . ,u (1) , ( ~ - ~ using ), Gaussian elimination (with pivoting, if necessary). 2. Solve AiiR = A12 using the LU-factorization from the previous item, and compute S = A22 - A21R, which is the Schur complement of All in A. Now let us notice that when we solve (2) it is necessary to divide by u,!l) for i = 1,. . . , s ( k - j ) . In this case if the blocks Bi (one or more) are singular, then at least one of the quantities u!') becomes very small or zero and the algorithm can break down, or behave poorly. In order to avoid this dangerous situation we propose to perturb them with 6, where b is sufficiently small. The implementation of this idea is presented in the next section. Stage 2. Solve Ly = d by using the LU-factorization of All (Stage 1). Stage 3. Solve Ux = y by applying Gaussian elimination (with pivoting, if necessary) to the block S.

3

The Stabilized Algorithm

As was noticed in the previous section the algorithm can break down, or behave poorly, when ujl), for i = 1,. . . , s(k - j ) and are zero or small. So, we can perturb them in such a way that it would be away from zero. The stabilization step can be summarized as follows:

if (1u!')1 < 6) if (\u!')l = 0) "jl) = 6 ; else ujl) = ujl) end end

+ sign(uj1))6;

In this way we shift ujl) away from zero. Hence, the algorithm ensures that we do not divide by a small number.

From the other side the obtained solution is perturbed. Then we apply the usual iterative refinement from [3], with some modification:

= 2; f o r m = 1,2, . . . ?-(m-1) = b - A X ( m - 1 ) ; (A + A) y(m) = r ( m - l ) . x(m) = x(m-') y(m); end x(0)

+

The difference here is that instead of A we solve perturbed systems with the matrix A+A, where A is a diagonal matrix with all such perturbations, and li: is the result of the perturbed algorithm before the iterative refinement is applied. We note that, when 6 = 6LZ (in double precision), in practice the perturbed solution is very close to the exact one and we need usually only one or two steps of iterative refinement, depending on what accuracy we require. Here by po we denote the machine roundoff unit. Taking into account [7] the condition of convergence of iterative refinement is Ccond(A)6 < 1, where cond(A) is a condition number of matrix A and C is a constant of the following kind

4

Numerical Experiments

Numerical experiments in this section are done in MATLAB,where the roundoff unit is po zz 2.22 x 10-16. The exact solution in our examples is x = (1,1,.. . , I ) ~by , Nit we denote number of iterations, and we measure two types of errors: 1. The relative forward error

where P is the computed solution; 2. The componentwise backward error (see [ 5 ] )

BE = max l
l(A5 ([All21 Idl); '

+

864

Finite volumes for complex applications

BE FE

Nit

6=0 cu cu 0

~5=lO-~ 6= 2.61 x 4.22 x 10-l5 2.13 x 10-l2 9.01 x 10-l4 2 2

6 = lo-' 1.42 x 10-l4 4.02 x 10-l3 2

Table 1: The forward and backward error and the number of iterations of Example 1, when E = 0

We stop the iterations when at least one of the following conditions is fulfilled:

2. Number of iterations is

> 10;

Let us consider the following examples. Example 1. Let A be a matrix of the following kind:

This matrix is very well conditioned. In our tests we set E equal to 0 and 10-12. Hence, the blocks B, become singular. We report the results with different values of 6 in Table 1 (when E = 0) and in Table 2 (when E = lo-''). When S = 0 we obtain the original algorithm without stabilization. We see that the original algorithm can break down (when E = 0) on account of dividing by zero, or behave poorly (when E = 10-12). At the same time the stabilized algorithm gives much better results. Let us note that at most two steps of iterative refinement are enough and FE, BE are sufficiently small when S = and that only a few perturbations are necessary. Example 2. In this example we generate 1000 random matrices for each version of the algorithm (original and stabilized), where k = 7, s = 15,j = 2.

Table 2: The forward and backward error and the number of iterations of Example 1, when E = 10-l"

Orig. Stab.

Average F E Worst FE Average F E Worst F E Average N;t Worst Nit

6=0 0.12 2.77

6=

6 = lop8

6 = lo-'

x lo-13 2.42 x 10W1".22 x 10-1".23 3.11 x lo-' 2.52 x 10-l2 6.72 x lo-'' 1.24 1.12 1.41 2 2 2

Table 3: The forward error and the number of iterations for the random matrices of Example 2.

Then the blocks Ba are matrices of the following kind:

where i is a random integer = Now let us fix ,!$> = pr;J' = 3/!2 = ,8g) which belongs t o the range [I,151) to make the block Bi singular and hence the original algorithm numericaly unstable. We report the results in Table 3 for different values of 6, again. The second and the fourth rows of Table 3 present the average forward error for the two versions of the algorithm, and the third and the fifth rows contain the worst forward error which is obtained during the series of experiments. The average and the worst number of iterations we present in the sixth and the seventh rows of Table 3. The tests with random matrices show that the optimal value of 6 is again. So, if we compare both algorithms, we see that the stabilized algorithm is significantly better than the original one. In conclusion, the algorithm behaves well in pracrice, and we need to do more research for its theoretical justification.

