Analysis and Numerical Computation of Conversation Laws

Gerald Warnecke (Ed.) Analysis and Numerics for Conservation Laws

Gerald Warnecke Editor

Analysis and Numerics for Conservation Laws With 236 Figures and 18 Tables

123

Editor

Gerald Warnecke Institut für Analysis und Numerik Otto-von-Guericke-Universität Magdeburg Postfach 4120 39016 Magdeburg, Germany e-mail: [email protected]

Library of Congress Control Number: 2005922932

Mathematics Subject Classification (2000): 35L65, 65M99, 65Z05, 76N15

ISBN-10 3 -540-24834-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-24834-7 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the author using a Springer TEX macro package Cover design: Erich Kircher, Heidelberg, Germany Printed on acid-free paper

46/3142/sz - 5 4 3 2 1 0

Preface

What do a supernova explosion in outer space, flow around an airfoil and knocking in combustion engines have in common? The physical and chemical mechanisms as well as the sizes of these processes are quite different. So are the motivations for studying them scientifically. The supernova is a thermo-nuclear explosion on a scale of 108 cm. Astrophysicists try to understand them in order to get insight into fundamental properties of the universe. In flows around airfoils of commercial airliners at the scale of 103 cm shock waves occur that influence the stability of the wings as well as fuel consumption in flight. This requires appropriate design of the shape and structure of airfoils by engineers. Knocking occurs in combustion, a chemical process, and must be avoided since it damages motors. The scale is 101 cm and these processes must be optimized for efficiency and environmental considerations. The common thread is that the underlying fluid flows may at a certain scale of observation be described by basically the same type of hyperbolic systems of partial differential equations in divergence form, called conservation laws. Astrophysicists, engineers and mathematicians share a common interest in scientific progress on theory for these equations and the development of computational methods for solutions of the equations. Due to their wide applicability in modeling of continua, partial differential equations are a major field of research in mathematics. A substantial portion of mathematical research is related to the analysis and numerical approximation of solutions to such equations. Hyperbolic conservation laws in two or more space dimensions still pose one of the main challenges to modern mathematics. This is due to the fact that the fundamental question of an existence theory for solutions to these equations has remained an open problem for many decades, despite intensive efforts of some of the leading mathematicians during the last fifty years. Due to their importance in applications the development of efficient numerical methods had to proceed despite the fact that mathematics could not prove that the solutions, to be approximated computationally, actually exist. The preferred practice is to seek numerically objects that are mathematically proven to exist and whose properties are well understood.

VI

Preface

In this situation it is important that analysis and numerical computations, including the work in fields of application, are studied in close cooperation. By numerical experiments we can explore the structure and properties of solutions in the hope of inspiring the search for the right analytical framework for these equations. Researchers developing computational methods or applying numerical method need as much analytical information on solutions as possible in order to gain confidence in their computations. Especially applied researchers have to be aware of this situation in order to make the right assessments in their comparisons with experimental or observational data. Hyperbolic conservation laws for fluid flows in conjunction with other mechanisms such as material elasticity, magneto-hydrodynamics or combustion were at the center of a major research effort in Germany in recent years. The priority research program Analysis and Numerics for Conservation Laws was funded by the German research foundation Deutsche Forschungsgemeinschaft (DFG) for a period of six years from 1997. The program consisted of three periods of two year grants for about 25 projects in each period including funds for visitor programs and workshops. The participating groups came from astrophysics, fluid mechanics, mathematical analysis and numerical mathematics. The span of research interests was from specific applications to fundamental mathematical questions. A number of workshops were organized in each year and in 2000 the 8th International Conference on Hyperbolic Problems: Theory, Numerics and Applications was held in Magdeburg, Germany. Another special event was a Symposium on Entropy jointly organized with two other DFG priority research programs Interacting Stochastic Systems of High Complexity as well as Ergodic Theory, Analysis and Efficient Simulation of Dynamical Systems (DANSE). Together with the two co-organizers of the symposium Andreas Greven and Gerhard Keller specially selected contributions to the topic were compiled in a book titled Entropy. The diversity of topics, represented in the volume you are looking at, was one of the strengths of the research program. It brought together research groups of very different background most of which were interacting with each other for the first time. This volume can be seen as a research report of the priority research program. It contains contributions from most participating projects and gives an overview of some results achieved in the program. The authors intend to convey their results to readers outside of their own particular field. They cover a wide range of conservation laws modeling, for instance, bubbly flows, retrograde and BZT fluids, detonation waves in combustion, magnetoplasmadynamic propulsion of space craft, solar physics, chemotaxis in mathematical biology, and type Ia supernovae. Kinetic models using the Boltzmann equation or the Boltzmann-Peierls equation are explored in relation to conservation laws. The numerical approaches range over finite volume methods, central schemes, the method of transport, discontinuous Galerkin methods, kinetic schemes to meshless particle methods. The efficient use of level-set methods for problems needing better interface resolution was a research topic for a number of groups. The majority of the contributions deal

Preface

VII

with the development and study of properties of numerical schemes. Additionally, mathematicians present results on error estimates for scalar conservations laws, existence of traveling wave solutions, structural stability, relaxation dynamics and scaling limits, as well as the value of dual a posteriori error estimates for adaptive algorithms. For some computational results a color scale is a convenient way to plot field variables. Therefore, a number of color plates have been included in the back. I would like to take this opportunity to thank those who helped in many ways. Robert Paul K¨ onigs and Bernhard Nunner were in turn the program officers of the DFG for mathematics during the conception and implementation of the program. They were exceptionally helpful with very sound advice and contributed substantially to the success of this program. A big thanks also to all participating colleagues for their input into devising the program and their continual support in running the program, this especially to the members of the coordinating board Dieter H¨ anel, Wolfgang Hillebrandt, Rupert Klein, Dietmar Kr¨ oner and Willi J¨ ager. Other colleagues supported this effort by taking on the difficult task of refereeing the project proposals that always outnumbered the budgetary possibilities of the DFG. They are Wolfgang Hackbusch, Rolf Jeltsch, Wilhelm Kegel, Wilhelm Kordulla, Stephan Luckhaus, Hans Ruder, Martin Sommerfeld, Peter Szmolyan and Harry Yserentant. Most important to me was the success of a quite substantial number of doctoral students funded within the program or taking part in activities generated by the program. I enjoyed their lively interaction very much and would like to thank them for their contributions to the activities and the scientific results of this program. The manuscripts of this book were handled by my secretary Stephanie Wernicke and the final draft for the publisher was expertly compiled by R¨ udiger M¨ uller. Magdeburg December 2004

Gerald Warnecke

Contents

Wave Processes at Interfaces S. Andreae, J. Ballmann, S. M¨ uller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Numerics for Magnetoplasmadynamic Propulsion J. Heiermann, M. Auweter-Kurtz, C. Sleziona . . . . . . . . . . . . . . . . . . . . . . . 27 Hexagonal Kinetic Models and the Numerical Simulation of Kinetic Boundary Layers H. Babovsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 High-resolution Simulation of Detonations with Detailed Chemistry R. Deiterding, G. Bader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Numerical Linear Stability Analysis for Compressible Fluids A.S. Bormann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Simulation of Solar Radiative Magneto-Convection M. Sch¨ ussler, J.H.M.J. Bruls, A. V¨ ogler, P. Vollm¨ oller . . . . . . . . . . . . . . . 107 Riemann Problem for the Euler Equation with Non-Convex Equation of State including Phase Transitions W. Dahmen, S. M¨ uller, A. Voß . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Radiation Magnetohydrodynamics: Analysis for Model Problems and Efficient 3d-Simulations for the Full System A. Dedner, D. Kr¨ oner, C. Rohde, M. Wesenberg . . . . . . . . . . . . . . . . . . . . . 163 Kinetic Schemes for Selected Initial and Boundary Value Problems W. Dreyer, M. Herrmann, M. Kunik, S. Qamar . . . . . . . . . . . . . . . . . . . . . 203

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Contents

A Local Level-Set Method under Involvement of Topological Aspects F. V¨ olker, R. Vilsmeier, D. H¨ anel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen, K.P. Hadeler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Travelling Waves in Systems of Hyperbolic Balance Laws J. H¨ arterich, S. Liebscher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 The Role of the Jacobian in the Adaptive Discontinuous Galerkin Method for the Compressible Euler Equations R. Hartmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 The Multi-Scale Dust Formation in Substellar Atmospheres Ch. Helling, R. Klein, E. Sedlmayr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Meshless Methods for Conservation Laws D. Hietel, M. Junk, J. Kuhnert, S. Tiwari . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Simulations of Turbulent Thermonuclear Burning in Type Ia Supernovae W. Hillebrandt, M. Reinecke, W. Schmidt, F.K. R¨ opke, C. Travaglio, J.C. Niemeyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Hyperbolic GLM Scheme for Elliptic Constraints in Computational Electromagnetics and MHD Y.J. Lee, R. Schneider, C.-D. Munz, F. Kemm . . . . . . . . . . . . . . . . . . . . . . 385 Flexible Flame Structure Modelling in a Flame Front Tracking Scheme H. Schmidt, R. Klein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Riemann-Solver Free Schemes T. Kr¨ oger, S. Noelle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Relaxation Dynamics, Scaling Limits and Convergence of Relaxation Schemes H. Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Multidimensional Adaptive Staggered Grids S. Noelle, W. Rosenbaum, M. Rumpf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 On Hyperbolic Relaxation Problems W.-A. Yong, W. J¨ ager . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 Appendix: Color Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Wave Processes at Interfaces Sigrid Andreae1 , Josef Ballmann1 , and Siegfried M¨ uller2 1

2

Lehr- und Forschungsgebiet f¨ ur Mechanik der RWTH Aachen sigrid.andreae|[email protected] Institut f¨ ur Geometrie und Praktische Mathematik, RWTH Aachen [email protected]

Summary. We investigate the interaction of shock waves in a heavy gas with embedded light gas bubbles next to a rigid wall. This may give insight regarding cavitation processes in water. Due to the highly dynamical, unsteady processes under consideration we use an adaptive FV scheme for the computations to resolve accurately all physically relevant effects. The results are validated by comparison with tube experiments.

1 Introduction The formation and collapse of vapor bubbles in a liquid is called cavitation. Lord Rayleigh discovered that pressure waves emitted during the process of cavitation [Ray17] may damage solids, e.g., marine screw propellers. Since then, the mechanism of cavitation damaging has been subject of experimental [Lau76, LH85] and analytical research. However, it is still unclear whether the shock and rarefaction waves or the liquid jet onto the solid is the main reason for the erosion of the material. The loading on an elastic-plastic solid exposed to shock and rarefaction waves in water was investigated by Specht in [ASB00]. Hanke and Ballmann showed one-dimensional results for a bubble collapse in water in [HB98]. Cavitation is induced by a pressure drop in the liquid below vapor pressure. Such a pressure decrease may occur due to local acceleration of the liquid flow caused by geometrical constraints, e.g., if the liquid flows through a narrow orifice or around an obstacle. In case the pressure drops below vapor pressure, the liquid bursts and creates a free surface filled with gas and vapor – the bubble. Due to changes in the flow field, the pressure in the liquid may increase afterwards causing the bubble to collapse. The collapse is accompanied by strong shock and rarefaction waves running into the bubble and the surrounding liquid. The shock wave focuses in the center of the bubble. This leads to extreme physical states in the interior. In addition, the shrinking of the bubble leads to a compression of the vapor. Both effects evoke an increase

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S. Andreae, J. Ballmann, S. M¨ uller

of pressure which bulges the bubble. Hereby, a dynamic oscillation process is initiated which finally leads to the collapse of the bubble. If the collapse takes place next to a solid, the pressure distribution becomes asymmetric and a liquid jet develops [PL98] which is either directed towards or away from the solid boundary. The direction of the jet depends on the elasticity of the solid and on a ratio determined by the initial distance between the boundary and the center of the bubble divided by the maximum extension of the bubble. This has been shown experimentally by Brujan et al. in [BN+ 01], [BN+ 01a]. In order to investigate experimentally the dynamics of a bubble collapse, the bubble is produced by a laser pulse. Thereby, the fluid is heated in the focus of the laser and forms a small, hot gas bubble at very high temperature. This experimental setup provides an exact positioning of the bubble. The processes taking place in the interior of the collapsing and oscillating bubble and the prediction of onset and extent of the cavitation damaging are still subject of theoretical and experimental research. However, small time and space scales as well as the complicated dynamics make an experimental approach difficult. Therefore numerical investigations are developed to reveal information about the wave dynamics in the fluid as well as the damaging of the solid. Of particular interest are pressure contours and velocity vectors in the liquid phase as requested in [BN+ 01a]. The primary objective of the present work is to provide an accurate prediction of all occurring wave phenomena. This concerns wave interactions among each other, with phase boundaries or neighboring solids. Of particular interest is the occurrence of instabilities as, e.g., the Richtmyer-Meshkov instability. Since all present methods for simulating two-phase flows suffer from pressure oscillations at the phase boundary, we use a very dense and heavy gas instead of water. The occurring wave phenomena are expected to be qualitatively comparable with those in water. For two-phase flow problems with different equations of state the phase boundary can be tracked or treated in a Lagrangian manner as a sharp, interior boundary using two meshes [Dick96]. The latter suffers from the drawback that the mesh has to be updated in every step which is expensive and time consuming and may result in a poor mesh quality for large displacements of the interface. Instead, we track the phase boundary using a level set method. Consequently, the phase boundary is represented as a mathematically sharp boundary as we will explain in Section 2. We do not implicate surface tension and mixing of the two fluids. For the sake of completeness, it is to be mentioned that for a two phase flow of one fluid a homogenized approach is possible, as will be presented by Voß in [Vos04]. There, one equation of state is used for the liquid as well as the gaseous phase and the so-called mixture region. Here, even states consisting of gas and vapor fractions can be modeled. The small time scales of the unsteady problem require the numerical scheme to be highly efficient regarding computational time and memory requirements. This is realized by a local grid refinement strategy. Furthermore,

Wave Processes at Interfaces

3

all physically relevant phenomena have to be reliably detected and adequately resolved. Moreover, the scheme has to be robust and must not exhibit numerical oscillations, e.g., pressure oscillations at the phase boundary. The details of the numerical scheme are presented in Section 3. In Section 4 numerical results for a bubble collapse near a rigid wall are presented and the arising dynamic wave pattern is discussed.

2 Level Set For modeling two-phase flows, there are mainly two different approaches to treat the two media, a fitting of the phase boundary with two separate grids connected by interface conditions, [Dick96], or one grid with a suitable algorithm to track the phase boundary. Since in the first case the grid has to be redesigned in every time step which is very time consuming, we use only one grid and the level set method to distinguish the two fluids. The level set method, proposed by Osher and Sethian in [OS88], is a tool to track propagating interfaces without an explicit description like a function of the interface under consideration. Instead a scalar field given in the domain is used to represent the motion of the interface. Consider the case of two domains Ω1 , Ω2 separated by a contact surface Γ . Now, a scalar field φ = φ(x, t) is introduced which is φ(x, t) < 0 for Ω1 x ∈ Ω1 and φ(x, t) > 0 for x ∈ Ω2 , see Fig. 1. The Γ Ω2 interface Γ is evolved in time by the fluid velocity v. Therefore, we may describe the time evolution of Fig. 1. Domains. the scalar field φ by ∂φ + v · ∇φ = 0 . (1) ∂t There are two different methods to exploit the evolution of the scalar field for tracking a moving interface. The most common approach suggested by Osher and Sethian in [Set96] is to define the scalar field as a smooth, signed distance function to the front under consideration, whereby the material interface corresponds to φ = 0. Sussman et al. used this approach in computing incompressible two phase flows in [SSO94, SA+ 99, SF99]. The smoothness of φ has to be sustained by a reinitialization after each time step. This way, it is guaranteed that the level set itself will not steepen and develop shocks. A disadvantage of this method is the loss of conservativity. Nguyen et al. suggested methods to recover the conservativity, see [NGF02]. Here, we follow an idea of Mulder et al. [MO92] where the level set function is not a smooth but a discontinuous scalar field. Initially, we assign φ(x, t) = −1 for x ∈ Ω1 and φ(x, t) = +1 for x ∈ Ω2 . This notation for φ is sometimes called “color”-function. The sign (color) of φ decides which fluid occupies which domain. Thereby, the choice of the equation of state is

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S. Andreae, J. Ballmann, S. M¨ uller

controlled. It has to be noticed that we track the jump in φ and not the zero level set. Multiplying equation (1) with the density ̺ and employing the continuity equation of fluid dynamics yields a conservative form of the transport equation for φ, i.e., ∂̺φ + ∇ · (̺φv) = 0 , (2) ∂t Written in this form equation (2) can be added to the system of conservation equations (1) as an additional equation. The main advantage of this approach is the preservation of conservativity.

3 Governing Equations and Method of Solution The fluid flow is modeled by the time-dependent 2D Euler equations for compressible fluids. Appending the evolution equation (2) this leads to the system of conservation equations 
∂ U dV + F · n dS = 0 with (3) ∂t V ∂V ⎛

⎞ ̺ ⎜ ̺v ⎟ ⎟ U=⎜ ⎝̺E ⎠, ̺φ

⎛

⎞ ̺v ⎜ ̺ v ◦ v + p1 ⎟ ⎟ F=⎜ ⎝ v (̺ E + p) ⎠ . φv

Here, U is the array of the mean conserved quantities: density of mass, momentum, specific total energy and level set. p is the pressure and v the fluid velocity. The quantity V denotes a time-independent control volume with the boundary ∂V and the outer normal n. The flux F contains only the convective terms. Since the two fluids under consideration are gaseous both, there is no need to deal with the surface tension at their contact surface. The system of equations is closed by the perfect gas equations of state for the fluid i = 1, 2 present in the domain Ωi , i.e. the affiliated thermal equation, p = Ri ̺ T , and the caloric equation, e = cvi T . Herein, e is the internal energy and T the temperature. cvi and Ri are the heat capacity at constant volume and the special gas-constant for fluid i = 1, 2, respectively. The material properties, cvi and Ri , are listed in Table 1. The evaluation of the equations of state is governed by the scalar field φ, i.e.,

p1 = p (R1 , ̺, T ) : φ < 0 p= . (4) p2 = p (R2 , ̺, T ) : φ > 0 The conservation equations (3) are discretized by a finite volume method. The convective fluxes are determined by solving quasi–one dimensional Riemann problems at the cell interfaces. For this purpose we employ a two-phase

Wave Processes at Interfaces

5

Roe Riemann solver designed for the coupled system of the 2D Euler equations and the evolution equation (2) of the level set φ. For the construction of this solver we proceed similarly to [LV89] for real gases. In order to avoid non-physical expansion shocks we use Harten’s entropy fix. The spatial accuracy is improved by applying a quasi one-dimensional second order ENO reconstruction. Due to the strong dynamic behavior of the considered flow problems the time integration is performed explicitly. In order to properly resolve all physical relevant phenomena we need a very fine discretization of the computational domain. Due to the heterogeneity of the flow field, this high resolution is not needed throughout the entire computational domain but only locally near discontinuities. For this purpose we employ a dynamic local grid adaptation strategy to resolve the physically relevant phenomena at the expense of possibly few degrees of freedom and correspondingly reduced storage demands. The main distinction from previous work in this regard lies in the fact that we employ here recent multi-resolution techniques, see [M¨ ul02]. The starting point is to transform the arrays of cell averages associated with any given finite volume discretization into a different format that reveals insight into the local behavior of the solution. The cell averages on a given highest level of resolution are represented as cell averages on some coarse level where the fine-scale information is encoded in arrays of detail coefficients of ascending resolution. This requires a hierarchy of meshes. The multiscale representation is used to create locally refined meshes. For details we refer to [M¨ ul02]. Following Mulder [MO92] we chose φ as a color function in our computations. Mulder observed in [MO92] that using this formulation of φ the pressure shows spurious oscillations at the phase boundary. To reduce these oscillations we use averaged pressure and energy equations near the interface, i.e., ⎫ ⎧ : φ < −ǫ ⎬ p1 ⎨ , (5) p = (1 − αǫ (φ)) p1 + αǫ (φ) p2 : |φ| < ǫ ⎭ ⎩ p2 : φ > ǫ ⎧ ⎫ cv1 : φ < −ǫ ⎬ ⎨ (1 − αǫ (φ)) cv1 + αǫ (φ) cv2 : |φ| < ǫ e = T cv = T . (6) ⎩ ⎭ : φ > ǫ cv2 Here, the function αǫ (φ) is chosen as a linear interpolation between 0 and 1 in the interval [−ǫ, ǫ], i.e., αǫ (φ) = (φ/ǫ + 1) /2 for |φ| < ǫ. For our computations we chose ǫ = 0.5. Results of a comparison between the approximate Riemann solver using this averaging method and an exact Riemann solver are given in Figs. 2(a) and 2(b). Since φ is initialized by +1 for fluid 1 and by −1 for fluid 2, ǫ has to be chosen less than 1 to make sure that the modification of the pressure law is only applied in the vicinity of the interface. Note that the initial jump of φ is smeared by the FV scheme. The width of this numerical transition layer depends on the underlying grid resolution. It becomes smaller with finer

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S. Andreae, J. Ballmann, S. M¨ uller

grids. In particular, when the material boundary is a phase boundary and different equations of state have to be applied on both sides, a grid adaption strategy is strongly recommended to provide a high resolution of the interface. From this point of view, equation (5) describes not really a physically relevant phase transition, it serves more as a numerical stabilizer of the phase boundary.

4 Numerical Results The current work focuses on the ability of the scheme to accurately resolve the dynamics and wave pattern occurring in the presented test configurations for two-fluid flow. At first, we validate our scheme using experiments performed by Haas and Sturtevant, [HS87]. Herein, a shock runs across a bubble filled with helium in the one case and with R22 gas in the other. R22 is the heavy refrigerant chlorodifluromethane (CHClF2 ). In Table 1 the physical properties of the gases under consideration are given. The surrounding fluid is air in both cases. We compare our numerical results with the Schlieren photographs taken by Haas and Sturtevant. The third configuration is a helium bubble surrounded by R22 and placed next to a rigid wall, with initial conditions corresponding to an explosion problem. The main focus lies on the interaction of the emitted waves with the wall. Since we expect the occurring wave phenomena have something in common with the formation and collapse of a cavitation bubble, this configuration indicates possible causes for the damaging mechanisms accompanying cavitation in the related experiments. The Table 1. Molecular weight umol , special gas-constant R, ratio of specific heats γ and speed of sound c (at 293.15 K, 101, 35 kPa) for air, helium and R22. fluid

umol [103 kg/mol]

R [J/kg/K]

γ

c [m/s]

air helium R22

28.964 4.003 864.687

287.0 2077.0 96.138

1.4 1.66 1.178

343.3 1007.4 184.0

characteristic physical quantity in dealing with wave interactions with boundaries is the acoustic impedance ̺c. Herein, c is the speed of sound. The ratio of the acoustic impedances of two fluids governs what happens to a shock wave traveling through fluid 1 and impinging on the phase boundary between fluid 1 and 2. According to the acoustic wave theory, the impinging shock wave is split up in a transmitted part traveling through Fluid 2 and a reflected part. The larger the jump of the acoustic impedance the more energy is reflected. In case, (̺c)1 >> (̺c)2 most of the energy is reflected with a phase change of 180◦ . If (̺c)1 << (̺c)2 most of the energy is reflected, too, but without change

Wave Processes at Interfaces

7

of phase angle. The results for shock impacts on the helium and the R22 bubble, presented in Section 4.2, show the influence of the acoustic impedance on the wave pattern. This will be discussed later in this chapter. 4.1 Validation To validate the solver the solution of a 1D Riemann problem is compared to the exact solution, see Figure 2(a), evaluated with an exact Riemann solver by Colella and Glaz [CG85] which is capable to deal with two phases. Apart from some slight pressure wiggles at the phase boundary, our approximate solver gives satisfying results. For time t = 0 the membrane at x = 0 cm bursts; on the left hand side in the high pressure region is helium, on the right hand side R22. A shock runs into the low pressure region. Behind the shock the density jumps from 10 kg/m3 to 67.2107 kg/m3 . The shock is followed by a very fast contact discontinuity. Over the contact discontinuity the density drops to 10.5365kg/m3 which is nearly its right initial value. Into the high pressure region runs a rarefaction wave. Table 2. Initial conditions for two phase Riemann problem with helium and R22.

3

̺ [kg/m ] p [N/m2 ] T [K] ̺ c [kg/s/m2 ] vx , vy [m/s]

helium bubble

R22 surrounding

20.0 1.217 × 107 293.0 20101.85 0.0 , 0.0

10.0 281684.3 293.00 1821.78 0.0 , 0.0

3

5

ρ [kg/m ]

2

p [10 N/m ]

60

100

40 50

20 0

-4

-2

0

(a)

2

4 x [m]

0

-4

-2

0

2

4 x [m]

(b)

Fig. 2. Exact and approximative solution for the two-phase Riemann problem helium (left) and R22 (right); density (a) and pressure (b).

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S. Andreae, J. Ballmann, S. M¨ uller

4.2 Shock Bubble Interaction Haas and Sturtevant performed experiments with shocks passing gas inhomogeneities of helium or R22 in an air surrounding to clarify the mechanisms of turbulence and mixing caused by shock waves, see [HS87]. They carried out two-dimensional and three-dimensional (for consistency) experiments. In the two-dimensional experiments, we compare with a cylindrical volume that is first enclosed by a 0.5 µm thick nitrocellulose membrane which bursts under the impact of the shock wave. With shadowgraph photography wavefronts and the topology of the bubble were made visible. The geometrical setup shown in Figure 3 is the same for both problems. The initial mesh has 125 × 10 cells and 5 levels of refinement are used. Due to the mirror symmetry of the problem, only the upper half was computed. A shock coming from left impinges on a gas bubble. Initially, the bubble and the surrounding pre-shocked air are at rest and in thermal and mechanical equilibrium. The initial conditions for the problems discussed here are given in Tables 3 and 4. pre−shocked

r = 0.025m at 0.4895 m

shocked

air

shock position at 0.02225 m

air

vshock

helium

r

0.0445 m

post−

axsis of symmetry 0.445 m

Fig. 3. Computational domain for subsection 4.2.

Helium Bubble in Air In the first computation the setup is the helium filled gas bubble surrounded by air. Figure 8 shows the time evolution of the density gradient in the domain given in Figure 3. In Figures 9(a) and 9(b) the pressure p is plotted in zdirection over the x-y-plane and the phase boundary is marked as a black line. In the x-y-plane the pressure gradients are shown (shock fronts visible as black lines). These figures show results for the same instant as Figures 8(b) and 8(d). Figure 4 shows a comparison between the experimental and our computational results. Two clippings of the adapted mesh are shown in Figures 5(a) and 5(b). In Figure 8(a) the incoming shock (marked as i) has already crossed the most left part of the bubble boundary. It is partly transmitted as a refracted shock (rr) and partly reflected as a rarefaction wave (rw). This behavior is governed by the ratio of the acoustic impedances, see values given in Table 3. Inside the bubble the transmitted shock runs ahead since the speed of sound

Wave Processes at Interfaces

9

Table 3. Initial conditions for shock interaction with helium bubble.

̺ p ̺c vx , vy

post-shocked air

pre-shocked helium

pre-shocked air

1.376 1.575 1.742 0.396 , 0.0

0.138 1.0 0.479 0.0 , 0.0

1.0 1.0 1.183 0.0 , 0.0

in helium is higher than in air at the same temperature. The shock-front is curved due to the spherical shape of the undisturbed phase boundary. The fore-running shock in helium arches as a thin, black line from x = 0.039 m to x = 0.0315 m at the phase boundary where the shock just hits the boundary. Outside the bubble the incident shock is visible as a straight, black, vertical line. The density jump at the phase boundary is a thin, opaque line marked as (pb). Behind the shock, the reflected rarefaction wave appears as a dark area. Since in the very beginning of the shock bubble interaction the shock front is parallel to the phase boundary, all the waves travel in x-direction. Later on, the shock impinges on the phase boundary under an increasing angle, see Fig. 8(b). Due to the laws of geometrical optic the rarefaction wave is reflected under the same angle as the shock impinges on the helium surface. Since the shock inside is faster than outside, a shock wave (marked s as “ side” shock) emanates where the refracted shock meets the phase boundary. A complicated four shock configuration develops which Henderson explained in [HCP91] and called twin regular reflection refraction. In Figure 8(c) the refracted shock is just passing the most right boundary of the bubble at x = 0.073 m, whereas the incident shock is at x = 0.043 m. The acoustic impedance in the post-shocked helium is only 0.542 kg/m2 /s but in the pre-shocked air 1.183 kg/m2 /s. Therefore, the air acts in the sense of a rigid boundary which makes that the reflected part of the shock wave hitting this boundary is a shock. This reflected shock (rl) focuses on the x-axis at 0.059 m which is visible as a small, light dot in the density gradients of Fig. 8(d) (marked by an arrow). The focus is more distinct in the corresponding pressure gradients, see Fig. 9(b). Here, the pressure is plotted in z-direction over the x-y-plane. Below, in the same figure the pressure gradient isolines are shown in gray-scale. As a result of the higher shock speed the helium near the x-axis is stronger accelerated than the air above it. Thereby, an anti-clockwise rotation of the bubble content is induced and at the symmetry axis the bubble constricts and develops a small throat. The helium volume remains rotating, splits up at the x-axis and travels circulating upstream, see Figs. 8(h)–8(j). We compared our results to photographs taken by Haas and Sturtevant, see [HS87], and found a good agreement see Figs. 4. In particular, the numerical results exhibit all waves visible in the Schlieren photographs. However, since the numerical results do neither include the complete experimental setup, e.g., the support for cylindrical membrane in Fig. 4(c), nor perturbations due

10

S. Andreae, J. Ballmann, S. M¨ uller

Fig. 4. Comparison between numerical (a,b) and experimental (c,d) results for helium bubble in air. Experimental pictures scanned from [HS87]. The ring is part of the experimental setup. For the indices at the waves, see Fig. 8.

