Complex Effects in Large Eddy Simulations (Lecture Notes in Computational Science and Engineering)

Lecture Notes in Computational Science and Engineering Editors Timothy J. Barth Michael Griebel David E. Keyes Risto M. ...

Author: Stavros C. Kassinos | Carlos A. Langer | Gianluca Iaccarino | Parviz Moin (Editors)

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Lecture Notes in Computational Science and Engineering Editors Timothy J. Barth Michael Griebel David E. Keyes Risto M. Nieminen Dirk Roose Tamar Schlick

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Stavros C. Kassinos Carlos A. Langer Gianluca Iaccarino Parviz Moin (Eds.)

Complex Effects in Large Eddy Simulations With 233 Figures, 51 Colour Plates and 17 Tables

ABC

Editors Stavros C. Kassinos Carlos A. Langer

Gianluca Iaccarino Parviz Moin

Department of Mechanical and Manufacturing Engineering University of Cyprus Kallipoleos Street 75 1678 Nicosia, Cyprus E-mail: [email protected] [email protected]

Department of Mechanical Engineering Stanford University Escondido Mall 488 94305-3035 Stanford, USA E-mail: [email protected] [email protected]

Library of Congress Control Number: 2006933936 Mathematics Subject Classification (2000): 76F65, 80A32, 76F55, 65C20, 76F50, 76M28, 65M15, 65M50 ISBN-10 3-540-34233-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-34233-5 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 springer.com c Springer-Verlag Berlin Heidelberg 2007 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 authors and techbooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

SPIN: 11754695

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This LNCSE volume marks the launching of UCY-CompSci, an initiative in the Computational Sciences supported by a Marie Curie Transfer of Knowledge Fellowship of the European Community’s Sixth Framework Programme, under contract number MTKD-CT-2004-014199.

Preface

This volume contains a collection of expert views on the state of the art in Large Eddy Simulation (LES) and its application to complex ﬂows. Much of the material in this volume was inspired by contributions that were originally presented at the symposium on Complex Eﬀects in Large Eddy Simulation held in Lemesos (Limassol), Cyprus, between September 21st and 24th, 2005. The symposium was organized by the University of Cyprus together with the Center for Turbulence Research at Stanford University and NASA Ames Research Center. Many of the problems that must be tackled in order to advance technology and science increasingly require synergetic approaches across disciplines. Computational Science refers to interdisciplinary research aiming at the solution of complex scientiﬁc and engineering problems under the unifying theme of computation. The explosive growth of computer power over the last few decades, and the advancement of computational methods, have enabled the application of computational approaches to an ever-increasing set of problems. One of the most challenging problems to treat computationally in the discipline of Computational Fluid Dynamics is that of turbulent ﬂuid ﬂow. Turbulent ﬂow contains eddies, representing seemingly chaotic zig-zagging or swirling motion, that extend over many orders of magnitude in size. One can attempt to simulate turbulent ﬂow by faithfully representing motion at all scales, but then even with the most powerful supercomputers available today, our simulations would be limited to low speeds and geometries that are far too simple for engineering application. One of the most accurate and elegant alternatives, while striving to keep a reasonable solution cost, is LES. Large Eddy Simulation has its origins in a simulation approach ﬁrst used for weather prediction. In many engineering applications, as in Meteorology, the large-scale turbulent motions are of primary interest, so in LES they are simulated in their entirety. Smaller-scale eddies are not of direct interest, and are thus not simulated directly, but since they do aﬀect the large-scale turbulence, a model has to account for their presence. The ﬁeld of Large Eddy Simulations is now reaching a level of maturity that brings this approach to

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the mainstream of engineering computations while it opens new opportunities and brings new challenges for further progress. The symposium that inspired this volume was held in Cyprus to mark the launching of a new initiative in the Computational Sciences at the Engineering School of the University of Cyprus. This initiative became possible through a generous grant from the European Community under a Marie Curie Transfer of Knowledge fellowship of the Sixth Framework Program (contract number MTKD-CT-2004-014199). The organization of the Symposium was made possible by contributions from several organizations. We are glad to acknowledge our gratitude to the following institutions: the University of Cyprus, the Center for Turbulence Research, and the Cyprus Energy Regulatory Authority. In addition, the organization of the event would not have been possible without the help of Thanasis Vazouras, Elena Takoushi, Filippos Filippou and Ria Demosthenous who made an incredible job behind the scenes ensuring an eﬃcient yet warm atmosphere during the symposium. The symposium brought together experts from diﬀerent countries to discuss the state-of-the-art and the emerging approaches in treating complex eﬀects in LES and the role of LES in the context of multi scale modeling and simulation. The workshop provided an opportunity for open discussion on current issues in LES, strengthened existing collaborations and developed new contacts between participants that we believe will help in advancing the state of the art in LES. With this volume we share with the community developments inspired by the symposium.

Nicosia, May 2006

Stavros C. Kassinos Carlos A. Langer Gianluca Iaccarino Parviz Moin

Contents

Complex Eﬀects in Large Eddy Simulations P. Moin, G. Iaccarino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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On the Relation between Subgrid-Scale Modeling and Numerical Discretization in Large-Eddy Simulation N. A. Adams, S. Hickel, T. Kempe, J. A. Domaradzki . . . . . . . . . . . . . . . . 15 Space-Time Error Representation and Estimation in Navier-Stokes Calculations T. J. Barth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Multiresolution Particle Methods M. Bergdorf, P. Koumoutsakos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 LES Computation of Lagrangian Statistics in Homogeneous Stationary Turbulence; Application of Universalities under Scaling Symmetry at Sub-Grid Scales M. Gorokhovski, A. Chtab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Anisotropic Subgrid-Scale Modelling: Comparison of LES with High Resolution DNS and Statistical Theory for Rapidly Rotating Turbulence L. Shao, F. S. Godeferd, C. Cambon, Z. S. Zhang, G. Z. Cui, C. X. Xu

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On the Investigation of a Dynamic Nonlinear Subgrid-Scale Model I. Wendling, M. Oberlack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Three Problems in the Large–Eddy Simulation of Complex Turbulent Flows K. Mahesh, Y. Hou, P. Babu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

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Filtering the Wall as a Solution to the Wall-Modeling Problem R. D. Moser, A. Das, A. Bhattacharya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 A Near-Wall Eddy-Viscosity Formulation for LES G. Kalitzin, J. A. Templeton, G. Medic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Investigation of Multiscale Subgrid Models for LES of Instabilities and Turbulence in Wake Vortex Systems R. Cocle, L. Dufresne, G. Winckelmans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Numerical Determination of the Scaling Exponent of the Modeled Subgrid Stresses for Eddy Viscosity Models M. Klein, M. Freitag, J. Janicka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 A Posteriori Study on Modelling and Numerical Error in LES Applying the Smagorinsky Model T. Brandt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Passive Scalar and Dissipation Simulations with the Linear Eddy Model C. Papadopoulos, K. Sardi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Lattice-Boltzmann LES of Vortex Shedding in the Wake of a Square Cylinder P. Mart´ınez-Lera, S. Izquierdo, N. Fueyo . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 LES on Cartesian Grids with Anisotropic Reﬁnement G. Iaccarino, F. Ham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Towards Time-Stable and Accurate LES on Unstructured Grids F. Ham, K. Mattsson, G. Iaccarino, P. Moin . . . . . . . . . . . . . . . . . . . . . . . . 235 A Low-Numerical Dissipation, Patch-Based AdaptiveMesh-Reﬁnement Method for Large-Eddy Simulation of Compressible Flows C. Pantano, R. Deiterding, D. J. Hill, D. I. Pullin . . . . . . . . . . . . . . . . . . . 251 Large-Eddy Simulation of Richtmyer-Meshkov Instability D. J. Hill, C. Pantano, D. I. Pullin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 LES of Variable Density Bifurcating Jets A. Tyliszczak, A. Boguslawski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Large-Eddy Simulation of a Turbulent Flow around a Multi-Perforated Plate S. Mendez, F. Nicoud, T. Poinsot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

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Simulation of Separation from Curved Surfaces with Combined LES and RANS Schemes F. Tessicini, N. Li, M. A. Leschziner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Highly Parallel Large Eddy Simulations of Multiburner Conﬁgurations in Industrial Gas Turbines G. Staﬀelbach, L. Y. M. Gicquel, T. Poinsot . . . . . . . . . . . . . . . . . . . . . . . . 325 Response of a Swirled Non-Premixed Burner to Fuel Flow Rate Modulation A. X. Sengissen, T. J. Poinsot, J. F. Van Kampen, J. B. W. Kok . . . . . 337 Investigation of Subgrid Scale Wrinkling Models and Their Impact on the Artiﬁcially Thickened Flame Model in Large Eddy Simulations T. Broeckhoven, M. Freitag, C. Lacor, A. Sadiki, J. Janicka . . . . . . . . . . . 353 Analysis of Premixed Turbulent Spherical Flame Kernels R. J. M. Bastiaans, J. A. M. de Swart, J. A. van Oijen, L. P. H. de Goey 371 Large Eddy Simulation of a Turbulent Ethylene/Air Diﬀusion Flame D. Cecere, G. Gaudiuso, A. D’Anna, R. Verzicco . . . . . . . . . . . . . . . . . . . . 385 Energy Fluxes and Shell-to-Shell Transfers in MHD Turbulence D. Carati, O. Debliquy, B. Knaepen, B. Teaca, M. Verma . . . . . . . . . . . . . 401 Color Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Complex Eﬀects in Large Eddy Simulations Parviz Moin and Gianluca Iaccarino Center for Turbulence Research Stanford University Stanford, CA, 94305 - USA [email protected] Summary. The Large Eddy Simulation technique is enjoying widespread success in the engineering analysis as a result of the recent advances in computer performance. Initially limited to the simulation of turbulent ﬂows in simple geometries, current LES tools are being applied to multidisciplinary problems involving a variety of physical processes. Several examples of recent advances in LES methodology and complex multi-physics applications are presented.

1 Introduction Computer power has increased by two orders of magnitude over the past ten years. This has led to increased applications of the large eddy simulation (LES) technique to multi-physics simulations of realistic engineering ﬂows. Complex eﬀects in LES include consideration of complex geometry, multiple ﬂow phases, chemical reactions, compressibility eﬀects and shock waves, aeroacoustics, aero-optics, non-Newtonian ﬂuids and transitional ﬂows. There have been several signiﬁcant advances in the elements of LES for complex ﬂows. The most signiﬁcant enabling technology has been in the development of robust, non-dissipative numerical methods for unstructured meshes [1,2]. A similar advance has been made for compressible ﬂows, where it has been shown that a primary cause of numerical instabilities is the violation of the Second Law of Thermodynamics due to discretization of the non-linear terms in the governing equations [3]. LES of high Reynolds number attached boundary layers requires lower ﬁdelity modeling of the region near the wall due to high computational cost if LES were to extend to the wall. Wall modeling in LES dates back nearly forty years to the work of Deardorﬀ, and has always been considered a pacing item in application of LES to aeronautical ﬂows. However, the existing wall models are not satisfactory. Cabot and Moin [4] have shown that proper wall modeling should include the eﬀects of numerical truncation and subgrid scale modeling errors. Optimal control theory was recently formulated for the development

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of numerical wall models that account for the above errors [5]. The early results show a remarkable degree of robustness to mesh resolution, subgrid scale models and Reynolds number. LES of most complex engineering ﬂows requires treatment of multiphase ﬂows. For example, liquid fuel jets in the combustion chamber of gas-turbine engines undergo atomization into droplets, evaporation, collision/coalescence of droplets, and turbulent mixing of fuel and oxidizer giving rise to spray ﬂames. Multi-phase ﬂows pose additional challenges for load balancing on parallel computer platforms. Recently, considerable progress has been made in computation of multiphase ﬂows using LES [6]. However, the development of subgrid scale models for primary atomization remains an important pacing item in such simulations [7]. Practical simulation of complex engineering systems may require more than one computer code and diﬀerent modules representing the multi-physics nature of the problem. In a recent simulation of a gas-turbine engine at Stanford, an unstructured mesh LES code was used for the combustor and a structured multi-block compressible RANS code was used for the computation of the turbomachinery components (fan, compressor and turbine). Within the LES code a separate two-phase ﬂow module was used for Lagrangian tracking of fuel droplets. Proper integration of multiple codes and modules, which involves rigorous mathematical treatment of the interfaces is an important area for research [8,9]. In future computations of complex ﬂows, particular attention has to be paid to computational engineering, and the development of numerical technology and programming environment for integration. In the same way that integration technology is critical for simulation of complex systems, rigorous methods are required for validation and veriﬁcation of the outputs of the resulting complex codes. Validation of numerical results by comparison to experiments has always been an integral part of the development of LES. With the development of large complex codes containing sub-ﬁlter models representing complex eﬀects, which are not derived from ﬁrst principles, new emphasis is being placed on the development of methods for error estimation and uncertainty quantiﬁcation of numerical results. The objective is to have uncertainty bars on numerical output that reﬂect uncertainties in computational parameters such as turbulence models, boundary conditions, geometry deﬁnition and numerical errors [10]. In this paper we will expand on the topics outlined in this introduction and present supporting results, mostly obtained from the CTR programs in these areas. In section 6 of the manuscript we will highlight some results from two novel applications of LES in aero-optics and shape optimization for noise mitigation. LES is particularly suitable for these applications as the proper representation of the unsteady dynamics of turbulence is necessary.

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2 Numerical Methods Computational ﬂuid dynamics for highly complex conﬁgurations is carried out using unstructured meshes. It has been shown that numerical dissipation can have a detrimental eﬀect on the resolution of turbulence structures [11]. Therefore, the development of non-dissipative numerical methods for LES on hybrid unstructured grids by Mahesh et al. [1] is considered an enabling technology for LES in complex ﬂows. Their approach was based on control volume collocated discretization and fractional step method for the incompressible Navier-Stokes equations. The algorithm was designed to enforce the kinetic energy conservation principle, by ensuring that the convective and pressure terms have no spurious contributions to the volume integral of the kinetic energy. Ham and Iaccarino [2] improved on this algorithm by analyzing the treatment of the pressure terms. Their modiﬁcations were shown to substantially improve the accuracy and conservation properties of the algorithm in the presence of mesh skewness, which is generally unavoidable in grid generation for complex domains. The superior performance of the modiﬁed algorithm was demonstrated in simulation of Taylor vortices on Cartesian and highly skewed meshes (Figure 1). The Taylor vortices shown are the exact solutions of the Navier-Stokes equations; they are stationary in space and simply decay. The modiﬁed algorithm seems to preserve this property on skewed meshes in contrast to the original algorithm which distorts the structure of the vortices. For shock-free compressible turbulence simulations, Honein and Moin [3] developed a numerical formulation for the mitigation of non-linear instabilities. Numerical stability was enhanced through satisfaction of global conservation properties stemming from the Second Law of Thermodynamics and the entropy equation. The robustness of the method was demonstrated by performing unresolved numerical simulations and LES of compressible isotropic turbulence at very high Reynolds numbers. Figure 2 shows the evolution of turbulent kinetic energy at several Reynolds numbers obtained from 323 calculations without subgrid scale models. The stability of the scheme is clearly demonstrated. They also tested several other schemes [12], and demonstrated that numerical instability originates as soon as the Second Law is violated (Figure 3).

3 Wall Modeling Cabot and Moin’s discovery [4] that successful wall modeling must take into account the eﬀects of numerical discretization and subgrid scale modeling, has led to the use of optimal control theory (as opposed to phenomenological modeling) for the modeling of the near wall region in turbulent boundary layers. Templeton et al. [13] have developed a wall model based on optimal control theory. Reynolds averaged Navier-Stokes equations coupled to both the LES and control, is used near the wall to provide a target mean velocity proﬁle.

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Fig. 1. Computational grid and contours of u-velocity from the inviscid Taylor problem. Reconstruction of Mahesh et al. [1] (center) – Ham and Iaccarino’s reconstruction [2] (right). Results have been copied in periodic directions for clarity. (See Plate 1 on page 413)

Fig. 2. Turbulent kinetic energy in compressible isotropic turbulence at various Reynolds numbers.

Fig. 3. Time evolution of ρs − ρ0 s0 in under-resolved compressible isotropic turbulence using the high order ﬁnite diﬀerence scheme of Nagarajan et al. [15]. Numerical instability occurs when entropy decreases.

Complex Eﬀects in Large Eddy Simulations

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The control minimizes a cost function consisting of the diﬀerence between the LES mean velocity and the target velocity. The cost function is minimized by modifying the wall stresses used as boundary conditions by the LES. This work generalized the previous work of Nicoud et al. [14] by not requiring a priori target proﬁles, thus making wall modeling truly predictive. The wall model was tested successfully in turbulent channel ﬂow at high Reynolds numbers with coarse grids (Figure 4). The control based method shows a remarkable insensitivity to grid resolution, numerical errors and subgrid scale model used in the outer ﬂow LES. The method should be tested in more complex ﬂows where conventional wall models have not performed satisfactorily.

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Fig. 4. Left: Mean velocity proﬁles computed on a 32 × 33 × 32 grid using LES with : u+ = 2.41 log(y + ) + 5.2, : Reτ = 20000, control-based wall model. : Reτ = 4000, : Reτ = 640. Right: Mean velocity proﬁles for Reτ = 4000 : u+ = 2.41 log(y + ) + 5.2, obtained using LES with control-based wall model. : 64 × 65 × 64 cells, : 32 × 33 × 32 cells.

4 Two-Phase Flows Complex engineering ﬂows and natural phenomena involve multiphase ﬂows. Capturing the topologically complex interface between the phases is critical to the accuracy of the computations. Geometrical parameters of the interface such as radius of curvature and the surface area are directly related to important physical properties such as surface tension and heat release from pre-mixed ﬂames. The level set method is an attractive Eulerian approach for tracking interfaces as it allows straightforward computation of the geometrical properties of the interfaces, and it handles topology changes of the interfaces automatically. The main drawback of the method is that it does not preserve the volumes of the ﬂuids on either side of the interface. Herrmann [7] has

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introduced the Reﬁned Level Set Grid method (RLSG) to address the volume errors and establish the grid convergence of the interface geometry. In RLSG one solves the level set equations on a ﬁner Cartesian mesh overlaid on the main computational mesh and highly reﬁned in the neighborhood of the interface (Figure 5). The primary break-up of liquid sheets into drops is dependent on the interface topology and hydrodynamic instabilities. The problem has signiﬁcant practical consequences in many applications, and in particular in combustors where the primary atomization of liquid fuel jets into drops and their subsequent evolution is critical to the prediction of the combustion phenomenon. Herrmann [7] has also proposed an interface for the RLSG method to a Lagrangian drop model, which provides a framework for the simulation of the breakup process of liquid jets and sheets. Figure 6 shows the results from his calculation of a spherical liquid drop subjected to a periodic deformation ﬁeld. Iso-surfaces of volume fraction and the generated Lagrangian drops are shown.

Fig. 5. Reﬁned Level Set Grid approach.

Fig. 6. Deformation of a spherical interface in a strain ﬁeld and formation of drops.

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Fig. 7. Load balancing for multiphase ﬂow calculations of a jet engine fuel injector. Sprays are highly localized in the vicinity of the injector. A gas-phase cell-based partition leads to high particle counts on only few processors (left); a dual partition constraint provides nearly optimal load balancing for both the gas and liquid-phase (right). Top ﬁgures show the grid partition boundaries, whereas the cell and particle counts per processor are reported in the histograms. (See Plate 2 on page 413)

Once the liquid drops are introduced in the computational domain, they are tracked using a Lagrangian scheme. Parallel computer implementation of dual Lagrangian/Eulerian simulations requires additional consideration of load balancing among the processors. A domain partition scheme that is based on the computational eﬀort required on both gas and particle phases was implemented by Ham et al. [16]. Single phase and dual based domain decomposition and the corresponding processor loads are shown in Fig. 7.

5 Integration Flexible computational infrastructures that consist of several independent but integrated codes provide a natural framework for developing large scale multidisciplinary simulation capabilities. Each component application performs a speciﬁc task and addresses a speciﬁc physical aspect of the problem; the integration framework allows for a timely and accurate exchange of information and ensures that new features or updated models can be included without disrupting the entire infrastructure.

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Within the ASC program, Stanford developed CHIMPS (Coupler for Highperformance Integrated Multi-Physics Simulations). It consists of a scripting interface written in Python and a series of modules that can perform a wide range of functions, from ﬂow simulations using RANS or LES, to aero-structural analysis, to multiphase interface tracking. In Fig. 8 a typical multi-code integrated simulation of the ﬂow in a jet engine is illustrated. In this case a RANS method (for the turbomachinery components) is coupled with an LES solver, used to simulate the reactive, multiphase ﬂow within the combustor.

Fig. 8. Schematic representation of the multi-code coupling for the simulation of the ﬂow in a jet engine (left). Flow simulation of the high-spool of a Pratt & Whitney engine. Contours of entropy are reported in the compressor and turbine, isosurfaces of temperature in the combustor (right). (See Plate 3 on page 414)

The multicode simulations illustrated in Fig. 8 involve transitions between structured grids and unstructured meshes, compressible and incompressible ﬂow assumptions and RANS vs. LES modeling. The problem of interfacing a compressible and incompressible ﬂow solver has been addressed in [8] through the use of a penalty based interface formulation. The results are promising and demonstrate that large scale ﬂow structures can be advected through an interface with minimal distortion. Cross-sectional views of the Pratt & Whitney combustor simulations are shown in Fig. 9. This is a snap-shot of a multiphase chemically reacting ﬂow in a real combustor. The physical richness of the ﬂow is striking: observe the swirling liquid jet emanating from the injector, the Kelvin-Helmholtz instability in the jets from the dilution holes and the hot temperature streaks at the exit of the combustor. Comparison of the mean temperature proﬁle at the exit of the combustor with the experimental data is sown in Fig. 10.

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Fig. 9. Temperature distribution in cross-sections of a Pratt & Whitney combustion chamber. (See Plate 4 on page 414)

Fig. 10. Temperature proﬁle at the combustor exit.

6 Novel Applications of LES In this section we will present two recent applications of LES which are beyond the capability of traditional RANS-based CFD methods. 6.1 Aero-Optics The distortion of an optical wavefront by a turbulent, variable index-ofrefraction ﬁeld poses a serious problem for many optical systems [17]. To

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Fig. 11. Schematic of optical propagation through a turbulent wake behind a circular cylinder. (See Plate 5 on page 415)

accurately predict the aero-optical phenomenon, a wide range of spatial and temporal ﬂow scales needs to be captured. A new application of LES at CTR involves the prediction and analysis of aero-optical distortions by a turbulent wake. Since the ﬂuctuating index-of-refraction ﬁeld is linearly related to the density of the ﬂuid, LES can be used to obtain the optically-relevant spatial scales of density ﬂuctuations. The optical computation is then performed using ray tracing through the near-aperture turbulence ﬁeld and Fourier optics for far-ﬁeld propagation. The compressible ﬂow over a circular cylinder at Re = 3900 and M a = 0.4 is computed by LES using a 6th order, energy conservative, compact ﬁnite difference scheme [15] with the dynamic SGS model. An instantaneous vorticity snap shot is shown in Fig. 11 along with a schematic of the optical beam. As an example of the induced optical aberrations, Fig. 12 shows the spatial evolution of the far-ﬁeld intensity of a distorted beam and contrasts it with that of an undistorted one, for an optical wavelength of 2.5 × 10−6 D where D is cylinder diameter. The distorted beam shows a highly irregular intensity with interference patterns. After a suﬃcient distance of transmission, the distortion leads to a signiﬁcant loss of intensity. The beam starts to diverge at 4000D. In contrast, the undistorted beam remains focused up to a distance of about 32000D. In other words, the eﬀective range of the optical beam is reduced by an order of magnitude due to turbulence-induced distortions. More details of this study can be found in Mani et al. [18]. 6.2 Aero-Acoustics The use of LES for aero-acoustic predictions has grown dramatically in recent years. Compared to alternative approaches, LES oﬀers the best compromise between accuracy (level of acoustic source details) and computational expense. Recent work at CTR in this area included jet noise and trailing-edge noise, among others.

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Fig. 12. Instantaneous far-ﬁeld intensity patterns for an aberrated beam (top) and a non-aberrated beam (bottom) at diﬀerent distances of propagation. The optical wavelength is 2.5 × 10−6 D. The intensity levels are normalized by the peak intensity at the aperture where a Gaussian proﬁle is assumed. (See Plate 6 on page 415)

An interesting and useful extension of LES-based noise prediction is the passive noise control using shape optimization. This approach was explored recently [19] using a combination of LES with a derivative-free optimization technique known as the surrogate management framework [20]. The cost function, targeted for reduction by the control, represented the radiated acoustic power based on Lighthill’s theory and was evaluated from LES data. To reduce the computational expense, a method was developed that used RANS calculations for constraint evaluation and LES for cost function evaluation. Starting from the model airfoil used in the Notre Dame experiment [21] at a chord Reynolds number of 1.9 × 106 , the optimal solution produced a shape which reduced the trailing-edge noise power by 89% while maintaining lift and drag at desirable levels. As shown in Fig. 13, the acoustic spectrum showed a large decrease in low frequency noise because large-scale vortex shedding was suppressed. In addition, the higher frequency noise was reduced as well.

7 Conclusions Numerical simulation of turbulence using LES has reached new heights. Signiﬁcant progress has been made in multi-physics simulations of complex engineering ﬂows. Such complex computations require integration of multiple

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-7

10-8 -9

Φpa

10 10

-10

10-11 10

-12

10-13 0 10

10

1

10

2

ω Original shape

Optimal shape

Fig. 13. Dimensionless sound pressure spectra as a function of frequency before (solid line) and after (dashed line) trailing-edge shape optimization. Dependence on observer location has been absorbed into the spectra based on the far-ﬁeld directivity of a half-plane. Insert: Snapshots of streamwise velocity contours in a spanwise plane for the original and optimized trailing edges.

computer codes and modules. Development of programming environments for integration and the associated scientiﬁc issues such as prescription of boundary conditions are active areas of research. There is also an important need for the development of methods for uncertainty quantiﬁcation and error estimation in the numerical results. This is a pacing item for LES in complex engineering systems and therefore is an active and growing area for research.

Complex Eﬀects in Large Eddy Simulations

13

Acknowledgments This work was supported by the U.S Department of Energy ASC Program, AFOSR, NASA and ONR. The authors wish to thank Prof. M. Wang for his help in the preparation of this manuscript.

References [1] K. Mahesh, G. Constantinescu, and P. Moin. A numerical method for large-eddy simulation in complex geometries. J. Comput. Phys. 197:215– 240, 2004. [2] F. Ham and G. Iaccarino. Energy conservation in collocated discretization schemes on unstructured meshes. Annual Research Briefs, Center for Turbulence Research, Stanford Univ./NASA Ames, Stanford, California, pp. 3–14, 2004. [3] A.E. Honein and P. Moin. Higher entropy conservation and numerical stability of compressible turbulence simulations. J. Comput. Phys. 201:53– 545, 2004. [4] W. Cabot and P. Moin. Approximate wall boundary conditions in the large-eddy simulation of high Reynolds number ﬂow. Flow Turb. Combust. 63:26, 2000. [5] J. Templeton, M. Wang, and P. Moin. An eﬃcient wall model for large-eddy simulation based on optimal control theory. Phys. Fluids 18(2):025101, 2006. [6] P. Moin and S. Apte. Large eddy simulation of realistic turbine combustors. AIAA J. 44(4):698–708, 2006. [7] M. Herrmann. Reﬁned level set grid method for tracking interfaces. Annual Research Briefs, Center for Turbulence Research, Stanford Univ./NASA Ames, Stanford, California, pp. 3–14, 2005. [8] K. Mattsson, I. Iourokina, and F. Ham. Towards a stable and accurate coupling of compressible and incompressible ﬂow solvers. Annual Research Briefs, Center for Turbulence Research, Stanford Univ./NASA Ames, Stanford, California, pp. 31–41, 2005. [9] J.U. Schluter, X. Wu, E. Weide, S. Hahn, J.J. Alonso, and H. Pitsch. Multi-code simulations: A generalized coupling approach. AIAA paper 2005-4997, 2005. [10] S.F. Wojtkiewicz, M.S. Eldred, R.V. Field, A. Urbina and J.R. RedHorse. Uncertainty quantiﬁcation in large computational engineering models. AIAA Paper 2001-1455, 2001. [11] R. Mittal and P. Moin. Suitability of upwind biased schemes for largeeddy simulation of turbulent ﬂows. AIAA J. 35(8):1415–1417, 1997. [12] A.E. Honein and P. Moin. Numerical aspects of compressible turbulence simulations. Report TF92, Dept. of Mech. Eng., Stanford, CA, 2005.

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[13] J. Templeton, M. Wang, and P. Moin. A predictive wall model for largeeddy simulation based on optimal control techniques. Report TF98, Dept. of Mech. Eng., Stanford, CA, 2006. [14] F. Nicoud, J.S. Baggett, P. Moin, and W. Cabot. Large eddy simulation wall modeling based on suboptimal control theory and linear stochastic estimation. Phys. Fluids 13:2968, 2001. [15] S. Nagarajan, S.K. Lele and J.H. Ferziger. A robust high-order compact method for large eddy simulation. J. Comput. Phys. 191:392, 2003. [16] F. Ham, S. Apte, G. Iaccarino, X. Wu, M. Herrmann, G. Constantinescu, K. Mahesh, and P. Moin. Unstructured LES of reacting multiphase ﬂows in realistic gas turbine combustors. Annual Research Briefs, Center for Turbulence Research, Stanford Univ./NASA Ames, Stanford, California, pp. 139–160, 2003. [17] E.J. Jumper and E.J. Fitzgerald. Recent advances in aero-optics. Prog. Aerospace Sci. 37:299, 2001. [18] A. Mani, M. Wang, and P. Moin. Computational study of aero-optical distortion by turbulent wake. AIAA Paper 2005-4655, 2005. [19] A.L. Marsden, M. Wang, J.E. Dennis, and P. Moin. Trailing-edge noise reduction using large-eddy simulation and derivative-free optimization. J. Fluid Mech., 2006, in press. [20] A.J. Booker, J.E. Dennis, P.D. Frank, D.B. Seraﬁni, V. Torczon, and M.W. Trosset. A rigorous framework for optimization of expensive functions by surrogates. Structural Optimization 17(1):1–13, 1999. [21] S. Olson and T.J. Mueller. An Experimental Study of Trailing Edge Noise. Tech. Report UNDAS-IR-0105. Dept. of Aerospace and Mechanical Engineering, Univ. of Notre Dame, Notre Dame, Indiana, 2004.

On the Relation between Subgrid-Scale Modeling and Numerical Discretization in Large-Eddy Simulation N. A. Adams1 , S. Hickel1 , T. Kempe2 , and J. A. Domaradzki3 1

2

3

Technische Universit¨ at M¨ unchen Institute of Aerodynamics D-85747 Garching, Germany [email protected], [email protected] Technische Universit¨ at Dresden Institute of Fluid Mechanics D-01062 Dresden, Germany [email protected] University of Southern California Department of Aerospace and Mechanical Engineering Los Angeles, CA 90089-1191, U.S.A. [email protected]

Summary. Subgrid-scale models in LES operate on a range of scales which is marginally resolved by the discrete approximation. Accordingly, the discrete approximation method and the subgrid-scale model are linked. One can exploit this link by developing discretization methods from subgrid-scale models, or vice versa. Approaches where SGS models and numerical discretizations are fully linked are called implicit SGS models. Diﬀerent approaches to SGS modeling can be taken. Mostly, given nonlinearly stable discretizations schemes for the convective ﬂuxes are used as main element of implicit SGS models. Recently we have proposed to design nonlinear discretization schemes in such a way that their truncation error functions as SGS model in regions where the ﬂow is turbulent and as a second-order accurate discretization in regions where the ﬂow is laminar. In this paper we review the current status on this so-called adaptive local deconvolution method (ALDM) and provide some application results.

1 Introduction The original intention of subgrid-scale modeling was to stabilize underresolved ﬂow simulations while preserving reasonable accuracy on the resolved scales. Theoretical SGS-model development is mainly based on the ﬁltering approach [1] to Large-Eddy Simulation (LES) where ﬁltering of the underlying conservation law and the subsequent discretization of the ﬁltered con-

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N. A. Adams, S. Hickel, T. Kempe, and J. A. Domaradzki

servation law are separated. We call explicit SGS models such models which provide explicit approximations or estimations of the unclosed SGS terms obtained after ﬁltering. Explicit SGS models require the explicit computation of SGS-stress approximations during time advancement. In particular for compressible ﬂows the computational overhead can be signiﬁcant. Recent developments of explicit SGS models include attempts to reconstruct directly a part of the unﬁltered ﬁeld such as the estimation model [2] and the approximate deconvolution model [3]. A review of these approaches is given in [4]. First theoretical analyses of the mutual interference of SGS model and truncation error of the numerical discretization led to the conclusion that for SGS-model terms to dominate over the truncation error for standard ﬁnitediﬀerence schemes at least 4th-order accuracy is required [5]. These theoretical results were corroborated by numerical simulations [6]. For LES of ﬂows in complex geometries the use of higher-order schemes leads to implementational complications and computational overhead which one tries to avoid whenever possible. The interference of SGS model and truncation error can also be beneﬁcial, however. First indications that the truncation error of a linear upwind scheme in some cases may function as implicit SGS model, i.e. a SGS model whose terms are not explicitly modeled or computed, were reported by Kawamura and Kuwahara [7]. More generally, the use of nonlinearly stable schemes for implicit LES, i.e. LES with implicit SGS model, was proposed by Boris et al. [8]. Originating from the use of monotone schemes this approach has been dubbed MILES for Monotonically Integrated LES, although in practice schemes satisfying less restrictive stability constraints are used. For the latter reason the term implicit LES (ILES) appears to be more appropriate. Noteworthy are in particular the application of the piecewise parabolic method to turbulence by Porter et al. [9] and the so-called MPDATA (Multidimensional Positive Deﬁnite Advection Transport Algorithm) method of Smolarkiewicz and Margolin [10], see also [11]. Most intensely the ﬂux-corrected transport (FCT) method was used in the recent past for which considerable success in predicting wall-bounded turbulence was reported in [12] (see also references therein). On the other hand, the application of oﬀ-the-shelf non-oscillatory schemes to isotropic turbulence is less than straight-forward, as reported by Garnier et al. [13]. This uncertainty has stimulated a deeper analysis of nonoscillatory discretizations. The most suitable vehicle for analyzing a numerical scheme with respect to its implicit SGS modeling capabilities appears to be the modiﬁed-diﬀerential equation analysis [14, e.g.]. The FCT scheme was analyzed with this method by Fureby et al. [15], the MPDATA method by Margolin and Rider [16]. Following an earlier suggestion [17] we have introduced an approach to design a nonlinear discretization based on standard approaches in such a way that the truncation error provides a suitable implicit SGS model. Computations with the constructed implicit SGS model should produce results at

On the Relation between SGS Modeling and Numerical Discretization

17

least as good as common explicit models [18, 19]. The beneﬁt lies mainly in the implicit character of the model which removes the computational and a signiﬁcant part of the implementational complications of explicit SGS models. An appropriate framework for connecting ﬁltering and discretization of the underlying conservation law is available by the ﬁnite-volume method [14]. We consider now for simplicity the initial-value problem for a generic scalar nonlinear transport equation for the variable v ∂v ∂F (v) + =0. ∂t ∂x

(1)

On a mesh xj = jh with equidistant spacing h and j = . . . , −1, 0, 1, . . . the grid . function vN = {vj } represents a discrete approximation of v(x) by vj = v(xj ). A spectrally accurate interpolant of the grid function with the same Fourier transform can be constructed using the Whittaker cardinal function [20]. For ﬁnite h the representation of the continuous solution v(x) by the grid function vN results in a subgrid-scale error GSGS =

∂FN (vN ) ∂FN (v) − ∂x ∂x

(2)

which arises from the nonlinearity of F (v). The modiﬁed diﬀerential equation for vN is ∂vN ∂FN (vN ) + = GSGS . (3) ∂t ∂x A ﬁnite-volume discretization of eq. (1) corresponds to a convolution with a top-hat ﬁlter returning the ﬁltered solution in terms of a grid function uj at xj xj+1/2 1 u(x′ )dx′ . (4) u ¯j = G ∗ u = h xj−1/2

The resulting ﬁnite-volume approximation of eq. (1) is given by ∂ F˜N (˜ uN ) ∂u ¯N +G∗ =0, ∂t ∂x

(5)

. where u ˜N = uN results from an approximate inversion of the ﬁltering u ¯N = G ∗ u. Although the inverse-ﬁltering operation is ill-posed, an approximation u ˜N of u on the grid xN can be obtained by regularized deconvolution [4]. For a brief summary on the deconvolution concept for subgrid-scale modeling it is ˆ illustrative to consider the Fourier-transform G(ξ) of G. The ﬁltering operation damps each wavenumber and truncates the ﬁltered data at the Nyquist wavenumber ξh (constant grid spacing assumed). An inverse operator can be ˆ −1 (ξ). Since G(ξ) ˆ written in Fourier space as G = 0 for |ξ| ≥ ξh only wavenumber contributions |ξ| < ξh to the ﬁltered solution can be inverted. This is the

18

N. A. Adams, S. Hickel, T. Kempe, and J. A. Domaradzki

regularized deconvolution obtained by singular-value decomposition. In this paper we propose a regularization based on adjusting nonlinearly the local interpolation polynomials to the solution properties. By this extension of the ˜ −1 ∗ u deconvolution operation u ˜N = G ¯N to a solution-adaptive nonlinear formulation it is expected that an additional regularization such as used in [21] and [22] can be avoided. Another utensil which can be exploited is the choice of an appropriate and consistent numerical ﬂux function F˜N which approximates FN . In summary, the construction of an implicit SGS model which we propose here amounts to develop an adaptive approximate deconvolution ˜ −1 and to devise a suitable numerical ﬂux function F˜N . operator G Once deconvolution operation and numerical ﬂux function are determined, the modiﬁed-diﬀerential-equation analysis of eq. (5) leads to an evolution equation of u ¯N in the form of ∂u ¯N ∂FN (uN ) +G∗ = GN , ∂t ∂x where GN = G ∗

∂ F˜N (˜ uN ) ∂FN (uN ) −G∗ ∂x ∂x

(6)

(7)

is the truncation error of the discretization. If GN approximates G¯SGS in some sense for ﬁnite h we obtain an implicit subgrid-scale model contained within the discretization. Note that this requirement is diﬀerent from the requirement of GN approximating G¯SGS for h → 0, which would dictate a spectrally convergent discretization as optimum.

2 The ALDM Approach With the adaptive local deconvolution method (ALDM) the local approxima N is obtained from a solution-adaptive combination of deconvolution tion u polynomials. Numerical discretization and SGS modeling are merged entirely. This is possible by exploiting the formal equivalence between cell-averaging and reconstruction in ﬁnite-volume discretizations and top-hat ﬁltering and deconvolution in SGS-modeling. Instead of maximizing the order of accuracy, deconvolution is regularized by limiting the degree of local interpolation polynomials and by permitting lower-order polynomials to contribute to the truncation error. Adaptivity of the deconvolution operator is achieved by weighting the respective contributions by an adaptation of WENO smoothness measures. The approximately deconvolved ﬁeld is inserted into a consistent numerical ﬂux function. Flux function and nonlinear weights introduce free parameters. These allow for controlling the truncation error which provides the implicit SGS model. The eﬃciency of this approach is demonstrated in [18] for 1D conservation laws on the example of the viscous Burgers equation. The extension for the three-dimensional Navier-Stokes equations is detailed by Hickel

On the Relation between SGS Modeling and Numerical Discretization

19

et al. [19]. Similarly as with the preferred-stencil weights in smooth-ﬂow regions for WENO schemes [23] there are preferred stencil weights for ALDM which are active in ﬂow regions of developed turbulence. In the following we brieﬂy summarize a key element in deriving these weights which constitute the implicit model. This key element is a modiﬁed-diﬀerential equation (MDE) analysis in spectral space. Further details can be found in [19]. We consider the discretization of a (2π)3 -periodic domain. N is the number of grid points in one dimension and ξC = N/2 is the corresponding cut-oﬀ wave number. Using Fourier transforms the MDE can be written in spectral space as ¯ C ∂u iξ · N C ( C ¯ C = G +G uC ) + νξ 2 u (8a) ∂t ¯ C = 0 iξ · u (8b) The hat denotes the Fourier transform, i is the imaginary unit, and ξ is the wave-number vector. A physical-space discretization √ covers contributions to the numerical solution to wavenumbers up to |ξ| = 3ξC . On the represented wave-number range the kinetic energy of the deconvolved velocity is 1 C (ξ) · u ∗C (ξ) E(ξ) = u 2

(9)

∗C of u C we obtain Multiplying equation (8a) by the complex-conjugate u TC (ξ) + 2νξ 2 G C (ξ) E(ξ) ∂ E(ξ) − G(ξ) ∗C (ξ) · G =u G ∂t

(10)

The nonlinear energy transfer TC (ξ) is the Fourier transform of the nonlinear term. Finally, we deconvolve eq. (10) by multiplication with the inverse ﬁlter −1 (ξ) which is deﬁned on the range of represented scales |ξ| ≦ ξC coeﬃcient G and obtain ∂ E(ξ) −1 (ξ) C (ξ) − TC (ξ) + 2νξ 2 E(ξ) =G u∗C (ξ) · G ∂t

(11)

The right-hand side of this equation is the numerical dissipation C (ξ) −1 (ξ) εnum (ξ) = G u∗C (ξ) · G

(12)

εSGS (ξ) = 2νSGS ξ 2 E(ξ)

(13)

implied by the discretization of the convective term. Now we investigate how to model the physical subgrid dissipation εSGS by εnum . An exact match between εnum and εSGS cannot be achieved since εSGS involves interactions with non-represented scales. For modeling it is therefore necessary to invoke theoretical energy-transfer expressions. Employing an eddy-viscosity hypothesis the subgrid-scale dissipation is

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N. A. Adams, S. Hickel, T. Kempe, and J. A. Domaradzki

Similarly, the numerical dissipation can be expressed as νnum =

εnum (ξ) 2ξ 2 E(ξ)

.

(14)

In general νnum is a function of the wavenumber vector ξ. For isotropic turbulence, however, statistical properties of eq. (11) follow from the scalar evolution equation for the 3D energy spectrum ∂ E(ξ) − TC (ξ) + 2νξ 2 E(ξ) = εnum (ξ) . ∂t

For a given numerical scheme νnum (ξ) can be computed from −1 (ξ) G N (ξ)dξ ∗C (ξ) · G νnum (ξ) = − u 2ξ 2 E(ξ)

(15)

(16)

|ξ|=ξ

Convenient for our purposes is a normalization by ξ ξC + + νnum (ξ ) = νnum C) ξC E(ξ

(17)

with ξ + = ξξC . The concept of a wavenumber-dependent spectral eddy viscosity was ﬁrst proposed by Heisenberg [24]. For high Reynolds numbers and under the assumption of a Kolmogorov range Chollet [25] proposes the expression + −3/2 + (18) 1 + 34.47e3.03ξ (ξ + ) = 0.441CK νChollet as best ﬁt to the exact solution. The objective is to adjust the model parameters in such a way the implicit model’s dissipative properties are consistent with analytical theories of turbulence. For this purpose we consider freely decaying homogeneous isotropic turbulence in the limit of vanishing molecular viscosity. Filtered and truncated highly resolved LES data are used as initial condition for trial computations with ALDM on a (2π)3 -periodic box, discretized by 32 × 32 × 32 uniform ﬁnite volumes. Each trial computation for a certain parameter set is advanced for a small number of time steps and followed by an a-posteriori analysis of the data which allows to identify the spectral eddy viscosity of the implicit SGS model [19]. This procedure is the kernel of an evolutionary optimization employed for model parameter identiﬁcation. For further details the reader should refer to the mentioned publications.

3 Isotropic Turbulence As ﬁrst validation example we consider decaying grid-generated turbulence for which also the correct representation of the energy-containing range of the

On the Relation between SGS Modeling and Numerical Discretization 10

E 10

10

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-1

-2

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10 1

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ξ 3

Fig. 1. Instantaneous 3D energy spectra for LES with 64 cells and for measurements of Comte-Bellot – Corrsin ; ·−·−·− Smagorinsky model, −··−··− dynamic Smagorinsky model, −−−−−−− ALDM ; t′ = 42 , t′ = 98 and △ t′ = 171 experimental data of [27].

spectrum is important [26]. Computations are initialized with spectrum and Reynolds numbers adapted to the wind-tunnel experiments of Comte-Bellot and Corrsin [27], denoted hereafter as CBC. Among other space-time correlations CBC provides streamwise energy spectra for grid-generated turbulence at three positions downstream of a mesh with a width M = 5.08cm. In the simulation this ﬂow is modeled by matching the energy distribution of the initial velocity ﬁeld to the ﬁrst measured 3D energy spectrum of CBC. The SGS model can now be veriﬁed by comparing computational and experimental 3D energy spectra at later time instants which correspond to the subsequent measuring stations. In order to create the initial velocity ﬁeld a random ﬁeld was allowed to develop for about one large-eddy turnover time according to Navier-Stokes dynamics while maintaining the 3D energy spectrum as given for the ﬁrst measuring station. Results of ALDM are compared with those obtained with a 4th-order central discretization scheme and an explicit Smagorinsky SGS model. The Smagorinsky model is used in its conventional and in its dynamic version. For the conventional model [28] the parameter is set to CS = 0.18. Lilly [29] derived this value for suﬃciently large Reynolds numbers and a sharp spectral cut oﬀ in the inertial range assuming CK ≈ 1.4. The dynamic algorithm was proposed by Germano et al. [30]. Here, CS is computed according to Lilly [31] and averaged in space. Examining the computed energy spectra, ﬁgure 1, we note that ALDM performs just as well as the dynamic Smagorinsky SGS model. It should be emphasized that the prediction of the dynamic Smagorinsky model can be considered as a reference for isotropic turbulence. The conventional Smagorinsky

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N. A. Adams, S. Hickel, T. Kempe, and J. A. Domaradzki

model requires an ad-hoc adjustment of CS . Choosing CS somewhat smaller than the theoretical value gives better results which are close to that of the dynamic Smagorinsky model. For the decay of total kinetic energy K, ﬁgure 2, we ﬁnd ∂K/∂t ∼ t−n with n = 1.25. This corresponds to ε = ∂K/∂t ∼ t−2.25 or ε ∼ K 1.8 . The exponent n = 1.25 is in a reasonable agreement with published experimental data [27, 32, 33] which range from n = 1.2 to n = 1.3.

10

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Fig. 2. Contributions to energy dissipation in ALDM for LES of decaying homogeneous isotropic turbulence according to the Comte-Bellot – Corrsin experiment; ·−·−·− molecular dissipation, −−−− implicit SGS dissipation, −−−−−−− total dissipation, ·········· ε ∼ t−2.25 .

4 Plane-Channel Flow with Periodic Constrictions In this section we consider the case of a channel ﬂow with periodic constrictions in the streamwise direction which has been investigated intensely in the past [34, e.g.]. The geometry is sketched in ﬁgure 3. The Reynolds number based on the bulk velocity U0 within the narrow section and the constriction height h is Re = 10595. All dimensional units are normalized by h and U0 . The streamwise and wall-normal extents are shown in ﬁgure 3, the spanwise extent of the computational domain is 4.5h. The grid resolution of 190 × 140 × 170 (streamwise × wall-normal × spanwise) grid points corresponds roughly to the 196 × 128 × 186 of [34]. However, our computations are based on a Cartesian mesh using a cut-cell technique at the solid boundaries not aligned with the mesh lines. An overall impression of the mean ﬂow ﬁeld and a qualitative assessment of diﬀerent SGS models can

On the Relation between SGS Modeling and Numerical Discretization

23

3

y/ h

2.5 2 1.5 1 0.5 0 0

2

4

6

8

x/ h

Fig. 3. Geometry and computational mesh for the constricted channel.

be drawn from ﬁg. 4. The agreement with ﬁg. 7 of [34] is good for both the ALDM and the dynamic-Smagorinsky prediction. A quantitative comparison of mean velocity data is provided in ﬁg. 5. The mean velocity proﬁles computed with ALDM are consistently in between the reference data and data obtained with the dynamic Smagorinsky model on the same mesh as used for ALDM. (a)

3

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(b)

3

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2.5 2 1.5 1 0.5 0 0

2

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Fig. 4. Streamlines of the mean ﬂow: (a) ALDM, (b) dynamic Smagorinsky model.

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N. A. Adams, S. Hickel, T. Kempe, and J. A. Domaradzki (b)

3

z/h

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<W> <W> <W>

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DSM DSM ILES ILES FML FML

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(d)

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DSM DSM ILES ILES FML FML

1.5 1

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Fig. 5. Mean-velocity proﬁles at positions: (a) x = 0.005, (b) x = 0.05, (c) x = 2, (d) x = 4, (e) x = 6, (f) x = 8. FML data of [34], DSM data with dynamic Smagorinsky model, ILES data with ALDM

5 Conclusions We have presented an approach for the design of implicit SGS models which is based on the main components of ﬁnite-volume discretizations. The resulting implicit model allows for a full merge of discretization and subgrid-scale model. Although model parameters are determined by isotropic turbulence in the

On the Relation between SGS Modeling and Numerical Discretization

25

limit of inﬁnite Reynolds number we have shown that the resulting model can be applied also to complex separated turbulent boundary layers.

Acknowledgments The research is supported by the German Research Council under contract AD 186/2. JAD was supported by NSF and the Alexander von Humboldt foundation.

References [1] A. Leonard. Energy cascade in large eddy simulations of turbulent ﬂuid ﬂows. Adv. Geophys., 18A:237–248, 1974. [2] J. A. Domaradzki and E. M. Saiki. A subgrid-scale model based on the estimation of unresolved scales of turbulence. Phys. Fluids, 9:2148–2164, 1997. [3] S. Stolz and N. A. Adams. An approximate deconvolution procedure for large-eddy simulation. Phys. Fluids, 11:1699–1701, 1999. [4] J. A. Domaradzki and N. A. Adams. Modeling subgrid scales of turbulence in large-eddy simulations. J. Turb., 3:24, 2002. [5] S. Ghosal. An analysis of numerical errors in large-eddy simulations of turbulence. J. Comput. Phys., 125:187–206, 1996. [6] A. Kravchenko and P. Moin. On the eﬀect of numerical errors in largeeddy simulation of turbulent ﬂows. J. Comput. Phys., 130:310–322, 1997. [7] T. Kawamure and K. Kuwahara. Computation of high Reynolds number ﬂow around a circular cylinder with surface roughness. AIAA-paper, 840340, 1984. [8] J. P. Boris, F. F. Grinstein, E. S. Oran, and R. L. Kolbe. New insights into large eddy simulation. In Fluid Dynamics Research, volume 10, pages 199–228. North Holland, 1992. [9] D. H. Porter, P. R. Woodward, and A. Pouquet. Inertial range structures in decaying compressible turbulent ﬂows. Phys. Fluids, 10:237–245, 1998. [10] P. K. Smolarkiewicz and L. G. Margolin. MPDATA: a ﬁnite-diﬀerence solver for geophysical ﬂows. J. Comput. Phys., 140:459–480, 1998. [11] J. A. Domaradzki, Z. Xiao, and P. K. Smolarkiewicz. Eﬀective eddy viscosities in implicit large eddy simulations of turbulent ﬂows. Phys. Fluids, 15:3890–3893, 2003. [12] C. Fureby and F. F. Grinstein. Large eddy simulation of high-Reynoldsnumber free and wall-bounded ﬂows. J. Comput. Phys., 181:68–97, 2002. [13] E. Garnier, M. Mossi, P. Sagaut, P. Comte, and M. Deville. On the use of shock-capturing schemes for large-eddy simulation. J. Comput. Phys., 153:273–311, 1999.

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[14] R. J. LeVeque. Finite volume methods for hyperbolic problems. Cambridge UP, 2002. [15] C. Fureby, G. Tabor, H. G. Weller, and A. D. Gosman. A comparative study of subgrid scale models in homogeneous isotropic turbulence. Phys. Fluids, 9:1416–1429, 1997. [16] L. G. Margolin and W. J. Rider. A rationale for implicit turbulence modeling. Int. J. Numer. Meth. Fluids, 39:821–841, 2002. [17] N. A. Adams. The role of deconvolution and numerical discretization in subgrid-scale modeling. In Direct and Large-Eddy Simulation IV, 2001. [18] N. A. Adams, S. Hickel, and S. Franz. Implicit subgrid-scale modeling by adaptive deconvolution. J. Comput. Phys., 200:412–431, 2004. [19] S. Hickel, N. A. Adams, and J. A. Domaradzki. An adaptive local deconvolution method for implicit LES. J. Comput. Phys., page (in press), 2005. [20] R. Vichnevetsky and J. B. Bowles. Fourier Analysis of Numerical Approximations of Hyperbolic Equations. SIAM, Philadelphia, PA, 1982. [21] S. Stolz, N. A. Adams, and L. Kleiser. An approximate deconvolution model for large-eddy simulation with application to incompressible wallbounded ﬂows. Phys. Fluids, 13:997–1015, 2001. [22] N. A. Adams and S. Stolz. A deconvolution approach for shock-capturing. J. Comput. Phys., 178:391–426, 2002. [23] C.-W. Shu. Essentially non-oscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws. In B. Cockburn, C. Johnson, C.-W. Shu, E. Tadmor, and A. Quarteroni, editors, Advanced numerical approximation of nonlinear hyperbolic equations, volume 1697 of Lecture Notes in Mathematics, pages 325–432, Berlin, 1998. Springer. [24] W. Heisenberg. Zur statistischen Theorie der Turbulenz. Z. Phys., 124:628–657, 1946. [25] J.-P. Chollet. Two-point closures as a subgrid-scale modeling tool for large-eddy simulations. In F. Durst and B. E. Launder, editors, Turbulent Shear Flows IV, pages 62–72, Heidelberg, 1984. Springer. [26] S. B. Pope. Turbulent Flows. Cambridge University Press, 2000. [27] G. Comte-Bellot and S. Corrsin. Simple Eulerian time correlation of full and narrow-band velocity signals in grid-generated ‘isotropic’ turbulence. J. Fluid Mech., 48:273–337, 1971. [28] J. Smagorinsky. General circulation experiments with the primitive equations. Mon. Weath. Rev., 93:99–164, 1963. [29] D. K. Lilly. The representation of small-scale turbulence in numerical simulation experiments. In H. H. Goldstein, editor, Proc. IBM Scientiﬁc Computing Symposium on Environmental Sciences, pages 195–201. IBM, 1967. [30] M. Germano, U. Piomelli, P. Moin, and W. H. Cabot. A dynamic subgridscale eddy viscosity model. Phys. Fluids, A 3:1760–1765, 1991. [31] D. K. Lilly. A proposed modiﬁcation of the Germano subgrid-scale closure model. Phys. Fluids A, 4:633–635, 1992.

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[32] AGARD. A Selection of Test Cases for the Validation of Large-Eddy Simulations of Turbulent Flows. Technical Report AGARD-AR-345. NATO, 1998. [33] H. S. Kang, S. Chester, and C. Meneveau. Decaying turbulence in an active-grid-generated ﬂow and large-eddy simulation. J. Fluid Mech., 480:129–160, 2003. [34] J. Fr¨ ohlich, C.Mellen, W. Rodi, L. Temmerman, and M. Leschziner. Highly resolved large-eddy simulation of separated ﬂow in a channel with streamwise periodic constrictions. J. Fluid Mech., 526:19–66, 2005.

Space-Time Error Representation and Estimation in Navier-Stokes Calculations Timothy J. Barth NASA Ames Research Center, Moﬀett Field, CA 94035 USA [email protected]

Summary. The mathematical framework for a-posteriori error estimation of functionals elucidated by Eriksson et al. [1] and Becker and Rannacher [2] is revisited in a space-time context. Using these theories, a hierarchy of exact and approximate error representation formulas are presented for use in error estimation and mesh adaptivity. Numerical space-time results for simple model problems as well as compressible Navier-Stokes ﬂow at Re = 300 over a 2D circular cylinder are then presented to demonstrate elements of the error representation theory for time-dependent problems.

1 Introduction For better or worse, our physical world is constantly evolving in time. Many important physical phenomena depend fundamentally on time either deterministically or through dynamical system behavior. As the complexity of numerical time-dependent ﬂuid ﬂow simulations continues to rapidly increase with advancements in computer hardware, the ability to represent, estimate and control numerical errors occurring in these time-dependent simulations becomes paramount so that the uncertainty and risk associated with using these results in engineering designs becomes acceptable. This task is especially diﬃcult for systems such as the Navier-Stokes equations where the prospect of controlling pointwise errors is intractable at suﬃciently large Reynolds numbers. In this latter case, a pointwise error bound exists but the bound grows too rapidly with Reynolds number thus rendering the result useless. Even so, it still may be feasible to represent, estimate, and control (via adaptivity) the error in certain space and space-time integrated quantities such as body forces, averages, and other ﬂuid statistics that can mathematically represented as functionals, the mapping of a function space to a single real number. In these special cases, the integration process “averages out” the sensitivity of pointwise errors to solution perturbations so that useful error bounds are obtained. For a given solution data, u, in m dependent variables, these scalar

30

Timothy J. Barth

output functionals will be denoted by J(u) : IRm → IR. For example, an output functional considered in the next section is the space-time average of the i-th solution component in the hypercube centered at (x0 , y0 , z0 , t0 ) J(ui ) =

t1

t0

d×d×d

ui (x − x0 , y − y0 , z − z0 , t − t0 ) dx3 dt .

(1)

1.1 Computability of Functional Outputs To illustrate the beneﬁts of estimating integrated functional outputs, Hoﬀman and Johnson [3] have computed solutions of the backward facing step problem shown in Fig. 1 at Re = 2000 using a conforming ﬁnite element method with linear elements for incompressible ﬂow, see [3] for details. Space-time error estimates were then constructed for the functional (1) for various values of the box width d together with the time averaging interval t0 = 9, t1 = 10. In velocity and pressure variables (V, p), the following error estimate for the streamwise velocity component functional have been obtained in terms of the backwards in time dual solution (ψ, φ) and the element residuals rh,0 and rh,1 for the (V, p) system

˙ h0 rh,0 (Vh , ph ) + C1 D2 φ h2 rh,0 (Vh , ph ) |J(V, p) − J(Vh , ph )| ≤ C0 ψ ˙ h0 rh,1 (Vh , ph ) + C3 Dφ h rh,1 (Vh , ph ) + C2 φ where h0 and h are the temporal and spatial element lengths. Motivated by this estimate, stability factors for boxwidths d = 1/8, 1/4, 1/2 have been computed by Hoﬀman and Johnson as summarized in Table 1. The growth in stability factor magnitude with decreasing boxwidth size clearly shows a deterioration in the error estimate for the functional (1) as a pointwise error estimate is approached. In that same work, Hoﬀman and Johnson also use a drag force functional for incompressible ﬂow over a cube geometry attached to a solid wall. By using error estimates of the drag functional in mesh adaptation, the mesh is only locally reﬁned in a small portion

1 d 1/2

1

Fig. 1. Schematic of backward facing step problem showing J(u) averaging box position used by Hoﬀman and Johnson.

Space-Time Error Representation and Estimation in CFD

31

Table 1. Stability factors for the backward facing step problem for the averaging functional (1) for d = 1/8, 1/4, 1/2. d 1/8 1/4 1/2

˙ ψ 124.0 39.0 10.5

∇ψ 836.0 533.4 220.3

∇φ 138.4 48.9 16.1

˙ φ 278.4 46.0 25.2

of the domain so that numerical errors in the drag force are controlled using a relatively small number of degrees of freedom. For other problems, the overall challenge is not only to ﬁnd convenient ways to estimate and control the numerical error in these functionals, but also to ﬁnd suitable functionals that are both physically meaningful and lead to the resolution of the essential physics in the simulation. 1.2 Estimation of Functional Outputs Let u denote the exact solution data for a given problem and uh a numerical approximation of this exact solution. The task at hand is the representation, estimation, and control of errors in the numerically approximated functional so that given a user speciﬁed tolerance TOL |J(u) − J(uh )| < TOL is achieved using minimal resources. At ﬁrst glance, the inclusion of time derivatives into the formulation appears to signiﬁcantly complicate the construction of error bounds for functionals. Fortunately, for the class of uniform space-time ﬁnite-element methods this is not the case since the time dimension is discretized in precisely the same way that any spatial dimension is discretized. Speciﬁcally, we consider herein a space-time variant of the discontinuous Galerkin (DG) method due to Reed and Hill [4] and LeSaint and Raviart [5] for hyperbolic problems. In the space-time DG method, the ﬁnite element discretization in time naturally yields time causality so that discretizations can be marched forward in time on a time-slab by time-slab basis for both hyperbolic and parabolic problems. Thus conceptually, the addition of time into the DG method does not introduce any new analysis complications. As a practical matter, full spacetime formulations of DG are quite computationally expensive but extremely accurate methods. 1.3 Preview of Space-Time Error Bounds For an introduction to a posteriori error analysis of functionals see the articles by Eriksson et al. [1], Becker and Rannacher [2], Giles et al. [6, 7], Johnson

32

Timothy J. Barth

et al. [8], Prudhomme and Oden [9, 10], S¨ uli [11], the collected NATO lecture notes [12] and the multitude of additional references contained therein. In Sect. 2, a brief introduction to these general theories is given with special attention given to space-time generalizations of previous results for the discontinuous Galerkin method previously given in Larson and Barth [13] as well as Hartmann and Houston [14]. Our main goal is to present the formulation of error estimates for output functionals in space-time. To illustrate the general form of these estimates, consider the space-time hyperbolic problem with initial data u0 (x) and spacetime boundary data g(x, t): L u − s(x, t) ≡ u,t + div(f (u)) − s(x, t) = 0, −

p (u − g(x, t)) = 0, u(x, t = 0) − u0 (x) = 0,

in Ω × [0, T ]

on Γ × [0, T ] initial data

(2) (3) (4)

where p± is a characteristic space-time projector such that p− = 1 at spacetime inﬂow and p+ = 0 at space-time outﬂow. Let K be a partition of a polygonal domain Ω with spatial boundary Γ into non-overlapping shape regular elements (or control volumes) denoted by K, let I n denote an interval in time n n I n ≡ [tn+ , tn+1 − ], and Q ≡ K × I a single space-time element with space-time dimension h. Throughout, we always consider time integration to a ﬁxed ﬁnal N −1 n time T using

N discrete time slabs such

that [0, T ] = ∪n=0 I . Further, let (a, b)Q ≡ Q (a · b) dx dt and a, b ∂Q ≡ ∂Q (a · b) dx dt. In Sect. 2, the abstract theory is given for space-time error representation and estimation of functionals which has historically served as the basis for constructing weighted and unweighted estimates of the form (trivially extended here to include time): • Weighted error estimates for functionals, Becker and Rannacher [2] |J(u) − J(uh )| ≤

N

n=0 Qn

|(rh , φ − πh φ)Qn +

∂Qn

jh , φ − πh φ ∂Qn |

(5)

where rh denotes the residual on element interiors rh ≡ Luh − s(x, t) .

(6) n

Similarly, jh denotes an interface residual for a space-time element, Q ≡ K×I n , due to jumps occurring at element interfaces and inﬂow boundaries, viz.  xn  p− [f (uh ) · n]x+ ∂K\Γ × I n n ,  −   − p (f (g) · n − f (uh ) · n), ∂K ∩ Γ × I n (7) jh ≡ tn +  [u ] , K, n = 0  h tn  −  K, n = 0 (uh (t0+ ) − u0 (x)),

In the basic error estimate, φ represents an auxiliary function obtained from solving a (backwards in time) dual problem and πh φ is any projection

Space-Time Error Representation and Estimation in CFD

33

of φ into the approximation space containing uh as discussed in Sect. 2.

Suppose J(u) = Ω×[0,T ] (Ψ · u) dx dt for a suﬃciently smooth weighting function Ψ , then the dual problem is related to the locally linearized adjoint problem L∗ φ = Ψ together with appropriate boundary conditions. Observe that the functional error is a weighted combination of numerical residuals with weighting function φ − πh φ. • Error estimates for functionals via stability factors, Johnson et al. [3, 8]. The [4] computation of the weighted error estimate requires the resolution of a dual problem φ which becomes computationally expensive when performed in space-time. In the unweighted estimate of Johnson et al. [8], basic approximation theory results are applied to the dual problem appearing in (5) thus yielding   N −1 (hs rh Qn + hs−1/2 jh ∂Qn )2  |J(u)−J(uh )| ≤ C0 φ H s (Ω×[0,T ]) n=0 Qn

for 1 ≤ s ≤ p + 1. The dual problem is then replaced with a stability constant satisfying φ H s (Ω×[0,T ]) ≤ Cstab

thereby producing the computable estimation formula 

|J(u) − J(uh )| ≤ C0 Cstab 

N −1 n=0 Qn

1/2

(hs rh Qn + hs−1/2 jh ∂Qn )2 

.

The basis idea is that it may be feasible to catalog stability constants Cstab for various ﬂow problems so that computation of the dual problem can be avoided. The simplifying assumptions utilized here lead to an unavoidable loss in sharpness when compared to the weighted error estimate. Next, we brieﬂy review the relevant theory for representing the space-time error in functionals which serves as the primogenitor for both of the previously given a-posteriori estimates.