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Finite volumes for complex applications

References [I] Balle, S., Hansen, P., Higham. N.: A Strassen-type matrix inversion algorithm for the connection machine. APPARC PaA2 Deliverable, Esprit BRA Contract # 6634; Report UNIC-93-11, UNIaC, October 1993. [2] Conroy, J.: Parallel Algorithms for the solution of narrow banded systems. Appl. Numer. Math. 5 (1989) 409-421. [3] Golub, G., Loan, C. van: Matrix Computations. The John Hopkins University Press, 1996. [4] Hansen, P., Yalamov, P.: Stabilization by perturbation of a 4n2 Toeplitz solver, SIMAX, 1999. (to appear) [5] Higham, N.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia, 1996. [6] Meier, U.: A parallel partition method for solving banded linear systems. Parallel Comput. 2 (1985) 33-43. [7] Skeel, R.: Iterative refinement implies numerical stability for Gaussian elimination. Math. Comp. 35 (1980) 817-832.

[8] Wang, H.: A parallel method for tridiagonal linear systems. ACM Transactions on Mathematical Software 7 (1981) 170-183. [9] Yalamov, P.: Convergence of the iterative refinement procedure applied to stabilization of a fast Toeplitz solver. Proc. Second IMACS Symposium on Iterative Methods in Linear Algebra, Eds. P. Vassilevski and S. Margenov, IMACS (1996) 354-363. [lo] Yalamov, P., Pavlov, V.: On the Stabilty of a Partitioning Algorithm for Tridiagonal Systems. SIAM J . Matrix Anal. Appl. 20 (1999) 159-181.

[ l l ] Yalamov, P., Pavlov, V.: Stability of a partitioning algorithm for special classes of banded linear systems. Preprint N 38, University of Rousse, March 1998. (submitted to LAA)

On Jeffreys Model of heat conduction

by Maksymilian Dryja and Krzysztof Moszyliski University of Warsaw Banacha 2, 02-097 Warszawa Poland

ABSTRACT We present differential equations describing the Jeffreys Model of heat conduction, and certain corresponding numerical models. We define some suitable weak form of Jeflreys equations with boundary conditions of low regularity, (defined by L2 functions), and formulate existence and uniqueness theorem for this generalized differential problem. We give also theorem on unconditional stability of the numerical model and propose its finite-volume interpretation. Some results of numerical computations, compared with experiments with heat waves are presented.

1.

Introduction

Classical Fourier law relates a heat flow Q and temperature T in a very simple way: Q = -kAT, where % is a positive constant. However, it is well known that the classical heat conduction equation based on this law does not describe well enough all phenomenons observed in experiments. Such phenomenon is, for example, so called heat wave incitated in thin metallic layer by a short heat impulse. Jeffreys model is based on a somewhat more complicated relation than that given by the Fourier law: it is certain integro-differential relation which involves so called relaxation time ([I],[2], [3],[4]). Due t o the modified Fourier law, Jeffreys model fits better to experiments with heat waves: numerical computations based on this model show quite good consistency with experimental results. The heat impulses imitating the heat waves, which we observe in numerical experiments, are represented in the numerical model by the boundary conditions for Jeffreys differential equations. We apply very sharp, and even discontinous heat impulses. Differential equations of the Jeffreys model are in fact very

Finite volumes for complex applications

868

simple, and the only difficulty in present problem was caused by the assumed low regularity of the boundary conditions.

2. Jeffreys Model and its weak form

General equations of Jeffreys Model can be presented in the following form:

where the scalar function T ( t , x) is the temperature, the vector-function Q(t, a) is the heat flow, t E R+, x E R , R C R3, D and ti are positive coefficients (in general constant). In fact we are interested in one-dimensional model, because it corresponds to known results of experiments with the heat waves. The one-dimensional equations are of the following form:

with t E [0, t,,,], x E [L, PI, T(0, x) = To(x), Q(0, x) = Qo(x), T(t , L) = 4 ( t ) , Q ( t ,L) = $(t), T(t, P) = Q(t, P ) = 0, where To, Qo, 4 and $ are given functions. Functions To and Qo are initial conditions, while I$ and $, involved in the left boundary condition, define the heat impulse. Since we assume that 4 and $J are in L'(L, P), it is clear that we have to formulate the boundary value problem for equation (1) in a weak form, which makes possible to prove the existence (and uniqueness) of solution of assumed regularity. Our boundary value problem is defined on the bounded interval [L, PI, therefore it is easy to avoid non-homogenity of the boundary conditions. Let us

and

and let ?(t, x) = T(t, x) - d ( t , x) and &(t, x) = Q(t, x) - $(t, x). Functions T and Q satisfy nonhomogeneous Jeffreys equations and homogenous boundary conditions. Further, we introduce new functions

Varia

and

1

869

t

s ( t , x) =

Q(S, x)ds.

Using these two functions we have the following boundary value problem:

St

+ DR, + S - KS,, = g

(21,

with zero initial and boundary conditions, where f and g are in terms of 4 , $, To and Qo. In order to state the weak formulation for equations (2), we are proceeding in quite standard way. We introduce two bilinear forms: a : (L2(L,P) x H ~ ( LP, ) ) x (L2(L,P ) x H ~ ( LP, ) )

b : (L'(L, P ) x L2(L,P ) ) x (L2(L, P ) x L2(L, P ) )

-+

R,

R,

where:

b(R, S; V, W ) =

LP

[DRV + SWIdx.