0.04 y [m]

0

0.1

0.15

0.2

x [m]

0.25

0.3

(a) Resolution of bubble (left) and shock (right).

0.04

y [m]

0

0.1

0.15

x [m]

(b) Grid resolution in the bubble region. Fig. 5. Parts of adapted mesh corresponding to Fig. 8(i).

Wave Processes at Interfaces

11

to the rupture of the membrane, the comparison can only be qualitative. Nevertheless, as indicated by the same labels as in Fig. 8 all the waves from the experiment are resolved in the computation. In particular, the topology of the bubble is excellently reproduced. Notice that the ring which was necessary to fix the bubble in the experiment must not be confused with a wave surface. To show qualitatively the grid refinement, a part of the adapted mesh is presented in Fig. 5(a). The shock at x = 0.27 m as well as the bubble contour are well resolved. The re-coarsening of the grid inside the bubble is visible in Fig. 5(b). R22 Bubble in Air For the case of an R22 bubble in air, physical data are given in Table 4. The density gradients corresponding to the experimental Schlieren images are shown in Fig. 10. Figure 6 shows the absolute value of the velocity and integralcurves of the instantaneous velocity field. In Figure 11(a) the density is plotted in z-direction over the x-y-plane. In the x-y-plane the density gradient isolines are plotted in gray-scale; Figure 11(b) shows a similar plot for the pressure in z-direction, but here the phase boundary is marked with a black line, and in the x-y-plane the pressure gradient isolines are presented. Since the acoustic impedance of R22 is only slightly higher than the acoustic impedance for the post-shocked air, see Table 4, the incident shock (i) is mainly transmitted and only a small portion is reflected as a shock (rl), see Fig. 11(a). There, the transmitted and thereby refracted shock (rr) is visible as a concave curved black line extending from x = 0.037 m to x = 0.048 m. Compared to the incident shock (i) the refracted shock (rr) is slower, because the sound speed in R22 is lower than in air. This is also the reason for the higher acceleration of the air above the bubble compared to that of the gas R22. This fact leads to a clock-wise rotation of the material in the R22 bubble later on. The outwards running shock diffracts since it is decelerated at the phase boundary, see Fig. 10(c). Between the incident shock (i) and the inside running shock (rr) develops a compression wave (cw). By the compression wave the flow direction is turned by 90◦ towards the symmetry axis as indicated in Fig. 6. The front of the refracted shock bends more and more until it focuses on the x-axis at the phase boundary, see box on left side in Table 4. Initial conditions for shock interaction with R22 bubble.

γ ̺ p ̺c vx , vy

post-shocked air

pre-shocked R22

pre-shocked air

1.4 1.376 1.575 1.742 0.396 , 0.0

1.178 2.985 1.0 1.875 0.0 , 0.0

1.4 1.0 1.0 1.183 0.0 , 0.0

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S. Andreae, J. Ballmann, S. M¨ uller

Fig. 6. Absolute value of velocity with integral-curves for vx and vy (arrows) and phase boundary (φ = 0-level) as a black dashed line.

Fig. 10(e). The arising pressure peak is depicted in Fig. 11(b). The shock is reflected after focusing and runs outwards, see wave (rf) in Fig. 10(f). Again the shock (rf) is traveling slower in the R22 than in the helium. When the incident shock has passed the bubble it crosses its symmetric counterpart, see Fig. 10(f) at x = 0.06 m. Thereby, a reflected shock (s) running upstream is induced. These two shocks (s and rf) pass across the bubble in upstream direction and cause reflected and refracted waves inside the R22 bubble visible in the density gradients of Figs. 10(g)–10(i). Since they do not produce new physical effects they are not discussed further. The bubble migrates downstream and thereby it prolongates and rolls up its top (t). In Figure 10(j) the phase boundary represented by the zero level φ = 0 is indicated by a solid, black line. Obviously, there are growing instabilities on the top of the structure. It is assumed that these are Rayleigh-Taylor instabilities due to the shock passing across a curved phase boundary. The comparison with the experiment is given in Fig. 7. Again, the ring from the experimental setup is visible. The wave (w) at the bottom of Fig. 7(c)

Fig. 7. Comparison between numerical (a,b) and experimental (c,d) results for R22 bubble in air. Experimental pictures scanned from [HS87]. The ring is part of the experimental setup. For the indices at the waves, see Fig. 10.

Wave Processes at Interfaces

13

is a reflection from the shock tube wall. A good agreement between the phase boundaries (pb), the incident and refracted shocks (i and rr) is visible in in Figs. 7(a) and 7(c). Note that even the compression wave (cw) is resolved. In Figures 7(b) and 7(d) the phase boundary (pb) as well as the shock (rf) – reflected from the focus of the refracted shock – match perfectly their experimental counterparts. At the left border the reflected shock (s) from the crossing of the incident shocks at the symmetry axis is visible. 4.3 Explosion Problem Helium in R22 Next to a Rigid Wall This computation is motivated by the idea of a laser induced vapor bubble in a liquid next to a rigid surface. The resulting bubble expands very fast and causes a pressure wave in the liquid which interacts with the wall and after reflection again with the bubble. In the case discussed in the sequel, we study the sudden expansion of a hot, high pressure helium bubble with cylindrical shape surrounded by the heavy gas R22 at low temperature. The midpoint of the bubble of diameter 0.01 m is placed at x = 0.14 m , y = 0.08 m at the right of the computational domain which has the size 0.16 m × 0.32 m. Since the problem is mirror-symmetric with respect to the x-axis we computed only the upper half. Both fluids are treated as perfect gases. The chemical data is given in Table 1. The zero level set lies on the phase boundary between helium and R22; it is indicated as a white, dashed line in the Figs. 12–17. The initial conditions correspond to a the two-phase Riemann problem in Section 4.1. The computational results for the two-dimensional explosion problem are presented in Figures 12–21. The series of Figs. 12–17 show pressure (a), density (b) and the corresponding grid (c) at different times. The phase-boundary which is evaluated from the level set function as its zero level is indicated as a white dashed line in every picture. In the very beginning a circular shock wave (s) runs undisturbed outwards, see Figs. 12(a) and 12(b). It is closely followed by the contact discontinuity (white dashed line). Inwardly, a rarefaction wave runs concentrically into the center of the bubble. There, it is reflected as a now outwards running rarefaction wave which follows the shock and the contact discontinuity, see Figs. 13(a) and 13(b). In Figures 12(c) and 13(c) it is clearly visible that the mesh is refined close to the waves and is re-coarsened in the center after the reflection of the rarefaction wave. The first wave hitting the rigid wall is the shock. It is then reflected as a shock wave and the pressure level increases due to the superposition, see pressure and density legend in Fig. 13. The reflected shock, see rs in Figure 14, passes over the contact discontinuity which is decelerated thereby. Due to the gas R22 present in the high pressure region behind the shock, the contact discontinuity does not reach the wall, but is repelled before it, see Fig. 21(a) at 20 µs. The outwards running waves set the material inside the bubble in an outwards directed motion, see Fig. 18. In the center, the rarefaction wave is reflected as an outwards running rarefaction wave which decelerates the outwards running material producing an inverse

14

S. Andreae, J. Ballmann, S. M¨ uller

pressure gradient. This gradient steepens forming a shock inside the helium domain, see s1 in Fig. 14(a) and Fig. 21(a) at time t = 29 µs and x = 0.13 m x = 0.15 m, respectively. The shock that was reflected at the solid wall catches up the right branch of shock s1 in the helium region at x = 0.149 m and y = 32 µs and merges with it. Due to the interaction a weak contact discontinuity appears, see Fig. 21(b) at x = 0.147 m and t = 35 µs. In theses pictures, the wave dynamics on the x-axis in a range from x = 0.08 − 0.16 m is plotted as pressure (Fig. 21(a)) and density (Fig. 21(b)) isolines in an x-t plane. Later on, the shock reflected from the wall also interacts with left branch of the shock s1 in the interior of the helium bubble. They cross each other mainly undisturbed and a contact discontinuity emerges from the interaction, see x = 0.137 m and t = 45 µs in Fig. 21(a) (the phase boundary is marked as a black, dotted line). Since the initial shock front is cylindrical, it hits the wall under a continuously increasing angle. At first, the shock impinges onto the wall under an angle of 0◦ and is reflected regularly. With increasing angle, however, a regular reflection becomes impossible as explained by, e.g., BenDor in [BD91] and a Mach stem (ms) develops, see Fig. 17 on the x-axis at y = 0.072 m. The slip line (sl) coming from the Mach stem is visible in Fig. 19. In the case of a cavitation bubble collapsing next to a elastic or elastic-plastic wall in water the Mach stem of the reflected shock produces surface waves in the wall which could be one reason for cavitation damaging, see Specht in [Spe00]. The pressure distribution at the rigid wall in Fig. 20 displays the loading jump evoked by the Mach stem – visible at y = 0.0517 m – on the solid.

5 Conclusion and Future Work We presented results for highly dynamical two-fluid flow problems with wave interactions at material boundaries. The comparison with experiments in Section 4.2 verifies that our solver is adequate for computing two-fluid flow problems for different perfect gases. The advanced grid refinement strategy automatically detects all appearing waves and provides a perfect resolution of those waves. For the test case of a strongly expanding high pressure cylindrical gas bubble next to a rigid wall, the complicated wave interactions are well resolved and give hope for further computations with gas and liquid. Acknowledgement. The authors would like to thank Dr. Alexander Voß, Institut f¨ ur Geometrie und Praktische Mathematik, RWTH Aachen, for helpful discussions on Riemann problems and programming issues. In particular, the discussions on the physical modeling have been very helpful. This work was financially supported by the DFG Priority Research Program “Analysis and Numerics for Conservation Laws”.

Wave Processes at Interfaces

i

(e)

rr

rw pb rl

Fig. 8. Shock-bubble (helium) interaction, (a)-(e): density gradients.

15

(j)

Fig. 8. Shock-bubble (helium) interaction, (f)-(j): density gradients.

Wave Processes at Interfaces

17

i rw Z

pb

fo c u

pb

s rl

rr

(b) Fig. 9. x-y-plane: pressure gradients, z-axis: Pressure over x-y-plane with phase boundary marked as a black, solid line. For annotations, see Fig. 8.

18

S. Andreae, J. Ballmann, S. M¨ uller

(e)

Fig. 10. Shock-bubble (R22) interaction, (a)-(e): density gradients.

Wave Processes at Interfaces

(j)

t

pb

pb

Fig. 10. Shock-bubble (R22) interaction, (f)-(j): density gradients.

19

20

S. Andreae, J. Ballmann, S. M¨ uller

pb

pb

cw Z

fo c u

s

s rr

i

(b) Fig. 11. x -y-plane: (a) density gradients, z -axis: density over x -y-plane. (b) pressure gradients, z -axis: pressure over x -y-plane. For annotations, see Fig. 10.

Wave Processes at Interfaces

s

s

0

0

(a) Pressure.

(b) Density. Fig. 12. t = 7.0µs, φ = 0 white dashed line.

s

s

0

0

(a) Pressure. (b) Density. Fig. 13. t = 18.53µs, φ = 0 white dashed line.

(c) Grid.

s

s

rs

rs s1

0

s1

0

(b) Density. (a) Pressure. Fig. 14. t = 29.00µs, φ = 0 white dashed line.

(c) Grid.

s

s

rs

rs s1 0

(c) Grid.

s1 0

(b) Density. (a) Pressure. Fig. 15. t = 40.00µs, φ = 0 white dashed line.

(c) Grid.

21

22

S. Andreae, J. Ballmann, S. M¨ uller

s

s rs

s1

rs

s1

0

0

(b) Density. (a) Pressure. Fig. 16. t = 59.99µs, φ = 0 white dashed line.

ms

ms s

s

rs 0

(c) Grid.

s1

rs

s1

0

(a) Pressure. (b) Density. Fig. 17. t = 90.01µs, φ = 0 white dashed line.

(c) Grid.

rs

Fig. 18. Absolute value of velocity with integral-curves.

Fig. 19. Isolines of Mach number; mach stem (ms), slipline (sl).

Wave Processes at Interfaces

ms s

rs s1 0

Fig. 20. Pressure distribution at wall for t = 90.01µs.

t [µ s]

t [µ s]

x [m]

x [m]

(b) Density isolines. (a) Pressure isolines with φ = 0 as black dashed line. Fig. 21. Wave dynamics at x-axis for all computed time steps.

23

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S. Andreae, J. Ballmann, S. M¨ uller

References [ASB00]

[BD91] [BN+ 01] [BN+ 01a]

[CG85] [Dick96]

[HS87]

[HB98] [HCP91] [Lau76] [LH85]

[LV89]

[MO92] [M¨ ul02]

[NGF02]

[OS88]

[PL98] [Ray17] [Set96]

Andreae, A., Ballmann, J., Specht, U.:Wave phenomena at liquid-solid interfaces. In: Proceedings of Eighth International Conference on Hyperbolic Problems - Magdeburg, Birkh¨ auser (2000) Ben-Dor, G.: Shock Wave Reflection Phenomena. Springer (1991) Brujan, E.-A., Nahen, K., Schmidt, P., Vogel, A.: Dynamics of laserinduced cavitation bubbles near an elastic boundary. J. Fluid Mech., 433, 251–281 (2001) Brujan, E.-A., Nahen, K., Schmidt, P., Vogel, A.: Dynamics of laserinduced cavitation bubbles near elastic boundaries: influence of the elastic modulus. J. Fluid Mech., 433, 283–314 (2001) Colella, Ph., Glaz, H. M.: Efficient solution algorithms for the Riemann problem for real gases. J. Comp. Phys., 59, 264–289 (1985) Dickopp, Ch.: Ein Navier Stokes L¨oser zur Simulation kollabierender Kavitationsblasen in der N¨ ahe elastischer Festk¨ orperoberfl¨ achen. Shaker Verlag, Aachen (1996) Haas, J. F., Sturtevant, B.: Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J. Fluid Mech., 181, 41–76 (1987) Hanke, M., Ballmann, J: Strong changes of state in collapsing bubbles. ZAMM, 78, 453–453. (1998) Henderson, L. F., Colella, P., Puckett, E. G.: On the refraction of shock waves at a slow-fast gas interface. J. Fluid Mech., 224, 1–27 (1991) Lauterborn, W.: Numerical Investigation of Nonlinear Oscillations of Gas Bubbles in Liquids. J. Acoust. Soc. Am., 59 2,283–293 (1976) Lauterborn, W., Hentschel, W.: Cavitation bubble dynamics studied by high speed photography and holography: Part One. Ultrasonics, 23, 260–268 (1985) Liu, Y., Vinokur, M.: Nonequilibrium flow computations. I. An analysis of numerical formulations of conservation laws. J. Comp. Phys., 83, 373–397 (1989) Mulder, W., Osher, S.: Computing interface motion in compressible gas dynamics. J. Comp. Phys., 100, 209–228 (1992) M¨ uller, S.: Adaptive Multiscale Schemes for Conservation Laws, Lecture Notes on Computational Sciences and Engineering, Vol. 27, Springer, 2002 Nguyen, D., Gibou, F., Fedkiw, R.: A fully conservative ghost fluid method & stiff detonation waves. 12th Int. Detonation Symposium, San Diego, CA (2002) Osher, S., Sethian, J. A.: Fronts Propagating with CurvatureDependent Speed: Algorithms Based on Hamilton-Jacobi Formulations. J. Comp. Phys.,79, 12–49 (1988) Philipp, A., Lauterborn, W.: Cavitation erosion by single laser-produced bubbles. J. Fluid Mech., 361, 75–116 (1998) Lord Rayleigh: On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag., 34, 94–98 (1917) Sethian, J. A.: Level Set Methods, Cambridge Monographs on Applied and Computational Mathematics (1996)

Wave Processes at Interfaces [Spe00] [SSO94] [SA+ 99]

[SF99]

[Vos04]

25

Specht, U.:Numerische Simulation mechanischer Wellen an Fluid-Festk¨ orper-Mediengrenzen. Fortschritt-Berichte VDI, Reihe 7, 398 (2000) Sussman, M., Smereka, P., Osher, S.: A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow. J. Comp. Phys.,114, 146–159 (1994) Sussman, M., Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H., Welcome, M. L.: An Adaptive Level Set Approach for Incompressible Two-Phase Flows. J. Comp. Phys., 148,81–124 (1999) Sussman, M., Fatemi, E.: An efficient, interface-preserving Level Set Redistancing Algorithm and its Application to interfacial incompressible Fluid Flow. SIAM J. Sci. Comput.,20/4, 1165–1191 (1999) Voß, A.: Exact Riemann Solution for the Euler Equations with Nonconvex and Nonsmooth Equation of State, PhD thesis, RWTH Aachen, 2005

Numerics for Magnetoplasmadynamic Propulsion J¨ org Heiermann1 , Monika Auweter-Kurtz2 , and Christian Sleziona3 1

2

3

Institute of Space Systems, University of Stuttgart [email protected] Institute of Space Systems, University of Stuttgart [email protected] IHI, Heidelberg [email protected]

Summary. A finite volume method has been developed in this work for solving the conservation equations of argon plasma flows in magnetoplasmadynamic self–field accelerators. These accelerators can be used for interplanetary spaceflight missions because of their high specific impulse and high thrust density. Calculations show in agreement with the experiment that a primary reason for plasma instabilities at high current settings – which are limiting the operational envelope and the thruster lifetime – is the depletion of density and charge carriers in front of the anode because of the pinch effect. The calculated thrust data agree well with experimental values, so that the newly developed method can be used for the design and optimization of new thrusters.

1 Introduction Manned interplanetary spaceflight is the next milestone after the Apollo missions on the moon, the introduction of the partly reusable space transportation system Space Shuttle, the commercialization of Earth–observing, navigation and telecommunications satellites and the building of the International Space Station. The choice of the propulsion system is most crucial to achieving reasonably short flight durations for interplanetary missions. Electric propulsion systems [Jah68, Auw92] are particularly suitable because they produce considerably higher specific impulses than chemical propulsion systems [BWB90]. In magnetoplasmadynamic (MPD) self–field thrusters the fuel is heated and ionized by an electric arc (Fig. 1). The plasma is accelerated by thermal expansion and magnetic forces caused by the self–induced magnetic field so that exhaust velocities above 20 km/s and thrust levels of 100 Newton can be achieved. Depending on the thruster design, the input power for MPD self–field engines is between 100 kW and 1 MW, which must be delivered by

28

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

big solar generators or nuclear energy sources. Because nuclear reactors have a compact design and because of the great distance from the sun, nuclear power is the only reasonable energy source for interplanetary flights to the outer solar system so that MPD self–field engines – besides nuclear thermal rockets – represent an advantageous propulsion concept due to their high exhaust velocity and high thrust density. Since the 1980s MPD thrusters have been investigated experimentally [Weg94, ABH98], theoretically [Wag94, Kae91] and numerically [Sle92, Sle98, Boi99] at IRS. Technology aspects for the thruster design [Bue86, Auw94] and basic plasmaphysical processes [Sch82, Sch87] have been considered in order to achieve higher efficiencies and to avoid power–limiting instabilities. Reasons for thruster instabilities, which are characterized by voltage oscillations and increasing anode losses [MAH88], are the depletion of charge carriers at the anode [Hue80], micro turbulence [Cho91, Cho99] and space charge, drift and gradient driven plasma instabilities [Sch82, WKA98]. Taking into account the high cost of experiments, the application of numerical methods is necessary for detailed basic investigations of MPD flows and for thruster optimization.

2 Conservation Equations The configurations of the IRS–built nozzle–type self–field MPD accelerators DT2 and HAT, which are investigated in this work, are axisymmetric so that the flow is rotationally symmetric. Hence, the conservation equations can be written in a two–dimensional form using cylindrical coordinates. The assumption of axisymmetry allows only instabilities to occur in an axisymmetric manner as well. Argon is the propellant considered because this noble gas is used at IRS. Assuming continuum flow and quasi neutrality of the plasma [Cap94, Sch93], conservation equations for a two–fluid flow in thermal and reaction non–equilibrium are formulated without explicitly treating the electrode sheaths. Heavy particle species (neutral, singly and multiply ionized argon) and electrons are considered. The heavy particle gas and the electron gas are treated as ideal gases. Thermal non–equilibrium between electrons and heavy particles is assumed because the densities and thereby the collision frequencies and the energy exchange between the particles can become relatively low for expanding supersonic flows, and because heavy particles are cooled on solid walls while the electrons behave adiabatically. Reaction non–equilibrium is assumed because of the high flow speeds. Because of their low mass and their high mobility the electrons are assumed to be carrying the electric current in the plasma. Hence, the electron gas is being ohmically heated, and part of the energy is transferred to the heavy particles by collisions.

Numerics for Magnetoplasmadynamic Propulsion

29

In order to be able to simulate turbulence in the interior of the thruster and in the free plasma jet, a single–equation model is employed. The arc discharge is described by the Maxwell equations of classical electrodynamics and by Ohm’s law for plasmas. Including boundary conditions and constitutive equations, one obtains a closed system of thirteen conservation equations for the species densities ni , the axial momentum ρvz , the radial momentum ρvr , the heavy particle energy eh , the turbulence ρR, the electron energy ee , and the stream function Ψ of the self–induced magnetic field: ∂ni = − div (ni v) − div jD,i + ωi ∂t

for i = 0...6 ,

(1)

∂(ρvz ) = − div f z,invisc − div f z,visc , ∂t

(2)

∂(ρvr ) = − div f r,invisc − div f r,visc + qr , ∂t

(3)

∂eh = − div f h,invisc − div f h,visc + qh , ∂t

(4)

    √ νT ∂(ρR) = − div(ρR v) + div ρ νh + ∇R + ρ(Cε2 f2 − Cε1 ) R P ∂t σε    νT ρ − ∇ ρ νh + · ∇R − ∇νT · ∇R , σε σε (5) 5k 1 ∂ee = − div(ee v) − pe div v + j · ∇Te − j · ∇pe ∂t 2e ene   6  3 kTe − div Zi jD,i + div(λe ∇Te ) 2 i=1 +

6  i=0

(6)

5

ne ni αei (Th − Te ) +

|j|2  − ωi+1 χi→i+1 , σ i=0

  Ψ Ψ vr 1 ∂Ψ = − div v + 2 r ∂t r r ⎤ ⎞ ⎡ ⎤ ⎡ ⎡ ⎤ ⎛ 0 0 0 β 1 rot⎣Ψ/r⎦ + rot⎣Ψ/r⎦ × ⎣Ψ/r⎦ − β∇pe ⎠ . (7) − rot⎝ µ0 σ µ0 0 0 0 ϕ

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J. Heiermann, M. Auweter-Kurtz, C. Sleziona

The system is hyperbolic–parabolic and includes source terms. The equations are formulated in such a way that by using fluxes a maximum of conservativity is achieved in the finite volume discretization. The details concerning the physical modelling, the complete system of equations, the boundary conditions and the definitions of all variables can be found in Ref. [Hei02].

3 Numerical Methods The solving of the conservation equations is performed on unstructured, adaptive triangular meshes with dual cells. Unstructured meshes can be produced and modified significantly easier and faster for the complex geometries of self–field accelerators than structured meshes. By appropriate adaption, the computational time for approximating a solution can be reduced considerably. For the calculation of the inviscid fluxes the robust flux vector splitting scheme of Eberle is modified and extended [HAE99, HAK99, Ebe00]. With the heavy particle pressure ph , the electron pressure pe and the magnetic field B the magnetoacoustic speed is defined as  γ(ph + pe ) B2 + . (8) c = ρ µ0 ρ One now defines a new reference speed    q2 s = αc2 + q 2 1 − 2α + α 2 , c

(9)

with q 2 = min(c2 , qn 2 ) . α is chosen as α =

2 γπ

(10) (11)

with γ = 5/3 being the adiabatic coefficient. The velocity normal to the cell face of two neighbouring dual cells is qn = vz nz + vr nr .

(12)

In the case of qn vanishing, the choice of α leads to the mass flux from one side being equal to the effusion flux. The eigenvalues of the heavy particle flow are now defined as λ0 = qn , λ1 = λ0 + s , λ2 = λ0 − s .

(13)

Numerics for Magnetoplasmadynamic Propulsion

31

With the physical states left (l) and right (r) of a cell face the following equations are chosen for the splitting: 1 (λ1l + |λ1l |), 4 1 h1r = (λ1r − |λ1r |), 4 1 h2l = (λ2l + |λ2l |), 4 1 h2r = (λ2r − |λ2r |). 4 The upwind flux function for the species densities ni is now h1l =

(14) (15) (16) (17)

ni Finvisc = nil (h1l + h2l ) + nir (h1r + h2r ) .

(18)

With r1 = ρh1

and r2 = ρh2

(19)

the axial and the radial inviscid momentum fluxes are given by ρvz Finvisc ⎡⎛

⎢⎜ ⎜ =⎢ ⎣⎝vz + ⎡⎛

⎢⎜ ⎜ +⎢ ⎣⎝vz +

 

B2 ph + pe + 2µ0 ρs

ph + pe +

B2 2µ0

ρs

 

⎞

⎛

nz ⎟ ⎜ ⎟ r1 + ⎜vz − ⎝ ⎠



B2 ph + p e + 2µ0 ρs



⎞ ⎤ nz ⎟ ⎥ ⎟ r2 ⎥ ⎠ ⎦

  ⎞ ⎤l ⎛ ⎞ B2 p + p + nz ⎟ nz ⎟ ⎥ h e ⎜ 2µ0 ⎟ r1 + ⎜vz − ⎟ r2 ⎥ ⎝ ⎠ ⎠ ⎦ ρs

r

(20)

and ρvr Finvisc ⎡⎛

⎢⎜ ⎜ =⎢ ⎣⎝vr + ⎡⎛

⎢⎜ ⎜ +⎢ ⎣⎝vr +

 

B2 ph + p e + 2µ0 ρs

ph + pe + ρs

B2 2µ0

 

⎞

⎛

nr ⎟ ⎜ ⎟ r1 + ⎜vr − ⎠ ⎝



B2 ph + p e + 2µ0 ρs



⎞ ⎤ nr ⎟ ⎥ ⎟ r2 ⎥ ⎠ ⎦

  ⎞ ⎤l ⎛ ⎞ B2 p + p + nr ⎟ nr ⎟ ⎥ h e ⎜ 2µ0 ⎟ r1 + ⎜vr − ⎟ r2 ⎥ . ⎝ ⎠ ⎠ ⎦ ρs

The heavy particle energy flux is

r

(21)

32

J. Heiermann, M. Auweter-Kurtz, C. Sleziona eh Finvisc

= +





ph B2 ρ + (vz 2 + vr 2 ) + ph + pe + γ−1 2 2µ0 2

ph ρ B + (vz 2 + vr 2 ) + ph + pe + γ−1 2 2µ0





r1 + r2 ρ r1 + r2 ρ





l

(22) .

r

For the turbulence, the electron energy and the magnetic field, the upwind flux functions are constructed like equation (18): ρR Finvisc = (ρR)l (h1l + h2l ) + (ρR)r (h1r + h2r ) ,

(23)

ee Finvisc = eel (h1l + h2l ) + eer (h1r + h2r )

(24)

B = Bl (h1l + h2l ) + Br (h1r + h2r ) . Finvisc

(25)

and

This upwind formulation is also being used for the discretization of the term div v, because a central discretization would lead to numerical oscillations: div v Finvisc = h1l + h1r + h2l + h2r .

(26)

The upwind algorithm is very robust with respect to strong shocks, and it is simple and requires only a few calculations to obtain a good approximation of the solution of the Riemann problem. The variables which are needed at the cell faces for the approximate solution of the Riemann problem are linearly reconstructed by a Weighted Essentially Non–Oscillatory (WENO) scheme [Fri98] so that the discretization achieves a second–order accuracy in space. The WENO algorithm is explained in the following for the density ρ. A dual cell i is surrounded by J triangles. On each triangle Tj the gradient ⎤ ⎡ ρz,j ⎦ (∇ρ)Tj = ⎣ (27) ρr,j

can be computed with Cramer’s rule. The weight for the gradient is chosen as ωj =

(ε + ρ2z,j + ρ2r,j )−2 J  (ε + ρ2z,k + ρ2r,k )−2

(28)

k=1

so that the contributions of the biggest gradients are dampened. ε is a small number (10−20 ) which helps to avoid divisions by zero. Furthermore, J  j=1

ωj = 1 ,

(29)

Numerics for Magnetoplasmadynamic Propulsion

33

so that the WENO algorithm is linear preserving. The WENO gradient for a dual cell is now J  (∇ρ)WENO (30) ωj (∇ρ)Tj . = i j=1

The non–linear weights ωj can be computed easily. They assure oscillation– free solutions in the vicinity of discontinuities and very smooth solutions for regions with smooth solutions.

The discretization of the viscous fluxes is done by a central scheme using average fluxes of neighbouring triangles. For all other derivatives which cannot be written in divergence or curl form a least–squares discretization is chosen. All conservation equations are discretized using the theorems of Gauß and Stokes. For the discretization of the right–hand–side of the species conservation equation (1),  K  1 ni ni FWENO (xωk ) ∆Aωk − RHS = ∆Vω k=1  (31) K  ! 1 (jD,i )Tωjk + (jD,i )Tωkl ∆Aωk nωk + ωi − 2 k=1

ni (xωk ) for a dual cell ω, which is surrounded by K neighboring cells, FWENO denotes the upwind flux through a cell surface ∆Aωk at the Gauss point xωk , where the left– and right–hand values at the cell face are computed by WENO reconstruction. The flux vector splitting already includes the normal surface vector nωk . It has to be written here, though, in the central discretization of the diffusion fluxes jD,i on the adjacent left and right triangles Tωjk and Tωkl . The spatial discretization of all other equations follows the same way.