2 Error Estimation of Functionals Consider once again the model space-time problem (4) in a domain Ω ⊂ IRd B with boundary Γ . In this domain, let Vh,p denote a mesh dependent broken space of piecewise polynomials of complete degree p in each Qn with no continuity between space-time elements

34

Timothy J. Barth N −1 n B Vh,p = {w : w|Q ∈ Pp (Q) , ∀Q ∈ K × ∪n=0 I } .

(8)

To illustrate the abstract framework for error estimation of functionals, we consider the discontinuous Galerkin (DG) ﬁnite element method introduced by Reed and Hill [4] and LeSaint and Raviart [5] as analyzed by Johnson and Pitk¨ aranta [15] and further reﬁned for nonlinear conservation laws by Cockburn et al. [16, 17]. In presenting the abstract theory, one quickly sees that the techniques used here can be easily applied to any method that can be written in weighted residual form. B Space-Time Discontinuous Galerkin FEM. Find uh ∈ Vh,p such that

BDG (uh , w) = F (w),

B ∀w ∈ Vh,p

(9)

where BDG (uh , w) − F (w) =

N −1 n=0

n

Q

Qn

(−w,t · uh − w,xi · fi (uh )) dx dt

w− · h(n; (uh )− , (uh )+ ) ds dt I n (∂K∩∂Qn )\Γ w− · h(n; (uh )− , g) ds dt + I n (∂K∩∂Qn )∩Γ n+1 n n (w(tn+1 + − ) uh (t− ) − w(t+ ) uh (t− )) dx Ω,n=0 +

+

Ω,n=0

(w(t1− ) uh (t1− ) − w(t0+ ) u0 )) dx

where h(n; u− , u+ ) is a numerical ﬂux function such that h(n; u, u) = ni fi (u) and h(n; u− , u+ ) = −h(−n; u+ , u− ). 2.1 DG FEM Error Representation: The Linear Case The objective is to estimate the error in a user speciﬁed functional J(u), the mapping of a function space to a single real number. In this work, we consider functionals that can be expressed as a weighted integration over the space-time domain T Jψ (u) = ψ · N (u) dx dt 0

Ω

or a weighted integration on the boundary of the space-time domain Jψ (u) = ψ · N (u) ds ∂(Ω×[0,T ])

for some user speciﬁed weighting function ψ(x) : IRd → IRm and linear/nonlinear function N (u) : IRm → IRm . By an appropriate choice of ψ(x)

Space-Time Error Representation and Estimation in CFD

35

and N (u), it is possible to devise functionals of practical engineering use, e.g. lift and drag forces on a body, stress intensity factors, average quantities, etc. Let BDG (·, ·) denote a bilinear form for the discontinuous Galerkin method and J(·) a linear functional. In the following derivations, πh denotes any suitB able projection operator (e.g. interpolation, L2 projection) into Vh,p . Begin by introducing the primal numerical method assuming all boundary conditions are weakly enforced. B Primal numerical problem: Find uh ∈ Vh,p such that

BDG (uh , w) = F (w)

B ∀ w ∈ Vh,p

with the Galerkin orthogonality condition B BDG (u − uh , w) = 0 ∀ w ∈ Vh,p .

Next, we introduce the auxiliary dual problem utilizing inﬁnite-dimensional trial and test spaces. Dual problem: Find Φ ∈ V B such that BDG (w, Φ) = J(w)

∀ w ∈ VB .

An exact error representation formula for a given functional J(·) results from the following steps J(u) − J(uh ) = J(u − uh ) = BDG (u − uh , Φ)

(linearity of M ) (dual problem)

= BDG (u − uh , Φ − πh Φ) (orthogonality) = BDG (u, Φ − πh Φ) − BDG (uh , Φ − πh Φ) (linearity of B)

= F (Φ − πh Φ) − BDG (uh , Φ − πh Φ)

(primal problem)

so in summary J(u) − J(uh ) = F (Φ − πh Φ) − BDG (uh , Φ − πh Φ) .

(10)

Notably absent from the right-hand side of this equation is any dependence on the exact solution u. 2.2 DG FEM Error Representation: The Nonlinear Case Let BDG (·, ·) denote a semilinear form for the discontinuous Galerkin method and J(·) now a nonlinear functional. To cope with nonlinearity, mean-value linearization is employed BDG (u, w) = BDG (uh , w) + BDG (uh , u; u − uh , w) J(u) = J(uh ) + J(uh , u; u − uh ) .

∀ w ∈ VB

(11) (12)

36

Timothy J. Barth

For example, if B(u, w) = (Lu, w) for some nonlinear diﬀerential operator L then for w ∈ V 1 B(u, w) = B(uh , w) + L,u (˜ u(θ)) d θ (u − uh ), v 0

= B(uh , w) + (L,u (u − uh ), w) = B(uh , w) + B(uh , u; u − uh , w).

with u ˜(θ) ≡ uh + (u − uh ) θ. For brevity, the dependence of B on the path integration involving the exact solution u will be notationally suppressed. We then proceed in the same fashion as in the previous example. B Primal numerical problem: Find uh ∈ Vh,p such that

BDG (uh , w) = F (w)

B ∀ w ∈ Vh,p

(13)

with orthogonality condition for the linearized form B BDG (u − uh , w) = 0 ∀ w ∈ Vh,p .

A mean-value linearized dual problem is then introduced which utilizes inﬁnite-dimensional trial and test spaces. Linearized dual problem: Find Φ ∈ V B such that B DG (w, Φ) = J(w)

∀ w ∈ VB .

(14)

An exact error representation formula for a given nonlinear functional J(·) then results from the following steps J(u) − J(uh ) = J(u − uh )

= B DG (u − uh , Φ)

(mean-value J) (dual problem)

= B DG (u − uh , Φ − πh Φ)

(orthogonality)

= F (Φ − πh Φ) − BDG (uh , Φ − πh Φ),

(primal problem)

= BDG (u, Φ − πh Φ) − BDG (uh , Φ − πh Φ) (mean-value B)

so in summary J(u) − J(uh ) = F (Φ − πh Φ) − BDG (uh , Φ − πh Φ) .

(15)

Note that although Eqns. (10) and (15) appear identical, mean-value linearization introduces a subtle right-hand side dependency on the exact solution in Eqn. (15).

Space-Time Error Representation and Estimation in CFD

37

3 Computable Error Estimates and Adaptivity Computationally, the error representation formulas (10) and (15) are not suitable for obtaining computable a posteriori error estimates and use in mesh adaptation. • The function Φ − πh Φ is unknown where Φ ∈ V B is a solution of the inﬁnite-dimensional dual problem. • The mean-value linearization used in the linearized dual problems (14) requires knowledge of the exact solution u. Various strategies which address the numerical approximation of Φ are discussed in Barth and Larson [18], e.g. postprocessing, higher order solves, etc. Due to Galerkin orthogonality, the dual problem in the discontinuous Galerkin ﬁnite element method must be approximated in a larger space of functions than that utilized in the primal numerical problem. For purposes of the present study, this is achieved in the discontinuous Galerkin method by solving the dual problem using a polynomial space that is one polynomial B then Φ ≈ degree higher than the primal numerical problem, viz. if uh ∈ Vh,p B Φh ∈ Vh,p+1 . In the present study, the mean-value linearization depending on the states u and uh is replaced by the simpler Jacobian linearization evaluated at the numerical state uh . This is not the only practical choice. In Barth and Larson [18], a more sophisticated technique involving the postprocessing of primal data and the approximation of the mean-value linearization by numerical quadrature is employed in computations. 3.1 Direct Estimates When written in global abstract form, the error representation formula does not indicate which elements in the mesh should be reﬁned to reduce the measured error in a functional. By applying a sequence of direct estimates, error bounds suitable for adaptive meshing are easily obtained. The goal in constructing these estimates is to estimate the local contribution of each element in the mesh to the functional error. This local cell contribution will then be used as an error indicator for choosing which elements to reﬁne or coarsen in the adaptive mesh procedure. |J(u) − J(uh )| = |BDG (uh , Φ − πh Φ) − F (Φ − πh Φ)| =| ≤

N −1 n=0 Qn

N −1 n=0 Qn

(error representation)

BDG,Qn (uh , Φ − πh Φ) − FQn (Φ − πh Φ) |

BDG,Qn (uh , Φ − πh Φ) − FQn (Φ − πh Φ)

(element assembly)

(triangle inequality) (16)

38

Timothy J. Barth

where BDG,Qn (·, ·) and FQn (·) are restrictions of BDG (·, ·) and F (·) to the partition element Qn . The basic deﬁnition of the discontinuous Galerkin semilinear form given in (10) shows one possible element assembly form but this is not a unique representation. For example strong and weak forms of the semilinear operator BDG (·, ·) yield diﬀering assembly representations. For the discontinuous Galerkin method, the error representation formula together with (10) for a single element Qn yields BDG,Qn (uh , Φ − πh Φ) −FQn (Φ − πh Φ) = − uh · (Φ − πh Φ),t dx dt In K fi (uh ) · (Φ − πh Φ),xi dx dt − I n K (Φ − πh Φ)− · h(n; (uh )− , (uh )+ ) ds dt + I n ∂K\Γ + (Φ − πh Φ)− · h(n; (uh )− , g) ds dt I n ∂K∩Γ n+1 + (Φ − πh Φ)(tn+1 − ) · u(t− ) dx K − (Φ − πh Φ)(tn+ ) · u(tn− ) dx K,n = 0 − (Φ − πh Φ)(t0+ ) · u0 ) dx (17) K,n=0

or a weighted residual (strong) form can be obtained upon integration by parts BDG,Qn (uh , Φ − πh Φ) − FQn (Φ − πh Φ) = (Φ − πh Φ) · Luh dx dt Qn (Φ − πh Φ)− · (h(n; (uh )− , (uh )+ ) − f (n; (uh )− ) ds dt + I n ∂K\Γ + (Φ − πh Φ)− · (h(n; (uh )− , g) − f (n; (uh )− ) ds dt I n ∂K∩Γ tn (Φ − πh Φ)(tn+ ) · [u]t+ + n dx − K,n=0 + (Φ − πh Φ)(t0+ ) · (uh (t0+ ) − u0 ) dx (18) K,n=0

This latter weighted residual form Qn BQn (·, ·) − FQn (·) is preferred in the error estimates (16) since the individual terms represent residual components that vanish individually when the exact solution is inserted into the variational form and a slightly sharper approximation is obtained after application of the triangle inequality in (16). 3.2 Adaptive Meshing Motivated by the direct estimates (16), we deﬁne for each partition element Qn the space-time element quantity

Space-Time Error Representation and Estimation in CFD

ηQn ≡ BDG,Qn (uh , Φ − πh Φ) − FQn (Φ − πh Φ) ,

39

(19)

so that |ηQn | serves as an adaptation element indicator such that |J(u) − J(uh )| ≤

N −1 n=0 Qn

|ηQn |

(20)

and the sum of ηQn over space-time provides an accurate adaptation stopping criteria N −1 ηQn | . (21) |J(u) − J(uh )| = | n=0 Qn

These quantities suggest a simple mesh adaptation strategy in common use with other indicator functions: Mesh Adaptation Algorithm (1) Construct an initial space-time mesh. (2) Compute a numerical approximation of the primal problem on the current space-time mesh. (3) Compute a suitable numerical approximation of the inﬁnite-dimensional dual problem on the current mesh. (4) Compute error indicators, ηQn . N −1 (5) If( | n=0 Qn ηQn | < T OL) STOP (6) Otherwise, reﬁne and coarsen a speciﬁed fraction of the total number of elements according to the size of |η|nQ , generate a new mesh and GOTO 2 3.3 Numerical Example: Error Estimates for Time-Dependent Hyperbolic Problems Numerical error estimation results for stationary hyperbolic problems using the discontinuous Galerkin method and the error estimates described above are given by the present author in [13, 18] and later revisited by Hartman and Houston in [14]. To demonstrate the application of the error representation formula (10) and the error bound (16) to time-dependent hyperbolic problems, discontinuous Galerkin solutions have been obtained for the following 2-D advection problem in the square domain Ω ∈ [−1, 1]2  in Ω × [0, T ] , u,t + λ · ∇u = 0 u(x, t) = 0 on ∂Ω × [0, T ] (22)  u(x, 0) = u0 (x) initial data

with circular advection ﬁeld λ = (−y, x)T , ﬁnal time T = 1.15, and C ∞ initial data u0 (x) = σ(2/10; (x − x0 )2 + (y − y0 )2 ) , (x0 , y0 ) = (7/10, 0)

40

Timothy J. Barth

with σ(r0 ; r) ≡

0 r ≥ r0 2 2 2 . e1+r0 /(r −r0 ) r < r0

(23)

An output functional was also devised consisting of a C ∞ space-time averaging ball of radius 1/4 located at (xf , yf , tf ) = (.30, .26, .64), i.e. J(u) =

0

T

Ω

σ(1/3;

(t − tf )2 + (x − xf )2 + (y − yf )2 ) dx dt .

The numerical experiment then consists for the following steps 1. Numerical solution, uh , of the primal (forwards in time) problem using the discontinuous Galerkin method (9) with linear space-time elements, see Fig. 2(left). 2. Numerical solution, φh , of the dual (backwards in time) problem using the discontinuous Galerkin method (9) with quadratic space-time elements, see Fig. 3. 3. Approximation of the inﬁnite-dimensional dual problem by the quadratic space-time element approximation, φ ≈ φh . 4. Accumulation of the functional error, |J(u)−J(uj )|, using the error representation formula (10) and the error bound formula (16), see Fig. 2(right). Note that although we plot in Fig. 2 the accumulation of terms appearing in the error estimation formulas (10) and (16) during the backwards in time integration, it is only the ﬁnal termination value at time zero that provides a bound for the desired functional error. As expected, the error representation formula very accurately reproduces the exact functional error for this linear diﬀerential equation since the only source of approximation in this formula is the replacement of φ by a higher order quadratic space-time element approximation φh . The graph in Fig. 2 corresponding to Eqn. (16) lies strictly 0.0002 primal problem

| J(u) − J(u_h)|

0.00015 Error representation, Eqn (10) Error indicator bound, Eqn (14) Exact functional error (t=1.15)

0.0001

5e−05 dual problem

0

1

0.5

0

time

Fig. 2. Primal solution of the circular advection problem at t = .45. Shown are a carpet plot of the primal solution using linear space-time elements (left) and a graph of the accumulated functional error, J(u) − J(uh ), during the backwards in time dual solution integration (right). (See Plate 7 on page 416)

Space-Time Error Representation and Estimation in CFD

41

Fig. 3. Dual solution of the circular advection problem at t = .45. Shown are a carpet plot of the dual solution φh using quadratic space-time elements (left) and a carpet plot of the dual solution defect φh − πh φh (right) with πh a projection to space-time linear functions. (See Plate 8 on page 416)

above the error representation formula since the absolute value occurring in the Eqn. (16) precludes any interelement cancellation of error terms. In this particular example, this lack of interelement cancellation in Eqn. (16) also gives rise to a diﬀerent slope and a continuous deterioration in Eqn. 16 with increasing functional measurement time.

4 Extension to the Time-Dependent Navier-Stokes Equations Next, we consider the error estimation of functionals applied to the timedependent compressible Navier-Stokes equations for a perfect gas with ﬂuid density, velocity, pressure, and temperature denoted by ρ, V, p and T inv vis = Fi,x u,t + Fi,x i i

in Ω × [0, T ]

(24)

with implied summation on repeated indices, 1, . . . , d, and       0 ρVi ρ ρVi V1 + δi1 p ρV1    τ1i      1     ..    . . vis inv .. .. u =  .  , Fi =   , Fi =        Re  ρVi Vd + δid p ρVd    τ3i γ µ τij Vj + γ−1 T Vi (E + p) ρE ,x i Pr (25) µ 2 τ= (∇V + (∇V )T − (∇ · V )I) (26) Re 3 where γ, Re, and P r denote the non-dimensional ratio of speciﬁc heats, Reynolds number and Prandtl numbers, respectively. The present implementation of the discontinuous Galerkin method utilizes a change of dependent variables, u → v, where v are the so-called symmetrization variables for the Navier-Stokes equations as described in [19]. Using this

42

Timothy J. Barth

change of variables, viscous ﬂuxes are written in chainrule form Fivis = Mij v,xj ,

Mij ∈ IRm×m .

The extension of the discontinuous Galerkin method to the time-dependent Navier-Stokes equations follows the original symmetric interior penalty (SIP) work of Douglas and Dupont [20] for elliptic and parabolic problems. The SIP formulation has been recently applied to the steady Navier-Stokes equations in [21, 22]. The present SIP formulation and subsequent error estimation calculations accommodates the full time-dependent compressible Navier-Stokes equations. For brevity, the method is stated here in strong form. In practice, the method should always be implemented in weak integrated-by-parts form so that exact discrete conservation is assured even in the presence of inexact numerical quadrature. B Space-Time Discontinuous Galerkin SIP Navier-Stokes FEM. Find vh ∈ Vh,p such that N −1 B (27) BDG,Qn (vh , w) = 0 , ∀w ∈ Vh,p n=0 K

with B

DG,Qn

(v, w) =

inv vis dt dx w · u,t + Fi,x − Fi,x i i K I n w(x− ) · (h(n; v+ , v− ) − ni Fiinv (v− )) dt dx + n ∂K\Γ I + w(x− ) · ni (Fiinv wall − Fiinv )(v− )) dt dx n ∂K∩Γ I wall w(x− ) · (h(n; g∞ , v− ) − ni Fiinv (v− )) dt dx + n ∂K∩Γ I farfield 1 w(x− ) · [ni Fivis ]xx+ dt dx − − n 2 ∂K\Γ I w(x− ) · ni (Fivis (gN ) − Fivis (v− )) dt dS − ∂K∩ΓN I n 1 x+ + [v] · ni Mij (x− )w,xj (x− ) dt dx 2 x− n ∂K\Γ I + (gD − v(x− )) · ni Mij (x− )w,xj (x− ) dt dx ∂K∩ΓD I n − κp2 /h xx+ w(x− )) · ni nj Mij [v]xx+ dt dx − − n ∂K\Γ I − (κp2 /h)x− w(x− ))· ni nj Mij (gD − v(x− ))) dt dx ∂K∩ΓD I n tn + w(tn+ ) · [u(v)]t+ n dx − K,n=0 + w(t0+ ) · (u(v(t0− )) − u0 ) dx K,n=0

Space-Time Error Representation and Estimation in CFD

43

x

where κ is an O(1) parameter, · x+ − ≡ ((·)x− + (·)x+ )/2 and g∞ , gN and gD denote farﬁeld, Neumann, and Dirichlet data, respectively. This formulation immediately gives a convenient error representation formula of the same form as (15) N −1 J(u(vh )) − J(u(v)) = (28) BDG,Qn (vh , φ − πh φ) n=0 K

and an error estimate suitable for mesh adaptivity of the same form as (16) |J(u(vh )) − J(u(v))| ≤

N −1 n=0 K

|BDG,Qn (vh , φ − πh φ)| .

(29)

4.1 Numerical Example: Time-Dependent Compressible Navier-Stokes 2D Cylinder Flow at Re = 300 The accurate prediction of lift and drag forces on a cylinder undergoing vortex shedding remains an important and diﬃcult test problem in numerical timedependent Navier-Stokes simulations [23–25]. In the present study, subsonic ﬂow at Mach=1/10 and Re=300 has been computed over a 2D cylinder for a length of time suﬃcient to establish periodic vortex shedding. A numerical solution suitable for assessing error estimates and adaptation was computed using 12K linear space-time elements and a time step size resulting in approximately 48 time steps per oscillation period. Another extremely accurate “reference” solution was also computed using 40K quadratic space-time elements and 50% reduced timestep, see Fig. 4. A measurement functional was then devised consisting of the the nondimensional drag force integrated over the cylinder surface and averaged over

Fig. 4. Navier-Stokes solution at the non-dimensional time t = 635 computed on the reference 40K element mesh using P2 space-time elements. Presented here are velocity contours (left) and logarithmically scaled vorticity magnitude contours (right). (See Plate 9 on page 417)

44

Timothy J. Barth

four drag oscillation periods, t ∈ [605, 700], determined from the reference solution with time non-dimensionalized here using freestream sound speed and cylinder diameter. tb 1 D J(u) = Fdrag (u) dx dt . (30) 2 t −t 1/2ρ∞ V∞ b a ta cyl wall Following the same procedure as outline earlier in Sect. 3.3, a dual solution corresponding to the time-averaged drag functional was obtained on the 12K mesh using quadratic space-time elements. This solution serves as an approximation of the inﬁnite-dimension dual solution φ. Figure 5 provides a time snapshot of the quadratic space-time element dual solution, φh , and the defect dual solution, φh − πh φh , where πh is the L2 space-time projection from quadratic elements to linear elements. A rather striking feature of the dual solution as is illustrated in Fig. 5 is the localized support of the defect φ − πh φ when compared to φ itself. Since only the defect φ − πh φ appears in the error representation formula (28), the resulting mesh adaptivity becomes very localized as well. Although it is possible to adapt the mesh during every time step of the calculation based on error indicators derived from (29), in the present calculations the mesh indicators have been averaged over four oscillation periods so that a single (static) mesh adaptation is performed, see Fig. 6. Figure 7 shows the lift and drag coeﬃcient histories on all meshes. A comparison of drag coeﬃcient histories shown in this ﬁgure reveals a dramatic improvement in oscillation amplitude and mean value with only modest mesh adaptivity. Table 2 quantiﬁes the accuracy of the computed functional error using the 40K element reference solution as the “exact” solution together with the estimated values from (28) and (29). The tabulated results show some lack of

Fig. 5. Locally linearized dual Navier-Stokes solution corresponding to the drag coeﬃcient functional with local linearization about P1 primal space-time data on a 12K mesh. Presented here are dual solution contours at the same approximate nondimensional time as Fig. 4 showing the x-momentum component of the dual solution φ (left) and the x-momentum component of the dual solution defect φ − πh φ (right). (See Plate 10 on page 417)

Space-Time Error Representation and Estimation in CFD

45

Fig. 6. Closeup of the original 12K element mesh (left) and reﬁned 15K element mesh (right) obtained from error indicators averaged over four oscillation periods.

1.45

1 Drag Coefficient

Lift and Drag Coefficients

2

0

1.35

−1 Drag, 12K linear elements Drag, 15K adapted linear elements Drag, 40K quadratic elements

Lift & drag, 12K linear elements Lift & drag, 15K adapted linear elements Lift & drag, 40K quadratic elements −2

0

200

400 Non−dimensional time

600

1.25 605

655 Non−dimensional time

Fig. 7. Lift and drag coeﬃcient history for 2D Navier-Stokes cylinder ﬂow at Re=300 and Mach=0.1 using 12K P1 space-time elements, 15K P1 adapted spacetime elements and 40K P2 space-time elements. Shown are the lift and drag coeﬃcient histories (left) and an extreme closeup of the drag coeﬃcient histories in the functional interval (right). Table 2. Error estimates for the 2D cylinder on meshes with 12K and 15K adapted elements. #elements |J(u) − J(uh )| 12K .0107 15K (adapted) .0020

Eqn. (28) .0183 .0024

Eqn. (29) .1098 .0139

sharpness in the computed error representation formula (28) that would be further ampliﬁed in the estimate (29). The improvement with adaptive mesh reﬁnement may indicate that this source of error originates from linearization error resulting from the use of the Jacobian linearization as a replacement for the mean-value linearization in (11) and (12). Another source of error may originate from the speciﬁcation of the four period averaging interval in the

46

Timothy J. Barth

functional deﬁnition. These issues will be addressed in future expanded work for this problem.

5 Concluding Remarks As one might expect, the extension of a-posteriori error estimation to spacetime is almost transparent for ﬁnite element methods that permit discretization in time using the same ﬁnite element method. Even so, the implied added computation is signiﬁcant since for time-dependent problems a backwards in time dual problem must be solved or approximated. Many open problems exist in this problem area: • Eﬃcient techniques for approximating the dual in space-time problem. • Computable functionals for time-dependent problems that do not lead to dual problems that become large or unbounded in reverse time. • Techniques for estimating the error in general Lp norms.

References [1] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson. Introduction to numerical methods for diﬀerential equations. Acta Numerica, pages 105– 158, 1995. [2] R. Becker and R. Rannacher. Weighted a posteriori error control in FE methods. In Proc. ENUMATH-97, Heidelberg. World Scientiﬁc Pub., Singapore, 1998. [3] J. Hoﬀman and C. Johnson. Adaptive ﬁnite element methods in incompressible ﬂuid ﬂow. In Barth and Deconinck, editors, Error Estimation and Adaptive Discretization Methods in CFD, volume 25 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Heidelberg, 2002. [4] W. H. Reed and T. R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos National Laboratory, Los Alamos, New Mexico, 1973. [5] P. LeSaint and P.A. Raviart. On a ﬁnite element method for solving the neutron transport equation. In C. de Boor, editor, Mathematical Aspects of Finite Elements in Partial Diﬀerential Equations, pages 89– 145. Academic Press, 1974. [6] M. Giles, M. Larson, M. Levenstam, and E. S¨ uli. Adaptive error control for ﬁnite element approximations of the lift and drag coeﬃcients in viscous ﬂow. preprint NA-97/06, Comlab, Oxford University, 1997. [7] M. Giles and N.A. Pierce. Improved lift and drag estimates using adjoint Euler equations. Technical Report 99-3293, AIAA, Reno, NV, 1999.

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[8] C. Johnson, R. Rannacher, and M. Boman. Numerics and hydrodynamics stability theory: towards error control in CFD. SIAM J. Numer. Anal., 32:1058–1079, 1995. [9] J. T. Oden and S. Prudhomme. Goal-oriented error estimation and adaptivity for the ﬁnite element method. Technical Report 99-015, TICAM, U. Texas, Austin,TX, 1999. [10] S. Prudhomme and J.T. Oden. On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comp. Meth. Appl. Mech. and Eng., pages 313–331, 1999. [11] E. S¨ uli. A posteriori error analysis and adaptivity for ﬁnite element approximations of hyperbolic problems. In Kr¨ oner, Ohlberger, and Rohde, editors, An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, volume 5 of Lecture Notes in Computational Science and Engineering, pages 122–194. Springer-Verlag, Heidelberg, 1998. [12] T. J. Barth and H. Deconinck(eds). Error estimation and adaptive discretization methods in CFD. In Barth and Deconinck, editors, Error Estimation and Adaptive Discretization Methods in CFD, volume 25 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Heidelberg, 2002. [13] M.G. Larson and T.J. Barth. A posteriori error estimation for adaptive discontinuous Galerkin approximations of hyperbolic systems. In Cockburn, Karniadakis, and Shu, editors, Discontinuous Galerkin Methods, volume 11 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Heidelberg, 1999. [14] R. Hartmann and P. Houston. Adaptive discontinuous Galerkin methods for the compressible Euler equations. J. Comp. Phys., 182(2):508–532, 2002. [15] C. Johnson and J. Pitk¨ aranta. An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp., 46:1–26, 1986. [16] B. Cockburn, S.Y. Lin, and C.W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin ﬁnite element method for conservation laws III: One dimensional systems. J. Comp. Phys., 84:90–113, 1989. [17] B. Cockburn and C.W. Shu. The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems. Technical Report 201737, Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley R.C., 1997. [18] T. J. Barth and M.G. Larson. A-posteriori error estimation for higher order Godunov ﬁnite volume methods on unstructured meshes. In Herbin and Kr¨ oner, editors, Finite Volumes for Complex Applications III, pages 41–63. Hermes Science Pub., London, 2002. [19] T. J. R. Hughes, L. P. Franca, and M. Mallet. A new ﬁnite element formulation for CFD: I. symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Comp. Meth. Appl. Mech. Engrg., 54:223–234, 1986.

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[20] J. Douglas and T. Dupont. Interior penalty procedures for elliptic and parabolic galerkin methods. In Lecture Notes in Physics, volume 58 of Lecture Notes in Physics. Springer-Verlag, Heidelberg, 1976. [21] R. Hartman and P. Houston. Symmetric interior penalty DG methods for the compressible Navier-Stokes equations I: Method formulation. Int. J. Numer. Anal. Model., to appear. [22] R. Hartman and P. Houston. Symmetric interior penalty DG methods for the compressible Navier-Stokes equations II: Goal-oriented a posteriori error estimation. Int. J. Numer. Anal. Model., to appear. [23] C. Farell and J. Blessman. On critical ﬂow around smooth circular cylinders. J. Fluid Mech., 136, 1983. [24] R. Mittal and S. Balachandar. Eﬀect of three-dimensionality of the lift and drag on nominally two-dimensional cylinders. Physics of Fluids, 7(8):1841–1865, 1995. [25] C. Norberg. Eﬀects of Reynolds Number and Low-Intensity Free-Stream Turbulence on the Flow Around Circular Cylinders. PhD thesis, Department of Applied Thermosciences and Fluid Mechanics, Chamers University of Technology, Sweden, 1987.

Multiresolution Particle Methods Michael Bergdorf and Petros Koumoutsakos Computational Science and Engineering Laboratory, ETH Z¨ urich, Switzerland [email protected], [email protected]

Summary. We present novel multiresolution particle methods with extended dynamic adaptivity in areas where increased resolution is required. The inherent Lagrangian adaptivity of smooth particle methods is complemented by adaptation of the particle size based on criteria such as ﬂow strain and wavelet-based decomposition. In this context we present two particle multiresolution techniques: one based on globally adaptive mappings and one employing a wavelet-based multiresolution analysis of the transported quantities to guide the allocation of computational elements. Results are presented from the application of these methods to level sets and two-dimensional vortical ﬂows.

1 Introduction Most problems in ﬂuid dynamics, interface capturing, quantitative biology and physics are inherently dynamic and multiscale and the methods employed to simulating these problems are faced with the problems of adaptivity and a compromise between adequate resolution and the employment of models that describe the eﬀect of the non-resolved scales on the numerically represented scales. Particle methods present an adaptive, eﬃcient, stable and accurate computational method for simulating transport phenomena ranging from the evolution of vortex ﬁelds to the evolution of level sets. In particle methods the transported continuum quantities are discretized using smooth particles with a ﬁnite core size. In smooth particle methods, in contrast with point particle methods, each particle is associated with a smooth core function. Field quantities can always be reconstructed as a linear superposition of neighboring particles. Particle methods discretize the Lagrangian form of the governing equations, circumventing the stability issues associated with the Eulerian discretization of convective terms while introducing very little numerical dissipation. The accuracy of the method is determined by the size of the particle and by the – often overlooked – requirement that particles overlap. As particles follow

50

Michael Bergdorf and Petros Koumoutsakos

the streamlines of the ﬂow map their locations eventually become distorted and the ﬁeld quantities cease to be well represented by the particles. The requirement for particle overlap is closely linked with the accuracy of the methods and it introduces the need for a periodic regularization of particle properties. The introduction of a mesh detracts to a certain degree from the meshless character of particle methods. A number of works (see [1] and references therein ) have demonstrated that it is not possible to construct grid-free particle methods and expect an apriori speciﬁed degree of accuracy. The introduction of a mesh to regularize the particle properties does not detract from the adaptive character of the method. On the contrary, it enables the development of particle methods that can take advantage of advances made in mesh-based schemes such as mesh embedding and multiresolution to provide a robust and accurate computational tool for the simulation of diverse physical systems. Using an underlying mesh also allows to couple particle methods with grid-based solvers, in particular for ﬁeld calculations, and enables the introduction of multiresolution based adaptivity. In particle methods, as in grid-based methods with uniform grids, resolving all scales with uniformly sized particles can become prohibitively computationally expensive. It is hence necessary to employ particles with diﬀerent resolution to eﬃciently represent the diﬀerent scales of the physics of the problem. The convergence of particle methods with variable core sizes was proven by Hou [2] for the Euler equations. This proof was extended to the viscous case by Ploumhans & Winckelmans [3]. In [4], Cottet, Koumoutsakos, and Ould-Salihi employed mappings from a reference space with uniform core sizes to a “physical” space with cores of varying size. In our recent work [5] we have extended this method by introducing two dynamically adaptive particle methods. The ﬁrst method is based on adaptive global mappings (AGM); it uses a ﬁnite-dimensional evolving mapping from reference to the physical space, distributing the multiscale character of the physics among the ﬁeld quantities represented by the particles and the mapping. The second method is based on adaptive mesh reﬁnement (AMR): it consists of distributing local reﬁnement patches, on which ﬁner sized particles are created, in areas where small scales are present. The ﬁne and coarse particles exchange information through a consistent remeshing scheme. While both methods have displayed improved accuracy, their adaptation mechanisms do not inherit the Lagrangian character of the underlying particle method, i.e. the distribution of numerical scales is not transported in a Lagrangian way, but for example in the AGM, by a nonlinear diﬀusion equation. Note that this does not detract from the stability or the Lagrangian character of the adaptive particle method itself, but accuracy (for the AGM) and eﬃciency (for the AMR) limit the time step to C δx u −1 ∞ where C is a constant. One part of the present work will therefore be concerned with the introduction of an adaptation mechanism that handles the distribution of numerical scales in a Lagrangian fashion.

Multiresolution Particle Methods

51

For more than a decade, wavelets have been used to eﬃciently solve PDEs that exhibit multiscale solutions. Two diﬀerent classes of wavelet-based methods for solving PDEs can be distinguished: wavelet-Galerkin type methods [6] which use a weak formulation approach, and wavelet-collocation methods [7]. Both classes have been applied to problems from electromagnetism, acoustics and ﬂuid mechanics, e.g. the two-dimensional (2D) simulations of incompressible ﬂow around bluﬀ bodies; see Schneider & Farge [8] for the wavelet-Petrov-Galerkin approach and Kevlahan & Vasilyev [9] for waveletcollocation. In our present work we employ a wavelet-collocation scheme to guide the adaptation of the grid and the particles. We choose this approach over the AMR-based reﬁnement introduced in [5] as it oﬀers a natural scale decomposition - or multiresolution analysis, and does not rely on heuristics to guide this decomposition.

2 Approximations using Particles The development of particle methods is based on the integral representation of functions and diﬀerential operators. The integrals are discretized using particles as quadrature points. 2.1 Function Approximation The approximation of continuous functions by particle methods starts with the equality q(x) ≡ q(x − y) δ(y) dy . (1)

Using N particles we discretize above equality by numerical quadrature and get the “point-particle” approximation of q: q h (x) = Qp (t) δ(x − xp ) . (2) p

Point particle methods based on the approximation (2) yield exact weak solutions of conservation laws. A drawback of point particle approximations is that the function q h can only be reconstructed on particle locations xp . This shortcoming is addressed by mollifying the Dirac delta function in (1) resulting on a molliﬁed approximation: ε q (x) = q(x − y) ζ ε (y) dy , (3) where ζ ε = ε−d ζ(x/ε), x ∈ Rd , and ε being a characteristic length scale of the kernel. For consistency of the approximation the kernel ζ has to fulﬁll the following moment conditions:

52

Michael Bergdorf and Petros Koumoutsakos

ζ xα dx = 0α for 0 ≤ |α| < r .

(4)

The kernel ζ is of order r and the following error bound holds: q − q ε ≤ Cεr q ∞ .

(5)

Now again, we get a discrete but smooth function approximation by approximating the integral in (3) by a mid point quadrature rule yielding Qp ζ ε (x − xp ) , (6) q ε,h (x) = p

The error of (6) can be assessed by splitting q − q ε,h into q − q ε,h ≤

q − q ε +

q ε − q ε,h m ≤ C1 εr q ∞ + C2 hε q ∞ .

(7)

We conclude from this, that (h/ε) must be smaller that 1, i.e. smooth particles must overlap1 . 2.2 Diﬀerential Operator Approximation In smooth particle methods diﬀerential operators can be approximated by discrete integral operators. Degond & MasGallic developed an integral representation of the diﬀusion operator - isotropic and anisotropic - which was later extended to diﬀerential operators of arbitrary degree in [11]. The integral operator for the 1D Laplacian for instance takes the form 1 [q(y) − q(x)] η ε (x − y) dy , (8) ∆ε q = 2 ε

where the kernel η(x) has to fulﬁll x2 η(x) dx = 2. This integral is discretized by particles using their locations as quadrature points: ε,h ∆ q (xp′ ) = ε−2 (9) [qp − qp′ ] η ε (xp′ − xp ) vp . p

3 Solving Transport Problems with Particle Methods Particle methods discretize the Lagrangian form of the governing equation, ∂q + ∇ · (u q) = L(q, x, t) , ∂t 1

(10)

for certain kernels, an r-th order approximation can be achieved even with ε = h [10]

Multiresolution Particle Methods

53

resulting in the following set of ODEs: dxp = u(xp , t) , dt dvp = vp (∇ · u) (xp , t) , dt dQp = vp Lε,h (q, xp , t) . dt

positions volumes

(11)

weights

Particle positions are usually initialized as a regular lattice with spacing h, volumes are thus initially set to vp = hd and Qp = qo (xp ) hd . The ODES (11), are now advanced using a standard explicit time stepper and the transported quantity q can be reconstructed as q(x, t) = Qp (t) ζ ε (x − xp (t)) . (12) p

However, as the particles follow the ﬂow map u(x, t) their positions eventually become irregular and distorted, and the function approximation (12) ceases to be well-sampled. To ascertain convergence, it is therefore necessary to periodically regularize the particle locations; this process is called “remeshing”. 3.1 Remeshing Remeshing involves interpolation of particle weights from irregular particle locations onto a regular lattice. New particles are then created on the lattice, replacing the old particles. This interpolation process takes the form = (13) W (xp′ − xp ) Qold Qnew p′ p , p

where Qnew are the new particle weights, and xp′ are located on a regular p′ lattice. The interpolation function W (x) is commonly chosen to be a tensor product of one-dimensional interpolation function which for accuracy have to be suﬃciently smooth and moment-conserving. The M4′ function [12] is commonly used in the context of particle methods; it is in C 1 (R) and of third order. The introduction of a grid clearly detracts from the meshless character of particle methods. The use of a grid in the context of particle methods does not restrict the adaptive character of the method and provides the basis for a new class of “hybrid” particle methods with several computational and methodological beneﬁts 3.2 Hybrid Particle Methods The introduction of a grid enables fast evaluation of diﬀerential operators using compact PSE kernels, enables the use of fast grid-based Poisson solvers

54

Michael Bergdorf and Petros Koumoutsakos

[13], facilitates parallelization and is a key component in adaptive particle methods, which we will present in section 4. Hybrid particle methods make heavy use of these computational advantages [1, 14, 15]. Recently, we have developed a generic hybrid particle method framework [16], enabling eﬃcient, parallel simulations of large-scale transport problems as diverse as the DNS of turbulent ﬂows and diﬀusion processes in complex biological organelles. Figure 1 shows visualizations of the Crow instability and the elliptic instability of two counter-rotating vortex tubes employing a maximum of 33 million particles. The simulations were performed on a 16 cpu Opteron cluster. One time step for 1 million particles took less than 30 seconds. Current implementations using the fast multipole method which retain the meshless character of the particle method require approximately 2400 seconds per time step [17]. This clearly demonstrates the advantages of hybrid methods.

Fig. 1. Crow (left) and short-wave or elliptic instability (right). (See Plate 11 on page 418)

4 Adaptive Particle Methods The accuracy of smooth particle methods is determined by the core size ε of the kernel ζ ε (x). For computational eﬃciency this core size needs to be spatially variable to resolve small scales in diﬀerent parts of the ﬂow, such as the boundary layer and the wake of bluﬀ body ﬂows. As particles need to overlap, varying core sizes imply spatially varying particle spacings. This can be achieved in two ways: • remeshing particles on a regular grid corresponding to variable size particles in a mapped using a global (adaptive) mapping • remeshing particles by combining several simple local mappings in a domain decomposition frame. In the context of vortex methods, Hou [2] ﬁrst introduced spatially varying particle sizes and proved the convergence of the method in the case of the 2D Euler equations. This proof was extended in [3] to the viscous case and the

Multiresolution Particle Methods

55

method was used for the simulation of wakes with stretched particle resolution. In [4] Cottet, Koumoutsakos, and Ould-Salihi formulated a convergent variable core size vortex method for the Navier-Stokes equations by using ˆ ⊆ Rd with uniform core size εˆ to the mappings from a reference space Ω d “physical” space Ω ⊆ R with cores of varying size ε(x) in conjunction with an anisotropic diﬀusion operator, i.e. ˆ , x = f (x)

ˆ = g(x) , x

Φ

=

ij

∂x ˆi ∂xj

and

|Φ| = det Φ

(14)

Like in the uniform core size method (11), we convect the particles in physical space, but diﬀusion is performed in reference space, so that with N particles, ˆ j )}N located in {xj (t)}N j=1 = {f (x j=1 we ﬁnd an approximate solution to (10) by integrating the following set of ODEs: dxj = u(xj , t) , dt m (x ˆk ) ˆ ν ǫˆ dQj pq ˆ j ) + mpq (x ˆj ] , ˆj − x ˆk ) [ˆ vj Qk − vˆk Q = 2 ψpq (x dt ǫˆ 2

(15)

k

dˆ vj ˆ · Φu (xj , t) vˆj . =∇ dt

ˆ j denote the particle strength in physical and In the above equation Qj and Q reference space, respectively, related by ˆ j = Qj |Φ|(xj ) . Q In [4] analytic, invertible mappings have been employed. Albeit being a simple and robust way to eﬃciently resolve the range of length scales in the ﬂow, this method requires prior knowledge about the ﬂow physics. In [5] we extended this method by introducing two diﬀerent approaches to dynamical adaptivity in particle methods; One approach makes use of a global adaptive mapping (AGM, see section 4.1), and one employing dynamically placed patches of smaller sized particles, reminiscent of adaptive mesh reﬁnement in ﬁnite volume methods (AMR). 4.1 Particle Method with Adaptive Global Mappings ˆ × [0, T ] → Ω represented by We introduce a transient smooth map f : Ω particles: ˆ t) = f (x, ˆ t) = ˆ − ξj ) , x(x, χj (t) ζ εˆ(x (16) j

where ξp are ﬁxed at grid point locations. The parameters in the map that are changed in the process of adaptation are the node values χj . As the map (16) is not easily invertible, we require it to be smooth in both space and

56

Michael Bergdorf and Petros Koumoutsakos

time. Given this property, the governing equation (10) can be entirely cast into reference space, again yielding a transport equation: ∂ qˆ′ ˆ q ′ , x, ˆ · (ˆ ˜ = L(ˆ ˆ t) , +∇ q ′ u) ∂t

(17)

where qˆ′ = (|Φ|)−1 qˆ and ˜ = Φ(u ˆ − U) , u

and

U=

∂χj ∂f ˆ − ξj ) . = ζ εˆ(x ∂t ∂t j

(18)

What remains is to chose a U , such that particle core sizes in physical space are small where small scale features are present in the ﬂow. In [5] this was accomplished by setting U to be the solution of a moving mesh partial diﬀerential equation (MMPDE), ˆ · M (x, ˆ (x, ˆ t)∇f ˆ t)T , (19) U =∇

ˆ t) is a so-called monitor function: a positive measure which takes where M (x, great values where numerical resolution should be increased, e.g. ˆ t) = 1 + α|Bqˆ|2 , M (x, (20)

B being a high-pass ﬁlter. We applied this method in [5] to the evolution of an elliptical vortex governed by the 2D Euler equations. Figure 2 depicts the adaptation of the underlying grid, and thus the particle core sizes ε(x).

4.2 Wavelet-Based Multiresolution Particle Method We employ a wavelet-based multiresolution analysis (MRA) using L + 1 levels of reﬁnement to guide the creation of particles on the grid. The function q(x, t) can be represented as

Fig. 2. Simulation of the evolution of an inviscid elliptical vortex using the AGM particle method: vorticity (left), particle sizes (middle, dark areas represent coarse particle sizes) and grid (right). (See Plate 12 on page 418)

Multiresolution Particle Methods

qL =

qk0 ζk0 +

k

dlk ψkl ,

57

(21)

0≤l
where qk0 are the weights, ζk0 are the scaling functions (or kernels) on the coarsest level, dlk are the so-called detail coeﬃcients and ψkl are the wavelets. The MRA here is based on an iterative interpolation scheme as introduces by Deslauriers and Dubuc [18], thus we do not have explicit scaling functions ζ and wavelets ψ. As illustrated in Figure 3 each detail coeﬃcient is associated y dl,2 k dl,3 k dl,1 k clk

x

dl,m k ,

Fig. 3. Each detail coeﬃcient with m = 1, . . . , 2d − 1 corresponds to a speciﬁc grid point on the next higher level.

with a grid point on the next ﬁner grid. Let child(k, m) be the grid point associated with dl,m and let ancs(k) denote the set of grid points k′ needed k to interpolate the value qkl from values qkl−1 and detail coeﬃcients dl−1 ′ k′ of the next coarser level. Then an adapted grid is constructed by discarding all those grid points whose |dl,m k | are smaller as a prescribed threshold, i.e. ! " K> = K0 ∪ k′ = child(k, m) ∪ ancs(k′ ) |dl,m k | > ε , l ∈ [0, L − 1] . (22)

Note that K0> ≡ K0 and that ancs(k′ ) are added to maintain proper nestedness of the grids (see for instance [19] for details). In order to be able to capture small scales that may emerge between two subsequent MRAs we follow the conservative approach of Liandrat and Tchamitchian [6] and additionally activate all children of the active grid points, i.e. Kl> ← Kl> ∪ { child(k, m) | k ∈ Kl−1 > , m = 1, 2, 3 } for l = L, . . . , 1 . (23) Multilevel remeshing interpolates particles created on a set of grid points Kl> onto a set of grid points Kl× . This is accomplished in the following way: let M denote the kernel used for remeshing the particles, then (i) horizontally extend the set of source grid points Kl> by Bl , where " ! (24) Bl = k′ min |k′ − k| ≤ ⌈ 12 supp(M ) + LCFL⌉ , k∈Kl>

where the “Lagrangian CFL” LCFL ≡ δt ∇ ⊗ u ∞ , (ii) create particles on Kl> ∪ Bl1 , i.e.

58

Michael Bergdorf and Petros Koumoutsakos

Qlp = clk (hl )d ,

vpl = (hl )d ,

xlp = xlk ,

(iii) after convection, interpolate these particles onto a new set of grid points Kl× . Clearly, for consistency Kl× cannot be chosen arbitrarily. We propose the following method: Introduce and indicator function χl deﬁned as # 1 , k ∈ Kl> l χk = (25) 0 , k ∈ Bl , and convect the particles, i.e. solve the following set of equations dχlp dxlp dvpl dQlp = L(q, x, t) , = 0, = u(xlp , t) , = vpl (∇ · u) (xlp , t) . (26) dt dt dt dt The particle weights and the indicator are then interpolated onto the grid ˜l denotes and grid points with χ ˜lk > 0 are selected to constitute Kl× , where χ the remeshed indicator function. Using this technique, the scale distribution {Kl> }L l=0 is naturally convected with the ﬂow and we obtain and adaptation mechanism which is independent of the CFL number. To demonstrate the Lagrangian character of the adaptation we considered the convection of a passive scalar in 2D, subject to a vortical velocity ﬁeld [20]. The problem involves strong deformation of a initial circular “blob” which at the end of the simulation returns to the initial condition. Figure 4 illustrates the adaptation of the grid/particles at two diﬀerent times. We measure the L2 and L∞ error of the ﬁnal solution for diﬀerent choices of ǫ and observe second order convergence, corresponding to fourth order convergence in h, as depicted in Figure 6. The maximum CFL measured during the course of the simulation was 40.7.

Fig. 4. Active grid points/particles at two diﬀerent times of the simulation of a passive scalar subject to a single vortex velocity ﬁeld.

Fig. 5. Active grid points/particles at two diﬀerent times of the simulation of a propagating interface subject to a single vortex velocity ﬁeld.

We also applied the presented method to the simulation of a propagating interface using a level set formulation. A “narrow band” formulation is easily accomplished with the present method by truncating the detail coefﬁcients that are far from the interface. We consider the well-established 2D

Multiresolution Particle Methods

59

deformation test case which amounts to the propagation of a circle subject to the same velocity ﬁeld as above. Figure 5 depicts the grid adaptation and comparing to Figure 4, one can clearly see the restriction of the reﬁnement to a small neighborhood around the interface. We measure the error of the area encompassed by the interface at the ﬁnal time and compare it against a nonadaptive particle level set method [21] and against the “hybrid particle level set method” of Enright et al. [22]. Figure 7 displays this comparison and we ﬁnd that our adaptive approach performs favorably, which may be attributed in part to the adaptive character and in part to the high order of the method.

-1

e∞ , eL2

10

-2

10

-3

10

-4

10

-5

10

3

10

4

10

N

5

10

Fig. 6. ε-reﬁnement study; the data points correspond to ε = 2−p × 10−3 for p = 0 , . . . , 10. The triangle represents 2nd-order convergence. N is the number of active grid points/particles.

Fig. 7. Plot of relative error of the area enclosed by the interface against degrees of freedom: Hieber , parti& Koumoutsakos [21] ( cles at time t=0), Enright et al.[22] , auxiliary particles at time ( , grid points) and t=0 and , active grid present method ( , active grid points at time t=0, points at the ﬁnal time).

5 Conclusions We present multiresolution particle-mesh methods for simulating transport equations. We outline two methods introducing enhanced dynamic adaptivity and multiresolution capabilities for particle methods. The ﬁrst method is based on an adaptive global mapping from a reference space to physical space for the particle locations; it has been successfully applied to the evolution of an elliptical vortex in an inviscid incompressible ﬂuid. The second method is based on a wavelet multiresolution decomposition of the particle function

60

Michael Bergdorf and Petros Koumoutsakos

representation. It is equipped with a Lagrangian adaptation mechanism that enables the simulation of transport problems and interface capturing problems independent of the CFL number. We have presented results of an interface tracking problem where the method has shown to have superior volume conservation properties. We are currently working on the application of this method to the Navier-Stokes equations.

References [1] P. Koumoutsakos. Multiscale ﬂow simulations using particles. Annual Review of Fluid Mechanics, 37(1):457–487, 2005. [2] T. Y. Hou. Convergence of a variable blob vortex method for the Euler and Navier-Stokes equations. SIAM Journal on Numerical Analysis, 27(6):1387–1404, 1990. [3] P. Ploumhans and G. S. Winckelmans. Vortex methods for highresolution simulations of viscous ﬂow past bluﬀ bodies of general geometry. Journal of Computational Physics, 165:354–406, 2000. [4] G.-H. Cottet, P. Koumoutsakos, and M. L. Ould Salihi. Vortex methods with spatially varying cores. Journal of Computational Physics, 162:164– 185, 2000. [5] M. Bergdorf, G.-H. Cottet, and P. Koumoutsakos. Multilevel adaptive particle methods for convection-diﬀusion equations. Multiscale Modeling and Simulation, 4(1):328–357, 2005. [6] J. Liandrat and P. Tchamitchian. Resolution of the 1D regularized Burgers equation using a spatial wavelet approximation. ICASE Report 90-83, NASA Langley Research Center, 1990. [7] A. Harten. Adaptive multiresolution schemes for shock computations. Journal of Computational Physics, 115:319–338, 1994. [8] K. Schneider and M. Farge. Adaptive wavelet simulation of a ﬂow around an impulsively started cylinder using penalization. Applied and Computational Harmonic Analysis, 12:374–380, 2002. [9] N. K.-R. Kevlahan and O. V. Vasilyev. Ad adaptive wavelet collocation method for ﬂuid-structure interactions at high Reynolds numbers. SIAM Journal on Scientiﬁc Computing, page submitted, 2005. [10] A.-K. Tornberg and B. Engquist. Numerical approximations of singular source terms in diﬀerential equations. Journal of Computational Physics, 200:462–488, 2004. [11] J. D. Eldredge, A. Leonard, and T. Colonius. A general deterministic treatment of derivatives in particle methods. Journal of Computational Physics, 180:686–709, 2002. [12] J. J. Monaghan. Extrapolating b-splines for interpolation. Journal of Computational Physics, 60:253–262, 1985. [13] F. H. Harlow. Particle-in-cell computing method for ﬂuid dynamics. Methods in Computational Physics, 3:319–343, 1964.

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[14] J. H. Walther and P. Koumoutsakos. Three-dimensional particle methods for particle laden ﬂows with two-way coupling. Journal of Computational Physics, 167:39–71, 2001. [15] G.-H. Cottet and P. Poncet. Advances in direct numerical simulations of 3D wall-bounded ﬂows by vortex-in-cell methods. Journal of Computational Physics, 193:136–158, 2003. [16] I. F. Sbalzarini, J. H. Walther, M. Bergdorf, S. E. Hieber, E. M. Kotsalis, and P. Koumoutsakos. PPM – a highly eﬃcient parallel particle-mesh library. Journal of Computational Physics, 215(2):566–588, 2006. [17] Q. X. Wang. Variable order revised binary treecode. Journal of Computational Physics, 200(1):192 –210, 2004. [18] G. Deslauriers and S. Dubuc. Symmetric iterative interpolation processes. Constructive Approximation, 5:49–68, 1989. [19] O. V. Vasilyev. Solving multi-dimensional evolution problems with localized structures using second-generation wavelets. International Journal of Computational Fluid Dynamics, 17(2):151–168, 2003. [20] R. J. Leveque. High-resolution conservative algorithms for advection in incompressible ﬂow. SIAM Journal on Numerical Analysis, 33(2):627– 665, 1996. [21] S. E. Hieber and P. Koumoutsakos. A Lagrangian particle level set method. Journal of Computational Physics, 210(1):342–367, 2005. [22] D. Enright, R. Fedkiw, J. Ferziger, and I. Mitchell. A hybrid particle level set method for improved interface capturing. Journal of Computational Physics, 183(1):83–116, 2002.

LES Computation of Lagrangian Statistics in Homogeneous Stationary Turbulence; Application of Universalities under Scaling Symmetry at Sub-Grid Scales Mikhael Gorokhovski1 and Anna Chtab2 1

2

LMFA / UMR 5509 CNRS / Ecole Centrale de Lyon 36 avenue Guy de Collongue 69131 Ecully cedex, France [email protected] CORIA / UMR 6614 CNRS / University of Rouen Site Universitaire du Madrillet BP12 76801 St Etienne du Rouvray cedex, France [email protected]

Summary. In this paper, the turbulent cascade with intermittency is presented in the framework of universalities of eddy fragmentation under scaling symmetry. Based on these universalities, the stochastic estimation of the velocity increment at sub-grid scales is introduced in order to simulate the response of light solid particle to inhomogeneity of the ﬂow at small spatial scales. The LES of stationary box turbulence was performed, and the computed Lagrangian statistics of tracking particle was compared with measurements. The main eﬀects from recent experimental study of high Reynolds number stationary turbulence are reproduced by computation. For the velocity statistics, the numerical results were in agreement with classical Kolmogorov 1941 phenomenology. However the distribution of velocity increment, computed at diﬀerent time lag, revealed the strong intermittency: at time lag of order of integral time scale, the velocity increment was distributed as Gaussian, at small time lags this distribution exhibited the long stretched tails.

1 Introduction This paper reports the results of computation of Lagrangian statistics of light solid particle in a high-Reynolds number stationary turbulence. The study was motivated by recent experimental observations of Mordant et al. [1]. In this experiment, the solid polystyrene sphere (d = 250 µm; τSt = 3.67 ms; ρp = 1.06 g/cm3 ) was immersed into turbulent ﬂow of water. The ﬂow was generated by two counter-rotating discs. Remote from solid boundaries, such a ﬂow was considered as a model of homogeneous isotropic turbulence, in which, across the inertial range of scales, the Lagrangian velocity variation of particle, was studied at Reλ = 570 − 1200 (λ is Taylor micro-scale). The

64

Mikhael Gorokhovski and Anna Chtab

measurements showed that the velocity auto-correlation function and the time spectrum agreed to Kolmogorov, 1941 phenomenology. However, distribution of the velocity increment revealed the strong intermittency of the ﬂow at small scales: the periods of weak velocity increment alternated with moments, when the particle was subjected to intense accelerations. Along with fundamental interest, the eﬀects of intermittency can be also of crucial importance in many engineering applications, including rocket engine, diesel, HCCI engine and LPP gas turbines. In these applications, the Reynolds number is high and, consequently, the ﬂow is very intermittent at small scales. Then the strong local gradient of velocity may induce the large ﬂuctuations in distribution of droplets, variations of their size at each spray location, ﬂuctuations in vaporization rate, “spontaneous” ignition or extinction sites, etc. In order to predict these eﬀects, one needs to compute correctly the particle/droplet dynamics. During the past years, it has been recognized that in computation of highReynolds turbulent gas ﬂow, the large eddy simulation (LES) approach provides accurate local estimates of statistical quantities [2]. Recently, the grouptheoretical model of turbulence proposed in [3], allowed to formulate this approach in the frame of renormalization-group invariance of averaged turbulent ﬁelds (by this model, the turbulent viscosity appeared not as a result of averaging of the nonlinear term in the Navier-Stokes equation, but from the molecular viscosity term with the help of renormalization-group transformation). Applying LES of turbulent ﬂow with immersed heavy particles, one can expect that the dynamics of such particles can be predicted accurately. However when the particle is light, its response to the ﬂow may be controlled by the length (or time) scales, which are not resolved. Such interaction needs to be modeled. Presuming universality of the ﬂow structure at unresolved scales is one of the way in the modeling of this interaction. A number of subgrid closures were proposed in the literature: the similarity model [4]; stretched vortex model [5]; unresolved scales estimator [6]; linear eddy model [7]; fractal model [8] (a comprehensive review of these models is given in [8]). These and others models (subgrid Langevin model [9], combination with DNS [10], deﬁltering procedure [11]) allowed to retrieve the subgrid information from resolved velocity ﬁeld and to improve the computation of particle motion in the framework of LES approach [12–14]. In this paper, the velocity “seen” by particle is also reconstructed using resolved velocity ﬁeld. However, our approach diﬀers from the above-mentioned models in two aspects. First, the long range interaction in the spectrum of turbulent scales is introduced, allowing to transfer the energy directly from resolved scale to smallest one (which is the case when Re is high). Second, the fractal properties of subgrid velocity ﬁeld are not presumed statically but appear progressively when the turbulent scale gets smaller than resolved one. This was formulated as turbulent cascade under scaling symmetry. The stochastic model based on universalities, appearing in such fragmentation, is introduced in order to simulate the response of solid light particle to the ﬂow inhomogeneity at sub-grid

LES Computation of Lagrangian Statistics

65

scales. Along with LES computation, can this model match the observations from [1] - was the question raised before computation.

2 Universalities of Fragmentation under the Scaling Symmetry and Turbulent Cascade with Intermittency Fragmentation plays an important role in a variety of physical, chemical, and geological processes. Although each individual action of fragmentation is a complex process, the number of these elementary actions per unit time is large. In this situation, it is natural to abstract a simple scenario of fragmentation and to represent its essential features. One of the model is the fragmentation under the scaling symmetry. Here each breakup action reduces the typical length of fragments, r ⇒ αr, by an independent random multiplier α, which is governed by the fragmentation intensity spectrum, say,

1 q(α), q(α)dα = 1. Introducing the normalized distribution of size, f (r, t)

∞

0

( f (r)dr = 1) evolving with time in such a fragmentation, the following 0

question raised: how does evolve this distribution to the ultimate steady-state solution f (r) = δ(r) ? This question can not be completely answered since such evolution requires the knowledge of the spectrum q(α), which is principally unknown function. However, as it has been shown in [15], due to scaling symmetry r ⇒ αr, the evolution of f (r, t) goes, at least, through two intermediate asymptotic distributions. Evaluating these distributions does not require the knowledge of entire spectrum q(α) - only its ﬁrst two logarithmic moments (ﬁrst universality), and further only the ratio of these moments in the long-time limit (second universality), determine the shape of f (r, t). The ﬁrst universality leads to the Kolmogorov’s [16] log-normal distribution of f (r, t), and the second one veriﬁes to be power (i.e. fractal) distribution (so far, the delta-function distribution is never achieved). In turbulence, due to intermittency, the energy of large unstable eddy is transferred to the smaller one at ﬂuctuating rate. Here the fragmentation under scaling symmetry can be formulated as follows (according to Castaing’s et al. papers [17–21]): when the turbulent length scale r gets smaller, the velocity increment, δl v(x) = |v(x + l) − v(x)|, is changed by independent positive random multiplier, δl2 v = αδl1 v, with l2 ≤ l1 . In [22], making use the analogy between this last statement and the fragmentation under scaling symmetry, r ⇒ αr, the evolution of the velocity increment distribution along with decreasing of the length (or time) scale was formulated in terms of universalities of fragmentation under scaling symmetry. To this end, ﬁrstly, an evolution parameter τ = ln(LLES /l) of penetration towards smaller scales (LLES is typical scale of ﬁnite-diﬀerence cell and l is a progressively decreasing eddy scale) has been introduced; if l = LLES , τ = 0; if l → 0, τ → ∞. Such a parameter was earlier used in [23]. It stems from the Batchelor’s expression

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Mikhael Gorokhovski and Anna Chtab

for stretched length scales l(t) = L exp −γt. Secondly, instead of f (r, t), for the fragment length, the normalized distribution of the velocity increment f (δv, τ ) was introduced, and the evolution equation for f (δv, τ ) with growing τ was formulated. This equation governs the evolution from distribution f (δv, τl1 ) at scale l1 towards distribution f (δv, τl2 ≻ τl1 ) at smaller scale l2 . The long-time limit of this equation is the Fokker-Planck equation and its solution is ∞ (1) f (δv, τ ) = dβB(β, τ )f (βδv; τ0 ) 0

where f (βδv, τ0 ) is the initial distribution and (ln β+ < ln α > τ )2 1 B(β, τ ) = exp − 2 < ln2 α > τ 2π < ln2 α > τ

(2)

If the ﬂow is computed by LES approach, the velocity increment, δVLES , is resolved at LLES exactly and hence the initial distribution of the velocity increment (l = LLES ⇒ τ = 0) is the Dirac delta function, f (βδv; τ = 0) = δ(βδv − δVLES ). In this case, solution (1) yields < ln α >2 1 1 f (∆v, τ ) = exp − τ × ∆VLES 2π < ln2 α > τ 2 < ln2 α > 1− LES ln2 ∆V∆v ∆VLES × exp − (3) 2 ∆v 2 < ln α > τ When time progresses (the turbulent length scale gets progressively smaller), this expression shows the evolution from the log-normal distribution towards power function with growing stretched tails. The intermittency is manifested in the same way: highest strain appears in narrow (dissipative) regions of ﬂow. At largest times, only one universal parameter, < ln α > / < ln2 α >, determines (3). According to discussion given in [22], this parameter is taken here in the following form : < ln2 α > / < ln α >= ln(λT /LLES ), where λT is referred to as local Taylor micro-scale. The ﬁrst logarithmic moment we assumed as < ln α >= A · ln(λT /LLES ), which is equivalent to assumption that < ln2 α > is proportional to < ln α >2 . The unknown constant A was adjusted $ % $ %2 comparing the evolution of the excess in (1), K(τ ) = (δv)4 / (δv)2 − 3, with values measured in [1]. To perform this, the distribution, which was measured in [1] at integral turbulent time (Gaussian with variance of 1 m/s), has been introduced as initial distribution in Eq. (1). The value λT = 0.076 cm taken from [1], also was used. The best ﬁt was obtained when A = 0.008.

3 Computational Model In this work, the ﬁltered Navier-Stokes equations were integrated by numerical code developed in [24]. Implementing the forcing scheme from [25], this

LES Computation of Lagrangian Statistics

67

code was adopted for simulation of stationary homogeneous gaseous turbulence in 3D box with periodic conditions. Two test computations, decaying turbulence and stationary “synthetic” turbulence, are demonstrated in Fig. 1 and Fig. 2, respectively. Figure 1 shows the evolution of energy spectrum in a “free” turbulence computed on 323 grid points mesh at three times: t = 0; 0.284; 0.655 s. These distributions are compared with measurements of Comte-Bellot and Corrsin [26]. In the computation, the upper and lower boundaries in Fourier space were kmin = 0.1015 cm−1 and kmax = 3.147 cm−1 , respectively and the ﬂuid viscosity was 0.15 cm2 /s. For better ﬁtting of measured decay of turbulence, the Smagorinsky constant was taken 0.2 instead of 0.17. For the forced stationary turbulence, Fig. 2 shows the kinetic energy spectrum compared to the presumed spectrum from [25]. Here, the Reynolds number, the kinetic energy, the micro-scales of Kolmogorov and Taylor were chosen close to those mentioned in et al. [1] (Reλ = 740; σv = 0.98 m/s; ηk = 14 µm; λT = 0.076 cm; Tint = 22.4 ms; τk = 0.2 ms). It is seen that the presumed spectrum is represented up to k = 2.5 cm−1 . At smaller length scales, the computed spectrum deviates from the presumed one; the contribution of these scales to the light particle dynamics need to be simulated. Along with LES computation of the ﬂow, the rigid particle, same as in [1], was tracked (dp > ηk and τSt > τk ). The particle interacts with turbulent structure around its diameter. Therefore the gas velocity “seen” by the particle at sub-grid scale was assumed to be a sum of resolved velocity, VLES , and increment of the velocity at spatial scales of order of the particle diameter, dp . The estimation of the last one can be obtained from distribution (3) by 3

10

E, cm3s-2

exp. Comte-Bellot&Corrsin LES N=323; Smagorinsky; CS=0.2

2

10

1

10 -1 10

100 -1 k, cm

101

Fig. 1. Evolution of computed energy spectrum in “free” turbulence at t = 0; 0.284; 0.655 s; lines - measurements of Comte-Bellot and Corrsin [26]

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Mikhael Gorokhovski and Anna Chtab

10

5

presumed spectrum computed spectrum

E, cm3s-2

104

103

10

2

101

1

k, cm-1

2

3

4 5 6

Fig. 2. Computed kinetic energy spectrum compared to the presumed by Overholt and Pope [25]

setting τp = ln(LLES /dp ). The equation for dragging particle is then used in the following form VLES + δv − Up dUp = (4) dt τSt where δv is sampled from (3) conditionally to δVLES (δVLES is the resolved velocity increment), i.e. δv = rnd {f (δv|δVLES , τp )}. The sampling was performed each time after passage of local time |S|−1 , where |S| = (2S ij S ij )1/2 is the norm of resolved velocity gradients tensor. The second order RungeKutta schema is chosen for computing particle motion. The velocity of gas at the particle position is obtained by the second order Lagrange interpolating polynomial. The time step ∆t in the LES and in (4), is the same, equal to 0.05 ms (∆t < τk ). This allowed to obtain the velocity statistics at time lag identical to experimental one.