Let:

-

1 = -[D P-L

L-P -1L , + ( s ) ~ s Jt

-

(P - x)rn(t)~+TO(X),

t

J +(s)ds 0

- ( P - x)($(t)

+

1

d(s)ds)l+ QO(z).

We define now the space H of pairs of functions (R, S), in which we want to find solutions of our problem. Let H be the space of all pairs of functions ( I Z , S),

such that:

870

Finite volumes for complex applications

Here

11 . 110

is the L2(L, P)-norm.

Weak formulation of Jeffreys Model We are looking for a pair ( R , S ) E H such that for almost all t E [0, t,,,], , all V E L2(L, P ) and W E H ~ ( LP):

for

holds. The following theorem establishes an unique solution of the problem (3)

Theorem 1. Assume that:

TO E L2(L, P ) ,

then in H there exists a solution ( R , S ) of the variational equation (3). Let U c H be the subspace of H of such pairs ( R , S) that Rl S x E C(0, tmax, L ~ ( LP, ) ) ,

s E C(O1 tmaz, H,'(L,

P)).

If a solution ( R , S ) of (3) is in U , then it is unique in U Proof of this theorem will appear in [5]. Now, let us give some comments on the above result. First of all it implies that the solution ( R , S ) of equation (3) is such that: Rt = T - 4 E ~ ' ( 0 t,,,, , L 2 ( ~P)) , and

= Q - II E L~(o,~,,,, H , ~ ( Lp, ) ) , exactly what we need for our solution (TI&).Moreover, the function T with respect to 2, is in L2(L, P ) , hence it should only weakly depend on the function 4 defining the boundary condition for it. This fact was observed during experimental computations. St

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871

3. Numerical model In fact the used numerical model is an implicit finite difference scheme on the rectangular grid of points (tn,xk), where t, = rn for n = 0, I , . . . , N with t,,,=Nr,x~=L+khfor k=O,l,...,M+l,andP=L+(M+l)h . On this grid is defined the following finite difference scheme:

with the following initial and boundary conditions:

where T p , Q; are values of the grid functions in the point (t,, xk), Afk = f k t l - f k , Vfk = f k - f k - 1 , X = and p = &. The next step is to prove the stability of the scheme (4). In order to avoid strong regularity hypotheses concerning functions q5 and $ such as, for instance, boundedness of divided differences and so on, we proceede in similar way as in the section 2. We begin with functions ? ; = T; - 4(tn, xk), and Q; = Qp - $(in, xk), where d ( t , z) and $(t, x) were defined in the section 2. Further, we define:

Using that, the scheme is defined as:

where:

+X

= T o 4 ( ~ % +-1 Q!-')+

872

Finite volumes for complex applications

and

The grid functions R; and Sr are subject to homogeneous Dirichlet boundary conditions, and the following initial conditions:

and

si

= T ( Q O ( x k ) - $(o,

zk)).

Let

Rn = [Ry,R;, . . . , R

M ] ~ ,

and

For the scheme (5), the following stability theorem can be proved:

Theorem 2

+-

If p = is arbitrary, but constant, independent of T and h , then for any m, 0 5 rn 5 N - 1, solution ( R n ,Sn) of the problem ( 5 ) satasjies the inequality:

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873

where K depend only on 4, $, T o , and Q O , and does not depend on divided differences of these functions. Proof of this theorem will appear in [5] together with proof of theorem 1.

Since

Theorem 2 implies in particular that if ,u is arbitrary, but constant, then Tc and Q; are bounded by Ii' in the sense of discrete L~ norm. This means unconditional stability and, by Lax Theory, convergence follows.

4. Numerical experiments There have been done a lot of numerical experiments involving equations (4). Further we compared graphs obtained by computation with those obtained by pl~ysicalexperiments. We succeeded t o choose good enough values for coefficients D and K . We made computations with Fourier model as well. Conclusions are as follows:

1. Jeffreys model fits quite well t o experimental results. T h e heat waves appear in both cases: when the impulse is introduced by initial or by boundary conditions. 2. If we see some kind of waves in Fourier model (it is possible only when the impulse is introduced by boundary condition), they travel with visibly too great speed, and their amplitude vanishes very quickly.

3 . Coefficient D plays the most important role from the point of view of the speed of the heat waves. For more informations see [4]

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Finite volumes for complex applications

References [l] Brorson S.D. Fujimoto J.G. Ippen E.P. "Femtosecond electronic heattransport dynamics in thin gold films" Phys. Rev. Lett. 59 (1987),1962-1965.

[2] Joseph D.D. Preziosi L. "Heat waves" Rev. Mod. Phys. 61 (1989),41-73 [3] Joseph D.D. Preziosi L. Addendum to the paper "Heat waves7' Rev. Mod. Phys. 62 (1990),375-391 [4] Moszyriski K . Palczewski A. "Assymptotic analysis of heat propagation models" submitted to Archives of mechanics

[5] Dryja M. Moszynski K. "On Jeffreys model of heat conduction" Prepared for publication

Investigation some method of cavitating jets

S.A. Ocheretyany and V.V. Prokof'ev Department of unsteady hydrodynamics, Institute of mechanics Moscow State University, 11 9899, Michurinski pr. 1, Moscow, Russia