Since a steady–state solution is wanted, an explicit first order time stepping scheme is employed for time–stabilization. For example, the calculation of the explicit local time step for equation (7) takes into account the magnetic diffusion because of the finite conductivity σ which is in some sense comparable with a classical heat conduction problem. With xm,k denoting the center of mass of cell k, all K neighbouring cells are included:   2 K K  1  µ0 1 σω + ∆tB = CFLB · σk |xm,k − xm,ω | . (32) 2 K +1 K k=1

k=1

The CFLB number dampens the other mechanisms in Ohm’s law which are not taken into consideration in equation (32). Before the actual time stepping is performed, all local time steps are multiplied with different random numbers RND between 0 and 1:

34

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

∆tB,RND = ∆tB · RND B . The time integration for the stream function Ψ ⎡

⎢  ⎢ 1 1 ⎢ n+1 n Ψ −Ψ = ∆tB,RND ⎢ 1 − RHS B ω rm,ω + ⎢ K K ⎣

(33)

K 

⎤

RHS B k rm,k ⎥ ⎥ ⎥ k=1 ⎥ . (34) ⎥ K ⎦

includes local residual smoothing. The time integration of all other equations follows the same way. The requirement of robustness is the reason for the choice of first–order time– stepping. Higher–order schemes like Runge–Kutta schemes generally accelerate convergence as long as the right hand sides are relatively simple. In the case of the complex non–linear right hand side presented here, they tend to produce numerical oscillations. The random time stepping also adds stability. It allows an increase of the CFL number by about an order of magnitude. Although its mathematical properties are not yet understood, it is clear that it introduces a non–linear, dampening dissipation which disappears once the steady state solution is achieved. It should be mentioned here that results obtained with and without random time steps are the same. For adapting the meshes, a combination of an analytically obtained error indicator [SW97, IWH00] and a gradient indicator is used. The local and weighted indicator νω for a dual cell ω with the volume ∆Vω is computed from a discrete solution of the equations (1) to (4). The basic concept is to calculate the jumps of the fluxes across the cell surfaces which is denoted by [ ]E : max {|[ρv · n]E | ∆AE } E⊂∂ω νω(1) = , (35) ρ ∆Vω # $ "# max #[(f z,invisc + f z,visc ) · n]E # ∆AE E⊂∂ω % νω(2) = , (36) (ρvz )2 + (ρvr )2 ∆Vω # # #
# # # "# $ # max #[(f r,invisc + f r,visc ) · n]E # ∆AE + ## qr dV ## E⊂∂ω # # ω % νω(3) = , (37) 2 2 (ρvz ) + (ρvr ) ∆Vω # # #
# # # $ # "# # # # max [(f h,invisc + f h,visc ) · n]E ∆AE + # qh dV ## E⊂∂ω # # ω , (38) νω(4) = eh ∆Vω

Numerics for Magnetoplasmadynamic Propulsion

1&

35

'

(39) ν (1) + νω(2) + νω(3) + νω(4) . 4 ω Choosing the conserved variables as weights proved to be very helpful in numerous numerical experiments. The new local mesh size g1 is now defined as ( Cε1 g1 = . (40) νω νω :=

The mathematically based indicator is combined with the gradient indicator ) * Cε2 *

. g2 = * (41) + |∇Th | |∇vz | |∇vr | |∇ρ| + + + |vz | |vr | ρ Th

The constants Cε1 and Cε2 are specified by the user so that the arithmetic average 1 g = (g1 + g2 ) (42) 2 is computed for the new local mesh size g which is required by the advancing front algorithm for remeshing the domain. Boundary layers, shocks and regions with strong source terms are refined reasonably from a physical point of view so that the computational time can be reduced.

4 Results The two IRS–developed MPD thrusters HAT and DT2 have been investigated with the above described finite volume method. Both thrusters have an exit nozzle diameter of 100 mm. The results presented here are for an argon mass flow rate of 0.8 g/s. Fig. 2 shows the thruster HAT being fired in the laboratory. A typical discretization of the computational domain is shown in Fig. 3. In the nozzle area, the mesh is highly refined for a good resolution of the boundary layers and for capturing the oblique shock (Fig. 4) which is also visible in the experiment (Fig. 2). The current distributions at high currents of 4000 A (Fig. 5) and 5000 A (Fig. 6) show that the electric arc starts to constrict downstream of the nozzle throat because of the pinch effect. This effect also becomes slightly visible in the density distributions shown in Figs. 7 and 8. The density is being lowered a little bit downstream of the nozzle throat at the anode wall. However, the

36

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

density stays at a fairly high level at the inner anode surface so that there are enough charge carriers available for the electric arc. So far, no plasma instabilities were observed for the HAT at 5000 A in the laboratory experiment. However, the situation is completely different for the DT2. The arc also constricts at 4000 A (Fig. 9) and 5000 A (Fig. 10). Due to the slightly different nozzle shape of the DT2, the density depletion in front of the inner anode wall (Figs. 11 and 12) is significantly stronger than in the HAT. In the laboratory experiment, a current of 5000 A cannot be achieved for a mass flow rate of 0.8 g/s. Instead, plasma oscillations start at 4650 A. This can be linked to the depletion of density, and thus, charge carriers, in front of the anode surface because of the pinch effect. The experimental instability at 4650 A is not directly predicted by CFD because it is a 3–D effect. Experimental and numerical data for the thrust are given in the tables 1 and 2. The experimental data for the HAT show a slight drifting of the data which is caused by thermal heating of the pendulum thrust balance [ABH98, Weg94]. Nevertheless, the comparison of numerical and experimental data shows a good agreement.

Table 1. Thrust HAT (num.: numerical, exp.: experimental [ABH98, Weg94]). I [A]

m ˙ [g/s]

2000 2500 3000 3500 4000 4500 5000

0,8 0,8 0,8 0,8 0,8 0,8 0,8

F [N] num. 4,3 5,2 5,9 6,8 8,5 9,2 10,1

F [N] exp. 3,7-4,7 5,3 5,7-7,0 7,4 7,8-9,4 10,0 11,1

Table 2. Thrust DT2 (num.: numerical, exp.: experimental [ABH98, Weg94]). I [A]

m ˙ [g/s]

2000 2500 3000 3500 4000 4500 5000

0,8 0,8 0,8 0,8 0,8 0,8 0,8

F [N] num. 4,3 4,8 5,8 6,5 7,1 7,7 8,2

F [N] exp. 4,6 5,2 6,2 7,1 8,3 9,1 -

Numerics for Magnetoplasmadynamic Propulsion

37

5 Summary For the numerical solution of the conservation equations for argon plasma flows in magnetoplasmadynamic self–field thrusters a new finite volume method for unstructured, adaptive meshes has been developed. It has been shown for the first time by numerical simulations of the MPD thruster DT2 that a strong depletion of density in front of the anode is caused by the pinch effect at high current settings. The correlation of the numerical simulations with experimental data at instability–producing current settings clearly points to the fact that a primary reason for plasma instabilities in nozzle–type self–field MPD thrusters is the depletion of density and charge carriers in front of the anode caused by the pinch effect, which has been predicted theoretically for a long time. The prediction of thrust for the MPD thrusters DT2 and HAT agrees well with experimental data. Also, the simulations show that the depletion of charge carriers in front of the anode is delayed for HAT because its nozzle expansion angle is slightly less than that of DT2. This is probably the reason why no plasma instabilities can be observed for HAT at high current settings during experiments. The numerical method can therefore already be used for the design and development of new MPD thrusters. It particularly makes an optimization of the nozzle with respect to thrust possible, and it helps to preselect a nozzle geometry so that the depletion of density in front of the anode can be shifted to higher current settings. In addition, the numerical data can be used for further instability analysis. Acknowledgement. We gratefully acknowledge the decisive support by the German Research Foundation DFG (Deutsche Forschungsgemeinschaft) through the project “Conservation Equations and Numerical Solutions for Technically Relevant Magneto–Plasmas” (Au 85/9-1,2,3), which belongs to the German Priority Research Program (Schwerpunktprogramm) “Analysis and Numerics for Conservation Laws”. We thank L. Cassady (Princeton, USA), E. Choueiri (Princeton, USA), C. Coclici (Bosch, Stuttgart), A. Eberle (EADS, Munich), U. Iben (Bosch, Stuttgart), S. Jardin (Princeton, USA), H. Kaeppeler (Stuttgart), A. Kodys (Princeton, USA), A. Kolesnikov (Moskau, Russia), A. Meister (L¨ ubeck), G. Moro¸sanu (Ia¸si, Romania), C.–D. Munz (Stuttgart), P. Nikrityuk (Dresden), K. Sankaran (Princeton, USA), T. Sonar (Braunschweig), G. Warnecke (Magdeburg) and W. Wendland (Stuttgart) for their dedication, their patience and their support of our project.

38

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

Anode C a th o d e

Insulator

Fig. 1. MPD thruster: Technical drawing (left), principle of its functionality (right).

Fig. 2. HAT firing in the laboratory, 2000 A, 0.8 g/s. (See also color figure, Plate 1.)

Numerics for Magnetoplasmadynamic Propulsion

39

Anode

•

m

Fig. 3. Partial view of the adapted mesh for the MPD self–field thruster HAT (28034 nodes, 2000 A, 0.8 g/s).

Fig. 4. HAT: Heavy particle temperature Th , 2000 A, 0.8 g/s. (See also color figure, Plate 2.)

40

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

Anode Insulator •

m Cathode

Fig. 5. HAT: Current distribution Ψ , 4000 A, 0.8 g/s. (250 A between 2 isolines)

Anode Insulator •

m Cathode

Fig. 6. HAT: Current distribution Ψ , 5000 A, 0.8 g/s. (250 A between 2 isolines)

Numerics for Magnetoplasmadynamic Propulsion

41

Fig. 7. HAT: Density log10 (ρ), 4000 A, 0.8 g/s. (See also color figure, Plate 3.)

Fig. 8. HAT: Density log10 (ρ), 5000 A, 0.8 g/s. (See also color figure, Plate 4.)

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

Insulator

42

Anode Insulator •

m Cathode

Insulator

Fig. 9. DT2: Current distribution Ψ , 4000 A, 0.8 g/s. (250 A between 2 isolines)

Anode Insulator •

m Cathode

Fig. 10. DT2: Current distribution Ψ , 5000 A, 0.8 g/s. (250 A between 2 isolines)

Numerics for Magnetoplasmadynamic Propulsion

43

Fig. 11. DT2: Density log10 (ρ), 4000 A, 0.8 g/s. (See also color figure, Plate 5.)

Fig. 12. DT2: Density log10 (ρ), 5000 A, 0.8 g/s. (See also color figure, Plate 6.)

44

J. Heiermann, M. Auweter-Kurtz, C. Sleziona

References [ABH98] Auweter-Kurtz, M., Boie, C., Habiger, H., Kaeppeler, H.J., Kurtz, H.L., Sleziona, P.C., Wegmann, T., Winter, M.W.: Numerische Simulation von MPD–Triebwerken und Vergleich mit durchzuf¨ uhrenden experimentellen Untersuchungen. Endbericht zum DFG–Forschungsvorhaben Au85/5-2, Institut f¨ ur Raumfahrtsysteme, Universit¨ at Stuttgart (1998) [Auw92] Auweter–Kurtz, M.: Lichtbogenantriebe f¨ ur Weltraumaufgaben, B.G. Teubner Stuttgart (1992) [Auw94] Auweter–Kurtz, M.: Plasma Thruster Development Program at the IRS. Acta Astronautica, 32, No. 5, 377–391 (1994) [BWB90] Bennett, G.L., Watkins, M.A., Byers, D.C., Barnett, J.W.: Enhancing Space Transportation: The NASA Program to Develop Electric Propulsion. IEPC–91–004, 21st AIAA/DGLR/JSASS International Electric Propulsion Conference, Orlando, Florida (1990) [Boi99] Boie, C.: Numerische Simulation magnetoplasmadynamischer Eigenfeldtriebwerke mit hochaufl¨ osenden adaptiven Verfahren. Dissertation, Institut f¨ ur Raumfahrtsysteme, Fakult¨ at Luft- und Raumfahrttechnik, Universit¨ at Stuttgart (1999) [Bue86] B¨ uhler, R.D.: Plasma thruster development: magnetoplasmadynamic propulsion, status and basic problems, AFRPL TR–86–013, Air Force Rocket Propulsion Laboratory, Edwards AFB, California (1986) [Cap94] Cap, F.: Lehrbuch der Plasmaphysik und Magnetohydrodynamik. Springer Verlag (1994) [Cho91] Choueiri, E.Y.: Electron–Ion Streaming Instabilities of an Electromagnetically Accelerated Plasma. Ph.D. thesis, Princeton University, New Jersey, USA (1991) [Cho99] Choueiri, E.Y.: Anomalous resistivity and heating in current–driven plasma thrusters. Physics of Plasmas, 6, No. 5, 2290–2306 (1999) [Ebe00] Eberle, A.: A Nonlinear Flux Vector Split Defect Correction Scheme for Fast Solutions of the Euler and Navier Stokes Equations. In: Freist¨ uhler, H., Warnecke, G. (eds.) Hyperbolic Problems: Theory, Numerics, Applications. Birkh¨ auser (2001) [Fri98] Friedrich, O.: Weighted Essentially Non–Oscillatory Schemes for the Interpolation of Mean Values on Unstructured Grids. Journal of Computational Physics 144, 194–212 (1998) [Hei02] Heiermann, J.: Ein Finite–Volumen–Verfahren zur L¨osung magnetoplasmadynamischer Erhaltungsgleichungen. Dissertation, Institut f¨ ur Raumfahrtsysteme, Fakult¨ at Luft– und Raumfahrttechnik und Geod¨ asie, Universit¨ at Stuttgart (2002) [HAE99] Heiermann, J., Auweter–Kurtz, M., Eberle, A., Iben, U., Sleziona, P.C.: Robuste hochaufl¨ osende Methoden zur Simulation magnetoplasmadynamischer Raketentriebwerke. DGLR-JT99-034, DGLR Jahrbuch, ISSN 0070–4083, 923–929 (1999) [HAK99] Heiermann, J., Auweter-Kurtz, M., Kaeppeler, H.J., Eberle, A., Iben, U., Sleziona, P.C.: Recent Improvements of Numerical Methods for the Simulation of MPD Thruster Flow on Adaptive Meshes. IEPC–99–169, 26th International Electric Propulsion Conference, Kitakyushu, Japan (1999)

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[Hue80] H¨ ugel, H.: Zur Funktionsweise der Anode im Eigenfeldbeschleuniger. Habilitation, Fakult¨ at f¨ ur Luft- und Raumfahrttechnik, Universit¨ at Stuttgart, DFVLR Forschungsbericht 80–20, (1980) [IWH00] Iben, U., Warnecke, G., Heiermann, J., Auweter-Kurtz, M.: Adaptive Numerics for the Simulation of Magneto–Plasmadynamic Rocket Thrusters. In: Neittaanm¨ aki, P., Tiihonen, T., Tarvainen, P. (eds.) Proceedings of The Third European Conference on Numerical Mathematics and Advanced Applications, Jyv¨ askyl¨ a, Finland. World Scientific, Singapore (2000) [Jah68] Jahn, R.G.: Physics of Electric Propulsion. McGraw–Hill Series in Missile and Space Technology (1968) [Kae91] Kaeppeler, H.J.: Basic equations and elements of the dispersion relation for a four–fluid formalism of magneto–plasmadynamics with non–equilibrium ionization. IEPC–91–059, Proceedings of the 22nd International Electric Propulsion Conference, Viareggio, Italy (1991) [MAH88] Merke, W.D., Auweter–Kurtz, M., Habiger, H., Kurtz, H.L., Schrade, H.O.: Nozzle Type MPD Thruster Experimental Investigations. IEPC–88– 028, Proceedings of the 20th International Electric Propulsion Conference, Garmisch–Partenkirchen, Deutschland (1988) [Sch82] Schrade, H.O.: Basic Processes of Plasma Propulsion. Interim Scientific Report, Air Force Office of Scientific Research Grant 82–0298, Institut f¨ ur Raumfahrtsysteme, Universit¨ at Stuttgart (1982) [Sch87] Schrade, H.O.: Basic Processes of Plasma Propulsion. Interim Scientific Report, Air Force Office of Scientific Research Grant 86–0337, Institut f¨ ur Raumfahrtsysteme, Universit¨ at Stuttgart (1987) [Sch93] Schumacher, U.: Fusionsforschung. Hanser Verlag (1993) [Sle92] Sleziona, P.C.: Numerische Analyse der Str¨omungsvorg¨ ange in magnetoplasmadynamischen Raumfahrtantrieben. Dissertation, Institut f¨ ur Raumfahrtsysteme, Fakult¨ at Luft- und Raumfahrttechnik, Universit¨ at Stuttgart (1992) [Sle98] Sleziona, P.C.: Hochenthalpiestr¨omungen f¨ ur Raumfahrtanwendungen. Habilitationsschrift, Fakult¨ at Luft- und Raumfahrttechnik, Universit¨ at Stuttgart (1998) [SW97] Sonar, T., Warnecke, G.: On Finite Difference Error Indication for Adaptive Approximations of Conservation Laws (revised 2nd printing). Hamburger Beitr¨ age zur Angewandten Mathematik, Reihe A, Preprint 122, Hamburg (1997) [Wag94] Wagner, H.P.: Theoretische Untersuchung drift- und gradientengetriebener Instabilit¨ aten in MPD–Triebwerksstr¨ omungen. Dissertation, Institut f¨ ur Raumfahrtsysteme, Fakult¨ at Luft– und Raumfahrttechnik, Universit¨ at Stuttgart (1994) [WKA98] Wagner, H.P., Kaeppeler, H.J., Auweter–Kurtz, M.: Instabilities in MPD thruster flows: 1. Space charge instabilities in unbounded and inhomogeneous plasmas. 2. Investigation of drift and gradient driven instabilities using multi–fluid plasma models. J. Phys. D: Appl. Phys. 31, 519–541 (1998) [Weg94] Wegmann, T.: Experimentelle Untersuchung kontinuierlich betriebener magnetoplasmadymamischer Eigenfeldtriebwerke. Dissertation, Institut f¨ ur Raumfahrtsysteme, Fakult¨ at f¨ ur Luft- und Raumfahrttechnik, Universit¨ at Stuttgart (1994)

Hexagonal Kinetic Models and the Numerical Simulation of Kinetic Boundary Layers Hans Babovsky Institute for Mathematics, Ilmenau Technical University, P. O. Box 100565, D-98684 Ilmenau, Germany, [email protected] Summary. The paper deals with the transition regime of gas flows between the mesoscopic and the macroscopic levels. We survey theoretical results and provide numerical tools. As the basic numerical scheme for the solution of the Boltzmann equation we use a hexagonal model proposed in [1]. Key words: Boltzmann equation, discrete velocity model, kinetic boundary layer. MSC: 76N20, 76P05.

1 Introduction The evolution of a gas flow may be formulated on different levels of description. The most intuitive is the microscopic level, where a gas is considered as a huge N -point system of particles which move and interact as a Hamiltonian system (with short range interaction potentials). Of course from a numerical point of view this is certainly not the level of choice, since the number of particles is extremely too large for a simulation. However, the main method in practice for the simulation of rarefied gas flows is that of Monte Carlo systems, where features of the microscopic view point are taken over. In the mesoscopic level, particles as individuals are given up and replaced by density functions in the physical phase space. The corresponding equation is the Boltzmann equation. The influence of particle interactions (collisions) is balanced in the Boltzmann collision integral. Finally, on the macroscopic level, the densities in velocity space are given up and replaced with a system of moments. The resulting equations are the Euler resp. the Navier Stokes systems. Passing over from the microscopic to the macroscopic level results in a system reduction and thus in diminished complexity. For numerical purposes, reduced systems are to be preferred. However, at times the more complex description is necessary for an appropriate simulation. For example, calculating the Boltzmann collision

48

H. Babovsky

operator (in a fully six-dimensional phase space) requires in each time step the evaluation of a five-dimensional in each point of the discretized calculational domain. (See section 2 for the two-dimensional version.) This is a highly time and memory consuming task. To cope with this complexity, one may try to get rid of the collision operator and pass to simplified systems like relaxation schemes (like BGK; see [2]) or truncated moment hierarchies [16]. When there is a need for the collision operator, one may try to reduce the Boltzmann regime to a domain as small as possible or to find kinetic model systems which are more efficiently solvable than the original (discretized) operator. Progress in both directions is the subject of this paper. More specifically, • The evolution of the collision integral requires for any velocity pair (v, w) the integration over the surface of the ball (in 3D) resp. the circle (in 2D) between v and w (see Fig. 1). This lower dimensional manifold is not ”compatible” with a regular rectangular grid discretization which leads to an extremely low consistency order for classical integration schemes (see [10]). We propose the replacement of the circular structure with a hexagonal system as described in section 3. In [1], it was shown that kinetic equations based on hexagonal models possess all properties of classical kinetic systems (H-theorem, conservation laws, linearized collision operators etc.). From a numerical point of view, their efficiency has been demonstrated in [6]. • In flows around bodies, the only domain requiring the kinetic description by the Boltzmann equation is in many cases a very thin zone along the boundary of the body, called the kinetic boundary layer. The coupling of these layers to the macroscopic fluid flow has been e.g. investigated in [15]. The most promising way of coupling (for small Knudsen numbers) is the calculation of jump conditions which provide a kinetic correction to the macroscopic formulation. We develop an algorithm for this. The scope of the paper is as follows. In section 2, we survey some basic facts and some most recent results concerning the relationship between mesoscopic and macroscopic equations as far as they concern the topic of this paper. Section 3 shortly introduces the hexagonal system as proposed in [1]. This part also includes some demonstrations of the flexibility of the system when applied to different areas of fluid flows, and a continuum limit for the discretization parameter going to 0. Section 4 investigates boundary layers and proposes an algorithm for the calculation of jump conditions. From now on we switch exclusively (except section 2.2) to two-dimensional velocity models. Three-dimensional models exhibit similar properties. Their investigation is on progress; a publication is in preparation.

Hexagonal kinetic models and simulation of kinetic boundary layers

49

2 Structure of fluid flows 2.1 Boltzmann equation versus Navier Stokes system The Boltzmann equation is an equation for the density function f = f (t, x, v) for particles in the four-dimensional phase space. (In the following we use the conventions x = (x, y)T and v = (vx , vy )T .) Its stationary, spatially onedimensional version reads vx ∂x f (x, v) = J(f , f )(x, v)

(1)

where J(f , f ) is the Boltzmann collision integral

J(f , f )(v) = k(|v − w|, η)[f (v′ )f (w′ ) − f (v)f (w)] dη d2 w lR2

(2)

S1

The velocity pairs (v′ , w′ ) and (v, w) are related via momentum and energy conservation and are parametrized by a unit vector η ∈ S 1 , v′ = v − η, v − w · η,

(3)

′

w = w + η, v − w · η

(4)

For given (v, w), the set of all possible ”pre-collision” velocities is located on the circle between v and w, as illustrated in Fig. 1.

V’ W

V W’ Fig. 1. Collision pairs.

For details concerning elementary properties of the Boltzmann equation we refer the reader to standard literature, e.g. [4, 11]. The only collision invariants of the collision operator are the four functions φ(v) = 1, vx , vy , |v|2
φ(v)J(f , f )(v) d2 v = 0 (5) lR2

50

H. Babovsky

Given a density function f (v), define mass ρ and the bulk velocities u = (ux , uy )T by
ρ := f (v) d2 v (6) lR2
ρu := vf (v) d2 v (7) lR2

internal energy e and heat flux vector q = (qx , qy )T by
1 ρe = |v − u|2 f (v) d2 v 2 lR2
1 (v − u)|v − u|2 f (v) d2 v q= 2 lR2 and the components of the stress tensor Π = (pij )i,j∈{x,y} by
pij := (vi − ui )(vj − uj )f (v) d2 v .

(8) (9)

(10)

lR3

Multiplying the Boltzmann equation (1) with the four abovementioned functions φ and integrating yields the four moment equations

∂x

&

∂x (ρux ) = 0 ' ρux u + p(1) = 0

(11) (12)

    1 2 (1) ∂x ρux |u| + e + p , u + q1 = 0 2

(13)

where p(1) = (pxx , pxy )T denotes the first column of Π. Finally, the pressure p is defined by  pii . (14) 2p = ρe = i∈{x,y}

Equations (11) – (13) represent a non-closed system of four equations for the nine unknowns ρ, ux , uy , pxx , pxy , pyy , qx , qy , e. This system may be closed by introducing some phenomenological relations expressing pij and qx in terms of the other quantities. For a Navier-Stokes-Fourier fluid these relations are (see [11, Sections II.8, IV.7]): pxx = p + µ(−∂x ux + ∂y uy ) pyy = p + µ(+∂x ux − ∂y uy ) pxy = −µ(∂x uy + ∂y ux ) q = −κ∇x e .

,

, ,

(15) (16) (17) (18)

Hexagonal kinetic models and simulation of kinetic boundary layers

51

with viscosity and heat conduction coefficients µ and κ. Inserting these yields the following Navier-Stokes equations. ∂x (ρux ) = 0    1 2 ∂x ρ u x + e = µ ∂x2 ux 2 ∂x



∂x (ρux uy ) = µ ∂x2 uy ,  , 2 1 µ ρux |u| + 3e = κ ∂x2 e + ∂x2 |u|2 2 2

(19) (20) (21) .

(22)

Assuming a non-permeable wall at x = 0 and with this ux (0) = 0, equation (19) yields ux ≡ 0

,

(23)

and a linear profile for uy , uy (x) = uy (0) + ∂uy · x .

(24)

In case of uy = 0, we also find a linear profile for the internal energy e. The problem of determining slip boundary conditions is to find relations between the boundary values at x = 0 and the constant gradients. Furthermore, equations (10) and (11) turn into ∂x (p11 ) = ∂x (p12 ) = 0 , ∂x (p12 uy + q1 ) = 0 .

(25) (26)

In numerical simulations, on may use these equations as a test which parts of the half space system belongs to the kinetic boundary layer and which to the fluid system. 2.2 Flows along walls Here. we switch to three-dimensional flows. Sone et al. [15] have studied the structure of a fluid flow governed by the Boltzmann equation in the presence of a physical wall. Consider a compressible steady fluid flow adjacent to some smooth wall ∂Ω, which is given by the Boltzmann equation v · ∇x f (x, v) = 1/ǫ2 · J[f (x, ·), f (x, ·)](v)

(27)

supplemented by the diffuse boundary condition at a ∈ Ω, f (a, v) = c(a) · mT (a) (v) for

n(a), v > 0

.

(28)

Here, J[f , f ] is the Boltzmann collision integral as described in the preceding section, and ǫ2 is proportional to the Knudsen number, i.e. to the mean free

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H. Babovsky

path; n(a) is the normal vector on ∂Ω at a pointing into the fluid domain; mT is a Maxwellian with zero mean velocity and temperature T , mT (v) = (2πT )−3/2 exp[−|v|2 /(2T )]

(29)

(where the Boltzmann constant has been for simplicity set equal to 1); c(a) is the normalizing constant guaranteeing that there is no flux through the wall, i.e.
n(a), vf (a, v)d3 v = 0 . (30) lR3

In this paper we are interested in the case of small Knudsen number ǫ2 , with the curvature of ∂Ω being large compared to ǫ. As was analyzed in [15], the flow may be decomposed into a thin kinetic boundary layer (with a thickness ∼ ǫ2 ) which is appropriately described by a low order Hilbert expansion (and thus by the linearized Boltzmann equation), a viscous boundary layer (with thickness ∼ ǫ), and an outer domain well described by the Euler equations. One of the objectives of the paper is the coupling of the kinetic boundary layer to the viscous layer. We are going to investigate numerical tools for the calculation of slip boundary conditions as kinetic corrections to the viscous part. E.g., if the wall temperature is T (a), then the fluid temperature at a is in general different from T (a), with a correction term of the order of ǫ and dependent on the temperature gradient. Similar statements hold for the fluid velocity components tangential to ∂Ω. Sone et al. [15] pointed out that an appropriate scaling for the boundary region around a ∈ ∂Ω is given by a space-coordinate transformation taking into account the strong anisotropy of the flow in the boundary layer. For this we assume a bijection ξ(x) = (ξ1 , ξ2 , ξ3 )T such that x(0, ξ2 , ξ3 ) ∈ ∂Ω

,

(31)

and for x ∈ ∂Ω, the system {n, ω1 , ω2 } defined by ǫ·n=

∂x (0, ξ2 , ξ3 ), ∂ξ1

ω1 =

∂x (0, ξ2 , ξ3 ), ∂ξ2

ω2 =

∂x (0, ξ2 , ξ3 ) ∂ξ3

(32)

forms an orthonormal system. The stationary system then easily transforms into 1 ∂ ∂ 1 ∂ · vn · f + vω1 · f + vω2 · f = 2 · J[f , f ] . (33) ǫ ∂n ∂ω1 ∂ω2 ǫ In the kinetic boundary layer, the partial derivatives concerning the tangential vectors ω1 and ω2 may be neglected, and we end up with the half space problem 1 ∂ f = · J[f , f ] . (34) ∂n ǫ The analysis of this problem yields the details of the jump conditions mentioned above. vn ·

Hexagonal kinetic models and simulation of kinetic boundary layers

53

3 Hexagonal kinetic models 3.1 Hexagonal discretization of lR2 For h > 0, consider the partition Ph of lR2 which is given by regular hexagons with side length h as demonstrated in Fig. 2. In the sequel we will denote these as basic hexagons. We assume that the partition is numerated by some index set I.

Fig. 2. Hexagonal discretization of lR2 .

The above constellation gives rise to two regular grids on lR2 . The grid Ch is given by the set {cz |z ∈ I}

(35)

of all centers of basic hexagons. As one can check quickly, Ch is spanned by two vectors: Ch = c0 + {k · (c(1) − c0 ) + l · (c(2) − c0 )|k, l ∈ Z} .

(36)

Furthermore, Ch exhibits hexagonal symmetry. Denote by R the (e.g. clockwise) rotation around the angle π/3. Then c0 , c ∈ Ch

=⇒

c0 + Rk (c − c0 ) ∈ Ch

for k = 0, . . . , 5

.