4 Numerical Results and Discussion L In & the Kolmogorov '1941 scaling, the second-order structure function, D2 (τ ) = 2 (v(t + τ ) − v(t)) , is expected to behave as D2L (τ ) = C0 ετ . According to Taylor 1921 theory D2L (τ ) = 2σv2 1 − e−τ /TL , where σv2 is variance of the Eulerian velocity ﬁeld and TL provides the measure of the Lagrangian velocity

LES Computation of Lagrangian Statistics

69

Fig. 3. Second order structure function at diﬀerent time lags (τk - Kolmogorov’s time): upper part - measurements of Mordant et al. [1]; bottom part - computation with and without sub-grid model

2σv2 τ 2σv2 and then C0 = . At very TL εTL small times, the Lagrangian particle is assumed to be ballistic one: D2L (τ ) ≈ ε3/2 ν −1/2 τ 2 [27]. Figure 3 shows the behavior of D2L (τ )/ετ computed and measured in [1] for diﬀerent time lag. The last on was scaled by Kolmogorov’s

memory. For τ ≪ TL , one gets D2L (τ ) =

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Mikhael Gorokhovski and Anna Chtab

103

10

1/ω

1

2

EL (ω)

10-1 1/ω4

10-3 10-5 10-7

1

10

10

2

3

ω 10

10

4

10

5

Fig. 4. Computed Lagrangian velocity spectrum

time τk . Similar to experimental observation [1], D2L (τ )/ετ has bell-like shape with maximum C0 = 2.82 against measured C0 = 2.9 and from Taylor’s expression C0 = 3.06 (computation with zero velocity increment at subgrid scales yields C0 = 1.6). It is to note that the plateau with maximum C0 in this ﬁgure (Kolmogorov’s scaling) is very short. The computation showed that at small scales, the second-order structure function has, practically, quadratic dependency on τ (particle does not respond to very small scales due to inertia) and at very large times, it tends to 2σv2 . From Fig. 4, where the computed velocity spectrum, presented as Fourier transform of autocorrelation function, it is seen that the proportionality to ω −4 is dominant and the behavior ω −2 takes place in short band of frequency (short plateau in Fig. 3). To some extend, one may observe an agreement with classical theory: proportionality to ω −4 at small scales and proportionality to ω −2 , at large scales. At the same time, the computed statistics of the velocity increment displayed a strong nonGaussianity. In Fig. 5, we show the computed and measured distributions of particle velocity increment at diﬀerent time lag. It is seen that at integral times, the velocity increment is normally distributed. This conﬁrms the result of Oboukhov’s 1959 Lagrangian model of turbulence [28]. However at smaller time lag, the distributions exhibit a growing central peak with stretched tails. Such a manifestation of intermittency at small scales is seen also from the computed distribution of the particle acceleration (Fig. 6): the accelerations 2 of particle may attain very large values (going up to 4500 m/s ). In the same ﬁgure, we present the measured distribution and the distribution, which was computed without sub-grid modeling (zero δp v). One can see that in this

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0 -1

log10 (PDF)

-2

τ

-3 -4 -5 -6 -7 -20

-10

0 ∆τv / <(∆τv)2>1/2

10

20

Fig. 5. Velocity increment distribution at diﬀerent time lags (upper part - measurements of Mordant et al. [1]; bottom part - computation at same time lags)

case, the computed acceleration is of order of Kolmogorov’s scaling ak = 3 1/4 2 ε /ν ≈ 300 m/s , which is substantially less than the measured one.

5 Conclusions In Obukhov’s 1959 theory, the turbulence is represented by Markov process under space-translation and rotation symmetries. These symmetries allowed to reduce the generalized Fokker-Planck equation to the diﬀusion-type equation, corresponding to the Gaussian statistics of the velocity ﬂuctuation at large turbulent scales. At the same time, the recent experimental studies of high Reynolds number turbulence showed a highly intermittent structure of the ﬂow with a strong non-Gaussianity at small scales. Following Castaign’s

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-3

log10(PDF)

-4

-5

-6

-7 LES with sub-grid model LES without sub-grid model

-8 -5000

-2500

0

2

a (m/s )

2500

5000

Fig. 6. PDF of acceleration (upper part - measurements of Mordant et al. [1]; bottom part - computation)

papers, this motivated us to introduce another symmetry, namely the scaling symmetry, for the velocity increment at small sub-grid scales. The LES of a stationary box turbulence was performed with reconstruction of a synthetic velocity ﬁeld, in which the light (of the same density as ﬂuid) solid particle was dragged. The computation was compared with measurements. To some extend, the Kolmogorov’s 1941 phenomenology for the velocity of particle was conﬁrmed: maximum value of C0 agreed with Taylor’s expression; the velocity spectrum was proportional to ω −4 at small scales and proportional to ω −2 (in the short band of frequency) at large scales. To demonstrate the strong intermittency eﬀects, the distribution of velocity increment, was computed at diﬀerent time lag. Here again, similar to measurements, at time lag of order of integral time scale, the velocity increment was distributed as Gaussian, while at small time lags, this distribution displayed long stretched tails. The strong ﬂuctuations of acceleration, of the same order, as in experiment, have been

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obtained by computation. Nevertheless it should be noted that the model proposed in this paper, is only the ﬁrst very simple step to represent the complex real physics of intermittency. In this model, the two spatial points are correlated only through resolved velocity ﬁeld and hence the longer persistence of two points correlations should be introduced in the model. This may explain why a more narrow distribution of acceleration and of the velocity increment were obtained in computation, compared to measurements. Another weak point of proposed model may be also noted; it concerns the requirement to know the constant A. Is this constant, with the value found to match the experimental results [1], an universal constant, remains an open question. Recently, using the same value of this constant, we computed the statistics of two particles separation in a high Reynolds number ﬂow and compared with experiment, performed in the group of Bodenschatz [29]. The experimental eﬀects were well predicted.

References [1] N. Mordant, E. Leveque, and J.-F. Pinton. Experimental and numerical study of the Lagrangian dynamics of high Reynolds turbulence. New Journal of Physics, 6:116, 2004. [2] P. Moin and J. Kim. Numerical investigation of turbulent channel ﬂow. J. Fluid Mech., 118:341–377, 1982. [3] V.L. Saveliev and M.A. Gorokhovski. Group-theoretical model of developed turbulence and renormalization of Navier-Stokes equation. Physical Review E, 72:016302, 2005. [4] J. Bardina, J.H. Ferziger, and W.C. Reynolds. Improved subgrid scale models for large eddy simulation. AIAA paper 80-1357, 1980. [5] A. Misra and D.I. Pullin. A vortex-based subgrid stress model for largeeddy simulation. Phys. Fluids, 8:2443–2454, 1997. [6] J.A. Domaradzki and E.M. Saiki. A subgrid-scale model based on the estimation of unresolved scales of turbulence. Phys. Fluids, 9:2148–2164, 1997. [7] A.R. Kerstein. A linear-eddy model of turbulent scalar tsransport and mixing. Combust. Sci. and Tech., 60:391, 1988. [8] A. Scotti and C. Meneveau. A fractal model for large eddy simulation of turbulent ﬂow. Physica D 127:198–232, 1999. [9] J. Pozorski, S.V. Apte, and V. Raman. Filtered particle tracking for dispersed two-phase turbulent ﬂows. In: Proceedings of the Summer Program 2004, Center for Turbulence Research, 329–340, 2004. [10] N. Okong’o and J. Bellan. A priori subgrid analysis of temporal mixing layers with evaporating droplets. Phys. Fluids, 12:157, 2000. [11] S. Stolz and N.A. Adams. An approximate deconvolution procedure for large-eddy simulation. Phys. Fluids, 11(7):1699, 1999.

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[12] I. Vinkovic, C. Agguire, S. Simoens, and M. Gorokhovski. Large eddy simulation of droplet dispersion for inhomogeneous turbulent wall ﬂow. Int. J. of Multiphase Flow, 32(3):344–364, 2005. [13] J.G.M. Kuerten and A.M. Vreman. Can turbophoresis be predicted by large-eddy simulation? Phys. Fluids, 17:011701, 2005. [14] B. Shotorban and F. Mashayek. Modeling sub-grid eﬀects on particles by approximate deconvolution. Phys. Fluids, 17:081701, 2005. [15] M.A. Gorokhovski, V.L. Saveliev. Further analyses of Kolmogorov’s model of breakup and its application in air blast atomization. Phys. Fluids, 15(1):184–192, 2003. [16] A.N. Kolmogorov. On the log-normal distribution of particles sizes during break-up process. Dokl. Akad. Nauk. SSSR, XXXI 2, 99–101, 1941. [17] B. Castaing, Y. Gagne, and E.J. Hopﬁnger. Velocity proprobability density functions of high Reynolds number turbulence. Physica D, 46:177– 200, 1990. [18] B. Castaing, Y. Gagne, and M. Marchand. Log-similarity for turbulent ﬂows? Physica D, 68:387–400, 1993. [19] B. Castaing. Temperature of turbulent ﬂows. Journal De Physique II, 6(1):105–114, 1996. [20] H. Kahalerras, Y. Mal´ecot, Y. Gagne, and B. Castaing. Intemittency and Reynolds number. Phys. Fluids, 10(4):910–921, 1998. [21] A. Naert, B. Castaing, B. Chabaud, B. Hebral, and J. Peinke. Conditional statistics of velocity ﬂuctuations in turbulence. Physica D: Nonlinear Phenomena, 113(1):73–78, 1998. [22] M. Gorokhovski. Scaling symmetry universality and stochastic formulation of turbulent cascade with intermittency. Annual Research Briefs, Center of Turbulence Research, Stanford University, NASA, 197–204, 2003. [23] R. Friedrich and J. Peinke. Description of a turbulent cascade by a Fokker-Planck equation. Phys. Rev. Lett., 113(5):1997. [24] C.D. Pierce. Progress-variable approach for large-eddy simulation of turbulent combustion. PhD thesis, Stanford University, USA, 2001. [25] M.R. Overholt and S.B. Pope. A deterministic forcing scheme for direct numerical simulations of turbulence. Computers & Fluids 27(1):11–28, 1998. [26] G. Comte-Bellot and S. Corrsin. Simple Eulerian time correlation of fulland narrow-band velocity signals in grid-generated, “isotropic” turbulence. J. Fluid Mech., 48(2):273–337, 1971. [27] A.S. Monin and A.M. Yaglom. Statistical ﬂuid mechanics: mechanics of turbulence. The MIT press, 1981. [28] A.M. Obukhov. Description of turbulence in terms of Lagrangian variables. Adv. In Geophys., Atmospheric diﬀusion and air pollution, 6:113– 115, 1959.

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[29] N.T. Ouellette, H. Xu, M. Bourgoin, and E. Bodenschatz. Lagrangian velocity statistics in high Reynolds number turbulence. Submitted in Phys. Rev. Lett., 2006.

Anisotropic Subgrid-Scale Modelling: Comparison of LES with High Resolution DNS and Statistical Theory for Rapidly Rotating Turbulence L. Shao1 , F. S. Godeferd1 , C. Cambon1 , Z. S. Zhang2 , G. Z. Cui2 , and C. X. Xu2 1

2

´ LMFA UMR 5509, Ecole Centrale de Lyon, France [email protected] Dep. of Mechanical Eng., University Tsinghua, Beijing 100084, China

Summary. Rotating homogeneous turbulence is known to exhibit strongly anisotropic features at low Rossby number, as well as a modiﬁed dynamics. These are the result of the presence of inertial waves due to the Coriolis force. The spectral distribution of kinetic energy in spectral space also reﬂects this strong anisotropy through a dependence of the energy on the wavevector orientation at almost all wavenumbers. Up to now, subgrid scale models for large eddy simulation are not adapted to describing strongly anisotropic dissipative scales, therefore we introduce a means of deriving an eddy viscosity from the orientation-dependent structure function equation. Only at the end of the derivation of a ﬁrst version, the model is simpliﬁed by neglecting directional anisotropy in the ﬁnal eddy viscosity, in a ﬁrst stage for assessing the method. In order to illustrate qualitatively the relevance of the approach, and to prepare improved anisotropic subgrid-scale modelling, we compare the results of high Reynolds number large eddy simulations based on this model to energy density spectra obtained at lower Reynolds number by direct numerical simulations, and to spectra obtained by a recent two-point statistical model. The subgrid-scale model is shown to perform well, all the more when looking at the reduction of interscale energy transfer due to rotation, quantiﬁed here by the dependence of the velocity derivatives skewness on the micro-Rossby number.

1 Introduction Strong anisotropy of small scales was recently shown and investigated for turbulence in a rotating frame and/or in a stably stratiﬁed ﬂuid. For instance, the anisotropy of the vorticity components correlation tensor can be stronger than that of the Reynolds stress tensor. This reﬂects a speciﬁc anisotropic distribution of kinetic energy in 3D Fourier space, as obtained by high resolution DNS [1], and better understood from statistical theory models, ranging

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from Wave-Turbulence to EDQNM [2] (see also Galtier 2003, reference in [2]). It is important to stress that a strong shear, induced or not by a wall, is not required to create anisotropy at small scales. On the contrary, it is commonly agreed that in such ﬂows subjected to anisotropic ‘production’, anisotropy concerns the largest scales with no extension towards the smallest ones, especially at high Reynolds number. This simple viewpoint is radically questioned in the case of rotating ﬂows, in which there is no production, but a regime of nonlinearly interacting dispersive waves—the inertial waves—, which alter the cascade process and render it strongly anisotropic at the smallest scales. Modelling such ﬂows using LES is a challenge in such applications as geophysical ﬂows (atmosphere, ocean), and eventually industrial ﬂows dominated by strong swirl (combustors, etc.). In this context, we evaluate the recent LES approach by Cui et al. [3], which oﬀers exciting perspectives when applied to rotating turbulence. A key quantity in this model is the energy transfer across the spectrum, captured by the equation for the second order structure function, and information from the resolved third-order structure function is incorporated for modulating subgrid-scale (SGS) eddy viscosities. In this sense, SGS diﬀusivity becomes self-adaptive and time-dependent in statistically homogeneous unsteady turbulence. Finally, it is also rendered scale-dependent in Fourier space by adding a conventional cusp (Chollet 1983, reference in [4]). We describe the SGS model in section 2, and give comparisons with direct numerical simulations and a statistical two-point model in section 3. The following tasks for comparison and extension of the SGS model are given in section 4.

2 Description of the Subgrid Scale Model We describe shortly hereafter the new dynamic subgrid eddy viscosity model derived from the Kolmogorov equation for ﬁltered velocity. The isotropic model (CZZS) has been shown to perform very well in large eddy simulations of isotropic turbulence as well as for the channel ﬂow, adapting itself automatically to account for the walls. [3] Its application by Shao et al. [4] to the case of rotating homogeneous turbulence has demonstrated that an anisotropic derivation of the model accounts very well for the elaborate anisotropy of the ﬂow, and especially for the reduced interscale energy transfer due to the presence of inertial waves. We denote large-scale ﬁltered quantities by an overbar. For incompressible rotating homogeneous turbulence, the ﬁltered ﬂuctuating velocity equations are ∂u ¯i ∂ p¯ ∂2u ¯i ∂τik ∂u ¯i +u ¯k =− +ν + 2ǫik3 Ω u ¯k + (1) ∂t ∂xk ∂xi ∂xk ∂xk ∂xk ∂u ¯i =0 ∂xi

(2)

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in which rotation is chosen along the x3 axis at a rate Ω. The subgrid stress tensor components are τij = u ¯i u ¯ j − ui uj . We note the second order structure function Dii (r) =< (∆ui )2 > for the increment ∆ui = ui (x + r) − ui (x) of the ﬁltered velocity ﬁeld u. The contracted third order structure function is denoted Diik =< (∆ui )2 ∆uk >. The dynamic equation for Dii is derived from (1)–(2) expressed at points x and x − r to form the velocity increment, multiplied by ∆ui and ensemble ′ averaged (see [3] and [4] for details). We call the second point x′ , at which τik is evaluated, in agreement with the evaluation of two-point structure functions in this equation, and the Einstein summation convention applies throughout this paper. This yields ∂Dii ∂τik ∂Diik ∂ 2 Dii ∂τ ′ + ui = 2ν − 4ǫ + 2 ∆¯

− 2 ∆¯ ui ik′ ∂t ∂rk ∂rk ∂rk ∂xk ∂xk )* + (

(3)

Tii

where ǫ = 2ν∇2 u is the contribution of the resolved scales to molecular dissipation. Unlike the Kolmogorov equation, equation (3) retains the anisotropic dependence of the structure functions upon the separation vector r. The last two terms of equation (3) introduce a third order tensor, of which the contraction Tii plays a role in the dynamics of the resolved scales. In this correlation tensor, we shall assume that the subgrid stresses may be modelled using an eddy viscosity νt spatially uniform although eventually time-varying: τij = −2νt S¯ij + τkk δij /3 .

(4)

In the above equation, S¯ij is the deformation tensor of the resolved scales, and the isotropic term that contains the trace τkk may augment the pressure ﬁeld. Therefore, Tii can be recast using (4) as Tii = 2νt

¯i ∂ 2 Dii ∂u ¯i ∂ u + 4νt

∂rk ∂rk ∂xk ∂xk

(5)

and used in equation (3) to provide the modelled structure function equation ∂Dii ∂Diik ¯i ∂ 2 Dii ∂u ¯i ∂ u + = 2(ν + νt ) + 4(ν + νt )

. ∂t ∂rk ∂rk ∂rk ∂xk ∂xk

(6)

From this equation, one can extract the eddy viscosity. But for the resulting model to be tractable, a single-valued νt should be obtained, so that we rather use a volume averaged equation (6), by integrating over the volume of a sphere iik of radius r. The integral of the derivative ∂D ∂rk can be transformed in a surface integral by use of the Green-Ostrogradsky relation. We denote quantities integrated over this sphere with a V superscript, and integration over the sphere’s surface with Sr . Using these notations, in anisotropic homogeneous turbulence, the subgrid eddy viscosity νt appears in the spherically integrated structure function equation:

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L. Shao, F.S. Godeferd, C. Cambon, Z.S. Zhang, G.Z. Cui, and C.X. Xu V

∂Dii 3 Sr ¯i V 6(ν + νt ) ∂Dii Sr ∂u ¯i ∂ u + Diir = ( ) − 4(ν + νt )

. ∂t r r ∂r ∂xk ∂xk

(7)

Neglecting molecular viscosity for high Reynolds number turbulence and dropping the time derivative term for small r, one gets the eddy viscosity for anisotropic turbulence as νt =

Sr 3Diir

S

∂Dii − 4r 6 ∂r

,

∂u ¯i ∂ u ¯i ∂xk ∂xk

-−1

.

(8)

The novelty of this eddy viscosity with respect to previous structure functionbased models, is the presence of the third order structure function through the Sr contraction Diir , which plays a crucial role since it is related to the energy transfer. The obtained eddy viscosity is scale-independent by construction, but a cusp is introduced by a correction a` la Chollet (see [4]). This deﬁnition is then implemented in a pseudo-spectral code for rotating homogeneous turbulence, and used to perform large eddy simulations of freely decaying rotating turbulence, starting from random initial conditions. At each timestep, the available velocity ﬁeld is used to compute the structure functions needed in equation (8), averaged in accordance to the symmetries of the problem. A similarity hypothesis is used to obtain the power law behaviour of the third order structure function term, which, being a high order moment, is quite noisy. The resulting νt is then injected in the ﬁltered ﬂuctuating velocity equations that are solved with the pseudo-spectral Fourier method. From the resolved velocity ﬁeld, one extracts statistics that reﬂect the anisotropy in the ﬂow, keeping the dependence on the vertical or the horizontal directions, or on the wavevector orientation. Thus, we obtain the energy density spectra as a function of the wavenumber K and its orientation θ, as well as more classical one-point statistics such as the Reynolds stress tensor components.

3 Results We compare the results obtained from three numerical approaches to decaying rotating turbulence, starting from consistent isotropic initial data with a narrow-band energy spectrum for the isotropic pre-computation, so as not to impose a given inertial range scaling: • LES with typical 962 × 384 resolution adapted for the vertical elongation of structures. • High resolution pseudo-spectral full DNS, typically 5123 or above. The pseudo-spectral method is classical enough not to be detailed here (see e.g. [5] for details). As for LES, the rotation is applied on a fully developed turbulent ﬁeld coming from a pre-computation without rotation.

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• The asymptotic vanishing Rossby two-point statistical model AQNM [2]). The AQNM model consists of equations for anisotropic energy density spectra, and the closure assumes high Reynolds numbers and inﬁnite rotation rate, as a simpliﬁcation to the more complex rotating EDQNM model. AQNM is a quasi-normal two-point closure for rotating wave turbulence, based on the dynamical equations for inertial waves, and resonant waves theory. As such, and it being formulated in spectral space, it is especially suitable for reproducing anisotropic energy density spectra. The latter are smooth functions with respect to spectra extracted from velocity ﬁelds, as in LES and DNS. The performance of the SGS model is evaluated against DNS and AQNM over the spectral two-point statistics, in order to assess the ability of the LES to reproduce the structuring of the ﬂow into vertically elongated vortices. Another key feature of rotating turbulence is the reduction of the downscale energy cascade due to a diﬀerent dynamics from the presence of inertial waves. This energy balance is quantiﬁed by computing the evolution of the skewness of the velocity derivatives, which is directly linked to the energy transfer, or the triple correlations of velocity. 3.1 Skewness We quantify the reduction of the cascade of energy, hence of the decay rate of turbulence, due to rotation by computing the velocity derivative skewness Sk [6]; the higher the rotation rate, the slower the decay. This trend is well observed also in the LES computations performed with the present SGS model [4]. For decaying rotating turbulence, it was shown that the dependence of the normalized skewness with the micro Rossby number Ro ω —i.e. based on vorticity—can be analytically represented by the function: [6] Sk 1 = Sk (t = 0) (1 + 2Ro ω

−2 )0.5

.

(9)

In LES, Ro ω has to be estimated using the resolved scale dissipation ǫ instead of the full turbulent dissipation. Figure 1 plots the evolution of the skewness using the analytical expression (9), LES with the isotropic formulation of the SGS model based on the structure function (CZZS [3]), of which the AISM (asymptotic skewness model) is the asymptotic limit at very large Rossby number and amounts to the structure function model of M´etais and Lesieur, and LES with the current anisotropic formulation of this model [4]. One observes that the isotropic models are way oﬀ when the Rossby number decreases, i.e. they fail to reduce the dissipation enough when the rotation is increased. The present SGS model’s results follow the analytical curve closely, which has been shown to describe very accurately the universal eﬀect of rotation on homogeneous turbulence in both DNS and statistical models (see ﬁgure 3 in [6], p. 317). From the

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Fig. 1. Skewness variation versus micro Rossby number with rotation rate Ω = 10. A comparison between the analytical curve [6], the AISM model, the isotropic CZZS one, and the present model, in a 64x64x256 LES run.

dynamical point of view, our new SGS model is therefore very well adapted, the only one among those plotted on ﬁgure 1. 3.2 Energy Spectra The reduction of the cascade of kinetic energy of rotating homogeneous turbulence is also necessarily related to a change in the spectrum. It is known that the energy spectrum E(K) of rotating turbulence exhibits a K −3 slope (see e.g. [7]), even though whether this corresponds to complete or partial two-dimensionalization is still an object of debate. Still, as shown on ﬁgure 2, we observe that the SGS model presented here allows our LES to undergo a transition from the spectral slope of isotropic turbulence (close to K −5/3 at high Reynolds number) to a much steeper K −3 inertial range slope when time evolves. This is again a trace of the diﬀerence between isotropic turbulence dynamics and inertial waves dynamics. This slope is consistently recovered by the statistical model AQNM, and the 5123 resolution DNS simulations, which present the same trend towards an increase of the slope of the spectrum (ﬁgure 2 right); further quantitative comparison is needed to accurately evaluate the LES model against DNS.

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Fig. 2. Left: Time evolution of E(k) (increasing times downwards), showing the spontaneous appearance of the k−3 scaling, consistent with previous analyses of rotating turbulence (962 × 384 LES data). Right: Same with a 5123 DNS (over one turnover time only) showing the same trend.

3.3 Directional Energy Density Spectra The most elaborate anisotropic statistical quantity for a thorough comparison is the angle-dependent energy spectrum, denoted e(K, θ, t) since it depends not only on the wavenumber K but also on its polar angle θ to the vertical. Together with a polarization spectrum Z(K, θ, t)—only marginally relevant here—, it allows to reconstruct any second-order, single or two-point statistical quantity, including of course Reynolds stress and vorticity components correlation tensors, as well as the structure tensors deﬁned in [8]. The spectrum e is the main output of the statistical models (see also [6]), whereas it comes from speciﬁc post-processing of DNS and LES ﬁelds by averaging over circular bands. The more spherically averaged energy spectrum is

1 common 2 recovered as E(K, t) = 0 4πK e(K, θ, t) d cos θ. We analyze here the anisotropy of the velocity ﬁeld by using energy spectra as a function of wave number K and the azimuthal angle θ measured with the vertical (polar) Fourier direction [5]. The energy is integrated over cuts of spherical shells along the polar angle direction. In a discrete ﬂow ﬁeld, for n shells of width ∆K and radius Kn divided into m equal polar sectors of angle ∆m and angle θm we can write down an expression for the directional kinetic energy spectrum . /−1 θm +∆θ/2 m ˆ ∗ (K) · u(K). ˆ u cos θdθ (10) E(Kn , θm ) = 2 θm −∆θ/2 |K|∈In ,θK ∈Jm

It corresponds to an integration of the blue surface in ﬁgure 3(a) over a shell width of ∆K. The spectra represent classically integrated spherical spectra,

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L. Shao, F.S. Godeferd, C. Cambon, Z.S. Zhang, G.Z. Cui, and C.X. Xu

Fig. 3. (a) Integrating surface for angular dependent spectra. (b) Direction of integration of an angular spectrum along K.

but instead of interpreting them along a ray from the center of the sphere, they represent spectra along a side of cones with diﬀerent opening angles, schematically sketched in ﬁgure 3(b). Recent results for e from LES, DNS and AQNM are shown on ﬁgures 4(b), (a) and (c), exhibiting the strong anisotropy at the small scales predicted by the three approaches. The plots 4(a), (b) and (c) result from independent calculations, made with diﬀerent parameters. AQNM is derived assuming inﬁnitely large rotation rate, and a Reynolds number normalized in a speciﬁc way, not easily comparable to that of DNS, for which the Taylor based Reynolds number is around 150 and a macro Rossby number of the order of 10−2 . For LES, the Reynolds number is very large—at least if conventionally based on laminar viscosity, since νt ≫ ν ≃ 0—, and the micro Rossby number is observed to be of order unity (larger initially, smaller at the end of the computation). Figure 4(a) shows that the energy density spectra corresponding to quasihorizontal wavevectors is highest whereas for quasi-vertical wavevectors the spectrum has the smallest energy, as in the other two plots. This indicates the tendency of rotating homogeneous turbulence to accumulate energy in the equatorial (horizontal) spectral plane, and this translates into vertically elongated vortices. Note that the anisotropy in this simulation goes all the way down to the smallest scales, where it is largest. In these DNS of course, the Reynolds number is quite low, in addition to the fact that we have retained a signiﬁcant small wave number region, so that the inertial range is not easily discernable. On ﬁgure 4(b), the LES model exhibits the same strong anisotropic features, but this time in a very high Reynolds number context, with a clear inertial subrange. Finally, the AQNM model results on ﬁgure 4(c) shows that the spectral distribution, and its anisotropy in the small scales, is very

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much like that obtained in the two other approaches. The analogy is therefore also valid for the very low Rossby number regime of the AQNM model. The agreement with high-Reynolds number AQNM proves that strong anisotropy of small scales is not a low Reynolds number artefact, as could be the case in shear ﬂows. The Reynolds number and the initial conditions of the DNS and LES runs are very diﬀerent, but one can attempt to compare a little better the amount of anisotropy—the diﬀerent energy of the directional spectra—by normalizing the energy by the total kinetic energy at the peak wavenumber, and the wavenumbers by the maximum one. This is shown on ﬁgure 5. One observes that the anisotropy is much larger in LES than in DNS, and probably too large to represent the actual anisotropy at a cut-oﬀ scale which is not too small. The qualitative role of eddy viscosity is good when distributing energy between the vertical and the horizontal spectral directions, but the dissipation is too low at the equator and too large in the vertical. Of course, the fact that the spectra at the cut-oﬀ scale are anisotropic shows that there is no such single value of the eddy viscosity for all orientations.

4 Discussion and Perspectives We have shown in the above that the new SGS model based on an anisotropic derivation of the structure function equation, proposed by Shao et al. [4], reproduces the two important features of freely decaying rotating homogeneous turbulence: (a) the reduction of the energy cascade is well-predicted as shown by a very good agreement of the evolution of the skewness with the microRossby number, compared with previous DNS and statistical models; (b) the structure of the ﬂow in elongated vertical columns is retrieved, as indicated by the anisotropy of the directional energy density spectra (which otherwise would collapse, in isotropic turbulence). Conventionally, people expect visualizations of iso-vorticity surfaces. Such visualizations however are only qualitative, whereas angle-dependent spectra give the ﬁnest multiscale quantitative evaluation for this trend. These very encouraging results suggest to follow with a more complete set of comparisons, looking at other statistical quantities for quantitative assessment of the anisotropy. It is more problematic, but even more crucial to compare statistical quantities reﬂecting the small scale distribution. This information is essentially lost in LES, and would have to be reconstructed by statistically extrapolating the resolved scales. For example regarding one-point quantities, in axisymmetric turbulence, the anisotropy of the Reynolds stress tensor and the vorticity components correlation tensor are interesting indicators, among which b33 and bω 33 respectively are highly relevant to anisotropy in rotating turbulence. In the LES computations presented above, we observe that the initial value of b33 is zero on average, corresponding to the isotropic turbulence value (ﬁgure 6). At about unit

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Fig. 4. Energy density spectra—weighted to be dimensionally equivalent to a spherically integrated spectrum—4πK 2 e(E, θ) of freely decaying rotating turbulence at a given time, obtained from (a) 5123 DNS (computation by Liechtenstein); (b) 962 ×384 LES; (c) asymptotic statistical model AQNM with 3003 points in spherical coordinates wave space (computation by Bellet). The lowest curves on each set are for θ close to π/2, i.e. the equatorial or horizontal direction, evolving upwards to the highest curves which represent θ close to 0, the polar or vertical direction, aligned with the rotation vector. (The angular sectors given in degree on (b) are only applicable to LES data.) The straight lines are K −3 .

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∗ ) as a function Fig. 5. Normalized energy density spectra 4π(k∗ )2 e(k∗ , θ)/E(kpeak of the normalized wavenumber k∗ = K/KM ax . Open squares indicate the anisotropy range of the DNS spectra ; black squares for LES.

Ro Ω , it then decreases towards the value 1/3 which is expected for exactly twodimensional turbulence, that would correspond to inﬁnite vertical vortices. However, note that one does not expect exact two-dimensionalization from the dynamics—to be distinguished from numerical conﬁnement eﬀects—, so that b33 will eventually saturate. This conventional Reynolds stress anisotropy tensor can also be split using alternately the similar decompositions by [6] and [8]. Hence a future work will be to study the deviatoric part of the dimensionality tensor y33 = −2be33 . A next step in this work will be to derive an anisotropic subgrid scale model based on a truly anisotropic eddy viscosity. This is now rendered possible due to the consistent method for deriving the eddy viscosity with the help of the anisotropic structure function equation. In so doing, one should retain the orientation θ to the vertical direction as an additional dependency variable for the eddy viscosity. Short term perspectives also include applying this LES model to rotating and stratiﬁed turbulence, rendering it all the more relevant for geophysical applications.

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Fig. 6. Component b33 of the anisotropy of the Reynolds stress tensor deﬁned as bij = /q 2 − δij /3.

References [1] L. Liechtenstein, F. S. Godeferd, and C. Cambon. Nonlinear formation of structures in rotating stratiﬁed turbulence. J. of Turb., 6:1–18, 2005. [2] F. Bellet, F. S. Godeferd, J. F. Scott, and C. Cambon. Wave-turbulence in rapidly rotating ﬂows. J. Fluid. Mech., to appear, 2005. [3] C. Guixiang, H. Zhou, Z. Zhang, and L. Shao. A dynamic subgrid eddy viscosity model with application to turbulent channel ﬂow. Phys. Fluids, 16:2835–2842, 2004. [4] L. Shao, Z.S. Zhang, G.X. Cui, and C.X. Xu. Subgrid modelling of anisotropic rotating homogenenous turbulence. Phys. Fluids, 17:115106, 2005. [5] F. S. Godeferd and C. Staquet. Statistical modelling and direct numerical simulations of decaying stably-stratiﬁed turbulence: Part 2: Large and small scales anisotropy. J. Fluid. Mech., 486:115–150, 2003. [6] C. Cambon, N. N. Mansour, and F. S. Godeferd. Energy transfer in rotating turbulence. J. Fluid Mech., 337:303–332, 1997. [7] X. Yang and J.A. Domaradzki. Large eddy simulations of decaying rotating turbulence. Phys. Fluids, 16:4088, 2004. [8] S. C. Kassinos, W. C. Reynolds, and M. M. Rogers. One-point turbulence structure tensors. J. Fluid Mech., 428:213, 2001.

On the Investigation of a Dynamic Nonlinear Subgrid-Scale Model Ingmar Wendling and Martin Oberlack Fluid- and Hydromechanics Group Technische Universit¨ at Darmstadt Petersenstr. 13, 64287 Darmstadt, Germany [email protected], [email protected] Summary. An anisotropic subgrid-scale model with ﬁve terms depending on strain and rotation rate is investigated. Single terms and the combinations of two of the ﬁve terms are tested in a turbulent channel ﬂow at a turbulent Reynolds number of Reτ = 395. The model constants, one for each term, are determined dynamically. Some double term models showed signiﬁcant improvement compared to the dynamic Smagorinsky model. Determination of two coupled dynamic constants reduces the stochastic behaviour of the constants, which may preserve local stability and no averaging is needed.

1 Introduction Eddy-viscosity type models are often used for Reynolds averaged (RANS) modelling and subgrid-scale (SGS) modelling in large-eddy simulations (LES). Here a linear relation between the unknown stresses and the resolved strain rate is assumed based on the Boussinesq hypothesis. The advantages of these models are the easy implementation and the high numerical stability. The most commonly used SGS model is the Smagorinsky model [1], which is an eddy-viscosity type model. It is known that this model gives a rather inaccurate estimation of the subgrid-scale stresses. Therefore Lund and Novikov [2] proposed an anisotropic SGS model based on strain rate and rotation rate as a new ingredient. The reason for this is that the molecular transport analogy of the Boussinesq hypothesis based only on strain rate is diﬀerent from the physical turbulent transport of the unresolved scales. Moreover it is believed that vortex stretching is the main process for energy transfer from larger to smaller scales [3]. Similar approaches are known from RANS modelling [4, 5]. Kosovi´c [6] and Wang and Bergstrom [7] proposed nonlinear SGS-models based on Speziale’s quadratic constitutive relation [8]. These models contain only three of the ﬁve terms, which are included in Lund and Novikov’s model. While Kosovi´c determined the model constants analytically, the constants in Wang

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and Bergstrom’s model are calculated dynamically. This model exhibits desirable numerical robustness, possibility of incorporating backscatter and a more realistic geometrical representation of the SGS stress compared to the conventional dynamic model [9, 10]. Parts of the anisotropic SGS model by Lund and Novikov have been employed for the calculation of a turbulent channel ﬂow. The single terms and permutations of two of the ﬁve terms are taken into account at a time, because the computation of more than two coupled dynamic constants is not feasible, due to very high computational costs.

2 Numerical Procedure The simulations are accomplished with the ﬁnite volume code FASTEST 3D. The governing equations, which are the ﬁltered Navier-Stokes equation and the continuity equation for incompressible ﬂuids, are discretized on a blockstructured grid with a second order spatial discretization. The solution is updated in time using the second order implicit Crank-Nicholson method. For the pressure-velocity coupling the SIMPLE algorithm is used. The considered SGS model is an anisotropic SGS model based on the assumption that the SGS stress is expressible as a tensor function of the strain and rotation rates and the unit isotropic tensor (δ): τij = fij (S, R,δ) where the strain and rotation rate tensors are respectively deﬁned as 1 ∂ui ∂uj + , Sij = 2 ∂xj ∂xi ∂uj 1 ∂ui − , Rij = 2 ∂xj ∂xi

(1)

(2) (3)

with ui as resolved velocity. Expressing τij as an integrity basis [11] of Sij and Rij and neglecting tensor products of order four and higher the model consists of ﬁve terms with a distinct model constant for each term: 0 ∗ τij = − C1 ∆2 |S|Sij + C2 ∆2 (Sik Skj )∗

+ C3 ∆2 (Rik Rkj )∗ + C4 ∆2 (Sik Rkj − Rik Skj ) 1 1 (Sik Skl Rlj − Rik Rkl Slj ) . + C5 ∆2 (4) |S| Here ∆ is the grid ﬁlter width, |S| = (Sij Sij ) and the asterisk denotes the trace-free part of a tensor. Here only the oﬀ-diagonal part of the stresses are

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modelled, because it is common practice for incompressible ﬂows to combine the isotropic part with the pressure. In the following the ﬁve terms are called term 1, term 2, etc. Lund and Novikov [2] tested the model only a priori on ﬁltered DNS data of homogeneous, isotropic turbulence. The Smagorinsky model (term 1) was found to be the dominant term. Including the remaining terms did not signiﬁcantly improve the results. They never used the nonlinear model in an LES and a priori analysis cannot provide detailed quantitative but only qualitative estimates of the performance of the SGS model [12]. The dynamic procedure proposed by Germano et al. [9] with the modiﬁcation introduced by Lilly [10] is used to determine the model constants C1 − C5 dynamically for the single term models, where one single term of (4) is taken as a model and all other terms are set to zero. For a reduced model of the form (4) with two terms and two constants τij = − [Cα αij (u) + Cβ βij (u)] ,

(5)

Eij = Lij + Cα Mij + Cβ Nij

(6)

where αij and βij are single terms of equation (4) without constants, and Cα and Cβ are the corresponding constants, the SGS error tensor is given by:

where Lij = u2 i uj − ui uj ,

(7)

Nij

(9)

) − α Mij = αij (u ij (u), − β = βij (u) ij (u).

(8)

2 The hat designates a test ﬁltered quantity. Minimizing the error density Eij leads to two coupled equations for Cα and Cβ : 3 43 4 3 4 Mij Mij Mij Nij Cα −Lij Mij = (10) Nij Mij Nij Nij Cβ −Lij Nij

With this set of equations the constants Cα and Cβ can be determined. The calculated dynamic constants are highly ﬂuctuating. Due to the assumption (11) C α αij = Cα α ij

in the derivation and for numerical stability reasons the constant has to be a smooth function of space and time. Therefore an averaging over homogeneous directions is applied to the parameter. Usually negative values for the model coeﬃcient are set to zero for eddy viscosity type of models (term 1) to avoid negative turbulent viscosity. To allow negative values for the remaining terms, no clipping is applied to any term in this study, including term 1.

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20

u/uτ

15

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DNS Coarse DNS

5

0 0,001

0,01

0,1

y

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Fig. 1. Mean velocity proﬁle in streamwise direction for LES without SGS model in comparison to DNS at Reτ = 395. 5

DNS Coarse DNS

kturb/uτ²

4

3

2

1

0 0

0,2

0,4

y

0,6

0,8

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Fig. 2. Turbulent kinetic energy for LES without SGS model in comparison to DNS at Reτ = 395.

3 Results The present test case is a turbulent channel ﬂow at a turbulent Reynolds number of Reτ = 395. In x− and z−directions we use periodic boundary conditions, while at the bottom and top walls no slip boundary conditions are employed. The size of the computational domain is 2π x 2 x π and the grid contains 96 x 64 x 96 points in x−, y− and z−directions. In x− and z−direction the grid is uniform. In wall normal direction the grid cells are reﬁned using a geometric law. The grid point next to the wall is positioned at y + = 1. The ﬂow is driven by a constant mass ﬂow. The results of the simulation are compared to results of the DNS of a channel ﬂow by Moser et al. [13]. The DNS results are not ﬁltered to the LES grid. Figure 1 and 2 show the result of a simulation without subgrid-scale model, which is called

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a coarse DNS here. The relatively high resolution leads to a good agreement with the DNS data. If the wall is not well resolved, the results of a LES of a turbulent channel ﬂow are far away from the DNS which makes the results hardly comparable and an evaluation of the subgrid-scale models is almost not possible. 3.1 Single Term Models Additional computational costs of the single term models compared to the Germano model are negligible. Figure 3 shows the distribution of the dynamic parameter for all ﬁve models. The constant of model term2 is negative while the constant of all other models are in the positive range. The mean velocity proﬁle in streamwise direction is shown in ﬁgure 4. The model including the ﬁrst term which is known as Germano model was expected to achieve the best results. As one can see the best performing model for this test case is term 5, for which the data agrees very good with the DNS also for the kinetic energy shown in ﬁgure 5. For the sake of clarity here only the best performing models are plotted. Model term 3 gave results very close to the Germano model. Term 3 is not included in Speziale’s quadratic constitutive relation mentioned above due to production of erroneous results in isotropic turbulence subjected to solid body rotation. The models of Kosovi´c and Chang and Bergstrom also do not include term 5, the best performing term in this study. 3.2 Double Term Models The permutation of two model terms in equation (4) leads to ten diﬀerent models. In the following the models are called according to the included terms: term 1+2, term 1+3, etc. The models with two of the ﬁve terms needed additional computation time of about 25% for one iteration compared to the

0,05

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y

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Fig. 3. Distribution of the dynamic constant for one term models.

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20

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15

10

DNS Term 1 Term 2 Term 3 Term 4 Term 5

5

0 0,001

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Fig. 4. Mean velocity proﬁle in streamwise direction for single term models in comparison to DNS at Reτ = 395. 5

DNS Term 1 Term 3 Term 5

kturb/uτ²

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3

2

1

0 0

0,2

0,4

y

0,6

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Fig. 5. Turbulent kinetic energy for single term models in comparison to DNS at Reτ = 395.

single term model. The reason of these additional costs is the determination of two coupled dynamic parameters. The models including term 5 as well as the models term 1+3, and term 2+3 underperformed and were not plotted in the ﬁgures 6 and 7. Lund and Novikov found the combination of term 1 and term 4 with variable coeﬃcients and the combination of term 1 and 2 with constant coeﬃcients to be best for a combination of two terms. They tested this model on ﬁltered DNS data of homogeneous isotropic turbulence. They determined the correlation coeﬃcient between the exact and the modelled SGS stresses. Herewith they deﬁned the best and the worst groupings. As one can see in ﬁgure 6 the mean velocity proﬁles of these two models also show the best agreement with the DNS data. In reference to the kinetic energy shown in ﬁgure 7 the model term 1+2

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20

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15

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5

0 0,001

0,01

0,1

y

1

Fig. 6. Mean velocity proﬁle in streamwise direction for double term models in comparison to DNS at Reτ = 395. 5

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kturb/uτ²

4

3

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1

0 0

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0.4

y

0.6

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1

Fig. 7. Turbulent kinetic energy for double term models in comparison to DNS at Reτ = 395.

shows the best performance, only at the center of the channel the energy is lower than the DNS data. Combining two poor performing single term models like models term 2 and term 4 can result in good model performance, while combining two good performing models must not lead to a good results. As mentioned above the dynamic constants were averaged over homogeneous directions in this study. It was observed that determining two coupled dynamic constants reduces the ﬂuctuations. In ﬁgure 8 the probability density functions (PDF) of the constant at the center of the channel belonging to the Smagorinsky term of single and double term models are plotted. The standard deviation reduces from 0.2 for the single term model to 0.008 and 0.009 for C1 of the double term models term 1+2, term 1+4 respectively. It is known that large negative values cause numerical instability. The minimum value for the single term model is −19.2, while the minimum value for C1 of

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C1 of Term 1 C1 of Term 1+2 C1 of Term 1+4

25

PDF

20 15

10 5 0 -0.2

-0.1

0

0.1

0.2

C1

Fig. 8. Probability density functions of the constant C1 in single term model and double term models. 0,06 0,04

C

0,02 0

C1 of Term 1+2 C2 of Term 1+2 C1 of Term 1+4 C4 of Term 1+4

-0,02 -0,04 -0,06 -0,08 0

0,2

0,4

y

0,6

0,8

1

Fig. 9. Distribution of the dynamic constant for double term models term 1+2 and term 1+4.

model term 1+2 is only −2.6. Also the rate of values lower than −0.1 reduces from 6.3% to 3.4%. For the double term models an accumulation close to zero can be observed. But despite the fact that the maximum of the PDF is close to zero, the mean value is still in the positive range, as ﬁgure 9 shows. This means there is still a contribution to the SGS dissipation. It must be stated that the PDF are correctly normalized. The PDFs of the double term models tends faster to zero for large negative and positive values than the PDF for the single term model. Preliminary tests show that the models term 1+2, and term 1+4 run stable without averaging the constants for the present test case.

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4 Conclusion and Discussion Parts of the anisotropic SGS model proposed by Lund and Novikov were investigated. Simulations with single term and double term models were conducted. The single term model with term 5 performed signiﬁcantly better than the dynamic Smagorinsky model (term 1). The double term models term 1+2 and 1+4 show a very good agreement with the DNS data. It is unclear why the double term models including term 5 perform worse than the single term model. In addition to the improved performance the double term models exhibit a second advantage. The determination of two coupled dynamic constants reduces the ﬂuctuations of the constants, which lead to a reduced modeling error and numerical stability. The models run stable in the channel ﬂow when no averaging is applied. Further investigations will show if the double term models run stable without averaging in more complex geometries.

References [1] J. Smagorinsky. General Circulation Experiments with the Primitive Equations. Mon. Weath. Rev. 91, 99–164, 1963. [2] T.S. Lund and E. Novikov. Parameterization of Subgrid-Scale Stress by the Velocity Gradient Tensor, Annu. Res. Briefs 1992, 27–43, 1992. [3] S. Pope. Turbulent Flows, Cambridge University Press, 2000. [4] S. Pope. A More General Eﬀective-Viscosity Hypothesis, J. Fluid Mech., Vol. 72 part 2, 331–340, 1975. [5] C.G. Speziale. On Nonlinear k-l and k-e Models of Turbulence, J. Fluid Mech., Vol. 178, 459–475, 1987. [6] B. Kosovi´c. Subgrid-scale Modelling for the Large-Eddy Simulation of High-Reynolds-Number Boundary Layers, Phys. Fluids, Vol. 336, 151– 182, 1997. [7] B.C. Wang and D. Bergsrom. A Dynamic Nonlinear Subgrid-Scale Stress Model, Phys. Fluids, Vol. 17, No. 3 , 035109, 2005. [8] T.B. Gatski and C.G. Speziale. On Explicit Algebraic Stress Models for Complex Turbulent Flows, J. Fluid Mech., Vol. 254, 59–78, 1993. [9] M. Germano, U. Piomelli, P. Moin, and W. Cabot. A Dynamic SubgridScale Eddy Viscosity Model, Phys. Fluids, Vol. 3, No. 2, 1760–1765, 1991. [10] D. Lilly. A Proposed Modiﬁcation of the Germano Subgrid-Scale Closure Method, Phys. Fluids, Vol. 4, No. 3, 633–635, 1992. [11] A. Spencer. Theory of Invariants, Academic Press, Part III, 239–352, 1987. [12] U. Piomelli, P. Moin, and J.H. Ferziger. Model Consistency in Large Eddy Simulation of Turbulent Flows, Phys. Fluids, Vol. 31, No. 7, 1884–1891, 1988. [13] R.D. Moser, J. Kim, and N.N. Mansour. Direct Numerical Simulation of Turbulent Channel Flow up to Reτ = 590, Phys. Fluids, Vol. 4, 943-945, 1999.

Three Problems in the Large–Eddy Simulation of Complex Turbulent Flows Krishnan Mahesh, Yucheng Hou, and Pradeep Babu Aerospace Engineering and Mechanics University of Minnesota Minneapolis, MN 55455 [email protected] Summary. This paper: (i) discusses an algorithm that addresses the problems posed by low Mach numbers and high Reynolds numbers in large–eddy simulation of compressible turbulent ﬂows, (ii) uses numerical solutions of the RDT equations to suggest the possibility that the linear eﬀects of pressure might be more important to model in the near–wall problem, than nonlinear transfer, and (iii) a simple kinematic model that possibly explains why large–eddy simulation predicts turbulent mixing accurately, even though the viscous processes are not being represented.

1 An Algorithm for Large–Eddy Simulation of Compressible Turbulent Flows 1.1 Introduction A key issue in turbulence simulation is ensuring robustness without the use of numerical dissipation. Considerable attention has been devoted to this problem for incompressible ﬂows, where algorithms that discretely conserve kinetic energy have been found to be very successful in reliably performing large–eddy simulation (LES). However, the compressible equations do not conserve kinetic energy; energy is exchanged between kinetic and potential energy. Also, small Mach numbers result in the compressible equations becoming very stiﬀ. A solution that addresses both problems is discussed below. The basic idea is that a robust algorithm for compressible turbulence may be derived by requiring that the discrete equations reduce to the incompressible equations at low Mach numbers, and that the discretization conserves kinetic energy in the inviscid incompressible limit. Hou & Mahesh [1] discuss the approach in detail; this paper summarizes the results.

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1.2 Governing Equations The governing equations are the compressible Navier–Stokes equations which are non–dimensionalized as follows: ρd ud td µd , ui = i , t = , µ= , ρr ur L/ur µr Td ur ur pd − p r , T = , and Mr = =√ p= . 2 ρr ur Tr ar γRTr ρ=

(1)

ur , L, ρr , Tr are the reference velocity, length, density and temperature respectively, and the superscript ‘d’ indicates dimensional variables. The reference pressure, pr = ρr RTr . Note that the pressure has been non– d r dimensionalized as p = pρr−p ur 2 . This yields the following non–dimensional equations: ∂ρ ∂ρuj + = 0, ∂t ∂xj ∂ρui ∂ρui uj ∂p 1 ∂τij + =− + , ∂t ∂xj ∂xi Re ∂xj

Mr2

3

∂ ∂t

p+

4 ∂uj ∂ γ−1 γ−1 ρui ui + ρui ui uj + γp + 2 ∂xj 2 ∂xj 2 µ∂T ∂ 1 (γ − 1)Mr ∂τij ui . + = Re ∂xj ReP r ∂xj ∂xj

(2a) (2b)

(2c)

where Re = ρr ur Lr /µr , and the non–dimensional equation of state is: ρT = γMr2 p + 1.

(3)

Note that as Mr tends to zero, the energy equation (2c) reduces to µ∂T 1 ∂ = ReP r ∂xj ∂xj . Along with the continuity equation, this shows that the velocity ﬁeld is divergence–free if the density and temperature are constant. On the other hand, if the Boussinesq approximation is made, an advection–diﬀusion equation is obtained for temperature. The equation of state similarly reduces to ρT = 1. The above non–dimensional equations therefore naturally yield the incompressible equations in the limit of very small Mach number. Also, all spatial derivatives in the above equations are in divergence form, and hence conservative. The above set of governing equations are therefore very attractive in that at high Mach numbers, they would yield the proper jump in variables across shock waves, and at very small Mach numbers, variations on the fast, acoustic time–scale would be projected out at time–steps larger than the acoustic time–scale. Bijl & Wesseling [2] and van der Heul et al. [3] use a similar set of equations to obtain a staggered ∂uj ∂xj

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algorithm on structured grids for the Euler equations, and inviscid MHD equations respectively. While Bijl & Wesseling solve the energy equation in non–conservative form, a fully conservative energy equation is used by van der Heul et al. [3]. 1.3 Discretization The Cartesian velocities, pressure and density are colocated in space at the centroids of the control volumes. Also, density, pressure and temperature are staggered in time from velocity. This feature makes the discretization symmetric in space and time, and is essential to ensuring zero dissipation at ﬁnite time-steps. The face normal velocity is located at the face centers and denoted by vN in this paper. At every time step, the velocity components ui , and vN are advanced from time t to t + 1 and the thermodynamic variables, p, ρ and T are advanced from t + 21 to t + 32 . Integrating the governing equations over a control volume, and using Gauss’ theorem to transform volume integrals into surface integrals yields the following discrete equations: 3

1

t+ t+ 1 t+1 t+1 ρcv 2 − ρcv 2 + ρf ace vN Af ace = 0. ∆t V

(4)

f aces

t+1 t − gi,cv gi,cv 1 t+ 12 t+ 21 ∂ t+ 21 + pcv gi,f ace vN Af ace = − ∆t V ∂xi f aces 1 1 t+ 12 τij + Nj Af ace . Re V f ace

(5)

f aces

Here, gi = ρui denotes the momentum in the i direction and (τij )f ace is the t+ 1

stress tensor at the face. Nj is the outward normal vector at the face. pcv 2 is obtained by applying the trapezoidal rule to integrate the pressure–gradient term. The discrete energy equation is given by t+ 21 t+ 12 γ−1 Mr2 γ−1 2 ∂ Mr ρui ui ρui ui + γpcv + pcv + ∂t 2 V 2 f ace f aces

1 t+ 12 (γ − 1)Mr2 1 t+ 12 vN Af ace = (τij ui )f ace Nj Af ace ·vN Af ace + V Re V f ace f aces 1 ∂T t+ 2 1 1 (6) µ + Af ace . ReP r V ∂N t+ 21

f aces

The central diﬀerences in time and space make the algorithm second order on regular grids. Also, the algorithm is fully implicit, and hence not limited

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by viscous, convective or acoustic stability limits. The discrete energy equation shows that Mr and ∆t determine whether high frequency acoustics are captured in a time–accurate manner. At small Mach numbers, ∆t of the order of Mr2 allows acoustic waves to be represented in a time–accurate manner. When high frequency acoustics are not of physical importance, the ∆t may be of the order of the convective time–scale, and the energy–equation discretely projects out acoustic eﬀects and yields zero–divergence for the velocity ﬁeld. A pressure–correction method is used to solve the above equations. A notable feature is that the face–normal velocities are projected to satisfy a constraint on the divergence that is determined by the energy equation. This is in contrast to most approaches that project the momentum to be constrained by the continuity equation. A result of using the energy equation to project the velocity is that at small Mach number, the projection step ensures that the velocity ﬁeld is discretely divergence–free. Also as will be seen below, there is no odd–even decoupling in the incompressible limit. An iterative approach is used to solve the continuity, momentum and energy equations. Let k denote an iteration level in an outer loop which seeks to advance the the velocities from t to t + 1 and pressure, temperature and density from t + 12 to t + 23 . The discrete energy equation yields an equation for the pressure correction, δp. The solution procedure is as follows. 1. Initialize the outer loop; i.e. 3

1

1

3

t+1 t uit+1,0 = uti , ρt+ 2 ,0 = ρt+ 2 , T t+ 2 ,0 = T t+ 2 , vN = vN . 3

2. Advance the continuity equation to get ρt+ 2 ,k+1 by using the face normal t+1,k velocity vN . 3. Advance the momentum predictor equation to get a provisional value of gi∗ by using pressure and velocity at current iteration step. ∗ ∗ t+1,k+1 4. Obtain velocities at the control volume centers using u∗i = gi /ρ ∗ t+1,k+1 t+3/2,k+1 t+1/2 /2. Interpolate ui to obtain vN at where ρ = ρ +ρ the faces. 5. Solve the pressure correction equation to get δp. 6. Update the pressure, momentum and the velocities at center of the control volumes, and update the face normal velocity using the pressure correction. 7. Check convergence for the pressure correction, density and momentum between outer loop iterations. 1.4 Results The algorithm is applied to simulate decaying isotropic turbulence on a very coarse grid without a subgrid model. This problem poses a severe test of the capability of the algorithm to ensure robustness at high Reynolds numbers without numerical dissipation. The initial ﬂuctuation Mach number Mt = q/a

Three Problems in LES (a) 0.02

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k

Fig. 1. Long–time evolution of inviscid, isotropic, compressible turbulence. (a) turbulence kinetic energy at Mt = 0.4, rms density at Mt = 0.4, turbulence kinetic energy at Mt = 0.01, rms density at Mt = 0.01. The range of kinetic energy is shown on the right of the plot, while the left of the plot shows the range of rms density. (b) Three–dimensional energy (E(k)) spectra as a function of time for inviscid, isotropic, compressible turbulence (Mt = 0.01. The ent/τ = 0, t/τ = 5, ergy spectrum is normalized by the initial value of q 2 t/τ = 10, t/τ = 70.

√ λ and turbulent Reynolds number, Reλ = urms . Here, q = u′i u′i , a = γRT0 ν is the mean speed of sound, and λ denotes the initial Taylor microscale. Simulations are performed for compressible (Mt = 0.4), and nearly incompressible (Mt = 0.01) conditions. Results are shown for Rλ = ∞; (i.e. inviscid). The domain is (2π)3 , and the computational grid has 32 points in each direction. The time–step is ﬁxed at 0.025 τ where τ is an ‘eddy–turnover’ time–scale which is equal to the initial value of λ/urms . Note that no subgrid model is used for this simulation. Figure 1(a) shows the solution over a long time. Note that at Mt = 0.4, the solution is stable up to about 30τ , following which the kinetic energy and density ﬂuctuations increase rapidly. This behavior is due to the formation of shock waves in the domain. In contrast, the Mt = 0.01 ﬂow maintains its initial kinetic energy over this length of time. Figure 1(b) shows three–dimensional spectra of the turbulence kinetic energy for the Mt = 0.01 ﬂow. Note that the nearly incompressible ﬂow exhibits equipartition; i.e. its spectrum at long times varies as k 2 . These results therefore show that the algorithm is stable even in the inviscid limit on the convective time-scale, and is unstable only when shock waves form. This behavior is in contrast to other commonly used non–dissipative schemes which become unstable at very short times even at very low Mach numbers; i.e. they are unstable on the convective time–scale (t/τ ∼ 1) at high Reynolds numbers (e.g. see ﬁgure 11 in [4]). Dissipative methods would cause the solution to decay even in the inviscid limit.

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1.5 Summary The algorithm addresses the problems caused by low Mach numbers and under–resolved high Reynolds numbers without numerical dissipation. It colocates variables in space to allow easy extension to unstructured grids, and discretely conserves mass, momentum and total energy. The discrete divergence is constrained by the energy equation. As a result, the discrete equations analytically reduce to the incompressible equations at very low Mach number. The algorithm discretely conserves kinetic energy in the incompressible inviscid limit, and is robust for inviscid compressible turbulence on the convective time-scale. These properties make it well-suited for DNS/LES of compressible turbulent ﬂows. A limitation, which needs to be overcome in the future is lack of shock–capturing ability.

2 RDT Applied to the Near–Wall Modeling Problem 2.1 Introduction Large–eddy simulations of attached boundary layers at high Reynolds numbers require very ﬁne near–wall resolution when the LES equations are integrated down to the wall. We consider the question of whether common subgrid models are modeling the dominant physical/numerical eﬀect of the subgrid scales in the inner–layer region. Most subgrid models are required to model the net non–linear transfer of energy from the resolved scales to the subgrid scales. However, Kline et al. [5], Uzkan & Reynolds [6], and Lee et al. [7] suggest that streaks, which dominate the near–wall region, are produced by the linear mechanism of rapid straining of turbulent ﬂuctuations. Lee et al. in particular, establish a close connection between turbulence in the viscous sub– layer, and homogeneous turbulence that is sheared at very high shear–rates. They also show that the evolution of rapidly sheared homogeneous turbulence is well described by linear rapid distortion theory (RDT) and that the RDT can reasonably predict the Reynolds stress anisotropy and structural features of near–wall turbulence. We therefore consider the possibility that the errors involved when numerically solving the RDT equations on a coarse mesh might correspond to the errors in the near–wall region on coarse meshes. The discretized RDT equations, thus obtained, can be solved analytically using the notion of ‘modiﬁed wave–number’. The RDT equations are then analyzed to explain the observed trends. 2.2 Inviscid Rapid Distortion Theory Figure 2 shows a schematic of the problem where initially isotropic turbulence is subjected to mean shear. The rate of shear is assumed rapid as compared

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Fig. 2. Schematic of the problem.

to characteristic time-scales of the turbulence. The mean velocity for homogeneous shear is (7) U1 = Sx2 , U2 = U3 = 0. and the corresponding coordinate transformation which yields constant coefﬁcient equations [8] is ξ1 = x1 − Stx2 , ξ2 = x2 , ξ3 = x3 , τ = t. The coordinate transformation yields linear, constant–coeﬃcient equations which are then solved using conventional Fourier representation. Knowledge of the Fourier coeﬃcients enables computation of the energy spectrum tensor which is then integrated over all wavenumbers to determine the Reynolds stresses. The wavenumbers vary continuously from −∞ to +∞ in an analytical representation of the turbulent ﬁeld. Two important diﬀerences arise when the same problem is numerically solved. First, ﬁnite spatial resolution implies that the wavenumbers are ﬁnite, and given by ki = 2πj/L, where j varies from −N/2 to N/2 − 1. Here, N denotes the number of grid points, and L denotes the domain size in the ith direction. Second, discretization error results in the spatial derivatives in the linear equation not being correctly represented. Both factors result in the evolution of the numerical solution being diﬀerent from the analytical solution. An analytical solution to the discrete RDT equations can be obtained. The solution is identical to the classical analytical solutions (e.g. [9,10]) with the exception that wavenumbers in the analytical solutions are replaced by the modiﬁed wavenumber (see [11] for a discussion of modiﬁed wave–numbers). Note that the solution is completely general in the choice of numerical scheme. 2.3 Results 2

Two initial spectra are considered, E(k) ∼ k −2 and E(k) ∼ (k/k0 )4 e−2(k/k0 ) . The ﬁrst choice of spectrum corresponds to the situation where the grid is so coarse that even the largest energy–containing motions are not resolved. Even a Fourier method would yield incorrect solutions under these conditions. When direct numerical simulation is performed in a channel, the minimal channel

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notion of Jimenez & Moin [12] suggests that the essential dynamics of the nearwall region and outer region can be thought of as being independent and so resolved small scale turbulence near the wall is rapidly sheared. But when the near–wall region is severely under–resolved, outer layer motions that are larger than the near-wall region will experience near-wall shear. The k −2 spectrum attempts to model this situation. The second choice of spectrum corresponds to the situation where the energy–containing motions are resolved by the grid. A Fourier method would be expected to yield reasonable results under these conditions while less accurate numerical schemes would show the eﬀects of discretization error. Lee et al. [7] establish the relation between homogeneous shear and near– wall turbulence using statistics at St = 8. In assessing the impact of numerical error, this paper therefore uses the RDT solution at St = 8. 2.4 k−2 Initial Spectrum Figure 3 shows the eﬀect of truncation error when the RDT equations are numerically solved. The initial three–dimensional energy spectrum, E(k) varies as k −2 . The computational domain is 8π × 2π × 2π, which corresponds to that used by Lee et al. [7]. The RDT equations are solved using a second–order central diﬀerence scheme; i.e. kα′ = sin kα ∆α /∆α . The ﬁnite–diﬀerence results are contrasted to those obtained using Fourier diﬀerentiation to isolate the eﬀects of discretization error and truncation. Four diﬀerent grids are considered – 16 × 128 × 16, 32 × 128 × 32, 16 × 16 × 16, and 32 × 32 × 32, respectively. The resolution of 128 in the y direction is chosen to approximate the channel simulation where the near–wall normal direction is nearly resolved (∆y + ≤ 1), while vertical resolutions of 16 and 32 assume that the corresponding channel simulation is not resolved in the near–wall direction. Around St = 8, the streamwise intensities are higher, while the vertical and spanwise intensities are smaller than their actual values. The Reynolds shear stress is closer to the exact solution than the intensities. While it does not show smaller magnitudes at all St, it oscillates about the exact solution at longer times, and is smaller in magnitude for small St. Fourier diﬀerentiation yields essentially the same results as the ﬁnite–diﬀerence scheme for R22 and R33 , although R11 and R12 are closer to the exact solution. This behavior is in contrast to that described below when the initial spectrum is resolved on the computational grid, and is explained in section 2.6. 2.5 Resolved Initial Spectrum Figure 4 shows results from RDT computations where the initial spectrum corresponds to turbulence that is resolved on the computational grid prior to being sheared. The computational domain is 2π × 2π × 2π, and the initial 2 E(k) ∼ (k/k0 )4 e−2(k/k0 ) . Here k0 is the wavenumber at which the spectrum peaks, and is chosen to be 10. Grids of size, 163 , 323 and 643 are considered.