The problem of multiphase liquid flow after ventilating cavity with internal pressure exceeding the ambient pressure and with low concentrations of gas (or vapor) bubbles i n liquid is considered. The two-dimensional Bow considered corresponds i n one case to Chaplygin-Kolscher scheme, and i n another one-to the problem of two symmetric jets collision with a cavity formation. The velocity profile at large distances behind the cavity is determined. The most intensive disturbances in the gas-liquid mixture Pow are introduced by the gas injection into liquid i n the neighbourhood of the tail point of the cavity. And the injection of small gas bubbles is most eficient there. The vapor injection into a cavity is not eficient for increasing the liquid flow momentum, but i t can be used for generation of a cavitating jet. The account of multi-velocity effects i n the two-phase medium equations is essential for bubbles of supercritical size, when the viscous friction effect is small. ABSTRACT

Key Words: cavity, bubbles, heat and mass transfer, two-phase flow, multivelocity effects

1. Introduction

We shall model such a flow in a plane-parallel statement, considering, for example, the flow over a body (Figure l a ) , the rear critical point of which is substituted by a finite stagnation zone (cavity) with smooth closure of jets. Such a cavity can be generated by means of artificial injection (ventilation) of vapor or gas into the cavity. Since the boundary of such a cavity is Taylor instable, the flow will be accompanied by intensive outflow of gas (or vapor) in form of bubbles and by the formation of a two-phase tail. The first example corresponds to flow over a convex cylinder according to

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Finite volumes for complex applications

the Chaplygin-Kolscher scheme [GUR 651 with formation of finite stagnation zone behind the cylinder with pressure exceeding the pressure in a free stream. Here one can use the well-known solution for a circular cylinder [GUR 651; we, however, shall use a simpler solution for a flow over a plate (Figure l a ) [PRO 981. Though such a scheme can not be applied for description of flow over a plate (with singularities at points E and F), it is quite suitable for studying the flow in the cavity neighbourhood. The second example presents the collision of two symmetric jets with formation of the stagnation region [GUR 651. We shall suppose for simplicity, that the jet-leading channels are stretched up to infinity. In this case, while moving upwards along the stream, the flow becomes non-univalent, but we shall ignore this fact, since we consider the flow in the neighbourhood of jets' closure point B (Figure l b ) and in the outgoing jet. If the flow is reversed, then we arrive at the problem of "splitting" of the jet flowing onto the region with heightened pressure, which was considered in [YAK 871. Near the cavity the flow is highly inhomogeneous, and the bubbles fall into the region of high pressure gradients; as a result, the bubbles motion with respect to the liquid becomes essential. When the multi-velocity effects are taken into account, the model is more complicated: the medium becomes anisotropic [VOI 751. 2. The main relations and results.

2.1 Two-Phase liquid flow with low concentration of bubbles formulations.

To model for a rarefied bubble mixture [GAR 731, [VOI 751, which takes into account the two-velocity effects: dpu dpv -+-10,

ax

ay

F, 2 F1 = --nu 3

3

dnul

dnvl

ax

ay

-+--

+ Fz = 0,

-

0,

PI

PI

dlVl dV pl(- 3-) - 2 n ~ ~ ~ l-aV () , ~ F2 l = k.rrpla(V - V1), dt dt

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Here p = p l ( l - a) is the mixture density under the neglection of vapor (gas) density in a bubble (pl=const is the liquid density), V(u, v) is the massaverage mixture velocity, which is equal to the mean liquid velocity under the neglection of bubbles mass, V l (ul ,vl) is the mean velocity of bubbles in an elementary volume, cx = 4/37rna3 is the volume concentration of bubbles (n is the numerical density of bubbles), a is the radius of bubbles, which are supposed to be spherical. For a steady motion, supposing the slipping-related stress tensor -rij to be a tensor function of the slipping vector W(W,, Wy) = V1-V one can write [VOI 751

Further, F1 is the hydrodynamic force acting on a bubble of variable radius moving with variable velocity [VOI 731, [YAK 731; F2 is the viscous drag force. In the first example, where the bubble size is small and the relative velocity is low (the Reynolds numbers of relative motion Re zz l ) , the k = 4 value was accepted for the viscous drag force F2 (the Adamar-Rybchinskii formula). For the second example, where larger bubbles are considered and the Reynolds numbers of relative motion reach the values of the order of lo3, the asymptotic Levich formula was applied [LEV 621 (k = 12). Further, ~1 is the liquid viscosity coefficient (the viscosity is taken into account in the bubbleslliquid interaction process only), pk is the gas pressure in a cavity, u = 1 - for the isothermal process and v = y (y is the Poisson adiabatic index) - for the adiabatic expansion-compression of gas, a is the bubble boundary velocity, C is the surface tension factor. We shall suppose that the bubbles only weakly disturb the liquid flow. We << denote: u = Uo + u', u = Vo v r , p = po + p', where ur,ur << po, and Uo, Vo,po are the parameter of undisturbed potential flow of liquid. Similarly, for bubbles: u l = U1 u i ; u l = Vl vi, where Ul; VI is the field of bubbles' velocities in the undisturbed liquid (described by equations [3], if u , v is replaced by Uo; Vo). It was shown in paper [OCH 981, that in the linear approximation the continuity equation for bubbles [I] can be integrated. Transferring in equation [I] to the Lagrangian coordinate system I, t (where I is the coordinate of the point of bubbles injection a t the cavity boundary, t is time), we obtain:

+

dmlp'