(37)

All six-tupels of the form (37) are the nodes of a regular hexagon in Ch . A second grid Gh is given by the nodes gz of the basic hexagons. Obviously all nodes can be represented in the form   1 g = c + h · Rk (38) 0

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H. Babovsky

for some c ∈ Ch and for k ∈ {0, . . . , 5}. Gh has triangular symmetry: g0 , g ∈ Gh

=⇒

g0 + R2k (g − g0 ) ∈ Gh

for k = 0, 1, 2

.

(39)

We introduce a sign-function sign : Gh → {−1, 1}

(40)

such that two neighboring points have different sign: sign(g1 ) · sign(g2 ) = −1 if |g1 − g2 | = h .

(41)

Then it turns out that the two subsets of Gh of elements with definite sign have hexagonal symmetry. Following the arguments above, there are a lot of regular hexagons contained in the grid Gh . Those with the center belonging to Ch are denoted as ”class A”, and those with center in Gh as ”class B”. See Fig. 3 for the illustration of some examples (Class A on the left and Class B on the right). Let (g0 , . . . , g5 ) describe any regular hexagon in G, numbered in clockwise order. Then it turns out that for class A hexagons sign(gi ) · sign(gi+1 ) = −1 ,

(42)

while for class B hexagons sign(gi ) · sign(gi+1 ) = 1 .

(43)

In the following we parametrize the hexagons described above by tuples (η, ξ), where η ∈ C ∪ G denotes the center of the hexagon, and η + Rk ξ, k = 0, . . . , 5 its nodes. For short, we write H(η, ξ). Notice that H(η, ξ) = H(η, Rn ξ) for n ∈ Z. The set of hexagons H(η, ξ) is denoted as H.

( a) Fig. 3. Some regular hexagons in Gh .

Hexagonal kinetic models and simulation of kinetic boundary layers

55

3.2 A hexagonal collision model Assume H ∈ H to be given by the six-tupel H = H(η, ξ)=(g ˆ 0 , . . . , g5 )

(44)

with gi = η + Ri ξ

.

(45)

A binary hexagonal collision model is given by the collision relation (gi , gi+3 ) → (gj , gj+3 )

(46)

which maps opposite pairs (gi , gi+3 ) of velocities with equal probability into one of the other opposite pairs. The corresponding collision operator reads ⎞ ⎛ 2  1 (47) JH,bin [fH , fH ]i = γH,bin ⎝ · fj fj+3 − fi f(i+3) mod 6 ⎠ , 3 j=0 where the indices 0, . . . , 5 have been identified with the positions g0 , . . . g5 . Its weak version reads for any test function πH on H . 2 2  1 fi fi+3 πH , JH,bin [fH , fH ] = γH,bin · (πH,j + πH,j+3 ) 3 j=0 i=0 (48) / −(πH,i + πH,i+3 )

According to the rules of classical kinetic theory we assume shift invariance of the collision model and thus that γH,bin depends only on the diameter diam(H) = 2|ξ| of the hexagon. The binary collision operator Jbin on the whole grid Gh is given by summing up over all H ∈ H. Given v, w ∈ Gh , denote η := 0.5 · (v + w) as well as

,

0 0 kh (v, w) := 2γH,bin

ξ := v − η if H(η, ξ) ∈ / H, else .

(49)

(50)

Furthermore, the point measure with support on the nodes of the hexagon between two vectors is denoted as 5

µh (v, w) :=

1 i δ 6 i=0 η+R ξ

.

(51)

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H. Babovsky

Then π, Jbin [f , f ] =



v,w∈Gh

kh (v, w) (π, µ(v, w) − π(v)) f (v)f (w)

(52)

Unfortunately, Jbin contains – besides the physically correct invariants mass, momenta and kinetic energy – a further invariant sign, f . To suppress this, we are forced to introduce an additional ternary collision operator. Its restriction onto one single hexagon H ∈ H reads JH,ter [fH , fH , fH ]i = γter · (−1)i (f1 f3 f5 − f0 f2 f4 ) ,

(53)

and its weak version is πH , JH,ter [fH , fH , fH ] = γH,ter ·sign(g0 )·πH , sign·(f1 f3 f5 −f0 f2 f4 ) .

(54)

Since in general this term does not vanish for πH = sign, sign is no longer an artificial invariant. In [1] a complete kinetic theory on hexagonal grids has been developed proving that all of the main features of the classical Boltzmann collision operator are still valid. This includes the correct number of collision invariants as well as the set of equilibria (which stay close to the equilibria given by Maxwellians), the H-theorem and the linearization around an equilibrium. This theory is based on the interplay between class A and class B hexagons. The ternary part of the collision operator is cubic in f and thus dominates the binary operator, if the density ρ is large. Thus in numerical simulations we choose γter ∼ 1/ρ if necessary. 3.3 Grid refinement and continuum limit The refinement of the grid Gh with discretization parameter h to the grid Gh/2 may be achieved in two ways. On one hand, we may shrink the grid by a factor 0.5; i.e. we define – for some fixed g0 ∈ Gh – Gh/2 := g0 + {0.5 · (g − g0 )|g ∈ Gh }

.

(55)

Alternatively, we may replace each element of Gh with some 4-stencil as is indicated in Fig. 4. In what follows we make use of this second interpretation. Our aim is to construct the limiting collision operator after successive refinement of the grid. The arguments are presented on a formal level but can be made rigorous using the classical theory of convergence of measures as described e.g. in [9]. Passing from Gh ⊂ Gh/2 to Gh/2 we can easily extend the function sign =: signh such that the restriction signh/2 |Gh = signh . For given h > 0, each g ∈ Gh is a node of three adjacent basic hexagons. Connect the centers of these to form a regular triangle ∆h = ∆h (g). Its area is given by

Hexagonal kinetic models and simulation of kinetic boundary layers

57

Fig. 4. Grid refinement.

λ(∆h ) =

√ 3 · h2

.

(56)

A function µ : Gh → lR+ may be considered as a discrete measure on lR2 with support in Gh . However, we may reinterprete it as an absolutely continuous measure on lR2 with density which is piecewise constant on the triangles ∆h . In order to compare signed measures on Gh with (e.g. absolutely continuous) measures on lR2 we define for functions πh on Gh the discrete signed measures  πh (g) · δg (57) πh , µh  := g∈Gh

With the special function 1(g) l := 1 we define the measure µh := l 1, µh  . Given a function f ∈ L1loc , we define the mapping Ph f : Gh → lR by
Ph f (g) := f (v) dv .

(58)

(59)

∆h (g)

The following statements are well-known results in measure theory. 3.1 √Lemma: Denote by →w the weak convergence of measures on lR2 . Then (a) 3h2 µh →w dλ (Lebesgue measure) (b) Ph f , µh  →w f (v)dλ(v) (c) signh , µh  →w 0 . From Property (c) follows that the artificial invariant of the binary collision operator vanishes in the continuum limit. Thus there is no need to consider the ternary collision part. Now assume that kh (., .) is bounded (an assumption which may be weakened). Then by the usual techniques of measure theory it is straightforward to derive in the weak limit an integral expression for the right hand side of (52). From this we can deduce a ”classical” formulation for

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H. Babovsky

the continuous hexagonal model. (See, e.g. [3, 4] for the weak formulation of collision integrals.) We summarize the results. 3.2 Theorem: Under the above assumptions, there exists a collision kernel k0 (|v − w|) such that the binary collision operator Jbin [f , f ] converges weakly to the kinetic collision operator

k0 (|v − w|) · [f , µ(v, w) − f (v)] f (w)dvdw . (60) J[f , f ](v) = lR2

lR2

This operator has beyond the physical invariants mass, momenta and kinetic energy no further (in particular no artificial) invariants. 3.3 Remark: The main difference to classical two-dimensional collision operators is the summation f , µ(v, w) over the regular hexagon between v and w (i.e. over six angles) rather than the integration over the whole circle between v and w shown in Fig. 1. 3.4 Numerical examples: Channel flows We study several details of the collision model in the most simple twodimensional rectangular geometry. For numerical simulations we have to restrict to finite (i.e. truncated) grids. In all the calculations presented here we use a socalled ”two-layer” model. This is a 54-velocity model generated by attaching to a single central basic hexagon two ”layers” of further basic hexagons (see Fig. 5). It turns out that this model is rich enough to resolve a couple of interesting detailed structures of kinetic flows. The range of applications includes interior as well as exterior flows, extends from slow to fast (supersonic) flows and covers a wide range of Knudsen numbers. Numerical simulations of time dependent problems are easy to implement if they are based on an operator splitting scheme, where the free-flow part is evaluated by a standard upwind difference scheme. We have used such a scheme to obtain the results demonstrated below. (However, we should mention that for steady boundary value problems there are more efficient ways; in [7], a combination of FEM techniques with Newton methods has been proposed.) As an illustration, we present two numerical examples. Fig. 9 shows the propagation of a supersonic jet into a channel. We recognize a shock front in the neighborhood of the lower plate with a dense regime beyond and a rarefaction zone below the plate. Furthermore, the flow field between the shock front and the stagnation point (lower plate) is demonstrated. A second example is given by the convection problem demonstrated in Fig. 10. It exhibits a flow in a gravitational field generated by heating of the center of the lower plate. The flow field shown in the second part of Fig. 10 demonstrates the existence of convection rolls. In detail, we investigate features of channel flows concerning the boundary slip conditions and the flow parameters.

Hexagonal kinetic models and simulation of kinetic boundary layers

59

Fig. 5. The 2-layer model.

Boundary conditions Kinetic theory allows a variety of models for the design of gas-surface interactions which is much larger than for macroscopic models. ”Typical” models (linear, local in space and time) are described by so called reflection laws prescribing how a particle is reflected from a wall if it hits the wall at some point a with some velocity v. There are two main classes (and linear combinations of these). Deterministic reflection laws are given by mappings v → Rv telling that the velocity after the wall contact is Rv. If R is one-to-one, then the corresponding boundary condition for f is n(a), Rv · f (a, Rv) = |n(a), v| · f (a, v),

(61)

where n(a) denotes the inner normal on the wall surface at a. The main representatives are specular reflection: Rv = v − 2n(a), v · n reverse reflection: Rv = −v Stochastic reflection laws describe the result of the contact with the wall by some probability law P (.|v). The most prominent one is diffuse reflection, where particles leave the wall thermally distributed. The corresponding boundary condition is given by (2.28). Reflection laws highly influence the fluid flow along a boundary. In the following we restrict to the three laws described above. In terms of macroscopic flow parameters, one can find immediately some fundamental differences. Common to all of these is that there is no flux through the boundary, i.e. v⊥ = 0.

(62)

The only law guaranteeing the no-slip condition, i.e. v = 0

(63)

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H. Babovsky

is reverse reflection. (This is true in a more general setting. As was shown in [13], reverse reflection is the only ”no-slip” law in a large class of physically reasonable laws.) Both deterministic laws above are adiabatic (i.e. there is no energy exchange between fluid and wall), while the diffuse law is not. The only law with vanishing shear stress is specular reflection. We have calculated a plane channel flow under each of these conditions. The results presented in Fig. 8 illustrate the differences. We find e.g. a non-vanishing velocity at the boundary in the case of diffuse reflection (in contrast to reverse reflection) which has to be described by a jump condition. For specular reflection, shear stress is constant zero. Flow parameters In the fluid dynamic limit, the viscosity coefficient and the heat conduction are two mayor parameters to characterize a fluid flow. We investigate these for the hexagonal model. Since both are coupled via the Prandtl number (which is fixed in our monatomic model) we restrict to the calculation of the viscosity coefficient. The derivation of hexagonal models with various Prandtl numbers (via internal energies) is under investigation. In the classical kinetic theory, flow parameters depend on the collision model. E.g. Maxwellian molecules (with bounded collision kernel) behave different than molecules with kernels given by some other law. Here we test how the flow is influenced by the choice of collision frequencies γH for different hexagons H. As expected, it turns out that the viscosity coefficient does not depend on the flow density. However, it strongly depends on the distribution of collision frequencies γH for the hexagons. As is usual in the classical theory, γH should depend only on the diameter of H. We assume a parameter dependent law γH = λ + (1 − λ) · 0.32 · diam(H)

.

(64)

For λ ∈ [0, 1.5], the corresponding collision kernels are strictly positive. For λ = 1, all collision frequencies are equal. For λ = 0, they vary from 0.640 (basic hexagons) to 2.790 (largest hexagons). For λ = 1.5, they change from 1.180 to 0.105, i.e. interactions within the largest hexagons are almost suppressed. The corresponding viscosity coefficients are linearly dependent on λ as is shown in Fig. 6. Here, we do not go into details concerning ”artificial” viscosity, i.e. the influence of the discretization parameters on the viscosity coefficient (cf. e.g. [14] in the context of lattice Boltzmann models). This will be due to further work. (For our calculation we used a 128 × 16- grid in position space, however doubling the grid points in each direction does not lead to significant changes.)

Hexagonal kinetic models and simulation of kinetic boundary layers

61

7

fi

0 0

ti

1.51

Fig. 6. Viscosity coefficient versus collision model.

4 Kinetic boundary layers 4.1 Structure of linearized boundary layers We consider kinetic flows with finite (i.e. truncated) velocity space which we again denote by G. As pointed out in section 2.2, the coupling of kinetic boundary layers to fluid flows can well be modeled by linearized half space problems. For discrete velocity systems, these have been analyzed in [5]. (The continuous case has been treated in [12].) For a small perturbation φ(x, v) around an equilibrium, the linearized system of equations can be formulated in the form ∂x φ = Lφ

,

(65)

The operator L acts only in the velocity space and is (in 2D) similar to the matrix in Jordan normal form, J = diag(0, 0, α1 · J1 , α2 · J1 , Λ, −Λ) where J1 is the Jordan block given by   01 J1 = 00

(66)

,

(67)

and Λ is a positive diagonal matrix. In terms of the corresponding basis {b1 , . . . , bn } of the n-point discretization of the velocity space, the general solution of (3.1) is of the form φ(x) = β1 b1 + β2 b2 + (β3 + x · α1 β4 )b3 + β4 b4 + (β5 + x · α2 β6 )b5 + β6 b6 + exp(xΛ)b+ + exp(−xΛ)b−

.

(68)

Here, the basis vectors b1 , b2 , b3 and b5 are related to the invariants mass, momenta and energy. b4 and b6 are macroscopically not observable fluctuations which are responsible for the relation between the gradients of b3 and

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H. Babovsky

b5 and their values at the boundary x = 0 (jump conditions). The term exp(xΛ)b+ is a short notion for k  i=0

exp(λi x) · b7+i .

A similar abbreviation is exp(−xΛ)b− . To formulate the boundary conditions, we denote  +   φ P+ φ φ= =: P− φ φ−

(69)

(70)

where P+ (resp. P− ) denotes the restriction of φ to the velocity points pointing to the right (resp. to the left). We call a solution (3.4) a kinetic boundary layer in [0, ∞), if P+ φ(0) = RP− φ(0),

(71)

for some linear (totally absorbing) reflection law R, and in addition for large x φ(x)/x is bounded .

(72)

(70) is satisfied if the exponentially increasing term vanishes, i.e. if b+ = 0

(73)

In order to obtain a unique boundary layer, we have to introduce two further conditions; e.g. we may fix the constant gradients α1 β4 and α2 β6 . Under these restrictions, the construction of a kinetic boundary layer is a well-posed problem [5]. Analyzing the algebraic structure of the problem it is possible to find analytic expressions for the jump conditions – at least for discrete models of no more than moderate size. Examples are given in [8]. For larger systems it is more convenient to replace the half space problem with an appropriate two-point boundary value problem. For its preparation, we introduce some properties of the eigenspace of L which have been derived in a quite general setting in [5] and which are valid also in the case of hexagonal models. We call a vector φ on G even, if P+ φ = P− φ, and odd, if P+ φ = −P− φ. 4.1 Properties of L: (a) The eigenvectors b1 , b3 and b5 are even, and b2 , b4 and b6 odd. (b) If for some λ > 0 and (b+ , b− )T = 0 L(b+ , b− )T = λ(b+ , b− )T

(74)

L(b− , b+ )T = −λ(b− , b+ )T

(75)

then

Hexagonal kinetic models and simulation of kinetic boundary layers

63

A straightforward consequence of this is − T 4.2 Lemma: Suppose φ(1) = (φ+ (1) , φ(1) ) is a solution of (3.1), (3.6) and + T (3.7) in [0, ∞). Define φ(2) (−x) := (φ− (1) (x), φ(1) (x)) . Then φ(2) is a kinetic boundary layer in (−∞, 0] satisfying the boundary condition P+ φ(2) (0) = RP+ φ(2) (0)

(76)

4.2 The two-point BVP We want to numerically construct the linearized kinetic boundary layer on [0, ∞)  + b− φ(x) = φ(1) (x) = (β3 + x · α1 β4 )b3 + β4 b4 + exp(−xΛ) · (77) b− − In particular, we want to find out the relation between the gradient g1 := α1 β4 and the jump condition β3 for b3 . From theoretical arguments we easily find a linear relationship s1 g1 + s2 β3 = 0.

(78)

For the calculation we define for some H > 0 the shift operators SH φ(x) := φ(x − H)

,

S−H φ(x + H)

(79)

and the function ψ(x) :=

1, S−H φ(1) (x) − SH φ(2) (x) 2

(80)

on [−H, H], where φ(2) is defined as in Lemma 5. ψ has the representation ψ(x) = x · α1 β4 b3 + β4 b4  −    + 1 b− b− + exp(−(x + H)Λ) · − − exp(−(H − x)Λ) · b− b+ 2 −

(81)

We assume H large enough and thus neglect terms of the form exp(−H · Λ). Then ψ satisfies ψ(0) = β4 b4 , ′

ψ (0) = α1 β4 b3 , ψ ′′ (0) = 0.

(82) (83) (84)

The boundary values are  φ− (2H) φ+ (2H)  +  φ (2H) ψ(H) = −φ(2) (0) + φ− (2H)

ψ(−H) = φ(1) (0) −



(85) (86)

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H. Babovsky

where φ(2H) = β3 b3 + 2Hψ ′ (0) + ψ(0) = (β3 + α1 β4 )b3 + β4 b4 =: πb3 + β4 b4 .

(87)

The boundary conditions for ψ read ψ + (−H) + φ− (2H) = R[ψ − (−H) + φ+ (2H)] ψ − (H) − φ− (2H) = R[ψ + (H) − φ+ (2H)]

(88) (89)

Suppose the gradient ψ ′ (0) for b3 is prescribed. Because of the linear dependence (77) we may solve the linearized differential system with the boundary conditions (84), (85) and with φ(2H) given by (86) with some fixed π. The only unknown is β4 . Once this is known, all other quantities of interest can be reconstructed. Given a fixed β4 , a solver of the corresponding BVP is readily written applying an upwind technique as in section 3.4. The correct value of β4 is found using an iteration scheme to construct the correct solution ψ. This is found by exploiting the symmetry property of ψ with respect to the origin. A typical result of such calculations is shown in Fig. 7. which demonstrates the temperature distribution of a heat layer problem. Finally we want to mention that two-point BVPs are also successfully applied to calculate boundary layers on the basis of the nonlinear collision operator. The only problem is to formulate boundary conditions at the artificial boundary x = H. This can be managed with an iterative scheme which minimizes the curvature close to the right boundary.

1.2

1.1

T 1 i

1

0.9

0.8 0.8

0 0

20

40

60

80

100

i

Fig. 7. Linearized heat layer problem.

120 128

Hexagonal kinetic models and simulation of kinetic boundary layers

(A,1)

(A,2)

(B,1)

(B,2)

(C,1)

(C,2)

65

Fig. 8. Channel flow with different boundary conditions. A: diffuse, B: reverse, C: specular reflection; 1: velocity, 2: shear stress.

References 1. L. S. Andallah, H. Babovsky. A discrete Boltzmann equation based on hexagons. Math. Models Methods Appl. Sci, 13:1537–1563, 2003. 2. P. Andries, J.-F. Bourgat, P. Le Tallec, B. Perthame. Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases. Preprint no. 3872, INRIA Rocquencourt, 2000. 3. H. Babovsky. Convergence proof for Nanbu’s Boltzmann simulation scheme. Europ. J. of Mech. B/Fluids, 1:41–55, 1989. 4. H. Babovsky. Die Boltzmann-Gleichung. Teubner, Stuttgart, 1998. 5. H. Babovsky. Kinetic boundary layers: on the adequate discretization of the Boltzmann collision operator. J. Comp. Appl. Math., 110:225–239, 1999. 6. H. Babovsky Design of numerically efficient kinetic models. Preprint 05/02, Inst. f. Math., TU Ilmenau. 2002 (zur Ver¨ offentlichung eingereicht).

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Fig. 9. Supersonic flow propagating into a channel. (a) density; (b) flow field around the stagnation point. (See also color figure, Plate 7.)

7. H. Babovsky, D. G¨ orsch und F. Schilder. Steady kinetic boundary value problems in: Lecture Notes on the Discretization of the Boltzmann Equation, N. Bellomo and R. Gatignol Eds., pp. 131–156, World Scientific 2002. 8. H. Babovsky and T. Platkowski. Kinetic boundary layers for the Boltzmann equation on discrete velocity lattices. Preprint 07/03, Inst. f. Math., TU Ilmenau. 2003 (zur Ver¨ offentlichung eingereicht). 9. P. Billingsley. Convergence of probability measures. John Wiley & Sons, New York, 1968. 10. A.V. Bobylev, A. Palczewski and J. Schneider. On the approximation of the Boltzmann equation by discrete velocity models. C. R. Acad. Sci. Paris I 320(5):639–644, 1995.

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Fig. 10. Thermally driven flow. (a) density; (b) convection cells (flow field). (See also color figure, Plate 7.)

11. C. Cercignani. The Boltzmann equation and its applications. Springer, New York, 1988. 12. F. Golse, B. Perthame and C. Sulem. On a boundary layer problem for the nonlinear Boltzmann equation. Arch. Rational Mech. Anal., 103:81–96, 1988. 13. J. Schnute, M. Shinbrot. Kinetic theory and boundary conditions for fluids. Can. J. Math., 25:1183–1215, 1973. 14. V. Sofonea and R. F. Sekerka. Viscosity of finite difference lattice Boltzmann models. J. Comput. Phys., 184:422–434, 2003. 15. Y. Sone, C. Bardos, F. Golse and H. Sugimoto. Asymptotic theory of the Boltzmann system, for a steady flow of a slightly rarefied gas with a finite Mach number: General theory. Eur. J. Mech. B-Fluids, 19:325–360, 2000. 16. H. Struchtrup and M. Torrilhon. Regularization of Grad’s 13 moment equations: Derivation and linear analysis. Phys. Fluids, 15:2668–2680, 2003.

High-resolution Simulation of Detonations with Detailed Chemistry Ralf Deiterding1 and Georg Bader2 1

2

California Institute of Technology, 1200 East California Blvd., Mail-Code 158-79, Pasadena, CA 91125, [email protected] Institut f¨ ur Mathematik, Technische Universit¨ at Cottbus, Universit¨ atsplatz 3-4, 03044 Cottbus, [email protected]

Summary. Numerical simulations can be the key to the thorough understanding of the multi-dimensional nature of transient detonation waves. But the accurate approximation of realistic detonations is extremely demanding, because a wide range of different scales need to be resolved. This paper describes an entire solution strategy for the Euler equations of thermally perfect gas-mixtures with detailed chemical kinetics that is based on a highly adaptive finite volume method for blockstructured Cartesian meshes. Large-scale simulations of unstable detonation structures of hydrogen-oxygen detonations demonstrate the efficiency of the approach in practice.

1 Introduction Reacting flows have been a topic of on-going research since more than hundred years. The interaction between hydrodynamic flow and chemical kinetics can be extremely complex and even today many phenomena are not very well understood. One of these phenomena is the propagation of detonation waves in gaseous media. A detonation is a shock-induced combustion wave, which internally consists of a discontinuous hydrodynamic shock wave followed by a smooth region of decaying combustion. The adiabatic compression due to the passage of the shock rises the temperature of the combustible mixture above the ignition limit. The reaction results in an energy release that drives the shock wave forward. In a self-sustaining detonation, shock and reaction zone propagate essentially with an identical speed dCJ that is approximated to good accuracy by the classical Chapman-Jouguet (CJ) theory, cf. [37]. But up to now, no theory exists that describes the internal flow structure satisfactory. The Zel’dovich-von Neumann-D¨ oring (ZND) theory is widely believed to reproduce the one-dimensional detonation structure correctly, but already early experiments [7, 30] uncovered that the reduction to one space dimension is not

70

R. Deiterding, G. Bader B

Trajectory D

A

G

Mach stem Transverse wave 

C

Head of reaction zone

F E

Triple point

Incident shock

L

Fig. 1. Left: regular detonation structure at three different time steps on triple point trajectories, right: enlargement of a periodical triple point configuration. E: reflected shock, F: slip line, G: diffusive extension of slip line with flow vertex.

even justified in long combustion devices. It was found that detonation waves usually exhibit non-neglectable instationary multi-dimensional sub-structures and do not remain planar. The multi-dimensional instability manifests itself in instationary shock waves propagating perpendicular to the detonation front. A complex flow pattern is formed around each triple point, where the detonation front is intersected by a transverse shock. Pressure and temperature are increased remarkable in a triple point and the chemical reaction is enhanced drastically giving rise to an enormous local energy release. Hence, the accurate representation of triple points is essential for safety analysis, but also in technical applications, e.g. in the pulse detonation engine. Some particular mixtures, e.g. low-pressure hydrogen-oxygen with high argon diluent, are known to produce very regular triple point movements. The triple point trajectories form regular “fish-scale” patterns, so called detonation cells, with a characteristic length L and width λ (compare left sketch of Fig. 1). Fig. 1 displays the hydrodynamic flow pattern of a detonation with regular cellular structure, how it is known since the early 1970s, cf. [29, 23]. The right sketch shows the periodic wave configuration around a triple point in detail. It consists of a Mach reflection, a flow pattern well-known from non-reactive supersonic hydrodynamics [4]. The undisturbed detonation front is called the incident shock, while the transverse wave takes the role of the reflected shock. The triple point is driven forward by a strong shock wave, called Mach stem. Mach stem and reflected shock enclose the slip line, the contact discontinuity. The Mach stem is always much stronger than the incident shock, what results in a considerable reduction of the induction length lig , the distance between leading shock and measurable reaction. The shock front inside the detonation cell travels as two Mach stems from point A to the line BC. In the points B and C the triple point configuration is inverted nearly instantaneously and the front in the cell becomes the incident shock. Along the symmetry line AD the change is smooth and the shock strength decreases

High-resolution Simulation of Detonations with Detailed Chemistry

71

continuously. In D the two triple points merge exactly in a single point. The incident shock vanishes completely and the slip line, which was necessary for a stable triple point configuration between Mach stem and incident shock, is torn off and remains behind. Two new triple points with two new slip lines develop immediately after D. But the experimental analysis of the described transient sub-structures is difficult. Direct numerical simulation of the governing equations can be an alternative here [24, 8, 11, 16, 36, 33]. In the following, we will demonstrate that recent parallel computers of moderate size, e.g. clusters of standard personal computers, allow detonation structure simulations in two and three space dimensions even for detailed non-equilibrium chemistry that provide deep insight into the internal flow structure far beyond previous experimental results [29, 23].

2 Detonation Modeling The governing equations for detonation propagation in premixed gases with realistic chemistry are the Euler equations for multiple thermally perfect species with chemically reactive source terms [10, 37]. In d-dimensional Cartesian coordinates these equations can be written as an inhomogeneous conservation law of the structure d  ∂ ∂ q(x, t)+ fn (q(x, t)) = s(q(x, t)) , x = (x1 , . . . , xd )T ∈ Rd , t ∈ R+ 0 , ∂t ∂x n n=1 (1) where q = q(x, t) ∈ S ⊂ RM denotes the vector of conserved quantities. The functions fn (q) ∈ C1 (S, RM ), n = 1, . . . , d are the hydrodynamic fluxes, s(q) ∈ C1 (S, RM ) is the source term.

2.1 Euler Equations for Gas-Mixtures For the multi-component Euler equations with K species the vector of conserved quantities has M = K + d + 1 components. We choose the form q(x, t) = (ρ1 , . . . , ρK , ρu1 , . . . , ρud , ρE)T .

(2)

The partial density of the ith species denoted by ρi , where i = 1, . . . , K. is K The total density of the mixture ρ = i=1 ρi is a conserved quantity, too. The K ratios Yi = ρi /ρ are called mass fractions. They satisfy the relation i=1 Yi = 1. We denote the nth component of the velocity vector u = (u1 , . . . , ud )T by un and E is the total energy per unit mass. The flux functions are fn (q) = (ρ1 un , . . . , ρK un , ρu1 un + δ1n p, . . . , ρud un + δdn p, un (ρE + p))T (3)

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for n = 1, . . . , d. Herein, p is the hydrostatic pressure and δjn denotes the Kronecker-Symbol. We assume that all species are ideal gases in thermal equilibrium. Under this assumption the same temperature T can be used to evaluate the partial pressure of all species as pi = RT ρi /Wi with R denoting the universal constant and Wi the molecular weight, respectively. According to Dalton’s law the total pressure is given by p=

K  i=1

pi = RT

K  ρi . Wi i=1

(4)

Each species is assumed to be thermally perfect and has a temperaturedependent specific heat cpi (T ). The functions cpi (T ) are usually approximated by polynomials of degree 4 which are valid within a restricted temperature range, e.g. from 300 K to 5000 K [31, 20]. The enthalpies per unit mass are written as
T

hi (T ) = h0i +

cpi (σ)dσ

T0

with h0i called the heat of formation at the reference temperature T 0 . For the K enthalpy of the mixture h = i=1 Yi hi (T ) holds true. Inserting this into the thermodynamic relation ρh−ρE +ρ u2 /2−p = 0 and inserting (4) for p yields ϕ(q, T ) :=

K  i=1

ρi hi (T ) − ρE + ρ

K  ρi u2 − RT =0. 2 Wi i=1

(5)

It can be proven rigorously [5], that for each q in the space of admissible states S a unique temperature T exists that satisfies (5). Unfortunately, a closed form of the inverse can only be derived under simplifying assumptions and the computation of T from (5) is in general unavoidable, whenever the pressure p has to be evaluated. The appropriate speed of sound for the described model is the frozen speed of sound, which is given by 2

c =



∂p ∂ρ



s,Y1 ,...,YK

=

K  i=1

, Yi φi − (γ − 1) u2 − H

(6)

& u2 ' ∂p R u2 and φi := − hi (T ) + γ = (γ − 1) T . The 2 ∂ρi 2 Wi coefficient γ = γ(Y1 , . . . , YK , T ) can be calculated from the mixture quantities '−1 & K K Y c (T ) and W = by employing the relation Y /W cp = i pi i i i=1 i=1

with H = h +

γ = cp (cp − R/W )−1 . By inserting the previous expression for φi into (6) and by applying the ideal-gas law (4) it can be shown, that the frozen speed of sound of a thermally perfect gas-mixture satisfies the relation c2 = γp/ρ.