Three Problems in LES 35

1

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-3

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Fig. 3. The evolution of the Reynolds stresses (normalized by the initial value of Rii /3) as a function of grid resolution when the RDT equations are numerically solved. The ﬁrst and second rows correspond to the second-order central diﬀerence scheme, and Fourier derivatives respectively. The ﬁrst and second ﬁgures in the third row correspond to the second-order central diﬀerence scheme, and Fourier Exact, 16 × 128 × 16 derivatives respectively. The initial E(k) ∼ k−2 . 32 × 128 × 32 mesh, 16 × 16 × 16 mesh, 32 × 32 × 32 mesh. mesh,

Since the turbulence is initially resolved, noticeable diﬀerence is observed between results obtained using a Fourier spectral scheme, and the second order ﬁnite diﬀerence scheme. Note that Fourier diﬀerentiation shows good agreement with analytical solution on the 323 and 643 grids for which the peak in the initial spectrum is less than the grid cut–oﬀ wavenumber (k = 32 and 16 respectively). Deviation from the analytical solution is only seen for the 163 grid for which the cut–oﬀ wavenumber (k = 8) is less than the peak in the spectrum. In contrast, the ﬁnite–diﬀerence solutions deviate from the exact solution at all resolutions. Note that R22 , R33 and R12 are all less in magnitude than their exact values. However, R11 is now less in magnitude than the exact solution. This behavior is in contrast to that observed when the initial spectrum was not resolved, and is a result of the ﬁnite–diﬀerence scheme rep-

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Krishnan Mahesh, Yucheng Hou, and Pradeep Babu 22

1.2

20

1.1

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Fig. 4. The evolution of the Reynolds stresses (normalized by the initial value of Rii /3) as a function of grid resolution when the RDT equations are numerically solved. The ﬁrst and second rows correspond to the second-order central diﬀerence scheme, and Fourier derivatives respectively. The ﬁrst and second ﬁgures in the third row correspond to the second-order central diﬀerence scheme, and Fourier derivatives 2 Exact, respectively. The initial E(k) ∼ (k/k0 )4 e−2(k/k0 ) where k0 = 10. 64 × 64 × 64 mesh, 32 × 32 × 32 mesh, 16 × 16 × 16 mesh.

resenting the smallest resolved scales inaccurately. The modiﬁed wavenumber shows that diﬀerencing error in the ﬁnite–diﬀerence scheme is signiﬁcant beyond k∆ ∼ 1. The peak in the initial spectrum corresponds to k0 ∆ = k0 2π/N where N is the number of grid points in each direction. The 163 , 323 and 643 grids yields values of k0 ∆ of 3.93 (unresolved), 1.96 and 0.98 respectively. The resulting truncation error has the eﬀect of ignoring the energetic scales in the initial condition, the growth in energy in those scales due to mean shear is not represented, and the net result is an underprediction of the Reynolds stresses.

Three Problems in LES (b) 0.5

0.4

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∂p -u′1 ∂ξ 1

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∂p -u′1 ∂ξ 1

(a)

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Fig. 5. The pressure-strain correlation (normalized by the initial value of SRii /3) in the R11 equation as a function of grid resolution when the RDT equations are numerically solved using (a) : the second-order central diﬀerence scheme, (b): Fourier Exact, 16 × 128 × 16 mesh, diﬀerentiation. E(k) ∼ k−2 initially. 32 × 128 × 32 mesh, 16 × 16 × 16 mesh, 32 × 32 × 32 mesh.

2.6 An Explanation The RDT equations show that the only variable being spatially diﬀerentiated is the ﬂuctuating pressure. Truncation error would therefore entirely result from errors in approximating the spatial derivatives in pressure, and then projecting the velocity ﬁeld to ensure the divergence–free condition. Note that this error includes the eﬀects of both excluding high wavenumber modes, and discretization error in diﬀerentiating the resolved modes. The analytical derivatives are exact for the resolved wavenumbers when Fourier methods are used, but exclusion of high wavenumber information implies that the derivatives in physical space are incorrect. Using ﬁnite–diﬀerence or ﬁnite–volume schemes to compute the spatial derivatives increases the error since even the resolved wavenumbers are not diﬀerentiated exactly. The evolution equations for the Reynolds stresses in the RDT limit allow further clariﬁcation. We have ∂p′ d R11 = −SR12 − u′1 , dt ∂ξ1 d ∂p′ ∂p′ R22 = −u′2 + St u′2 , dt ∂ξ2 ∂ξ1 ∂p′ d R33 = −u′3 . dt ∂ξ3

and

(8)

The Reynolds shear stress, R12 combines with mean shear to ‘produce’ R11 . ∂p′ The pressure–strain correlation, u′1 ∂ξ acts to redistribute energy from R11 1 to the other two components.The RDT results show that R11 is higher than expected, although R12 is smaller or even equal to the correct value. In other

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Krishnan Mahesh, Yucheng Hou, and Pradeep Babu

words, R11 is higher although the production term is smaller. This is only possible if the pressure–strain correlation is not large enough. Figure 5 shows the pressure-strain correlation in the R11 equation in the RDT limit. The pressure–strain term is indeed smaller at coarse resolutions for all resolutions considered. Suppression of the transfer from u′1 to the other components may be considered a result of constraining the velocity ﬁeld to be divergence–free in the presence of truncation error. Each of the individual derivatives, ∂uα /∂xα is incorrect due to truncation and discretization. The sum of the three gradients is still constrained to be zero. In terms of the Reynolds stresses, this error shows up in the pressure–strain correlation. 2.7 Summary This suggests that, near the wall, it is probably more important to account for the eﬀect of the subgrid scales on the non–local eﬀects of pressure than it is to model their nonlinear eﬀects due to advection.

3 Passive Scalar Mixing 3.1 Introduction There is reasonable evidence to suggest that large–eddy simulation yields better predictions than RANS–based approaches for turbulent mixing, and is therefore more suited to applications such as turbulent combustion. However, one objection that is commonly raised, is that turbulent mixing requires molecular mixing at the diﬀusive scales, and therefore LES, which does not directly represent the diﬀusive scales should not yield accurate predictions. This paper attempts to answer this concern by proposing a kinematic model for the scalar ﬂuctuations, which predicts the ﬂuctuations in scalar without detailed knowledge of the diﬀusive processes. 3.2 Kinematic Model for Scalar Fluctuations The Reynolds number dependence of scalar ﬂuctuations in turbulent jets is summarized by Dimotakis ([13], ﬁgure 6 of his paper, reproduced in ﬁgure 7b), who shows crms /c at the centerline as a function of jet Reynolds number, for liquid–phase and gas–phase jets. The data show that crms /c at the jet centerline decreases with increasing Reynolds number, and reaches an asymptotic value at suﬃciently high Reynolds number. Gas phase jets are observed to be less sensitive to Reynolds number than liquid phase jets, over the range of Reynolds numbers shown. A simple kinematic model is proposed below, that predicts the experimentally observed variation of scalar ﬂuctuations at the jet centerline with

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Reynolds number and Schmidt numbers. The model is also used to show that the radial variation of scalar ﬂuctuations in turbulent jets may be interpreted using the same arguments. The model assumes that scalar ﬂuctuations at a ﬁxed location in the jet result from the oscillation of scalar fronts, whose thickness depends on Reynolds number, Schmidt number, and radial position in the jet, and whose oscillation amplitude depends on the level of turbulent ﬂuctuations. The main idea behind the model is that the oscillation of sharp gradients produces high levels of ﬂuctuations. The scalar concentration in the vicinity of a measurement location in the jet is assumed to vary as follows. The ﬂuctuations about the local mean (c) are represented by fronts of thickness, δscalar , which are oscillated with amplitude δosc . Consider πx πx cmax sin( ) if sin( ) > 0, c δscalar δscalar πx = 0 if sin( ) ≤ 0. δscalar

g(x) =

(9)

The domain is assumed to be equal to length 2π for convenience. Each period of the sine function in g(x), or equivalently, each blip in g(x) is assigned an equal probability of being positive or negative. Furthermore each blip is separated by a mean distance, δsep , and is randomly translated over a distance bounded by ±δosc . The problem is statistically homogeneous, and so statistics are computed by generating an ensemble, and averaging over x and time. Note that the scalar proﬁle is completely described by the non–dimensional parameters, cmax /c and δosc /δscalar . It is readily seen that cmax /c aﬀects the absolute levels of scalar intensities, but not their variation with δosc /δscalar . Eﬀect of Reynolds and Schmidt Numbers The parameter, δosc /δscalar represents the ratio of the amplitude of oscillation of scalar fronts to their thickness. Its value will therefore depend on jet Reynolds number, Schmidt number, and radial position. As the Reynolds number increases, the thickness of the scalar fronts is expected to decrease, which implies that δosc /δscalar increases. Similarly, increasing the Schmidt number will decrease the thickness of the scalar fronts, and therefore increase the ratio, δosc /δscalar . The turbulence levels decrease with increasing radius. This means that δosc /δscalar will decrease away from the jet centerline towards the edges. This dependence can be represented as δosc δscalar

∼

Ren · Scp , rm

(10)

where m, n and p are positive. Figure 6 shows the variation of crms /c with varying δosc /δscalar as predicted by the model. The parameter, cmax /c was set equal to one, while δosc /δscalar was varied from 3 × 10−3 to 103 . Note that the intensity of the scalar ﬂuctuations decreases, and asymptotes to a constant value for suﬃciently high values

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Krishnan Mahesh, Yucheng Hou, and Pradeep Babu 1

crms /c

0.8

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Increasing δsep /δscalar 10-2

10-1

100

101

102

103

δosc /δscalar Fig. 6. Predicted values of scaled scalar intensity (cmax /c = 1) plotted against the ratio of the oscillation amplitude of scalar fronts to their thickness for two ratios of ) and 1.5 ( ). δsep /δscalar . δsep /δscalar = 0.5 ( 0.3

0.2 0.4

crms 2 /c2

∼ crms 2 /c2

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∼ log10 Re

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Fig. 7. Comparison of model prediction to experiments. Note that the axes in the model prediction are proportional to the Reynolds number and scalar intensity. Also, only the regime close to asymptotic state is shown in (a) for comparison with (b). (a): Model (at given Sc). The non–dimensional ratios δsep /δsc = 1.5 and cmax /c = 1.0. ) suggested (b): experimental data from [13]. (◦ ) water jet, ( ) air jet, and ( data ﬁt by [13] for water jets.

of the ratio δosc /δscalar . This behavior is used below to explain the eﬀect of the Reynolds number, Schmidt number and radial position in the jet. At ﬁxed radial distance and Schmidt number, equation 10 shows that δosc /δscalar ∼ Ren . Increasing Re is therefore equivalent to increasing the magnitude of δosc /δscalar , which according to ﬁgure 6 results in the scalar intensity decreasing and reaching a constant value at suﬃciently high Reynolds numbers. The experimental data of Dimotakis [13] in ﬁgure 7(b) show that gas jets appear less sensitive to Reynolds number than liquid jets at low Reynolds numbers. At suﬃciently high Reynolds numbers, both liquid as well as gas phase jets seem to reach an asymptotic value. Note that the model results

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(b)

(a) 0.4

3 2.5

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crms /cc

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Fig. 8. rms value of ﬂuctuations in scalar concentration normalized with (a): mean scalar concentration at jet centerline, (b): mean local scalar concentration. Note: , Re = 5000, Sc ∼ 600) [14], x–axis in (b) is plotted in inverse log scale. ( ◦ , Re = 5000, Sc = 1.0) [15] and ( , Re = 2400, Sc = 1.0) DNS of [16]. (

resemble behavior observed in the higher Schmidt number ﬂow. This behavior is explained as follows. cmax /c was assumed to not vary with Reynolds number in ﬁgure 7a. This assumption is valid only for high Schmidt number ﬂow. The instantaneous scalar fronts at any location are the result of advection of scalar from another location by outer–scale motions. As the scalar advects it will diﬀuse, depending upon the magnitude of 1/(ReSc). Increased diﬀusion of the scalar will result in lower values of cmax . It is readily seen that at low Re, a higher Schmidt number ﬂow will experience less diﬀusion. cmax in a lower Schmidt number ﬂow will therefore be smaller at low Reynolds numbers. At high enough Reynolds numbers, diﬀusion of the scalar will be negligible, regardless of the Schmidt number. The observed diﬀerence between experimental data in water and air jets, may be explained in these terms, within the context of the model. This argument is consistent with experiments in water jets [14] which show that unmixed free–stream ﬂuid is found to reach and cross the jet axis, while experiments [15] in gas phase jets at low Reynolds number, show that the free–stream ﬂuid mixes before reaching the jet centerline. 3.3 Eﬀect of Radial Distance Figure 8a shows scalar ﬂuctuations non–dimensionalized with mean local centerline value, and plotted against the self–similarity variable η = r/(z − z0 ). The radial variation of scalar ﬂuctuations can also be interpreted in terms of δosc /δscalar . At ﬁxed Re, equation 10 yields, δosc /δscalar ∼ Scp /rm . Also, the model predictions are for crms normalized by the local value of c. Equation 10 implies that

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Krishnan Mahesh, Yucheng Hou, and Pradeep Babu

log(δosc /δscalar ) ∼ m · log(

1 ) + p · log(Sc) r + r0

(11)

where r0 prevents the singularity at r = 0. In logarithmic coordinates, δosc /δscalar is therefore equivalent to 1/η. Figure 8b shows the same data as in ﬁgure 8a, but non–dimensionalized with the local mean scalar value, and plotted against 1/η. Note that the observed trends are similar to that predicted by the model in ﬁgure 6. Also, note that the eﬀect of Schmidt number is the same as that discussed in the preceding section, and is more pronounced near the edges of the jet. 3.4 Summary It is signiﬁcant that the kinematic model makes no assumption about the diﬀusive processes inside the scalar fronts; it only requires that the variation of their thickness be represented. The model therefore suggests an explanation for the success of methods such as large–eddy simulation in predicting scalar mixing. Large–eddy simulation, by deﬁnition does not represent viscous dissipation; it only captures large–scale convective motions. According to the model, any modeling approach which accurately captures the energy– containing convective motions, and the approximate thickness of the scalar fronts will yield good predictions for the scalar ﬁeld. Furthermore, predictions in gas–phase ﬂows, or high Reynolds number ﬂows are likely to be easier than those in liquid phase, low Reynolds number ﬂows, given the signiﬁcantly weaker dependence of gas phase scalar ﬂuctuations on Reynolds number.

Acknowledgments This work was supported by the Air Force Oﬃce of Scientiﬁc Research under grant/contract FA9550-04-1-0341, Department of Energy through the Stanford ASC Alliance Program, and the Oﬃce of Naval Research under Grant N00014-02-1-0978. Computing time was provided by the Minnesota Supercomputing Institute, San Diego Supercomputer Center and National Center for Supercomputing Applications.

References [1] Y. Hou and K. Mahesh. A robust colocated implicit algorithm for direct numerical simulation of compressible turbulent ﬂows, J. Comput. Phys., 205(1): 205–221, 2005. [2] H. Bijl and P. Wesseling. A uniﬁed method for computing incompressible and compressible ﬂows in boundary-ﬁtted coordinates. J. Comput. Phys., 141: 153–173, 1998.

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[3] D.R. van der Heul, C. Vuik, and P. Wesseling. A conservative pressurecorrection method for the Euler and ideal MHD equations at all speed. Intnl. J. Num. Methods Fluids, 40: 521–529, 2002. [4] S. Nagarajan, S.K. Lele, and J.H. Ferziger. A robust high-order compact method for large eddy simulation. J. Comput. Phys., 191: 392–419, 2003. [5] S.J. Kline, W.C. Reynolds, F.A. Schraub, and P.W. Rundstadler. The structure of turbulent boundary layers. J. Fluid Mech., 30: 741–773, 1967. [6] T. Uzkan and W.C. Reynolds. A shear-free turbulent boundary layer. J. Fluid Mech., 28: 803–821, 1967. [7] M.J. Lee, J. Kim, and P. Moin. Structure of turbulence at high shear rate. J. Fluid Mech., 216: 561–583, 1990. [8] R.S. Rogallo. Numerical experiments in homogeneous turbulence. NASA Tech. Mem., 81315, 1981. [9] A.A. Townsend. The structure of turbulent shear ﬂow. (2nd edn.) Cambridge University Press, Cambridge, 1976. [10] M.M. Rogers. The structure of a passive scalar ﬁeld with a uniform mean gradient in rapidly sheared homogeneous turbulence. Phys. Fluids A, 3: 144–154, 1991. [11] P. Moin. Fundamentals of engineering numerical analysis. Cambridge University Press, 2001. [12] J. Jimenez and P. Moin. The minimal ﬂow unit in near–wall turbulence. J. Fluid Mech., 225: 213–240, 1991. [13] P.E. Dimotakis. Mixing transition in turbulent ﬂows. J. Fluid Mech., 409: 69–98, 2000. [14] W.J.A. Dahm and P.E. Dimotakis. Measurements of entrainment and mixing in turbulent jets. AIAA J., 25: 1216–1223, 1987. [15] D.R. Dowling and P.E. Dimotakis. Similarity of the concentration ﬁeld of gas–phase turbulent jets. J. Fluid Mech., 218: 109–141, 1990. [16] P. Babu and K. Mahesh. Direct numerical simulation of passive scalar mixing in spatially evolving turbulent round jets. AIAA Paper, 2005– 1121, 2005.

Filtering the Wall as a Solution to the Wall-Modeling Problem Robert D. Moser1 , Arup Das2 , and Amitabh Bhattacharya2 1

2

University of Texas at Austin 1 University Station, Stop C2200, Austin, TX 78712, USA [email protected] Department of Theoretical and Applied Mechanics University of Illinois, Urbana, IL 61801 USA

Summary. In this paper, a possible solution to the long-standing problem of nearwall modeling in Large-Eddy Simulation (LES) is presented. It is observed that in an LES, in which resolution is ﬁnite, it is inconsistent to locate the wall with precision. Instead we propose ﬁltering through the wall using homogeneous or nearly homogeneous ﬁlters, eﬀectively smearing it. The resulting ﬁltered equations then have an explicit wall term, which is modeled using a novel optimization technique. To test the validity of this approach, simulations were done with optimal LES models for the subgrid stress term which were derived from DNS statistical data, with very good results. The properties of subgrid models that appear to be important for this application are discussed.

1 Introduction One of the pacing problems in the development of reliable large eddy simulation (LES) models for use in turbulent ﬂows of technological interest is the so-called LES wall-modeling problem [1]. It arises because the length-scale associated with the wall layer of a turbulent wall-bounded shear ﬂow (wall units) gets smaller relative to the shear layer thickness approximately like the −7/8 in the channel ﬂow). The “large-scale” inverse Reynolds number (like Reτ turbulence in this thin layer also scales in wall units. If the cost of an LES of wall-bounded ﬂows is to remain ﬁnite in the limit of inﬁnite Reynolds number, then this wall layer and the large-scale turbulence it supports cannot be represented directly, and so must be modeled. However, current LES models are generally not valid for this near-wall layer because underlying assumptions such as small-scale homogeneity and isotropy are not valid. The alternative is to resolve the near-wall turbulence. The most successful LES of wall-bounded shear ﬂows employ this technique, though this is clearly not viable for arbitrarily large Reynolds number. In this paper, we propose a possible solution to this wall-modeling problem.

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Our approach is motivated by the observation that in an LES, locating anything, including the wall, to more precision than the ﬁlter width is inconsistent with the representation. This leads us to a formulation in which the wall is ﬁltered as well as the turbulence. This allows the use of homogeneous or nearly homogeneous ﬁlters normal to the wall, avoiding commutation error. Also, the dynamics of the sublayer can be eﬀectively ﬁltered out, which we will see is advantageous. Filtering the wall, however, also introduces an extra term in the ﬁltered equations which must be modeled, along with the usual subgrid stress term. A modeling approach for this extra term is proposed here. To allow the ﬁltered boundary approach to be evaluated with minimum uncertainty arising from the subgrid stress model, optimal LES models will be used [2–5]. In optimal LES, the subgrid force term (or the subgrid stress) is approximated using stochastic estimation. Optimal LES is a formal approximation to what we have called the ideal LES evolution [2], which can be shown to produce one-time statistics that are exact, and minimum mean-square variation in the instantaneous large-scale evolution. The optimal LES formalism has the advantage in this context of being valid even in the absence of smallscale isotropy or homogeneity; that is, it is valid for near-wall turbulence. As input, optimal LES requires detailed two-point correlation data. For the purposes of testing the viability of the proposed wall-modeling approach, this data has been obtained from the direct numerical simulation data of Moser et al [6]. In the remainder of this paper, the ﬁltered boundary formulation is introduced and a test of it’s capabilities is presented. The optimal LES models used here are brieﬂy described. The results of ﬁltered boundary LES of the turbulent channel at Reτ = 590 are then presented followed by a brief discussion of the implications of these results, particularly the properties of the subgrid stress model needed in this formulation.

2 Filtered Wall Formulation In the ﬁltered boundary LES formulation, the wall-bounded domain is embedded in a larger domain, with the Navier-Stokes equations applied to the interior, and u = 0 applied to the exterior domains. However, since u = 0 is a solution to the Navier-Stokes equations, we can also extend Navier-Stokes into the exterior domain, with the wall boundary supporting a stress discontinuity just suﬃcient to maintain u = 0 in the exterior. A ﬁlter is then applied to this larger domain. The resulting ﬁltered equations are then: ∂u ˜i ˜i ∂u ˜5 ˜i ∂ p˜ 1 ∂u ju + =− + + bi + M i ∂t ∂xj ∂xi Re ∂xj ∂xj

(1)

where Mi = −∂τij /∂xi is the usual LES model term and bi is the boundary term. The subgrid stress is given by τij = u5 ˜5 i uj − u i u˜j and the boundary term (bi ) can be written

Filtering the Wall as a Solution to the Wall-Modeling Problem

bi (x) =

∂R

119

σij (x′ )nj G(x − x′ ) dx′

where σ is the stress at the boundary, including pressure and viscous stress, ∂R is the boundary of the ﬂuid region R and nj is the unit normal to the surface. If a suﬃciently ﬁne ﬁlter width is used then Mi is negligible and the only eﬀect is the ﬁltering of the boundary (i.e. a ﬁltered boundary “DNS”). Such a “DNS” was used as a test case (see below). In many LES of wall bounded ﬂows, approximate boundary conditions are used to model the eﬀect of the wall layer [7]. The approximate boundary conditions are prescribed in terms of the wall shear stress, so wall stresses must be determined in terms of the resolved velocities. In the present formulation, the unﬁltered wall stresses are also required, and for analogues reason. In the current description, in which the unﬁltered velocity is zero in the buﬀer domain external to the walls, the wall stress is the surface forcing required to ensure that momentum and energy are not transferred to the buﬀer domain. That is, that the velocity remains zero. This suggests a technique for determining the wall stress. It would be inappropriate to deﬁne a force to make the velocity zero at the boundary as in embedded boundary numerical methods [8, 9], since the ﬁltered velocity is not zero at the wall. Instead, we choose σwall to minimize the transport of momentum to the exterior domain. To this end, the wall stresses at each time step are deﬁned by minimizing 2 ∂˜ u 2 |˜ u| + α dx E= (2) ∂t B

where the integral is over the buﬀer domain. The |˜ u|2 term forces the energy 2 u term ensures that large in the buﬀer domain to be small, and the α ∂˜ ∂t errors are not incurred in the stress to instantaneously correct small errors in the buﬀer domain. The constant α controls the balance between these two competing requirements and is set to a value of order ∆t2 . In this paper, we consider turbulent ﬂow in a channel and apply a Fourier cut-oﬀ ﬁlter and Fourier spectral method in all three spatial directions, though the wall ﬁltering approach described here is applicable for general ﬁlters. In this case, the minimization of E required to determine the wall stress σ is straight forward since it can be done independently for each (kx , kz ) wavenumber, resulting in a 6-parameter optimization in (σxy , σyy , σzy ) [10]. To evaluate this approach, we consider two test cases: propagation of an Orr–Sommerfeld wave and low Reynolds number turbulence in a channel. In both cases, the Fourier cut-oﬀ ﬁlter is ﬁne enough to make the model term Mi negligible. In the Orr-Sommerfeld case, in which y resolution (ﬁlter width) of δy = 0.03δ (64 Fourier modes) is used, the simulated growth rate was within 0.25% of the exact value for the case considered. More interesting is the role the boundary term plays. Consider the exact unﬁltered pressure ﬂuctuations.

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(b)

4e−04

0e+00

−4e−04

v˜

u ˜

0e+00

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−9e−05

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−2e−04

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6e−05 0

−2e−03

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−5e−04

bv

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1

−d˜ p/dy + bv

−1

−2e−03 −3e−03 −1

0

y

1

−3e−03

1e−04 6e−05 0e+00 −1

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1

y

Fig. 1. Eﬀect of the boundary terms in the evolution of small disturbances in a channel ﬂow. (a) ﬁltered u velocity, (b) ﬁltered v velocity, (c) ﬁltered pressure, (d) pressure gradient, (e) boundary term for v equation, (f) pressure gradient + boundary term. — real part, - - - imaginary part.

They are formally zero in the exterior, resulting in a discontinuity in pressure, and the resulting Gibbs phenomenon in the ﬁltered pressure is shown in ﬁgure 1c. The wall normal pressure gradient appears in the v-momentum equation, and this quantity is dominated by the ﬁltered delta function at the boundary and the resulting Gibbs phenomenon (ﬁgure 1d). Yet the Gibbs phenomenon in velocity perturbations in ﬁgure 1a and b is imperceptible. The reason is that the term bv (ﬁgure 1e) has exactly the same structure as the pressure gradient and cancels the Gibbs phenomenon (ﬁgure 1f). The role of the boundary terms in the momentum equation is thus to regularize the stress discontinuities at the wall (both pressure and viscous stresses). To assess the applicability of this technique in simulating turbulent ﬂow, a fully developed channel ﬂow is computed on a 128 × 256 × 128 grid in x, y and z respectively, with 20 y-points in the buﬀer region. The friction Reynolds number (Reτ = 180), the domain size and the horizontal resolution are the same as in the DNS of Moser et al [6]. In the wall-normal direction, the eﬀective resolution (ﬁlter width) is ∆y + = 1.5 throughout the channel, which is ﬁner than the DNS resolution , except for the region y + < 5, where the

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3

20

+ + u+ rms , vrms , wrms

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u+

15

10

2

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5

1

0.5

0

1

10

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0

0

50

y+

100

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y+

Fig. 2. Mean (left) and rms (right) velocities in turbulent channel ﬂow at Reτ = 180. present using ﬁltered boundary formulation, Moser et. al. (1999)

DNS grid is strongly clustered with 9 points. Thus, this is essentially a DNS with a ﬁltered wall. The mean and rms velocities from this simulation are in excellent agreement with those from the DNS (see ﬁgure 2), and the near wall turbulence exhibits the familiar structures, such as streaks and inclined shear layers.

3 Optimal Large Eddy Simulation Optimal LES is based on the observation that there is an ideal LES model, which guarantees correct single time statistics and minimum error is shorttime dynamics [2, 11]. This ideal model given by mi = Mi |˜ u = w ,

(3)

where mi is the model for the term Mi in (1), w is the LES ﬁeld, and u ˜ is the ﬁltered real turbulence. In essence, this is the average of Mi over all turbulence ﬁelds that map to the LES ﬁeld through the ﬁlter. Unfortunately this model is intractable, so in optimal LES we approximate this model using stochastic estimation [12–14]. In the LES performed here, the stochastic estimation formulation is simpliﬁed by the homogeneity of the channel ﬂow in directions parallel to the wall, and the formulation must be further simpliﬁed to avoid problems of over generalization [3]. For each wavenumber, the linear stochastic estimate used here can thus be written: ˆ i + K ˆ ij (y)E ˆj (y) m ˆ i (y) = M ˆ i′ (y)E ˆk∗ (y) = K ˆ ij (y) E ˆj (y)E ˆk∗ (y) M

(4) (5)

ˆ ij is the optimal LES model kernel, where ˆ· indicates the Fourier transform, K and the event vector Ej is a vector consisting of the ﬂuctuating LES velocities wj′ and their y derivatives. The correlations appearing in (5) must be determined to complete the model. For the purposes of the test described below,

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30 DNS Filtered DNS LES

25

2.0

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0.5

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100

1000 +

0.0 0

100

200

300

400

500

600

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y y Fig. 3. Mean velocity (left) and rms streamwise velocity ﬂuctuations (right) in a turbulent channel at Reτ = 590. The LES was performed using the ﬁltered boundary optimal LES formulation.

the correlations were evaluated using the DNS data from Moser et al [6] at ˆ ij was thus determined once from the DNS Reτ = 590. The model kernel K and then used in the LES reported here. Using DNS data allows the optimal LES formulation to be evaluated without uncertainties introduced by further modeling of the correlations, or other modeling approximations.

4 Filtered-Wall LES Results The ﬁltered boundary formulation and the optimal LES model were used to perform an LES of turbulent channel ﬂow with bulk Reynolds number Reb = 10, 950 corresponding to a channel with Reτ = 590. Periodic boundary conditions were used in streamwise (x) and spanwise (z) directions, with domain sizes Lx = 2πh and Lz = πh (h is the channel half width). DNS of this case was performed by Moser et al [6], and optimal LES were performed by Volker et al [3]. To accommodate the ﬁltered boundary formulation, a buﬀer region is added outside the channel and periodic boundary conditions are used in the extended wall-normal (y) domain. Fourier cut-oﬀ ﬁlters with eﬀective ﬁlter widths of ∆x+ = 116, ∆y + = 37 and ∆z + = 58 are used in the three spatial directions. In x and z, these are the same ﬁlters used in Volker et al [3]. Note that these ﬁlter widths are suﬃciently large to eliminate the structure of the near-wall viscous and buﬀer layers. The ﬁltered boundary model and the optimal LES model were used to perform an LES of the channel ﬂow. The statistical correlations required as input to the optimal LES formulation were determined from the DNS of Moser et al [6]. Sample results from this simulation are shown in ﬁgure 3. Note that despite the fact that the wall layer was not resolved, both the mean velocity and the rms velocities are in remarkably good agreement with the ﬁltered DNS. Furthermore, the one-dimensional spectra in the x and z directions are in good agreement with the ﬁltered DNS data (see ﬁgure 4).

Filtering the Wall as a Solution to the Wall-Modeling Problem

10

10

0

10

y + = 147

-1

-2

10

(a)

y + = 442

E(kz )

E(kx )

10

10 10

(b) y + = 147

0

-1

y + = 442

-3

-4

-3

10 1

1

-2

10

123

10

10 1

10

Fig. 4. One-dimensional streamwise (a) and spanwise (b) spectra in the channel; LES ( ), ﬁltered DNS ( )

It should be noted that the optimal LES form used here is the simplest of those proposed by Volker et al [3] for the channel, and that in that study models of this form performed poorly. The reason was that this form does not properly represent the wall-normal transport of energy and Reynolds stress. As pointed out by Haertel et al [15], in the absence of wall-normal ﬁltering, the contribution of the subgrid term to the resolved-scale energy equation is positive near the wall (see ﬁgure 5), which is due to the subgrid contribution to the transport of energy from the production peak to toward the wall. Volker et al [3] found that a more complicated form that did represent the wall normal transport produced a model that performed very well. However, with coarse wall-normal ﬁltering, the subgrid term is everywhere dissipative, as is also shown in ﬁgure 5. There is therefore no instability introduced by using the simple model form used here, with the result that the model produces high quality simulations, as shown above. Also notice the large ﬂuctuations (wiggles) in the proﬁle of the subgrid dissipation shown in ﬁgure 5. The a posteriori dissipation from the LES reproduces these features, and is otherwise in general agreement with the a priori values. The reason for these oscillations is the Gibbs phenomenon introduced by ﬁltering through the wall, and sublayer. These ﬂuctuations are apparently important in the simulation, because they cancel similar Gibbs phenomena appearing in other terms, as with the Orr-Sommerfeld test case described in section 2.

5 Discussion The results described above are intriguing because they suggest that it is not necessary to resolve the near-wall layer in an LES to obtain an accurate simulation of a wall-bounded ﬂow. However, because the simulations reported here were based on knowledge of statistical correlations obtained from DNS, the work presented here does not constitute a practical broadly-applicable

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no y–ﬁlter y–ﬁlter a posteriori

20

˜ ui Mi

10 0

-10 -20 -30

0

100

200

300

y+ Fig. 5. Subgrid energy transfer ˜ ui Mi in turbulent channel ﬂow, as measured from DNS,[6] with and without wall-normal ﬁltering, and a posteriori from the ﬁlteredwall LES.

LES model. For this, the need for DNS statistical data must be overcome. None-the-less, the current results do demonstrate the value of the wall-ﬁltering approach, and the “no-leakage” optimization model for the wall stresses. It would appear that this approach may form the foundation of a solution to the well-known LES wall modeling problem. The results of this study are also curious, because the “no-leakage” model does not incorporate any information regarding the properties of the near-wall sublayer that has been ﬁltered away. Whereas it has been commonly assumed that the wall boundary treatment must account for the presence of the sublayer. Further, the “no-leakage” model cannot by its nature account for the presence or absence of roughness. This raises the question of how information regarding near-wall properties of the turbulence (which are directly aﬀected by roughness) is incorporated into the LES. The answer is that the nature of the near-wall turbulence is reﬂected in the subgrid ﬂuctuations, which aﬀect the LES through the subgrid model term Mi = −∂τij /∂xj . In the simulation reported here, the optimal LES model of the subgrid term is formulated based on statistical data obtained from a smooth-wall DNS. Thus, the simulation produced results consistent with a smooth-walled channel. We propose that an optimal model formulated for a particular wall roughness would also produce an LES consistent with that rough wall. It appears then, that the LES wall boundary condition problem may not be as diﬃcult as has been commonly believed. The ﬁltered boundary, “noleakage” formulation, appears to provide an adequate boundary condition. Instead, the challenge is the modeling of the volumetric subgrid stress term. However, at high Reynolds number, and with the sublayer eﬀectively ﬁltered out, the model need only represent the subgrid terms in the log-layer, where similarity scaling makes the modeling easier. As indicated above, it is also necessary that the eﬀect of roughness on the log-layer subgrid model be characterized.

Filtering the Wall as a Solution to the Wall-Modeling Problem

125

To relieve the need for DNS data, which was used for subgrid modeling here, a model for the subgrid term with minimal empirical input and which retains the good properties of the optimal LES model is needed. But, what properties of the subgrid model used here are important for LES performance in wall-bounded ﬂows? The formulation of the optimal model provides guidance, because it was designed to accurately represent only the transfer of energy and Reynolds stress to the small scales, and the direct contribution of the subgrid to the mean Reynolds stress. More precisely, it provides an accurate representation of the dependence of the subgrid dissipation on the x and z wavenumbers and y location, as well as the anisotropy of the dissipation. Whether this much detail in the representation of the subgrid dissipation is needed, is not at this point clear, and is being investigated. Perhaps, if less detail is suﬃcient, standard models (e.g. Smagorinsky) may be applicable. However, such standard models are not likely to be able to produce the oscillatory behavior in the dissipation (ﬁgure 5), as our experience with dynamic Smagorinsky in this context shows. This suggests that the Fourier cut-oﬀ ﬁlter would be a poor choice for wall ﬁltering, if standard subgrid models are to be used. Finally, we note that the tests performed here were particularly arduous for the ﬁltered boundary formulation because discontinuities in derivatives (as in the velocity at the wall) are poorly represented by Fourier spectral methods, with Gibbs phenomena as the result. It is particularly remarkable, then, that the wall stress model used here is able to treat and largely cancel this Gibbs phenomenon.

Acknowledgments This work was supported by the United States National Science Foundation, Air Force Oﬃce of Scientiﬁc Research and Department of Energy. This support is gratefully acknowledged. We are also indebted to Prof. Javier Jimenez and ´ Dr. Juan Carlos del Alamo for helpful discussion.

References [1] U. Piomelli and E. Balaras. Wall-layer models for large-eddy simulations. Ann. Rev. Fluid Mech., 34:349–374, 2002. [2] J. Langford and R. Moser. Optimal LES formulations for isotropic turbulence. Journal of Fluid Mechanics, 398:321–346, 1999. [3] S. Volker, P. Venugopal, and R. D. Moser. Optimal large eddy simulation of turbulent channel ﬂow based on direct numerical simulation statistical data. Physics of Fluids, 14:3675, 2002. [4] P. S. Zandonade, J. A. Langford, and R. D. Moser. Finite volume optimal large-eddy simulation of isotropic turbulence. Physics of Fluids, 16:2255– 2271, 2004.

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[5] J. A. Langford and R. D. Moser. Optimal large-eddy simulation results for isotropic turbulence. Journal of Fluid Mechanics, 521:273–294, 2004. [6] R.D. Moser, J. Kim, and N.N. Mansour. Direct numerical simulation of turbulent channel ﬂow up to Reτ = 590. Phys. Fluids, 11(4):943–945, April 1999. [7] E. Balaras, C. Benocci, and U. Piomelli. Two-layer approximate boundary conditions for large-eddy simulations. AIAA Journal, 34:1111–1119, 1996. [8] R. Verzicco, J. Mohd-Yusof, P. Orlandi, and D. Haworth. LES in complex geometries using boundary body forces. Technical Report Annual Research Briefs, Center for Turbulence Research, Stanford University, 1998. [9] J. Mohd-Yusof. Development of immersed boundary methods for complex geometries. Technical Report Annual Research Briefs, Center for Turbulence Research, Stanford University, 1998. [10] Arup Das and R. D. Moser. Filtering boundary conditions for LES and embeded boundary simulations. In L. Sakell C. Liu and T. Beutner, editors, Proc. International conference on DNS/LES, 3rd. Greyden Press, 2001. [11] S. B. Pope. Turbulent Flows. Cambridge University Press, 2000. [12] R. Adrian. On the role of conditional averages in turbulence theory. In J. Zakin and G. Patterson, editors, Turbulence in Liquids, pages 323–332. Science Press, Princeton, New Jersey, 1977. [13] R. Adrian, B. Jones, M. Chung, Y. Hassan, C. Nithianandan, and A. Tung. Approximation of turbulent conditional averages by stochastic estimation. Physics of Fluids, 1(6):992–998, 1989. [14] R. Adrian. Stochastic estimation of sub-grid scale motions. Applied Mechanics Review, 43(5):214–218, 1990. [15] C. H¨ artel and L Kleiser. Analysis and modelling of subgrid-scale motions in near-wall turbulence. J. Fluid Mech., 356:327–352, 1998.

A Near-Wall Eddy-Viscosity Formulation for LES Georgi Kalitzin1 , Jeremy A. Templeton2 , and Gorazd Medic3 1 2 3

637 Alvarado Row, Stanford, CA 94305, [email protected] P.O.Box 19565, Stanford, CA 94309, [email protected] P.O.Box 19803, Stanford, CA 94309, [email protected]

Summary. A near-wall eddy-viscosity formulation for LES is presented. This formulation consists of imposing a RANS eddy-viscosity dynamically corrected with the resolved turbulent stress in the near-wall region. The RANS eddy-viscosity is obtained from an averaged velocity proﬁle of a resolved LES of channel ﬂow at Reτ = 395 and stored in a look-up table. Results are presented for channel ﬂow at Reτ = 395 with no-slip boundary conditions, and up to Reτ = 1, 000, 000 using a wall model.

1 Introduction Large-eddy simulations (LES) of attached ﬂows are currently limited to moderate Reynolds numbers due to the need to resolve the small, energy containing scales near the wall. It has been estimated that the cost of resolving such scales adequately with sub-grid scale (SGS) models goes as Re2τ [1]; almost as expensive as direct numerical simulation (DNS). Due to the signiﬁcance of this problem, many solutions have been proposed. One such solution called wall modeling dates back to some of the earliest LES [2]. Wall models are formulations that provide approximate wall boundary conditions for LES when the wall layer is not resolved. Model variations range from imposing a log-law type relation between velocity and shear stress [3], full solution of a set of near-wall equations [4] and the use of suboptimal control techniques [5]. The diﬃculty with using a wall model is that it must compensate not only for unresolved physics, but also for numerical and SGS modeling errors [6]. Another solution is to solve the RANS equations in the near-wall layer while forcing the ﬂow into an unsteady LES-like mode away from the wall. One such approach is detached eddy simulation (DES) in which the modiﬁed Spalart-Allmaras turbulence model provides the SGS eddy-viscosity for the LES away from the wall. This approach creates an artiﬁcial buﬀer layer at the location where the RANS model switches to LES [7]. A recent review of similar techniques can be found in [8].

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Georgi Kalitzin, Jeremy A. Templeton, and Gorazd Medic

In this paper, a near-wall eddy-viscosity formulation for LES is presented. A RANS eddy-viscosity is imposed in the near-wall region and is corrected with the resolved turbulent stress. The RANS eddy-viscosity is obtained from a resolved LES channel ﬂow at Reτ = 395 and stored in a look-up table. This approach can be used with both a no-slip boundary condition or a wall stress model. Results are presented for channel ﬂow at Reτ = 395 with a wall-resolved grid, and up to Reτ = 1, 000, 000 using coarse grids.

2 Near-Wall Treatment 2.1 Look-up Tables Look-up tables were generated for both the wall shear stress, τw = ρu2τ , and eddy-viscosity, νt with a method similar to what was used in [9] for RANS wall functions . The tables are constructed using an averaged velocity proﬁle obtained from the resolved LES of a fully developed channel ﬂow at Reτ = 395, which has been extrapolated in the logarithmic region. The averaged velocity proﬁle u+ (y + ) is transformed to u+ (y + · u+ ) = u+ (Re), where Re = yu/ν. The friction velocity, uτ , follows explicitly from u1 and y1 in the ﬁrst cell center above the wall. The look-up table for the RANS eddy-viscosity, νt+ (y + ), is obtained from the non-dimensional RANS equation for channel ﬂow (1 + νt+ ) du+ /dy + = 1 − y + /Reτ , where du+ /dy + is the gradient of the averaged velocity taken from the LES. When the tables are used only for y + values that lie in the universal region, the eddy-viscosity can be created without the last term on the right hand side. 2.2 Dynamic Procedure for Near-Wall Eddy-Viscosity Correction The basic idea behind the derivation of a dynamic procedure for near-wall eddy-viscosity correction consists of comparing the averaged LES equation for the streamwise velocity component in channel ﬂow d dˆ u dp¯ˆ les (1) uvˆ + (ν + νt ) −ˆ = dy dy dx with the corresponding RANS equation d du d¯ p (ν + νtrans ) = dy dy dx

(2)

ˆ denotes the LES ﬁlter and () ¯ a plane- or time-average. where () If the same mean pressure gradient is imposed in both equations, then −ˆ uvˆ + (ν + νtles )

du dˆ u = (ν + νtrans ) dy dy

(3)

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Assuming that the eddy-viscosity and the wall-normal mean velocity gradient ¯ u ˆ les du are independent, the subgrid stress can be written as νtles dˆ dy = ν t dy . This assumption has been widely used in LES (e.g. [5]) and was veriﬁed to be a good approximation in the present computations. For an ideal LES and RANS that predict exactly the same mean ﬂow, the velocity gradients in equation ¯ (3) are equal: du ˆ/dy = d¯ u/dy. With these assumptions, equation (3) provides a relation between the LES and RANS eddy-viscosity. rans ν les +u ˆvˆ / t = νt

d¯ u . dy

(4)

Since u ˆvˆ and d¯ u/dy have opposite signs, the second term on the right hand side of equation (4) is negative. Thus, in the presence of turbulent ﬂuctuations, the mean LES viscosity is always less than the RANS viscosity with the diﬀerence being a dynamic correction for the resolved ﬂuctuations. Note that as the grid is reﬁned, more ﬂuctuations are resolved and ν¯tles becomes smaller. In the limit of a grid with DNS resolution all turbulent stress is resolved and ν¯tles → 0. In our approach, the eddy-viscosity in the LES region, νtsgs , is obtained from the dynamic Smagorinsky SGS model [10] and the instantaneous eddyviscosity in the near-wall region, νtsgs,nw , is computed using the equation (4). The velocity gradient and the RANS eddy-viscosity are obtained from lookup tables and the turbulent stress, u ˆvˆ, comes from the LES. The averaging operator used in equation (4) can either be plane- or time-averaging. When using this approach, it is necessary to clip the near-wall eddyviscosity, νtsgs,nw . As is standard practice when using the dynamic model, the eddy-viscosity, νtsgs , is clipped whenever its value drops below zero. In the near-wall region the eddy-viscosity is clipped at the level of the dynamic model.

3 Numerical Results Results presented in this paper have been computed using a second-order centered ﬁnite diﬀerence LES code [11]. As a proof of concept, simulations have ﬁrst been performed for channel ﬂow at Reτ = 395 on a wall-resolved grid. The streamwise, wall-normal and spanwise dimensions of the channel are 2πh × 2h × πh. The grid is uniform in the streamwise and spanwise directions with 128 and 96 cells, respectively. In the wall-normal direction, a hyperbolic tangent distribution is used with 129 cells. At the wall, the ﬁrst cell center is at y1+ ≈ 1 while in the channel center the grid spacing is ∆y + ≈ 10. This grid resolution is about twice as coarse in each direction as the grid used in [12] for a DNS using a pseudo-spectral method.

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+

switch at y =23 + switch at y =69 + switch at y =115 LES

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Fig. 1. Fine grid, Reτ = 395, mean velocity, using νtrans (left) and corrected eddyviscosity νtcorr (right). (See Plate 13 on page 419) 3

3

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Fig. 2. Fine grid, Reτ = 395, rms velocities, using νtrans (left) and corrected eddyviscosity νtcorr (right). (See Plate 14 on page 419) 10

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Fig. 3. Fine grid, Reτ = 395, spanwise energy spectra of streamwise velocity near the wall (left) and at the center of channel (right). (See Plate 15 on page 419)

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Fig. 4. Fine grid, Reτ = 395, streamwise vorticity, isosurfaces of ωx = ±45; LES (left) and LES with corrected near-wall eddy-viscosity (right). (See Plate 16 on page 420)

Fig. 5. Fine grid, Reτ = 395, streamwise vorticity, isosurfaces of ωx = ±45; LES with corrected near-wall eddy-viscosity (left) and LES with RANS eddy-viscosity in near-wall region (right). (See Plate 17 on page 420)

The near-wall eddy-viscosity, νtsgs,nw , was applied up to various y + locations (y + = 23, 69 and 115), above which the viscosity from the dynamic model was used. νtsgs,nw is set to either the RANS eddy-viscosity νtrans or the proposed corrected eddy-viscosity νtcorr deﬁned by equation (4). In both cases, the RANS eddy-viscosity is obtained from the look-up table based on the local value of y + . Results are presented in Figures 1-8. The corresponding mean and rms velocity proﬁles, Figures 1 and 2, show that the ﬂuctuations in the near wall region need to be accounted for. The rms velocities are signiﬁcantly damped when νtrans is used and the damping is larger for a wider near-wall region. The correction, deﬁned by equation (4), captures well the amount by which the eddy-viscosity needs to be reduced. Thus, the results obtained using νtcorr agree signiﬁcantly better with the resolved LES than those using νtrans . The spanwise energy spectra of the streamwise velocity are shown in Figure 3. In the center of the channel (y + = 395), all simulations show similar be-

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1

1

using νrans t modeled resolved total theoretical

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Fig. 7. Fine grid, Reτ = 395, switch at y + = 115, stress balance, using νtrans (left) and νtcorr (right)

havior to the resolved LES. However, closer to the wall (y + = 23), the results using νtrans with the switch at y + = 115 have signiﬁcantly more damping at higher wave-numbers. This can also be shown in physical space by visualizing the ﬂow structures using the streamwise vorticity component of the velocity. The inﬂuence of the near-wall eddy-viscosity on the near-wall structures is illustrated in Figures 4-5. In Figure 4, the vortices computed with νtcorr used up to y + = 23 appear similar to the resolved LES. If the switching occurs at y + = 115, the same is true for νtcorr , whereas when νtrans is used many fewer turbulent structures are retained. Stress balances for simulations with the switch occurring at y + = 23 and y + = 115 are presented in Figures 6 and 7, respectively. Clearly, the use of νtrans in the near-wall region reduces the resolved stress while increasing the modeled stress (the modeled stress plotted in the ﬁgures also includes the viscous component). Note that νtrans is an order of magnitude greater than νtcorr , as shown in Figure 8.

A Near-Wall Eddy-Viscosity Formulation for LES

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Fig. 8. Fine grid, Reτ = 395, eddy-viscosity computed using νtrans (left) and νtcorr (right) in the near-wall region. (See Plate 18 on page 421) 12

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Fig. 9. Coarse grid, Reτ = 395, eddy-viscosity computed using νtcorr in near-wall region. (See Plate 19 on page 421)

Computations have also been performed on a coarse grid using 16×17×16 cells by applying a wall stress model. The ﬁrst cell center is at y1+ = 23 and νtcorr is applied in 1, 2 and 3 cells above the wall with y + = 23, 69 and 115, respectively. The eddy-viscosity proﬁles computed using the corrected eddyviscosity, νtcorr , are presented in Figure 9. A comparison to Figure 8 reveals that the eddy-viscosity νtcorr adjusts to the coarseness of the grid, i.e. it is much larger on the coarser grid. The mean velocity proﬁles are presented in Figure 10. Note that the velocity is signiﬁcantly underpredicted when only the viscosity from the dynamic model is used. Examples of the spanwise energy spectra of the streamwise velocity in the middle of the channel ( at y + = 395) and the ﬂow structures are presented in Figure 11. On grids this coarse, only the very large structures are resolved. These structures look qualitatively similar to the large structures computed on ﬁne grids. Similarly to the wallresolved grid computations, the resolved stress is signiﬁcantly larger than the

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+

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Fig. 10. Coarse grid, Reτ = 395, using corrected eddy-viscosity νtcorr in 1, 2 and 3 wall-adjacent cells. (See Plate 20 on page 422) 10

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Fig. 11. Coarse grid, Reτ = 395, spanwise energy spectra of streamwise velocity (left); streamwise vorticity, using νtcorr , isosurfaces of ωx = ±5 (right). (See Plate 21 on page 422) 1

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Fig. 12. Coarse grid, Reτ = 395, stress balance, using νtcorr in 1 cell with y + = 23 (left) and in 3 cells up to y + = 115 (right)

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modeled stress. This is shown by the stress balances computed using the corrected eddy-viscosity with the switch at y + = 23 and 115 presented in Figure 12. Obviously, when νtcorr is used over a wider near-wall region, the modeled stress increases. Additional simulations were performed for Reτ = 1000, 2000, 4000 and 10, 000 using grids consisting of 32 × 33 × 32 cells uniformly distributed in each direction. The mean velocity proﬁles are shown in Figure 13, which summarizes the results for the corrected eddy-viscosity method (with νtcorr applied in cell 1) for these moderate Reynolds numbers. Unlike in RANS computations, the ﬂow structures are present and the stress balance shows that the resolved stress is signiﬁcantly larger than the modeled stress (see Figure 14 for Reτ = 4000).

LES, Reτ = 395 Reτ = 395 Reτ = 1000 Reτ = 2000 Reτ = 4000 Reτ = 10000

30 25

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Reτ = 4,000 0.8

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Fig. 14. Coarse grid, Reτ = 4, 000, streamwise vorticity, isosurfaces of ωx = ±5 (left) and stress balance (right). (See Plate 23 on page 423)

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4 Application to Very High Reynolds Numbers So far, the near-wall eddy-viscosity formulation has been applied to ﬂow computations on coarse grids at moderate Reynolds numbers. This section explores the ability of the proposed formulation (as well as of the dynamic Smagorinsky model) to compute channel ﬂow at very high Reynolds numbers, up to Reτ = 1, 000, 000. Reynolds numbers that high are usually found in aeronautical ﬂows. The computations at Reτ = 100, 000 have been carried out on grids with 32 × 33 × 32, 48 × 49 × 48, 64 × 65 × 64 and 96 × 97 × 96 cells. Figure 15 shows the mean velocity proﬁles compared to the logarithmic law. Two additional velocity proﬁles are also included: one from a resolved LES at Reτ = 395 computed with the same code with a no-slip boundary condition and the other, at Reτ = 100, 000, where the wall stress, τw , was imposed as the boundary condition, but no modiﬁcations were made to the eddy-viscosity (the eddyviscosity from the dynamic Smagorinsky model is used in the entire domain). The logarithmic law is well predicted on all four grids using the eddy-viscosity correction. This is not trivial, as illustrated by the result where only τw is applied with no correction to the SGS viscosity which signiﬁcantly underpredicts the velocity proﬁle. The computations at Reτ = 1, 000, 000 have been carried out on grids with 64×65×64 and 96×97×96 cells. The mean velocity proﬁles are shown on the right of Figure 15. These indicate that for this extremely high Reynolds number, a larger value of κ might be more appropriate since the velocity proﬁle drops below the logarithmic law as the grid is reﬁned. The eddy-viscosities for diﬀerent grids at both Reynolds numbers are presented in Figure 16. Clearly, the near-wall formulation increases the eddy-viscosity in the wall adjacent cells. Note that both the subgrid scale eddy-viscosity and the near-wall eddy-viscosity increase as the grid is coarsened (for both

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Fig. 15. Reτ = 100, 000 and Reτ = 1, 000, 000, various grids, mean velocity proﬁles. (See Plate 24 on page 424)

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Reynolds numbers). Like for moderate Reynolds numbers the method adapts to the diﬀerent Reynolds numbers and resolutions.

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Fig. 17. Reτ = 100, 000 and Reτ = 1, 000, 000, 96 x 97 x 96, streamwise vorticity, isosurfaces of ωx = ±55 (left) and ωx = ±5 (right). (See Plate 25 on page 424)

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An important question is what features are retained in simulations on grids this coarse. To investigate this issue, isosurfaces of the instantaneous streamwise vorticity are plotted in Figure 17 for both Reynolds numbers on the 96 × 97 × 96 grid. Turbulent structures are clearly visible, conﬁrming that large eddies are indeed present (obviously, as the grid is reﬁned smaller structures are being resolved). The presence of large eddies is also supported by the energy spectra shown in Figure 18. The results also indicate that with such coarse grids no inertial range is present in the computations for these Reynolds numbers. However, an energy cascade is observed for these large scales. Interestingly, at these low wave numbers the energy scales as k −2/3 . This observation is also interesting with regard to the well-known fact that Smagorinsky model is derived based on the assumption of resolution down to the inertial range, which the spectra clearly show is not the case with these simulations. Stress balances for both Reynolds numbers on the 96 × 97 × 96 grid are presented in Figure 19. These demonstrate that most of the stress is

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being resolved and the turbulent ﬂuctuations are captured. They also indicate how the proposed near-wall method adapts to the grid resolution.

5 Conclusions A near-wall eddy-viscosity formulation for LES is proposed. A RANS eddyviscosity is imposed in the near-wall region and is corrected with the resolved turbulent stress. This approach recovers the correct mean velocity by properly accounting for the ﬂuctuations in the near-wall region. The use of a wall stress model extends the method to coarse grids allowing the computation of channel ﬂow over a wide range of Reynolds numbers. In addition, the proposed approach is computationally inexpensive and simple to implement. The application to complex ﬂows will be investigated in future work.

References [1] J.S. Baggett, J. Jimenez, and A.G. Kravchenko. Resolution requirements in large-eddy simulation of shear ﬂows. CTR Annual Research Briefs, 51–66, 1997. [2] U. Schumann. Subgrid scale model for ﬁnite diﬀerence simulations of turbulent ﬂows in plane channels and annuli. J. Comp. Phys. 18, 376– 404, 1975. [3] U. Piomelli, J. Ferziger, P. Moin, and J. Kim. New approximate boundary conditions for large eddy simulations of wall bounded ﬂows. Phys. Fluids A (1), 1061–1068, 1989. [4] M. Wang and P. Moin. Dynamic wall modeling for large-eddy simulation of complex turbulent ﬂows. Phys. Fluids 14(7), 2043–2051, 2002. [5] F. Nicoud, J.S. Baggett, P. Moin, and W. Cabot. LES wall-modeling based on suboptimal control theory and linear stochastic estimation. Phys. Fluids 13(10), 2968–2984, 2001. [6] W. Cabot. Wall models in large eddy simulation of separated ﬂow. CTR Annual Research Briefs, 97–106, 1997. [7] N.V. Nikitin, F. Nicoud, B. Wasistho, K.D. Squires, and P.R. Spalart. An approach to wall modelling in large-eddy simulations. Phys. Fluids. Letters 12(7), 1629, 2000. [8] P. Sagaut. Large Eddy Simulation for Incompressible Flows. Springer, Berlin, 2002. [9] G. Kalitzin, G. Medic, G. Iaccarino, and P.A. Durbin. Near-wall behavior of RANS turbulence models and implications for wall functions. J. Comp. Physics 204(1), 265–291, 2005. [10] M. Germano, U. Piomelli, P. Moin, and W. Cabot. A dynamic subgridscale eddy-viscosity model. Phys. Fluids A (3), 1760–1765, 1991.

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[11] J.A. Templeton. Wall Models for Large-Eddy Simulation based on Optimal Control Theory. PhD thesis, Stanford University, Stanford, 2005. [12] R.D. Moser, J. Kim, and N.N. Mansour. Direct numerical simulation of turbulent channel ﬂow up to Reτ = 590. Phys. Fluids 11(4), 943–945, 1999.

Investigation of Multiscale Subgrid Models for LES of Instabilities and Turbulence in Wake Vortex Systems R. Cocle, L. Dufresne, and G. Winckelmans Universit´e catholique de Louvain (UCL) Mechanical Engineering Department, Division TERM and Center for Systems Engineering and Applied Mechanics (CESAME) 1348 Louvain-la-Neuve, Belgium [email protected] Summary. This paper investigates the capabilities of diﬀerent subgrid scale (SGS) models, including the recent “multiscale” models, for large-eddy simulation (LES), here in a vortex-in-cell (VIC) method, of complex wake vortex dynamics. More speciﬁcally, we here consider the multiscale dynamics developing in a counterrotating four-vortex system, that evolves from a simple state to a turbulent state. The various SGS models are tested and compared on this complex and transitional ﬂow. Comparisons are also made with results obtained using a pseudo-spectral method. Energy diagnostics (global and modal) and spectra are provided and used to support the comparisons. A discussion on the applicability of the various models to LES of complex wake vortex ﬂows is made. The multiscale models are seen to be the most appropriate.

1 Introduction The present investigation is concerned with LES of complex and transitional ﬂows at very high Reynolds number, in aircraft wake vortex systems and involving multiscale instabilities. The ﬂow problem considered here provides potential for rapid changes in ﬂow topology as well as for the generation of small scale structures and turbulence, through the strong interaction between counter-rotating vortices. To simulate such high Reynolds number ﬂows, one needs LES models that are inactive during the initial, well-resolved, phase of the dynamics, and become active only when the strong interactions between the vortices create small scales. We here consider the dynamics of a longitudinally periodic counterrotating four-vortex system and we investigate diﬀerent SGS models. More speciﬁcally, two types of SGS eﬀective viscosity models are investigated: 1) models acting on the complete LES ﬁeld, τij = 2 νsgs Sij : the Smagorinsky (SMAG) model, the “small-complete” Filtered Structure Function (FSF)

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model, the “small-complete” version of the Smagorinsky model (SMAG2) and the Wall-Adapting Local Eddy-viscosity (WALE) model; 2) models acting on a s : the “complete-small” Regularized Varismall-scale LES ﬁeld, τij = 2 νsgs Sij ational Multiscale (RVM) model, the “small-small” Regularized Variational Multiscale (RVM2) model and the “small-small” Filtered Structure Function (FSF2) model. The notation “small/complete - small/complete” means that νsgs is computed from the small-scale or the complete LES ﬁeld, and that the diﬀusion operator acts on the small-scale or the complete LES ﬁeld. A simulation is also carried out using a low order hyper-viscosity (HV) SGS model, thus also pushing the SGS dissipation towards the smallest scales. Comparisons are also made with results obtained using a Fourier-based pseudo-spectral method with a high order k 8 hyper-viscosity SGS model. The numerical method and the SGS models are ﬁrst presented brieﬂy. The deﬁnition of the ﬂow problem is presented next, together with some deﬁnitions for the energy diagnostics. We then show results on the global evolution of the vortex system, followed by the more formal comparison between the diﬀerent models. A discussion on the applicability of the diﬀerent models to complex wake vortex ﬂows, transitional and turbulent, concludes the paper.

2 Numerical Method We here consider LES of incompressible vortex ﬂows, using an eﬀective SGS viscosity νsgs and in the vorticity-velocity formulation of the Navier-Stokes equations. Depending if the LES model acts on the complete LES ﬁeld or on the small-scale LES ﬁeld, the equations are respectively: Dω = ∇ · (u ω) + ν∇2 ω + ∇ · νsgs ∇ω + (∇ω)T , Dt Dω = ∇ · (u ω) + ν∇2 ω + ∇ · νsgs ∇ω s + (∇ω s )T , Dt

(1) (2)

in which ω = ∇ × u . As the problems of interest are at very high Reynolds number and in unbounded domain, we here furthermore restrict ourselves to LES where the molecular viscosity ν is set to zero. The numerical solution of (1) or (2) is obtained following the vortex-in-cell (VIC) approach, see [1–3]. The vorticity ﬁeld is constructed from a sum of regularized vortex particles: 1 |x − xp (t)|2 αp (t) √ , exp − ω σ (x, t) = σ2 ( π σ)3 p

with αp = ω dx = ω p h3 the strength of particle p, xp its position, h the discretization size (grid size, also used for particle redistribution), √ and σ the regularization parameter (overlapping parameter, taken as σ = 2 h). Interpolations between particles and grid, as well as particle redistribution, are all done using the M4′ scheme. The vector streamfunction ψ is obtained

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143

by solving the Poisson equation ∇2 ψ = −ω on the grid, using 2nd order ﬁnite diﬀerences (FD). The velocity ﬁeld (needed for convection and stretching) is then obtained from u = ∇ × ψ, also using FD. The convective part (i.e., the lhs of (1) or (2)) is done using the Lagrangian approach: dxp /dt = u(xp ); this ensures good convection (i.e., negligible dispersion errors). The time variation of the particle strengths (i.e., the rhs of (1) or (2) including both the vorticity stretching and the dissipation terms) is evaluated on the grid using FD. The global time marching procedure is carried out using the Leap Frog scheme for the convection and the Adams-Bashford scheme for the diﬀusion. Finally, the intrinsic divergence-free character of the vorticity ﬁeld is maintained by a proper projection of the discrete vorticity ﬁeld (which also requires solving a Poisson equation). Particular to our VIC implementation is the treatment of the boundary conditions on the grid, required for solving the Poisson equation. The boundary conditions for ψ are here obtained using the Green’s function approach, via a Parallel Fast Multipole (PFM) method [3–5]. The unbounded domain condition can then be ensured accurately while using a small grid: a grid only as large as the vorticity ﬁeld itself. This allows for very signiﬁcant reduction of the computational cost. Furthermore, the method can be parallelized, using the domain decomposition method: the PFM code, which has a global view of the whole ﬁeld, is used to obtain the proper boundary conditions on each subdomain [3, 4]. Such VIC-PFM combination amounts to a very eﬃcient parallelized Poisson solver: one without iteration between the subdomains. 2.1 SGS Models Investigated Concerning the LES modelling, we investigated two approaches: νsgs computed using the complete LES ﬁeld or using the small-scale LES ﬁeld. When (1) is used, we refer to the ﬁrst approach as “complete-complete” and to the second as “small-complete”. Equivalently, when (2) is used, the notation used is: “complete-small” for the ﬁrst approach and “small-small” for the second. For all models, the LES cut-oﬀ length ∆ is taken equal to the grid size h (here uniform). We begin with the models used in (1). The Smagorinsky (SMAG) model is the most common “complete-complete” model: νsgs = CS ∆2 ( 2 Sij Sij )1/2 ,

(3)

where Sij is the strain rate tensor: Sij = 12 ( gij + gji ) with gij = ∂ui /∂xj . The coeﬃcient CS was set to its theoretical value in isotropic turbulence at high Reynolds number: CS = (0.3)3 = 0.027. We also recall that, for ﬂow with homogeneous directions (isotropic turbulence, channel ﬂow), the coeﬃcient can also be obtained dynamically, using the “dynamic procedure” [6, 7]: this is then the “dynamic” version of the Smagorinsky model. In fact, any model can be made dynamic (e.g., a

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R. Cocle, L. Dufresne, and G. Winckelmans

a dynamic mixed model as in [8]). Notice that the dynamic procedure does not change the spectral behavior of the Smagorinsky model: it remains a “complete-complete” model. Moreover, unless one uses ad hoc “clipping”, the dynamic procedure is not usable for ﬂows without homogeneous directions, such as the ﬂows investigated here. Thus, the present paper investigates SGS models from the point of view of their spectral behavior; recognizing that any model can be made dynamic. A second model considered here is the FSF model [9] (a “small-complete” model). Here, the subgrid viscosity is related to the “ﬁltered structure func(n) tion” F2s : (n)

νsgs = CF ∆

F2s

(n)

(n)

(x) 2

,

(4)

where

F2s

(n)

=

&

us

(n)

(x + x′ ) − us

'

|x′ | = ∆

.