+

+

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Finite volumes for complex applications

where f(1) is determined from the boundary conditions. For the function of liquid phase's flow disturbance P = U'UO vlVO+ p'lpl along the stream line of undisturbed liquid flow I)= const one can write (transferring in equations [2] from variables (x, y) to variables (x, Q) [OCH 981):

+

In equation [5] functions F, and F, are determined for the known potential field of flow Uo, Vo and from the velocities of motion of bubbles U1, Vl in this field of flow. Further, after determining Ul, Vl and a equation [5] is solved by finding a quadrature along the stream line Q = const. We shall suppose that bubbles have the same initial radius ao, the initial pressure is pg = p k and that the velocities of liquid and bubbles at the cavity boundary (0 5 1 5 L k ) are equal. Then the initial conditions for the system of ordinary differential equations, governing the trajectories of bubbles in the flowfield of the liquid Uo, Vo take the form at t = 0:

where xk, y k are t.he cavity boundary coordinates. Under such initial conditions the Jacobian of transformation of Eulerian coordinates to Lagrangian ones J zeroes at the initial time moment, since the cavity boundary is an envelope of trajectories of bubbles. By this reason at the cavity boundary the numerical concentration n (or a ) can not be specified by a finite quantity. Nevertheless, the arbitrary function f (1) can be determined [OCH 981. Indeed, for x -i KI the undisturbed flow tends to uniform one, the velocities of phases become equal. Hence, for x + rn VI 0,i Ul + V, and J V,dy/dl. i Then we shall have from 141

where q, is the specific numerical flux of bubbles through the cavity boundary. This relation uses the fact that the flux of bubbles in a stream tube of bubbles' medium is conserved. Thus, instead of specifying the distribution n(1) we shall determine function f (1) by specifying the distribution q,(l). The integral quantities are conveniently used for normalizing the distribution. These quantities are:

where N is the number of bubbles carried away per time unit from the whole cavity, Qgk is the integral volume gas flow rate from a cavity. Further, the numerical density and volume concentration will be referred to characteristic quantities: n, = N ffc = 5 V , L5, ~where Q, is the volume gas flow rate normalized with respect to pressure p,, L is the characteristic linear dimension of the problem. So, by solving a simple sequence of problems-the equation of dynamics of individual bubbles, the equation for concentration [4] and the equation for disturbances [5], we find in the linear approximation the field of the disturbance function P . In order to determine the disturbances themselves, one must determine the vorticity from function P , then introduce the stream function of undisturbed flow and then solve the complicated boundary-value problem for the Poisson equation taking into account the free boundary deformation in the linear approximation. However, if we make use of the conditions for x -+ oo (in motion along the escaping jet) Uo + V, Vo + 0, p' -+ 0, then P + V,ul, i.e. in the linear approximation, along with the distribution of concentration of bubbles, we find the profile of a longitudinal disturbed velocity far behind the ca.vity. This profile can be used for calculating the integral characteristics of a flow-the force acting on a body flown over and the jet momentum variation. Jf, however, the velocity of bubbles motion relative to the fluid is low and the terms of the order of square of relative velocity may be neglected, then it follows from [5],that the disturbance P is determined by the distribution of a volume concentration of bubbles and by undisturbed pressure variation along the liquid stream line. If, however, the bubbles are so small, that they are "frozen" into a liquid and their dynamics may be neglected, then cr becomes a function of a local pressure and stream function. For example, for the isothermal law of bubbles expansion a = ao(+)p,/po, where function cro($) characterizes the volume concentration distribution at some injection cross-section or in a free stream. Then we obtain from [5]:

m,

where pol($) is the undisturbed pressure distribution at the injection crosssection (it is supposed that in a free stream P = 0). 2.2 The Chaplygin-Kolscher flow disturbance by bubbles escaping the cavity. Within the framework of the model, which takes into account the motion of bubbles relative to a liquid, we shall consider a symmetric flow over a plate with formation of stagnation region according to the Chaplygin-Kolseher scheme, rather than a rear critical point [PRO 981. The exterior of a unit circle in the right half-plane is chosen as a parametric region (the correspondence of points is shown in Figure la). The field of velocities is specified in a complex form by

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Finite volumes for complex applications

formulae:

+ +

Vk 1 u2n2 u n2 u2 '

Uo - iVo = -

Here u is a complex parameter, Vk is the velocity at the cavity boundary, quantity kl is related with the cavitation number a by expressions

All velocities are referred to the free stream velocity, the linear dimensions-to the width L of plate AB. In virtue of smallness of gas bubbles as compared to the characteristic dimension of the problem, we assume the pressure in a bubble to be equal to the ambient liquid pressure (the bubble pulsations are neg!ected) and the relative motion velocity t o be low (the term of the order of aW2 is absent in the equations for disturbances). The isothermal law of bubble expansion is accepted. Possessing information on the disturbance of a longitudinal component of liquid velocity in a wake far behind a plate, we may determine the force acting on a plate in connection with gas rejection from a cavity. The calculations will be performed for the following values of parameters: Re = poLV,/pl = lo4, E u = 2p,lpoV& = 1 and 3, a = -0,25 and --0,585; the size of bubbles was chosen to be close to supercritical one (therefore, the relative velocities are low): a o / L = 0,01 and 0,08. For bubble size lower than the critical value the viscous drag forces are dominating. Two versions of bubbles' flux distribution over the cavity boundary were used: 1) the specific numerical flux of bubbles q, is proportional to xh (q, = 0 a t the cavity head and maximum-at the tail point xk = L k ) , and 2) q, is proportional to y k (q, is maximum a t the cavity head and zero-at the tail point).