High-resolution Simulation of Detonations with Detailed Chemistry

73

Mathematical Properties The Euler equations for thermally perfect gas-mixtures inherit most mathematical properties of the standard Euler equations , - for a single polytropic gas with equation of state p = (γ − 1) ρE − ρ u2 /2 , cf. [28]. Utilizing expression (6) for the speed of sound the hyperbolicity of (1) with vector of state (2) and flux functions (3) can easily be proven [5]. For d = 2, 3 the proof uses the validity of the rotational invariance property that carries over from the standard Euler equations almost directly [32] and requires just the diagonalization of the Jacobian A1 (q) = ∂f1 (q)/∂q. For instance, in two space dimensions the matrix of right eigenvectors R1 (q) = (r1 | . . . |rK+d+1 ) that diagonalizes A1 (q) with R−1 1 (q) A1 (q) R1 (q) = Λ1 (q) for all q ∈ S with Λ1 (q) = diag(u1 − c, u1 , . . . , u1 , u1 + c) takes the form ⎡ ⎤ Y1 1 0 ... 0 0 Y1 ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ . . . . . . .. .. .. .. .. .. ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ R1 (q) = ⎢ Y ⎥ . (7) 0 ... 0 1 0 YK K ⎢ ⎥ ⎢ u1 − c u1 ... u1 0 u1 + c ⎥ ⎢ ⎥ ⎢ u2 ⎥ u2 ... u2 1 u2 ⎢ ⎥ ⎣ ⎦ φK φ1 ... u2 − u2 H + u1 c H − u1 c u2 − γ−1 γ−1 Furthermore, it can be shown that the flux functions fn (q) and their Jacobians An (q) satisfy the homogeneity property fn (q) = An (q)q for all q ∈ S. The Homogeneous Riemann Problem The profound understanding of the Riemann Problem (RP) in the nonreactive case provides the theoretical basis for the construction of a reliable Godunov-type method in Sec. 3.2 as a key ingredient for detonation simulation. For s ≡ 0 the solution structure of a quasi-one-dimensional RP with discontinuous initial data ql , xn < 0 , (8) q(x, 0) = qr , xn > 0 can be shown to be in principle identical to the standard case of a single polytropic gas that is discussed in detail for example in [28, 12, 32]. The first and last characteristic field with the eigenvalues un −c and un +c are genuinely nonlinear, provided that the condition ∂γ γ(γ + 1) = (1 − γ)T ∂T

(9)

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R. Deiterding, G. Bader

is satisfied for all q ∈ S [5]. All other characteristic fields are associated to the eigenvalue un and are linearly degenerate. If we assume that condition (9) is satisfied, it can be concluded that the solution of the RP consists of admissible shocks or smooth rarefaction waves in the first and last characteristic field, while the contributions in all other characteristic fields sum up to a single contact discontinuity. Unlike standard Euler equations, no complete set of Riemann invariants can be found for the Euler equations for thermally perfect gases [22]. The only Riemann invariants that can be derived are the mass fractions Yi , which are constant across the first and last characteristic field [5], and the velocity un and the pressure p, which are invariant across the contact discontinuity. 2.2 Reactive Source Terms We write the source term of detailed chemical reaction in the form s(q) = (W1 ω˙ 1 , . . . , WK ω˙ K , 0, . . . , 0, 0)

T

.

The chemical production of each species is expressed as product of its production rate in molar concentration per unit volume ω˙ i = ω˙ i (q) ∈ C 1 (S, R) and Wi . The rates ω˙ i (ρ1 , . . . , ρK , T ) are derived from a reaction mechanism of J chemical reactions K 

f νji Si ⇋

K 

r νji Si ,

j = 1, . . . , J ,

i=1

i=1

f r are the stoichiometric coefficients of species Si appearing as where νji and νji a reactant and as a product. The entire molar production rate of species Si per unit volume is then given by

ω˙ i =

f r   1 νjl νjl K  K  1 ρl ρl f r (νji −νji ) kjf −kjr , Wl Wl j=1

J 

l=1

i = 1, . . . , K , (10)

l=1

with kjf (T ) and kjr (T ) denoting the forward and backward reaction rate of each chemical reaction [37]. The reaction rates are calculated by the Arrhenius law f /r

kj

f /r

f /r

(T ) = Aj T βj

f /r

exp(−Ej

/RT ) .

(11)

Some backward reaction rates might be derived by assuming the corresponding chemical reaction to be in chemical equilibrium, but especially detonations usually require mechanisms that utilize non-equilibrium backward reaction rates at least for some of the reactions. A chemical kinetics package, e.g. Chemkin [19], can be utilized to evaluate (10), (11) according to the reaction mechanism and given thermodynamic data.

High-resolution Simulation of Detonations with Detailed Chemistry

75

3 Numerical Methods We use the time-operator splitting approach or method of fractional steps [17] to decouple hydrodynamic transport and chemical reaction numerically. This technique is most frequently used for time-dependent reactive flow computations. The homogeneous partial differential equation d

H(∆t) :

∂q  ∂ + fn (q) = 0 , ∂t n=1 ∂xn

∆t ˜ κ+1 IC: Qκ =⇒ Q

(12)

and the usually stiff ordinary differential equation S (∆t) :

∂q = s(q) , ∂t

∆t ˜ κ+1 =⇒ IC: Q Qκ+1

(13)

are integrated successively with the data Q from the preceding step as initial condition (IC). 3.1 Finite Volume Schemes The appropriate discretization technique for conservation laws with discontinuous solution is the finite volume (FV) approach. For simplicity, we assume an equidistant discretization in two space dimensions with mesh widths ∆x1 , ∆x2 and a constant time step ∆t. A conservative time-explicit finite volume scheme for (12) has the formal structure 3 3 2 2 ˜ κ+1 = Qκ − ∆t F1 1 −F1 1 − ∆t F2 1 −F2 1 (14) H(∆t) : Q jk jk j− 2 ,k j,k− 2 ∆x1 j+ 2 ,k ∆x2 j,k+ 2  ˜ κ+1 and the important discrete conservation property = j,k Qjk  satisfies κ j,k Qjk for vanishing boundary fluxes. Such a scheme can easily be constructed by applying the idea of operator splitting also to (12) and by using two quasi-one-dimensional FV schemes successively, e.g. 2 3 1 κ 1 κ κ ˜ κ+ 2 = Qκ − ∆t F1 (Qκ , . . . , Q )−F (Q , . . . , Q ) , Q jk j−ν+1,k j+ν,k j−ν,k j+ν−1,k jk ∆x1 2 3 1 1 1 1 1 ˜ κ+ 2 )−F2 (Q ˜ κ+ 2 , . . . , Q ˜ κ+ 2 ˜ κ+ 2 − ∆t F2 (Q ˜ κ+1 = Q ˜ κ+ 2 , . . . , Q ) , Q jk jk j,k−ν+1 j,k+ν j,k−ν j,k+ν−1 ∆x2 (∆t)

(∆t)

i.e. H(∆t) = X2 X1 . With this definition the entire splitting method reads (∆t) (∆t) Qκ+1 = S (∆t) X2 X1 (Qκ ). The later method is formally only first-order accurate, but it usually gives very satisfactory results, if high-resolution shock(∆t) capturing schemes are employed for the operators Xn . Formally secondorder accurate splitting methods are possible [32], but they lead to similar results in most practical cases and we have observed only minor improvements for typical detonation structure simulations [5]. For the upwind method

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R. Deiterding, G. Bader

formulated in Algorithm 1, the described splitting is stable under the CourantFriedrichs-Levy (CFL) condition & ∆t ' ∆t ≤1, (15) CCFL := max Sj+ 12 ,k , Sj,k+ 12 j,k ∆x1 ∆x2 where Sj+ 12 ,k , Sj,k+ 12 denote the maximal signal speeds in both space directions according to step (S12) in Algorithm 1. 3.2 High-resolution Upwind Method (∆t)

The operators Xn can be implemented effectively with a single quasi-onedimensional scheme that allows the canonical exchange of the velocities un and the indices j and k. The method should achieve a higher order of accuracy in smooth solution regions and should approximate discontinuities on the basis of the characteristic information (upwinding) without spurious overshoots. Today, several excellent textbooks are available, e.g. [32], which discuss the construction of quasi-one-dimensional high-resolution methods for supersonic hydrodynamics in great detail and we therefore sketch the basic components of our particular method only briefly. Our high-resolution scheme is built around a first-order Godunov-type method that solves the Riemann problem between two neighboring cell values Ql and Qr approximately, which we describe exemplary for the x1 -direction. The method is based on an extension of Roe’s linearized Riemann solver for Euler equations for a single polytropic gas for multiple thermally perfect species by Grossman and Cinella [13] that corresponds to the steps (S1) to (S7) in Algorithm 1. The structure of the Roe-averaged right eigenvectors ˆrm is given in (7). In (S8), (S9) the two intermediate states of the linearized RP are evaluated and the intrinsic problem of unphysical total densities and internal energies near vacuum due to the Roe linearization, cf. [9], is circumvented by switching in case of an unphysical approximation to the simple, but extremely robust Harten-Lax-Van Leer (HLL) Riemann solver. If Roe’s flux approximation is applied in step (S10), violations of the entropy condition are generally avoided by adding an appropriate amount of numerical viscosity [15]. A natural choice for the parameter η for Euler equations is η = 12 (|u1,r − u1,l | + |cr − cl |), cf. [27]. In one space-dimension, (16) need only be applied to ι = 1, 3 and s¯2 = s2 can be used, but two- and threedimensional detonation simulations usually require the extension of (16) to ι = 2. The shock of typical detonation waves is extraordinarily strong and its approximation is often corrupted by the carbuncle phenomenon, a multi-dimensional numerical crossflow instability that occurs at strong grid-aligned shocks or detonation waves Fig. 2. H-correc[26]. The carbuncle phenomenon can be avoided com- tion between the cells pletely by applying (16) to all characteristic fields and (j, k) and (j, k + 1).

High-resolution Simulation of Detonations with Detailed Chemistry (S1) Calculate ρˆ :=

√

ρl ρr and vˆ :=

√ √ ρl vl + ρr vr √ √ ρl + ρr

77

for un , Yi , H, hi , T .

K

cv with cˆ{p/v} = i=1 Yˆi cˆ{p/v}i and (S2) Compute γˆ := cˆp /ˆ
Tr 1 c{p,v}i (τ ) dτ . cˆ{p/v}i = Tr − Tl T l & 2 ' ˆ i + γˆ R Tˆ. γ − 1) uˆ2 − h (S3) Calculate φˆi := (ˆ Wi '1/2 & K ˆ ˆ ˆ (S4) Calculate cˆ := γ − 1)(ˆ u2 − H) . i=1 Yi φi − (ˆ

(S5) Use ∆Q = Qr − Ql and ∆p to compute the wave strengths

∆p ∆p ∓ ρˆcˆ∆u1 , a1+i = ∆ρi − Yˆi 2 , aK+n = ρˆ∆un . 2ˆ c2 cˆ K+d  rm , W3 = aK+d+1 ˆ rK+d+1 . r1 , W2 = am ˆ (S6) Calculate W1 = a1 ˆ a1,K+d+1 =

m=2

(S7) Evaluate s1 = u ˆ1 − cˆ, s2 = u ˆ1 , s3 = u ˆ1 + cˆ.

(S8) Evaluate ρ⋆l/r , u⋆1,l/r , e⋆l/r , c⋆1,l/r from Q⋆l = Ql + W1 and Q⋆r = Qr − W3 .

(S9) If ρ⋆l/r ≤ 0 or e⋆l/r ≤ 0 set s1 = min(u1,l − cl , u1,r − cr ), s3 = max(u1,l + cl , u1,r + cr ), use HLL flux ⎧ f (Ql ) , 0 < s1 , ⎪ ⎪ ⎪ ⎨ s3 f (Ql ) − s1 f (QR ) + s1 s3 (Qr − Ql ) F(Ql , Qr ) = , s1 ≤ 0 ≤ s3 , ⎪ s3 − s1 ⎪ ⎪ ⎩ 0 > s3 , f (Qr ) ,

and go to (S12). ,  (S10) Evaluate Roe flux F(Ql , Qr ) = 21 f (Ql ) + f (Qr ) − 3ι=1 |˜ sι |Wι with entropy enforcement formula |sι | , |sι | ≥ 2η , (16) |¯ sι | = |s2ι |/(4η) + η , |sι | < 2η . l K Yi , Fρ ≥ 0 , ⋆ (S11) With Fρ := i=1 Fi replace Fi by Fi = Fρ · Yir , Fρ < 0 . (S12) Evaluate maximal signal speed by S = max(|s1 |, |s3 |).

Algorithm 1. Hybrid Roe-HLL scheme for detonation simulation.

evaluating η in a multi-dimensional way. In all computations of Sec. 5 we have successfully utilized the “H-correction” of Sanders et al. [27] for this purpose. For instance in the x2 -direction it takes the form 6 5 η˜j,k+ 21 = max ηj+ 21 ,k , ηj− 21 , k , ηj, k+ 12 , ηj− 21 , k+1 , ηj+ 12 , k+1

in the two-dimensional case, see Fig. 2. Step (S11) ensures the positivity of the mass fractions Yi , if the Roe approximation is applied [21]. The HLL scheme can be proven to be positivity preserving in Yi and does not require this step.

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R. Deiterding, G. Bader

A detailed derivation of the entire Roe-HLL scheme and thorough numerical comparisons with various standard methods can be found in [5]. The hybrid Riemann solver is extended to a high-resolution method with the MUSCL-Hancock variable extrapolation technique by Van Leer [34]. The technique uses a five-point stencil with ν = 2. In contrast to the Euler equations for a single polytropic gas, the extrapolation for the Euler equations of Sec. 2.1 can not be formulated completely in conservative variables, because the solvability of the nonlinear equation (5) can not be guaranteed for an extrapolated vector of state. We recommend to apply the MUSCL extrapolation to ρ, p, Yi and ρun and to derive a thermodynamically consistent extrapolated vector of state from these. See [5] for details. 3.3 Integration of Reaction Terms The numerical treatment of chemical reaction terms with the method of fractional steps requires the solution of the ODE ∂ρi = Wi ω˙ i (ρ1 , . . . , ρK , T ) , ∂t

i = 1, . . . , K

with initial condition ρi (0) = ρYi0 , i = 1, . . . , K in every FV cell. The total density ρ in this cell, the specific energy E and the velocities un remain unchanged during the integration, what corresponds to a reaction in an adiabatic constant volume environment. ODEs arising from chemical kinetics are usually stiff and we employ a semi-implicit Rosenbrock-Wanner method by Kaps and Rentrop of fourth order with automatic step-size adjustment [18]. The computational expensive reaction rate expressions (10) are evaluated by a mechanism-specific routine, which is produced by a source code generator on top of the Chemkin-II library [19] in advance. The code generator implements the formulas of Sec. 2.2 without any loops and inserts the constants f /r f /r f /r νji , Aj , −Ej directly into the code. 3.4 Evaluation of the Temperature The FV method for thermally perfect gas-mixtures sketched in Sec. 3.2 and the reaction term integration described in Sec. 3.3 require the computation of the temperature T from a discrete vector of state Q by solving (5). As (5) has a unique temperature solution for all admissible vectors of state and ϕ(·, T ) can be shown to be a strict monotone function in T (see [5]), the efficient solution of (5) is straight-forward: We start the solution procedure with a standard Newton iteration that is initialized with the temperature value of the preceding time step. If the Newton method does not converge in a reasonable number of steps, a standard bisection technique is employed. In order to speed up further the polynomial evaluation of the temperaturedependent properties cpi (T ) and hi (T ), look-up tables for all species are constructed during the startup of the computational code. These tables store

High-resolution Simulation of Detonations with Detailed Chemistry 0.4

3000

6e-6

2000

4e-6

79

4 Pts/lig 1 Pts/lig

1000

0.1

0

2O2

0.2

YH

T [K]

3

ρ [kg/m ]

0.3

ρ (left axis) T (right axis) 15

10

298 5

0

2e-6

0 5

0

Fig. 3. ZND solution for a self-sustaining hydrogen-oxygen detonation (dCJ ≈ 1627 m/s, lig ≈ 1.404 mm) and representation of the mass fraction of H2 O2 on different meshes (right). The points represent the value in the center of a finite volume. The abscissas display the distance behind the detonation front in mm.

cpi (T ) and hi (T ) for all integers in the valid temperature range and intermediate values are interpolated. 3.5 Meshes for Detonation Simulation Numerical simulations of detonation waves require computational meshes, which are able to represent the strong local flow changes due to the reaction correctly. In particular, the shock of a detonation wave with detailed kinetics can be very sensitive to changes of the reaction behind and, if the mesh is too coarse to resolve all reaction details correctly, the Riemann Problem at the detonation front is changed remarkably leading to a wrong speed of propagation. We make a simple discretization test in order to illustrate, how fine computational meshes for accurate detonation simulations in fact have to be. The left graph of Fig. 3 displays the flow fields of ρ and T according to the ZND detonation model for the frequently studied H2 : O2 : Ar CJ detonation with molar ratios 2 : 1 : 7 at T0 = 298 K and p0 = 6.67 kPa for a hydrogenoxygen reaction mechanism extracted from a larger hydrocarbon mechanism assembled by Westbrook [35]. The mechanism uses 34 elementary reactions for the 9 species H, O, OH, H2 , O2 , H2 O, HO2 , H2 O2 and Ar. Throughout this paper only this mechanism has been employed. The right graph of Fig. 3 displays the exact distribution of YH2 O2 discretized with different FV grids. Even a resolution of 4 finite volumes per induction length (4 Pts/lig ) is not sufficient to capture the maximum of the intermediate product H2 O2 correctly. This requires at least 5 to 6 Pts/lig , but in triple points even finer resolutions are required. The discretization of typical combustors with such fine uniform grids can easily exceed 109 FV cells in the two- and 1012 cells in the three-dimensional case. As multi-dimensional detonations are intrinsically unstable (compare Sec. 1), numerical simulations have to be instationary and usually involve several ten thousand time steps. Consequently, uniform meshes are far too expensive and the application of a sophisticated dynamically adaptive mesh refinement technique is indispensable.

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R. Deiterding, G. Bader

Fig. 4. The AMR method creates a hierarchy of rectangular subgrids.

4 Adaptive Mesh Refinement In order to the supply the required temporal and spatial resolution efficiently, we employ the blockstructured adaptive mesh refinement (AMR) method after Berger and Colella [2, 3] which is tailored especially for hyperbolic conservation laws on logically rectangular FV grids (not necessarily Cartesian). We have implemented the AMR method in a generic, dimension-independent object-oriented framework in C++. It is called AMROC (Adaptive Mesh Refinement in Object-oriented C++) and is free of charge for scientific use [6]. The adaptive algorithm has been realized completely decoupled from a particular FV method. All what the algorithm requires are specific implementations of the operators H(·) and S (·) on a single rectangular grid G, where H(·) has to utilize ν auxiliary cells (ghost or halo cells) around G to define discrete boundary conditions. The entire framework has been validated extensively on a large number of hydrodynamic standard test cases. See [5] and [6] for results. 4.1 Berger-Collela AMR Method Instead of replacing single cells by finer ones, as it is done in cell-oriented refinement techniques, the Berger-Collela AMR method follows a patch-oriented approach. Cells being flagged by various error indicators (shaded in Fig. 4) are clustered with a special algorithm [1] into non-overlapping rectangular grids Gl,m that define the domain of an entire level l = 0, . . . , lmax by 7 Ml Gl := m=1 Gl,m . Refinement grids are derived recursively from coarser ones and a hierarchy of successively embedded levels is thereby constructed, cf. Fig. 4. All mesh widths on level l are rl -times finer than on level l − 1, i.e. ∆tl := ∆tl−1 /rl and ∆xn,l := ∆xn,l−1 /rl with rl ∈ N, rl ≥ 2 for l > 0 and r0 = 1, and a time-explicit FV scheme (in principle) remains stable under a condition like (15) on all levels of the hierarchy. The recursive integration order visualized in the left sketch of Fig. 5 is an important difference to usual unstructured adaptive strategies and is one of the main reasons for the high efficiency of the approach.

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Fig. 5. Left: Recursive integration order. Right: Sources of ghost cell values.

The numerical scheme is applied on level l by calling the single-grid routines H(∆t) , S (∆t) in a loop over all subgrids Gl,m . The execution of the numerical loop in UpdateLevel() in Alg. 2 requires the previous setting of the ghost cell values. Three types of boundary conditions have to be considered in the sequential case, see right sketch of Fig. 5. Cells outside of the root domain G0 are used to implement physical boundary conditions. Ghost cells in Gl have a unique interior cell analogue and are set by copying the data value from the grid, where the interior cell is contained (synchronization). On the root level no further boundary conditions need to be considered, but for l > 0 also internal boundaries can occur. They are set by a conservative time-space interpolation from two previously calculated time steps of level l − 1. Beside a general data tree that stores the topology of the hierarchy (cf. Fig. 4), the AMR method requires at most two regular arrays assigned to each subgrid which contain the discrete vector of state Q for the actual and updated time step. In the Algorithms 2 and 3 we denote by Ql (t) and Ql (t + ∆tl ) the unions of these arrays on level l. The regularity of the input data for the numerical routines allows high performance on vector and super-scalar processors and cache optimizations. Small data arrays are effectively avoided by leaving coarse level data structures untouched, when higher level grids are created. Values of cells covered by finer subgrids are overwritten by averaged fine grid values subsequently. The later operation leads to a modification of the numerical stencil on the coarse mesh and requires a special flux correction in cells abutting a fine grid. The correction replaces the coarse grid flux along the fine grid boundary by a sum of grid fluxes and ensures the discrete conservation property for the hierarchical method. See [2] or [5] for details. The basic recursive AMR algorithm is formulated in Alg. 2. Except the regridding procedure, all operations have already been explained. New refinement grids on all higher levels are created by calling Regrid() from level l. Level l by itself is not modified. To consider the nesting of the level domains already in the grid generation, Alg. 3 starts at the highest refineable level lc , where 0 ≤ lc < lmax . The refinement flags are stored in grid-based integer arrays N ι . A clustering algorithm [1] is necessary to create a new refinement

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Regrid(l) AdvanceLevel(l) For ι = lc Downto l Do Repeat rl times Flag N ι according to Qι (t) Set ghost cells of Ql (t) ˘ ι+1 from N ι If time to regrid Generate G Regrid(l) ˘ l+1 , . . . , G ˘ lc +1 Ensure nesting of G UpdateLevel(l) If level l + 1 exists For ι = l + 1 To lc + 1 Do Set ghost cells of Ql (t + ∆tl ) ˘ ι (t) from G ˘ι Create Q ι−1 AdvanceLevel(l + 1) ˘ ι (t) Interpolate Q (t) onto Q Average Ql+1 (t + ∆tl ) onto ι ι ˘ Copy Q (t) onto Q (t) Ql (t + ∆tl ) ˘ ι (t) Set ghost cells of Q l n,l+1 Correct Q (t + ∆tl ) with δF ι ι ˘ Q (t) := Q (t) t := t + ∆tl Alg. 2. Recursive AMR algorithm.

Alg. 3. Regridding procedure.

˘ ι+1 on basis of N ι until the ratio between flagged and all cells in every new G ˘ ι+1,m is above a prescribed threshold 0 < ǫtol < 1. grid G The reinitialization of the hierarchical data structures is done in the second ˘ ι (t). Cells in newly refined regions loop of Alg. 3 utilizing auxiliary data Q ˘ ˘ ι ∩ Gι are copied. Gι \Gι are initialized by interpolation, values of cells in G As interpolation requires the previous synchronized reorganization of Qι−1 (t), recomposition starts on level l + 1. 4.2 Parallelization In the AMROC framework, we follow a rigorous domain decomposition approach and partition the AMR hierarchy from the root level on. We assume a parallel machine with P identical nodes and split the root domain G0 into P non-overlapping portions Gp0 , p = 1, . . . , P by G0 =

P 8

p=1

Gp0

with Gp0 ∩ Gq0 = ∅ for p = q .

The key idea now is that all higher level domains Gi are required to follow the decomposition of the root level, i.e. Gpl := Gl ∩ Gp0 .

(17)

Condition (17) can cause the splitting of a subgrid Gl,m into multiple subgrids on different processors. Under requirement (17) we estimate the work on an arbitrary subdomain Ω ⊂ G0 by / . l l max 1 (18) rκ . Nl (Gl ∩ Ω) W(Ω) = l=0

κ=0

Herein, Nl (·) denotes the total number of FV cells on level l in the given domain. The product in (18) is used to consider the time step refinement.

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A nearly equal distribution of the work necessitates Lp :=

P · W(Gp0 ) ≈ 1 for all p = 1, . . . , P . W(G0 )

(19)

In AMROC, decompositions Gp0 with similar workload are found at runtime as the hierarchy evolves by a hierarchical partitioning algorithm based on a generalization of Hilbert’s space-filling curve [25]. The space-filling curve defines an ordered sequence on the cells of the root level that can easily be split in load-balanced portions. As such curves are constructed recursively, they are locality-preserving and therefore avoid an excessive data redistribution overhead in the final loop of Alg. 3. The second goal in designing an efficient parallelization strategy for distributed memory machines, the minimization of the communication costs, is already considered in condition (17) in a natural way. Together with the use of ghost cells this condition allows an almost local execution of Alg. 2. The only parallel operations that have to be incorporated are the parallel ghost cell synchronization and the application of flux correction terms across processor borders. See [5] for implementation details. Analogous to Alg. 2 the regridding procedure of Alg. 3 is hardly affected by the parallelization as long as a repartitioning of the hierarchy is only allowed at root level time steps, which is usually sufficient in practice. New refinements ˘ p can be found local, only a global concatenation of the new topology G ι+1 ˘ ι+1 = 7 G ˘p G p ι+1 is mandatory to ensure the correct proper nesting of the new hierarchy and to create a new load-balanced root level distribution Gp0 with the partitioner. Finally, the data distribution of parts of Qι (t) to other processors must be incorporated.

5 Numerical Results The self-sustaining CJ detonation of Sec. 3.5 is an ideal candidate for fundamental detonation structure simulations, because it produces extremely regular detonation cell patterns [29]. The application of the numerical methods of Sec. 3 within the parallel AMROC framework allowed a two-dimensional cellular structure simulation that is four-times higher resolved (44.8 Pts/lig ) than earlier calculations [24, 8, 11]. Only recently Hu et al. presented a similarly resolved calculation for the same CJ detonation on a uniform mesh [16]. Unfortunately, no technical details are reported for this simulation. In our case, the calculation ran on a small Beowulf-cluster of 7 Pentium III-850 MHz-CPUs connected with a 1 Gb-Myrinet network and required 2150 h CPU-time. On 24 Athlon-1.4 GHz double-processor nodes (2 Gb-Myrinet) of the HEidelberg LInux Cluster System (Helics) our approach allowed a sufficiently resolved computation of the three-dimensional cellular structure of a hydrogen-oxygen detonation. The maximal effective resolution of this calculation is 16.8 Pts/lig

84

R. Deiterding, G. Bader Slip line Shock Contact Head of reaction zone

M

Head of reaction zone

K

Mach stem shock

H L I

F

G

D B

C

A

Primary triple point moving downwards

Transverse wave Head of reaction zone

E

lig

Incident shock

Fig. 6. Flow structure around a triple point before the next collision. Left: isolines of YOH (black) on schlieren plot of u2 (gray).

and the run required 3800 h CPU-time. Our adaptive results are in perfect agreement with the calculations of Tsuboi et al. for the same configuration obtained on a uniform mesh on a super-scalar vector machine [33]. Further on, we present successful simulations of diffracting two-dimensional hydrogenoxygen detonations that reproduce the experimentally measured critical tube diameter of 10 detonation cells. These computations demonstrate the advantages in employing a dynamically adaptive method impressively and used approximately 4600 h CPU-time on the Helics. 5.1 Two-dimensional Cellular Structure We extend the one-dimensional ZND detonation of Fig. 3 to two space dimensions with u2 = 0 and initiate transverse disturbances by placing a small rectangular unreacted pocket with the temperature 2086 K behind the detonation front, cf. [24] or [5]. After an initial period of ≈ 200 µs very regular detonation cells with λ ≈ 3 cm and oscillation period ≈ 32 µs can be observed in computations with a resolution finer than 7 − 8 Pts/lig (see [5] for a mesh refinement study). We exploit this regularity and simulate only a single cell. The calculation is done in a frame of reference attached to the detonation and requires just the computational domain 10 cm × 3 cm. The adaptive run uses a root level grid of 200 × 40 cells and two refinement levels with r1,2 = 4. A physically motivated combination of scaled gradients and heuristically estimated relative errors is applied as adaptation criteria. See [5] for details. Two typical snapshots with the corresponding refinement are displayed in Fig. 10. The adaptive computation uses between 340, 000 and 390, 000 FV cells, while a uniform grids with the same effective resolution would require 2, 048, 000 cells. About 3554 root level time steps (CCFL ≈ 0.95) to tend = 800 µs were calculated.