(5)

Numerically, this is evaluated using the nearest neighbors (33 = 27) values. (n) The small-scale ﬁeld us is here obtained as in [9] (also for the FSF2 model below) by application (n times) of the 2nd order FD Laplacian: n (n) ∆2 2 us ∇ = − u. (6) 4 (n)

The present results were obtained using the coeﬃcients CF as proposed in [9], see Table 1. The model has already been shown to capture well the transition to turbulent ﬂow that results from the growth of instabilities in wake vortex simulations [10]. A third model is the SMAG2 model (also a “small-complete” model): (n)

s s 1/2 νsgs = CS2 ∆2 ( 2 Sij Sij ) ,

(7)

s where Sij is the strain rate tensor of the small-scale ﬁeld. The small-scale ﬁeld is here obtained (and also for all models presented after, except FSF2) using the compact (stencil 3) tensor-product discrete ﬁlter, and that is iterated n times to produce a order 2n ﬁlter [11, 12]:

us u

(n)

(n)

(n)

=u−u 0 n 1 n n u I − −δz2 /4 I − −δy2 /4 = I − −δx2 /4

(8)

where δx2 fi,j,k = fi+1,j,k − 2fi,j,k + fi−1,j,k , etc. In Fourier space, the ﬁltered ﬁeld is (n)

u

(k) = G(n) (kx hx ) G(n) (ky hy ) G(n) (kz hz ) u(k) (9) 2n 2n 2n = 1 − sin (kx hx /2) 1 − sin (ky hy /2) 1 − sin (kz hz /2) u(k) .

Investigation of Multiscale Subgrid Models

145 (1)

The model was used with the coeﬃcients given in Table 1. The coeﬃcient CS2 was ﬁxed using, as for the RVM models (see the explanation below), the same ratio between the SGS viscosity computed with a small-scale ﬁeld and with a (1) complete ﬁeld: 1.88/1.60 = 1.18 which gives CS2 = 1.18 CS . The coeﬃcients for n > 1 are computed so that each operator G(n) has the same value as G(1) at the medium wavenumber, k = kc /2 = π/(2h): 1 − sin2n (α(n) π/4) = G(n) (α(n) π/2) = G(1) (π/2) = 1 − sin2 (π/4) = 1/2. One obtains α(2) = 1.27 and α(3) = 1.40. A last model is the WALE model [13]: νsgs = CW ∆2

d d 3/2 ( Sij Sij )

d S d )5/4 ( Sij Sij )5/2 + ( Sij ij

,

(10)

d is the traceless part of the square of the velocity gradient tensor: where Sij

Sij =

1 (gik gkj + gjk gki ) = Sik Skj + Ωik Ωkj , 2

d Sij = Sij −

1 Skk δij , (11) 3

where Ωij = 12 (gij − gji ) is the rotation rate tensor. The coeﬃcient CW was d d Sij has set from isotropic turbulence to CW = (0.5)2 = 0.25. The term Sij been shown to be a good detector of turbulence. In pure shear (e.g. next to a wall), the model gives a zero νsgs ; in turbulent channel ﬂow, νsgs has the proper near-wall behavior. In pure rotation however, νsgs is found to be proportional to the vorticity (νsgs = CW ∆2 (2/3)1/4 |ω|/2); it is thus maximum in the center of a vortex : a non-desirable feature in our applications, as we wish to capture the development and growth of inviscid unstable modes while preserving the vortex cores as much as possible. We now present the models used in (2) where the SGS viscosity is applied on the small-scale vorticity ﬁeld (ω s ), here obtained using (8) acting on ω. The ﬁrst “complete-small” model is the “regularized” version, here using discrete ﬁlters, of the “variational multiscale” model (VM) of [14, 15]. The VM model used by Hughes et al. is a “large-small” model as the viscosity is computed using the large-scale LES ﬁeld (i.e. using u). Here, it is simply the Smagorinsky model but acting on the small-scale ﬁeld, it is thus a “complete-small” model. It was ﬁrst proposed and tested, in isotropic turbulence, by [11] who called it the “regularized variational multiscale” (RVM) model. It was then tested by [12] in channel ﬂow and using the dynamic procedure: thus a dynamic RVM model. The RVM model was then further proposed and tested by [16] and also (n) by [17, 18]. The theoretical (for isotropic turbulence) model coeﬃcients CR (1) are given in Table 1. CR is obtained knowing that |S|2 ≃ |S|2 +|S s |2 and using the values proposed in [14], with sharp Fourier cutoﬀ applied at the medium wavenumber. This choice is justiﬁed as the discrete n = 1 ﬁlter is symmetrical with respect to the medium wavenumber: G(1) (π/2) = 1/2. From the “largesmall” and “small-small” models coeﬃcients used by Hughes et al. (2.62 CS and 1.64 CS respectively), we can deduce that |S s |/|S| ≃ 2.62/1.64 = 1.60.

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It leads, using the previous relation, to |S|/|S| ≃ 1.88 and thus |S|/|S s | ≃ (1) 1.88/1.60 = 1.18. Finally, this gives CR ≃ (2.62/1.88) CS = 1.39 CS . The second model is the regularized version (RVM2) of the “small-small” variational multiscale (VM2) model of [14, 15]. It was proposed and evaluated by [16] and [17, 18]: (n)

s s 1/2 νsgs = CR ∆2 ( 2 Sij Sij ) .

(12) (1)

(n)

The coeﬃcients CR2 are given in Table 1 and are computed from the CR2 given in [14]. The VM and RVM models (also their VM2 and RVM2 counterparts) have been tested, with good success, on homogeneous isotropic turbulence and on channel ﬂow. Here, we want to test their ability to simulate complex (transitional and turbulent) vortex ﬂows. The last “small-small” model considered is the FSF2 model [18]: (n) (n) νsgs = CF2 ∆ F2s , (13) (n)

(1)

where the coeﬃcients CF2 are given in Table 1. The coeﬃcient CF2 was (1)

determined from CF , using the same ratio as between a SGS viscosity acting (1) on a small-scale ﬁeld (RVM) and on a complete ﬁeld (SMAG): it gives CF2 = (1)

(2)

(3)

(2)

1.39 CF . CF2 and CF2 are obtained, in the same way, from CF respectively.

(3)

and CF

Table 1. Coeﬃcients values for the diﬀerent SGS models, depending on the order of the ﬁlter used. The values in boldface are determined from theory or are taken as proposed in the literature (notice that, as previously computed, α(2) = 1.27 and α(3) = 1.40). CS

CS2

CF

CW

CR

CR2

CF2

n=0

(0.3)3

−

−

(0.5)2

−

−

−

n=1

−

1.18 CS

0.080

−

(2)

1.39 CS (2)

1.64 CS (2)

1.39 0.080 (2)

n=2

−

α

1.18 CS 0.069

−

α

n=3

−

α(3) 1.18 CS 0.054

−

α(3) 1.39 CS α(3) 1.64 CS α(3) 1.39 0.054

1.39 CS α

1.64 CS α

1.39 0.069

In addition to the eddy-viscosity models, we also investigated simple (uniform) hyper-viscosity models. With the VIC-PFM methodology, we used Dω = ∇ · (u ω) − νhv ∇4 ω Dt

(14)

Investigation of Multiscale Subgrid Models

147

where ∇4 was evaluated by two applications of the FD Laplacian and where the hyper-viscosity was taken as νhv = C4 ∆4 /T0 with T0 the global time scale. With the parallel pseudo-spectral code (velocity based), used for comparisons, we used a high order k 8 hyper-viscosity model: Du = −∇P − νhv ∇8 u Dt

(15)

with νhv = C8 ∆8 /T0 . The dealiasing is done using a phase shift procedure [19, 20], and the time marching is carried out using a 3rd order Runge-Kutta scheme. In a spectral method, such high order term is easily done. Even higher order terms (∇16 u) have been used before in the simulation of turbulent ﬂows [21, 22].

3 Simulation Results 3.1 Deﬁnition of the Problem The base ﬂow condition consists of a four vortex system with two pairs of counter-rotating vortices, see Fig. 1. The outer principal vortices are characterized by their circulation Γ1 , their core size radius a1 , and their inner spacing b1 . The inner secondary pair is characterized by Γ2 , a2 and b2 . Normalization of the dimensional quantities can be made using the natural reference scales: b0 = (Γ1 b1 + Γ2 b2 )/Γ0 with Γ0 = Γ1 + Γ2 , V0 = Γ0 /(2π b0 ) and T0 = b0 /V0 . In what follows, all quantities will be written in dimensionless form (for instance, τ = t/T0 ). The results were obtained considering the case where Γ2 /Γ1 = −0.3 , b2 /b1 = 0.3 , a1 /b1 = 0.075 , and a2 /a1 = 2/3 . This conﬁguration was shown to develop rapidly growing instabilities [4, 23]. This conﬁguration provides a challenging simulation problem with rapid change in ﬂow topology, generation of small scale structures and, ﬁnally, turbulence. Simpler conﬁgurations,

Fig. 1. Schematic transversal view of a four-vortex system in counter-rotating conﬁguration.

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R. Cocle, L. Dufresne, and G. Winckelmans

as the co-rotating vortex pair, have been studied by Crouch [27] (a simulation was also performed in [4]). The counter-rotating conﬁguration creates more rapid instabilities, strong interactions between vortices of opposite sign and diﬀerent strength, and turbulence; it constitutes a good conﬁguration to test the ability of SGS models in LES of complex (transitional and turbulent) vortex ﬂows. Notice that such ﬂow also develops short wavelength (elliptic type) instabilities (see Widnall [24]) and long wavelength (Crow type) instabilities. See also reviews by Spalart [25] and Jacquin et al. [26]. The longitudinal vorticity distribution of each vortex is here taken as: ωx =

a2 Γ , 2 π (r + a2 )2

(16)

with r2 = (y − y0 )2 + (z − z0 )2 , r being the distance to the vortex centerline of coordinates { y0 , z0 } . 3.2 Longitudinal Modal Energy Before presenting the comparisons between the results, we provide deﬁnitions regarding the characterization of the instabilities and the evaluation of the energy diagnostics. Taking into account the periodicity in the longitudinal direction, we write, for the velocity ﬁeld, Nx /2

u(x, y, z, t) =

m=−Nx /2

(kx , y, z, t) eikx x u

with

kx = m

2π , (17) Lx

with Nx the number of grid points; a similar development applies to ψ and ω. The longitudinal Fourier transform of u is given by Lx 1 (kx , y, z, t) = u u(x, y, z, t) e−ikx x dx . (18) Lx 0 The longitudinal modal energy Em (for a given domain length Lx , we make use of the modal index integer m instead of the wavenumber kx ) is accordingly deﬁned by 0 1 Lx Lx ∗ ∗ dydz , (19) ·ω ·ψ ∗ + ω ·u dydz = Em (t) = ψ u 2 4

with superscript ∗ denoting the complex conjugate. Using the symmetry (Em = E−m ), we consider only the modes with m ≥ 0 . Accordingly, the modal energy becomes 0 1 Lx ∗ ·ω ∗ dydz . (20) ·u dydz = ∗ + ω ·ψ u ψ Em (t) = Lx 2 The total energy is then

N/2

E =

m =0

Em .

(21)

Investigation of Multiscale Subgrid Models

149

3.3 Global Evolution The global evolution of the four-vortex system is illustrated in Fig. 2. The length of the domain was set to Lx /b0 = 8.53 to allow for the simultaneous development of one long wavelength “Crow” type instability, in addition to the medium and short wavelength instabilities. As the secondary vortices orbit around the primary ones, the fastest growing medium wavelength instability strongly distorts the secondary vortices to form “Omega-loops”, as also observed experimentally [28]. Furthermore, as the vortices come into contact, and only partially reconnect (they cannot fully reconnect as they have diﬀerent circulation), a signiﬁcant amount of small scale structures is being produced, eventually leading to a turbulent ﬂow. These results were obtained using VIC-PFM for 5800 time steps in 65 hours on 48 Xeon processors. The VIC grid grew from 720 × 128 × 64 to 720 × 310 × 312 points and the number of vortex particles from 3.3 to 18 millions. The low order hyper-viscosity SGS model was used (Eq. 14, and using C4 = 6.8). This run was done so as to compare with one carried using the spectral code, and using a high order k 8 hyper-viscosity SGS model (Eq. 15, and using C8 = 3.5). It was then later veriﬁed that the hyper-viscosity SGS models are not as performant as the multiscale models (see next section). The results presented here are thus for assessment of instability growth and global evolution. We show, in Fig. 3, the evolution of the modal energies Em of the base ﬂow (m = 0) as well those for the most unstable short (m = 77, elliptic instability), medium (m = 9, Omega-loops), and long (m = 1, Crow instability) modes; the values have been normalized by the initial base ﬂow energy Em=0 (τ = 0) (noted E0 ). The values obtained using the spectral code are also shown for comparison. Assuming an exponential growth (i.e., Em ∝ exp(2 σm τ )) for the early (linear) phase of the dynamics, we obtain, for both simulations, σ1 ≈ 26, σ9 ≈ 15 and σ77 ≈ 15 . Notice that both runs are not aﬀected by the SGS model so long as there are no small scales produced: thus up to τ ≈ 1. The small diﬀerences in the time at which the unstable modes start to grow comes from the slightly diﬀerent initial, low amplitude, random perturbation. The evolution of both the total and the base ﬂow energies is shown in Fig. 4. The agreement between the energy evolutions is seen as quite good. 3.4 Comparison of the SGS Models Due to the high computational cost of the simulations above, the detailed comparison of the SGS models was carried out on a computational domain corresponding to one single medium wavelength instability: Lx /b0 = 0.76 . To further standardize the evolution of the vortex system, the same structured low amplitude perturbation of the fundamental mode m = 1 (i.e., λ/b0 = 0.76) was used for all cases. The evolution of this short four-vortex system, obtained using the VIC-PFM with the RVM model (n = 3), is shown in Fig. 5.

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R. Cocle, L. Dufresne, and G. Winckelmans

Fig. 2. Evolution of the four vortex system as obtained using the VIC-PFM with hyper-viscosity model. Iso-surfaces shown are |ω|b21 /Γ1 = 10 (high opacity) and 2 (low opacity). Times shown are τ = 0.0, 0.52, 1.06, 1.16 and 1.58 .

Investigation of Multiscale Subgrid Models

151

m=0

0

10

−2

10

Em E0

m=9 −4

10

m = 77

−6

10

−8

10

m=1

−10

10

0

0.25

0.5

0.75

1

1.25

1.5

τ Fig. 3. Evolution of the modal energy Em for the base ﬂow mode m = 0, the equivalent long wavelength Crow mode (m = 1), the most unstable medium (m = 9) and short (m = 77) wavelength modes; VIC-PFM (solid ) and spectral (dash).

In Fig. 6, we show the evolution of the modal energies (base ﬂow, perturbed mode and its ﬁrst harmonic) for various SGS models. The modal energy values have been renormalized by the initial base ﬂow energy E0 . For all models, we obtain the same growth rate values σ1 ≈ 14 (similar to σ9 ≈ 15 for the long domain simulation) and σ2 ≈ 27. When reaching the nonlinear phase and saturation, small diﬀerences are seen to occur, but the overall modal energy behavior remains similar. In Figs. 7 and 8, we show the evolution of both the total and the base ﬂow energies. Before the instability signiﬁcantly develops, the ﬂow remains essentially 2D and both E and Em=0 remain nearly equal. When the ﬂow becomes 3D, the diﬀerence between the two curves becomes apparent. As the simulations are done without ν, one would expect energy conservation as long as there are no strong interactions between the primary and secondary vortices. As expected, the Smagorinsky model produces signiﬁcant eﬀective dissipation. If we associate its energy decrease between τ = 0 and 0.6 to an eﬀective Reynolds number (using dE/dt = −2 νeff E , in which E = 21 |ω|2 dV is the enstrophy), we obtain Re = Γ0 /νeff ≈ 1 104 : indeed quite low (recall that a vortex at Re = Γ0 /ν = 5 103 is already quite diﬀusive and is easily computed using DNS, as in [3, 5]). The results obtained using the WALE

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R. Cocle, L. Dufresne, and G. Winckelmans

1

E E0 0.8

Em=0 E0

0.6

0.4

0.2

0

0

0.25

0.5

0.75

1

1.25

1.5

τ Fig. 4. Evolution of the total energy E and of the base ﬂow energy Em=0 : VIC-PFM (solid ) and spectral (dash).

model are even worse: this is due to its high dissipation in vortex ﬂows. The SMAG and the WALE models are clearly not acceptable for LES of high Reynolds number vortex ﬂows. On the contrary, the multiscale models (RVM, RVM2, FSF, FSF2, SMAG2) all perform quite well: good energy conservation up to the time of strong vortex interactions, and then roughly same energy decay of the turbulent ﬂow. The low order HV model also produces good results, yet not as good as those obtained using the multiscale models. For the ﬁnal simulation time shown (τ = 1), the multiscale models lead to about the same energy: E/E0 ≈ 0.83 . The results shown are all n = 3; but even n = 1 produces results far superior to SMAG or WALE models. Finally, we also consider the spectral behavior of the models, see Figs. 9 and 10 which provide the spectrum of the turbulent ﬂow (at τ = 1.05). We conﬁrm the good behavior of the multiscale models, best using the available wavenumbers of the LES grid: they preserve the low to medium wavenumbers (inertial range) while providing dissipation at the high wavenumbers. The SMAG and WALE models dissipate at all wavenumbers, thus signiﬁcantly altering the inertial range dynamics of the LES. As a last observation, we note that the energy decay during the turbulent phase of the ﬂow is essentially independent of the SGS model details, as long as the model is “good” and thus does not aﬀect the large to medium scale

Investigation of Multiscale Subgrid Models

153

Fig. 5. Evolution of the four-vortex system obtained using VIC-PFM with the RVM model. Iso-surfaces shown are |ω|b21 /Γ1 = 10 (high opacity) and 2 (low opacity). Times shown are τ = 0.0, 0.61, 0.79 and 1.05 .

dynamics (here of order of the Omega-loops). In turbulence, it is indeed the dynamics of those scales that determine the dissipation rate.

4 Conclusions We have considered LES of transitional ﬂows involving multiscale instabilities developing in complex vortex systems and with small cores (as typical of aircraft wake vortices). First, it is essential that the numerical method used be of high quality: energy conserving (in absence of viscosity and SGS modelling) and with negligible dispersion; the VIC-PFM method and the spectral method are such quality methods. Second, it is also most important that the SGS model be active only during the complex phases of the ﬂow (here, the strong interactions between the opposite sign vortices, leading to partial reconnection, generation of small scales and turbulence), while remaining inactive during the “gentle” phases. The SMAG and the WALE models are not able to achieve that. The multiscale models (RVM, RVM2, FSF, FSF2, SMAG2) achieve that naturally: the eﬀect of the model remains negligible as long as

154

R. Cocle, L. Dufresne, and G. Winckelmans m=0

0

10

−2

10

−4

10

Em E0

m=1 −6

10

m=2

−8

10

−10

10

−12

10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ Fig. 6. Evolution of the modal energy Em for the base ﬂow mode m = 0, the perturbed mode m = 1 and its ﬁrst harmonic m = 2: SMAG (dot), RVM (dashdot), FSF (dash), and spectral method with k8 hyper-viscosity (solid ). Results with the other multiscale models (not shown) are similar to RVM.

there is no signiﬁcant energy content in the high wavenumbers of the LES ﬁeld. Hyper-viscosity models also do that; yet they do not perform as well: they tend to over-dissipate at the high wavenumbers. From a computational point of view, they are some diﬀerences in the CPU cost between the models. The “small-small” models (RVM2 and FSF2, here with n = 3) are about 40% more expensive than the basic SMAG model: this is so because we here need to ﬁlter twice (ﬁlter u and ω). The “smallcomplete” and “complete-small” models (RVM, FSF, SMAG2) are about 25% more expensive. For LES of complex (transitional and/or turbulent) vortex ﬂows, and using FD methods (thus also VIC; other validations being currently performed using a fourth-order FD code in the velocity-pressure formulation), the multiscale SGS models come out as good choices. They are easily implemented (here using iteration of eﬃcient, tensor product, stencil-3 discrete ﬁlters) and they have a good spectral behavior: they preserve the low to medium wavenumbers while they provide the SGS dissipation at the high wavenumbers.

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155

1

E E0 0.9

0.8

0.7

Em=0 E0 0.6

0

0.25

0.5

0.75

1

τ Fig. 7. Evolution of the total energy E and of the base ﬂow energy Em=0 : SMAG (dot), RVM (dash-dot), FSF (dash), WALE (solid with crosses), low order HV (solid with circles) and spectral method with k8 hyper-viscosity (solid ).

Acknowledgments Louis Dufresne was also funded as postdoctoral research fellow from Canada by NSERC/CRSNG. Part of the simulations were done using the facilities of the Centre for Research in Aeronautics CENAERO (www.cenaero.be). Some simulations were also done as part of a benchmark exercise developed during the AWIATOR project (IP of the European 5th Framework Programme). Part of this work was also done in support of the FAR-Wake project (STREP of the European 6th Framework Programme). Finally, we also thank G. Daeninck and L. Bricteux for their contribution concerning the theoretical analysis of the WALE model.

References [1] G.-H. Cottet and P.D. Koumoutsakos. Vortex Methods: Theory and Practice, Cambridge Univ. Press, 2000. [2] G.-H. Cottet and P. Poncet. Advances in direct numerical simulations of 3D wall-bounded ﬂows by Vortex-in-Cell methods, J. Comp. Phys., 193, 136–158, 2004.

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1

E E0 0.9

0.8

0.7

Em=0 E0 0.6

0

0.25

0.5

0.75

1

τ Fig. 8. Evolution of the total energy E and of the base ﬂow energy Em=0 : SMAG2 (dot), RVM (dash-dot), RVM2 (dash-dot with crosses), FSF (dash) and FSF2 (dash with circles).

[3] G.S. Winckelmans. Chapter 5: “Vortex Methods,” in Volume 3 (Fluids) of the Encyclopedia of Computational Mechanics, Erwin Stein, Ren´e de Borst, Thomas J.R. Hughes (Eds), John Wiley & Sons, 2004. [4] G. Winckelmans, R. Cocle, L. Dufresne, and R. Capart. Vortex methods and their application to trailing wake vortex simulations, C. R. Physique, 6, 467–486, 2005. [5] R. Cocle, G.S. Winckelmans, G. Daeninck, and F.Thirifay. An eﬃcient combination of vortex-in-cell and fast multipole methods, 2005, submitted to J. Comp. Phys.. [6] M. Germano, U. Piomelli, P. Moin, and W. Cabot. A dynamic subgridscale eddy-viscosity model, Phys. Fluids, A 3, 1760, 1991. [7] S. Ghosal, T.S. Lund, P. Moin, and K. Akselvoll. A dynamic localization model for large-eddy simulation of turbulent ﬂows, J. Fluid Mech., 286, 229, 1995. [8] G.S. Winckelmans, A.A. Wray, O.V. Vasilyev, and H. Jeanmart. Explicitﬁltering large-eddy simulation using the tensor-diﬀusivity model supplemented by a dynamic Smagorinsky term, Phys. Fluids, 13, 1385-1403, 2001.

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−1

10

k−5/3 −2

10

E(k) −3

10

−4

10

−5

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k Fig. 9. Energy spectrum at τ = 1.05: SMAG (dot), RVM (dash-dot), FSF (dash), WALE (solid with crosses) and low order HV (solid with circles).

[9] F. Ducros, P. Comte, and M. Lesieur. Large-eddy simulation of transition to turbulence in a boundary layer spatially developing over a ﬂat plate, J. Fluid Mech., 326, 1–36, 1996. [10] F. Laporte. Simulation num´erique appliqu´ee `a la caract´erisation et aux instabilit´es des tourbillons de sillage d’avions de transport, PhD thesis, INPT, Toulouse, 2002. [11] G.S. Winckelmans and H. Jeanmart. Assessment of some models for LES without and with explicit ﬁltering, Direct and Large-Eddy Simulation IV, edited by Geurts B.J., Friedrich R. and M´etais O., ERCOFTAC Series 8, Kluwer, 55–66, 2001. [12] H. Jeanmart and G.S. Winckelmans. Comparison of recent dynamic subgrid-scale models in turbulent channel ﬂow, Proc. Summer Program 2002, Center for Turbulence Research, Stanford University and NASA Ames, 105–116, 2002. [13] F. Nicoud and F. Ducros. Subgrid-scale stress modelling based on the square of the velocity gradient tensor, Flow Turb. Comb., 62(3), 183– 200, 1999. [14] T.J.R. Hughes, L. Mazzei, A.A. Oberai, and A.A. Wray. The multiscale formulation of large eddy simulation: decay of homogeneous isotropic turbulence, Phys. Fluids, 13, 505–512, 2001.

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k Fig. 10. Energy spectrum at τ = 1.05: SMAG2 (dot), RVM (dash-dot), RVM2 (dash-dot with crosses), FSF (dash) and FSF2 (dash with circles).

[15] T.J.R. Hughes, A.A. Oberai, and L. Mazzei. Large eddy simulation of turbulent channel ﬂow by the variational multiscale method, Phys. Fluids, 13, 1784–1799, 2001. [16] A.W. Vreman. The ﬁltering analog of the variational multiscale method in large-eddy simulation, Phys. Fluids, 15, L61, 2003. [17] S. Stolz, P. Schlatter, D. Meyer, and L. Kleiser. High-pass ﬁltered eddyviscosity models for LES, Direct and Large-Eddy Simulation V, edited by Friedrich R., Geurts B.J. and M´etais O., Kluwer, 81–88, 2004. [18] S. Stolz, P. Schlatter, and L. Kleiser. High-pass ﬁltered eddy-viscosity models for large-eddy simulations of transitional ﬂow, Phys. Fluids, 17, 065103, 2005. [19] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang. Spectral Methods in Fluid Dynamics, Springer-Verlag, New-York, 1988. [20] J.P. Boyd. Chebyshev and Fourier Spectral Methods, Dover Publ., Mineola, 2nd ed, 2001. [21] V. Borue and S.A. Orszag. Self-similar decay of three-dimensional homogeneous turbulence with hyperviscosity, Phys. Rev. E, 51(2), R856–R859, 1995. [22] V. Borue and S.A. Orszag. Kolmogorov’s reﬁned similarity hypothesis for hyper-viscous turbulence, Phys. Rev. E, 53(1), R21–R24, 1996.

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[23] D. Fabre, L. Jacquin, and A. Loof. Optimal perturbations in a fourvortex aircraft wake in counter-rotating conﬁguration, J. Fluid Mech., 451, 319–328, 2002. [24] S.E. Widnall. The structure and dynamics of vortex ﬁlaments, Ann. Rev. Fluid Mech., 7, 141–165, 1975. [25] P.R. Spalart. Airplane trailing vortices, Ann. Rev. Fluid Mech., 30, 107– 138, 1998. [26] L. Jacquin, D. Fabre, D. Sipp, V. Theoﬁlis, and H. Vollmers. Instability and unsteadiness of aircraft wake vortices, Aerosp. Sci. Tech., 7, 577–593, 2003. [27] J.D. Crouch. Instability and transient growth for two trailing-vortex pairs, J. Fluid Mech., 350, 311–330, 1997. ¨ Sava¸s. Experimental study of the insta[28] J.M. Ortega, R.L. Bristol and O. bility of unequal-strength counter-rotating vortex pairs, J. Fluid Mech., 474, 35–84, 2003.

Numerical Determination of the Scaling Exponent of the Modeled Subgrid Stresses for Eddy Viscosity Models Markus Klein, Martin Freitag, and Johannes Janicka Institute of Energy- and Powerplant Technology Technical University Darmstadt Petersenstr. 30, D–64287 Darmstadt, Germany [email protected] Summary. LES quality assessment is very important in view of predictive LES applications. Recently Klein [1] proposed to evaluate the numerical as well as the modeling error in a LES using an approach based on Richardson extrapolation, where it is assumed that the modeling error scales like a power law. In order to apply this approach, the scaling exponent for the numerical error with respect to the ﬁlter width has to be known in advance. This scaling law will be explored for three diﬀerent conﬁgurations: a channel ﬂow, a plane jet and a swirling recirculating ﬂow. Theoretical argumentation [2, 3] leads to a scaling of m = 2/3. The current ﬁndings suggest to use △4/3 for ﬂow conﬁgurations operating at moderate Reynolds numbers. The resulting scaling exponent will be used to assess the quality of LES simulations of these conﬁgurations.

1 Introduction LES has the potential to accurately predict complex ﬂow phenomena. But still there are some fundamental questions which have not been answered [4] and no single LES procedure has yet emerged as a standard [5]. The veriﬁcation of the simulation results is therefore extremely important. Uncertainty or quality assessment in RANS simulations has been investigated by several authors in the past [6] and there are guidelines available. In the context of LES, things become more complicated: In addition to the subgrid parameterization, the accuracy is strongly inﬂuenced by the numerical contamination of the smaller retained ﬂow structures [7]. Furthermore the numerical and the modeling error interact. This problem has recently been discussed in the literature [7–11]. Klein [1] proposed to evaluate the numerical as well as the modeling error using an approach based on Richardson extrapolation, where it is assumed that the modeling error scales like a power law. In order to apply this approach the scaling exponent for the numerical

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error with respect to the ﬁlter width has to be known in advance. The response of the modeled subgrid stresses with respect to a diﬀerent ﬁlter width is the focus of the present work.

2 Numerical Technique and Conﬁguration The governing equations for the problem to be investigated here, are the conservation equations of mass and momentum for an incompressible Newtonian ﬂuid in their instantaneous, local form. The LES equations are derived by applying a ﬁlter function to the conservation equations. The ﬁltering process separates the geometry dependent large scales from the more universal small scales which have to be modeled by a subgrid scale model. In this work, the well known Smagorinsky model and the corresponding dynamic variant have been used. The ﬁltering is performed implicitly assuming equal grid spacing and ﬁlter width. For the dynamic procedure a top hat test ﬁlter is used with = 2∆. The obtained model coeﬃcient C is not averaged. It is clipped in ∆ s order to satisfy the conditions ν + νt ≥ 0 and Cs ≤ 0.2. For details see [3]. The equations are solved by using a ﬁnite volume technique on a Cartesian or cylindrical mesh. The variables are located on a staggered grid. For spatial discretization second order central diﬀerences are used, whereas conservation of kinetic energy has been veriﬁed. Temporal discretization is an explicit third order, Runge-Kutta-method. The Poisson equation is inverted by using a direct fast elliptic solver [12]. The following sections give a short description of the conﬁgurations. More details are provided in [13–15]. 2.1 Plane Jet The turbulent plane jet is simulated with a Reynolds number of 4000 based on the inlet bulk velocity and the nozzle width D. The extension of the computational domain in the axial x, spanwise y and vertical z direction is 20D × 6.4D × 20D. The computational domain is resolved with 150 × 32 × 124 grid points. At the outﬂow, Neumann boundary conditions for the velocity and the pressure are prescribed, negative velocities are clipped. Setting the pressure to zero at the top and the bottom boundaries and using Neumann boundary conditions for the velocities, allows mass entrainment. Periodic boundary conditions are applied in the spanwise direction. More details can be found in [13]. At the inlet a hyperbolic-tangent proﬁle is used at the nozzle exit, with superimposed spatially and temporally correlated ﬂuctuations, generated by a procedure presented in [16]. The length scales at the inﬂow have been set to 0.4D, 0.125D, 0.125D in the x, y, z directions according to the channel ﬂow measurements reported in [17], the ﬂuctuation level has been set to a uniform level of 2%. No additional coﬂow has been applied.

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2.2 Channel Flow For the channel ﬂow LES the Reynolds number has been set to Reτ = 395 corresponding to the DNS data of [15]. Hereby, Reτ is based on the wall friction velocity, the channel half width and the kinematic viscosity. The extension of the computational domain in the streamwise x, spanwise y and vertical z directions is 12δ × 6δ × 2δ similar to [18], where δ is the channel half width. The computational domain is resolved with 128 × 64 × 32 grid points. The grid is stretched in the wall normal direction with a factor of 1.1. Periodic boundary conditions are applied in the axial and spanwise directions together with no slip conditions at the wall. A ﬁxed theoretical pressure gradient is prescribed in the simulations. 2.3 Swirl Flow The swirl ﬂow conﬁguration is based on the well-known TECFLAM burner. The setup has been investigated experimentally under isothermal and reacting conditions [19] thus providing a database for validation of the DNS data [14] used within this paper.

Fig. 1. Sketch of the swirler device.

The experimental setup consists of a moveable block type swirler, which feeds an annulus from where the airﬂow enters the measurement section at ambient pressure and temperature. The Reynolds number was 5000, calculated from the bulk velocity and the bluﬀ-body diameter. The geometrical (or theoretical) swirl number is calculated using the inlet velocities, rather than the velocity distribution [20]. In the case currently investigated it was set to S=0.75. A co-axial airﬂow of 0.25 m/s surrounds the swirler device. A sketch is given in ﬁgure 1. The extension of the computational domain in the axial x, and radial r direction is 12D × 8D. The computational domain is resolved with 360×128×120 grid points. The inlet boundary condition for the nozzle is a developed channel ﬂow mean velocity proﬁle together with a constant radial velocity and superimposed ﬂuctuations [16]. All other boundary conditions are similar to the plane jet conﬁguration.

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3 Evaluation of Modeling and Numerical Error A common way to assess the quality of CFD simulations in the context of RANS is to perform grid reﬁnement studies based on variants of Richardson extrapolation [6]. In the context of LES with implicit ﬁltering, modeling and numerical errors interact. Furthermore reﬁning the grid changes the model contribution and as a result of this, it is meaningless to speak of grid independent LES. Strictly speaking, implicit LES is not the solution to a set of diﬀerential equations since the model contribution depends on the chosen grid. This makes the veriﬁcation of LES based on grid reﬁnement studies diﬃcult. Recently a method has been proposed to study the sensitivity of the implicit large eddy simulation results on the two error contributions [1]. In addition to a grid variation, the fundamental idea is to change the model term, e.g. to reduce or increase the model parameter. Intuitively it is clear that from such a model variation, the inﬂuence of the SGS parameterization can be assessed at least qualitatively. To illustrate the idea, it is assumed in the following that the contribution from the numerical error (n) and the error contribution from the model term (m) are given by the right hand side of a Taylor expansion (see equation (1)) and that both contributions are independent (which is clearly not fulﬁlled in practice). Changing the model contribution by a certain factor α yields equation (2) and ﬁnally equation (3) represents the solution on a grid, coarsened by a factor of 2. Here u denotes the exact solution, u1 the standard LES solution, u2 the LES solution with a modiﬁed model contribution and u3 the coarse grid solution. u − u1 = cn hn + cm hm u − u2 = cn hn + αcm hm

u − u3 = cn (2h)n + cm (2h)m

(1) (2) (3)

Combining equations (1-3) the error contribution from the model (4) and the numerical error (5) is obtained. The total error is the sum of equation (4) and (5) which can, assuming m = n, be simpliﬁed to equation (6). This is for example the case when considering a second order numerical scheme and a second order dissipation error. (u2 − u1 )/(1 − α) = cm hm (u3 − u1 ) − (u2 − u1 )(1 − 2m )/(1 − α) = cn hn m = n 1 − 2n u3 − u1 = (cn + cm )hn m = n (1 − 2n )

(4) (5) (6)

The reasoning behind equation (1) is that discretization of the NavierStokes equations gives rise to a truncation error

Determination of the Scaling Exponent of the Modeled Subgrid Stresses

NS(u) − N S h (u) = cn hn + O(hn+1 )

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(7)

where the operator NS represents the diﬀerential equation, N S h represents the discrete equation system, h is the grid spacing and n is the formal order of accuracy. Using the usual assumptions of smoothness, as well as the assumption that the local error is indicative of the global error (see [6]), the exact solution u may be expressed in terms of the discrete solution uh and the grid spacing h: (8) u = uh + cn hn + O(hn+1 ). In the context of LES a model for the subgrid scale stresses is added to equation (7). Hereby, the model term for the subgrid scale stresses denotes the only diﬀerence between unﬁltered and ﬁltered Navier-Stokes equations. For implicit LES the only reference to the ﬁlter is through this model term which gives rise to a modeling error. It is pointed out that, although the right hand side of (4) is called model error, it is not the model error in common sense (e.g. [7]), but rather the contribution of the actually used model to the total error on this particular grid with respect to the DNS.

4 Results In this section, ﬁrst the response of the modeled subgrid stresses with respect to changing grid and ﬁlter width will be discussed. Then the error equations 4 and 5 will be applied to the test cases introduced in section 2. 4.1 Determination of the Scaling Exponent The three equations (1)-(3) contain 5 unknowns (u, n, m, cm , cn ). Hence, either the system has to be extended, or some assumptions have to be made with respect to the scaling exponent for the numerical error and the modeling error. Since the present approach should be applicable for general engineering LES applications the second option has been chosen. Due to the second order central diﬀerencing used within this work the exponent for the numerical error can be set to n = 2. Equation (1) assumes that the model error scales with the power of h and the focus of the present work is the determination of the scaling exponent m. Using four sets of simulation results from the literature [7], Celik et al. [21] determined the value of m to be approximately 2, i.e. a second order dissipation error. On the other hand theoretical arguments [2, 3] suggest, that the true subgrid stresses scale like ∆2/3 . Hence the expected range for the modeled subgrid stresses is m = 2/3 . . . 2. In the Smagorinsky model the eddy viscosity is modeled using the mixing length hypothesis τij = −2νt S¯ij

with

νt = (Cs ∆)2 S¯ .

(9)

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Markus Klein, Martin Freitag, and Johannes Janicka Germano log2 (νtc /νtf )

Smagorinsky 2 5/3 4/3 1 0

50 100 150 200 250 300 350 z+ x/D=12.5, Germano

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z/D Fig. 2. Logarithm of the ratio of the turbulent viscosities taken from simulations on two grids with a reﬁnement ratio of two. Channel ﬂow (top) and plane jet (bottom).

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Fig. 3. Logarithm of the ratio of the turbulent viscosities taken from simulations on two grids with a reﬁnement ratio of two (Smagorinsky model). Swirl ﬂow at Re=5000.

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Consistent with the observation τij ∼ ∆2/3 the scaling for the eddy viscosity is νt ∼ ∆4/3 . This can be easily checked a posteriori in a two grid study. Figures 2 and 3 show the scaling exponent of the turbulent viscosities calculated from simulations on two diﬀerent grids with a reﬁnement ratio of two, where the Smagorinsky as well as the dynamic Smagorinsky model have been used. Figure 2 (top) shows the results from a channel ﬂow simulation in wall normal direction, ﬁgure 2 (bottom) shows the lateral proﬁle from a plane jet simulation in the far ﬁeld. Finally ﬁgure 3 depicts radial proﬁles at diﬀerent axial positions from a much more complex swirling recirculating ﬂow. As a general feature the ratio of the turbulent viscosities is roughly constant in the regions where the turbulence can be considered as homogeneous, but ﬂuctuates close to the wall in the channel ﬂow simulation or in the outer regions of the two free shear ﬂows. The increasing value of νtc /νtf for large z or r values (Fig. 2 bottom and Fig. 3) can be attributed to the combined eﬀect of large mean gradients and slightly diﬀerent ﬂow spreading at this axial position. For the swirling ﬂow, one has to keep in mind that for large radii the ﬂow becomes almost laminar. Consequently, the absolute values for νtc and νtf are very small and small deviations can lead to a large ratio. The situation is to some extent similar to Richardson extrapolation using three grids, where large variations in the local exponent of the numerical error have been observed [6]. From ﬁgures 2 (top) and 3 (bottom) it can be concluded that νt ∼ ∆5/3 seems to be a reasonable approximation for the cases investigated in this work although the jet simulation gives slightly lower values (ﬁgure 2 bottom). This implies that S¯ ∼ ∆−1/3 and hence τij ∼ ∆4/3 , i.e. m = 4/3. This value is within the expected range as explained above. For higher Reynolds number ﬂows the scaling exponent seems to approach the theoretical value 2/3 (not shown here). 4.2 Application of the Approach In the ﬁrst part of this section the results from a two grid study will be shown (without model variation). This simply corresponds to grid coarsening or Richardson extrapolation with a presumed convergence rate. Figures 4-6 compare the estimated errors to the diﬀerence between LES and DNS. Clearly this estimate, having the wrong magnitude and the wrong sign, is not representative for the true error. This phenomenon is due to the interaction between numerical and modeling error as discussed in the introduction and shows that two grid studies are insuﬃcient to qualify LES results. The systematic grid and model variation will now be applied to the three test cases assuming m = 4/3 and n = 2. Strictly speaking setting n = 2 would need validation. However this validation is not possible in implicit LES, but changing the scaling exponent for example from n = 2 to n = 1.7 does not change the results drastically. In all model variations the value of α, respectively Cs , has been reduced although the opposite would be possible,

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6 Richardson from DNS

Richardson from DNS

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Fig. 4. Error estimated from Richardson extrapolation compared to the diﬀerence between LES and DNS for a channel ﬂow. Smagorinsky (left) and Germano model (right). 0.06

0.06 Richardson from DNS

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too. Figures 7, 8 and 9 show the individual error contributions calculated from equations (4) and (5). As expected the estimated as well as the true errors from the Germano computations are considerably smaller compared to the Smagorinsky results. In all cases, roughly speaking the modeling error (cm hm ) and the numerical error (cn hn ) are of the same order of magnitude but with opposite sign, leading to a total error (cm hm + cn hn ) smaller than its individual contributions. The results also clearly show the deﬁciencies from the Smagorinsky model predicting wall bounded and transitional ﬂows. As can be seen from ﬁgures 7, 8 and 9, still the estimated total error describes the true error in the ﬂow not satisfactory, but surprisingly the individual error components, especially the modeling error for the Smagorinsky computations, gives very useful information on the error behavior with respect to location and

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0.15 x = 120 mm 0.075 0.0 -0.075 -0.15

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Fig. 6. Error estimated from Richardson extrapolation compared to the diﬀerence between LES and DNS for a swirling recirculating ﬂow. Smagorinsky model has been used.

magnitude. It is the authors belief that, besides grid reﬁnement/coarsening, a model variation is an important ingredient for qualifying LES results. To evaluate the additional expenses using this procedure contrarily to a single LES calculation, the computational overhead is calculated. Naturally, a second LES on the same grid results in a numerical overhead of 100%. Coarsening the grid in every direction by a factor of two, leads to 1/8 grid nodes and assuming a time step twice as large leads to overall 1/16, supplementary costs. Hence, the computational overhead of this procedure is 17/16 and thus acceptable. Furthermore the method can be used with any CFD code. However in its present status the above methodology has rather the character of a sensitivity analysis than an error analysis.

5 Conclusions The scaling law for the turbulent viscosity with respect to a changing ﬁlter width has been investigated numerically. It has been found that the modeled subgrid stresses roughly scale like ∆4/3 for moderate Reynolds numbers. This exponent has then been used within a quality assessment procedure, recently proposed in the literature. The so called systematic grid and model variation has been applied to three diﬀerent conﬁgurations: a channel ﬂow, a plane jet and a swirling ﬂow. Consistent with previous ﬁndings the model variation gives useful information on the error, especially for the Smagorinsky computations, whereas simple Richardson extrapolation is unreliable due to the interaction

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6 model numerical total from DNS

2

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Fig. 7. Estimated fractional modeling (cm hm ), numerical (cn hn ) and total error (cm hm +cn hn ) together with the error deﬁned with respect to DNS data (uDN S −u1 ). Channel ﬂow results. Smagorinsky model (left) and Germano model (right).

model numerical total from DNS

0.06 δU/U0

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0.02 0 -0.02

-0.04 10 15 20 0 5 10 15 20 x/D x/D Fig. 8. Estimated fractional modeling (cm hm ), numerical (cn hn ) and total error (cm hm +cn hn ) together with the error deﬁned with respect to DNS data (uDN S −u1 ). Plane jet ﬂow results. Smagorinsky model (left) and Germano model (right). 0

5

of modeling and numerical error. A more conservative approach to estimate the LES uncertainty based on the present ﬁndings is discussed in [22] and yields according to the authors opinion surprisingly good results.

References [1] M. Klein. An attempt to assess the quality of large eddy simulations in the context of implicit ﬁltering. Flow, Turbulence and Combustion, 75: 131–147, 2005. [2] S.B. Pope. Turbulent Flows. Cambridge Universtiy Press, 2000.

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numerical modelling total DNS

20

30

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60

70

80

r [mm]

Fig. 9. Estimated fractional modeling (cm hm ), numerical (cn hn ) and total error (cm hm +cn hn ) together with the error deﬁned with respect to DNS data (uDN S −u1 ). Results from a swirling recirculating ﬂow.

[3] P. Sagaut. Large Eddy Simulation for Incompressible Flows. Springer, 1998. [4] S.B. Pope. Ten questions concerning the large-eddy-simulation of turbulent ﬂows. New Journal of Physics, 6(35), 2004. [5] A. Nakayama and S.N. Vengadesan. On the inﬂuence of numerical schemes and subgrid-stress models on large eddy simulation of turbulent ﬂow past a square cylinder. International Journal for Numerical Methods in Fluids, 38:227–253, 2002. [6] P.J. Roache. Veriﬁcation and validation in computational science and engineering. Hermosa Publishers, Albuquerque, 1998. [7] B.J. Geurts and J. Fr¨ ohlich. A framework for predicting accuracy limitations in large eddy simulations. Physics of Fluids, 14(6):L41–L44, 2002. [8] F.K. Chow and P. Moin. A further study of numerical errors in large-eddy simulations. Journal of Computational Physics, 184:366–380, 2003. [9] A.G. Kravchenko and P. Moin. On the eﬀect of numerical errors in large eddy simulation of turbulent ﬂows. Journal of Computational Physics, 131:310–322, 1997. [10] J. Meyers, B.J. Geurts, and M. Baelmans. Database analysis of errors in large-eddy simulation. Physics of Fluids, 15(9):2740–2755, 2003. [11] B. Vreman, B. Geurts, and H. Kuerten. Comparison of numerical schemes in large-eddy simulation of the temporal mixing layer. International Journal for Numerical Methods in Fluids, 22:297–311, 1996. [12] U. Schumann and R.A. Sweet. A direct method for the solution of Poisson’s equation with Neumann boundary conditions on a staggered grid of arbitrary size. Journal of Computational Physics, 20:171–182, 1976.

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[13] M. Klein, A. Sadiki, and J. Janicka. Investigation of the inﬂuence of the Reynolds number on a plane jet using direct numerical simulation. International Journal of Heat and Fluid Flow, 24(6):785–794, 2003. [14] M. Freitag and M. Klein. Direct numerical simulation of a recirculating, swirling ﬂow. Flow, Turbulence and Combustion, 75:51–66, 2005. [15] R.D. Moser, J. Kim, and N.N. Mansour. Direct numerical simulation of turbulent channel ﬂow up to Reτ = 590. Physics of Fluids, 11(4): 943–945, 1999. [16] M. Klein, A. Sadiki, and J. Janicka. A digital ﬁlter based generation of inﬂow data for spatially developing direct numerical or large eddy simulations. Journal of Computational Physics, 186:652–665, 2003. [17] J.O. Hinze. Turbulence. McGraw-Hill, 1959. [18] J. Kim, P. Moin, and R. Moser. Turbulence statistics in fully developed channel ﬂow at low Reynolds number. Journal Fluid Mechanics, 177: 133–166, 1987. [19] C. Schneider, A. Dreizler, and J. Janicka. Fluid dynamical analysis of atmospheric reacting and isothermal swirling ﬂows. Flow, Turbulence and Combustion, 74:103–127, 2005. [20] J.M. Beer and N.A. Chigier. Combustion Aerodynamics. Applied Science Publishers, London, 1972. [21] I.B. Celik, Z.N. Cehreli, and I. Yavuz. Index of resolution quality for large eddy simulations. ASME Journal of Fluids Engineering, 127:949– 958, 2005. [22] M. Freitag and M. Klein. An improved method to assess the quality of large eddy simulations in the context of implicit ﬁltering. Journal of Turbulence, in press, 2006.

A Posteriori Study on Modelling and Numerical Error in LES Applying the Smagorinsky Model Tellervo Brandt Laboratory of Aerodynamics Helsinki University of Technology P.O. Box 4400, FIN-02015 TKK, Finland [email protected] Summary. In this paper, numerical and modelling errors in large eddy simulation (LES) applying the standard Smagorinsky model and a second-order scheme are studied and compared a posteriori in a turbulent plane Poiseuille ﬂow. The gridindependent solution of the LES equations is approached keeping the product of the model coeﬃcient and the length scale constant as the grid is reﬁned. The aim is to clarify the choice of the model parameters via the two error components, and to study the possibility to control the numerical error via the built-in ﬁlter of the model. The shape of the mean-velocity proﬁle is mainly determined by the modelling error, while both modelling and numerical error aﬀect the mean bulk velocity and the Reynolds stresses. As the model length scale is increased while keeping the grid resolution constant, the modelling error grows faster than the numerical error diminishes. Thus it is not reasonable to decrease the numerical error by increasing this parameter. The best choice of the model length scale is a compromise between the two errors, and in this case the modelling error is of the same size as the numerical error related to the second-order scheme.

1 Introduction In large eddy simulation (LES), we solve numerically the ﬁltered Navier– Stokes equations. The scales of motion are divided into resolved and sub-ﬁlter scales (SGS), and to describe the eﬀect of the later to the resolved scales, an SGS model is applied (see e.g. [1]). In complex geometries, low-order ﬁnitediﬀerence-type schemes are usually applied. As the smallest apparent ﬂow structures are of the same size as the grid resolution, they are badly described by the discrete grid, and it has been noticed in a priori tests that the involved numerical error may be large in comparison to the eﬀect of the model [2]. Explicit ﬁltering can be applied to control the level of numerical error and the eﬀect of the SGS model [2]. However, explicit ﬁltering has the drawback

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of increasing the computational cost [3], and a common choice, especially in complex geometries, is to let the numerical method and the SGS model provide implicit ﬁltering. In most models, like in the Smagorinsky model [4], the form of the ﬁlter function or the width of the ﬁlter are not explicitly deﬁned. However, these models involve a built-in ﬁlter, and the amount of damping the model produces and the width of the ﬁlter can be controlled via the model parameters. In the standard Smagorinsky model, the ﬁlter width is controlled via the product of the model coeﬃcient and the model length scale. Usually, the model length scale is chosen proportional to the grid spacing, and the smallest resolved scales are of the same size as the grid resolution. If the ﬁlter width is increased, the solution becomes smoother and the numerical error will decrease. In this case, the modelling error may increase. In addition, the possible interaction of the two error components aﬀects the total error [5]. In the Smagorinsky model, the model constant and thus the eﬀective width of the built-in ﬁlter is ﬂow dependent, and the choice of the applied value is often based on the observation that in some sense it produces the best results. However, the reason behind this choice is not clear. In this paper, the approach to a posteriori studies presented in Ref. [5] is applied in a fully developed turbulent channel ﬂow between two parallel inﬁnite plates at the Reynolds number Reτ = 395 based on the friction velocity and the channel half-height. The approach has previously been applied in a turbulent mixing layer and in homogeneous isotropic turbulence [5, 6], where time development of the error components in kinetic energy and in the Taylor length scale were studied. Here, the error in mean-ﬂow quantities of the channel ﬂow are studied. The second-order central-diﬀerence scheme, that is still widely applied in complex geometries, is used for spatial discretization and the standard Smagorinsky model as the SGS model. The aim of this paper is to describe the eﬀect of the ﬁlter width of the Smagorinsky model on the modelling and numerical errors in a wall-bounded ﬂow and to clarify the choice of the model constant by means of the two error components. In Section 2, we discuss the applied approach to a posteriori study and to obtaining the grid-independent solution of the governing equations in LES. In Section 3, the applied numerical methods and the grid resolution are described. In Section 4, turbulent channel ﬂow simulations, where the grid resolution is varied, are discussed. The eﬀect of grid resolution on both numerical and modelling error is studied. In Section 5, we study the eﬀect of varying the model length scale on the two error components. Finally, the conclusions are drawn in Section 6.

2 Applied Modelling and Grid-Independent LES The equations being solved in LES of an incompressible ﬂow are written in the non-dimensional form as

A Posteriori Study on LES Applying the Smagorinsky Model

∂u ˜i ∂P ∂ p˜ ∂ =− − + ∂t ∂xi ∂xi ∂xj

−u ˜i u ˜j − τij +

1 Reτ

∂u ˜i ∂u ˜j + ∂xj ∂xi

175

,

(1)

u1 , u ˜2 , where (x1 , x2 , x3 ) = (x, y, z) refer to spatial coordinates, t to time, (˜ u ˜3 ) = (u, v, w) to resolved velocity vector, ρ to density, P to mean pressure, p˜ to ﬂuctuating resolved pressure, and τij is a model for the sub-grid scale stress ˜i u ˜j (see e.g. [1]). Here, the equations tensor u5 i uj − u are scaled by the channel half-height, 0.5h, and friction velocity uτ = τwall /ρ, and the Reynolds number is deﬁned as Reτ = 0.5huτ ρ/µ = 395. The Smagorinsky model [4], that is studied in this paper, is an eddy viscosity model, and the SGS stress tensor is modelled as ∂u ˜i ∂u ˜j 2 + −τij = µT 2Sij = (CS ∆f ) 2Sij Sij . (2) ∂xi ( )* + ∂xj =µT

The product of the Smagorinsky coeﬃcient CS and the model length scale ∆f has been shown to be the equivalent of the Kolmogorov dissipation length for the LES ﬂow generated by Equations (1) [7], and thus it controls the size of the smallest resolved ﬂow structures. Usually, when the LES equations (1) are discretized, ∆f is chosen to be proportional or equal to grid spacing ∆. If the grid is not isotropic, the measure for the size of the grid cells is often calculated as 1/3

∆ = (∆x ∆y ∆z)

.

(3)

The Smagorinsky coeﬃcient is usually chosen according to the simulated ﬂow geometry. In homogeneous turbulence, the value CS ≈ 0.2 is applied, and in shear ﬂows, the parameter is often reduced to CS ≈ 0.1 (see e.g. [1]). Also, the use of values close to CS ≈ 0.2 in the channel ﬂow has been suggested [8]. In this study, the coeﬃcient is nominally set to CS = 0.085. This value has been previously noticed to give good results with the applied code when the model length scale ∆f was set equal to grid spacing ∆ in Eq. (3). However, it is not necessary to tie the model length scale to the grid resolution. Equations (1) can also be interpreted as a closed system of equations with an externally deﬁned parameter CS ∆f , which controls the smoothness of the resolved ﬂow ﬁeld [6, 7]. This interpretation is the starting point of the applied approach to a posteriori study. In wall-bounded ﬂows, the size of the smallest length scales of the ﬂow ﬁeld reduces as the wall is approached, and thus it is necessary to reduce the model length scale and the length of the implicit ﬁlter provided by the model in the near-wall region [9]. With the standard Smagorinsky model the van Driest damping is usually applied. It was also applied in this study both in actual and grid-independent LES. The use of damping reduces the eﬀect of the model and thus the eﬀect of modelling error in the near-wall region. However, it was considered that studying the behaviour of the standard Smagorinsky

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model in a wall-bounded ﬂow without damping would give quite pessimistic results. As suggested in Ref. [5], the numerical error involved in LES is deﬁned here as the diﬀerence between the grid-independent LES and the LES results, and modelling error as the diﬀerence between the result of a direct numerical simulation (DNS) and the grid-independent LES: ǫnumerical error = φgrid indep. LES − φLES

ǫmodelling error = φDNS − φgrid indep. LES .

(4)

The total error is the sum of these two: ǫtot = φDNS − φLES . Since in the Smagorinsky model the ﬁlter is not explicitly deﬁned, the DNS solution instead of ﬁltered DNS is applied in the deﬁnition of the modelling error. This choice was applied in Ref. [6]. The grid-independent LES is approached when the grid resolution is increased and the ﬁlter width involved with the SGS model is kept constant. For the standard Smagorinsky model, this means that the product CS ∆f in Eq. (2) is kept constant. Thus, no explicit ﬁltering is applied. The obtained grid-independent solution is not approaching a DNS solution as happens in normal LES when the resolution is increased. In contrast, grid-independent LES approximates the smooth solution of Equations (1), where the model is not a priori tied to the grid spacing. When the Smagorinsky model is applied, this solution has also been referred to as the grid-independent solution for the “Smagorinsky ﬂuid” [7]. In the previous studies [5, 6], parameters called SGS resolution and SGS activity parameter have been applied to describe the diﬀerent combinations of the width of the built-in ﬁlter and the resolution. The SGS resolution, r, is deﬁned as the ratio of the model length scale to the grid spacing r = ∆f /∆. When r is large, numerical error has only a small eﬀect on the solution and the solution is approaching the grid-independent case. The SGS activity parameter, s, describes the amount of modelling in LES compared to DNS. It is deﬁned as the ratio of the turbulent dissipation to the total dissipation as s=

εt , ε t + εµ

(5)

where the overline refers to time average and the turbulent dissipation and the molecular dissipation are deﬁned as εt = τij

∂u ˜i ∂u ˜i = µT 2S˜ij ∂xj ∂xj

and

εµ =

˜i 1 ˜ ∂u Sij , Reτ ∂xj

respectively.

(6) In DNS, the value of s is zero, and the value of unity is approached in LES as the Reynolds number grows towards inﬁnity. The value of s is related to the eﬀective ﬁlter width, and it is almost independent of the grid resolution [5]. This was also noticed in the present simulations in the channel ﬂow.

A Posteriori Study on LES Applying the Smagorinsky Model

177

3 Applied Numerical Methods and Grid Resolutions In the present study, the second-order central-diﬀerence scheme was applied on a staggered grid system for spatial discretization. The non-dimensional mean-pressure gradient was ﬁxed to the value two, and the mean velocity was allowed to vary in time. The ﬂuctuating pressure was solved from the Poisson equation. Time-integration was performed using a third-order, threestage, low-storage Runge-Kutta method. DNS of the channel ﬂow obtained using this code have been compared to the DNS results of Ref. [10] in [11]. Now, to study the actual order of the code, four cases of the channel ﬂow at Reτ = 395 were run using no SGS modelling and grids of 1263 , 1503 , 1803 and 2163 cells. The actual numerical error was evaluated for each case using the DNS of Ref. [10] (uDNS ). The obtained error for the mean velocity is given Figure 1. For a second-order method, the numerical error should behave as uDNS − u ≈ c∆2 + O ∆3 , and in the asymptotic range the ﬁrst term in the expansion should dominate. The theoretical error obtained by scaling the error of the coarsest grid case by the square of the grid reﬁnement factor is included in the ﬁgure for each grid. The actual error diminishes faster than the theoretical one, and similar results were obtained also for the Reynolds stresses. Based on the present study, the behaviour of the obtained numerical error is close to the nominal error of a second-order scheme, and we can claim that the numerical error studied in the next sections is actually produced by a second-order code. To approach the grid-independent LES solution four grids, which are labelled as grid 1, grid 2, grid 3 and grid 4 (see Table 1), were applied. Grid 1 is the coarsest and grid 4 has the same resolution as one of the DNS grids studied in the previous section. In DNS at this resolution, the error in the

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2

3

uDNS − u

126 1503 18033 216 3 theor. 150 3 theor. 180 3 theor. 216

1

10

y+

100

Fig. 1. Numerical error in mean velocity of simulations with no modelling.

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mean velocity was everywhere below 3% when compared to the results of Ref. [10]. The applied LES grids were equally spaced in the homogeneous, i.e. streamwise and spanwise, directions and the hyperbolic tangent function was applied to the grid stretching in the wall-normal direction. For each studied case, the grid-4 approximates the grid-independent LES, and is used for evaluation of the numerical and modelling errors via Equation (4). The DNS results of Ref. [10] were applied in the evaluation of the numerical error. Table 1. Domain size and resolution of the applied LES grids. (x=streamwise, z=spanwise, y=wall-normal direction). grid 1 x z

y

grid 2 x z

y

extent of the domain (scaled by h) 3.0 1.6 1.0 3.0 1.6 1.0 number of grid points 36 36 40 54 54 60 size of the grid cell in wall units (∆+ ) 66 35 3,. . . ,41 44 23 2,. . . ,27 grid 3

grid 4

extent of the domain (scaled by h) 3.0 1.6 1.0 3.0 1.6 1.0 number of grid points 108 108 120 180 180 180 size of the grid cell in wall units (∆+ ) 22 12 1,. . . ,14 13 7 0.7,. . . ,9

4 Eﬀect of Varying Grid Resolution on Numerical and Modelling Error In this section, we study LES results and the involved numerical and modelling error from the two coarsest grid resolutions – grid 1 and grid 2. Here, the subgrid-scale resolution is ﬁxed to value 1, i.e. the ﬁlter width ∆f is set equal to the grid spacing ∆ (Eq. 3). The SGS activity parameter from the two cases is plotted in Figure 2. As the grid resolution is increased, the ﬁlter width of the SGS model decreases, and also the SGS activity decreases. The mean-velocity proﬁles from grid-1 and grid-2 cases are plotted in Figure 3. As the resolution is increased, the non-dimensional mean bulk velocity becomes less overpredicted and the length of the viscous sublayer improves. If the grid resolution is increased further with r = 1, also the slope of the proﬁle improves and the proﬁle approaches the deviatoric DNS result. The ∗ diagonal Reynolds stress (u′ u′ = u′ u′ − 1/3 u′ u′ + v ′ v ′ + w′ w′ ) is plotted in Figure 4. The deviatoric stress is studied because the Smagorinsky model is traceless [12]. As the grid resolution is increased, the total error in this quantity diminishes. Next, we ﬁx the eﬀective ﬁlter width and increase the grid resolution for both studied cases. The aim is to approach the grid-independent situation and to evaluate the numerical and modelling error involved with the two cases.

A Posteriori Study on LES Applying the Smagorinsky Model 0.7

∆f = ∆

0.6

179

∆grid 1 ∆grid 2

0.5 s

0.4 0.3 0.2 0.1 0 0

0.1

0.2

y/h

0.3

0.4

0.5

Fig. 2. Sub-grid scale activity parameter. Grid resolution is varied and r = ∆f /∆ = 1.

22 20 18 u/uτ

16 14 12 10 grid 1 grid 2 DNS

8 6 10

y

+

100

Fig. 3. Mean velocity proﬁle. The grid resolution is varied and the SGS resolution is kept constant: ∆f = ∆.

The mean velocity proﬁle from the grid-1 case and the corresponding results from the higher resolution cases with the ﬁlter width of ∆f = ∆grid 1 are plotted in the upper part of Figure 5, where the diﬀerence between the DNS curve and the LES case with the highest resolution (grid 4) approaches the modelling error, and the diﬀerence between grid 4 and grid 1 approaches the numerical error. The error components are plotted in the lower part of the ﬁgure. In grid-1 case, the viscous sublayer is too long, and thus the slope of the proﬁle is not correct. Since the shape of the proﬁle changes only slightly with grid resolution, this is due to the modelling error. Both numerical and modelling error aﬀect the mean bulk velocity. The modelling error in the mean bulk velocity is larger than the total error, and it is of the opposite sign to the numerical error. Thus, there is counteraction of these two error components in the mean bulk velocity.

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Tellervo Brandt

0 −1

−u’u’*/uτ

2

−2 −3 −4 −5 −6 grid 1 grid 2 DNS

−7 −8 0

0.05

0.1

y/h

0.15

0.2

0.25

Fig. 4. Deviatoric diagonal Reynolds stress. The grid resolution is varied and the SGS resolution is kept constant: ∆f = ∆.

The mean velocity from the grid-2 case together with results from higherresolution cases with ∆f = ∆grid 2 are plotted in the upper part of Figure 6 and the numerical and modelling errors in the lower part of the ﬁgure. Here, the modelling error is clearly smaller than in the grid-1 case. Both the shape of the mean-velocity proﬁle and the mean bulk velocity are better predicted. However, the numerical error in the mean bulk velocity was not that much aﬀected by the grid resolution, and it is almost the same in grid-1 and grid-2 cases. The deviatoric streamwise Reynolds stress and the error components from the grid-1 case are plotted in Figure 7. Here, the modelling error dominates the numerical error in most part of the channel. Numerical error is larger than the modelling error only in the middle of the channel. In grid-1 case, there is no counteraction of the two error components . In Figure 8, the deviatoric streamwise Reynolds stress is plotted from the grid-2 case with the ﬁlter width set to ∆f = ∆grid 2 . Here, the modelling error is clearly decreased as compared to the grid-1 results in Figure 7. However, as seen by comparing Figures 8 and 7, the numerical error is only slightly aﬀected when grid resolution is varied and the SGS resolution is kept constant. Based on the above ﬁndings, the improved results in Figures 3 and 4 that were obtained when the resolution was increased from grid-1 to grid-2 keeping the SGS resolution constant were mainly due to the decreased modelling error. As the SGS resolution was kept constant, the model length scale and SGS activity decreased with increased grid resolution. Thus, the eﬀect of the model was also decreased, which makes the decreasing of the modelling error natural. One would have assumed that increasing the resolution would have decreased also the numerical error. However, it had very little eﬀect on the numerical

A Posteriori Study on LES Applying the Smagorinsky Model

181

25 20

u/uτ

15 10

∆f = ∆grid 1 = 1.5∆grid 2 grid 1 grid 2 grid 3 grid 4 DNS

5 0 10

y

+

100

2 1.5 1 0.5 0 −0.5

∆f = ∆grid 1 = 1.5∆grid 2 modelling num. grid 1 −1.5 num. grid 2 tot grid 1 −2 tot grid 2 −2.5 1 10 + y −1

100

Fig. 5. Upper: Mean velocity proﬁle in the wall coordinates. Filter width is kept constant and the grid resolution increased. ∆f = ∆grid 1 = 1.5∆grid 2 . Lower: Involved numerical and modelling errors.

error. This is probably due to the SGS motions that became badly described resolved scales as the grid resolution was increased.