2.3 The disturbance o f f l o w in the outgoing jet at symmetric collision of jets.

The example considered above indicates that the greatest increase of liquid flow in a wake behind a cavity is achieved, when the gas bubbles are injected in the neighborhood of the tail point of a cavity. If the cavity has considerable size

(u + 0) and the flow disturbance mechanism is mainly related with the flow in the cavity tail region, then the wake behind the cavity can be investigated with using the model flow shown in Figure l b as an undisturbed flow. This flow can be considered as an asymptotics for a + 0 for the jet air screen [GUR 651, which is suitable for description of a flow in the neighborhood of jets' connection point (point B in Figure l b ) and in the outgoing jet. The flow can be described in the parametric form by formulae

Here w is a complex potential, N is a scale constant. The parametric variable u = J + i q varies in the half-band 0 J 5 7r/2; TI 0. Quantities in [ll]denote: vo = l/k/T/, 1, Vk is the velocity at the cavity boundary, V, is the velocity at the outer boundary of a jet. The coordinate origin z = x + i y = 0 corresponds to point u = 0. The cavity and jet surfaces are, respectively, [ = 0 and J = .7r/2 7 0 (with moving upwards along the stream the shape of stream lines in the physical plane tends to circles, and the flow becomes non-univalent). The part of the real axis x 2 0 corresponds to q = 0; 0 5 7r/2. We denote the jet half-width for x + +m by A (then the scale constant N = 2A/7r). Unlike the previous problem, the bubbles may reach here the outer boundary of a jet, and we shall suppose them to intersect the boundary without obstacle. In calculating the disturbance in a wake behind the cavity we shall use the model of [I]-[3] with regard to the terms quadratic in the relative velocity. We specify the flow parameters, which are typical for the problem of artificial generation of cavitating jets [OCH 981: the head pressure is p = 5 - - 1 0 , l MPa, the pressure in a cavity is pk = 2 MPa, the ambient space pressure is p, = 0 , 1 MPa, A = 2 mm. We shall suppose the gas bubbles to be injected uniformly along the cavity surface (( = 0,rj O), and the gas in bubbles expands adiabatically. Unlike the first problem, some part of bubbles intersect the outer boundary of a jet here, and, hence, we shall use for normalization the mean volume concentration of bubbles in the outgoing jet a , for x + +m. At the upper asymptotic section (q + co) the stream lines represent the circles, and the pressure on these lines is constant [OCH 981. Besides, in the uniform injection of bubbles all two-phase medium's characteristics (the volume concentration, the velocity cornponents in the coordinate system fixed with undisturbed stream lines of liquid) are constant along these stream lines. It can be shown that, under neglection of viscous terms, the right-hand side in equation [5] zeroes in this case, and, hence, the disturbance P for 7 >> 1 may be ignored. Due to this reason, in specific numerical examples we have considered both the case of finite injection region (0 1 Lk)and of infinite one ( I 0). In the latter case the upper boundary of calculated region (the has been selected numerically (based on the absence of maximum value l,, I,, influence on calculation results).

<

<

>

>

<<

>

>

< <

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Finite volumes for complex applications

2.4 Results and descussion. Figure 2. presents the profiles of longitudinal velocity far from a cavity (in calculations this is the cross-section x / L = 2) for various injection conditions. The zero fi value for y = 0 is a result of scattering effect of the field of flow in the cavity's tail point neighborhood on the bubbles medium (the pressure gradient becomes infinite here). The fact is, that in motion along the zero stream line the disturbance value is P = 0 a t the cavity surface, since pressure po is constant here, and a t the section behind the tail point (CD) P = 0, since the concentration of bubbles a is zeroing here. The latter fact takes place, if the distribution of injection a t the cavity boundary [7] has no singularity a t 1 + 0. Figure 3 shows the profiles of a disturbed longitudinal velocity u a t the crosssection x/A = 5 with the initial size of bubbles a0 = 0,3 mm (ao/A = 0,15) for the head pressure of 5 and 10,l MPa ( a = -0,387 and -0,19, respectively) and with infinite length of injection region. The dashed line shows the profiles associated with liquid's density disturbance only. The comparison indicates that the account of terms of the order of a W 2 decreases the flow velocity and, hence, these terms play a part of dissipative ones, their contribution being increased as the jet flow velocity lowers. (This is due to increase of bubbles slipping velocity in the region of high pressure gradients). Figure 4 shows similar profiles at p = 5 MPa ( a = -0,387) for various initial sizes of bubbles a0 = 0,3; 0,02; 0,004 mm ( a o l h = 0,15; 0,Ol; 0,002). The latter size is close to critical one and, as Figure 4 shows, the influence of dissipative terms is weak in this case. For larger bubbles, however, the influence of these terms is so great, that even the conclusion on efficiency of bubbles effect on the flow in a wake changes. Whereas in case of disregarding the dissipative terms the larger bubbles cause a greater increase of longitudinal velocity due to their inertial growth up t o larger size as compared t o initial one (and, hence, due t o stronger disturbance of the medium density), then in case of accounting the terms of the order of aW2 the situation is just opposite. Figures 5 and 6 present the profiles of a/a, and the case of gaseous bubbles, but only for a smaller injection region (0 5 1 5 A), at the initial size of a bubble a o / A = 0,15. Quantity a, was calculated from the volume flow rate of vapor through the cavity boundary, normalized with respect to pressure pa according t o the adiabatic law. The data of Figures 5 and 6 indicate that the vapor bubbles distribution in a jet differ from gaseous ones not only quantitatively, by qualitatively as well. In fact, inside the basic liquid jet the two-phase jet with width of about x 0, l A is formed, which gradually approaches the outer boundary of the basic jet and reaches its boundary a t the distance of 6-7 A from the cavity tail. Such a cumulation of bubbles is associated wit pulsation character of motion of vapor bubbles. Note that the effect of increase of liquid velocity in the bubbles accumulation region is revealed only in the case, if the terms of the order of aW2 are taken into account.