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The high resolution of the simulation now admits a remarkable refinement of the triple point pattern introduced in Sec. 1. As the two transverse waves form a perfectly regular flow, it suffices to zoom into a single triple point and to analyze the wave pattern between two triple point collisions in detail. Fig. 6 displays the flow situation around the primary triple point A that is mostly preserved during the last 7 µs before a collision. An analysis of the flow field uncovers the existence of two minor triple points B and C along the transverse wave downstream of A. While B can be clearly identified by a characteristic inflection, the triple point C is much weaker and very diffused. B is caused by the interaction of the strong shock wave BD with the transverse wave. The slip line emanating from B to K is clearly present. C seems to be caused by the reaction front (which can be interpreted as a diffused contact discontinuity) and generates the very weak shock wave CI. Downstream of BD a weaker shock wave EF shows up. It is refracted in the point F, when it hits the slip line BK. From F to G this minor shock is parallel and close to the transverse wave, what results in a higher pressure increase in the region FG than in the region EF. Unreacted gas crossing the transverse wave between B and C therefore shows a shorter induction length than gas entering through AB. The minor shock is refracted and weakened by the reaction front at point G and forms the shock GH that is almost parallel to CI. The downstream line of separation between particles passing through incident or Mach Stem shock is the slip line AD. Along its extension DEL the movement of A results in a shear flow between the reaction zones behind the Mach stem and downstream of BD. In the actual picture the contact discontinuity LM seems to originate in this shear flow region, but a complete instationary analysis uncovers that it propagates constantly downstream. The collision of two triple points in the reinitiation of a detonation cell leads to the formation of an unreacted region behind the detonation front that burns in less than a microsecond. The burning generates upstream traveling shock waves that prevent the appearance of the flow field in Fig. 6 at an earlier stage. The strongest of these shocks hits the Mach stem from behind and forms an additional triple point with the contact discontinuity LM. In Fig. 6 shock and intermediate triple point have already vanished in A, but the contact discontinuity LM is still left behind. A detailed hydrodynamic analysis of the intermediate phase from reinitiation to the almost stable situation of Fig. 6 can be found in [5]. 5.2 Three-dimensional Cellular Structure We utilize the regular oscillating solution of the preceding section as initial condition for the three-dimensional simulation and disturb the oscillation in the x2 -direction with an unreacted pocket in the orthogonal direction. We use a computational domain of the size 7 cm × 1.5 cm × 3 cm that exploits the symmetry of the initial data in the x2 -direction, but allows the development of a full detonation cell in the x3 -direction. The AMROC computation uses a

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3D 2D

1.8 d*/dCJ

1.6 1.4 1.2 1 0.8 670

675

680

685

690 t [µs]

695

700

705

710

Fig. 7. Comparison of the temporal development of the detonation velocity along the line of symmetry through the middle of the detonation cells in the two- and three-dimensional simulation, cf. Figs. 10 and 11.

two-level refinement with r1 = 2 and r2 = 3 on a base grid of 140 × 12 × 24 cells and utilizes between 1.3 M and 1.5 M cells, instead of 8.7 M cells like a uniformly refined grid (3431 root level time steps with CCFL ≈ 0.95 to tend = 800 µs). After a simulation time of ≈ 600 µs a regular cellular oscillation with identical strength in x2 - and x3 -direction can be observed. In both transverse directions the strong two-dimensional oscillations is present and forces the creation of rectangular detonation cells of 3 cm width. The transverse waves form triple point lines in three space-dimensions. During a complete detonation cell the four lines remain mostly parallel to the boundary and hardly disturb each other. The characteristic triple point pattern can therefore be observed in Fig. 11 in all planes perpendicular to a triple point line. Unlike Williams et al. [36] who presented a similar calculation for an overdriven detonation with simplified one-step reaction model, we notice no phase-shift between both transverse directions. In all our computations for the hydrogen-oxygen CJ detonation only this regular three-dimensional mode, called “rectangularmode-in-phase” [14], or a purely two-dimensional mode with triple point lines just in x2 - or x3 -direction did occur. A direct comparison of the temporal development of the detonation velocity in the two- and three-dimensional case in Fig. 7 along lines through the center of the graphics in Figs. 10 and 11 shows that both simulations reproduce the same oscillation period of approximately 32 µs, but the detonation appears to be remarkably higher overdriven during reinitiation in the three-dimensional case. 5.3 Structure of Diffracting Detonations Experiments have shown that the behavior of planar CJ detonations propagating out of tubes into unconfinement is determined mainly by the width of the tube. For square tubes the critical tube width has been found to be of the order of 10-times the cell width, i.e. 10λ [23]. For widths significantly below 10λ the process of shock wave diffraction causes a pressure decrease

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Fig. 8. Density distribution on four refinement levels at tend = 240 µs for w = 10λ.

at the head of the detonation wave below the limit of detonability across the entire tube width. Hydrodynamic shock and reaction front decouple and the detonation decays to a shock-induced flame. This observation is independent of a particular mixture. While the successful transmission of the detonation is hardly disturbed for tubes widths ≫ 10λ, a backward-facing re-ignition wave reinitiates the detonation in the partially decoupled region for widths of ≈ 10λ and creates considerable vortices. We are interested in the decoupling of shock and reaction and also in the reignition phenomenon. Therefore, we simulate the two-dimensional diffraction of the H2 : O2 : Ar CJ detonation for the tube widths w = 8λ and w = 10λ. A periodically reproduced snapshot of the regular oscillating detonation propagating into unreacted gas at rest is used as initial condition. This is a reasonable idealization for the flow situation in real detonation tubes directly before the experimental setup. The symmetry of the problem is exploited by simulating just one half. The adaptive simulations utilize a base grid of 508 × 288 cells and use four levels of refinement with r1,2,3 = 2, r4 = 4. The calculations correspond to a uniform computation with ≈ 150 M cells and have an effective resolution of 25.5 Pts/lig in the x1 -direction (with respect to the initial detonation). Both runs are stopped 240 µs after the detonation has left the tubes (730 root level time steps with CCF L ≈ 0.8), when the flow situations of interest are clearly established. The enormous efficiency of the refinement is visualized in Fig. 8 for the setup with w = 10λ. At tend the calculation shown in Fig. 8 uses ≈ 3.0 M cells on all levels, where ≈ 2.4 M cells are inside one of the 2479 grids of the highest level (ǫtol = 0.8). A comparison of the flow fields in both setups after 240 µs in Fig. 9 clearly shows the extinction of the detonation for w = 8λ and the re-ignition wave for w = 10λ. The reappearance of triple points at the detonation front in the

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Fig. 9. Schlieren plots of ρ for a detonation diffracting out of the two different tubes. Left: detonation failure for the width w = 8λ, right: reinitiation for w = 10λ.

lower left plot in Fig. 9 is a characteristic indicator for the preservation of the detonation throughout the diffraction. It is interesting to note, that the re-ignition wave by itself is a detonation. The triple point track for w = 10λ (not shown) uncovers that it has developed out of the transverse wave of an initial triple point.

6 Conclusions We have described an efficient solution strategy for the numerical simulation of gaseous detonation waves with detailed chemical reaction. All temporal and spatial scales relevant for the complex process of detonation propagation were successfully resolved. The achieved resolutions are significantly finer than in earlier publications [24, 8, 11] and provide similar insight into the formation and propagation of transient detonation structures like recent large-scale simulations on uniform meshes [16, 33]. Beside the application of the time-operator splitting technique and the construction of a robust high-resolution shock capturing scheme, the key to the high efficiency of the presented simulations is the generic implementation of the blockstructured AMR method after Berger and Collela [2] in the AMROC framework [6]. AMROC provides the required high local resolution dynamically and follows a parallelization strategy that is tailored especially for the emerging generation of distributed memory architectures. All presented results have been achieved on Linux-Beowulf-clusters of moderate size in a few days real time, what demonstrates that advances in computational fluid dynamics do not necessarily require large super-computers, but integrated approaches, which combine fast and accurate discretizations with sophisticated techniques from computer science. Acknowledgement. This work was supported by the DFG high priority research program “Analysis and Numerics of Conservation Laws”, grant Ba 840/3-3.

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Fig. 10. Temperature and schlieren plots of the density on refinement regions in the first (left) and second half (right) of a detonation cell. (See also color figure, Plate 9.)

Fig. 11. Schlieren plots of ρ (upper row) and YOH (lower row) in the first (left) and second (right) half of detonation cell, mirrored at x2 = 0 cm, 5.0 cm < x1 < 7.0 cm. The plots of YOH are overlaid by an isosurface of ρ that visualizes lig . (See also color figure, Plate 10.)

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References 1. J. Bell, M. Berger, J. Saltzman, and M. Welcome. Three-dimensional adaptive mesh refinement for hyperbolic conservation laws. SIAM J. Sci. Comp., 15(1):127–138, 1994. 2. M. Berger and P. Colella. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys., 82:64–84, 1988. 3. M. Berger and J. Oliger. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys., 53:484–512, 1984. 4. R. Courant and K. O. Friedrichs. Supersonic flow and shock waves. Applied mathematical sciences, volume 21. Springer, New York, Berlin, 1976. 5. R. Deiterding. Parallel adaptive simulation of multi-dimensional detonation structures. PhD thesis, Brandenburgische Technische Universit¨at Cottbus, Sep 2003. 6. R. Deiterding. AMROC - Blockstructured Adaptive Mesh Refinement in Objectoriented C++. Available at http://amroc.sourceforge.net, Mar 2004. 7. Y. N. Denisov and Y. K. Troshin. Structura gazovoi detonatsii v trubakh (Structure of gaseous detonations in tubes). Zh. Eksp. Teor. Fiz., 30(4):450–459, 1960. 8. C. A. Eckett. Numerical and analytical studies of the dynamics of gaseous detonations. PhD thesis, California Institute of Technology, Pasadena, California, Sep 2001. 9. B. Einfeldt, C. D. Munz, P. L. Roe, and B. Sj¨ogreen. On Godunov-type methods near low densities. J. Comput. Phys., 92:273–295, 1991. 10. W. Fickett and W. C. Davis. Detonation. University of California Press, Berkeley and Los Angeles, California, 1979. 11. T. Geßner. Dynamic mesh adaption for supersonic combustion waves modeled with detailed reaction mechanisms. PhD thesis, Math. Fakult¨ at, University Freiburg, 2001. 12. E. Godlewski and P.-A. Raviart. Numerical approximation of hyperbolic systems of conservation laws. Springer Verlag, New York, 1996. 13. B. Grossmann and P. Cinella. Flux-split algorithms for flows with nonequilibrium chemistry and vibrational relaxation. J. Comput. Phys., 88:131–168, 1990. 14. M. Hanana, M. H. Lefebvre, and P. J. Van Tiggelen. Pressure profiles in detonation cells with rectangular or diagonal structure. In Proc. of 17th Int. Colloquium on the Dynamics of Explosive and Reactive Systems, Heidelberg, Jul 1999. 15. A. Harten. High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49:357–393, 1983. 16. X. Y. Hu, B. C. Khoo, D. L. Zhang, and Z. L. Jiang. The cellular structure of a two-dimensional H2 /O2 /Ar detonation wave. Combustion Theory and Modelling, 8:339–359, 2004. 17. N. N. Janenko. Die Zwischenschrittmethode zur L¨ osung mehrdimensionaler Probleme der mathematischen Physik. Springer-Verlag, Berlin, 1969. 18. P. Kaps and P. Rentrop. Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations. Num. Math., 33:55–68, 1979. 19. R. J. Kee, F. M. Rupley, and J. A. Miller. Chemkin-II: A Fortran chemical kinetics package for the analysis of gas-phase chemical kinetics. SAND89-8009, Sandia National Laboratories, Livermore, California, Sep 1989.

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20. R. J. Kee, F. M. Rupley, and J. A. Miller. The Chemkin thermodynamic data base. SAND87-8215B, Sandia National Laboratories, Livermore, California, Mar 1990. 21. B. Larrouturou. How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comput. Phys., 95:59–84, 1991. 22. B. Larrouturou and L. Fezoui. On the equations of multi-component perfect or real gas inviscid flow. In Carasso et al., editor, Proc. of Second Int. Conf. on Nonlinear Hyperbolic Equations - Theory, Numerical Methods, and Applications, Aachen 1988, Lecture Notes in Mathematics 1402, pages 69–98. Springer-Verlag Berlin, 1989. 23. J. H. S. Lee. Dynamic parameters of gaseous detonations. Ann. Rev. Fluid Mech., 16:311–336, 1984. 24. E. S. Oran, J. W. Weber, E. I. Stefaniw, M. H. Lefebvre, and J. D. Anderson. A numerical study of a two-dimensional H2 -O2 -Ar detonation using a detailed chemical reaction model. J. Combust. Flame, 113:147–163, 1998. 25. M. Parashar and J. C. Browne. On partitioning dynamic adaptive grid hierarchies. In Proc. of the 29th Annual Hawaii Int. Conf. on System Sciences, Jan 1996. 26. J. J. Quirk. Godunov-type schemes applied to detonation flows. In J. Buckmaster, editor, Combustion in high-speed flows: Proc. Workshop on Combustion, Oct 12-14, 1992, Hampton, pages 575–596, Dordrecht, 1994. Kluwer Acad. Publ. 27. R. Sanders, E. Morano, and M.-C. Druguett. Multidimensional dissipation for upwind schemes: Stability and applications to gas dynamics. J. Comput. Phys., 145:511–537, 1998. 28. J. Smoller. Shock waves and reaction-diffusion equations. Springer-Verlag, NewYork, 1982. 29. R. A. Strehlow. Gas phase detonations: Recent developments. J. Combust. Flame, 12(2):81–101, 1968. 30. R. A. Strehlow and F. D. Fernandez. Transverse waves in detonations. J. Combust. Flame, 9:109–119, 1965. 31. D. R. Stull and H. Prophet. JANAF thermodynamical tables. Technical report, U. S. Departement of Commerce, 1971. 32. E. F. Toro. Riemann solvers and numerical methods for fluid dynamics. Springer-Verlag, Berlin, Heidelberg, 2nd edition, 1999. 33. N. Tsuboi, S. Katoh, and A. K. Hayashi. Three-dimensional numerical simulation for hydrogen/air detonation: Rectangular and diagonal structures. In Proc. of the Combustion Institute, volume 29, pages 2783–2788, 2003. 34. B. van Leer. Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method. J. Comput. Phys., 32:101–136, 1979. 35. C. K. Westbrook. Chemical kinetics of hydrocarbon oxidation in gaseous detonations. J. Combust. Flame, 46:191–210, 1982. 36. D. N. Williams, L. Bauwens, and E. S. Oran. Detailed structure and propagation of three-dimensional detonations. In Proc. of the Combustion Institute, pages 2991–2998. 26, 1997. 37. F. A. Williams. Combustion theory. Addison-Wesley, Reading, Massachusetts, 1985.

Numerical Linear Stability Analysis for Compressible Fluids Andreas S. Bormann Institut f¨ ur Verfahrenstechnik, Technische Universit¨ at Berlin, Germany [email protected]

Summary. The Rayleigh-B´enard problem and the Taylor-Couette problem are two well-known stability problems that are traditionally treated with linear stability analysis. In the vast majority of these stability calculations the fluid is considered to be incompressible [Cha61, DR81]. Only with this assumption and simplification is possible to conduct a linear stability analysis analytically. In order to calculate the stability limits of a compressible fluid by use of a linear stability analysis therefore in this work a numerical linear stability analysis is presented. The numerical stability analysis is based upon the equations of balance for mass, momentum and energy that are completed with the constitutive equations by Navier-Stokes and Fourier. The algorithm allows to calculate the regions of stability for arbitrary one-dimensional and stationary basic states. This numerical stability analysis is used to calculate the stability region for the Rayleigh-B´enard problem. The main result is that the critical Rayleigh number does not have a constant value, as calculations involving the Boussinesq approximation suggest misleadingly, but that the value of the critical Rayleigh number depends strongly on the thickness of the fluid layer. Furthermore, an empirically found relationship between the critical Rayleigh number and the thickness of the fluid layer is presented (14). Its efficiency is successfully verified with the results of the numerical linear stability analysis. The results for the critical Rayleigh number show clearly that the compressibility of a fluid must not be neglected in the stability analysis of the Rayleigh-B´enard problem. Secondly, the more complicated Taylor-Couette problem is treated with the numerical linear stability analysis. In contrast to the traditional stability analysis by Taylor [Tay23], the fluid is considered to be compressible and includes the temperature as a field variable. The effectiveness of the numerical linear stability analysis is manifested by the good agreement of the comparison with experimental results. In addition to that, temperature effects are studied and are compared with experiments.

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1 Numerical linear stability analysis The approach to the numerical stability analysis is documented in detail in [Bor03] and therefore will be presented here in a shortened form. The approach consists of five steps: • Calculation of the one-dimensional and stationary basic state • Linearization of the compressible and viscous Navier-Stokes and Fourier balance equation system with respect to the basic state • Discretization of the linearized system • Establishment of a characteristic eigenvalue problem • Calculation of the eigenvalues and determination of the stability properties First, the one-dimensional and stationary basic state has to be calculated. For this purpose, the constitutive equations of Navier-Stokes and Fourier are inserted into the balance equations for mass, momentum and energy which results typically in a system similar to

∂ψA (xi , t) ∂ 2 ψA (xi , t) ∂ψA (xi , t) = f ψA (xi , t), , , ... (1) ∂t ∂xi ∂xi ∂xj with

ψA (xi , t) = {ρ(xi , t), v1 (xi , t), v2 (xi , t), v3 (xi , t), T (xi , t)} .

(2)

Usually, the boundary conditions are combined up in a non-dimensional number α, e.g. the Rayleigh number Ra or the Taylor number T a, which may be treated as a parameter for the basic state. The calculation of the basic state gives ψ A = ψ A (x1 , α). (3) Linearization of (1) with respect to (3) is done in the typical manner by the use of the ansatz (4) ψA (xi , t) = ψ A (x1 , α) + ψ9A (xi , t). The term taking account for the disturbance is defined by ψ9A (xi , t) = ψ:A (x1 ) exp {i(k2 x2 + k3 x3 ) + σt} ,

(5)

where k2 and k3 are positive real wave numbers and σ is a complex constant that governs the temporal behavior of the disturbance. The resulting system of differential equations reads dψ:B (x1 ) d2 ψ:B (x1 ) , + DDAB σ ψ:A (x1 ) = AAB ψ:B (x1 ) + DAB dx1 dx21

with

AAB = f1 (ψ A (x1 , α),

dψ A (x1 , α) d2 ψ A (x1 , α) , , α, k2 , k3 ), dx1 dx21

DAB = f2 (ψ A (x1 , α),

dψ A (x1 , α) d2 ψ A (x1 , α) , , α, k2 , k3 ), dx1 dx21

DDAB = f3 (ψ A (x1 , α),

dψ A (x1 , α) d2 ψ A (x1 , α) , α, k2 , k3 ). , dx1 dx21

(6)

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This set of equations is often too complicated to be solved analytically. Mainly, this is due to the coefficients A, D and DD that depend on x1 . One possibility to deal with that problem is to discretize (6) within the domain x1 ǫ{0, d} over ASt points which will lead to an algebraic system. The discrete values of the fields are defined by ρ:(i) = ρ:(x1 )|x1 =(i−1)h ,

v:1

(i)

= v:1 (x1 )|x1 =(i−1)h ,

i = {1, .., ASt},

= v:3 (x1 )|x1 =(i−1)h , # # = T:(x1 )# ,

i = {1, .., ASt},

v:2 (i) = v:2 (x1 )|x1 =(i−1)h , v:3

(i)

T:(i)

i = {1, .., ASt},

x1 =(i−1)h

with h =

i = {1, .., ASt}, i = {1, .., ASt},

d . ASt − 1

With these definitions and the use of difference quotients for the derivations in (6) one obtains the eigenvalue problem (MY Z − σδY Z )φ:Z = 0, with

Y, Z = {1, .., 5ASt},

(7)

2

MY Z = f (ψ A , dψ A , d ψ A , α, k2 , k3 ).

where the vector φ:Z is given by 5 (1) (1) (1) φ:Z = ρ:(1) , v:1 , v:2 , v:3 , T:(1) , . . . ,

(8)

6 (ASt) (ASt) (ASt) :(ASt) , ρ:(ASt) , v:1 , v:2 , v:3 , T

with Z = {1, .., 5 ∗ ASt).

The matrix MY Z in (7) may be given conveniently by the following relations # # DAB ## DDAB ## M5(i−1)+A, 5(i−2)+B = + , 2h #x1 =(i−1)h h2 #x1 =(i−1)h # DDAB ## , M5(i−1)+A, 5(i−1)+B = AAB |x1 =(i−1)h − 2 h2 #x1 =(i−1)h (9) # # DDAB ## DAB ## + , M5(i−1)+A, 5i+B = − 2h # h2 # x1 =(i−1)h

with

x1 =(i−1)h

A, B = {1, .., 5} and i = {2, .., ASt − 1} .

The inclusion of the boundary conditions for the discretization points 1 and ASt is not trivial and therefore must be elaborated for a given problem individually. Details for the Rayleigh-B´enard problem and the Taylor-Couette problem are explained in [Bor01, Bor03].

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The eigenvalues σ of (7) must now be calculated in order to describe the stability behavior of the chosen basic state in (3). Of particular interest are the real parts of the eigenvalues. If one of the real parts of the calculated eigenvalues is greater than zero then instability is to be expected because a disturbance would grow in time (see ansatz (5)). Stability, on the other hand, can only be achieved if for all combinations of k2 and k3 the real parts of the eigenvalues are less than zero. In order to find the critical non-dimensional number αcrit for a given stability problem one has to calculate the eigenvalue problem (7) for a fixed combination of k2 and k3 in an iterative way where α has to be varied until αkrit (k) ⇔ max(Re(σ = f (α, k2 , k3 ))) = 0.

(10)

Then, this calculation has to be repeated for other combinations of k2 and k3 in order to find the minimum of αkrit (k) αkrit = min(αkrit (k))

∀ k.

(11)

αkrit is the typical critical parameter of a stability problem. It describes the point of neutral stability; if the basic state has an α that is less than αkrit then stability of the basic state is maintained. In the other case where α is greater than αkrit stability of the corresponding basic state is lost. It may be easily seen that the main problem of this numerical linear stability analysis is the calculation of the eigenvalues σ. Especially when the number of discretization points ASt is high, there is a need for a reliable and fast eigenvalue solver. The ARPACK routine package allows a very fast and efficient calculation of some eigenvalues of large scale eigenvalue problems [LSY98]. It can therefore be used successfully for the calculations of the results in the next two sections.

2 The compressible Rayleigh-B´ enard problem 2.1 Introduction and motivation The Rayleigh-B´enard problem concerns a fluid layer that is bounded by two plates and the lower plate is heated. If the temperature gradient in the fluid at rest exceeds a critical value, convection will occur and stability is lost. For the determination of the state of stability a non-dimensional parameter was introduced [Ray16], that is called the Rayleigh number Ra =

gα|∆T |d3 . κc ν

(12)

Here g denotes the gravitational acceleration, d the thickness of the fluid layer, |∆T | the temperature difference between top and bottom of the layer. α, κc ,

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and ν are the coefficients of volume expansion, thermometric conductivity and kinematic viscosity. One way to determine the critical value of the Rayleigh number is to use a linear stability analysis [Cha61], where, in order to simplify the equations for the linear stability analysis, usually the Boussinesq approximation is applied. According to this approximation the density of a fluid is assumed to depend on the temperature only, while the dependence on the pressure, i.e. the compressibility κT , is neglected. The result of such a linear stability analysis is a constant value for the critical Rayleigh number and critical wave length kcrit that depends only on the boundary conditions. Table 1. Critical Rayleigh number and critical wave length kcrit . rigid-rigid boundaries:

I RaBouss = 1707.8 crit

I kcrit = 3.177

rigid-free boundaries:

II RaBouss = 1100.7 crit

II kcrit = 2.682

free-free boundaries:

III RaBouss = 657.5 crit

III kcrit = 2.221

Unfortunately, from a thermodynamic point of view the Boussinesq approximation contradicts the second law of thermodynamics because thermodynamic stability requires the inequality [Mue85] α2 ≦

ρcp κT . T

(13)

If κT is neglected as it is in the Boussinesq approximation, then also the volume expansion coefficient α must vanish. Another hint that the results of Table 1 should not be used for compressible fluids is given by recent experiments of Kogan& Meyer [KM01]. They confirm experimentally that the value of the critical Rayleigh number is dependent on the layer thickness. Furthermore a quite simple relation for the calculation of the value of the critical Rayleigh number of a compressible fluid may be found by adding the critical Rayleigh number for an incompressible and viscous fluid to the critical Rayleigh number of a compressible and inviscid fluid according to the Schwarzschild criteria ⎧ ⎫ ⎨ 1707.8 ⎬ α2 ρ2 0 T0 g 2 d4 . (14) Racrit = 1100.7 + ⎩ ⎭ κη 657.5 Although this semi-empirical relation seems to be too simple at first sight, it coincides very well with the experimental results of [KM01] and the results of the numerical linear stability analysis that are presented in the next section. The derivation of (14) is explained in detail in [Bor03]

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Fig. 1. Comparison of critical Rayleigh number of the numerical linear stability analysis (symbols) with the equation (14) (drawn through lines). The dashed lines indicate the results of the critical Rayleigh number involving the Boussinesq approximation.

2.2 Results A stability analysis for compressible fluids was performed as outlined in Section 1 for both, liquids and gases. A detailed description of the calculations is found in [Bor01, Bor03]. The selected liquids were water, ether (DEE), and mercury. As a gas argon was chosen. All calculations refer to a temperature T1 = 298 K and a pressure of p1 = 1 bar for the liquids, pressures of 10 bar down to 0.001 bar for argon. In Figure 1 the results of the stability analysis for the selected liquids and the gas argon are presented for the different boundary conditions. The abscissa represents the layer thickness d in meters while the ordinate gives the critical Rayleigh number. The marked points represent the results of the numerical linear stability analysis while the dashed lines indicate the constant value for the critical Rayleigh number for a given boundary problem that is obtained by use of the Boussinesq approximation. The drawn out lines are calculated from equation (14) For small layer thicknesses the critical Rayleigh number approximates asymptotically the constant value of the Rayleigh number of the Boussinesq case, while the critical Rayleigh number for large layer thicknesses is strongly dependent on the layer thickness.

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This behavior may be understood from the fact that a large fluid layer exerts a non negligible pressure on the fluid layer at the bottom. The fluid becomes compressed and density is larger than in the Boussinesq case where a change of density due to pressure is neglected. Now, in order to achieve a critical density gradient for instability, the temperature gradient - and with it the thermal expansion - has to be increased beyond the one of an incompressible fluid. This implies that the Rayleigh number must increase with an increasing layer thickness. The critical wave numbers for the different boundary cases resulted in the same values as already calculated by use of the Boussinesq approximation in Table 1. This observation is confirmed by [JH99] The results of the linear stability analysis for compressible fluids show clearly that the stability behavior of a the Rayleigh-B´enard problem is characterized by two parameters instead of only the constant Rayleigh number for the Boussinesq case. For a linear stability analysis for compressible fluids the value of the critical Rayleigh number becomes dependent on the thickness d of the fluid layer. The simple ansatz (14) agrees perfectly with the results of the numerical linear stability analysis and should therefore be used instead of the Rayleigh number obtained by use of the Boussinesq approximation. One important remark concerning the value of the isothermal compressibility: the value of the compressibility κT does not affect at all the deviation of the critical Rayleigh number for a compressible fluid from the critical Rayleigh number calculated with the Boussinesq approximation at all. This can be seen clearly in equation (14). Even for a vanishing small value of the compressibility κT the value of the critical Rayleigh number will depend strongly on the layer thickness. This means that the often read justification for the use of the Boussinesq approximation for water - its low compressibility - is not valid.

3 The compressible Taylor-Couette problem 3.1 Introduction and motivation The geometry of the Taylor-Couette problem consists of two coaxial, freely rotating cylinders. Up to a critical angular velocity of the cylinders the fluid flow between these cylinders is a one-dimensional shear flow. If the angular velocities exceed the critical limit, the stability of the shear flow is lost and another stationary flow can be observed - the Taylor vortices. The task of a stability analysis for the Taylor-Couette problem is to find the critical angular velocity for the inner cylinder Ω1,crit for a given angular velocity of the outer cylinder Ω2 . Another important question is the determination of the critical wave number with which the height of the Taylor vortices may be calculated.

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Taylor calculated the critical Ω1 − Ω2 relations for a incompressible and isothermal fluid and compared them successfully with his own experimental results [Tay23]. Because of the very good agreement of analytical and experimental results his work is often taken as a paradigm for analytical fluid dynamics. For his calculations he had to restrict himself to the ”narrow-gap” geometry. In that case the distance between the two cylinders has to be much smaller than the radii of the cylinders. Chandrasekhar overcame this restriction for the case that the outer cylinder has twice the radius of the inner cylinder. For this ”wide-gap” geometry he calculated the critical Ω1 − Ω2 relations for an incompressible and isothermal fluid. Donnelly and Fultz experimentally confirmed the calculations by Chandrasekhar [Cha58, DF60]. In contrast to the Rayleigh-B´enard problem, where the critical nondimensional wave is a constant, numbers are strongly dependent on the angular velocities. Another difficulty arises from the fact that the basic state of the compressible and temperature dependent Taylor-Couette problem cannot not be calculated analytically but only numerically. These facts complicate the linear stability analysis and therefore the Taylor-Couette problem may be used to prove the efficiency of the numerical linear stability analysis. Furthermore the numerical linear stability analysis is capable of calculating the critical Ω1 − Ω2 relations not only for a narrow-gap and wide-gap geometry of the two cylinders but for any arbitrary geometry. Another important advantage of the numerical linear stability analysis is the consideration of the temperature field. Effects due to viscous heating or applied temperature gradients may be investigated more conveniently. 3.2 Results A numerical linear stability is performed as outlined in Section 1. For details about the exact algorithm and entries in the matrix of the characteristic eigenvalue problem see [Bor03]. In order to compare the results with experiments, for the narrow-gap geometry the experimental data of Taylor was used [Tay23]. He performed experiments with water at room temperature for three different sets of geometry of the cylinders: the radius of the outer cylinder R2 was fixed to 4.035 cm while the radius of the inner cylinder R1 had the values 3 cm, 3.55 cm and 3.8 cm. Taylor carried out his experiments in the following manner: First the angular velocity of the outer cylinder Ω2 was adjusted. Then he increased the angular velocity of the inner cylinder Ω1 very slowly until the basic state of the shear flow lost its stability and the Taylor vortices were present. The transition of the one dimensional shear flow into the vortex flow marks the critical Ω1 − Ω2 relation. In Figure 2 the results of the numerical linear stability analysis, Taylor’s experiments and Taylor’s calculations are compared for the three different geometries of the cylinders.