5 Eﬀect of Varying Filter Width on Numerical and Modelling Error The eﬀect of the built-in ﬁlter width was studied using grid 2. The ﬁlter widths ∆f = 0, 0.5∆, ∆ and 1.5∆ were applied, and the corresponding SGS resolutions were r = 0, 0.5, 1 and 1.5, respectively. The SGS activity parameter s (Eq. 5) from these cases is depicted in Figure 9. The largest values (s ≈ 0.5) are found in the case with the largest ﬁlter width, and in the case with no

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25 20

u/uτ

15 10 ∆f = ∆grid 2 grid 2 grid 3 grid 4 DNS

5 0 10

y

+

100

2 1.5 1 0.5 0 −0.5

∆f = ∆grid 2 modelling num. grid 2 tot grid 2

−1 −1.5 −2 −2.5 1

10

y

+

100

Fig. 6. Upper: Mean velocity proﬁle in the wall coordinates. Filter width is kept constant and the grid resolution increased. ∆f = ∆grid 2 . Lower: Involved numerical and modelling errors.

model, s has the value zero. The case with the largest ﬁlter width 1.5∆grid 2 at grid 2 has the same SGS activity as the grid-1 case with r = 1 in Figure 2. These cases have also the same ﬁlter widths, and thus s seems to be rather independent of the grid resolution as has been previously noticed in turbulent mixing layer [5]. The mean velocity proﬁles from cases with diﬀerent ﬁlter widths on grid 2 are plotted in Figure 10. If no SGS model is applied, i.e. ∆f = 0, the mean velocity is underpredicted. As the ﬁlter width is increased, the situation ﬁrst improves but ﬁnally, the mean velocity becomes overpredicted. In addition, as the ﬁlter width is increased, the length of the viscous sublayer increases. In all cases, the slope of the velocity proﬁle in the log-layer is too low, and applying the SGS model does not improve this. In Figure 11, we have the deviatoric

A Posteriori Study on LES Applying the Smagorinsky Model

183

0 −1

−u’u’*/uτ2

−2 −3 −4 ∆f = ∆grid 1 = 1.5∆grid 2 grid 1 grid 2 grid 3 grid 4 DNS

−5 −6 −7 −8 0

0.05

0.1

y/h

0.15

0.2

0.25

1 0.5 0 −0.5 −1 ∆f = ∆grid 1 = 1.5∆grid 2 modelling num. grid 1 num. grid 2 tot grid 1 tot grid 2

−1.5 −2 −2.5 −3 0

0.05

0.1

y/h

0.15

0.2

0.25

Fig. 7. Upper: Deviatoric part of the streamwise Reynolds stress. Filter width is kept constant and the grid resolution increased. ∆f = ∆grid 1 = 1.5∆grid 2 . Lower: Involved numerical and modelling errors.

diagonal streamwise Reynolds stress. The SGS model does not improve the prediction. When the ﬁlter width is increased, the Reynolds stress becomes more overpredicted and the peak value starts to move towards the middle of the channel. Next, we study the numerical and modelling error involved with the cases discussed above. The grid resolution was again increased for each case while the eﬀective ﬁlter width was kept constant. The mean velocity proﬁles from cases with the ﬁlter widths equal to ∆f = ∆grid 2 and ∆f = 1.5∆grid 2 were already presented in Figures 6 and 5, respectively. Since the modelling error of the case grid-2 with the ﬁlter width of ∆f = 1.5∆grid 2 is the same as the modelling error of the grid-1 case with ﬁlter width ∆f = ∆grid 1 , the same discussion applies here for the diﬀerences between the modelling errors as for

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0 −1

−u’u’*/uτ

2

−2 −3 −4 −5

∆f = ∆grid 2 grid 2 grid 3 grid 4 DNS

−6 −7 −8 0

0.05

0.1

y/h

0.15

0.2

0.25

1 0.5 0 −0.5 −1 −1.5 ∆f = ∆grid 2 modelling num. grid 2 tot grid 2

−2 −2.5 −3 0

0.05

0.1

y/h

0.15

0.2

0.25

Fig. 8. Upper: Deviatoric part of the streamwise Reynolds stress. Filter width is kept constant and the grid resolution increased. ∆f = ∆grid 2 . Lower: Involved numerical and modelling errors.

the grid-1 and grid-2 cases. The modelling error decreases as the ﬁlter width is decreased. This error aﬀects the shape of the proﬁle and the value of the mean bulk velocity. The mean velocity proﬁle together with the error components from the case ∆f = 0.5∆grid 2 is plotted in Figure 12, and we notice that the same trend continues. The numerical error mainly aﬀects the mean bulk velocity, and it ﬁrst remains almost constant when the ﬁlter width is increased from ∆grid 2 to 1.5∆grid 2 (Figs. 6 and 5), and it seems to decrease slightly when the width 0.5∆grid 2 is applied (Fig. 12). This could be due to not grid-converged mean bulk velocity. However, the DNS results at this resolution are grid converged. As the ﬁlter width is decreased, the numerical error has a small eﬀect also on

A Posteriori Study on LES Applying the Smagorinsky Model 0.7

∆ = ∆grid 2

185

∆f = 0.5 ∆ ∆f = ∆ ∆f = 1.5 ∆

0.6 0.5 s

0.4 0.3 0.2 0.1 0 0

0.1

0.2

y/h

0.3

0.4

0.5

Fig. 9. Sub-grid scale activity parameter. Grid resolution is constant (grid 2) and r is varied via the ﬁlter width ∆f .

22 20

u/uτ

18 16 14 ∆f=0 ∆f=0.5 ∆ ∆f=∆ ∆f=1.5 ∆ DNS

12 10 8 10

y

+

100

Fig. 10. Mean velocity proﬁle in the wall coordinates. The grid resolution is constant (grid 2) and SGS resolution is varied via the ﬁlter width ∆f .

the slope of the mean-velocity proﬁle. The eﬀect increases as the ﬁlter width is decreased. The deviatoric streamwise Reynolds stress obtained using ﬁlter widths ∆f = ∆grid 2 and ∆f = 1.5∆grid 2 were already depicted in Figures 8 and 7, respectively. The modelling error clearly increases as the ﬁlter width is increased. The numerical error is almost zero when the ﬁlter width is ∆f = 1.5∆, and it increases as the ﬁlter width is decreased to ∆f = ∆. The Reynolds stress and the error components from the case with the ﬁlter width ∆f = 0.5∆grid 2 is plotted in Figure 13, and there the overall trend continues. However, the increase in the numerical error is not as large between cases ∆f = ∆grid 2 and ∆f = 0.5∆grid 2 as between ∆f = 1.5∆grid 2 and ∆f = ∆grid 2 . The situation in Figure 11, where increasing the ﬁlter width while keeping the grid resolution constant, did not improve the results, is due to both numerical and modelling error. As the ﬁlter width is increased, the numerical error is dimin-

186

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0 −1

−u’u’*/uτ

2

−2 −3 −4 ∆f=0 ∆f=0.5∆ ∆f=∆ ∆f=1.5 ∆ DNS

−5 −6 −7 0

0.05

0.1

y/h

0.15

0.2

0.25 ∗

Fig. 11. Deviatoric part of the streamwise Reynolds stress −u′ u′ . The grid resolution is constant (grid 2) and SGS resolution is varied via the ﬁlter width ∆f .

ished, but at the same time the modelling error increases quite rapidly and thus the total error increases.

6 Conclusions In this paper, the numerical and modelling errors in LES using the standard Smagorinsky model were studied applying the approach presented in Ref. [5]. The aims of the paper were to clarify the reasoning behind the choice of the eﬀective ﬁlter width CS ∆f in the channel ﬂow by means of numerical and modelling error and to describe the eﬀect of the ﬁlter width on the two error components. In Section 4, we studied the eﬀect of varying the grid resolution on the total, modelling and numerical errors. The diﬀerence between the two coarse grid cases, grid 1 and grid 2, of the present simulations, was mainly due to the modelling error, and increasing the grid resolution did not decrease the numerical error very eﬃciently. As the grid resolution is increased, some scales that were SGS scales in the coarser grid become badly described resolved scales on the ﬁner grid, and thus grid reﬁnement does not necessarily decrease the total numerical error. In Section 5, we studied the eﬀect of varying the width of the built-in ﬁlter of the standard Smagorinsky model on the mean ﬂow quantities. Numerical error had a small eﬀect on the slope of the mean-velocity proﬁle, but the shape of the proﬁle was mainly determined by the modelling error. It has been previously noticed that the two error components can have diﬀerent signs and they can partially cancel out each other [5, 6]. In the present study, this was noticed in only in the mean bulk velocity. In the Reynolds stress, the

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modelling error grew much faster than the numerical error diminished, and the prediction of this quantity was not improved when the model was applied. Here, we nominally ﬁxed the model constant of the standard Smagorinsky model to the value CS = 0.085, and thus the studied eﬀective ﬁlter widths were CS ∆f = 0.045∆, 0.085∆ and 0.1275∆. It has been noticed in simulations with the applied code, that the middle value is a some sort of optimal value. Based on the present results, it can be stated that this choice is a compromise between the numerical and modelling error. If one considers the choice of the model length scale from the point of view of the two error components, the model length scale becomes dependent also on the chosen numerical method. Thus, from the point of view of this study, modelling and numerics are not two separate issues. If the numerical error is controlled via the implicit ﬁltering

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of the model, the numerical scheme has to be taken into account when the model parameters are chosen. Previously, it has been noticed in a priori tests that when a second-order scheme is used for spatial discretization, the numerical error is large in LES [2]. However, based on the present results it seems that the modelling error involved with the standard Smagorinsky model in the channel ﬂow is of the same size as the numerical error related to the second-order scheme. As the width of the built-in ﬁlter of the model is increased, the modelling error will grow faster than the numerical error diminishes and the total error will also grow. Thus, controlling the numerical error via the built-in ﬁlter of the Smagorinsky model does not seem like a very promising approach.

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Acknowledgments This work has been funded by the Finnish Graduate School in Computational Fluid Dynamics. The computer capacity was partly provided by CSC, Scientiﬁc Computing Ltd. The applied channel-ﬂow code is based on a code written by Dr. Boersma from TU Delft. These contributions are gratefully acknowledged.

References [1] P. Sagaut. Large eddy simulation for incompressible ﬂows. Springer, 2001. [2] S. Ghosal. An analysis of numerical errors in large-eddy simulations of turbulence. Journal of Computational Physics, 125:187–206, 1996. [3] T. S. Lund and H.-J. Kaltenbach. Experiments with explicit ﬁltering for LES using a ﬁnite-diﬀerence method. Center for Turbulence Research, Annual Research Briefs, pages 91–105, 1995. Stanford University. [4] J. S. Smagorinsky. General circulation experiments with the primitive equations, part I: The basic experiment. Monthly Weather Review, 91:99– 152, 1963. [5] B. Geurts and J. Fr¨ ohlich. A framework for predicting accuracy limitations in large-eddy simulation. Physics of Fluids, 14(6):L41–L44, June 2002. [6] J. Meyers, B. Geurts, and M. Baelmans. Database analysis of errors in large-eddy simulation. Physics of Fluids, 15(9):2740–2755, September 2003. [7] A. Muschinski. A similarity theory of locally homogeneous and isotropic turbulence generated by a Smagorinsky-type LES. Journal of Fluid Mechanics, 325:239–260, 1996. [8] P.J. Mason and N.S. Callen. On the magnitude of the subgrid-scale eddy coeﬃcient in large-eddy simulations of turbulent channel ﬂow. Journal of Fluid Mechanics, 162:439–462, 1986. [9] P. Moin and J. Kim. Numerical investigation of turbulent channel ﬂow. Journal of Fluid Mechanics, 118:341–377, 1982. [10] R. D. Moser, J. Kim, and N. N. Mansour. Direct numerical simulation of turbulent channel ﬂow up to Reτ = 590. Physics of Fluids, 11(4):943–945, April 1999. [11] T. Brandt. A priori tests on numerical errors in large eddy simulation using ﬁnite diﬀerences and explicit ﬁltering. International Journal for Numerical Methods in Fluids, 51(6):635–657, June 2006. Published Online: 20 Dec 2005. [12] G. S. Winckelmans, H. Jeanmart, and D. Carati. On the comparison of turbulence intensities from large-eddy simulation with those from experiment or direct numerical simulation. Physics of Fluids, 14(5):1809–1811, May 2002.

Passive Scalar and Dissipation Simulations with the Linear Eddy Model C. Papadopoulos1 and K. Sardi2 1

2

Mechanical Engineering Department, National Technical University of Athens, Greece Regulatory Authority for Energy (RAE), Athens, Greece [email protected]

Summary. The purpose of this work was to extend knowledge on the potential of the Linear Eddy Model (LEM) to reproduce scalar statistics that are relevant to the modelling of turbulent combustion. In this context, emphasis was placed on the evolution of the scalar dissipation and its conditional moments. To allow for a systematic and quantitative analysis on the sensitivity of model to its input parameters, LEM has been employed as a stand alone model with prescribed ﬂow ﬁeld properties rather than being incorporated in an LES code as a subgrid model. Results from twelve parametric LEM computations are compared to experiments and DNS data from non-combusting and reactive turbulent mixing layers. It is shown that the LEM model reproduces a number of important features of scalar ﬁelds. However, input parameters are shown to signiﬁcantly inﬂuence the model behaviour so that care is required when employing the model particularly at early times.

1 Introduction The Linear Eddy (LEM) and the One Dimensional Turbulence (ODT) models proposed by Kerstein [1–14] are receiving increasing attention as stand alone models to compute and analyse the statistical properties of velocity and scalar ﬁelds in isothermal and combusting ﬂows [15–17] and also as potential high ﬁdelity subgrid closures for large eddy simulations [18, 19]. Although initially formulated as one-dimensional, so that they were applicable only in simple ﬂows (i.e. jets and mixing layers), both models resolve all turbulent scales and processes, readily accommodate complicated source conﬁgurations with ﬁnite rate chemistry and oﬀer a temporal and spatial description of turbulencechemistry interactions. LEM and ODT are based on a mechanistic description between advective (stirring) processes and molecular transport. Molecular processes are computed deterministically by solving the one-dimensional unsteady reactiondiﬀusion equations. Advective processes are implemented stochastically. The principal distinction between the two models is that LEM is a ‘mixing model’

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limited in the representation of the scalar ﬁelds with built-in ﬂow ﬁeld properties while ODT is a self-contained turbulence model. The purpose of this work is to investigate the potential of the LEM model to reproduce scalar statistics that are relevant to the modeling of turbulent combustion and emphasis is placed on the evolution of the scalar dissipation and its conditional moments. Here LEM has been employed as a stand alone model with prescribed ﬂow ﬁeld properties (thus it has not been included in an LES code as a subgrid model). Although this approach inevitably introduces a number of simpliﬁcations, it allows for a systematic and quantitative analysis on the sensitivity of model to its input parameters. Results are compared to experimental and DNS data from four turbulent mixing layers in the presence of passive and reactive scalars and are reported in terms of mean and rms scalar values, mean unconditional and conditional scalar dissipation proﬁles, probability and joint probability density functions and mean reaction rates.

2 Model Formulation Linear-eddy modelling involves the simulation of the time evolution of any spatially developing scalar ﬁeld cj (y,t) such as species concentration or temperature, on a one-dimensional spatial domain. Two processes that represent the eﬀects of molecular diﬀusion and turbulent advection are concurrently implemented. The former is implemented by numerically solving the onedimensional unsteady reaction-diﬀusion equation along the direction, y, of the largest scalar gradient: ∂ 2 cj ∂ cj = Dj 2 + w˙ j ∂t ∂y

j = 1, ...n

(1)

where Dj is the molecular diﬀusivity, ωj is the reaction rate and n the number of scalars. The latter is simulated by considering a random sequence of instantaneous ‘rearrangement events’ that punctuate the ongoing deterministic solution of (1) according to a predeﬁned ‘mapping rule’ which represents the eﬀect of individual eddies on the scalar ﬁeld. Three random variables govern each event: the time t0 at which the rearrangement will occur, the starting location y0 of the rearrangement and the size l of the aﬀected segment[6]. Following [8], the time t0 will be here referred to as epoch. The probability distributions of the three random variables and the ‘mapping rule’ are input parameters to the model and are discussed in the next paragraphs. Kerstein [1–12] proposed a mapping rule based on the so called ‘triplet map’. The eﬀect of this rule on a simple scalar ﬁeld c(y)=y is shown in Fig. 1. In detail, the scalar ﬁeld within a chosen segment of size l at time t0 is compressed by a factor of three and then replaced by three adjacent copies of the compressed ﬁeld with the middle copy mirror inverted. The map has a physical basis as it qualitatively resembles the distortion caused by a clockwise

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eddy on a two-dimensional concentration ﬁeld with a uniform scalar gradient. At time t0 +∆t the scalar gradients are smoothened by the action of the molecular diﬀusion through (1). Formally, application of the triplet map on a segment [y0 , y0+1 ] at time t0 transforms cj (y,t0 ) to cˆj (y,t0 ) according to  cj (3y − 2y0 , t0 ) y0 ≤ y ≤ y0 + 13 l    cj (−3y + 4y0 + 2l, t0 ) y0 + 13 l ≤ y ≤ y0 + 32 l cˆj (y, t0 ) = (2) cj (3y − 2y0 − 2l, t0 ) y0 + 23 l ≤ y ≤ y0 + l    cj (y, t0 ) otherwise Clearly, the model output is dependent on the mapping rule. To this end, Kerstein [8, 10–12] also examined the properties of two alternative maps: the ‘block inversion’ and the ‘quintuplet’ map. In the former, the scalar ﬁeld in a given segment is simply mirror imaged. The latter is similar to the triplet map but the scalar ﬁeld is compressed by a factor of ﬁve. Kerstein [10] concluded that the block inversion map induces strong discontinuities in the scalar ﬁeld leading to unrealistic scalar predictions. Use of the quintuplet map was shown [9] to induce only minor diﬀerences in the multifractal dissipation spectrum so that the triplet and the quintuplet maps were considered equivalent. Here, the triplet map was adopted and applied to all simulations. LEM assumes that rearrangement events are instantaneous and statistically independent with respect to segment size, location and time of occurrence. The location y0 of an event is chosen randomly from a uniform distribution over the spatial domain of the simulation [0, Y ⌉. (i)

(ii)

(iii)

eddy

Fig. 1. (i-ii) Triplet-map of length l applied on an initially linear scalar ﬁeld at a randomly selected location y0 at time t0 ; (iii) molecular diﬀusion at time t0 +∆t smoothens the scalar gradients.

An aspect of the turbulent ﬂow ﬁeld microstructure is incorporated into the model by requiring that rearrangement events involving segments of size l or less to obey the transport scaling that govern eddies in that size range and that the eddy (and segment) sizes are conﬁned to the range LK ≤l≤Lt , where Lt and LK are the model analogues of the integral, l t , and the Kolmogorov, ν k , length scales of a turbulent ﬂow. Note that Lt and LK are not necessarily equal to the respective ﬂow length scales. On the contrary in most applications of LEM as a stand alone model [7, 12, 14, 16], Lt was related to lt by

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Lt =Ct lt where Ct is a constant achieving values between 2 to 6.28. The model analogue of the Kolmogorov length scale was accordingly adjusted from [20] Lt /LK =Re3/4 . A random sequence of the rearrangement events introduces a random walk of a ﬂuid element that is interpreted as the turbulent diﬀusivity DT . This interpretation leads to a relationship between an event-frequency parameter λ, with units (length x time)−1 , the segment size probability density function f(l) and the physical quantity DT . ∞ 2 λ DT = l3 f (l)dl (3) 27 0 Epochs t0 are governed by a Poisson process such that the frequency of occurrence of events with segment centres in any interval [0,Y] is λY. Segment (eddy) sizes are sampled from a probability density function f(l) that incorporates the Kolmogorov cascade picture of inertial range turbulence and is given by l−8/3 5 (4) f (l) = 3 (L−5/3 − L−5/3 ) t K

Equation (4) is logarithmic in shape promoting the occurrence of segments of size comparable to LK . From (3) and (4) it is easy to deduce the value of λ from 54 DT Lt 5/3 ( )( ) (5) λ= 5 L3t LK

Flow ﬁeld properties such as the integral and the Kolmogorov length scales and the turbulent diﬀusivity are fed into the model either from experimental data (as is the case of a stand alone LEM calculation) or from the solution of the momentum equations in a LES simulation. In the latter case, LEM attains a role more critical than a subgrid model as it practically replaces the scalar transport equations [19]. The advantage of LEM as an LES submodel is that it allows for a direct description of small scale dynamics that are critical to the overall mixing and combustion processes and dispenses with the need to describe the subgrid scalar ﬁeld in terms of an eddy diﬀusivity concept.

3 Results and Discussion In the present work, LEM has been initially used to simulate the experiments of [21] downstream a single line source (heated wire) in homogeneous grid turbulence (Case 1) and [22] in a reactive mixing layer created by two air streams doped with nitric oxide and ozone respectively undergoing an irreversible reaction to nitrogen dioxide and molecular oxygen with negligible temperature rise (Case 2). To gain further insight into the behaviour of the model two additional test cases have been also considered and results were compared to DNS simulations [23] of the decay of a passive scalar in statistically steady

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homogeneous turbulence (Case 3) and measurements [24] at the ﬁrst two diameters downstream a slightly heated turbulent jet (Case 4). Note that data sets [21–23] on the evolution of the mean and rms scalar have been considered in previous work [7, 13, 16, 17] analysing the performance of the LEM model. Here some of these results were reproduced to provide a benchmark for code validation. Focus however was placed on the evolution of the unconditional and conditional scalar dissipation predicted by LEM rather than on mean and rms scalar values. Table 1 summarises key parameters of the experimental and DNS datasets. Table 1. Key parameters of the experimental and DNS cases used in this work for comparison with the LEM model Cases 1[20] 2[22] 3[23] 4[21] 2 −1 DM [cm s ] 0.20 NO:0.22 O3 :0.18 0.035 0.22 C(y=0 or t=0) 1 NO:4 O3 :4 1 1 Da na 1.81 na na Re 10900 11700 0 8300 Ret 120 700-970 107 73 lt [m] 1.35x10−2 2.60x10−1 1.01x10−2 2.80x10−3 νK [m] 3.53x10−4 1.91x10−3 na 1.12x10−4

Table 2. Input parameters to the LEM model for Case 1 Case Variants (I) (II) (III) (IV) (V) Lt 0.7lt 1.5lt lt 5.6lt 5.6lt LK Lt Lt Lt Lt 5.9νK DT [m2 s−1 ] 8.89x10−4 8.89x10−4 8.89x10−4 8.89x10−4 8.89x10−4 Ret 120 120 120 120 120 λ[m−1 s−1 ] 11377 1156 3902 22 8825 f(l) δ(l-Lt ) δ(l-Lt ) δ(l-Lt ) δ(l-Lt ) Eq. (4) Initial Condition Step Function

LEM was applied in a total of twelve parametric test cases summarised in Tables 2-4. As shown in the tables, parametric studies have been performed as a function of Lt and LK , the probability distribution of the rearrangement segment, the event-frequency parameter, the Reynolds and the Damk¨ ohler numbers and the initial scalar proﬁle at t=0. The Damk¨ ohler number, that is the ratio of a characteristic time for chemical reactions to a characteristic ﬂow time, was computed from [22] as Da=kM(CN O,1 +CO3 ,1 )/U where k is the reaction rate, M is the mesh spacing of the turbulence generating mesh and U and Ci,1 are the mean streamwise velocity, assumed constant, and the

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initial reactant concentrations. The LEM model was solved at the transverse direction in space and the distance between neighbouring nodes was set to LK /6 so as to ensure that all scales down to the Kolmogorov microscale were suﬃciently resolved. The streamwise evolution of the scalar ﬁeld was estimated by use of Taylor’s hypothesis for a time of the order of several Lagrangian time scales. A total of 5,000-15,000 realizations of the scalar ﬁeld were performed for each parametric case (variant) in order to ensure statistical convergence. Table 3. Input parameters to the LEM model for Case 2. Case Variants (I) (II) (III) Lt 2.5lt 2.5lt 2.3t LK 1.15νK 1.15ν K 1.05ν K DT [m2 s−1 ] 1.50x10−2 1.50x10−2 1.50x10−2 Re 11700 11700 5300 Ret 1968 1968 1969 λ[m−1 s−1 ] 7731 7731 10128 f(l) Eq. (4) Eq. (4) Eq. (4) Da 1.813 0.34 1.813 Initial Condition Step Function

Figure 2a shows that computed temperature ﬂuctuations for Case 1 are in good agreement with experiments [21] provided that the eddy segment size is randomly sampled from a distribution described by (4) and the model integral and Kolmogorov length scales are increased in comparison to their measured values by empirical factors of 5.6 and 5.9 respectively [7] (Table 2, variant V). A parametric study on the eﬀect of rearrangement segment size on LEM predictions shows that use of a constant segment (instead of one varying via (4)) results in a sharp increase of up to about 50% in the values of temperature ﬂuctuations at the vicinity of the line source. A similar behaviour has also been reported [7] in previous LEM and‘ﬂapping model’ computations and indicates that even at locations close to the source were turbulence is ‘young’, large scale motions are not solely responsible for scalar mixing. To this end, it is of interest to note that predictions using a constant segment size improved when the event-frequency parameter was increased but that the behaviour of the model was non-monotonic (compare variants I to III and IV and I to II in Table 2 and Fig. 2a) with distance from source. Figure 2b presents computed values of the mean scalar half-width, y1/2 , i.e. the transverse location at which the mean scalar is half its centerline value, again for Case 1. Note that the evolution of y1/2 is of particular importance in terms of the model’s predictive ability, as it is a measure of turbulent advection in the direction along which LEM is applied. As shown in Fig. 2, 3 4

CN O,1 =4, CO3 ,1 =4 CN O,1 =0.7, CO3 ,1 =0.7

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application of (4) and use of the parameters of variant V under-predicts y1/2 by almost a factor of three, possibly due to the assumptions involved in the selection of the eddy size distribution, (4), and also due to the fact that the action of the eddy is assumed instantaneous. Underprediction is pinpointed to occur in the convective region between the Kolmogorov and the Langrangian timescales. Use of a constant eddy size of value comparable to lt is shown to enhance predictions with an underestimation penalty at times smaller than the Kolmogorov. Thus, the need to account for adequate diﬀusion of the centerline scalar values towards the outer transverse regions at times smaller than the Lagrangian time scale justiﬁes the selection of larger values for Lt and LK than the ones provided by the experiment. Increasing the event-frequency parameter was also shown to enhance predictions but its eﬀect was secondary in comparison to segment size. Table 4. Input parameters to the LEM model for Cases 3 and 4. Case 3 Case 4 Case Variants (I) (II) (I) (II) Lt 2πlt 2πlt lt 5.6lt LK 5.93ν K 5.93ν K νK 5.6ν K DT [m2 s−1 ] 2.62x10−4 2.62x10−4 6.44x10−3 6.44x10−3 Re 0 0 8300 8300 Ret 93 93 73 73 λ[m−1 s−1 ] 71 71 676913515 3850010 f(l) Eq. (4) Eq. (4) Eq. (4) Eq. (4) Da na na na na Initial Condition Sinusoidal5 Sinusoidal6 Step Function

An extension of the previous study to the reacting ﬂow of Case 2 is shown in Figs. 3 and 4. Computed mean and rms concentrations of the two reactants are shown to be in good agreement with the measurements (Figs. 3a and b) provided that the eddy segment size is randomly sampled from a distribution described by (4) and the model integral and Kolmogorov length scales are increased [7] in comparison to their measured values by empirical factors of 2.5 and 1.15 respectively (Table 3, variants I and II). Further, the LEM model is shown to predict remarkably well the mean reaction rate as a function the Damk¨ ohler number (Fig. 3c). Similar ﬁndings have also been reported by [7]. Bilger et al.[22] did not measure scalar dissipation, χ. However, due to its importance in the modeling of combusting ﬂows [25], it is instructive to investigate the eﬀect of both the Damk¨ohler number and the ﬂow Reynolds number on LEM predicted values of χ. Figure 4a shows mean unconditional scalar dissipation proﬁles, computed in terms of the squared gradient of a 5 6

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mixture fraction estimated from C = (CN O − CO3 + CN O,1 )/(CN O,1 + CO3 ,1 ). Note that since the model is one dimensional, results reported here correspond to the transverse component of χ. It is clear that the model reproduces the quasi-linear dependence of χ with Reynolds number [23, 26, 27] and that predicted values are almost independent of Da. The model also predicts a quasi-log normal pdf of the scalar dissipation that is shifted towards smaller dissipation values when the Reynolds number was reduced (Fig. 4b). The ability of the model to reproduce the statistical moments of the unconditional χ and also of the scalar dissipation conditional on selected scalar values was further investigated by comparing results from LEM calculations to DNS simulations [23] and additional measurements [24]. The data of [23] have also been considered by [15]. Here their work is extended to include an

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(b) LEM - Case 2 (I) (II) (III)

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investigation on the ability of LEM to model joint statistics. Following [15] the LEM integral and Kolmogorov scales were adjusted by factors of 6.22 and 5.93 respectively. Figure 5a shows, as a function of initial conditions, that the time evolution of the scalar dissipation agrees remarkably well with the DNS data. Figures 5b and c assess statistical independence between the scalar and its dissipation by comparing the joint scalar-scalar dissipation pdf, P(c,χ) to the product of the two individual pdfs P(c)P(χ) at a time t equal to about 22% of the Lagrangian timescale. It can be seen that the model correctly depicts diﬀerences between P(c,χ) and P(c)P(χ), known to exist at early times [26], and thus a dependence between the scalar and the scalar dissipation. At later times, computations revealed that the scalar pdf relaxes to Gaussian and the scalar and its dissipation were statistically independent in accordance to theory. Figure 6 presents the mean and rms transverse normalized temperature proﬁles and the mean unconditional scalar dissipation for Case 4 at one and two diameters downstream a heated turbulent jet [24]. Consistent to previous

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C. Papadopoulos and K. Sardi (a) z/D=1 exp (I) (II) z/D=2 exp (I) (II)

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(r-r*)/r*

Fig. 6. (a) Mean and (b) rms mixture fraction proﬁles and (c) mean scalar dissipation as a function of downstream distance for Case 4. The radial distance from the centreline has been normalised as r*=(r-rd )/(rd /2), where rd corresponds to =0.5.

results from Cases 1-3, Figs. 6a and b show that use of LEM scales larger than the ﬂow scales, generally improves predictions in terms of the maximum rms values at the vicinity of the source. However use of larger rearrangement segments is shown to lead to a severe over prediction in the width of the mixing region and to a faster decay of the scalar ﬂuctuations with distance from source regardless of the values of Lt and LK in the range examined. LEM is shown to overpredict the mean scalar dissipation in both variants although results improve with distance from source and with use of larger eddy sizes. It is of interest to note that in Case 4, the event frequency parameter λ achieved values several orders of magnitude larger than the ones adopted in the previous cases. As a consequence, advection, via the rearrangement events, rather than molecular diﬀusion is the dominant mechanism for scalar dispersion. Figure 7 compares LEM predictions of the mean scalar dissipation conditional on the scalar ﬂuctuations with experimental data at an arbitrary selected location oﬀ the jet centerline corresponding to a mean scalar value (a)

10

1

/

/

10

0.1 exp. [22] = 0.52 = 0.22 0.01 LEM = 0.3 (I) (II)

1E-3

-5

-4

-3

-2

1

0.1 exp. 0.01

1E-3

-1

0

1

(c-c!)/c'

2

3

4

5

(b)

= 0.4 = 0.22 LEM = 0.3 (II) -5

-4

-3

-2

-1

0

1

2

3

4

5

(c-c!)/c'

Fig. 7. Transverse proﬁles of the y-component of the mean scalar dissipation conditional on the scalar ﬂuctuations at (a) z/D = 1 and (b) z/D = 2 for Case 4.

Statistical Properties of LEM

201

of 0.30. Measurements are presented for two locations corresponding to mean scalar values of 0.5 and 0.2. LEM is shown to generally maintain the experimentally observed trends of independence between scalar dissipation and scalar ﬂuctuations at small scalar rms values and to identify the existence of scalar dissipation values lower than the mean at ﬂuctuations of value of ±2c′ . The use of LEM scales larger than the ﬂow scales is shown to lead to conditional dissipation values that are symmetrically distributed around the mean scalar value with a tendency to underestimate the presence of the rare large scalar ﬂuctuations, a shortcoming that could have implications for predictions of limiting phenomena in combusting ﬂows. This symmetric distribution of the mean conditional dissipation around the mean scalar value is consistent to the presence of a symmetric scalar pdf, in this case quasi-Gaussian (corresponding to a fully mixed mixture), earlier than experimentally observed.

4 Conclusions The Linear Eddy Model was used to predict the statistical moments of nonreactive and reactive scalars in mixing turbulent layers. Results show that the LEM model reproduces a number of important features of reacting and conserved scalars. However input parameters are shown to signiﬁcantly inﬂuence the model behaviour so that care is required when employing the model particularly at early times both as a stand alone model and as an LES submodel.

References [1] W.T. Ashurst, A.R. Kerstein, L.M. Pickett, and J.B. Ghandhi. Phys. Fluids, 15:579-582, 2003. [2] J.C. Hewson and A.R. Kerstein. Combust. Sci. Technol., 174:35-66, 2002. [3] A.R. Kerstein, W.T. Ashurst, S. Wunsch, and V. Nilsen. J. Fluid Mech., 447:85-109, 2001. [4] T.D. Dreeben and A.R. Kerstein. Int. J. Heat Mass Transf., 43:3823-3834, 2000. [5] A.R. Kerstein and T.D. Dreeben. Phys. Fluids, 12:418-424, 2000. [6] A.R. Kerstein. J. Fluid Mech., 240:289-313, 1992. [7] A.R. Kerstein. Combust. Sci. Technol., 81:75-96, 1992. [8] A.R. Kerstein. J. Fluid Mech., 231:361-394, 1991. [9] A.R. Kerstein. Phys Fluids A-Fluid Dynam., 3:1110-1114, 1991. [10] A.R. Kerstein. J. Fluid Mech., 216:411-435, 1990. [11] A.R. Kerstein. Combust. Flame, 75:397-413, 1989. [12] A.R. Kerstein. Combust. Sci. Technol., 60:391-421, 1988. [13] T. Echekki, A.R. Kerstein, T.D. Dreeben, and J.Y. Chen. Combust. Flame, 125:1083-1105, 2001.

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[14] M.A. Cremer, P.A. McMurthy, and A.R. Kerstein. Phys. Fluids, 6:21432153, 1994. [15] P.A. McMurthy, T.C. Gansauge, A.R. Kerstein, and S.K. Krueger. Phys.Fluids A-Fluid Dynam., 5:1023-1034, 1993. [16] A.M. Debruynkops and J.J. Riley . Combust. Flame, 112:253-260, 1998. [17] P.A. McMurthy, S. Menon, and A.R. Kerstein. A Linear-Eddy Mixing Model for Large Eddy Simulation of Turbulent Combustion . In Galperin, B., Orzag, S.A. (eds) Large Eddy Simulation of Complex Engineering and Geophysical Flows. Cambridge University Press, Cambridge, 1993. [18] A.R. Kerstein. Comp. Phys. Comm., 148:1-16, 2002. [19] S. Menon. Subgrid Combustion Modelling for the Next Generation National Combustion Code (http://www.ccl.gatech.edu/reports/ paper info?pub id=236). Annual Report to NASA Glenn Research Centre, 2003. [20] H. Tennekes and J.L. Lumley. A First Course in Turbulence MIT Press, Cambridge, 1972. [21] Z. Warhaft. J. Fluid Mech., 144:363-387, 1984. [22] R.W. Bilger, L.R. Saetran, and L.V. Krishnamoorthy. J. Fluid Mech., 233:211-242, 1991. [23] V. Eswaran and S.B. Pope. Phys. Fluids, 31:506-520, 1988. [24] K. Sardi and A.M.K.P. Taylor. Joint scalar-scalar dissipation measurements at the exit of a turbulent jet In Durst, F., Launder, B.E., Schmidt, F.W., Whitelaw, J.H. (eds) Eleventh Symposium on Turbulent Shear Flows. Volume I, pp. 8/19-8/24. [25] N. Peters. Combust. Sci. Tech., 30:1-17, 1983. [26] K. Sardi, A.M.K.P. Taylor, and J.H. Whitelaw. J. Fluid Mech., 361:1-24, 1998. [27] W.T. Ashurst, A.R. Kerstein, R.M. Kerr, and C.H. Gibson. Phys. Fluids, 30:2343-2353, 1987.

Lattice-Boltzmann LES of Vortex Shedding in the Wake of a Square Cylinder Paula Mart´ınez-Lera, Salvador Izquierdo∗ , and Norberto Fueyo Fluid Mechanics Group and LITEC, University of Zaragoza Mar´ıa de Luna 3, 50018, Zaragoza, Spain [email protected] Summary. The success in the use of lattice-Boltzmann methods for the simulation of laminar ﬂows has prompted the interest in extending the technique for the simulation of turbulent ﬂows. In this paper, a square cylinder at Re = 21400 is used to investigate the suitability of lattice-Boltzmann methods to solve turbulent unsteady problems with open boundary conditions. Numerical simulations are performed with a single-relaxation-time lattice-Boltzmann method and with a large-eddy simulation turbulence model. These simulations present several not-fully-resolved issues for lattice-Boltzmann methods, which are brieﬂy discussed in this paper. Speciﬁcally, we propose a ﬁltered-density open-boundary condition for ﬂuid-ﬂow simulations with the lattice-Boltzmann equation. This ﬁlter modiﬁcation of inﬂow boundary condition is shown to improve simulations of unsteady ﬂows at high Reynolds numbers, in terms of calculation stability and speed, and quality of the results.

1 Lattice-Boltzmann Method for Turbulent Flows In recent years, the lattice-Boltzmann equation (LBE) method has emerged as a promising CFD tool for eﬃciently simulating laminar ﬂuid ﬂow (see for example Geller et al. [1]). The LBE is a mesoscopic kinetic approach for the solution of Navier-Stokes equations. Its fundamental idea is to build simpliﬁed kinetic models that have the essential information of the microscopic physics so as to construct averaged macroscopic properties obeying the desired macroscopic equations [2]. For further information about lattice-Boltzmann methods we refer to two comprehensive reviews [3, 4]. High-Reynolds-number simulations with lattice-Boltzmann methods combine the diﬃculties owing to the turbulent nature of the ﬂow and, additionally, several LBE-speciﬁc ones which have been only partially solved, as reviewed by Yu et al. [5]. Eﬀorts have been made mainly in the ﬁelds of: force evaluation over solid boundaries, grid reﬁnement, boundary conditions for plane and curved walls, inlet ∗

Supported by CSIC (Spanish Council for Scientiﬁc Research) under CSIC I3P Programme, ﬁnanced by the European Social Fund.

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Paula Mart´ınez-Lera, Salvador Izquierdo, and Norberto Fueyo

boundary treatment, numerical stability and turbulence models. The latter are required, as it is the case in other numerical methods for solving the NavierStokes equations, because of the impracticability of resolving the smallest ﬂow features at high Reynolds numbers, which for most practical ﬂows would require nowadays-inexistent computational resources. In this work, we use Large Eddy Simulation to predict the turbulent ﬂow around a square cylinder. We have chosen this ﬂow because it is largely dominated by the shedding of vortices in the wake of the cylinder, allowing us to focus our attention on the inﬂuence of the implementation of inlet and outlet boundary conditions on the transient behavior of the ﬂow. 1.1 The Lattice-Boltzmann Equation The lattice-Boltzmann-equation model can be derived from the Boltzmann equation, which is an equation for the particle-velocity distribution. Although historically the LBE method has its origins in the lattice gas automata [6], its derivation from the Boltzmann equation shows the potential developments of the method. Speciﬁcally, it is possible to develop other numerical methods, diﬀerent from the one employed in this work, to solve this equation and to simulate ﬂuid ﬂow. To obtain the lattice-Boltzmann equation from the Boltzmann equation, the latter is ﬁrst discretized in the velocity space using a ﬁnite set of velocity vectors: ∂fα + eα · ∇fα = Ωα , (1) ∂t where α = (0, 1, ..., N ) is the velocity index, fα is the particle distribution function, eα is the discrete set of velocities and Ωα is the collision operator. Here, the simpliﬁed collision operator BGK is used [7]. This collision operator is also called Single-Relaxation-Time (SRT) approximation, and generates the following SRT-LBE evolution equation for fα : 1 (2) fα (x + eα δt, t + δt) − fα (x, t) = − [fα − fαeq ] ; τ where τ is the relaxation coeﬃcient and fαeq are the equilibrium distribution functions (which will be speciﬁed below). The number of velocities used is N + 1, and depends on the model applied. The lattice-Boltzmann equation can be seen as a special ﬁnite-diﬀerence discretization of second order in space and ﬁrst order in time of (1). Macroscopic variables of the ﬂow ﬁeld, such as density, momentum, and the strain-rate tensor Sij , are obtained by integrating the distribution function f over the velocity space. Therefore, integrating in the discrete eα space, these quantities are, respectively:

LBM-LES of Vortex Shedding

ρ=

205

fα ,

(3)

eα fα ,

(4)

α

ρu =

α

Sij =

1 eαi eαj (fα − fαeq ) ; 2ρc2s τ α

(5)

with cs being the speed of sound (see below). To solve the evolution equation (2) for the particle distribution function, two steps are performed: collision and propagation. These are given by (6) and (7) below, where f˜α represents the post-collision state: 1 f˜α (x, t) = fα (x, t) − [fα − fαeq ] α = 0, 1, 2, . . . , N ; τ fα (x + eα δt, t + δt) = f˜α (x, t) α = 0, 1, 2, . . . , N .

(6) (7)

Two diﬀerent solvers have been used in this paper. The ﬁrst one was a twodimensional lattice-Boltzmann model with nine velocities (D2Q9), and the second one was the three-dimensional model with nineteen velocities (D3Q19). In these two LBE methods, the equilibrium distribution function fαeq has the following form: 4 3 9 3 3 2 eq (8) fα = wα ρ 1 + 2 (eα · u) + 4 (eα · u) − 2 u · u . c 2c 2c

The weighting factors wα are: w0 = 4/9, w1−4 = 1/9, and w5−8 = 1/36, for D2Q9 method; and w0 = 1/3, w1−6 = 1/18, and w7−18 = 1/36, for D3Q19. The set of velocities for D2Q9 and D3Q19 methods are given in (9) and (10), and the corresponding lattices are shown in Fig. 1. D2Q9 eα

D3Q19 = eα

#

=

#

(0, 0) α=0 (±1, 0)c, (0, ±1)c α = 1 − 4 , (±1, ±1)c α=5−8

(0, 0, 0) α=0 (±1, 0, 0)c, (0, ±1, 0)c, (0, 0, ±1)c α=1−6 . (±1, ±1, 0)c, (0, ±1, ±1)c, (±1, 0, ±1)c α = 7 − 18

(9)

(10)

In both cases, δx and δt, the spatial and temporal discretization steps (which determine the lattice units) can be taken as unity, and therefore the characteristic √ speed is c = δx/δt = 1. The speed of sound in these models is cs = c/ 3 , and, for pressure, an equation of state for ideal gas is used: p = ρc2s . Employing the Chapman-Enskog series expansion on (2), the NavierStokes continuity and momentum equations can be derived in the limit of small Knudsen and Mach numbers:

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Paula Mart´ınez-Lera, Salvador Izquierdo, and Norberto Fueyo

e15 e2

e6

e3

e10

e0

e1

e2 e12

e7

y

e4

e8

z x

e11

e5

e14 e3

e7

e17

e5

e1 e13

e6

e18

e8

e4

e9

e16

Fig. 1. D2Q9 (left) and D3Q19 (right) lattices

∂ρ ∂ρui + =0, ∂t ∂xi ∂(ρui ) ∂(ρuj ui ) ∂p ∂ ∂ui + =− + ρν . ∂t ∂xj ∂xi ∂xj ∂xj

(11) (12)

The kinematic viscosity in the Navier-Stokes equation is related to the relaxation coeﬃcient τ through the equation: ν = c2s (τ −1/2)δt. Since viscosity is positive, τ > 1/2 is required as a stability condition. 1.2 Lattice-Boltzmann Large-Eddy Simulation High-Reynolds-number direct simulations require too expensive a computational eﬀort for real applications. This leads to the use of turbulence models. In this paper the Large-Eddy Simulation (LES) approach is used. LES simulations are often explicit, which makes LBM well suited for their implementation. Recently, some SRT-LBE-LES for a mounted cube [8], a stirred tank [9], synthetic and free jets [10] and swirling ﬂows [11], have proved its applicability; but it is nevertheless clear that further research is required to improve the performance of the method in several respects, including: stability; wall boundary-conditions (wall models); non-reﬂecting boundary conditions; and better SGS models which can be computed locally. Recent works have focused on improving the stability of the LBM-LES simulations for high-Reynolds-number ﬂows using a multi-relaxation-times (MRT) latticeBoltzmann method, as shown by [12] and [13]. Among all the SGS models used in LES, the Smagorinsky one is perhaps the simplest and also the most commonly used one in the LBE literature (reference [10], which uses the Localized Dynamic Model, is an exception). It is nevertheless well known that its simplicity brings about some deﬁciencies, including an excessive dissipation or incorrect near-wall behavior. The main advantage of LES for LBM is that all the associated calculations are local,

LBM-LES of Vortex Shedding

207

as will be shown below. The implementation of Smagorinsky model for Large Eddy Simulation in the lattice Boltzmann equation was proposed by Hou et al. [14]. An eﬀective viscosity is deﬁned as: ¯ ; νtotal = ν0 + (CS ∆)2 |S|

(13)

where ν0 is the molecular viscosity, CS is the Smagorinsky constant, ∆ is the ¯ is the norm of the resolved strain-rate tensor, and the overbar ﬁlter width, |S| denotes ﬁltered values. In LBE methods, the norm of the resolved strain rate tensor can be computed as [14]: τ02 + 18(CS ∆)2 Q1/2 /ρ − τ0 ¯ ; (14) |S| = 6(CS ∆)2 ¯ neq , with Π neq = eαi eαj (fα − fαeq ). We use Cs = 0.1 ¯ neq Π where Q = Π ij ij ij α [15] and ∆ = 4δx. This choice for ∆ is based on the studies by Chow and Moin [16]. They found that, for a second order ﬁnite diﬀerence scheme, a ﬁlter-grid ratio of four should be used for the discretization errors to be smaller than the subgrid terms.

2 Square Cylinder Test Case The conﬁguration selected as a test case was the ﬂow around a square cylinder [17] for a Reynolds number of 21400 based on inﬂow velocity and on the cylinder width. The test problem was chosen because it is highly transient in nature, thus providing an appropriate test bench for the analysis of the inﬂuence of the open boundary conditions on the transient behavior. A schematic representation of the ﬂow geometry is given in Fig. 2. A free slip boundary condition was implemented at top and bottom limits of the domain, and periodic conditions at the lateral boundaries, whereas a no-slip condition was used on the walls of the square cylinder (bounce-back scheme). This scheme is considered second-order accurate when the wall is placed midway between two nodes, as it is the case. Regarding the inlet and outlet conditions, several alternatives were implemented in order to assess their performance (see Section 4). The drag (Cd ) and lift (Cl ) coeﬃcients were computed with the momentum exchange method. This approach to evaluate forces on a solid body is unique to the LBE method. Forces result from the momentum exchange between two opposite directions, say α and α ¯ of the wall node and its neighbors in the ﬂuid domain. The total force on a solid body with planar walls is: F=

N

xwall α=1

1 0 eα f˜α (xwall , t) + f˜α¯ (xwall + eα¯ δt, t) .

(15)

Further information on momentum exchange method implementation can be found in the work by Mei et al. [18]. The Strouhal number was calculated

208

Paula Mart´ınez-Lera, Salvador Izquierdo, and Norberto Fueyo y

Top

14D

x

D

Out

In Bottom 4.5D

D

15D

z Lateral 4D In

x Lateral

Out

Fig. 2. Square cylinder geometry

via a fast Fourier transform (FFT) of the vertical velocity on the x-centerline at a distance of one cylinder width downstream of the cylinder.

3 Square Cylinder Results The results shown in this section were computed with a D3Q19 model with a uniform grid of 265×169×49 nodes. Simulations were performed setting Re = 21400, M a = 0.17 (which results in an inlet velocity of U0 = 0.1) and ρ0 = 1. This (relatively coarse) LES is compared with numerical results [19] by other authors and with experimental data [20]. Figure 3 shows mean x-velocity proﬁles at diﬀerent x/D sections in the domain, together with Lyn’s experimental results [20]. Some discrepancies can be seen in the wake region; albeit an accurate reproduction of the experimental results is not the primary aim of this paper, it is nevertheless acknowledged that the mesh size employed may be insuﬃcient to properly resolve the ﬂow in the vicinity of the cylinder. Table 1 presents the values of the integral parameters from the present and other LES simulations, Lyn’s experimental data, and other experimental results. The present-simulation results are of a similar quality to those from other LES simulations. The integral parameters in the table are likely to be aﬀected by some of the features of the current simulation, the eﬀect of which should be studied in more detail, notably mesh resolution [1], the deﬁciencies of the Smagorinsky subgrid model, and the absence of

LBM-LES of Vortex Shedding

209

6 X/D=-3

X/D=-0.5

X/D=0.5

X/D=1.5

X/D=2.5

X/D=4.5

X/D=3.5

X/D=5.5

5

Y/D

4

3

2

1

0 0

0.5 1 U/U0

0

0.5 1 U/U0

0

0.5 1 U/U0

0

0.5 1 U/U0

0

0.5 1 U/U0

0

0.5 1 U/U0

0

0.5 1 U/U0

0

0.5 1 U/U0

1.5

Fig. 3. Mean velocity proﬁles at diﬀerent x/D: comparison with experimental data (Lyn et al. [20]) (squares)

turbulence in the inﬂow conditions (albeit this is a frequent assumption in simulations using other methods, [17]). Table 1. Comparison of time-averaged lift and drag coeﬃcient, Strouhal number and recirculation length

SRT-LBE-LES (This work) LES (Others [17][19]) Exp. (Lyn et al. [20]) Exp. (Others [17])

Cl

Cd

St

lr

0.002 -0.3–0.03 — —

1.9 1.66–2.77 2.1 1.9–2.1

0.16 0.066–0.15 0.132 —

2.7 0.89–2.96 — 1.38

In Fig. 4, instantaneous contours of isovorticity are plotted. This representation reveals the structures organized in von Karman vortex streets.

4 Computational Assessment of the Open Boundary Conditions A study of the LBM implementation of the inlet boundary-conditions was performed. Even though turbulence is of course a three-dimensional phenomenon, a D2Q9 (2D) model with a grid of 265×169 nodes was considered appropriate

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Paula Mart´ınez-Lera, Salvador Izquierdo, and Norberto Fueyo

Fig. 4. Instantaneous vorticity (ωz ) at Z/D = 0 xy-plane. Forty contour levels from ωD/U0 = −10 to ωD/U0 = +10 are plotted

for the purpose of assessing in an economical way the eﬀects of the boundarycondition implementation. As reported by Yu et al. [21], the interaction between the inlet and the nodes inside the domain may provide a mechanism for waves to be reﬂected back into the domain, spoiling the stability and quality of the solution. In the course of this study, it was found that most open-boundarycondition implementations limit the quality of the solution for the problem analyzed. This is an increasingly-important eﬀect as the Reynolds number is increased, since as the relaxation parameter gets smaller, the dissipation decreases, and errors due to boundary conditions become more apparent. Figure 5 shows the typical eﬀect of low frequency reﬂected waves on the simulation results. This simulation was performed using equilibrium values to ﬁx the macroscopic properties at the inlet (see Sect. 4.1). The ﬂuid in the domain is initially considered at rest, and thus the lift coeﬃcient is nearly zero until vortex-shedding starts at a later time. The reﬂecting waves cause an unphysical pressure ﬁeld and larger-than-expected oscillations of the integral parameters (such as Cl in Fig. 5). However, time-average ﬁelds are often sensible, and thus may be misleading as an indication of the quality of the LBE-LES solution for this type of ﬂows. 4.1 Analysis of Open Boundary Conditions: the Inlet Condition In order to prescribe a velocity or pressure at an open boundary, several methods have been proposed in the past. The most obvious one consists in setting the distribution functions at the boundary to their equilibrium values, since f neq << f eq . These values are calculated at the inlet from the given velocity values and from a pressure value that can be extrapolated or calculated from the known distribution functions at the inlet, as reported by Noble et al. [22] and Zou and He [23]. This condition has a leading error of ﬁrst order; to improve it, the value of the non-equilibrium functions must be estimated. Several methods have been proposed for estimating non-equilibrium values [23, 24]. A

LBM-LES of Vortex Shedding

211

120

100

Cl 80

60

20

40

60

80

100

120

140

160

Time steps

Fig. 5. Lift coeﬃcient (left) from equilibrium boundary condition with zero initial velocity and pressure ﬁelds; pressure contours and streamlines (right) around the square cylinder at time step 52,000

popular approach is to apply the bounce-back scheme to the non-equilibrium functions. However, as reported by Yu et al. [21], the standard bounce-back scheme creates a mechanism for pressure waves to be constantly reﬂected back to the domain from the inlet, degrading the solution. In general, boundary conditions involving pressure extrapolation or a dependence of the pressure on the values inside the domain suﬀer from a similar shortcoming. The simulations that were performed in this work conﬁrmed the importance of the problem of waves reﬂecting at the inlet. In these tests, the velocity was ﬁxed at the inlet and pressure at the outlet. In Fig. 6 the temporal evolution of the lift coeﬃcient for two diﬀerent inlet boundary-conditions is depicted. The solution is computed with density ﬁltering, as detailed in Sect. 4.2, until time step 80,000, when the corresponding boundary condition starts to be applied. At this point, the reﬂection of waves causes signiﬁcant errors in the calculated parameters. 10 8 6 4 2

Cl

Cl

0 2 4 6 8 10

Time steps

0

50000

100000

150000

Time steps

Fig. 6. Temporal evolution of lift coeﬃcient with equilibrium (left) and with Zou and He (right) boundary condition

212

Paula Mart´ınez-Lera, Salvador Izquierdo, and Norberto Fueyo 4 3.5 3 2.5

Cd

Cl

2 1.5 1 0.5 0

Time steps

0

50000

100000

150000

Time steps

Fig. 7. Temporal evolution of lift (left) and drag coeﬃcient (right) with Yu’s boundary condition

A new method was proposed by Yu et al. [21], whereby the non-equilibrium values are reﬂected back like in the bounce-back scheme and an interpolation scheme is included to account for a variable distance from to the ﬁrst node. The key idea of their method is to ﬁx the pressure as well as the velocity at the inlet, whereas at the outlet it implements just an extrapolation of the distribution functions. This inlet condition proposed by Yu et al. [21] was also tested. Although extrapolation at the outlet produced some undesired gradients and noticeable errors, the method proved to reduce the unphysical waves inside the domain. Figure 7 shows the calculated evolution of the lift and drag coeﬃcients, and indicates that a periodic stable value for both values is reached, thus improving the performance of the boundary-condition implementations previously described. It should be remarked that this boundary condition is more sensitive than others to initial conditions. Since just an equilibrium distribution function is used to set two variables at the inlet, the value of the density is not really ﬁxed, and this induces a variation also in the velocity. For example, when the initial condition for the simulation is a quiescent ﬁeld, the method provides an inlet velocity which is around 12% lower than required; this can be seen in Fig. 8. Yu et al. [21] do not indicate how their calculation is initialized to avoid this problem. Regarding initial conditions, it should be mentioned that the sudden start of an inﬂow on an initially quiescent ﬂuid can give rise to pressure waves that can be very slowly damped if a compressible method is used. To discard this eﬀect in the present ﬂow, some of the above simulations were repeated with a gradual increase in the inlet velocity with time, with very similar results to those presented. This suggests that the source of the pressure waves is not the initial condition, but the vortex shedding.

LBM-LES of Vortex Shedding 1

213

u=0 u0

0.8 0.6

U/U0 0.4 0.2 0 -0.2

-5

0

5

10

15

20

X/D

Fig. 8. Centerline average velocity proﬁle with Yu’s inlet condition and with two diﬀerent initial conditions

4.2 Density Filtering at Inlet In accordance with the aim of trying to reduce the interaction between the inlet and the inside of the domain, a modiﬁcation was introduced in those open-boundary methods, such as the equilibrium or the bounce-back condition, in which the pressure at the inlet depends on the values of distribution functions coming from the inside of the domain. The observation of the pressure evolution at the inlet led to the idea of implementing a low pass ﬁlter for the calculation of the density. The ﬁltering, while preventing the reﬂection of the higher frequency waves, allows for a gradual convergence to the solution. A ﬁrst order in time ﬁlter was chosen for this purpose, due to its simplicity of implementation. It has the following expression in the Fourier space: F (iω) =

1 ; 1 + Tc iω

(16)

where Tc is the inverse of the cut-oﬀ frequency ωc . This ﬁlter can be applied for all those boundary conditions where the velocity proﬁle is ﬁxed and a closure procedure is used to determine ρ and fα . Applying the ﬁlter to the density we obtain: ρi = ρi+1

1 . 1 + Tc iω

(17)

In the time domain, (17) results in an expression in the form: = ρn+1 i

n ρn+1 i+1 + Tc ρi , 1 + Tc

(18)

where i represents the inlet, i + 1 the ﬁrst ﬂuid node, n the previous time step and n + 1 the current time step. The implementation of the ﬁltered-density boundary condition involves the choice of an appropriate value for the cut-oﬀ frequency ωc or its inverse Tc . The larger Tc is, the less spurious oscillations will reﬂect into the domain.

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But, at the same time, the convergence of the density at the inlet to its ﬁnal value has an inertia of approximately Tc time steps. Therefore a large value for Tc should be used, but not so large as to make the convergence too slow. As shown in Fig. 9, the ﬁltering at the inlet proved to reduce the problem of unphysical waves in the domain, providing better results. Additionally, the ﬁltered-density boundary condition reduces the error produced by an arbitrary initialization, improving Yu’s boundary condition. Furthermore, it can reduce the transient time (e.g. towards a sought-after periodic solution) by breaking the symmetry of the initialization ﬁeld. This can be achieved by a spacedependent selection of Tc . 4 3.5 3 2.5

Cd

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Fig. 9. Lift (left) and drag coeﬃcient (right) for ﬁltered equilibrium boundary condition

5 LBE-LES Stability Discretization errors, compressibility errors and boundary conditions eﬀects determine the overall numerical quality of lattice-Boltzmann simulations. On the other hand, large-eddy simulations have two sources of numerical errors [16]: discretization and aliasing errors depending on the numerical scheme used, and subﬁlter-scale (SFS) model errors. This results in a complex accumulation of errors in the SRT-LBE-LES simulation, the interaction among which needs to be determined in order to minimize them. Even once the boundary-condition eﬀects have been minimized, the pressure ﬁeld may be aﬀected by high-frequency oscillations due to lack of stability. The eﬀect on the pressure ﬁeld of ﬁlter width and grid resolution can be seen in Fig. 10, which shows results for two successively-reﬁned meshes and ﬁlter-to-grid ratios (FGR) of 2 and 4. The ﬁgure indicates that a higher resolution does not result in a smoother ﬁeld, while increasing the FGR does.

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6 Conclusions A 3D SRT-LBE-LES simulation of the vortex shedding in a square cylinder has been performed and results have been presented. The diﬃculties to simulate turbulent ﬂow with the lattice-Boltzmann equation have been exposed and are supported by the results obtained in simulations. Some suggestions on how to improve the LBM-LES simulations are made. A key issue is the reﬂection of information at the domain open-boundaries. This is in part a problem shared with other numerical methods for incompressible ﬂows. Indeed, the migration of methods used successfully in the Finite Volume and Finite Diﬀerence ﬁelds to LBM is a route worth pursuing in earnest; however, two fundamental diﬃculties need to be born in mind; ﬁrstly, the fact that LBM methods require conditions on the distribution functions, which are not uniquely deﬁned in terms of the macroscopic variables; and, secondly, that the implementation should require only local calculations in order to retain one of the main computational advantages of LBM. As an alternative, a ﬁltered-density boundary condition has been proposed for simulating high-Reynolds-number unsteady ﬂows with the latticeBoltzmann equation. Two and three-dimensional simulations were performed

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to assess the improvements brought about by the new method. Three important issues which have been improved with this new boundary condition are: a reduction of the time required to reach the unsteady periodic ﬂow starting from the quiescent initial conditions; freedom to choose the initialization conditions; and partial elimination of the pressure waves arising out of the reﬂection on open boundary conditions. Additionally, it was checked that the mass error remained bounded, and that the ﬁlter yields the same ﬁnal timeaverage results. However, more work is needed to ﬁnd a priori the best value of the ﬁlter parameter Tc . Based on LBM-SRT-LES errors, further investigations are ongoing to improve the simulations of turbulent ﬂows. These include the determination of the eﬀect of the LES ﬁlter width on the results; and the eﬀect of using an MRT [25] collision term instead of the SRT one in unsteady conﬁgurations with open boundaries, as well as the improvement brought about by “incompressible” formulations [26] of the lattice-Boltzmann equation. However, these improvements are not likely to fully resolve the issue of wave reﬂection at the open boundary conditions.

References [1] S. Geller, M. Krafczyk, J. Toelke, S. Turek, and J. Hron. Benchmark computations based on lattice-Boltzmann, ﬁnite element and ﬁnite volume methods for laminar ﬂows. Computer & Fluids, 35:888–897, 2006. [2] S. Chen and G. Doolen. Lattice Boltzmann method for ﬂuid ﬂows. Annual Review of Fluid Mechanics, 30:329–364, 1998. [3] S. Succi. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford, 2001. [4] D. A. Wolf-Gladrow. Lattice-Gas Cellular Automata and Lattice Boltzmann Models. An Introduction. Lecture Notes in Mathematics 1725. Springer, 2000. [5] D. Yu, R. Mei, L.-S. Luo, and W. Shyy. Viscous ﬂow computations with the method of lattice Boltzmann equation. Progress in Aerospace Sciences, 39(39):329–367, 2003. [6] F. J. Higuera and J. Jim´enez. Boltzmann approach to lattice gas simulations. Europhysical Letters, 1989. [7] Y.H. Qian, D. d’Humi`eres, and P. Lallemand. Lattice BGK Models for Navier-Stokes Equation. Europhysics Letters, 17(6):479–484, 1992. [8] M. Krafczyk. Gitter-Boltzmann-Methoden: Von der Theorie zur Anwendung. Habilitation Thesis, 2001. [9] H. Hartmann, J. J. Derksen, C. Montavon, J. Pearson, I .S. Hamill, and H. E. A. van den Akker. Assessment of large eddy and RANS stirred tank simulations by means of LDA. Chemical Engineering Science, 59(12):2419–2432, 2004.

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[10] S. Menon and J.-H. Soo. Simulation of vortex dynamics in threedimensional synthetic and free jets using the large-eddy lattice Boltzmann method. Journal of Turbulence, 5(032), 2004. [11] J. J. Derksen. Simulations of conﬁned turbulent vortex ﬂow. Computer & Fluids, 34(3):301–318, 2005. [12] M. Krafczyk, J. Toelke, and L.-S. Luo. LES simulations with a multiple relaxation time LB model. International Journal of Modern Physics B, 17:33–39, 2003. [13] H. Yu and S. Girimaji. Near-ﬁeld turbulent simulations of rectangular jets using lattice boltzmann method. Physics of Fluids, 17:125106, 2005. [14] S. Hou, J. Sterling, S. Chen, and G. D. Doolen. A lattice Boltzmann subgrid model for high Reynolds number ﬂows. Fields Institute Communications, 1996. [15] H. Yu, S. S. Girimaji, and L.-S. Luo. DNS and LES of decaying isotropic turbulence with and without frame rotation using lattice Boltzmann methods. Journal of Computational Physics, 2005. [16] F. K. Chow and P. Moin. A further study of numerical errors in largeeddy simulations. Journal of Computational Physics, 184:366–380, 2003. [17] W. Rodi, J. H. Ferziger, M. Breuer, and M. Pourqui´e. Status of large eddy simulation: Results of a workshop. Transactions of the ASME, 119:248–262, 1997. [18] R. Mei, D. Yu, W. Shyy, and L.-S. Luo. Force evaluation in the lattice Boltzmann method involving curved geometry. Physical Review E, 65:041203, 2002. [19] ERCOFTAC. Test case LES2: Flow past a long square cylinder. http://www.ercoftac.org, 1997. [20] D. A. Lyn, S. Einav, W. Rodi, and J. H. Park. A laser-Doppler velocimetry study of ensemble-averaged characteristics of the turbulent near wake of a square cylinder. Journal of Fluid Mechanics, 1994. [21] D. Yu, R. Mei, and W. Shyy. Improved treatment of the open boundary in the method of lattice Boltzmann equation. Progress in Computational Fluid Dynamics, 5(1/2):3–12, 2005. [22] D. Noble, S. Chen, J. Georgiadis, and R. O. Buckius. A consistent hydrodynamic boundary condition for the lattice Boltzmann method. Physics of Fluids, 7:203–209, 1995. [23] Q. Zou and X. He. On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Physics of Fluids, 9:1591–1598, 1997. [24] P. A. Skordos. Initial and boundary conditions for the lattice Boltzmann method. Physical Review E, 48(6):4823–4842, 1993. [25] D. d’Humi`eres. Rareﬁed gas dynamics: Theory and simulations. In B. D. Shizgal and D. P. Weaver, editors, Generalized lattice Boltzmann equations, volume 159 of Progress in Astronautics and Aeronautics, pages 450–458. AIAA, Washington DC, 1992. [26] X. He and L.-S. Luo. Lattice Boltzmann model for the incompressible Navier-Stokes equation. Journal of Statistical Physics, 88:927–944, 1997.