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3. Bibliography

[GUR 651

M.I.GUREVICH,Theory of jets in Ideal Fluids, Academic Press, New York 1965.

[YAK 871

YU.L.YAKIMOV, "Limiting water flows", Mechanics and Scientific and Technical Progress. V.2 [in Russian], Nauka, Moscow 1987, p. 7-25.

[VOI 751

VOINOVO.V., PETROV A.G. On the Equations of Motion of Liquid with Bubbles. [in Russian], Izv. Akad. Nauk SSSR, Prikl. Matem. i Mekh., Vo1.39, N" 5, 1975.

[GAR 731

GARIPOVR.M. Closed Equations of Bubbly-Liquid Motion, [in Russian], Prikl. Matem. i Teor. Fiz.(1973), No 6.

[OCH 981

OCHERETYANY S.A., PROKOF'EV V.V. Multivelocity Effects in High-Pressure-Gradient Flow of Dilute Bubbly Media, Fluid Dynamics, Vo1.33, N" 1, 1998, p. 71-81.

[YAK 731

YAKIMOV Yu. L. Forces Acting on a Small Body in Arbitrary Flux of Incompressible Fluid and Equations of Motion of a Two-Phase Medium, [in Russian], Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza, N" 3, 1973.

[VOI 731

VOINOVO.V., PETROV A.G. Lagrangian Function of a Gas Bubble in a Non-Homogeneous Flux, [in Russian], Doklady Akad. Nauk SSSR, Vo1.212, N" 5, 1973.

[LEV 621

LEVICHV. G . Phisicochemical hydrodynamics, Prentice-Hall, Englewood Cliffs (NY), 1962.

[SED 871

SEDOVL.I. Metody podobiya i razmerenosti v mekhanike, [in Russian], Moscow, Nauka, 1987.

[OCH 951

OCHERETYANY S .A., PROKOF'EV V.V. Cavitation Initiation by Hot Vapor Injection into a Cold Liquid Jet, Fluid Dynamics, Vol.30, N" 1, 1995, p. 717-724.

[PRO 981

OCHERETYANY S.A., PROKOF'EV V.V. Rare Multiphase Liquid Flow after Ventilating Cavity, [in Russian], Summury edition anniversary academician L.LSedov, Institute of Mechanics Publishing, Moscow, 1998.

SUMMURY 1. I t is shown that in the linear approximation in volume concentration the velocity profile far behind a cavity can be determined without

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Finite volumes for complex applications

solving the full problem on liquid flow disturbance and without determining the disturbances of free boundaries. 2. The solutions of specific problems have shown that the most intensive disturbance of liquid flow by bubbles at fixed injected gas flow rate and cavitation number takes place, when the gas injection region is shifted toward the tail point of a cavity, and the injection of small bubbles is most effective. 3. The vapor injection into a cavity is not eficient for increasing the liquid momentum. The vapor is worth to be injected into the heightened pressure region with the purpose of jet saturation with cavitation bubbles, i.e. generation of a cavitating jet. 4. The account of multi-velocity effects i n the two-phase medium equations is essential for bubbles of supercritical size, when the viscous friction efect is small. The terms of the order of a square of relative velocity play a part of dissipative terms i n the case of gas bubbles, and the accounting of these terms considerable decreases the momentum of moderate-size bubbles effect on a jet.

+

Figure 1. Schemes of Aow in a physical plane (z = x i y ) and in the plane i ~ )for the Chaplygin-Kolscher flow (a) and for of parametric variable (u= a smooth closure of jets (b)

<+

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Figure 2. Curves 1 to1$ u = --0,2; 2% = -;, 585; f k n d 2 are the first and second versions of injection a o / L = 0 , l ; 3-ao/L = 0,08 0

00

, .' po=5 MIIa \

\

\

0.50

\ \

'--. /

0.00 0.00

0.02

0.04

0.06

0.08

u

0.1

Figure 3. Velocity disturbance profiles for x = 5, a0 = 0 , 3 mm

Figure 4. Velocity disturbance profiles for z = 5, po = 5 MPa

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Finite volumes for complex applications

Figure 5. Volume concentration profiles for vapor bubbles

Figure 6 . Profiles o f function P for vapor bubbles

Index des auteurs

Abgrall R. 3 Abouziarov M. 247 Achdou Y. 23 1 Afif M. 387 Agaphonov B .N. 743 Alcrudo F. 543 A1 Moatassime H. 595 Amaziane B. 387 Angermann L. 223 Angot P. 215 Aoufi A. 437, 825 Azarenok B.N. 795 Bailey C. 507, 585 Baines M. J. 787 Balalcrishnan N. 33 1 Ballmann J. 67 1 Benazzouz S. 663 Berkvens P.J.F. 499 Berthon C. 307 Berti G. 655 Bidtgaray B. 483 Blanc P. 117 Bleecke H. 753 Bojarevics V. 453 Bontoux P. 559 Botchev M.A. 499 Bouzouf B. 525 Bovolin V. 475 Brenner G. 753