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Fig. 2. Comparison of the results of the critical Ω1 −Ω2 -relations for the narrow-gap geometry of the numerical linear stability analysis (nLSA) with Taylor’s experimental and analytical results.

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It can be seen clearly that there is a very good agreement of the results for positive Ω1 − Ω2 relations for all three sets of experiments. Especially for faster angular velocities, the numerical linear stability delivers good results, better than those of Taylor’s own calculations. For negative Ω2 the results of the critical inner angular velocity Ω1 of the numerical linear stability analysis also show a good agreement but deviate with higher angular velocities from the experimental results. On the one side, this might be due to the high angular velocities, where the measurement of the experiments could only be accomplished with some difficulties. On the other side, the ansatz (5) only allows for axis-symmetric secondary flows while Taylor observed some slightly non-axis-symmetric flows at high angular velocities. After having proved the effectiveness of the numerical linear stability analysis for the narrow-gap geometry, that analysis was applied to the widegap geometry as well. While Chandrasekhar had to develop a linear stability analysis for this geometry from scratch [Cha58], in the numerical linear stability analysis only the radii of the cylinders had to be changed. Donnelly&Fultz [DF60] performed experiments with water and viscous oils for the wide-gap geometry. The inner cylinder was fixed to a radius of R1 = 3.1432 cm while the outer cylinder had the radius R2 = 6.2864 cm. The results of the numerical linear stability analysis, the experiments by Donnelly and Fultz and Chandrasekhar’s calculations are plotted in Figure 3. Once again there is a very good agreements of the results of the numerical linear stability analysis with the experimental results. Because of the low angular velocities a deviation from the experimental results was not expected and is not observed. The critical wave numbers have been calculated as well for both sets of geometries and they show a good agreements with the experiments for both cases. Details are to be found in [Bor03]. The influence of temperature on the critical Ω1 − Ω2 relation has been studied not as thoroughly as the isothermal basic state. Mainly, two scenarios were investigated. First, the influence of an applied temperature gradient was investigated by Snyder&Karlsson [SK64] and Sorour&Coney [SC79]. Snyder&Karlsson found that for the narrow-gap geometry an applied temperature gradient increases the stability region slightly, i.e. for a given Ω2 a slightly higher critical inner angular velocity Ω1 could be measured. This behavior could be verified qualitatively with the numerical linear stability analysis. Unfortunately the data base of the experiments did not allow for an exact quantitative comparison. For faster angular velocities both, Snyder&Karlsson and Sorour&Coney observed non-axi-symmetric secondary flows. The critical Ω1 − Ω2 relation for these flows could not be calculated with the numerical linear stability analysis because of its strictly axi-symmetric model for disturbances (5). Secondly, the influence of viscous heating is of some importance for the critical Ω1 − Ω2 relation in the Taylor-Couette problem. There has not been much research on this topic until recently. Al-Mubaiyedh et al. [ASK02] came

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Fig. 3. Comparison of the critical Ω1 − Ω2 -relation of the numerical linear stability analysis (nLSA) for the wide-gap geometry with the experiments by Donnely&Fultz and the calculations by Chandrasekhar.

to the theoretical result, that for fixed outer angular velocity Ω2 viscous heating should lead to a critical Ω1 that is less than the critical Ω1 for an isothermal flow. Also, according to them, the decrease of the critical inner angular velocity runs monotonically with the increase of the temperature in the fluid due to viscous heating. White&Muller [WM02] recently performed experiments on this topic with water, mixtures of water and glycerin and some oils. They could confirm the predictions made Al-Mubaiyedh et al. although not quantitatively. Unfortunately, also the experimental data of White&Muller do not contain the necessary information that would allow to perform a quantitative comparison of the numerical linear stability analysis with their experimental results. At least, qualitatively the numerical linear stability analysis shows the same behavior as reported by White&Muller and Al-Mubaiyedh et al.: With increasing viscous heating the critical inner angular velocity drops monotonically.

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4 Conclusion A numerical stability analysis is presented and proven to be a reliable and effective method for the calculation of stability of one-dimensional, stationary, compressible and viscous basic states. For the investigated stability problems - the Rayleigh-B´enard problem and the Taylor-Couette problem - the results are in good agreement with experimental results. For the Taylor-Couette problem the advantage of the numerical linear stability analysis over the traditional incompressible and isothermal calculation is to be found in the arbitrary choice of cylinder geometries and the consideration of the temperature field, which allows for a much broader range of applications. The results for the Rayleigh-B´enard problem show clearly that the often used Boussinesq approximation is not suitable for that problem. It could be proven that the value of the critical Rayleigh number is not a constant but a function that is strongly dependent on the thickness of the fluid layer. Furthermore a semi-empirical relation for the critical Rayleigh number for compressible fluids is presented (14), which should be used instead of the well know constant values of the critical Rayleigh number for the determination of the stability limit for that problem.

References [ASK02] Al-Mubaiyedh, U.A., Sureshkumar, R., Khomami, B.: The effect of viscous heating on the stability of the Taylor-Couette flow. J. Fluid Mech., 462, 111–132 (2002) [Bor01] Bormann, A.S.: The onset of convection in the Rayleigh-B´enard problem for compressible fluids. Cont. Mech. Therm., 13, 9–23 (2001) [Bor03] Bormann, A.S.: Lineare Stabilit¨ atsanalyse kompressibler Fluide. Diss., Technische Universit¨at Berlin, http://edocs.tuberlin.de/diss/2003/bormann andreas.pdf (2003) [Cha58] Chandrasekhar, S.: The stability of viscous flow between rotating cylinders. Proc. Roy. Soc. A, 246, 301–311 (1958) [Cha61] Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability. At the Clarendon Press, Oxford (1961) [DF60] Donnely, R., Fultz, D.: Experiments on the stability of viscous flow between rotating cylinders. II. Visual observations. Proc. Roy. Soc. A., 258, 101–123 (1960) [DR81] Drazin, P.G., Reid, W.H.: Hydrodynamic stability. Cambridge Universtity Press, Cambridge (1981) [JH99] Jeng, J., Hassard, B.: The critical wave number for the planar B´enard problem is unique. Int. J. Non Linear Mech., 34, 221–229 (1999) [KM01] Kogan, A.B., Meyer, H.: Heat transfer and convection onset in a compressible fluid: 3 He near the critical point. Phys. rev. E, 63, 4635–4638 (2001)

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[LSY98] Leboucq, R.B., Sorensen, D.C., Yang, C.: ARPACK Users guide: Solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM, Philadelphia(PA) (1998) [Mue85] M¨ uller, I: Thermodynamics. Pitman Advanced Publishing Program, Boston (1985) [Ray16] Strutt, J.W. (Lord Rayleigh): On the convection currents in a horizontal layer of fluid when the high temperature is on the under side. Phil. Mag., 2, 833–844 (1916) [SC79] Sorour, M.M., Coney, J.E.R.: The effect of temperature gradient on the stability of flow between vertical, concentric, rotating cylinders. J. Mech. Eng., 21, 403–409 (1979) [SK64] Snyder, H.A., Karlsson, S.K.F.: Experiments on the stability of Couette motion with a radial themal gradient. Phys. Fluids, 7, 1669–1706 (1964) [Tay23] Taylor, G.I.: Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Roy. Soc. A, 223, 289–343 (1923) [WM02] White, J.M., Muller, S.J.: Experimentals studies on the stability of newtonian Taylor-Couette flow in the presence of viscous heating. J. Fluid Mech., 462, 133–159 (2002)

Simulation of Solar Radiative Magneto-Convection M. Sch¨ ussler1 , J.H.M.J. Bruls2 , A. V¨ ogler1 , and P. Vollm¨ oller1,3 1

2

3

Max-Planck-Institut f¨ ur Sonnensystemforschung Max-Planck-Str. 2, 37191 Katlenburg-Lindau, Germany [email protected] Kiepenheuer-Institut f¨ ur Sonnenphysik Sch¨ oneckstr. 6, 79104 Freiburg, Germany [email protected] Present address: VAW, ETH-Zentrum, 8092 Z¨ urich, Switzerland

1 Introduction The term ‘magneto-convection’ summarizes the variety of processes arising from the dynamic interaction between convective motions and magnetic fields in an electrically conducting medium. Magneto-convective processes play an important role in many astrophysical systems; their effects can be best studied in the case of the Sun, where the relevant spatial and temporal scales of the phenomena can (in principle, at least) be observed. The generation of magnetic flux in the Sun by a self-excited dynamo process and the various spectacular phenomena of solar activity, like sunspots, coronal loops, flares, and mass ejections all are, directly or indirectly, driven by magneto-convective interactions. The large length scales of the typical convective flow structures on the Sun lead to high (hydrodynamic and magnetic) Reynolds numbers, so that the magneto-convective processes typically involve nonlinear interactions and formation of structures and patterns. Fig. 1 illustrates typical regimes of magneto-convection near the visible solar surface, differing mainly in the amount of magnetic flux per unit area (i.e., average magnetic field strength) and the orientation of the field. In the ‘quiet’ Sun, some magnetic flux is concentrated in bright magnetic elements, isolated patches with field strengths exceeding 1000 G (corresponding to 0.1 Tesla). In magnetically active regions, such magnetic elements densely populate the dark convective downflow network and decrease the size of convective upflows (‘granules’ in solar physics lingo). In the dark core of a sunspot (the so-called umbra), the strong vertical magnetic field is space-filling and largely suppresses the convective energy transport. The less dark, striated periphery of a sunspot (the penumbra) har-

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bors a magnetic field which is strongly inclined with respect to the vertical direction.

Umbra

Plage

Penumbra

Quiet Sun Fig. 1. Magnetic structure on the visible solar surface and different regimes of magneto-convection. There is only a small amount of magnetic flux in quiet regions and the convective pattern (hot plasma rising in bright ‘granules’, cooled by radiation and flowing back into the interior in the network of dark intergranular lanes) is best visible. In areas with more magnetic flux (plage regions), the flux becomes assembled in small flux concentrations, which appear bright because of locally enhanced transparency of the atmosphere. At even higher levels of magnetic flux density, the convection is largely suppressed by the magnetic field and sunspots form. They have a dark core (the umbra) with almost vertical magnetic field and a surrounding region of inclined field (the penumbra), whose striated appearance and mode of energy transport are not well understood (Image taken with the German Vacuum Tower Telescope on Tenerife, Spain, operated by the Kiepenheuer-Institut, Freiburg; courtesy O. von der L¨ uhe)

Realistic numerical simulations of solar magneto-convection represent a considerable computational challenge. There is an extended range of length scales between the dominant scale of the convective flow pattern (the granulation) of about 103 km and the dissipation scales of the order of a few km and less. The plasma is strongly stratified with pressure scale heights down

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to about 100 km and even a restricted simulation has to cover a density ratio of the order of 100. Convective velocities reach the sound speed, so that full compressibility is mandatory. Solar convection is strongly affected by partial ionization effects and over some height range near the surface the major part of the convective energy flux is transported in the form of latent heat. Consequently, the ionization state of the most abundant species (foremost hydrogen) has to be monitored in the course of the simulation and the related energetics have to be incorporated into the equation of state, which then cannot be written as a closed expression but has to be specified in the form of a numerical table. The energetics of the lower solar atmosphere is strongly affected by radiative energy transport. In this region, radiative heating or cooling of the plasma cannot be considered as a local (diffusive) process since the mean free path of photons is comparable to or larger than the dominant spatial scale of the flow patterns and the pressure scale height. Consequently, the radiative transfer equation for the specific intensity of radiation has to be integrated along a large number of rays of various angles in order to determine the radiation incident on each grid cell. In order to correctly represent the temperature field in the solar atmosphere, the frequency dependence of the radiation has to be taken into account. This further complicates the problem because about a million spectral lines contribute to the energy balance in the solar photosphere. Another complication is related to the boundary conditions. Because of the strong stratification and large size of the solar convection zone, the computational box for any realistic simulation of solar convection can only cover a tiny fraction of the whole convection zone and solar surface. Therefore, we have numerical boundaries where physically no boundaries are. For the side boundaries one can assume periodic conditions if the box is much larger than the dominant scale of the flow, but for the top and bottom boundaries the situation is less clear in a gravitationally stratified medium. Particularly at the bottom the assumption of a closed boundary would be quite unrealistic: in compressible convection, the downflows are narrow, fast and coherent over many scale heights. It is thus necessary to develop appropriate ‘open’ boundary conditions, which permit the free in- and outflow of matter while maintaining the total mass in the computational box and allowing for the correct amount of convective energy transport. In order to meet the computational and methodological challenges, our ANumE project had three major goals: 1. Develop and evaluate methods to treat frequency-dependent radiative energy transport in simulations of magnetohydrodynamic processes, 2. develop combined compressible MHD/radiative transfer codes with partial ionization and open boundaries, and

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3. perform realistic simulations and analyze them with spectral and polarization diagnostics in order to compare with observations and measurements of the Sun. The whole project is carried out in close cooperation with our colleagues A. Dedner, M. Wesenberg, Christian Rohde, and D. Kr¨ oner from the Institute for Applied Mathematics (IAM) of the University of Freiburg. It proved to be fruitful to develop two codes more or less in parallel. One development (mainly by the IAM group) concerned a code based upon MHD Riemann solvers for non-ideal gases, an unstructured grid with adaptive grid refinement and dynamical load balancing for parallel computation. The second code was developed in cooperation with F. Cattaneo, Th. Emonet, and T. Linde from the University of Chicago; it is less sophisticated (fixed structured grid, 4th-order finite-differences and time stepping) and thus was available in 3D somewhat earlier than the IAM code, so that the developed methods and modules for non-grey radiative transport (RT) could be incorporated and tested, including first full simulation runs. A 2D version of the IAM code has been already combined with the RT; for the 3D version this is currently been done and first results are expected in 2005. This paper is organized as follows. We briefly describe the basic equations of radiative MHD in Sec. 2. The developments of numerical methods for radiative transfer are discussed in Sec. 3: RT on unstructured grids in Sec. 3.2 and the treatment of the frequency dependence in Sec. 3.3. Results of simulations are presented in Sec. 4: first results of 2D simulations with the IAM code combined with RT are shown in Sec. 4.1 while 3D results with the MPAe-Chicago code and non-grey RT are given in Sec. 4.2. We conclude with a brief outlook in Sec. 5.

2 Equations of radiative magnetohydrodynamics The magnetohydrodynamic (MHD) approximation can be used to describe a collision-dominated, electrically well-conducting, quasi-neutral plasma. These conditions are fairly well fulfilled in the convection zone and lower atmosphere of the Sun. Starting from the Maxwell equations (in Gaussian units), ∇×B =

1 ∂E 4π j+ , c c ∂t

∇×E = −

1 ∂B , c ∂t

(1) (2)

∇ · E = 4πǫ ,

(3)

∇·B = 0,

(4)

with the electric field E, the magnetic field B, current density j, and electrical charge density ǫ, the MHD approximation is obtained under the assumptions

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that all material speeds and phase velocities are small compared to the speed of light and that the plasma is charge-neutral owing to a sufficiently high electrical conductivity. These conditions entail that the electrical field is small compared to the magnetic field and that the displacement current (second term on the right-hand side of (1)) can be neglected. Using Ohm’s law in a medium locally moving with velocity v, 1 j = σ(E + v×B) , c

(5)

where σ is the electrical conductivity, we derive a single equation for the time evolution of the magnetic field, the induction equation, viz. ∂B = ∇× (v × B) − ∇× (η∇ × B), ∂t

(6)

with the magnetic diffusivity defined as η = c2 /4πσ. The induction equation governs the time evolution of the magnetic field for a given velocity field, v. The first term on the r.h.s. describes the inductive effect of the velocity field, the second term accounts for diffusion of magnetic field due to the finite conductivity of the plasma. The order-of-magnitude estimate for the ratio of these terms gives the magnetic Reynolds number, Rm = vL/η, where v is a typical speed and L is a typical length scale of the flow under consideration. Estimates for the photosphere and upper convection zone give magnetic Reynolds numbers of the order of 105 − 106 , so that the diffusion term is almost negligible in these regions. In the high-Rm regime, Alfven’s theorem of flux-freezing applies: magnetic field lines are transported by the fluid as if frozen in and fluid motions relative to the magnetic field are possible only along the direction of field lines. The rest of the equations of the MHD approximation are the equations of hydrodynamics with appropriate magnetic terms in the momentum equation (the Lorentz force) and the energy equation (the Joule dissipation term in the case of a non-vanishing magnetic diffusivity). The continuity equation ∂̺ + ∇ · (̺v) = 0 ∂t

(7)

represents mass conservation. The equation of motion is written in momentum conservation form:     |B|2 BB ∂̺v + ∇ · ̺vv + p + (8) 1− = ̺g + ∇ · τ . ∂t 8π 4π Here p is the gas pressure and g the vector of gravitational acceleration. vv and BB are dyadic products and 1 is the 3 × 3 unit matrix. The magnetic force (Lorentz force) has been split into the gradient of the magnetic pressure, pmag = |B|2 /8π, and the term −∇·(BB/4π), which represents a tension along magnetic field lines. The last term on the r.h.s. of (8) is the viscous force, written as divergence of the viscous stress tensor, τ .

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The energy equation is written in the form     |B|2 ∂e 1 + ∇· v e+p+ B(v · B) − ∂t 8π 4π =

(9)

1 ∇ · (B × η∇ × B) + ∇ · (v · τ ) + ∇ · (K∇T ) 4π

+ ̺(g · v) + Qrad . for the total energy density per unit volume, e, which is the sum of internal, kinetic and magnetic energy densities. T is the temperature and K the thermal conductivity. Qrad is the radiative source term which accounts for heating and cooling of the plasma by radiation. This term is discussed in detail in Sec. 3. In order to close the system of MHD equations, we have to specify a relation between pressure, p, density, ̺, and internal energy of the gas, eint . At temperatures typically encountered in the photosphere and upper convection zone, the solar plasma is partly ionized and the simple thermodynamical relations for an ideal gas do not apply. Owing to the corresponding changes in the thermodynamical properties of the matter up to 2/3 of the enthalpy flux is transported by latent heat and buoyancy driving of convective motions is strongly enhanced. Under the given conditions in the solar photosphere and the uppermost part of the convection zone it is sufficient to consider only the first ionization of the eleven most abundant elements in the Sun. The internal energy per mass unit εint = eint /̺ can be written as 3 1 ∗ (ne + na ) kT + ni χi , 2̺ ̺

(10)

'  1  3kT & xi νi χi , 1+ xi νi + 2µa m0 µa m0

(11)

εint =

where the sum runs over the particle species, n∗i is the number density of ionized particles of type i, and χi is the corresponding ionization energy. na =  ni is the number density of nuclei, and ne the number density of electrons. Defining the ionization degree, xi = n∗i /ni , and the relative abundance, νi = ni /na , (10) can be rewritten as εint =

where µa is the mean molecular weight of the neutral gas (µa = 1.29 for solar composition) and m0 is the atomic mass unit. The ionization degrees, xi , are determined by the set of Saha equations 3/2 ui1 (T ) µa m0 2 (2πme kT ) xi  xi νi = exp (−χi /kT ) . 1 − xi ui0 (T ) ̺ h3

(12)

The temperature dependence of the partition functions ui1 , ui0 can be obtained from the literature. For temperatures exceeding about 16, 000 K, the

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elements are almost fully ionized and the temperature dependence can be neglected. In order to obtain the temperature from ̺ and eint , the nonlinear system of equations (11) and (12) needs to be solved iteratively. Once the temperature is known, the gas pressure follows from the perfect gas equation: '  ̺ & p = (ne + na ) kT = (13) 1+ xi νi kT. µa m0

3 Numerical methods for radiative transfer 3.1 The radiative source term The photosphere is the region where most of the radiation leaves the Sun and where radiation takes over from convection as the dominant mechanism of energy transport. The energy exchange between gas and radiation determines the temperature structure of the photosphere and is responsible for the entropy drop which acts as the main driver of convection. Therefore any realistic simulation must include the radiative energy exchange rate, Qrad , as a source term in the energy equation. Since the mean free path of photons increases strongly as the atmosphere becomes transparent in the photosphere, radiative transfer at this height becomes essentially non-local and a diffusion approximation of radiative energy transport as adequate for the solar interior cannot be applied. The starting point for determining Qrad is the (time- and frequencydependent) radiative transfer equation (RTE hereafter),   1 ∂ + Ω · ∇ Iν = κν ̺(Sν − Iν ) . (14) c ∂t The specific intensity, Iν , is defined such that the amount of energy dErad transported by radiation along direction Ω in the frequency interval (ν, ν +dν) across an area element dS into a solid angle dω in a time interval dt is dErad = Iν (x, Ω, t)(Ω · dS) dω dν dt .

(15)

Sν is the source function and κν is the frequency-dependent absorption coefficient of the material. Since the travel time of a photon through the photosphere is much shorter than any other relevant timescale, the radiation field can be assumed to adjust quasi-instantaneously to any change of the thermodynamical state of the gas, i.e. the time derivative in (14) can be neglected and we obtain: (16) Ω · ∇Iν = κν ̺(Sν − Iν ) . Defining the optical depth of a path element ds as dτν = κν ̺ ds the RTE for a given direction can be written in the form

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dIν (Ω) = Sν − Iν (Ω) , dτν which has the formal solution




Iν (τν ) = Iν (0) e−τν +

τν

Sν (tν ) e−(τν −tν ) dtν .

(17)

(18)

0

In the case of local thermal equilibrium (LTE), the source function is given by the Planck function Sν = Bν . The essential premise of LTE is that elastic collisions between particles represent the dominant interaction on a microscopic level. Then the velocity distribution of particles is Maxwellian and the ionization states and population numbers of atomic, ionic, and molecular energy levels are determined by Saha-Boltzmann statistics corresponding to the local temperature. Significant departures from LTE must be expected in the upper photosphere, especially in strong spectral lines, for which scattering dominates over thermal emission and disturbs the detailed energy balance of LTE. This effect can be neglected as long as these lines do not contribute significantly to the total (frequency-integrated) energy exchange rate, Qrad . Since in LTE the source function is independent of the radiation field, (18) can be integrated in a straightforward manner. The numerical treatment of radiative transfer is based on this formal solution. Once the radiation field is known, the radiative energy flux,
Fν = Iν (Ω) Ω dω , (19) and the average intensity, Jν =

1 4π




Iν (Ω) dω ,

can be calculated. The radiative heating rate then follows either from
Qrad = − (∇ · Fν ) dν

(20)

(21)

ν

or from the equivalent expression Qrad = 4πκ̺




ν

κν (Jν − Bν ) dν .

(22)

Consequently, for each cell in the computational grid, the numerical determination of the radiative heating rate, Qrad , requires a sequence of integrations: 1. Spatial integration of the equation of radiative transfer (16) along a number of directions to determine the respective specific intensity Iν (Ω), 2. angular integration of Iν (Ω) to determine the radiative energy flux, Fν , or the mean intensity Jν , and 3. frequency integration to determine the radiative heating rate from (21) or (22).

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3.2 Radiative transfer on unstructured grids In the course of our ANumE project in cooperation with the IAM of the University of Freiburg, we have carried out two studies on the spatial and angular integrations (steps 1 and 2 in the preceding section) in unstructured grids. The first study [1] concentrated on the modification of the short-characteristic formal solver [2] for the case of unstructured grids, while the second study [3] led to the development of a new class of adaptive solvers. Ignoring the frequency dependence (to be discussed in Sec. 3.3) for the time being and thus dropping the index ν in all frequency-dependent quantities, a short-characteristic solver starts from a discretized form of the formal solution (18) of the RTE. Consider the situation shown on the left panel of Fig. 2: given the incident intensity Ii+1 at grid point i + 1, characterized by optical depth τi+1 measured along the ray in the direction −Ω starting from τ (s = ∞) = 0, the intensity Ii is given by
τi+1 Ii = Ii+1 e−∆τi + S(τ ) e−(τ −τi ) dτ, (23) τi

where ∆τi = τi+1 − τi =




si

κ(s)ρ(s) ds.

(24)

si+1

Accurate numerical evaluation of the integral in (23) would in general be cumbersome, but after approximating S(τ ) by a linear or quadratic function of τ , it can be written analytically as ∆I =




τi+1 τi

S(τ ) e−[τ −τi ] dτ =

i+1 

Wj S j ,

(25)

j=i−1

where the coefficients Wj depend only on the optical depth intervals, ∆τi and ∆τi−1 . Approximating κ(s) by a linear or a quadratic function, (24) can also be replaced by a simple analytic expression in terms of the function values at the points i + 1, i, and i − 1. Details of the method are described in [1]. The next step towards the determination of QR is the integration of I(Ω) over 4π steradian of solid angle in order to obtain the mean intensity, J, or the radiative flux, F. This integration is expressed as a quadrature sum over a discrete set of directions. The choice of the directions Ωm and their weights wm is subject to a few mandatory normalization criteria for the  lowest moments of the intensity. In particular, the mean intensity J = wm Im  and the flux F = 4π wm Im Ωm should obtain their correct values for an isotropic radiation field. A sensible criterion is the invariance under rotation around the z−axis over multiples of 90◦ . Another desirable criterion, although difficult to achieve, is that the directions should be distributed as evenly as possible over the entire sphere. An exact construction procedure of the angular quadrature has been given in [4]. It can be shown that, for a given quadrature,

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the discretization errors in F are markedly smaller than those in J. Errors are typically in the percent range for a quadrature with 3 directions per octant; significant improvements can only be achieved at a high computational cost.

C

z,s

τ i

i+1 ∆τ i

Ω

E

i-1 ∆τ i-1

Ω

A D B

Fig. 2. Left: geometry for the short-characteristic solution of the radiative transfer equation in direction Ω. Right: radiative transfer on a cell from a triangular grid. The specific intensity on corner C can be computed once the values on A and B are available for interpolation on point D (from [1]).