LES on Cartesian Grids with Anisotropic Reﬁnement Gianluca Iaccarino and Frank Ham Center for Turbulence Research Stanford University Stanford, CA, 94305 - USA [email protected], [email protected] Summary. Cartesian grids with local anisotropic adaptation are combined with an Immersed Boundary (IB) method and used to perform large eddy simulations. The approach used to generate Cartesian grids with desired normal and tangential resolution to three-dimensional surfaces is described. Anisotropic reﬁnement result in a considerable reduction in grid size. An IB treatment based on solution reconstruction is proposed; it has the property of ensuring local mass conservation and its accuracy is investigated for laminar and turbulent ﬂows in channel not aligned with the grid. Mesh modiﬁcations that improve the grid quality for LES are proposed. The resulting mesh then requires a fully unstructured discretization and a parallel polyhedral-based ﬁnite-volume solver is applied to perform simulations of the ﬂow around several spheres.

1 Introduction Cartesian grids combined with the Immersed Boundary (IB) method were ﬁrst applied to problems in complex, moving geometries by [1]. The IB method allows the relatively simple Cartesian mesh generation and discretizations to be applied to problems with complex geometries by not requiring the grid to conform to the boundaries. Instead, the eﬀect of the boundaries on the ﬂow are modeled by adding source terms to the governing equations. The formulation of the IB forcing terms has been the subject of extensive research; [2] and, later, [3] have introduced a formally second-order approach which considerably improves the original, ﬁrst-order method. The major drawback of imposing boundary conditions through forcing terms (i.e. indirectly) is the lack of sharpness in capturing the non-conformal boundaries; this is mainly related to the spreading of the IB forcing on a cluster of neighboring cells. Another problem, seldom reported in the literature, is the inability of enforcing boundary conditions for strongly ﬂuctuating quantities, such as in turbulent ﬂows [4].

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Two classes of alternative treatments have been proposed in the literature; they both deﬁne boundary conditions directly on the non-grid-conforming boundaries: the cut-cell approach and the reconstruction method [5]. Cut-cell methods have been successfully developed for inviscid ﬂow simulations [6]. In this approach, the Cartesian cells that intersect the immersed surfaces are physically (or notionally) chopped, resulting in a polyhedron; one face or more coincide with the boundary and the desired conditions can be applied exactly. The extension of this approach to viscous ﬂow applications has been rather slow due to the inherent complexity in generalizing the cell-cutting algorithm for strongly stretched cells; [7] has successfully developed a two-dimensional method although applications are limited to low Reynolds numbers. The cut-cell approach relies on the use of irregular cells close to the boundaries thus introducing numerical complexity, loss of accuracy and potential stability limitations. In particular, the stability of the computational procedure is directly impacted by the presence of small cells obtained by cutting Cartesian cells near a vertex. Immersed boundary methods based on solution reconstructions in the neighborhood of the misaligned surfaces are gaining popularity because of their simplicity and wide range of applicability [4]. In this case, the desired boundary condition is used together with the computed solution in the ﬂuid cells away from the boundary to construct an interpolation stencil for the unknowns. The interpolant is used in lieu or in combination with the discretized governing equations in the cells intersected by immersed surfaces. [4] and [8] have used this method to simulate turbulent ﬂows within the ReynoldsAveraged Navier-Stokes framework, [9], [10] carried out Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES). Although these application were successful, a detailed study of the accuracy of the reconstruction method for turbulent ﬂow simulations has not been reported in the literature; the simple case of a channel ﬂow non aligned with the Cartesian directions has been extensively used for validation purposes only in the laminar regime. The objective of this paper is to illustrate the predictive capability of a newly developed immersed boundary approach for turbulent ﬂow simulations. The method is based on three main components: a fully unstructured lowdissipation LES solver, a conservative IB reconstruction, and a anisotropic adaptive mesh reﬁnement algorithm. Early on it was recognized that local adaptive mesh reﬁnement (AMR) could signiﬁcantly improve the performance of IB methods by introducing higher resolution in the proximity of the non-aligned boundaries [11-12]. AMR consists of a successive decomposition of the initial Cartesian cells until the desired resolution is locally achieved; parent cells are typically split in eight kids (in three-dimensions) and this naturally leads to a tree-like connectivity structure, OCTREE. The tree can be easily traversed to rebuild all the cell/neighbor information and grid coarsening and reﬁnement can be easily accomplished. The major limitation of this approach is its inherent isotropy: cells cannot be split according to a speciﬁc directional criteria (i.e. a gradient).

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The introduction of anisotropic mesh reﬁnement is a key factor in performing viscous ﬂow calculations eﬃciently. An extension of the tree-based AMR method to anisotropic reﬁnement has been introduced by [13] with the 2n tree where each parent cell can oﬀspring 2n kids with n = 1, 2, 3. However, the complexity of the resulting tree introduces substantial overhead in the computational procedure. The present approach is based on the use a fully unstructured solver: cells are not treated as Cartesian hexahedrons but as generic polyhedrons. No hanging nodes are present and cell faces are uniquely deﬁned by two associated control volumes. Additional advantages result when an unstructured solver is used to handle isotropic or anisotropic AMR grids: mesh smoothing and vertex motion can be employed to improve cell volume transitions, parallel decomposition can be very eﬃcient, ﬂexible multigrid agglomeration algorithms can be used, etc. In this paper only the ﬁrst aspect, related to improving mesh quality for locally reﬁned meshes, will be discussed. LES and DNS simulations of turbulent ﬂows on locally reﬁned grids are challenging even for boundary-conforming grids. [14] carried out simulations using a high-order bi-spline discretization and reported very accurate results for turbulent channels; further extensions of his methods to more complex ﬂows were hampered by extremely high computational cost of the numerical approach. [15] applied an AMR algorithm to perform DNS of the ﬂow around a sphere. One of the objectives of the present work is to present detailed statistics of the turbulent ﬂow in a channel with anisotropic local grid reﬁnement with and without mesh smoothing. In addition, the case of the channel non-aligned with the Cartesian direction is also studied In the following, ﬁrst a brief overview of the numerical method used in the unstructured solver is reported; the details of the mesh generation and node-motion are discussed. The immersed boundary treatment is analyzed for laminar and turbulent channel ﬂows and a conservative treatment in introduced. Last, the application of the current method to the ﬂow around a sphere is reported.

2 The Unstructured Solver: CDP The ﬂow solver CDP1 was used to perform all simulations reported in the present work. CDP is a parallel unstructured ﬁnite-volume-based solver that has been developed at Stanford University’s DOE-funded ASC center, the Center for Integrated Turbulence Simulations. The version of CDP used in the present contribution is based on a collocated fractional-step formulation to solve the ﬁltered incompressible Navier-Stokes equations with dynamic Smagorinsky subgrid scale closure. Details of CDP’s numerics are presented in [16] with some additional modiﬁcations presented in [17]. For clarity, the algorithm is described brieﬂy here. 1

CDP is named after Charles David Pierce (1969–2002)

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Uf ui,nb , pnb ui,p , pp

Fig. 1. Location of unknowns on a generic unstructured polyhedron grid.

Figure 1 shows the location of data used by CDP’s collocated formulation. The velocity vector ui and pressure p are stored at the control volume centers, and face-normal component of velocity Uf is stored at internal face centers. Here we refer to these locations as centers rather than centroids because they do not, in general, lie at the center of mass of the control volumes or faces, respectively (see section 3.2). On boundaries, the full velocity vector ub,i is stored at the boundary face centers. Time advancement of the velocity and pressure for the case of constant time step ∆t then proceeds as follows 1. Predict the face-normal velocity Uf using second-order Adams-Bashforth method Ufn+1 = 2Ufn − Ufn−1

(1)

where the superscripts represent the time level. 2. Solve the discretized momentum equation for velocity u ˆi,p associated with each cell p n−1/2 u ˆi,p − uni,p 1ˆ ∂p n + Hi,p + Hi,p =− ∆t 2 ∂xi p

(2)

where Hi represents the implicit discretization of convective and diﬀusive terms, and ∂p/∂xi the known cv-based pressure gradient from the previous time step. The details for Hi can be found in [17], and the pressure gradient uses the following Green-Gauss reconstruction 1 pp + pnb ∂p ni,f Af = ∂xi Vp 2

(3)

f

3. Remove the old pressure gradient to get u∗i

n−1/2 u∗i,p − u ˆi,p ∂p = ∆t ∂xi p

(4)

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4. Interpolate u∗i to internal faces using an equally-weighted average 1 ∗ ui,p + u∗i,nb ni,f (5) 2 where ni,f is the unit normal facing from cv p to cv nb. On boundary faces, Uf∗ is taken as the outward face-normal component of the known boundary velocity vector, i.e. Ufn+1 . 5. Solve the following Poisson system for the pressure at the mid-point of the current time step, pn+1/2 Uf∗ =

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(6)

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where the face-normal component of the pressure gradient ∂p/∂n is reconstructed as described in [17]. to account for the potential skewness of the face-normal relative to the straight line joining the two cv centers associated with the face. 6. Correct the cv-based velocity using the new pressure n+1/2 ∗ un+1 ∂p i,p − ui,p =− ∆t ∂xi p

(7)

3 Grid Generation and Smoothing The generation of Cartesian grids with anisotropic reﬁnement is carried out starting from a notional description of the domain of interest in (i,j,k) triplets. The immersed surfaces are described using a triangulated representation (STL, stereo-lithography, [4]); the surface nodes are converted into a (i,j,k) triplet, staggered with respect to the Cartesian grid, such that no conﬂicts can occur. Given a desired normal and tangential resolution (∆n, ∆t) on the STL surface the mesh control volumes are reﬁned in each Cartesian direction independently until they reach a target size deﬁned as: ∆n CV ; ∆t (8) ∆xi = M IN L |i |nST i where nST L is the local normal to the STL and i represents each Cartesian direction. A set of rules have been devised to drive the grid adaption procedure and are introduced in the following subsection. The grid reﬁnement algorithm proceeds in steps: 1. generate initial Cartesian grid covering the region of interest; 2. iterate until the mesh does not change: a) detect intersections between computational cells and the STL surfaces;

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b) compare current and target resolution for the cells cut by the immersed surfaces; c) set reﬁnement bits in the three Cartesian direction; d) enforce adaptation rules involving neighbors; e) perform cell splitting; f) rebuild proper unstructured mesh connectivity; (eventually) smooth the mesh; compute ﬁnal intersection between control volumes and STL surfaces; eliminate all cells fully contained within the STL surfaces (outside the computational domain); export mesh and IB intersections.

At each step in this procedure the mesh exists in two instances: a fully unstructured polyhedron grid with face-based connectivity and a notionally Cartesian-based mesh with elements deﬁned by two (i,j,k) triplets. These are the reﬁnement level representing the number of layers from the initial grids - and, thus, the cell size - and the index, deﬁning the position of the west, south, bottom (WSB) corner. The algorithm to detect and compute the intersection between the computational cells and the STL surfaces is based on Cartesian ray-tracing ([4]). Unique rays are identiﬁed and sorted according to their dimensions and location; an alternating digital tree is used to store the STL triangles and the ray-facet intersection checks are only carried out for a very limited number of cases, resulting in a very fast algorithm. The ray tracing method is currently implemented on a face-basis by identifying and storing intersections between the STL surfaces and the line connecting the face centroid and the control volume centroid. Each face can store two separate intersections and this allows to capture thin (but two-sided) surfaces as well as complex details not appropriately resolved by the grid (Fig. 2). This method produces a more consistent and robust way of handling complex geometry and also allows for a complete decoupling of the ﬂuid and solid regions. Ray tracing used in combination with a facet-vertex location algorithm guarantees that the control volumes intersecting all STL surfaces achieve the user-speciﬁed normal and tangential resolution. 3.1 Anisotropic Mesh Reﬁnement Anisotropic reﬁnement algorithms developed in two-dimensions do not extend in 3D in a straightforward manner (although this has often been incorrectly reported in the literature). In fact, for truly three-dimensional algorithms several additional constraints must be imposed; in the present implementation the following are used: 1. The maximum level diﬀerence in any direction is one - i.e. any neighbor can be only one level coarser or one level ﬁner. This rule is suﬃcient to deﬁne a 2-dimensional algorithm.

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b)

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Fig. 2. Examples of intersection for the face-based ray-tracing algorithm. a) thin surfaces b) under-resolved surfaces. I1 and I2 are the intersections stored at the face F corresponding to the left and right control volume, respectively.

2. all faces associated with a given direction (e.g. W,E, etc.) must have the same level in both tangential directions. This also makes the pairing required for coarsening more straight-forward. 3. Face-normal rule: a give face normal must pass through the centroid of at least one of its 2 cv neighbours. Alternatively stated, if a face must completely span at least one of its two cells. 4. Reﬁnement bias: If a requested reﬁnement will lead to a rule violation, neighbors are ﬁrst reﬁned to err on the side of reﬁnement. a)

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Fig. 3. Examples of reﬁned cells that violate the 3D rules: a) violation of rule 2; b) violation of rule 3.

3.2 Mesh Smoothing CDP’s numerical method, as described in section 2, is known to conserve kinetic energy ui ui /2 in the inviscid limit apart from a dissipative term associated with the pressure that must be present to ensure velocity-pressure coupling [17]. It is thus a potential candidate for LES where kinetic energy conservation is known to be an important property for accurate solutions. If implemented directly on the Cartesian adapted grids described previously, however, the algorithm will not be second-order accurate due to the equally weighted averages required to interpolate quantities to faces. In fact, the pressure gradient reconstruction described by equation 3 is not even consistent when the face location does not correspond to the mid-point between cell centers.

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To solve this problem of consistency, the mesh is modiﬁed by moving the nodes and face-center locations to the simple average of the associated cv centers. For the faces, this means the new face centers will be exactly half way between its two associated cv’s, ensuring consistency and improving the accuracy of any face-based interpolations that use equally-weighted averages. Figure 4 illustrates one such mesh before and after application of this mesh smoothing operation. Here the smoothing has been applied in 2 dimensions only for clarity.

Fig. 4. Comparison of Cartesian adapted mesh before (left) and after (right) mesh smoothing.

While such mesh smoothing algorithms are known to have problems with robustness and reliability when applied to general unstructured meshes (e.g. cvs falling outside the volume described by the modiﬁed node locations), our experience to date indicates that this algorithm always produces realizable grids when starting from a Cartesian adapted grid.

4 Immersed Boundary Reconstruction Before describing the IB reconstructions considered in the present work, it is useful to describe the manner in which boundary conditions are applied in the CDP code. This is important because it is desirable to implement the IB capability in a way that minimizes modiﬁcations, or ideally uses this same methodology for applying boundary conditions. For boundary conforming grids, the CDP solver requires 1) an expression for the momentum ﬂux at all boundary faces (as an implicit function of the nearest cv velocity value only), and 2) a speciﬁed face-normal velocity. This second requirement is critical to produce a well-posed Poisson system. For the IB faces, then, the boundary conditions can be applied by using a weighted least-squares reconstruction of the nearest velocity and its immediate neighbors, along with the known boundary velocity (i.e. ui = 0 for no-slip walls) and location (from the ray-tracing algorithm). This allows us to discretize and solve the Navier-Stokes equations in the cells cut by the boundary,

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treating the IB faces much like normal boundary faces. In the present work we have used equal weights in the least squares reconstructions, although one could argue that the known boundary velocity should have a larger weight because is is exactly known. To achieve mass conservation and formulate the Poisson system properly, the global ﬂow out of the boundary, including IB boundaries, must be zero. One way to achieve this is to introduce a constant global correction based on the net computed mass ﬂux. Such an approach will conserve mass globally but not locally. Alternatively, the correction can be applied locally, either through a mass source or sink introduced in each cell [18] or by modifying the facenormal component of the reconstructed IB velocity vectors such that mass is conserved in each cv. To test the accuracy of these IB reconstructions, a laminar channel ﬂow simulation with periodic boundaries was performed where the channel was oriented at angle α to the grid, and the channel walls were treated with the IB method. For a channel with walls at y = −1 and y = 1, the exact solution for this problem is of course the parabolic proﬁle. Figure 5 illustrates one such grid and solution.

y

1

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2

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Fig. 5. Laminar channel ﬂow grid with contours of streamwise velocity for angle α = 15◦ .

Figure 6 show the rate of error reduction in streamwise velocity and pressure respectively against relative grid size for this laminar channel problem. The most accurate approach is the “exact” reconstruction, where the exact laminar proﬁle is interpolated to the IB face locations as a Dirichlet velocity. The Navier-Stokes equations are still solved in the ﬂuid cells containing these faces. Because of the grid symmetry, the integrated global mass ﬂux through the IB wall is in fact zero, allowing us to solve the Poisson system without modifying these velocities. The rate of L∞ error reduction in both pressure and velocity is approximately 2nd -order for this case. In general, the exact reconstruction is not known, so this condition cannot be used. In addition, even if it were known, discrete conservation of mass cannot be guaranteed, so some sort of correction in the face-normal direction will be required. The least accurate approach – labeled “stairstep” – is where the no slip condition is applied directly to the stairstep boundary. This has the expected 1st -order behavior for both velocity and pressure.

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10

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stairstep IB local IB global exact

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Fig. 6. L∞ error against relative grid spacing for four diﬀerent IB reconstructions, steady laminar channel ﬂow with IB walls, angle α = 15◦ . (See Plate 27 on page 426)

The most instructive aspect of this investigation comes from comparing the two IB reconstructions. For the case “IB global”, the IB face velocities are interpolated using a linear least-squares reconstruction and then corrected by a single global correction to yield zero mass ﬂux through the boundary. For this case the rate of error reduction is 2nd -order for both velocity and pressure, with error levels similar to the exact reconstruction. While such a reconstruction is possible in general, it will not yield local mass conservation, and mass will tend to ﬂow into the IB boundary in certain regions and out of the boundary in other regions. Such an eﬀect may be undesirable. Additionally, if multiple boundaries are immersed in the same simulation, it does not necessarily make sense to correct all face-normal components with the same correction. This makes the IB boundary condition complex, forcing the management of diﬀerent corrections for each set of faces associated with a given boundary. a)

b)

Fig. 7. Computational grids for channel DNS; a) α = 0; b) α = 45.

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The case “IB local” uses the same linear least squares reconstruction to get the velocity vector at the IB faces, but the face-normal velocities are then corrected locally such that the net mass ﬂux through the IB faces of each cell is zero. This reconstruction thus has both local and global mass conservation. For this particular problem, the rate of error reduction is initially 2nd -order for velocity but 1st -order for pressure. Additionally, the velocity rate of error reduction falls oﬀ from 2nd -order, presumably as the solution is contaminated by the pressure errors. In conclusion, it appears that there is no free lunch: one cannot simultaneously achieve 2nd -order behavior and local mass conservation using the present IB approaches. Clearly the reconstruction is important in reducing the absolute level of velocity error relative to the stairstep approximation, however the local correction will eventually contaminate the velocity through the 1st -order behavior of pressure. It remains to investigate the importance of these 1st -order pressure errors in the context of turbulent simulations, which are the focus of this contribution.

5 Turbulent Channel Flow To investigate the ability of the IB method to model turbulent ﬂows with resolved wall boundary layers, a locally-adapted grid was generated based on the well known grid resolution requirements for the LES of turbulent channels at Reτ = 180.

25

4 Body Fitted U - 45 degrees

Body Fitted u’ - 45 degrees v’ - 45 degrees w’ - 45 degrees

20

Mean Velocity

Fluctuating Velocities

3

15

10

2

1 5

0

20

y+

40

60

20

y+

40

60

Fig. 8. Minimum channel DNS simulations. Left: mean velocity, right: RMS turbulence ﬂuctuations. (See Plate 28 on page 426)

Once again, the grid was generated at an angle to the wall and the IB method with local face-normal corrections used to simulate the walls. The following quadratic spacing function was used for the wall-normal direction:

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∆y + = 1 + 0.3124(y + ) − 0.0008(y + )2 /2

(9)

and periodic boundary conditions were applied in both the streamwise and spanwise directions. Figure 7 shows the resulting mesh for angles α = 0 and α = 45 representing the best and worst case scenario in terms of simulation cost; computational grid consisted of approximately 46,000 and 482,000 control volumes, respectively. DNS simulations were carried out for a minimum channel; the domain size was π and 0.289π in the streamwise and spanwise directions, respectively. Fig. 8 compares the mean velocity and the RMS turbulence velocity ﬂuctuations for the two cases. The mean velocity compares well, however the discrepancies in the RMS values are signiﬁcant. Some of this discrepancy might be a result of inadequate averaging for the signiﬁcant more expensive α = 45 case, and this is the subject of ongoing investigation. In any case, the expected near wall behavior is accurately captured in both cases.

6 Sphere Calculations The grid generation and reconstruction algorithm presented before are applied to the turbulent ﬂow around a sphere at a Reynolds number (based on the sphere diameter D) of 3,900. In this case the sphere surface is described as a collection of facets. To illustrate the potential of the present technique, two additional geometries are considered for comparison: a dimpled sphere (golf ball) and a hollow sphere, with an internal cavity connecting the frontal and the backward facing regions. The geometries are reported in Fig. 9. The mesh was generated automatically after specifying a surface resolution of 0.0002D and 0.015D in the normal and tangential direction respectively. The grid was also reﬁned in the area downstream of the spheres to capture the wake evolution (Fig. 10). The grids contain about two million grid points. Flow simulations were carried out for about 65 time units and the results are reported in Fig. 11 in terms of averaged streamwise velocity and instantaneous spanwise vorticity in the wake.

Fig. 9. Geometries considered; a) smooth sphere; b) dimpled sphere; c) hollow sphere.

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Fig. 10. Computational grid in the symmetry plane for the hollow sphere. (See Plate 29 on page 427)

Fig. 11. Left: Streamwise velocity. Right: instantaneous spanwise vorticity; a) smooth sphere; b) dimpled sphere; c) hollow sphere. (See Plate 30 on page 427)

7 Conclusions The IB method combined with Cartesian grid generation with local anisotropic reﬁnement has been investigated as a possible method of simulating turbulent ﬂows in complex geometry. A mesh smoothing algorithm has been proposed to improve the quality of the resulting grids by ensuring consistency of the unstructured discretization. The expected second-order behavior of the method has been demonstrated by performing laminar channel simulations with the channel boundaries treated using the IB method with a global correction to the normal velocity to achieve global mass conservation. When the correction was applied locally to achieve local conservation, however, the behavior was less than second order, although signiﬁcantly better than the ﬁrst-order stairstep boundary representation. Finally, to demonstrate the capability of the method, turbulent channel simulations at Reτ = 180 were performed with the grid aligned and at 45 degrees with respect to the chan-

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nel walls. Results were satisfactory in terms of mean velocity and diﬀered more signiﬁcantly for the RMS quantities. Calculations of ﬂows around several spheres are reported to illustrate the ability of the present technique to treat more complex geometries.

Acknowledgments This work was supported by the US Department of Energy under the Advanced Simulation and Computing (ASC) Program.

References [1] C.S. Peskin. Flow patterns around heart valves: a digital computer method for solving the equations of motion. Ph.D. thesis, Physiology, Albert Einstein College of Medicine. University Microﬁlms. 378:72-30. 1972. [2] R.P. Beyer and R.J. Leveque. Analysis of a one-dimensional model for the immersed boundary method. SIAM J. of Num. Anal. 29:332–364. 1992. [3] M.C. Lai and C.S. Peskin. An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comput. Phys. 160:705–719. 2000. [4] G. Iaccarino and R. Verzicco. Immersed boundary technique for turbulent ﬂow simulations. Appl. Mech. Rev. 56:331–347. 2003. [5] R. Mittal and G. Iaccarino. Immersed Boundary Method. Ann. Rev. Fluid Mech. 37:239–262. 2005. [6] M.J. Berger and M.J. Aftosmis. Aspects (and aspect ratios) of Cartesian mesh methods. Proc. 16th International Conf. on Num. Meth. in Fluid Dynamics. 1998. [7] R. Mittal, C. Bonilla, and H.S. Udaykumar. Cartesian grid methods for simulating ﬂows with moving boundaries. in Computational Methods and Experimental Measurements-XI, Ed. Brebbia, Carlomagno & Anagnostopoulous. 2003. [8] G. Iaccarino, G. Kalitzin, P. Moin, and B. Khalighi. Local grid reﬁnement for an immersed boundary RANS solver. AIAA Paper 2004-0586. 2004. [9] R. Verzicco, M. Fatica, G. Iaccarino, P. Moin, and B. Khalighi. Large eddy simulation of a road-vehicle with drag reduction devices. AIAA J. 40:2447–2455. 2001. [10] E. Balaras. Modeling Complex Boundaries using an external force ﬁeld on ﬁxed Cartesian Grids in large-eddy simulations. Comput. Fluids 33:375– 404. 2004. [11] D. Zeeuw and K. Powell. An adaptively-reﬁned Cartesian mesh solver for the Euler equations. AIAA Paper 1991-1542. 1991.

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[12] Z. Wu. Anisotropic Cartesian Grid Method for Viscous Flow Computation. in Computational Fluid Dynamics Review, Ed. M. Hafez, K. Oshima. 1998. [13] Z.J. Wang, R.F. Chen, N. Hariharan, A.J. Przekwas, and D. Grove. A 2n Tree Based Automated Viscous Cartesian Grid Methodology for Feature Capturing. AIAA Paper 1999-3300. 1999. [14] A.G. Kravchenko, P. Moin, and R. Moser. Zonal embedded Grids for Numerical Simulations of Wall-Bounded Turbulent Flows. J. Comp. Physics 127:412-423. 1996. [15] V. Seidl, S. Muzaferija, and M. Peric. Parallel Computation of Unsteady Separated Flows Using Unstructured, Locally Reﬁned Grids. Notes on Numerical Fluid Mechanics. Ed. R. Friedrich and P. Bontoux.64:37–50, 1998. [16] K. Mahesh, G. Constantinescu, and P. Moin. A Numerical Method for Large-Eddy Simulation in Complex Geometries. J. Comp. Physics 197:215–240. 2004. [17] F. Ham and G. Iaccarino. Energy Conservation in Colocated Discretization Schemes on Unstructured Meshes. Annual Research Briefs, Center for Turbulence Research 3–14. 2004. [18] J. Kim, D. Kim, and H. Choi. An immersed-boundary ﬁnite-volume method for simulations of ﬂow in complex geometries. J. Comput. Phys. 171:132–150. 2001.

Towards Time-Stable and Accurate LES on Unstructured Grids Frank Ham, K. Mattsson, Gianluca Iaccarino, and Parviz Moin Center for Turbulence Research Stanford University Stanford, CA, 94305 - USA [email protected] Summary. Control-volume (cv) and node-based (cell-vertex) ﬁnite volume discretizations of the incompressible Navier-Stokes equations are compared in terms of accuracy, eﬃciency, and stability using the inviscid Taylor vortex problem. An energy estimate is shown to exist for both formulations, and stable convective boundary conditions are formulated using the simultaneous approximation term (SAT) method. Numerical experiments show the node-based formulation to be generally superior on both structured Cartesian and unstructured triangular grids, displaying consistent error levels and nearly second-order rates of L2 velocity error reduction. The cv-formulation, however, out-performs the node-based for the case of Cartesian grids when the Taylor vortices do not cut the boundary.

1 Introduction The application of Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) to ﬂows involving complex geometry requires numerical methods capable of adequately resolving the broad spectrum of space and time scales. The substantial experience with structured grid methods has shown that the suitability of a numerical method for such broad spectrum simulations is not simply related to its order of accuracy. Of equal or perhaps greater importance are the discrete conservation properties of the method in terms of mass, momentum, and kinetic energy [1]. Expressed more mathematically, of equal or perhaps greater importance is the extent to which the discrete operators mimic the properties of the continuous operators. Finite volume methods for unstructured grids can be formulated with good discrete conservation properties, and represent a compromise between accuracy (typically second-order) and eﬃciency (can generally be developed and solved eﬃciently), and as such are the subject of this contribution. Motivated by the success of structured staggered grid methods for incompressible ﬂows based on the pioneering work of Harlow and Welch [2], several authors have attempted to develop staggered unstructured methods suitable

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for DNS/LES. The principal advantage of staggering is that the resulting formulation strongly couples velocity and pressure without the need for any numerical smoothing or dissipation. Perot [3] proposed a staggered unstructured method that discretely conserves mass, momentum, and kinetic energy (in the inviscid limit), however the method is ﬁrst-order accurate on non-uniform unstructured grids. Mahesh et al. [4] developed and tested similar unstructured staggerings, however ultimately concluded that the accuracy of these schemes was inadequate, particularly in 3D, despite their good conservation properties. Unlike staggered formulations, collocated formulations (with velocity and pressure collocated at either cvs or nodes) require special treatment to ensure velocity/pressure coupling for incompressible ﬂows. In the present contribution, we use the collocated fractional step formulation described by Mahesh et al. [4] with a consistent treatment of the pressure gradient reconstruction as described in Ham & Iaccarino [5]. The principle contribution of this work is the addition of a stable convective boundary condition based on the simultaneous approximation term (SAT) method [6], and the investigation of the overall method’s relative accuracy on cv and node-based ﬁnite volumes.

2 Consistency, Stability, & Accuracy A consistent and accurate scheme does not guarantee an accurate solution of the PDE. According to the famous paper by Lax and Richtmyer [7], a consistent approximation of a well-posed linear PDE will converge to the true solution if and only if the scheme is stable. This theorem is of great importance since it is in general easier to prove stability than proving convergence directly. Lax and Richtmyer’s theorem implies that convergence to the true solution at a ﬁxed time T is achieved as the grid size h → 0. With this deﬁnition of stability the error of the solution may, in fact, grow exponentially in time for realistic mesh sizes. As pointed out by Carpenter et al. [8] it is therefore important to devise schemes that do not allow a growth in time that is not called for by the diﬀerential equation. Such schemes are called strictly (or time) stable. One way to obtain a strictly stable and accurate scheme for a linear problem is to: i) approximate the derivatives of the initial boundary value problem with accurate, non dissipative operators that satisfy a summation by parts (SBP) formula [9], and ii) use a speciﬁc procedure, referred to as the Simultaneous Approximation Term (SAT) method [6] for implementation of boundary conditions. For nonlinear problems, stability by energy estimates can not be obtained in general. However, Strang [10] showed that if the solution of the nonlinear diﬀerential equation is suﬃciently smooth, convergence follows if the linearized approximation is stable, meaning that linear stability is a necessary and sometimes suﬃcient condition for nonlinear stability.

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2.1 Stability Deﬁnitions Consider the initial boundary value problem (IBVP), ut + P u = F (x, t) x ∈ Ω t ≥ 0 Lu = g(t) x∈Γ t≥0 u = f (x) x∈Ωt=0,

(1)

where P is the diﬀerential operator, L the boundary operator, F the forcing function, f the initial data and g the boundary data. Deﬁnition 1. (1) is said to be strongly well posed if an unique solution exists and the estimate t u(·, τ ) 2Γ dτ u 2Ω + 0 t F (·, τ ) 2Ω + g(τ ) 2Γ dτ , ≤Kc eηc t f 2Ω + 0

holds. Kc and ηc do not depend on F , f or g. · Ω and · Γ are suitable continuous norms.

When it comes to the discrete case, a simpliﬁcation is to consider the semidiscrete problem where time is left continuous. This approach is justiﬁed by Kreiss and Wu [11], where they show that a stable semidiscretisation can be discretised in time such that the fully discrete problem is also stable. The corresponding semi-discrete problem is vt + P v = F x ∈ Ω t ≥ 0 Lv = g(t) x∈Γ t≥0 v=f x∈Ωt=0,

(2)

where P and L are the corresponding discrete operators. Deﬁnition 2. (2) is said to be strongly stable if, for suﬃciently small ∆x, there is an unique solution that satisﬁes t 2 v 2Γ dτ v Ω + 0 t 2 ηd t 2 2 F Ω + g Γ dτ . f Ω + ≤Kd e 0

Kd and ηd do not depend on F , f or g. · Ω and · Γ are suitable discrete norms. This implies that convergence to the true solution at a ﬁxed time T is achieved as the grid size ∆x → 0. However, a strongly stable scheme may exhibit a non-physical growth in time [6] for realistic mesh sizes. It is therefore important to devise schemes that do not allow a growth in time that is not called for by the diﬀerential equation. Such schemes are called strictly (or time) stable.

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Deﬁnition 3. We call (2) strictly stable if the growth rate satisﬁes ηd ≤ ηc + O(∆x) In fact we can show that ηd ≤ ηc +O(∆xp ) holds for the (SBP+SAT) schemes we are about to describe, where p is the order of accuracy of the method. 2.2 Stability Analysis SBP operators arise naturally in various nodal based space discretizations. The main classes are spectral methods, ﬁnite elements, ﬁnite diﬀerences and ﬁnite volumes. High order accurate SBP operators for ﬁnite diﬀerences were derived by Kreiss & Scherer [9] and Mattsson & Nordstrom [12]. The SBP property for the unstructured node centered ﬁnite volume method was shown by Nordstrom et al. [13]. This is the scheme that we have adapted in the computations shown in this paper. To simplify the introduction of the SBP and SAT concepts we will restrict the analysis to one dimensional problems. The generalization to multi dimensional problems are straightforward but somewhat tedious and lengthy to describe in detail. Before we start describing the SBP property, some deﬁnitions are needed. 2 Let the inner

r product for real valued functions u, v 2∈ L [l, r] be deﬁned by (u, v) = l u v dx and the corresponding norm u = (u, u). The domain (l ≤ x ≤ r) is discretized using N+1 grid points. The numerical approximation at grid point xj is denoted vj , and the discrete solution vector v T = [v0 , v1 , · · · , vN ]. We deﬁne an inner product for discrete real valued vector-functions u, v ∈ RN+1 by (u, v)H = uT H v, where H = H T > 0, and a corresponding norm v 2H = v T H v. 2.3 SBP Property Consider the hyperbolic scalar equation, ut + ux = 0 (excluding the boundary condition). Notice ﬁrst that (u, ut ) + (ut , u) = d/dt u 2 . Multiplying the equation by u and integrating by parts leads to d u 2 = −(u, ux ) − (ux , u) = −u2 |rl , dt

(3)

where we introduce the notation u2 |rl ≡ u2 (r, t) − u2 (l, t). To simplify the notation for the continuous problem we will denote u(k, t) by uk . We introduce the following deﬁnition Deﬁnition 4. A diﬀerence operator D1 = H −1 Q approximating ∂/∂ x is said to be a ﬁrst derivative SBP operator if i) H = H T > 0 and ii) Q + QT = B = diag (−1, 0 . . . , 0, 1).

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By multiplying the semi discrete approximation Hvt + Qv = 0 by v T , adding the transpose and utilizing deﬁnition 4 we obtain v T H vt + vtT H T v =

d 2 v 2H = −v T (Q + QT )v = v02 − vN . dt

(4)

Equation (4) is a discrete analog to the integration by parts (IBP) formula (3) in the continuous case. Hence, an SBP operator mimics the behavior of the corresponding continuous operator with regard to the inner product mentioned above. 2.4 Frozen Coeﬃcient Technique A fruitful technique to analyze stability and to obtain well posed boundary conditions for non linear PDEs is to analyze the linearized problem. It follows that if the solution u to the nonlinear problem is suﬃciently smooth then convergence follows if the linearized problem is stable and consistent for all possible values of u. This is referred to as ”frozen coeﬃcient technique”, see Gustafsson et al. [14]. Consider the linearized problem ut + (au)x = 0, l ≤ x ≤ r, t ≥ 0 .

(5)

where a = u/2 is a frozen coeﬃcient. The energy method leads to d u 2 = −au2 |rl . dt Well posed boundary conditions are given by ul = gl if al = a+ l >0 ur = gr if ar = a− r <0 where

(6)

− 2 a+ l, r = al, r + |al, r | , 2 al, r = al, r − |al, r |

are the negative and positive part of a (at each boundary) respectively. If al, r > 0 this means that we only specify data at the left (inﬂow) boundary. The energy method leads to the following energy estimate d − 2 2 − 2 + 2 u 2 = a+ l gl + al ul − ar gr − ar ur dt

(7)

2.5 The SAT Method By using an SBP operator, a strict stable approximation for a Cauchy problem is obtained. Nevertheless, the SBP property alone does not guarantee strict stability for an initial boundary value problem, a speciﬁc boundary treatment is also required. To impose the boundary condition explicitly – i.e. to combine

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the diﬀerence operator and the boundary operator into a modiﬁed operator – usually destroys the SBP property. In general, this makes it impossible to obtain an energy estimate. This boundary procedure, often used in practical calculations, is referred to as the injection method and can result in an unwanted exponential growth of the solution, see for example Mattsson [15]. The basic idea behind the SAT method is to impose the boundary conditions weakly, as a penalty term, such that the SBP property is preserved and such that we get an energy estimate. The discrete approximation to (5) using the SAT method for the boundary conditions (6) leads to Hvt + Q Av = Plinv + Prinv

(8)

where A is the projection of a onto the diagonal and Plinv = −σl a+ ˆ0 {v0 − gl } , Prinv = σr a− ˆN {vN − gr } r e l e

(9)

are the inviscid penalties at the left and right boundaries respectively. Here eˆ0 = [1, 0, ..., 0]T and eˆN = [0, ..., 0, 1]T . The energy method leads to 2 2 a+ d σl 2 l σl v 2H =a+ − (1 − 2 σ ) v + g g 2 + a− l 0 l l v0 l dt 1 − 2 σl 1 − 2 σl l 2 a− σ 2 σr 2 − gr + r r gr2 − a+ − ar (1 − 2 σr ) vN + r vN . 1 − 2 σr 1 − 2 σr An energy estimate exists for σl, r > 21 . The choice σl, r = 1 yields d + − 2 2 − 2 − 2 + 2 2 v 2H = a+ l gl + al v0 − ar gr − ar vN − al (v0 − gl ) + ar (vN − gr ) . (10) dt Equation (10) is a discrete analog to the continuous energy estimate (7), where 2 − 2 the extra terms −a+ l (v0 − gl ) + ar (vN − gr ) introduce a small additional damping.

3 Unstructured Finite Volume Method In the present contribution, we want to apply the mathematical formalism described previously (including the SAT treatment for boundary conditions) to the ﬁnite-volume discretization of the incompressible Navier-Stokes equations. The basis for this work is the collocated fractional step discretization of the incompressible Navier-Stokes equations described by Mahesh et al. [4]. In addition, we limit the investigation to the case of inviscid ﬂows, where it is known that the internal scheme conserves mass, momentum, and kinetic energy apart from a pressure dissipation term required for velocity/pressure coupling [5]. We are particularly interested in the performance of the scheme

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ui,p ui,nb

ui,p Uf

Ue

ui,nb

Fig. 1. Comparison of control volume (left) and nodal, or dual (right) mesh topologies near a boundary control volume “p” and one of its internal neighbors “nb”.

when applying time-dependent boundary data. The interest is motivated by the longer-term goal of using the unstructured incompressible solver as part of a coupled simulation – coupled compressible/incompressible or coupled structured/unstructured for example. Figure 1 compares the control volume and nodal (sometimes called cellvertex, or dual) meshes used in the present study. Forming the mesh dual from a primal mesh of arbitrary polyhedral control volumes involves computing the edge, face, and cv centers, and then introducing triangular facets internal to the volumes that span between these edge-face-cv triplets. With these internal facets in place, the volumes can be then distributed to their associated nodes, and the normals and areas associated with each edge (i.e. pair of nodes) calculated. 3.1 Node-Based Discretization For the dual mesh with SAT boundary treatment, the convective term is discretized at node “p” as follows (for clarity we have dropped the time averaging): O (Conv.)N i,p =

e

Ue

ui,p + ui,nb Ae + Uf ui,p Af − σUf− (ui,p − gi,p )Af 2

(11)

where the subscript e refers to edges that connect node p with its neighbors nb, and the subscript f refers to the combined boundary faces associated with node p. Ue and Uf are the mass-conserving advecting velocity components at the edges and combined boundary faces respectively, and gi , p is the Dirichlet boundary condition. Areas Ae and Af must be treated with some care because these include several diﬀerent triangular facets (line segments in 2D) where the

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normals are not necessarily aligned. In the present work, these areas always appear multiplied by the face-normal velocity, so it is only important that they are deﬁned consistently with the discrete deﬁnition of continuity, Ue Ae + Uf Af = 0 (12) e

Note that the terms involving Af will only be present in boundary nodes. The third term in eq. (11), representing the SAT boundary treatment, will be non-zero only for incoming (negative based on the positive outward normal convention) advecting velocity Uf . An energy analysis similar to that done in section 2 can be used to determine appropriate values of σ. By contracting with velocity ui,p we can write:

O ui,p (Conv.)N i,p

ui,p ui,nb ui,p ui,p Ae + = Ue Ue Ae + Uf Af + 2 2 e e ui,p ui,p Uf Af − σ (ui,p ui,p − ui,p gi,p ) Uf− Af (13) 2

The ﬁrst term on the rhs is zero because of continuity. The second term represents the ﬂux of discrete kinetic energy to internal neighbors, and will cancel when we sum over the entire domain. This leaves the two terms on the second line, which we will call T : ui,p ui,p Uf Af − σ (ui,p ui,p − ui,p gi,p ) Uf− Af (14) 2 For strict stability, we must show that the terms in T involving the unknown boundary velocity ui,p are non-negative. For the case of positive Uf (outﬂow), that is Uf ≡ Uf+ , this is straightforward. Only the ﬁrst of these remaining terms will be non-zero, yielding: T =

ui,p ui,p + Uf Af (15) 2 which is always positive for any value that ui,p might take, and the scheme is strictly stable. For the case of an inﬂow boundary, where Uf ≡ Uf− , T can be rearranged as follows: T |Uf =U + = f

ui,p ui,p + σui,p gi,p Uf− Af T |Uf =U − = (1 − 2σ) f 2 2 2 σ 2 gi,p σgi,p 1 (1 − 2σ) ui,p + Uf− Af = − 2 1 − 2σ 1 − 2σ

(16) (17)

In this case, an energy estimate exists for σ > 1/2. In the simulations that follow, we choose σ = 1.

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3.2 Cv-Based Discretization The cv-based discretization of the convective term with SAT boundary treatment can be written for cv “p” as follows: (Conv.)CV i,p =

f ∈Γ /

Uf

ui,p + ui,nb Af + Uf ui,p Af − σUf− (ui,p − gi,f )Af 2 f ∈Γ

(18) Here we have diﬀerentiated between internal faces (f ∈ / Γ ) and boundary faces (f ∈ Γ ). Contraction with the cv-based velocity ui,p leads to a nearly identical energy estimate, with the important distinction that the Dirichlet boundary velocity, gi,f is now at a diﬀerent location than the cv-based unknown ui,p . This ﬁrst-order error cannot easily be avoided without developing a more complex reconstruction of the boundary face velocity used to close the convective term, and clearly has important implications (see the following section) on the accuracy of the overall cv-based method for arbitrary boundary velocity data.

4 Numerical Results As a test problem to compare the performance of the cv and nodal formulations described previously, we simulate the Taylor vortex problem, an array of counter-rotating 2-dimensional vortices that have the following analytic form: u = −cos(πx)sin(πy)e−

2π 2 t Re

2t − 2π Re

v = sin(πx)cos(πy)e 4π 2 t 1 p = − (cos(2πx) + cos(2πy))e− Re 4

(19) (20) (21)

where Re is the Reynolds number based on unit length and velocity scales. Solving the Taylor problem in the limit of inﬁnite Reynolds number should result in a steady solution of non-decaying vortices with constant kinetic energy, K = ui ui /2 = 1/4. 4.1 Cartesian Mesh Simulations As a starting point, the two formulations are veriﬁed by solving the inviscid Taylor problem on a uniform Cartesian mesh with domain −1 ≤ x ≤ 1, and −1 ≤ y ≤ 1 and periodic boundary conditions. In this case, the two meshes and thus formulations should be identical, apart from the small staggering of node and cv data points. Figure 2 compares the evolution of L2 error in velocity component u1 for three diﬀerent meshes, conﬁrming the identical

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behavior of the two schemes for this problem. Interestingly, the rate of error reduction is actually third-order, presumably because the ﬁner simulations are performed at smaller time step as well as mesh size so as to hold CFL approximately constant and near 1. This leads to the the observed third-order behavior because the error is dominated by the pressure dissipation term, with coeﬃcient ∆t∆x2 [5].

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The same Cartesian mesh simulations are repeated with Dirichlet velocities applied using the SAT formulation described previously and the analytic solution along the boundary. Figure 4 shows the results for the Taylor vortices exactly centered in the domain, as shown on the left of ﬁgure 3.Note that in this situation Uf = 0 and therefore no penalties are added. For this case,

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the error reduction rate for the dual simulations is reduced to approximately second-order, whereas the cv-based simulations exactly match the third-order fully periodic result. This is the only case where the cv-based result outperforms the node-based result in terms of error reduction. When the initial and boundary condition is shifted such that there are regions of inﬂow and outﬂow around the entire domain, the node-based formulation retains its approximately second-order behavior, however the cvbased result drops to ﬁrst order, presumably due to the inaccuracies in SAT boundary treatment for the CV-based formulation (ﬁgure 5).

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4.2 Unstructured Mesh Simulations On an unstructured triangular mesh transformed to produce regions of substantial skewness, inviscid Taylor simulations with Dirichlet boundary conditions were performed. For these cases, reﬁned grids were produced using a nested homothetic reﬁnement algorithm, where each parent triangle was divided into 4 identical child triangles. Figure 6 illustrates the coarsest mesh and velocity vectors for this case, and compares the computed solution at t = 3 for the two formulations. To illustrate the time stability of the method, even in the inviscid limit, the coarsest unstructured simulations were integrated out to t = 100. In both cases (cv and node), the L2 error increases and then plateaus near O(1),

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but does not increase beyond this. The kinetic energy for both cases (not shown) is initially constant at the initial value of 0.25, and then ﬂuctuates about this value but does not diverge. Finally, as seen in the Cartesian grid results reported in ﬁgure 7, the rate of error reduction (and absolution error levels over the entire range, for that matter) are signiﬁcantly better for the nodal-based discretization. In this case, the nodal based solve is also about 3 times faster than the cv-based solve because of the reduction in the number of unknowns on triangular grids.

5 Conclusions Control-volume (cv) and node-based ﬁnite volume discretizations of the incompressible Navier-Stokes equations were compared for accuracy and stability with boundary conditions applied using the simultaneous approximation term (SAT) method. The inviscid Taylor vortex problem was solved on structured and highly skewed unstructured grids to compare the two approaches.

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These numerical experiments show the node-based formulation to be generally superior on both structured Cartesian and unstructured triangular grids, displaying consistent error levels and nearly second-order rates of L2 velocity error reduction. The cv-formulation, however, out-performs the dual for the case of Cartesian grids when the Taylor vortices do not cut the boundary.

References [1] R. Mittal and P. Moin. Suitability of upwind-biased ﬁnite diﬀerence schemes for large-eddy simulation of turbulent ﬂows. AIAA J. 35(8):1415, 1997. [2] F.H. Harlow and J.E. Welch. Numerical calculation of time-dependent viscous incompressible ﬂow of ﬂuid with free surfaces. Phys. Fluids 8:2182, 1965. [3] B. Perot. Conservation properties of unstructured staggered mesh schemes. J. Comput. Phys. 159:58–89, 2000. [4] K. Mahesh, G. Constantinescu, and P. Moin. A numerical method for large-eddy simulation in complex geometries. J. Comput. Phys. 197:215– 240, 2004. [5] F. Ham and G. Iaccarino. Energy conservation in collocated discretization schemes on unstructured meshes. Center for Turbulence Research Annual Research Briefs Stanford University, Stanford, California 3–14, 2004. [6] M.H. Carpenter, D. Gottlieb, and S. Abarbanel. Time-stable boundary conditions for ﬁnite-diﬀerence schemes solving hyperbolic systems: Methodology and application to high-order compact schemes. J. Comput. Phys. 111(2), 1994. [7] P.D. Lax and R.D. Richtmyer. Survey of the Stability of Linear Finite Diﬀerence Equations. Comm. on Pure and Applied Math. IX, 1956.

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[8] M.H. Carpenter, D. Gottlieb, and S. Abarbanel. The stability of numerical boundary treatments for compact high-order ﬁnite diﬀerence schemes. J. Comput. Phys. 108(2), 1993. [9] H.-O. Kreiss and G. Scherer. Finite element and ﬁnite diﬀerence methods for hyperbolic partial diﬀerential equations. Mathematical Aspects of Finite Elements in Partial Diﬀerential Equations., Academic Press, Inc., 1974. [10] G. Strang. Accurate partial diﬀerence methods II. Non-linear problems. Num. Math. 6:37–46, 1964. [11] H.-O. Kreiss and L. Wu. On the stability deﬁnition of diﬀerence approximations for the initial boundary value problems. Appl. Num. Math. 12:212–227, 1993. [12] K. Mattsson and J. Nordstr¨ om. Summation by parts operators for ﬁnite diﬀerence approximations of second derivatives. J. Comput. Phys. 199(2), 2004. [13] J. Nordstr¨ om, K. Forsberg, C. Adamsson, and P. Eliasson. Finite volume methods, unstructured meshes and strict stability for hyperbolic problems. Appl. Num. Math. 45(4), 2003. [14] B. Gustafsson, H.-O. Kreiss, and J. Oliger. Boundary Procedures for Summation-by-Parts Operators, John Wiley & Sons, Inc., 1995. [15] K. Mattsson. Boundary Procedures for Summation-by-Parts Operators. J. Sci. Comput. 18, 2003.

A Low-Numerical Dissipation, Patch-Based Adaptive-Mesh-Reﬁnement Method for Large-Eddy Simulation of Compressible Flows C. Pantano1 , R. Deiterding2 , D. J. Hill1 , and D. I. Pullin1 1 2

Graduate Aeronautical Laboratories Applied and Computational Mathematics California Institute of Technology 1200 E. California Blvd. 205-45, Pasadena, CA [email protected]

Summary. This paper presents a hybrid ﬁnite-diﬀerence method for the large-eddy simulation of compressible ﬂows with low-numerical dissipation and structured adaptive mesh reﬁnement (SAMR). A conservative ﬂux-based approach is described. An explicit centered scheme is used in turbulent ﬂow regions while a weighted essentially non-oscillatory (WENO) scheme is employed to capture shocks. Several two- and three-dimensional numerical experiments and validation calculations are presented including homogeneous shock-free turbulence, turbulent jets and the strongly shockdriven mixing of a Richtmyer-Meshkov instability.

1 Introduction Compressible ﬂows of practical interest generally contain physically diﬀerent key features, e.g. complex shock structures and turbulence. Eﬃcient numerical simulation of these ﬂows typically requires varying degrees of spatial resolution. This need has made structured adaptive mesh reﬁnement (SAMR) techniques very popular for the Euler equations [1, 2]. Moreover, the regular data decomposition of SAMR enables eﬃcient load balancing on current distributed memory computers. For compressible turbulent ﬂows, it is often expedient to implement a solver appropriate to the locally dominant physics; for example switched or hybrid methods that become upwind-biased around shocks, but which revert to centered stencils in nominally smooth regions. This dual requirement in both resolution and algorithm switching poses a signiﬁcant challenge for the construction of numerical methods suitable for the large-eddy simulation of strongly compressible turbulence. In this context, ﬂux-based shock-capturing methods are employed to ensure weak convergence (prediction of the correct shock speeds) [3], whereas, in turbulent ﬂow-regions, methods with low numerical dissipation are preferred. Centered

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numerical schemes satisfy the latter requirement in a natural manner, but care must be taken to avoid non-linear instabilities since there is no intrinsic numerical stabilization. This problem can be alleviated to some extent by using energy conserving (skew-symmetric) formulations [4]. We describe a hybrid ﬁnite-diﬀerence solver for large-eddy simulation of compressible ﬂows with low-numerical dissipation with SAMR. The scheme is part of the AMROC (Adaptive Mesh Reﬁnement in Object-oriented C++) framework [5] that implements the Berger and Colella algorithm. Its primary applications are three-dimensional compressible turbulent ﬂows driven by shocks. A ﬂux-based approach is described that remains conservative at ﬁne-coarse mesh interfaces resulting from SAMR, and also in the presence of numerical scheme switching. An explicit, tuned centered discretization (TCD) is used in smooth and turbulent regions of the ﬂow. The TCD scheme transitions smoothly to a weighted essentially non-oscillatory (WENO) method for shock capturing. The entire approach is a further development and signiﬁcant improvement of the hybrid method of Hill and Pullin [6]. A number of numerical experiments and validations have been conducted, ranging from two- to three-dimensional problems that include homogeneous shock free turbulence, turbulent (reacting and non-reacting) jets and the strongly shock-driven mixing of a Richtmyer-Meshkov instability (RMI).

2 Numerical Method 2.1 Hybrid WENO-TCD Scheme We solve the compressible LES equations in conservation form. The stretched vortex subgrid model for momentum and scalar transport is used in the present simulations [7, 8]. Mass, momentum, total energy and internal energy variance are discretely conserved using a skew-symmetric formulation for compressible ﬂows [9]. We use an optimized centered second-order ﬁnite diﬀerence scheme (TCD) [6] given by ∂f 1 α(fj+2 − fj−2 ) + β(fj+1 − fj−1 ) , (1) = ∂x j ∆x where the derivative of a function f is evaluated at discrete uniform intervals with stepsize ∆x. The constants α, β are set to the values α = −0.197, β = 1/2 − 2α, which minimizes LES truncation errors in turbulent regions of Kolmogorov-type. A further aspect is that the SAMR approach is tailored speciﬁcally for ﬂux-based discretizations. Numerical ﬂuxes consistent with the centered ﬁnite diﬀerence stencil are therefore required at the cell edges and, given a stencil of the form of (1), one must derive the corresponding ﬂuxes such that Fj+1/2 − Fj−1/2 ∂f , (2) = ∂x j ∆x

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div is satisﬁed. A ﬂux Fj+1/2 which satisﬁes (2) for the TCD alone can be obtained readily and reads div = α fj+2 + fj−1 + (α + β) fj + fj+1 . (3) Fj+1/2

Additionally, skew-symmetric terms must be written in ﬂux-conservative form skew [10, 11]. Skew-symmetric ﬂuxes Fj+1/2 are sought satisfying skew skew Fj+1/2 − Fj−1/2 ∂b ∂a a +b ≡ , ∂x j ∂x j ∆x

(4)

when a and b are replaced by the Favre ﬁltered momentum ρ¯u ˜k , velocity u ˜i , internal energy e˜, species mass fraction Y˜i , and pressure p¯, depending on the transport equation being considered. It is easily veriﬁed that the choice skew = α aj+2 bj +aj−1 bj+1 +aj bj+2 +aj+1 bj−1 +β aj bj+1 +aj+1 bj , (5) Fj+1/2

then satisﬁes (4) for the TCD derivative deﬁned in (1). The total nondissipative skew-symmetric ﬂux is then given by T CD Fj+1/2 =

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for the momentum and scalar transport equations. This ﬂux-based approach simpliﬁes the implementation of a consistent and conservative scheme with SAMR considerably [12]. Finally, the ﬂuxes at the physical domain boundaries are computed consistently with the skew-symmetric formulation and the discrete boundary stencil speciﬁcally derived for the TCD. The second part of our hybrid approach involves the WENO scheme. In W EN O this case, upwind-biased ﬂuxes, Fi+1/2 , are calculated based on a convex weighting of candidate stencils designed to minimize diﬀerentiation across discontinuities. We use the 5-point stencil version of WENO [13] in which the WENO optimal stencil has been replaced by (1). This modiﬁcation minimizes dispersion errors produced by the mismatch of modiﬁed wavenumber behavior of the standard 5-point WENO scheme when used in hybrid mode together with (1), cf. [6]. 2.2 Resolution Boundaries We treat discontinuous ﬂow features and changes in mesh resolution similarly by performing upwind-biased diﬀerentiation. Scheme switching at ﬂow discontinuities is achieved through a detection criteria that is problem-dependent; ﬁne-coarse mesh interfaces are ﬂagged directly by AMROC. If ﬂagged regions are denoted by C, the hybrid ﬂux takes the form # W EN O Fi+1/2 , in C (7) Fi+1/2 = T CD Fi+1/2 , in C,

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Fig. 1. Example of a mesh hierarchy with three levels. (See Plate 35 on page 431)

where C is the complement of C. Currently, we employ a relative curvature criterion in pressure and density for shock detection. The SAMR block-structured embedded layout (within AMROC) is shown in Fig. 1 [5] for a three-level mesh hierarchy, with cells hypothetically reﬁned ﬂagged in red. The penalty of this structured approach, as opposed to a conforming unstructured technique, is that one looses the kinetic energy conservation property at ﬁne-coarse level interfaces. Finally, the solution is marched in time using a third-order Runge-Kutta method. Characteristic boundary conditions are applied at in- and outﬂow surfaces. When SAMR is used in conjunction with LES, purely numerical disturbances may result from the interaction between the ﬂow physics and adaptive reﬁnement, especially if the dynamic mesh adaptation is insuﬃciently sensitive to some ﬂow features. Spurious waves may arise if vortical unsteady structures, which may be marginally resolved in LES, travel through ﬁne to coarse mesh boundaries. In the traditional SAMR approach, important ﬂow features are always reﬁned appropriately [1, 2]. Transient features are kept within the reﬁnement by frequent mesh adaptation. In the case of problems involving shocks, one cannot resolve the ﬂow discontinuities and it is therefore necessary to choose an upper bound on the error of the numerically captured discontinuity depending on computational resources and precision needs. Similarly, in LES, one would predetermine a certain cutoﬀ scale based on the fraction of turbulence intensity that needs to be captured in the simulation and model the subgrid contributions. In principle, this cutoﬀ scale should be larger than the smallest grid resolution available and independent of the mesh resolution at all levels. This guarantees that the computed LES solution converges to the solution of the LES equations with increasing resolution.

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3 Veriﬁcation 3.1 Vortices in a Periodic Geometry A canonical problem with SAMR is the abrupt change in resolution at ﬁne-coarse mesh interfaces. To investigate further the features of LES and SAMR together, we consider the simpliﬁed ﬂow resulting from the interaction of vortices with mesh reﬁnement boundaries for the Euler equations. This model is interesting as it represents a worst-case scenario without physical dissipation or subgrid transfer of energy. We report on numerical dissipation/ampliﬁcation introduced at interfaces between meshes of diﬀerent sizes. Owing to the non-linear nature of WENO ﬂuxes at ﬁne-coarse mesh interfaces it is not possible to derive analytical estimates of energy conservation in these regions. Moreover, compressibility and time integration with the Runge-Kutta method contributes to small variation in kinetic energy. Presently, our parameters were chosen to make these eﬀects small in comparison to the kinetic energy dissipated/produced at mesh interfaces. We use a vortex solution of the Euler equations [14] which is given by the tangential velocity distribution uθ = uo η exp (−η 2 ),

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where ro is the vortex radius and p/ργ = po /ργo . For all tests presented throughout this section we use the values uo = π/4, ρo = 1, po = 2 and ro = 0.25. In our tests, two such vortices are separated a distance L apart. This conﬁguration is sketched in Fig. 2. The domain has an aspect ratio of 2 and periodic boundary conditions are used on all boundaries. The domain has been discretized using two meshes (ﬁxed in time) with a central region reﬁned by a factor of r = 2, shown in Fig. 2 as the shaded region. The vortices are equally spaced and they can be initially located at diﬀerent distance from the boundary, denoted xo for the ﬁrst vortex. Table 1. Parameters of the cases for the two vortex system. Cases (a) to (c) use adaptive time integration and case (d) uses synchronous time integration. (a) (b) (c) (d) xo 0 L2 0 0 2L uc 0 0 2L tf tf

The numerical experiment is set up such that we can also add a constant uniform velocity uc in the horizontal direction to make the vortices travel through the ﬁne-coarse mesh interface. We investigate four cases denoted from (a) to (d), whose parameters are listed in table 1. These cases were chosen to

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Fig. 2. Mesh conﬁguration and pressure distribution of the two vortex periodic system for cases (a), (b) and (c). 10

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be representative of the situations that could be encountered in practice when using SAMR for LES. The common scenario, (a), to the worst possible case of ﬁne-coarse-ﬁne mesh boundary traversal, (c), are considered. Case (d) is performed with uniform time integration (no time interpolation step between meshes) to estimate the eﬀect of time adaptation. For reasons to be discussed subsequently, scenarios where cases (b) and (c) arise in our simulations should be avoided. Except at reﬁnement interfaces, where WENO is employed, only the TCD stencil is used in the following to lead to results that are of direct relevance to LES simulations.

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3.2 Convergence and Conservation Figure 3 shows the behavior of the L1 norm of the density error as a function of the coarse mesh resolution in all cases. It is observed that the error is quite insensitive to whether the vortices are sitting at the mesh interface or are traveling through it; the magnitude of the error is larger in the latter case owing to the larger magnitude of the horizontal velocity. The convergence rate is of second order for case (a), (b), (d) and slightly lower for case (c). Use of synchronous time integration does contribute to a slight reduction of the larger error observed in case (c), as shown in case (d). To measure the mesh interface stability, we have determined the average ﬂow kinetic energy 1 L 2L 2 u (x, y, t) + v 2 (x, y, t) dxdy. (9) K(t) = 2 0 0

Figure 4 shows deviations of the average kinetic energy in terms of percentage of the total as a function of time for all cases. The normalization of case (a) and (b) is based on K(t), but that of case (c) and (d) is based on K(t) − L2 u2c to avoid artiﬁcially polluting the kinetic energy error by the large contribution to the kinetic energy which originates in the uniform translation velocity uc . It can be seen that in the coarse resolution cases, kinetic energy can decrease or increase depending on the ﬂow conditions. In the case that the ﬂow features are completely contained within the mesh, case (a), the kinetic energy varies very little with time. When the vortices are centered at the mesh interfaces or travel through them, kinetic energy can actually be produced initially, as shown in cases (c)-(d). Nevertheless, it appears that throughout an entire cycle kinetic energy is always dissipated. This is expected on the grounds of the nature of the WENO closure at the ﬁne-coarse mesh interface. We also observe that as resolution improves the amount of energy produced at the interface decreases consistently. The use of time interpolation at mesh interfaces gives results that are almost indistinguishable from those obtained with synchronous time integration; compare (c) and (d). The results discussed here were obtained for the Euler equations. In the case of LES, additional oﬀ-grid energy transfer takes place at rates that are typically much larger than the kinetic energy production rates observed in some of the coarse resolution test cases. This observation supports our previous discussion regarding the choice of subgrid cutoﬀ scale with respect to mesh resolution with the objective of minimizing the generation of spurious numerical noise at the interface and appropriately resolving the ﬂow features.

4 Comparison with Experiments 4.1 Low Mach-number Turbulent Jet The generic compressible solver described in Sec. 2 is being used to study a number of problems including shocks, turbulence and combustion. We simu-

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late a low Mach number turbulent jet [15] at a jet Reynolds number of 30, 000 in a computational domain of 2 × 2 × 0.5 meters (same as experiment); see Fig. 5a for iso-surfaces of mixture fraction. Owing to SAMR, this can be achieved with approximately 4 × 106 cells. Figure 5b shows the normalized 2 (x), where Uj = 35 m/s is the exit centerline streamwise velocity, as Uj2 /Um 2 (x) is the average centerline velocity. We expect a straight velocity and Um line when the ﬂow is self-similar. 4.2 Richtmyer-Meshkov Instability We have also simulated a planar re-shock experiment [16] in which a Mach 1.5 shock interacts with an Air-SF6 interface and generates turbulent mixing through the Richtmyer-Meshkov instability. The unshocked air has a density of 0.27885 kg/m3 and pressure of 23 kPa. Temperature is uniform in the

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unshocked side. These ﬂows exercise both the LES and the shock-capturing features of the solver with dynamically adaptive meshes. Visualizations of the Richtmyer-Meshkov instability are shown in Fig. 5c. Figure 5d shows the thickness of the mixing zone as a function of time and the respective experimental measurement of [16] (case IVe ). 4.3 Combustion Finally, the SAMR-LES methodology has been extended to combustion. For demonstration purposes we use a simple steady ﬂamelet model with an assumed Beta-pdf for a hydrogen turbulent diﬀusion ﬂame [17]. The fast chemistry approximation is reasonable for this hydrogen ﬂame. The stretched vortex model enables closure of the problem. Figure 6 shows iso-contours of tem-

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(a)

(b) Fig. 6. Iso-surfaces of temperature (a) and overlay of the mesh (b) for a turbulent hydrogen diﬀusion ﬂame. (See Plate 38 on page 433)

perature and a visualization of the mesh across a vertical plane. In this case, reﬁnement was allowed in the regions of large temperature gradients and the ﬁne mesh covers a fraction of the center of the domain.