Brigadnov I.A. 197 Brufau P. 255 BuSmann A. 607 Caltagirone J.P. 367 Caraeni D. 315 Castro-Diaz M. 809 Catalano L. 323 Chapin V.G. 663 Chassaing P. 663 Chen M. 803 Clerc S. 395 Cohen A. 841 Conway S. 315 Coquel F. 1 1,307 Cortes J. 445 Coudibre Y. 125 Croft N. 585 Croisille J.-P. 239 Cross M. 453,507,585 Darwish M. 339 Deconinck H. 27 Dedner A. 49 1 Degrez G . 27 Dervieux A. 63 1 , 8 17 Dick E. 709 Drikakis D. 725 Dryja M. 867 Dubois F. 133

888

Finite volumes for complex applications

Edelvik F. 14 1 El Ganaoui M. 559 Elmahi I. 525 Ewert R. 733 Eymard R. 149 Fadlun E.A. 693 Feistauer M. 647 Felcman J. 647 Forestier A. 7 17 Fort J. 701 Fournier L. 623 Fuchs L. 3 15 Fiirst J. 701 Gallouet T. 149, 155, 189, 215 Garcia-Navarro P. 255, 809 George P.L. 817 Ghidaglia J.-M. 69,483 Glinsky-Olivier N. 41 1 GIoth 0 . 6 2 3 Goryachev V.D. 743 Grahs T. 297 Greenshields C.J. 467 Greiner G. 753 Griin G. 205 Haaland S.E. 683 Hakim A. 595 Halama J. 701 Hanel D. 849 Helluy P. 833 HQard J.M. 7 17 Herbin R. 149, 155, 189,215 Hittinger J. 357 Holstad A. 163 Hubbard M.E. 787 Hughes M. 453 Ivanenko S.A. 795 Ivankovic A. 459,467 Japhet C. 23 1 Jolly B.A. 615

Kaber S.M. 84 1 Kaltenegger K. 533 Khoshyaran M.M. 55 1 Klein R. 41 Kolyvanov V.G. 743 Kozel K. 639,701 Kumbaro A. 445 Lau Man Wai J. 403 Lazarov R.D. 5 1 Leary S.J. 787 Lebacque J.P. 55 1 Leboucher L. 453 Lenke L.J. 419 Leonardi S. 693 Leservoisier D. 63 1, 8 17 Lie I. 163 Lioen W.M. 499 Lotstedt P. 769 Louda P. 639 Lu H. 585 Luk6cov6-Medvidova M. 289 Maday Y. 23 1 Majewski J. 725 Maneeratana K. 459 Marmignon C. 11 Mazhorova O.S. 559 Meese E.A. 683 Meinke M. 733 Meister A. 297 Merci B. 709 Michel A. 173 Moczala M. 607 Morton B. 347 Morton K.W. 289 Moszynski K. 867 Moukalled F. 339 Miiller L. 533 Nataf F. 23 1 Nkonga B. 575 Noelle S. 95 Ocheretyany S. 875

Index des auteurs

Ohlberger M. 761 Oldroyd A. 567 Orlandi P. 693 Ouazar D. 525 Pasciak J.E. 5 1 Pavlov V. 859 Penanhoat 0 . 8 17 Pericleous K. 453, 585 Perot B. 263 Perpeet S. 429 Peters N. 803 Postel M.84 1 Potapov S.V. 27 1 Praveen C. 33 1 Prihoda J. 639 Prokofiev V.V. 875 Rad M. 279 Raghay S. 595 Ris V.V. 743 Rister Th. 733 Rizk M. 615 Roe P. 279,347,357 Rohde C. 491 Rokicki J. 725 Rosenband V. 437 Rouy S . 833 Rumpf M. 205 Scanlon T.J. 567 Schall E. 41 1,631 Scheffler H.P. 181 Schmid 0 . 6 0 7 Schneider Th. 4 1 Shurina E. P. 377 Simon H. 4 19 Smirnov E.M. 743 Soares Frazao S. 403 Soderberg S. 769 Sonar T. 297

Stechbarth J. 533 Steelant J. 709 Taglialatela L. 475 Taylor G.A. 507,585 Thiele F. 777 Toumi I. 445 Tran L. 849 Tremel U. 753 van Keuk J. 67 1 Vanselow R. 1 8 1 Vassilevski P.S. 51 Verwer J.G. 499 Verzicco R. 693 Vignal M. H. 189 Villedieu P. 125 Vilsmeier R. 849 Vincent S. 367 Viozat C . 63 1 Vlasseva E. 5 17 Voi'tovich T.V. 377 von Lavante E. 429,607 Warnecke G. 289 Weller H.G. 467 Wenke P. 567 Wesenberg M. 49 1 Wheel M.A. 567 WheeIer D. 585 Wick A. 777 Xeuxet E. 7 17

Zachcial A. 429 Zaitsev D.K. 743 Zech Y. 403 Zhang X. 263 Zoltak J. 725

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