As an example for the adaptation of the short-characteristic method to a finite-volume scheme on an unstructured grid we consider the planar case with a triangular grid. The most obvious way to obtain QR would be through (20) by computing J directly for each cell center, but this straightforward approach requires excessive and poorly-defined interpolation. It is easier, more efficient, and more accurate to compute the intensities I at the vertices of the cells and afterwards use either J or F to compute the cell-average of QR required by a finite-volume method. First consider the radiative transfer problem within a single triangle (right panel of Fig. 2), where we want to compute IC (Ω). The procedure is as follows: starting from C, tracing the ray in the upwind direction, locate point D where the ray enters the cell. IC can be computed by means of a short-characteristic integration over the interval DC provided that the incident intensity ID is known. In general, point D does not coincide with a vertex of the grid and ID has to be interpolated from IA and IB , which must therefore be known beforehand: this requires that the intensities at the vertices are computed in the proper order. The incident intensities at the periodic side boundaries are obtained by iteration: from a suitable starting guess for the incident intensities at the vertices of the inflow boundary, compute the intensities at the corresponding vertices on the opposite boundary — which should be exactly

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the same — and use those as update for the incident intensities. This process is repeated until the intensities have converged. Using iteration for the initialization, however, is tricky because the convergence rate of this iteration procedure can be slow in the optically thin part of the atmosphere. However, once the incident intensities have been initialized correctly, iteration can well be used to update the incident intensities after each time step; the outflow intensities at time tn provide a good initial guess for the incident intensities at time tn+1 , so that few if any iterations are needed. The radiative transfer problem for the entire computational domain then simplifies to a sequence of radiative transfer problems on single triangles. The simplest short-characteristic integration method uses κC and κD (the values of the opacity at point C and D, respectively) to define a linear relation κ(s) on the interval DC. Together with the relation for ρ(s) this yields an analytic expression for ∆τDC . That, in turn, is used together with SC and SD to define a linear relation S(τ ) to evaluate ∆I from (23). This approach has limited applicability though: given the strongly non-linear dependence of κ on T , a small cell-to-cell variation of T already leads to significant nonlinear variation of κ, so that the optical path length ∆τDC is misrepresented. Straight application of the short-characteristic method with quadratic S(τ ) is not possible, since it requires a point downwind from point C in order to define the necessary quadratic functions for κ(s) and S(τ ). Such a point would be located outside of the triangle and violate the local character of the method. However, without significant adverse effects the method can be reformulated to use an auxiliary point E exactly halfway between D and C to define those quadratic functions. This quadratic approach significantly increases the maximum allowable cell-to-cell variation of T at a given accuracy level of I. We have implemented radiative transfer routines to compute the radiative heating rate, Qrad for various 2D model situations on a triangular grid [1]. Qrad can be computed from the mean intensity, J, or from the radiation flux, F. We have studied the accuracy of the short-characteristic radiative solver on such a grid and the accuracy of the angular integration required to compute J and F. It turned out that QJR has severe accuracy problems in the optically thick regions while QF R is stable and accurate there but may fail completely in optically thin layers. Therefore, the best solution is a combination of QJR and QF R with a selector based on the optical path lengths. In a subsequent study in the course of the ANumE project, Dedner and Vollm¨ oller [3] have compared a number of methods for the solution of the equation of radiative transfer on a triangular planar grid. These included the discontinuous-Galerkin finite element method [5], the long-characteristic method [6], and the short-characteristic method. All these methods can be generalized to the 3D case and to different grid geometries in a straightforward way. As a result of this study, a new class of methods was developed, the extended-short-characteristic (ESC) solvers, which combine the finite-element and short-characteristic approaches. In the case of triangles, the first-order version, ESC1, requires the determination of the intensity on the three node

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A

F

E

C

B

D

E

D

ff

A

cc

B

C

A

ee

cc

F dd

B

Fig. 3. To solve the RT equation on a single element using the extended-shortcharacteristic (ESC) method it is necessary to calculate the intensity for different points, depending on the desired order of the method (1 or 2) Left: in the ESC1 method the intensity at point C has to be computed on triangles with only one inflow edge along the characteristic starting at point cc. Middle and right: in the ESC2 method the intensities have to be computed at point F (two inflow edges) or E, C, D (one inflow edge) using the corresponding inflow intensities at point ff or at points ee, cc and dd respectively (from [3]).

points, while the second-order ESC2 method additionally includes the midpoints of the edges (see Fig. 3). The ESC methods combine the idea of local ansatz functions for the solution from the finite element framework with the idea of solving local 1D initial value problems along characteristics. Variation of the ansatz functions allows to develop schemes of higher order. The method can be adapted to specific applications by using ODE solvers for RT equation which depend on the stiffness of the underlying ODE. Indeed it was found that Runge-Kutta solvers are superior to the classical formal solution approach with respect to the error-to-runtime ratio. To speed up the higher order ESC method, an adaptation strategy including the variation of the order from grid cell to grid cell is relatively simple to implement because the computation of the intensity coefficients is independent of the chosen ansatz function. In three space dimensions, the coefficients for the discrete solution are also given by the same ODE as in 2D, so that this part of the module can be used without modifications. Details about this study and the ESC methods can be found in [3]. 3.3 Frequency-dependent radiative transfer Since the total absorption coefficient (opacity) in the solar atmosphere comprises the effect of the order of 106 atomic and molecular spectral lines, a number of roughly 106 -107 frequency points is required to model the detailed frequency dependence. While this direct approach is feasible in calculating 1D static models, the computational cost is intolerable in time-dependent 2D or 3D simulations. The most radical simplification of the problem is achieved by

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the grey approximation, replacing the frequency-dependent opacity by an averaged value, e.g. the Rosseland mean. The grey approach in combination with the diffusion approximation may be appropriate in the optically dense subphotospheric regions, where the radiation field is in local equilibrium with the gas, it is however unsatisfactory in the optically thinner regions where the radiation transfer becomes nonlocal. Here it is not only necessary to treat the full radiative transfer but also to incorporate line opacities, since line-blanketing effects have a considerable impact on both the photospheric dynamics and the emergent intensities. It is well known that the inclusion of line opacities in calculations of stellar model atmospheres strongly modifies the resulting temperature profiles, leading to considerably cooler outer layers, while the temperature is raised in deeper regions (the line cooling and backwarming effects). The effect of non-grey radiative transfer on the results of our MHD simulations has been studied in [7] and [8]. L

∆λ i-1

∆λ i

∆λ i+1 λ

L k+1

l

τ ref Lk

l -1

τ ref

∆λ i

λ

τ ref

Fig. 4. Left: Sorting of wavelengths according to a discretization of opacity. The hatched areas mark those parts of the wavelength interval ∆λi for which the opacity lies in the interval [Lk , Lk+1 ]. Right: Schematic illustration of the τ -sorting procedure. The wavelength intervals ∆λi are sorted to opacity bins according to the height where τ = 1 is reached for that wavelength interval, which is indicated by bold arrows (from [9]).

For reasons of computational feasibility in 2D/3D simulations, one has to resort to an appropriate statistical treatment of the line opacities that conserves the non-grey character of the radiation transport while drastically reducing the computational expense. In the context of time-dependent threedimensional simulations, for which the radiative transfer must be solved for every timestep, the only feasible approach – given the computing resources of today – is the opacity binning approach, also called multi-bin or multi-group method [10, 11, 12]. The basic idea of this method is to sort frequencies into 4–6 (non-contiguous) groups according to the geometrical depth in a 1D reference atmosphere at which optical depth unity at that frequency is reached. For each of these frequency groups, a separate RT equation with group-integrated source function and opacity is solved and the respective intensities are then

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added to obtain the total intensity. Under the assumption of local thermodynamic equilibrium (LTE), the (frequency-dependent) source function Sν is equal to the Planck function, Bν . The opacities can then be obtained from pre-compiled tables without having to solve the system of rate equations for each gridpoint and timestep during a simulation run. While the opacity binning approach has been tested in the case of 1D static model atmospheres [13], the application in the context of time-dependent 2D/3D simulations is accompanied by new sources of errors not encountered in the static 1D case like, for instance, strong lateral variations of the atmospheric properties and the occurrence of steep velocity gradients and shocks. In order to gain confidence in the applicability of this approach in simulations, we have performed in [9] a series of tests of the multigroup method for several 1D and 2D cases with the solution based on opacity distribution functions [14] serving as the reference solution. In a plane-parallel atmosphere, a substantial improvement of the radiative heating rates in comparison to the grey case can be achieved with the multigroup method already with a moderate number of frequency groups. In order to test whether this result also holds for situations with substantial lateral variations, we have considered two cases: a magnetic flux sheet embedded in a non-magnetic atmosphere and a snapshot from a 2D simulation of solar surface convection. Magnetic flux sheet We consider a simple model of a magnetic flux concentration, a 2D flux sheet. Using a 1D model atmosphere, the stratification in the interior of the sheet is shifted downwards by 200 km relative to the surrounding atmosphere, resulting in a strong lateral variation of the thermodynamic quantities. The width of the sheet as a function of height is determined by magnetic flux conservation together with the condition of total (magnetic plus gas) pressure equilibrium between the interior and the exterior of the sheet. At the height z = 0, corresponding to the visible surface (continuum optical depth unity) in the exterior, the flux sheet has a width of 150 km. The sheet is fanning out with increasing height as the magnetic pressure necessary to balance the jump in gas pressure decreases. At the interfaces between the interior and the exterior of the sheet, the atmospheric parameters are smoothed horizontally over a distance of a few tens of kilometers by way of a Gaussian error function. Owing to the mirror symmetry of the sheet, the calculations can be restricted to one half of the flux sheet, with symmetrical boundary conditions imposed on the sheets symmetry axis (located at x = 0). A Cartesian grid with 201 × 161 grid points and horizontal and vertical resolutions of 2.5 km and 5 km, respectively, was used. At equal geometrical height, density and temperature within the flux sheet are lower than the corresponding values in its surroundings. Consequently, the flux sheet is more transparent and thus subject to radiative heating from the

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Fig. 5. Horizontal profiles of Qrad at two different heights in a 2D atmosphere with an embedded flux sheet. x = 0 corresponds to the symmetry axis of the sheet. The curves indicate the reference solution (obtained using opacity distribution functions), the grey solution (1 bin, Rosseland opacity) and the solutions for opacity binning with 3 and 5 bins, respectively (from [9]).

hot ‘walls’ of surrounding plasma. This can be seen in Fig. 5, which shows the radiative heating rate, Qrad , as a function of the horizontal coordinate for two geometrical heights. The grey (Rosseland opacity) and multi-group results are compared with the reference solution based upon an opacity distribution function (ODF) description of the spectrum with effectively nearly 4000 frequency points. Corresponding to the horizontal temperature gradient, a heating peak inside the sheet and a stronger cooling region outside form near the sheet boundary. Below z ≃ −150 km, the outside atmosphere and most of the boundary region are optically thick and the radiative transfer is essentially grey, so that Qrad is well represented even without a detailed treatment of the frequency dependence. The maximum relative errors near the heating and cooling peaks range between 5 and 10 percent. At z = 100 km (left panel of Fig. 5), both multi-group solutions qualitatively reproduce the ODF case while the grey case is much less accurate. The heating peak at the boundary is shifted outwards by approximately 10 km, the peak value being reduced by 15 percent with respect to the ODF case, while the small cooling dip outside is not captured at all. At a still greater height of 250 km (right panel of Fig. 5), both the 3- and 5-bin approximations yield acceptable results, though neither captures the full extent of the cooling peak outside the sheet boundary. The grey solution, on the other hand, does not even approximately reproduce the reference solution. The heating peak inside the sheet has vanished; instead, heating takes place immediately outside the sheet, where the reference solution shows considerable cooling. This behavior can be explained by the fact that, on the basis of the Rosseland mean, the interior of the sheet is transparent at a height of 250 km; accordingly, the

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interaction between gas and radiation is very weak within the sheet, resulting in small values of Qrad . In the optically thicker regions immediately outside the sheet, radiation originating from deeper, hotter regions at the opposite sheet boundary and crossing the sheet almost unattenuated, leads to a net heating effect. This example clearly demonstrates how important a nongrey approach to radiative transfer can be in optically thin regions in order to obtain accurate values of Qrad . It also shows that the multi-group approach leads to a reasonably adequate description of the non-grey effects.

Fig. 6. Horizontal profiles of Qrad for a snapshot from a 2D simulation of solar convection at heights of 100 km (left panel) and 500 km (right panel) above the visible solar surface (from [9]).

Convection simulation As a further step towards more realistic situations we tested the opacity binning models with a snapshot from a 2D simulation of solar surface convection [15], which has no relation to the one-dimensional reference atmosphere used for sorting the frequencies into groups.. The vertical and horizontal extent of the computational domain is 1400 × 1400 km2 with a grid resolution of 35 km. Fig. 6 shows Qrad in horizontal cuts at heights of approximately 100 and 500 km above the visible solar surface, respectively. Similar to results for the flux sheet, all solutions (including the grey case) agree reasonably well with each other in the deeper layers, although the errors of the grey case become more pronounced towards the horizontal boundaries of the domain. The differences between the 3- and 5- bin solutions are only marginal. At 500 km height, the grey solution completely fails to reproduce the reference solution while the 5-bin solution excellently matches the reference curve. In summary we can conclude that our test calculations have shown that the multi-group approach yields a good approximation to the frequency-integrated

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radiative heating rate at moderate computational cost, which makes it the method of choice in multidimensional time-dependent MHD simulations. The advantage over the grey approximation is particularly pronounced in situations which deviate strongly from the one-dimensional plane-parallel case. In these cases the radiative transfer is complicated by lateral heating and cooling effects. Grey radiative transfer often fails to capture these effects, which leads to qualitatively wrong heating rates in the upper photosphere. This is particularly relevant if magnetic field concentrations are included, since the partial evacuation of these structures leads precisely to the kind of lateral inhomogeneities which are not well modeled by the grey approach. The test calculations with snapshots from numerical simulations have shown that the good performance of the multi-group method does not strongly depend on the choice of the reference atmosphere, which underlines the applicability of this method in realistic multi-dimensional simulations.

4 Simulations results for solar magneto-convection In the course of the ANumE project, we have developed two codes for the simulation of radiative MHD. The MHD part of the first code (called IAMMPAe code in what follows) is due to our partners at the IAM (University of Freiburg) and incorporates Riemann solvers for real gases and unstructured grids. We have implemented the ESC solvers for radiative transfer in the 2D version of this code, calculated the tables of the (non-ideal) equation of state, and introduced a realistic open boundary condition at the bottom of the computational box. This code has then be used to follow the formation of magnetic flux concentrations in the solar atmosphere and to analyze their oscillation properties [16]. Some of these results are summarized in Sec. 4.1. The MHD part of the second code has been jointly developed with colleagues from the University of Chicago. We have introduced a shortcharacteristic radiative transfer solver including a non-grey treatment based upon the multi-group method [7], partial ionization, and an open lower boundary. Results from simulations runs with this code (the MURAM1 code) are given in Sec. 4.2. 4.1 Convective intensification of magnetic flux Observations show that the majority of the magnetic flux through the solar atmosphere is assembled in magnetic field concentrations with a field strength of 1500 G and above [17]. Such field strengths exceed the value Beq corresponding to equipartition between magnetic energy density and kinetic energy density of the convective flows by at least a factor of three. Consequently, the concentration of the magnetic flux cannot be solely due to the passive advection of 1

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magnetic field lines by converging convective motions until the Lorentz force impedes further transport. It has been suggested that thermodynamic effects play an important role for the intensification of the magnetic field beyond the equipartition limit, Beq . The suppression of the horizontal convective motion once the field strength approaches Beq throttles the energy supply to the magnetic regions. Since at the solar surface the radiative energy loss into free space continues, this leads to a substantial cooling of the gas. Under the influence of gravity and pressure forces, the cool and dense gas sinks down. The downflow is further accelerated by the convectively unstable (strongly superadiabatic) stratification below the solar surface. This leads to a partial evacuation of the upper part of the forming flux concentration, which becomes laterally compressed by the external pressure until the field has grown strong enough to reestablish lateral pressure equilibrium. The combination of processes of flux advection by horizontal flow, suppression of convection, radiative cooling, downflow and compression is called convective collapse or convective intensification [18, 19, 20, 21]. We have used the 2D version of the adaptive IAM-MPAe code to study the convective intensification of magnetic flux in a region of 12, 000 km horizontal size extending in the vertical direction between 300 km above and 900 km below the visible solar surface. The computational setup is to start with a slightly perturbed plane-parallel convectively unstable stratification and let non-magnetic convection develop until a statistically stationary situation is reached. Then a homogeneous vertical magnetic field of 100 Gauss is introduced and its development followed in the course time. Fig. 7 (to be found in the section with color pages at the end of this volume) shows a sequence of snapshots of the magnetic field (field lines) and the temperature field (color coding), which clearly demonstrates the concentration of the magnetic field into a few intense flux sheets of kilogauss field strength located in convective downflow regions. The simulations nicely confirm the theoretical concepts of flux expulsion and convective collapse. Within a few minutes, most of the magnetic flux is transported by the converging horizontal flows to the cool downflow region (flux expulsion). Suppression of the convective energy transport and ongoing radiative cooling leads to downflow of the gas in the flux concentrations. The reduced internal pressure leads to lateral compression by the external gas pressure, resulting in a strong intensification of the field strength to kilogauss values. About 20 minutes after the introduction of the magnetic field, the flux concentrations have merged into three large flux sheets, which govern the surrounding flow pattern with strong downflows surrounding the flux sheets. After about 30 minutes this quasi-stationary situation has fully developed. Figure 8 shows the temporal evolution of the average field strength in magnetic flux sheet I in Fig. 7 near the visible solar surface. Two stages of field amplification can be seen: first, a rapid concentration of the field to about 1 kG by flux expulsion and radiative cooling within the first 3-4 minutes after the introduction of the magnetic field, which is followed by a second, slower

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Fig. 7. Time evolution of magnetic field (field lines are shown in black) and temperature (color coded) from a 2D simulation run of solar magneto-convection near the visible solar surface (located at z ≃ 100 km height). A homogeneous vertical field of 100 Gauss has been introduced at t = 0 after a statistically stationary convection pattern has evolved. Within a few minutes, most of the magnetic flux is transported by the converging horizontal flows to the cool downflow region (flux expulsion). Owing to the suppression of the convective energy transport, the gas in the flux concentrations cools and sinks; lateral compression by the external gas pressure then leads to a strong intensification of the field strength, which reaches kilogauss values. About 20 minutes after the introduction of the magnetic field, the flux concentrations have merged into three large flux sheets (labeled I,II,III), which start to determine the surrounding flow pattern with strong downflows surrounding the flux sheets. After about 30 minutes a quasi-stationary situation has developed. The velocity field in this state is shown in the form of velocity vectors in the last panel (from [16]). (See also color figure, Plate 11.)

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Fig. 8. Temporal evolution of the magnetic field strength for the magnetic flux sheet I in Fig. 7 at a geometrical height roughly corresponding to the visible solar surface. The initial field of 100 G is intensified to about 1 kilogauss within a few minutes by the combined action of flux expulsion by the convective flow and radiative cooling. The continuing downflow within the magnetic flux concentration leads to further evacuation and amplification of the field to values around 2 kilogauss.

increase of the field due to a persistent downflow and evacuation of the upper layers of the flux sheet. The second process leads to field strengths around 2 kG. 4.2 Formation of dynamic magnetic structure The MURAM code The study of the full dynamics of the solar magnetic structure requires realistic simulations in three dimensions. As an intermediate step before the full completion of the 3D version of the IAM-MPAe code by implementation of a non-grey radiative transfer module, we have developed the MURAM code. This code allowed us to carry out simulation runs with a full non-grey radiative transfer module in 3D. The MURAM code solves the MHD and RT equations on a three-dimensional regular Cartesian grid with constant grid spacing. The spatial derivatives are discretized with 4th-order centered differences on a 53 point stencil. Time stepping is explicit with a 4th-order RungeKutta solver. The scheme is stabilized by the application of shock-resolving diffusion and hyperdiffusivity [22], which prevent the build-up of energy at scales comparable to the size of the grid cells. These artificial diffusivities assume significant values only near discontinuities and in regions of unresolved waves while those regions which are well resolved remain largely unaffected by diffusion. For the equation of state the instantaneous ionization equilibrium for the first ionization of the 11 most abundant elements is considered.

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The relevant thermodynamic quantities are stored in tables from which the required values are interpolated during a simulation run. The bottom boundary conditions implemented in the MURAM code permit free in- and outflow of matter and maintain a constant mass in the box as well as a fixed energy flux through the system. In the present version of the code, the top of the domain is closed with stress-free boundary conditions for the horizontal velocity components; the implementation of a more realistic transmitting upper boundary is under development. The magnetic field is assumed to be vertical at the top and bottom boundaries, the footpoints of fieldlines are allowed to move freely. The horizontal directions are taken to be periodic in all variables. The code is parallelized by means of domain decomposition. The computational domain is divided into a three-dimensional array of subdomains, each of which is endowed with two layers of ghost cells at its boundaries as required by the 4th-order spatial discretization scheme. We use message passing (MPI) for parallel computers with distributed memory. The radiative transfer equation is solved for each frequency set determined by opacity binning and for each direction using the short characteristic scheme with linear or parabolic interpolation of opacity and source function as well as linear interpolation of density. In the context of the domain decomposition used for parallelization, the short characteristic scheme requires an iteration for each ray direction and each frequency set. For a given ray direction the scheme starts in each subdomain at those boundaries through which the radiation enters (the ‘upwind’ boundaries). The intensity values at these boundaries are assumed to be known. Then the traversal of the subdomain proceeds in the downwind direction, systematically moving away from the upwind boundaries, thus making sure that the upwind intensities required for the interpolation are always known. However, on those upwind boundaries of a subdomain which do not coincide with the top or bottom boundary of the computational box, the intensities are a priori unknown. Therefore, the scheme is iterated until convergence at the boundaries is obtained. After each iteration the intensities at a given upwind boundary are updated with the new values provided by the neighboring subdomain. We found that 2 to 3 iteration steps per frequency set and direction are usually sufficient, if one chooses as initial guess for the intensities on the upwind boundaries a linear extrapolation of the values of the previous two time steps. More details about and further results obtained with the MURAM code can be found in [7, 23, 24, 25]. Simulation of a solar plage region A plage region (cf. Fig. 1) is a strongly magnetized part of the solar atmosphere outside sunspots (but often in their vicinity) with a horizontally averaged field strength of about 200 G. We have carried out a simulation run in a

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computational box corresponding to a height of 1400 km and 6000× 6000 km2 width on the Sun, discretized with a spatial grid of 100 × 288 × 288 points. 600 km

τ=1

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Fig. 9. Geometrical setup of the simulation runs with the MURAM code. The vector B0 indicates the vertical homogeneous magnetic field introduced into the hydrodynamic convection at the beginning of the magnetic phase.

Similarly to the 2D runs described in the previous section, the simulations with the MURAM code were started as non-magnetic convection. After the convection had fully developed and reached a statistically stationary state (about one hour solar time after the start of the simulation), a homogeneous vertical initial magnetic field of field strength B0 was introduced. Here we show results from a run with B0 = 200 G, corresponding to a solar plage region. Within a few minutes of simulated time (approximately one turnover time of the convection) most of the magnetic flux has been transported to the downflow lanes of the convective granulation pattern and intensified to kilogauss field strength. For a snapshot taken about 2 hours solar time after the start of the magnetic phase, Fig. 10 shows the vertical magnetic field, vertical velocity, and temperature distributions on a horizontal plane corresponding roughly to the visible solar surface. In addition, the frequency-integrated intensity (brightness) is shown on the lower right panel. The magnetic map shows sheet-like magnetic structures extending along convective downflow lanes, while larger structures with diameters of up to 1000 km appear at the vertices where several downflow lanes merge. Typical field strengths in these field concentrations are between 1500 and 2000 G. The network of magnetic structures is organized on a ‘mesoscale’ which typically comprises 4–6 convective upflow regions (granules). While this magnetic pattern is rather stable (it evolves on a time scale of hours), the smallscale pattern of the field concentrations is highly time-dependent, with magnetic flux being constantly redistributed within the magnetic network. In the intensity map shown in Fig. 10, the larger flux concentrations appear dark owing to the reduced efficiency of convective energy transport. There is a considerable small-scale variation of the intensity within the pore-like flux concentrations, which is related to localized hot upflows in regions of reduced

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Fig. 10. Snapshot from the simulation of a solar plage region (B0 = 200 G) with the MURAM code. Brightness map (lower right) and horizontal cuts near the visible solar surface of vertical magnetic field component (upper left), vertical velocity component (upper right) and temperature (lower left). Light and dark shades indicate higher and lower values, respectively. The velocity plot shows convective upflows shaded in light grey separated by intergranular downflow lanes. In the magneticfield plot, the strong sheet- and pore-like magnetic field concentrations appear in white.

field strength. In the thin sheets, lateral heating effects in combination with the depression of the level of optical depth unity lead to a brightening with respect to the surrounding downflow regions (see Fig. 11). Fig. 12 shows some statistical properties of the simulation run from a series of statistically independent snapshots (i.e., with a temporal cadence exceeding the granule lifetime). A horizontal slice consisting of 8 grid layers (corresponding to a thickness of 112 km), which include the visible solar surface, served as the basis for the analysis.

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Fig. 11. Radiative properties of a sheet-like magnetic structure in a convective downflow lane. Left: vertical field strength (top) and brightness (bottom). The vertical line indicates the position of the cut shown on the right panel. Right: grey-shading of the temperature distribution in a vertical cut, together with the radiative flux vectors. There is an influx of radiation from the hot walls of the flux sheet into its partially evacuated cooler interior. The radiative heating leads to enhanced brightness (shown at the top), so that the flux sheet appears as a bright structure within the darker downflow lane.

The probability distribution function (PDF) for the magnetic field, signed with the orientation of its vertical component is shown in the upper left panel of Fig. 12. It shows a superposition of two components. Most of the volume considered is occupied by weak field, the probability density dropping off approximately exponentially with increasing field strength. The distribution reveals a pronounced local minimum at B = 0, indicating that the magnetic field, albeit being mostly weak, permeates the whole volume and field-free regions are nearly absent. Superimposed on this exponential distribution is a Gaussian “bulge” (the high field strength wing showing the characteristic parabolic shape on a logarithmic scale) with a maximum around 1500 G, which reflects the sheet- and pore-like structures in the network of concentrated magnetic field. The correlation diagram (joint PDF) of magnetic field strength and inclination angle of the field vector with respect to the horizontal plane given in the upper right panel of Fig. 12 shows that most of the strong field above the kilogauss level is vertical and upward directed (which is the orientation of the homogeneous initial field), presumably as the result of buoyancy forces acting on the partially evacuated magnetic structures. The inclination angle of weak

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Fig. 12. Statistical properties of a layer of about 110 km thickness around the visible solar surface. Upper left: probability distribution (PDF) of the field strength, signed with the vertical orientation of the field vector. Upper right: joint PDF of field strength and the inclination angle of B with respect to the horizontal, theta(B). Lower left: joint PDF of flow velocity, signed with its vertical orientation, and field strength. Lower right: joint PDF of the inclination angles of the flow, theta(v), and of the magnetic field, theta(B). The grey-scaling indicates the probability density on a logarithmic scale.

fields is much more evenly distributed. With decreasing field strength a slight preference for upward directed fields is observed. The joint PDF of the vertical magnetic field and the flow velocity signed with the vertical orientation of the flow in the lower left panel of Fig. 12 (positive velocities correspond to upflows) shows the effect of strong fields on the fluid motions: while flow velocities up to 8 km s−1 can be found in weak field regions, the amplitudes of fluid motions are reduced in magnetic structures with field strengths above 1000 G. Fluid motions are not completely suppressed, however, since the predominantly vertical fields leave vertical fluid motions largely unaffected. Downflows are preferred inside strong field features.

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The lower right panel of Fig. 12 shows the joint PDF of the inclination angles of magnetic field vector and flow vector with respect to the horizontal plane. The pronounced diagonal lobe indicates that in most of the volume considered flow field and magnetic field are more or less aligned. In addition to this component, one observes a strong correlation of (strong) vertical magnetic field with downflows. Further simulations and studies We have already carried out a number of further simulations with the MURAM code and used the results to study various physical processes on the Sun. Among these investigations are: 1. A study of the quantitative effects of the frequency-dependent radiative transport on the simulation results. It turns out that the non-grey treatment leads to significantly smaller temperature fluctuations, particularly in the upper photosphere. This leads to brightness contrasts in agreement with observational values [8]. 2. A parameter study of magneto-convection with different amounts of vertical magnetic flux in the box ranging from B0 = 10 G (‘quiet’ Sun), B0 = 50 G, B0 = 200 G (the plage region described in the foregoing section), to B0 = 800 G. As the average field strength increases, the magnetic flux concentrations become larger and the field increasingly affects and eventually dominates the convective motions (V¨ ogler et al., publication in preparation) 3. Simulations of larger magnetic structures in the solar photosphere. Such darkish ‘pores’ with diameters of a few Mm represent the transition from small, bright flux concentrations to dark sunspots. The questions investigated are: how is a pore held together, how is it affected by energy transport along the field, how does it interact with the surrounding convection? Further studies concern the atmospheric structure (temperature, pressure) within a pore and its observational signatures (Cameron et al., publication in preparation). 4. A comparison of the fractal dimension of the magnetic field pattern in the simulations with observations of the magnetic structure in the solar photosphere. There is a remarkable agreement between simulation and observation, increasing the confidence in the ‘realism’ of the simulations [26]. 5. A study of the brightness of magnetic flux concentrations when observed in spectral bands dominated by molecular lines, particularly in Fraunhofer’s ‘G band’ with many spectral lines from the CH molecule. Through the comparison of synthetic images from simulation results with actual observations, the physical mechanism leading to the brightness contrast of magnetic structures could be unveiled. The evacuated and heated magnetic flux concentrations show lower abundances of CH, leading to less

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absorption in spectral lines and thus larger brightness. These results put the indirect approach of studying of the magnetic field dynamics in the solar atmosphere by observing bright features in the G band on a firm physical basis [27]. 6. An investigation of the evolution of an initial field with mixed polarity, i.e., half of the initial magnetic field pointing upward and the other half downward. This permits the temporal decay of unsigned magnetic flux by reconnection of field lines and cancellation of opposite polarity flux. By running simulations with various horizontal distributions of the initially vertical field (half-by-half, two-by-two,...) and determining the decay rate, it is possible to determine values of the ‘turbulent’ magnetic diffusivity, which is a very important quantity for studies of the long-term evolution of the magnetic flux at the solar surface. Furthermore, the results of such simulation runs are compared with observations of the magnetic field dynamics in regions of mixed polarity on the Sun (V¨ ogler et al., publication in preparation). 7. The various tools developed for spectroscopic and polarimetric analysis to compare the simulation results with actual solar measurements have been applied to various magneto-convection simulations. The analysis of these results and the comparison with observations reveals rich diagnostic information in good agreement with measurements, which is used to identify the physical processes behind the observational phenomena (Shelyag et al., publication in preparation).

5 Summary & Outlook Our work on the ANumE project(s) has proven to be very fruitful. We have developed and tested methods to incorporate radiative transfer into MHD simulations and to take account of the frequency dependence of radiation in a stellar atmosphere. State-of-the-art codes have been developed and successfully applied to simulate magneto-convection in the solar atmosphere. With the MURAM code we have a fully working 3D code for realistic simulations, which has already produced a wealth of useful results. This code is our present ‘workhorse’ and several ongoing projects exploit its rich possibilities and analyze data produced by this code, often in direct comparison with observational results. Further development of the MURAM code will include introducing a transmitting boundary condition at the top and using a non-uniform grid in the vertical direction with a grid spacing proportional to the average pressure. This will allow us to extend the computational box in the vertical direction. Work on the 3D version of the IAM-MPAe code still continues. Comparison runs with the MURAM code will allow us to optimize the grid refinement strategy and will also provide a mutual test of both codes. The IAM-MPAe code will then be used to attack the challenging physical questions of the

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higher solar atmosphere, in particular the chromosphere, which is dominated by interacting MHD shock waves. The IAM-MPAe code with its Riemann solvers and adaptive mesh refinement is perfectly suited for such kind of environment. Acknowledgement. We would like to thank Andreas Dedner, Dietmar Kr¨ oner, Christian Rohde, and Matthias Wesenberg, our partners from the Institute of Applied Mathematics (IAM), University of Freiburg, for a fruitful cooperation during the ANumE programme. We are grateful to Fausto Cattaneo, Thierry Emonet, and Timur Linde from the University of Chicago for their contributions to the development of the MURAM code. We thank the coordinator of ANumE, Prof. G. Warnecke and his staff for much support and for the management of the programme. Last, but not least, we thank the Deutsche Forschungsgemeinschaft (DFG) for generous support of our project Schu 500/7 and all the DFG staff involved with ANumE and with our project in particular for their efficient, friendly, and non-bureaucratic dealing with our sometimes non-standard requests.

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Riemann Problem for the Euler Equation with Non-Convex Equation of State including Phase Transitions Wolfgang Dahmen, Siegfried M¨ uller, and Alex