5 Conclusions We have described an extension of the Berger-Colella SAMR method for compressible ﬂows to large-eddy simulation with a hybrid scheme. The formulation uses low-numerical dissipation, centered schemes in skew-symmetric form within patches of uniform resolution in turbulent regions, and upwind-biased diﬀerentiation at and around shocks and at ﬁne-coarse mesh interfaces. This operator is based on a modiﬁed version of the WENO method whose reference stencil matches the centered scheme. This property is crucial in order to minimize spurious reﬂections when the scheme transitions from centered to upwind form. Several veriﬁcation and validation computations have been analyzed and the order of accuracy, minimal LES-SAMR reﬁnement conditions, and energy generation and loss at ﬁne-coarse mesh interfaces of the overall method have been discussed. From the validation point of view, the presented LES-SAMR scheme allowed us to compare simulation results obtained on conventional distributed memory systems of moderate size directly to three-dimensional turbulent jet statistics and to the shock-induced mixing ﬂow produced by the Richtmyer-Meshkov instability.

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Acknowledgments This work was supported by the Advanced Simulation and Computing (ASC) Program under subcontract no. B341492 of DOE contract W-7405-ENG-48. The authors would like to acknowledge the many helpful conversations with P.E. Dimotakis and D.I. Meiron.

References [1] M.J. Berger and J. Oliger. Adaptive mesh reﬁnement for hyperbolic partial-diﬀerential equations. J. Comp. Phys., 53(3):484–512, 1984. [2] M.J. Berger and P. Colella. Local adaptive mesh reﬁnement for shock hydrodynamics. J. Comp. Phys., 82(1):64–84, 1989. [3] P. Lax and B. Wendroﬀ. Systems of conservation laws. Comm. Pure and Appl. Math., 13(2):217–237, 1960. [4] T.A. Zang. On the rotation and skew-symmetric forms for incompressible ﬂow simulation. Appl. Numer. Math., 7(1):27–40, 1991. [5] R. Deiterding. Parallel adaptive simulation of multi-dimensional detonation structures. PhD thesis, Brandenburgische Technische Universit¨ at Cottbus, 2003. [6] D.J. Hill and D.I. Pullin. Hybrid tuned center-diﬀerence-WENO method for large eddy simulations in the presence of strong shocks. J. Comp. Phys., 194(2):435–450, 2004. [7] A. Misra and D.I. Pullin. A vortex-based subgrid model for large-eddy simulation. Phys. Fluids, 9(8):2443–2454, 1997. [8] D.I. Pullin. Vortex-based model for subgrid ﬂux of a passive scalar. Phys. Fluids, 12(9):2311–2319, 2000. [9] A.E. Honein and P. Moin. Higher entropy conservation and numerical stability of compressible turbulence simulations. J. Comp. Phys., 201(2):531–545, 2004. [10] Y. Morinishi, T.S. Lund, O.V. Vasilyev, and P. Moin. Fully conservative higher order ﬁnite diﬀerence schemes for incompressible ﬂow. J. Comp. Phys., 143(1):90–124, 1998. [11] F. Ducros, F. Laporte, T. Souleres, V. Guinot, P. Moinat, and B. Caruelle. High-order ﬂuxes for conservative skew-symmetric-like schemes in structured meshes: Application to compressible ﬂows. J. Comp. Phys., 161(1):114–139, 2000. [12] A. Benkenida, J. Bohbot, and J.C. Jouhaud. Patched grid and adaptive mesh reﬁnement strategies for the calculation of the transport of vortices. Int. J. Numer. Meth. Fluids, 40:855–873, 2002. [13] G.-S. Jiang and C.-W. Shu. Eﬃcient implementation of weighted ENO schemes. J. Comp. Phys., 126(1):202–228, 1996.

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[14] D.S. Balsara and C.W. Shu. Monotonicity preserving weighted essentially non-oscillatory schemes with increasing high order of accuracy. J. Comp. Phys., 160(2):405–452, 2000. [15] E. Gutmark and I. Wygnanski. The planar turbulent jet. J. Fluid Mech., 73(3):465–495, 1976. [16] M. Vetter and B. Sturtevant. Experiments on the Richtmyer-Meshkov instability on a air/SF6 interface. Shock Waves, 4(5):247–252, 1995. [17] J.E. Rehm and N.T. Clemens. The large-scale turbulent structure of nonpremixed planar jet ﬂames. Combustion and Flame, 116(4):615–626, 1999.

Large-Eddy Simulation of Richtmyer-Meshkov Instability D. J. Hill, C. Pantano, and D. I. Pullin Graduate Aeronautical Laboratories California Institute of Technology 1200 E. California Blvd. 205-45, Pasadena, CA [email protected] Summary. We present results from large-eddy simulations (LES) of three-dimensional Richtmyer-Meshkov (RM) instability in a rectangular tube with reshock oﬀ the tube endwall. A hybrid numerical method is used that is shock capturing but which reverts to a centered scheme with low numerical viscosity in regions of smooth ﬂow. The subgrid-scale (SGS) model is the stretched-vortex (SV) model [1]. The shock strength, tube geometry, gas composition, initial conditions and initial interface disturbance were tailored to the experimental conditions of Vetter & Sturtevant [2] with shock Mach number Ms = 1.5, density ratio r = 5, and constituent gases air and SF6 . Use of the SV SGS model allows continuation of radial velocity spectra in the center-plane of the mixing layer, to subgrid scales, including the eﬀect of anisotropy and self-consistent calculation of the viscous cutoﬀ scale.

1 Introduction Richtmyer-Meshkov (RM) instability occurs when a shock impacts an interface separating two quiescent gases of diﬀerent densities [3, 4]. If the shock is parallel and the undisturbed interface are not co-planar, any misalignment between the pressure gradient across the shock and the density gradient produced by interfacial shape perturbations will result in the deposition of vorticity at the interface. Following the passage of the shock, the amplitude of the sheared interface exhibits ﬁrst a linear growth in time t, while the size of the perturbation remains small, followed by strongly nonlinear behavior. In the planar geometry studied presently, when the shock reﬂects oﬀ a wall and reshocks the interface, a rapid transition to turbulence results with the eﬀect that the gas interface is transformed into a turbulent mixing zone. Owing to its importance in astrophysical and inertial conﬁnement fusion phenomena, the RM instability has received much recent attention at the level of experiment, theory and simulation. Recent reviews are available [5, 6]. Presently we describe LES of a RM instability that is modeled on the experiments of Vetter and Sturtevant [2].

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2 LES: Numerical Method and SGS Model

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The numerical requirements for shock capturing and for LES using an explicit SGS model are essentially incompatible. For the former, automatically upwinding methods that are globally conservative, such as schemes of the Weighted Essentially Non-Oscillatory (WENO) type, are accurately shock capturing and oﬀer ﬂexibility in choice of order and ideal stencil, but, owing to their inherent numerical viscosity, are not suitable for LES. This is best performed with algorithms that exhibit low numerical dissipation such as spectral or centered ﬁnite-diﬀerence methods. The present approach is to utilize a hybrid scheme [7]; WENO is used in thin regions containing shock waves and is matched to a tuned, centered-diﬀerence (TCD) scheme in regions of smooth ﬂow where the SGS model is activated. This scheme has been extended to use an energy conserving skew-symmetric formulation away from shocks, and characteristic boundary conditions. The TCD stencil is optimized for good LES performance and is matched to the ideal WENO stencil. This eliminates the generation of spurious waves produced by dispersive impedance mismatch at cell boundaries, across which there is scheme switching based on measures of density and pressure gradients [7]. The hybrid WENOTCD scheme has been extensively tested for one-dimensional test problems and in LES of decaying three-dimensional compressible turbulence. Figure 1 shows a WENO/TCD solution of the one-dimensional Euler equations for the Riemann simple-wave problem [8]. TCD is selected until the onset of shock formation after which WENO is used in the shock region only. For times less than the shock-formation time, there is excellent agreement with the smooth exact solution.

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Fig. 1. Density proﬁles from numerical solution of Riemann simple-wave problem using WENO/TCD. Solid line, TCD. Circles, WENO.

We perform LES with modeling based on the Favre-ﬁltered Navier-Stokes equations. The SGS model is the stretched-vortex (SV) model [1]. This has been extended to LES of compressible ﬂows [9] and to the modeling of subgrid scalar transport [10]. It has been successfully applied to Rayleigh-Taylor

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driven mixing [11]. We view Favre-ﬁltering as strictly formal and as a guide to required resolved-scale and SGS modeling. No explicit ﬁltering of the resolvedscale ﬁelds is performed at any stage of the implementation of either the numerical algorithm for the solution of the LES equations or of the SGS model. The SGS ﬂuxes are τij = ρK(δij − evi evj ),

(1)

qiT = −ρ

(2)

△c 1/2 ∂(˜ cp T˜) K (δij − evi evj ) , 2 ∂xj ∂ ψ˜ △c 1/2 K (δij − evi evj ) qiψ = −ρ , 2 ∂xj

(3)

∞ where K = kc E(k)dk is the subgrid energy, evi is the unit vector aligned with the subgrid vortex axis, ν = µ/ρ is the kinematic viscosity and kc = π/△c represents the largest resolved wavenumber. The subgrid element is the stretched-spiral vortex [12] with local energy spectrum E(k) = K0 ǫ2/3 k −5/3 exp[−2k 2 ν/(3|˜ a|)],

(4)

where K0 is the Kolmogorov prefactor, ǫ is the local cell averaged dissipation (resolved ﬂow plus subgrid scale) and a ˜ = S˜ij evi evj is the axial strain along the subgrid vortex axis where S˜ij is the resolved rate-of-strain tensor. The composite prefactor K0 ǫ2/3 is calculated locally using resolved-scale, secondorder structure function information. The evj are assumed aligned with both the principal extensional eigenvector of S˜ij and the resolved-scale vorticity [9].

3 Initial and Boundary Conditions We simulate the air/sulfur hexaﬂuoride (SF6 ) RM experiment of Vetter and Sturtevant [2]. Here, a shock in air traveled down a rectangular tube with a Mach number strength ranging from 1.18 to 1.98 (depending on the experiment) where it impacted a contact surface consisting of a thin-membrane interface separating the air from the rest of the tube ﬁlled with SF6 . The initial density ratio, air:SF6 , was 1.18 : 5.97, with air on the shock side (lightheavy). Pressure and temperature are matched at the interface. Following the initial shock-interface interaction, the shock continued, reﬂected from a tube endwall, and then reshocked the interface. This “reshock” event deposited further vorticity at the evolving interface and generated both a transmitted shock and a reﬂected expansion. This, in turn, subsequently interacts with the evolving mixing layer and brings its mean location almost to rest in the laboratory frame of reference. In (x − y − z) Cartesian coordinates, we use a tube of square cross section in the (y − z)-plane with side 0.27m and of length L = 1.1m. The initially

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plane shock travels in the x-direction at Mach number 1.5. Periodic boundary conditions are implemented in the (y − z)-directions and a zero-stress boundary condition is used at the reﬂecting endwall, x = L. After reshock, the reﬂected shock exits the ﬂow domain and, subsequently, non-reﬂecting characteristic boundary conditions are implemented at the open end of the tube. In the mixture, transport properties were determined from standard binary mixing rules and pure-component transport properties with 0.76 power-law temperature variation. The initial interface shape was given by xI (y, z) = a0 | sin(πy/λ) sin(πz/λ)| + a1 h(y, z).

(5)

This is a combination of “egg-carton” modeling of the mesh used in the experiment to support the membrane and a symmetry-breaking perturbation with random phase, but with a prescribed initial power spectrum of the form k 4 exp (−(k/ko )2 ). Parameter values are λ = 0.02m, ko = 4 , a0 = 0.25cm, and a1 = 0.025cm. The LES reported here had a resolution of 776 × 2562 . It will be seen that this is two orders of magnitude too coarse for direct numerical simulation of the mixing layer (and many orders of magnitude less than that required for Navier-Stokes or Boltzmann resolution of the shock).

4 Results In Figure 2 we show isosurfaces of mixture fraction at three times during the LES. Figure 2(a) shows the initial condition, Figure 2(b) the interface shape after the ﬁrst shock encounter and Figure 2(c) illustrates the mixing layer well after the interaction of both the reﬂected shock and expansion with the mixing layer. We deﬁne the x-width of the mixing layer by δMZ (t) = 4 (1 − ψ ) ψ dx. (6) tube

where ψ is the scalar concentration, chosen initially to be ψ = 0, 1 on either side of the gas interface, and the angle bracket denotes an average over the (y − z)-plane. The time variation of δMZ (t) both before and after reshock is shown in Figure 3 compared to experiment [2]. Following the ﬁrst passage of the shock, during the single-mode phase, the density interface remains coherent and almost doubly periodic in shape. Its width can be characterized by growth in these perturbations which is nearly linear in time. At reshock, there is sudden kinematic compression and a second deposition of vorticity, followed by further growth and transition to a turbulent state where the two gases mix. The eﬀect of the reﬂected expansion on the growth of the mixing zone, at about t ≈ 5ms, can also be observed in Figure 3. Owing to its structured basis, the stretched-vortex SGS model allows multi-scale modeling of certain statistical quantities via estimation of the

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(a) t = 3.6ms

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Fig. 2. Images of the mixing zone at diﬀerent times. 0.12

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Fig. 3. Mixing width evolution compared with experiments [2]. The bold lines show their measured growth rates of the mixing layer width before and after reshock.

contribution to these statistics of scales below the resolved-scale cutoﬀ. These include predictions of subgrid mixing properties and estimates of the eﬀect of the Schmidt number. To illustrate results, Figure 4 shows velocity spectra in the y − z centerplane of the mixing layer some time after reshock. Twodimensional, circle-averaged (in ky − kz space) spectra are shown for both the component of the velocity normal to the y − z plane (u), and in the y − z plane. The subgrid extensions of the resolved-scale spectra can be calculated using the analysis of Pullin and Saﬀman [13] and are deﬁned by 2D (kr ) Eqq

2kr = π

|kr / cos αo |

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E(κ) dκ, (κ2 − kr2 )1/2 (kr2 − κ2 cos2 αo )1/2

(7)

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101 100

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Fig. 4. Components of the radial spectra at time t = 10ms computed in the center plane of the TMZ displaying resolved scale (solid line) and continuation (bro2D (k) (upper curve) and ken line). The spectra are divided into the anisotropic E33 2D 2D isotropic 1/2[E11 (k) + E22 (k)] (lower curve) directions. The Kolmogorov scale is deﬁned by η ≡ (< ν >3 / < ǫ >)1/4 where < ν > is the y − z plane-averaged kinematic viscosity and < ǫ > is the plane-averaged dissipation, including resolvedscale and SGS components. Approximately 80% of the energy resides in the resolved scales. 10

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Fig. 5. Resolved-scale cross spectra of the out-of-plane velocity u and the density ρ in the center-plane of the TMZ. Compensated by multiplying by k2 .

LES of Richtmyer-Meshkov Instability 2D E33 (kr ) =

2kr π

|kr / cos αo |

kr

(kr2 − κ2 cos2 αo )1/2 E(κ) dκ, κ2 (κ2 − kr2 )1/2

269

(8)

where α0 is the angle made by the vortex axis with the (y − z)-plane within a cell, and E(k) is given presently by (4). In (7) summation on q is implied. 2D (x ≡ “3”) is calculated by using the relaThe out-of-plane component E33 2 2 2 tion k3 = k − kr . The subgrid extended spectra in Figure 4 were obtained by averaging, for each k, the contribution from all cells in the center plane of the turbulent mixing zone (TMZ) at t = 10ms. Spectra are plotted so that they would coincide if isotropy was satisﬁed. The subgrid extension of the two spectra match reasonably smoothly onto the resolved-scale spectra and exhibit a similar degree of anisotropy in the out-of-plane (x ≡ ‘3’), compared to the in-plane directions (2D isotropy is expected in the y −z plane). Figure 4 also indicates that the viscous rolloﬀ of the continued subgrid spectra occurs at a wavenumber about two orders of magnitude higher than the resolvedscale cutoﬀ. The subgrid viscous cutoﬀ is computed from the plane average and feels, from (4) and (7)-(8), the eﬀects of both the temperature-dependent viscosity and the ﬂuctuating rate-of-strain at the resolved-scale cutoﬀ. The highest wavenumbers in the resolved-scale spectra do show the eﬀect of aliasing; no dealiasing strategy was attempted and no resolved-scale ﬁltering was used. Finally, Figure 5 shows cross spectra of the out-of-plane component of velocity u and the gas density ρ for three values of t during the mixing-layer evolution. The phenomenological prediction [14] is for a k −7/3 variation at high wavenumbers. There is some experimental evidence [15] supporting a k −2 inertial range for cross spectra and we have compensated by k 2 in Figure 5. At t = 6ms, which is after the expansion that is reﬂected oﬀ the tube end wall interacts with the mixing layer but just before the mixing-width growth saturates, there is no plateau in the compensated cross spectrum. At later times, the compensated spectra are approximately constant over about a decade of wavenumber space.

5 Concluding Remarks We have shown that successful LES of turbulent ﬂows driven by shocks can be achieved using a numerical method designed to be both conservative and shock capturing, but which reverts smoothly to a low-dissipation centered ﬁnitediﬀerence method away from shocks. The present LES appears to capture the salient features of the plane RM instability with endwall shock reﬂection. The interface passes brieﬂy through an initial linear growth phase, followed by nonlinear bubble-spike growth but without pairing/amalgamation. Reshock dramatically compresses the bubble-spike structures and causes a rapid transition to a turbulent mixing layer wherein the cascade process produces a broadening spectrum of turbulent scales. A secondary eﬀect of reshock is the

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production of an expansion wave which moves back to the endwall where it is also reﬂected. After reshock, the mixing layer grows at an increased rate. This is tending toward saturation but resumes when the mixing zone is stretched further in the out-of-plane direction by interaction with the reﬂected expansion. Further reﬂected expansions and compressions have diminished eﬀects and, within the time limits of the present LES, there is a halt to strong mixing-width growth. The stretched-vortex subgrid-scale models appears to provide a basis for continuation of second-order turbulent statistics to subgrid scales. The model correctly continues the anisotropy present in the components of the velocity (energy) spectrum near the resolved-scale cutoﬀ and also provides an estimate of the viscous cutoﬀ scale. Future work will address subgrid continuation for mixing statistics such as the density and scalar spectra and the probability density function of these quantities in planes spanning the mixing zone.

Acknowledgments This work was supported by the Advanced Simulation and Computing (ASC) Program under subcontract no. B341492 of DOE contract W-7405-ENG-48. The authors would like to acknowledge the many helpful conversations with P.E. Dimotakis and D.I. Meiron.

References [1] A. Misra and D. I. Pullin. A vortex-based subgrid stress model for largeeddy simulation. Phys. Fluids, 9:2443–2454, 1997. [2] M. Vetter and B. Sturtevant. Experiments on the Richtmyer-Meshkov instability of an air/SF6 interface. Shock Waves, 4:247–252, 1995. [3] R. D. Richtmyer. Taylor instability in shock acceleration of compressible ﬂuids. Commun. Pure Appl. Math., 13:297–319, 1960. [4] E. E. Meshkov. Instability of the interface of two gases accelerated by a shock wave. Sov. Fluid Dyn., 4:101–108, 1969. [5] N. J. Zabusky. Vortex paradigm for accelerated inhomogeneous ﬂows: Visiometrics for the Rayleigh-Taylor and Richtmyer-Meshkov environments. Annu. Rev. Fluid Mech., 31:495–536, 1999. [6] M. Brouillette. The Richtmyer-Meshkov instability. Annu. Rev. Fluid Mech., 34:445–468, 2002. [7] D. J. Hill and D. I. Pullin. Hybrid tuned center-diﬀerence-WENO method for large eddy simulations in the presence of strong shocks. J. Comput. Phys., 194:435–450, 2004. [8] L. D. Landau and E. M. Lifshitz. Fluid Mechanics. Permagon Press, New York, 1989.

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[9] B. Kosovic, D. I. Pullin, and R. Samtaney. Subgrid-scale modeling for large-eddy simulations of compressible turbulence. Phys. Fluids, 14:1511– 1522, 2002. [10] D. I. Pullin. A vortex-based model for the subgrid ﬂux of a passive scalar. Phys. Fluids, 12:2311–2319, 2000. [11] T. W. Mattner, D. I. Pullin, and P. E. Dimotakis. Large eddy simulations of miscible Rayleigh-Taylor instability. In 9th International Workshop on the Physics of Compressible Turbulent Mixing, 19-23 July, Cambridge, UK, 2004. [12] T. S. Lundgren. Strained spiral vortex model for turbulent ﬁne-structure. Phys. Fluids, 25:2193–2203, 1982. [13] D. I. Pullin and P. G. Saﬀman. Reynolds stresses and one-dimensional spectra for a vortex model of homogeneous anisotropic turbulence. Phys. Fluids, 6:1787–1796, 1994. [14] J. L. Lumley. Similarity and the turbulent energy spectrum. Phys. Fluids, 10:855–858, 1967. [15] L. Mydlarski and Z. Warhaft. Passive scalar statistics in high-P´ecletnumber grid turbulence. J. Fluid Mech., 358:135–175, 1998.

LES of Variable Density Bifurcating Jets Artur Tyliszczak and Andrzej Boguslawski Institute of Thermal Machinery, Czestochowa University of Technology Al. Armii Krajowej 21, 42-200 Czestochowa, Poland. [email protected], [email protected]

Summary. Three dimensional variable density jets at low Mach number conditions are analyzed by means of Large Eddy Simulations. The LES equations were derived starting from a low Mach number approximation of the continuity, NavierStokes and energy equations. The numerical method is based on the high-order compact/Fourier pseudospectral schemes. The superposition of axial and ﬂapping (helical) periodic disturbances at the jet outlet, forced the jet to bifurcate at certain excitation frequency. The LES calculations for non-isothermal case revealed that, for the case when the jet density is lower in comparison with the ambient ﬂuid, the excitation frequency needed for bifurcation is higher than for the isothermal jet.

1 Introduction Turbulent ﬂow control plays an important role in a wide variety of technological applications due to possible increase in process eﬃciency and safety. Turbulent jets are examples of the ﬂows for which ﬂow control in the fuel injection is needed in many applications, i.e. fuel injections in engines, aircraft propulsion systems or atomizers. Combustion, aerodynamic noise, jet/vortex interaction and vaporization are examples of physical processes related to these applications. In most cases these processes occur in variable density/variable temperature conditions which may considerably inﬂuence the character of the ﬂow. An example of a temperature dependent jet behavior in conditions of variable density/temperature is the absolute instability phenomenon occurring when the density ratio between the jet and the ambient is less than 0.7 approximately [1, 2]. In such cases the jets are characterized by self-excited oscillations leading to increased spreading rate, higher turbulence level and enhanced mixing. The ﬂow unsteadiness resulting from these phenomena is the source of aerodynamic noise, which can be decreased by controlling the turbulence and mixing intensity. On the other hand the enhanced mixing is desirable in the combustion processes as it increases the mixing of chemical species making combustion more eﬃcient. Furthermore, enhanced mixing decreases pollutants concentration while in the case of propulsion systems it

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decreases the exhaust gases temperature. For the liquid-droplet ﬂows occurring in the atomizers, the vaporization eﬃciency is directly related to the mixing process. In the case of liquid fuel injectors the vaporization process is responsible for gaseous fuel production which often determines the overall process eﬃciency. Many other examples may be cited to show the importance of the mixing eﬃciency in jet ﬂows. In natural (i.e. non-excited) jets the mixing eﬃciency is determined by the instability mechanism which in particular conditions (low jet density) may lead to the absolute instability phenomenon. Experimental studies have shown that this phenomenon is extremely sensitive to the distribution and level of turbulence at the jet nozzle exit [3] - high turbulence levels may totally suppress occurrence of the absolute instability. It was also observed that even the presence of a probe may change the ﬂow ﬁeld characteristics. It was demonstrated experimentally [4] that the mixing rate of the jet can be considerably increased by application of an external excitation at the jet nozzle exit. The ﬁrst important results on this active jet control were reported by Crow and Champagne [4], who observed that for the forcing frequency f corresponding to the Strouhal number equal to StD = f D/U1 = 0.3 (D is the inlet jet diameter, U1 is the inlet axial jet velocity) there exist two maxima of the turbulence intensity, the ﬁrst at the end of the potential core x/D = 4 (not observed in non excited jets), and the second one close to the maximum occurring in natural jets at x/D = 8. They noticed that this particular frequency (later on called jet preferred mode) corresponds to the peak frequency at the end of the potential core in the non excited case. From that time many experimental studies have focused on this aspect and on the enhancement of the jet spreading rate observed also in [4]. A very interesting phenomena occurring under particular excitation conditions are the bifurcating and blooming jets. In the former jets split into two separate yet well deﬁned streams, while in the latter jets split into “all” downstream directions producing nice vortex rings observed in experiments [5–7]. In the review paper [7] the authors summarize more than twenty years of their research on controlling jets by suitable external forcing. The necessary conditions to create bifurcating or blooming jets are formulated in terms of the type of forcing, its amplitude and frequency. Numerical simulations performed with direct [8–11] and large eddy simulations [11, 12] conﬁrm many experimental results. In particular, it was demonstrated that the spreading rate was very much dependent on the frequency ratio of the axial (fa ) to the helical (called also orbital or ﬂapping) (fh ) excitations. For example, in the experimental studies [5, 7] the necessary conditions to obtain bifurcating jets was formulated as fa /fh = 2 with the axial frequency corresponding to the Strouhal number in the range 0.3 − 0.7. For non-integer fa /fh in the range 1.5 − 3.2 blooming jets were observed. However, it was shown [10] using DNS, that for relatively high amplitude excitation (15% of the mean axial velocity), the bifurcation may occur with the helical forcing only.

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The LES simulations [11, 12] show that the excited jets spread more rapidly in the plane of helical excitation than in the remaining ones. However the results obtained in [11, 12] did not display patterns characteristic of the bifurcating jets even when the necessary conditions for the excitation were fulﬁlled. The role of controlling parameters (amplitude, fa , fh , fa /fh ) were analyzed in [13, 14] where the authors combined an optimization algorithm together with successive DNS solutions. It was shown that, by suitable modiﬁcation of the amplitudes, fa and fh , it was possible to achieve the spreading rate which fulﬁlled the assumed criteria. However, as stressed by the authors [14], the optimization procedure applied with DNS is very expensive computationally and limited to low Reynolds number ﬂows. The computations quoted above concern either incompressible constant temperature/constant density ﬂows or compressible ﬂows at Mach number equal to 0.9. In this work we perform a LES study of bifurcating jets for variable density/variable temperature in low Mach number conditions and additionally we focus on varying excitation parameters. The axial and helical excitations are superimposed on the inlet velocity proﬁle in the same manner as for the constant density jets. This should answer whether the excited variable density jets behave the same way as constant density ones and furthermore it should allow for direct comparison of the ﬂow structures resulting from the same type of excitation.

2 Governing Equations From the numerical point of view, in low Mach number ﬂows the Navier-Stokes equations for compressible ﬂuid become stiﬀ and therefore they are very diﬃcult to solve. This stiﬀness is caused by a large disparity between convective and acoustic waves which should be resolved with the allowable time step. This problem is mainly related to the steady computations performed with time marching methods, however, in the unsteady calculations one may observe that time-dependent phenomena, essential from the physical point of view, are solved at extremely high computational cost. The stability constraint for the explicit time integration method requires that an aﬀordable numerical time step scales proportionally with Mach number and it approaches zero as Mach number goes to zero. In this work, we applied the so-called low Mach number expansion [15–17] which makes the time step independent of Mach number and therefore allows for an eﬃcient solution. The idea of the method is based on the assumption that for low speed ﬂows, each ﬂow variable may be expressed as a power series of the small parameter ǫ = γM a2 ≪ 1 according to the following formula: ρ = ρ(0) + ǫρ(1) + ǫ2 ρ(2) + . . . . The symbol γ is the speciﬁc heat ratio and M a is the Mach number. Expressing the remaining variables in the same way and introducing these expansions into the governing equations and then performing LES ﬁltering, results in the following set of equations for the lowest order terms with respect to ǫ (except for the pressure, the superscripts have been omitted to simplify the notation):

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∂ ρ¯ ∂ ρ¯u ˜j + =0 ∂t ∂xj

(1)

∂ ρ¯u ˜i ∂ ρ¯u ˜i u ˜j ∂ p¯(1) 1 ∂ τ¯ij + =− + + TijSGS ∂t ∂xj ∂xi Re ∂xj . / ∂u ˜j ∂ 1 ∂ T 1 SGS ρ¯ µ + Kij = ∂xj ∂xj T ReP r ∂xj p(0) = ρ¯T

where the viscous stress tensor is given as: ∂u ˜i ˜k ∂u ˜j 2 ∂u τij = µ + − δij ∂xj ∂xi 3 ∂xk

(2) (3) (4)

(5)

The viscosity µ is computed according to Sutherland law. The above equations are in non-dimensional form derived with the procedure given in [17] for DNS of the reacting ﬂows. The variables ρ, ui , T, p are density, velocity, pressure and the temperature - these variables should be regarded as the zero order terms in the power series expansions. The pressure is split into the thermodynamic part p(0) , which is constant in space and also (in case of an open domain) constant in time, and the dynamic pressure p(1) which in our code is computed according to the projection method. The tilde (˜·) stands for the density weighted (Favre ﬁltered) variables: T = ρT /¯ ρ, and the bar (¯·) represents the LES ﬁltering. The symbols Re and P r stand for the Reynolds SGS are the subgrid tenand the Prandtl numbers. The symbols TijSGS and Kij sors modeled with the subgrid eddy viscosity assumption and using ﬁltered structure function model [18, 19]

3 Numerical Algorithm 3.1 Projection Method for Variable Density Flows The set of equations (1-4) is advanced in time with low-storage three stage Runge-Kutta method [20] which is of third order of accuracy. The scheme for the generic variable W is given in the form: W m = W n + αm ∆tRm W

n+1

=W

,

m = 1, 2, 3

3

(6)

where Rm represents spatially discretized terms. It is computed as: R1 = R(W n ) R2 = R(W 1 ) + β1 R1 2

R3 = R(W ) + β2 R2

(7)

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where α and β are Runge-Kutta coeﬃcients. Within each Runge-Kutta step the projection method is applied according to the following procedure: • (¯ ρu ˜i )m is computed as: (¯ ρu ˜i )m = (¯ ρu ˜i )∗ − ∆tαm

∂ p¯(1) ∂xi

(8)

where the term (¯ ρu ˜i )∗ is obtained from Eq.(2) without the pressure term. Substitution of the above formula to the continuity equation (1) results in the Poisson equation: m ∂ ρ¯ 1 ∂(¯ ρu ˜j )∗ 2 (1) + ▽ p¯ = (9) ∆tαm ∂t ∂xj • The density time derivative at stage m, is computed using the continuity equation (1): m ∂ ρ¯ = Rm (10) ∂t where Rm represents the spatial terms of continuity equation. In Eq.(10) Rm stands for the non-conservative form of the continuity equation, that allows to substitute the energy equation (3) directly to Eq.(10). For example R1 is given as: . # /6n ∂ 1 ∂ T ∂ ρ¯ 1 SGS µ + Kij R1 = −˜ uj − (11) ∂xj ∂xj T ReP r ∂xj

Eq.(10) is used in Eq.(9) and also to compute the density at the stage m. • Solution of Eq.(9) gives the pressure which is then used to compute (¯ ρu ˜i )m according to Eq. (8). The temperature is computed from the equation of state (4) as T m = p(0) /ρm . 3.2 Discretization Method

The spatial discretization is performed with a high-order compact scheme [21] in the direction of the jet axis and a Fourier approximation [22] in the plane perpendicular to the jet. The compact scheme for the I st derivative discretization is of the form (3 − 4 − 6 − 4 − 3) which denotes the V I th order discretization for the inner nodes 3, 4, . . . , N − 3, N − 2 where N is the number of nodes, the IV th order discretization for the near boundary nodes 2 and N −2 and the III th order for the boundary nodes. In preliminary computations we have also tested the scheme of the form (5−5−6−5−5) proposed in [23] which similarly to (3 − 4 − 6 − 4 − 3) was found to be stable in the sense of matrix stability analysis [24]. However, in this case we observed small oscillations in the pressure ﬁeld and therefore we continued computations with the basic

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scheme. In our computations the viscous terms were computed using the nonconservative approach, which is numerically superior in comparison with the two successive applications of the ﬁrst derivative approximation, because it provides ﬁnite damping of grid to grid oscillations [21]. The second derivatives resulting after expansion of the viscous terms into non-conservative form were also computed with the scheme (3 − 4 − 6 − 4 − 3). In the case of the Fourier ˜j ) were dealiased by analogy to 3/2 approximation the convective terms (¯ ρu ˜i u law [22], but for triple products the dealiasing method required expansion to 2N series. The numerical algorithm was stable without any numerical tricks and worked ﬁne for low Reynolds number (500 − 1000) ﬂows, even without a subgrid model (i.e. for DNS). When the Reynolds number was equal to 20000 the long time LES computations appeared to be unstable, which could be caused by various reasons. First, the ﬂow could be under-resolved and the dissipation coming from the subgrid model could be not large enough to damp high-frequency numerical oscillations. Second, the compact scheme applied is not conservative and the aliasing errors arising in the compact discretization could considerably contaminate the solution and lead to instability. To prevent instability we applied a low-pass compact ﬁlter of the IV th order [21] in the direction where the compact discretization was applied. The ﬁltering operation was used mainly for computational stability, following the suggestions from [17, 25]. However, this could also be regarded as explicit ﬁltering of the NavierStokes equations, which may be applied in LES to test the inﬂuence of the ﬁlter width explicitly determined by the ﬁltering procedure [26].

4 Boundary Conditions The computational domain was a rectangular box 10D × 10D × 16D, where D is the inlet jet diameter. Periodic boundary conditions were assumed on the lateral walls, and the inlet boundary conditions were speciﬁed in terms of instantaneous velocity, density and temperature. At the outlet we applied the convective type boundary conditions which allow the coherent structures to leave the computational domain with very small distortion. The inﬂuence of the outlet boundary on the upstream ﬂow was also very small. For each time step the instantaneous axial velocity was deﬁned as: u(x, t) = umean (x) + unoise (x, t) + uexcit (x, t) where the mean velocity was given by the hyperbolic-tangent proﬁle: 1R r U1 + U2 U1 − U2 R umean (x) = − tanh − 2 2 4θ R r

(12)

(13)

Here U1 and U2 denote the jet centerline velocity and the co-ﬂow velocity equal to U2 = 0.05U1 , the symbol r denotes x2 + y 2 , where x, y are the

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coordinates in the inlet plane. In this paper we present the results of computations performed with the momentum thickness θ of the initial shear layer characterized by the ratio R/θ = 20, where R = D/2. The ﬂuctuating component of velocity unoise (x, t) is the random Gaussian noise, which was adjusted to have turbulence level equal to 5% in the vicinity of the shear layer (0.8 < r/R < 1.2) and 1% in the region when r ≤ 0.8. The same level of turbulence was assumed for the remaining velocity components. The forcing component of the axial velocity was deﬁned as: πx U1 U1 π (14) uexcit (x, t) = Aa sin 2πSta t + Ah sin 2πSth t + sin D D 4 R

which is a superposition of axial forcing (the ﬁrst term) and helical/ﬂapping forcing (the second term). The forcing amplitudes are Aa and Ah , the Strouhal numbers are deﬁned as Sta = fa D/U1 and Sth = fh D/U1 . Both types of forcing were limited to the jet region only, and additionally to be closer to the ﬂow physics, the amplitude of helical forcing was multiplied by 0.2 for r ≤ 0.8, as suggested in [11]. Except for the random noise perturbations both the mean and excitation proﬁles corresponded to those applied in [11]. The density ratio between jet and ambient ﬂuid was adjusted using the assumed temperature proﬁle, which was identical to the mean velocity proﬁle: 1R r T1 − T2 R T1 + T 2 − tanh − T (x, t) = (15) 2 2 4θ R r

where T1 is the temperature in the jet axis and T2 is the ambient temperature and then the density was computed from the equation of state. It may be shown, that the formula (15) corresponds to the Busemann-Crocco relation for density.

5 Results The results presented concern computations performed for the natural and excited isothermal and non-isothermal jets for the Reynolds number equal to Re = 20000. The computational mesh consisted of 128 × 128 × 160 nodes stretched in the radial direction by a hyperbolic-tangent function - this ensured better and uniform resolution in the jet region. The main problem arising with the use of a Cartesian grid for the jet computations was related to the accurate description of the round inlet jet proﬁle. If the mesh is not dense enough, the jet may spread into preferential directions because of considerably diﬀerent grid resolution in particular directions. In order to test whether the mesh quality and resolution are suﬃciently good, we performed additional computations for the uniform mesh with the same number of nodes as for the basic mesh and also for the reﬁned, uniform mesh consisting of 256×256×256 nodes. These meshes were used for the selected cases only. Comparing the results for various meshes we assumed that any qualitative or relatively high

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Artur Tyliszczak and Andrzej Boguslawski

quantitative diﬀerences in the jet behavior indicate inﬂuence of the mesh. In such cases the results should be excluded from the analysis. 5.1 The Non-Excited Jet First we focus on the solution of natural jet without excitation. The main goal of these computations is to verify the numerical code by comparing the results obtained with experimental and numerical literature data. Furthermore, the spectral analysis of the non-excited jet should help us choose the proper Strouhal numbers Sta and Sth for the cases with excitation, which are presented in the next section. Figure (1) shows the instantaneous results obtained

Fig. 1. Instantaneous results for the non-excited jet for the density ratio 0.6. Figure on the left: 3D view of isosurface of Q = 0.1, temperature contours in x − z crosssection through the jet axis, vorticity modulus in z − y cross-section. Figures on the right: temperature in x − y cross-sections at z/D = 4 and z/D = 10.

for the density ratio of the jet to the ambient, S = ρ1 /ρ2 , equal to S = 0.6. The fully three dimensional ﬂow structures are visualized with Q criterion [27] indicating the coherent vortices. The comparison with the experimental data [4, 28] is presented in Fig.(2) on the left hand side. The results for isothermal case agree well with experiments: the mean axial velocity proﬁle obtained in

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281

the present computations and the proﬁle taken from [28] are very close for the entire range of z/D. In case of the ﬂuctuating axial velocity component we can observe that, at the beginning of the domain, i.e. z/D < 2, the disturbances decay - this is a typical behavior as they were generated without any space/time correlations and therefore a large part of disturbances, which are not physical, are damped. Starting from z/D > 2 velocity ﬂuctuations recover their proper behavior and up to z/D = 8 the proﬁle agrees very well with results from [28]. Further downstream we can see small discrepancies but the results are still in the acceptable range, which proves the correctness of our LES computations for constant density case. The results for the variable density jets are shown in Fig.(2) for qualitative comparison only, as the authors are not aware of detailed experimental data in the analyzed range of density ratio. Comparing to the constant density jet one may observe that as the density ratio decreases the mean velocity downstream from the potential core decreases faster. At the same time the ampliﬁcation of disturbances is higher starting already from z/D = 3. The maxima of ﬂuctuations increase

Crow and Champagne (1971) Zaman and Hussain (1980) present: S = 1.0 (isothermal) present: S = 0.8 present: S = 0.6

1.3 1.2 1.1 1

0.2

0.15

0.9

/U 1

1/2/U 1

0.8 0.7

0.1

0.6 0.5 0.4

Crow and Champagne (1971) Zaman and Hussain (1980) present: S = 1.0 (isothermal) present: S = 0.8 present: S = 0.6

0.05

0.3 0.2 0.1 0

0

2

4

6

8

z/D

10

12

14

16

0

0

2

4

6

8

10

12

14

16

z/D

Fig. 2. Mean and ﬂuctuating component of the axial velocity along the jet axis.

as the density ratio decreases, furthermore the maxima are shifted upstream when compared with constant density jet. The amplitude spectra are shown in Fig.(3) and one may see that, for constant density ﬂow, the dominating frequency corresponds to St = 0.5, which is in the range of the preferred mode. When the density ratio decreases to S = 0.8, the dominating frequency corresponds to St = 0.82 both at the axis and in the shear layer. The same frequency is observed in the shear layer for the density ratio S = 0.6. In this case there is no dominating mode at the axis. Additionally, in the shear layer one can ﬁnd peaks corresponding to St = 1.0 and St = 0.42. The existence of these peaks could be seen as a qualitative diﬀerence compared to the case S = 0.8 and this could be further interpreted as the absolute instability phenomena. The ampliﬁcation of ﬂuctuations seems to conﬁrm this assumption

Artur Tyliszczak and Andrzej Boguslawski 0.1

St = 0.5

0.04

0.02

0.5

1

St

1.5

0.1

(z/D=4.0) (r/D=0.0)

Amplitude/U 1

St = 0.82

0.06

0.04

0.5

0.1

1

St

1.5

St = 0.82

2

S=0.8 (z/D=4.0) (r/D=0.5)

0.08

0.06

0.04

0.02

0.02

0.5

1

St

1.5

0.1

0

2

(z/D=4.0) (r/D=0.0)

Amplitude/U 1

0.06

0.04

0.5

1

St

1.5

0.1

S=0.6

0.08

0.08

2

S=0.6

St = 0.42

(z/D=4.0) (r/D=0.5)

St = 0.82

0.06

St = 1.0

0.04

0.02

0

0

2

S=0.8

0.08

0

0.06

0.04

0.02

0

S=1.0 (z/D=4.0) (r/D=0.5)

Amplitude/U 1

(z/D=4.0) (r/D=0.5) (r/D=0.0)

0.08

Amplitude/U 1

Amplitude/U 1

0.06

0.1

S=1.0

0.08

Amplitude/U 1

282

0.02

0.5

1

St

1.5

2

0

0.5

1

St

1.5

2

Fig. 3. Amplitude spectra of the axial velocity at x/D = 4 in the jet axis r/D = 0 and in the mixing layer r/D = 0.5

but as we could not ﬁnd evidence of quantitative diﬀerences in the jet behavior (considerably faster centerline velocity decay, larger spreading rate) between results for S = 0.8 and S = 0.6, it is diﬃcult to prove that absolute instability was observed in our simulation. 5.2 The Excited Jet The computations for the excited jets were performed for the conditions for which we could expect the existence of the bifurcating jets, i.e. the Strouhal number of axial forcing (Sta ) in the range 0.3 − 0.7 and the ratio fa /fh = Sta /Sth equal to 2. The results presented in this paper concern computations performed mainly for the excitation amplitudes Aa = Ah = 0.05U1 and Sta /Sth = 2. At the beginning we assumed Sta = 0.5 for both constant and variable density jets. This value corresponds to the preferred mode obtained in our computations for the constant density case without excitation. The sample averaged results presenting contours and proﬁles of axial velocity are shown in Fig.(4). The results are presented in the plane of helical excitation (the bifurcation plane) and in the perpendicular one (the bisecting plane). The Y -shape, characteristic of the bifurcation is seen very clearly, the jet splits in the bifurcating plane into two separate streams, while in the bisect-

LES of Variable Density Bifurcating Jets 15

15

z/D 5

-4

-2

0

2

0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10

10

z/D

0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10

10

0

283

5

0

4

-4

-2

x/D

0

2

4

x/D

0.8

Present results: z/D = 6.5 Present results: z/D = 8.0 Lee & Reynols (1985): z/D = 6.5 Lee & Reynols (1985): z/D = 8.0

0.7

0.6

/U 1

0.5

0.4

0.3

0.2

0.1

0

-4

-2

0

2

4

x/D

Fig. 4. Mean axial velocity contours in the bifurcating plane (upper left ﬁgure) and the bisecting plane (upper right ﬁgure). Proﬁles in the radial direction in the bifurcating plane (lower ﬁgure).

ing plane starting from z/D ≈ 7 the jet becomes very narrow. The comparison with experimental data is shown also in Fig.(4). The lines represent numerical solutions along the jet radius in the bifurcating plane for two cross-sections located at z/D = 6.5 and z/D = 8.0 from the jet inlet, and the symbols are taken from the ﬁgures presented in [7]. Despite the fact that experimental data were obtained for the water jet at Re = 4300 for excitation amplitude 0.17U1 , the agreement with our results is more than satisfactory. The proﬁles of the mean axial velocity and its ﬂuctuating component along the jet axis for various density ratios are shown in Fig.(5). In the case of the constant density jet we can observe that there is no potential core; the mean velocity slowly decreases almost immediately after the jet inlet. At z/D ≈ 3 the mean velocity is about 0.95U1 and from this point it starts to decrease very fast. We relate this behavior with the ﬁrst maximum of ﬂuctuations which occurs

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Artur Tyliszczak and Andrzej Boguslawski S=1.0 S=1.0 S=0.8 S=0.8 S=0.6 S=0.6

1.2

1.0

(non-excited) (excited) (non-excited) (excited) (non-excited) (excited)

0.25

(non-excited) (excited) (excited) (excited)

0.20 1/2

/U 1

/U 1

0.8

0.6

0.15

0.4

0.10

0.2

0.05

0.0

S=1.0 S=1.0 S=0.8 S=0.6

0.30

0

2

4

6

8

z/D

10

12

14

16

0.00

0

2

4

6

8

10

12

14

16

z/D

Fig. 5. Mean and ﬂuctuating component of the axial velocity along the jet axis for S = 1.0, 0.8, 0.6.

exactly at this position. Close to the end of the domain at z/D = 16, the mean velocity reaches a value equal to the velocity of the co-ﬂowing stream. The second maximum of ﬂuctuations occurs in the region where the jet separates into two streams. The increase of the ﬂuctuations close to the end of the domain may be related to the ﬂuctuations in the co-ﬂowing stream. It may also result from the numerical implementation of the boundary conditions, or it may be caused by a too short averaging time. This issue cannot be fully explained so far. The velocity distribution for S = 1.0 is considerably diﬀerent from these, which were obtained for S = 0.8 and S = 0.6. In these latter cases we do not obtain a bifurcation. The mean velocity proﬁles are initially characterized by a very short potential core z/D < 2, then we observe a short plateau z/D = 3.5 − 4.5, and then the mean velocity decreases similarly to the non-excited jets. The proﬁles of the ﬂuctuating components for S = 0.8 and S = 0.6 diﬀer considerably from the ones obtained for the non-excited cases as well as from the results for the case S = 1.0 with excitation. The mean velocity proﬁle and a proﬁle of RMS (two smooth maxima) are characteristic for the jet with axial forcing only. Such a jet behavior is similar to that already observed in [4] for constant density ﬂow. Obtained results show that excitation with the same parameters as for the constant density jet does not cause bifurcation. As presented in the work of Danaila and Boersma [10] it is possible to obtain the bifurcating jet at high excitation amplitude with the helical excitation only. Knowing that, our ﬁrst idea was to increase the amplitude of excitation and that is why we performed computations for Aa = Ah = 0.10 − 0.20U1 . Unfortunately, the only eﬀect we observed was a larger spreading rate in the bifurcating plane than in the bisecting one. Furthermore, our results seemed to be in contradiction with [10]. Even for the constant density case we could not conﬁrm that helical forcing was suﬃcient for occurrence of the bifurcating jet. Anyway, we assumed that the helical forcing was crucial for bifurcation and we expected that by modifying either

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Fig. 6. Left and right-lower ﬁgures: instantaneous results for the excited jet for the density ratio 0.6. The quantities shown in 3D view are the same as in Fig.(1). Right-upper ﬁgure: mean and ﬂuctuating component of the axial velocity along the jet axis.

the excitation frequency or the ratio Sta /Sth we should get a bifurcating jet. Taking into account the diﬀerences in spectra shown in Fig.(2) we assumed that the excitation for S = 0.8 and S = 0.6 should have higher frequency than in the case for constant density ﬂow. This was suggested by the fact that for the density ratios S = 0.8 and S = 0.6 we observed peaks in the range of higher values of St. The next computations were performed for the same value of amplitude as for the constant density jet Aa = Ah = 0.05U1 , but for diﬀerent frequencies and also for diﬀerent frequency ratio. The computations with Aa = Ah = 0.05U1 , Sta = 1.0, Sta /Sth = 2.0 will be referred to as case I, the computations with Aa = Ah = 0.05U1 , Sta = 0.5, Sta /Sth = 2.4 will be referred to as case II and ﬁnally the computations mentioned earlier, Aa = Ah = 0.15U1 , Sta = 0.5, Sta /Sth = 2.0, will be referred to as case III. The value of Sta in the case I was assumed equal to twice the value of Sta for the constant density case, just to have excitation at higher frequency. However, we noted that the value Sta = 1.0 corresponded to the peak value in the non-excited case, see Fig.(2). The ratio Sta /Sth = 2.4 in case II was chosen because for this value the blooming jets were reported (see literature survey

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in [7]). We were interested in what was the inﬂuence (if any) of non-integer values of Sta /Sth on the results of computations for the variable density jet. In Fig.(6) the results for the density ratio S = 0.6 are presented. The 3D view shows the instantaneous solution obtained for case I which is the only case among the ones mentioned before for which we get the bifurcating jet. These results seem to conﬁrm that the generation of a bifurcating jet requires a higher excitation frequency when the jet density is lower than the ambient one. This is probably because the preferred mode frequency is also higher in these cases. We remind the reader that in the non-excited cases we observed a peak at St = 0.82, both for S = 0.8 and S = 0.6. The results for cases II and III show that in these conditions the jet behavior is approximately the same as for excitation with Aa = Ah = 0.05U1 , Sta = 0.5 and Sta /Sth = 2.0: there are no qualitative diﬀerences. Comparing the proﬁles presented in the upper right corner in Fig.(6) for case II with the results shown in Fig.(5), one may see that the diﬀerences in both the mean and the ﬂuctuating velocity proﬁles are quite small. On the other hand, the results for the bifurcating case (case I) are considerably diﬀerent from those for cases II, III and obtained previously, see Fig.(5).

6 Conclusions The results presented in this work show the ability of LES to compute the bifurcating jet. The low Mach number approximation of the governing equations applied with LES allowed for an eﬃcient solution of variable density jets. The results presented show that the jet behavior in such cases is considerably diﬀerent from the constant density case, even if we are not sure whether we observed absolute instability phenomena or not. It was shown that for density ratio equal to 0.6 the bifurcating jet occurred when the excitation frequency was much higher than for constant density ﬂow. On the other hand the computations performed with high amplitude excitation have shown that this parameter has practically no inﬂuence on generation of bifurcating jet.

Acknowledgments Support for the research was provided within statutory funds BS-1-103301/2004/P and the EU FAR-Wake Project No. AST4-CT-2005-012238. The authors are grateful to the TASK Computing Center in Gdansk (Poland) for access to the computing resources on the Holk PC Cluster.

References [1] P.A. Monkewitz and K.D. Sohn. Absolute instability in hot jets. AIAA Journal, 26(8):911–916, 1988.

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[2] P.A. Monkewitz, D.W. Bechert, B. Bariskow, and B. Lehmann. Selfexcited oscillations and mixing in a heated round jet. Journal of Fluid Mechanics, 213:611–639, 1990. [3] S. Russ and P.J. Strykowski. Turbulent structure and entrainment in heated jets: The eﬀect of initial conditions. Physics of Fluids, 5(12):3216– 3225, 1993. [4] S.C. Crow and F.H. Champagne. Orderly structure in jet turbulence. Journal of Fluid Mechanics, 48:547–691, 1971. [5] M. Lee and W.C. Reynolds. Bifurcating and blooming jets. Technical Report TF-22, Stanford University, 1985. [6] D. Parekh, A. Leonard, and W.C. Reynolds. Bifurcating jets at high Reynolds number. Technical Report TF-35, Stanford University, 1988. [7] W.C. Reynolds, D.E. Parekh, P.J.D. Juvet, and M.J.D. Lee. Bifurcating and blooming jets. Annual Review of Fluid Mechanics, 35:295–315, 2003. [8] J.B. Freund and P. Moin. Mixing enhancement in jet exhaust using ﬂuidic actuators: direct numerical simulations. ASME: FEDSM98-5235, 1998. [9] J.B. Freund and P. Moin. Jet mixing enhancement by high amplitude ﬂuidic actuation. AIAA Journal, 38(10):1863, 2000. [10] I. Danaila and B.J. Boersma. Direct numerical simulation of bifurcating jets. Physics of Fluids, 12(5):1255–1257, 1999. [11] C.B. da Silva and O. Metais. Vortex control of bifurcating jets: A numerical study. Physics of Fluids, 14(11):3798–3819, 2002. [12] G. Urbin and O. Metais. Direct and Large Eddy Simulations II, chapter Large-eddy simulations of three-dimensional spatially developing round jets. Kluwer Academic Publishers, 1997. [13] P. Koumoustakos, J. Freund, and D. Parekh. Evolution strategies for automatic optimization of the jet mixinng. AIAA Journal, 39(5):967– 969, 2001. [14] A. Hilgers and B.J. Boersma. Optimization of turbulent jet mixing. Fluid Dynamic Research, 29:345–368, 2001. [15] A. Majda and J. Sethian. The derivation of the numerical solution of the equations for zero Mach number Combustion. Combust. Sci. and Tech., 42:185–205, 1985. [16] P.A. McMurtry, W.H. Jou, J.J. Riley, and R.W. Metcalfe. Direct numerical simulations of a reacting mixing layer with chemical heat release. AIAA, 24:962–970, 1986. [17] W.C. Cook and J.J. Riley. Direct numerical simulation of a turbulent reactive plume on a parallel computer. Journal of Computational Physics, 129:263–283, 1996. [18] F. Ducros, P. Comte, and M. Lesieur. Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a ﬂat plate. Journal of Fluid Mechanics, 326:1–36, 1996. [19] O. Metais, M. Lesieur, and P. Comte. Transition, turbulence and combustion modelling, chapter Large-eddy simulations of incompressible and compressible turbulence. Kluwer Academic Publisher, 1999.

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[20] J.H. Williamson. Low-Storage Runge-Kutta Schemes. Journal of Computational Physics, 35:48–56, 1980. [21] S.K. Lele. Compact ﬁnite diﬀerence with spectral-like resolution. Journal of Computational Physics, 103:16–42, 1992. [22] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang. Spectral methods in ﬂuid dynamics. Springer-Verlag, 1988. [23] M.H. Carpenter, D. Gottlieb, and S. Abarbanel. The stability of numerical boundary treatments for compact high-order ﬁnite-diﬀerence schemes. Journal of Computational Physics, 108:272–295, 1993. [24] Ch. Hirsh. Numerical computation of internal and external ﬂows. John Wiley & Sons, Chichester, 1990. [25] C. Le Ribault, S. Sarkar, and S.A. Stanley. Large eddy simulation of a plane jet. Physics of Fluids, 11(10):3069–3083, 1999. [26] T.S. Lund and H.J. Kaltenbach. Experiments with explicit ﬁltering for LES using ﬁnite-diﬀerence method. Annual research briefs, Stanford University Center for Turbulence Research, 1995. [27] J.C.R. Hunt, A.A. Wray, and P. Moin. Eddies, stream, and convergence zones in turbulent ﬂows. Proceedings of the Summer Program, Center for Turbulence Research, NASA Ames/Stanford Univ. 193-208, 1988. [28] K.B.M.Q. Zaman and A.K.M.F. Hussain. Vortex pairing in a circular jet under controlled excitation. Journal of Fluid Mechanics, 101:449–491, 1980.

Large-Eddy Simulation of a Turbulent Flow around a Multi-Perforated Plate Simon Mendez, Franck Nicoud, and Thierry Poinsot CERFACS, 42, avenue Gaspard Coriolis. 31057 Toulouse cedex 1. France. [email protected]

Summary. The ﬁlm cooling technique is often used to protect the hot components in gas turbines engines by introducing cold air through small holes drilled in the wall. The hot products are mixed with the injected gas and the temperature in the vicinity of the wall is reduced. Classical wall functions developed for impermeable walls and used in Reynolds-Averaged Navier-Stokes methods cannot predict momentum/heat transfer on perforated plates because the ﬂow is drastically modiﬁed by eﬀusion. In order to obtain a better understanding of the ﬂow structure and predominant eﬀects, accurate simulations of a turbulent ﬂow around an eﬀusion plate are reported. Large-Eddy Simulations of the ﬂow created by an inﬁnite multi-perforated plate are presented. The plate is perforated with short staggered holes inclined at an angle of 30 deg to the main ﬂow, with a length-to-diameter ratio of 3.46. Injection holes are spaced 6.74 diameters apart in the spanwise direction and 11.68 diameters apart in the streamwise direction. Results for mean velocity and velocity ﬂuctuations are compared with measurements made on the LARA large-scale isothermal experiment [1].

1 Introduction In almost all the systems where combustion occurs, solid boundaries need to be cooled. One possibility often chosen in gas turbines is to use multi-perforated walls to produce the necessary cooling [2]. In this approach, fresh air coming from the casing goes through the perforations and enters the combustion chamber [3]. The associated micro-jets coalesce to give a ﬁlm that protects the internal wall face from the hot gases. The number of submillimetric holes is far too large to allow a complete description of the generation/coalescence of the jets when computing the 3D turbulent reacting simulation within the burner. Eﬀusion is however known to have drastic eﬀects on the whole ﬂow structure, noticeably by changing the ﬂame position. This is shown in Fig. 1, where the temperature ﬁelds from two RANS computations of a TURBOMECA conﬁguration are compared. The code used for these simulations is N3S-Natur, widely used by TURBOMECA for calculating the ﬂow in combustion cham-

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bers. The images shown are cuts of the combustion chamber with a perforated wall zone. Two diﬀerent models are used to reproduce the eﬀect of eﬀusion cooling on the main ﬂow: in Fig. 1.a, a simple model using a uniform injection is used to inject the correct mass ﬂow rate in the combustion chamber while in Fig. 1.b, an ad hoc model is used.

Fig. 1. Temperature ﬁelds from RANS computations using two diﬀerent models to reproduce eﬀusion cooling. a: simple model imposing the correct mass ﬂow rate without any additional treatment. b: ad hoc model developed by TURBOMECA. Results from TURBOMECA simulations.

It is obvious that the model used to mimic the eﬀect of eﬀusion cooling has a huge eﬀect on the general topology of the ﬂow: the position of the zone of high temperature is completely changed between Fig. 1.a and Fig. 1.b, showing that the position of the ﬂame is completely diﬀerent. As a consequence, new wall functions for turbulent ﬂows with eﬀusion are required to perform predictive full scale computations. The present study is a subset of a larger research activity whose main long term objective is to develop such appropriate and scientiﬁcally meaningful wall models. One major diﬃculty in developing wall functions is that the boundary ﬂuxes depend on the details of the turbulent ﬂow structure between the solid boundary and the fully turbulent zone. Such detailed description of the ﬂow in the near wall region can hardly be obtained experimentally because: • the perforation imposes small scales structures that are out of reach of current experimental devices • the thermal conditions in actual gas turbines make any measurement very challenging. This can explain the lack of data concerning full-coverage ﬁlm cooling through discrete holes. Most of the works concerning discrete-hole ﬁlm cooling treat the case of a single row of holes (cooling application for turbine blades), and only few studies on several rows of holes are available. Most available aerodynamic measurements deal with large-scale isothermal ﬂows [1, 3–5]. On the contrary, experimental studies about the thermal behavior (evaluation of cooling eﬀectiveness and heat transfer coeﬃcient at the wall) do not provide any ﬂow measurements [6] or too coarse ones [7].

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Numerical capabilities have increased during the last years: RANS simulations of full-coverage ﬁlm cooling with several rows (10 rows for Harrington et al. [8] and 7 rows for Papanicolaou et al. [9]) have been performed. These simulations have proved the ability of numerical codes to reproduce eﬀusion ﬂows. However, the results have never been analyzed in order to build a wall model for eﬀusion plates. Moreover, as long as only a few number of rows are considered, wall resolved LES can be performed in place of RANS calculations in order to gain insight about the jet-mainstream interaction. LargeEddy Simulations [10, 11] on single jets in crossﬂow have been performed but these studies do not give any information about the interaction between jets. Moreover, numerical simulations often study the case of large hole length-todiameter ratios, while eﬀusion cooling for combustion chamber walls is done through short holes, because of the small thickness of the plates in aeronautical applications. This makes it necessary to compute both the aspiration and the injection sides of the perforated plate. In order to obtain precise information about the behavior of the ﬂow near a perforated plate, Large-Eddy Simulations (LES) are performed: Direct Numerical or Wall-resolved Large-Eddy Simulations can be used to generate precise and detailed data of generic turbulent ﬂows under realistic operating conditions, with no limitations due to the size of the conﬁguration or to diﬃculties to realize measurements in a hot ﬂow. In this paper, a computational methodology is proposed to perform calculations of an eﬀusion ﬂow and results are presented. They are compared with the LARA experimental database, provided by TURBOMECA, that deals with a large-scale isothermal plate.

2 Computational Methodology 2.1 How to Study Eﬀusion Cooling The objective is to gain precise information about the behavior of the ﬂow around a multi-perforated wall. In a whole conﬁguration, the jets created by eﬀusion interact with the main ﬂow in the combustion chamber. This main ﬂow is turbulent and many eﬀects have to be taken into account to describe it. But since the aim is to focus on the ﬂow near the wall, it is not necessary to compute the main ﬂow as it is in a combustor. For practical studies of eﬀusion cooling, experimental test rigs are generally divided into two channels: one represents the combustion chamber, with a primary ﬂow of hot gases and the other represents the casing, with a secondary ﬂow of cooling air. Several rows of several holes are drilled into the plate that separates the two channels in order to observe the interaction between jets. Because the pressure is higher in the casing side, some air is injected through the perforated plate. For numerical simulations, like in experiments, this simpliﬁed conﬁguration is used.

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2.2 From the Experimental Studies to the Numerical Calculations Results in the open literature show that the eﬀusion ﬂow highly depends on the conﬁguration of interest: the ﬂow generated by a ten-row plate would be diﬀerent from the one generated by a twenty-row plate. This situation is hardly tractable from a modeling point of view and we decided to simplify the problem by considering the asymptotic case where the number of rows is inﬁnitely large. The simulation is then designed to reproduce this asymptotic case of a turbulent ﬂow with eﬀusion around an inﬁnite plate. This choice presents several advantages: • The inﬁnite plate can be reduced to a domain containing only one perforation, with periodic boundary conditions to reproduce the whole geometry of an inﬁnite plate, as it is suggested in Fig. 2. • The diﬃcult question of the inlet and outlet boundary conditions in turbulent simulations (see [12]) is avoided.

Fig. 2. From the inﬁnite plate to the ”bi-periodic” calculation domain. (a) Geometry of the inﬁnite perforated wall. (b) Calculation domain centered on a perforation; the bold arrows correspond to the periodic directions.

With such a periodic calculation domain, the objective is to have information about the structure of the ﬂow far from the ﬁrst rows, when the ﬁlm is established. However, this periodic option raises a problem: natural mechanisms that drive the ﬂow, such as pressure gradients, are absent. The ﬂow has to be generated artiﬁcially.

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2.3 Generation of a Periodic Flow with Eﬀusion Main Flows For a classical channel ﬂow, a volumetric source term is added to the momentum conservation equation in order to mimic the eﬀect of the mean streamwise pressure gradient that would exist in a non-periodic conﬁguration. This is a very classical method for channel or pipe ﬂow simulations. The source term is usually constant over space. For example, it can have the following form: S(ρ U ) =

(ρ Utarget − ρ Umean ) τ

(1)

The source term compares a target for momentum, ρ Utarget , with the average momentum in the channel, ρ Umean . A relaxation time τ characterizes the rapidity with which ρ Umean tends towards its target value. Naturally, this treatment is done for one channel. This approach can be generalized in the case of an eﬀusion conﬁguration, having a source term of the previous form for each channel: therefore, the source term on momentum is constant by part, with two distinct values for the cold and the hot sides. No source term is applied in the hole itself. Injection In experiments, channels are bounded by impermeable walls at the top and at the bottom. If used in conjunction with periodic boundary conditions in the tangential directions, this outer condition prevents the ﬂow from reaching a statistically steady state with eﬀusion, because the net mass ﬂux through the perforation tends to eliminate the pressure drop between the cold and the hot domains. In order to sustain the secondary eﬀusion ﬂow in periodic LES, two diﬀerent strategies were investigated [13]. • Constant Source Terms strategy: impermeable boundary conditions are kept at the top and the bottom of the computational domain, but source terms are added to the mass and energy equations in order to drive the mean pressure and temperature towards reference values, consistent with eﬀusion. • Boundary Conditions strategy: the Navier-Stokes equations are solved without any additional source term, but characteristic-based [14] freestream boundary conditions are used at the top and bottom ends of the domain in order to impose the appropriate mean vertical ﬂow rate. Both approaches are detailed in [13] and prove to give very similar results. Results from the second method (Boundary Conditions strategy) are discussed in this paper.

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3 Details of Numerical Simulations The conﬁguration that is considered in this paper corresponds to the geometry studied by Miron [1]. The study focuses on the case of a large-scale isothermal plate, with a hole diameter ﬁxed at d=5 mm (0.5 mm is the common value for gas turbines applications). The spacing between the holes correspond to the classical industrial applications: holes are spaced 6.74 diameters apart in the spanwise direction and 11.68 diameters apart in the streamwise direction. The thickness of the plate is 10 mm and holes are angled at 30 deg with the plate: they are short holes, with a length-to-diameter ratio of 3.46. The computational domain is designed as the smallest domain that can reproduce the geometry of an inﬁnite plate with staggered perforations, as it is shown in Fig. 2. All simulations are carried out with the AVBP code [15]. It is a fully explicit cell-vertex type code that solves the compressible multispecies Navier-Stokes equations on unstructured meshes for the conservative variables (mass density, momentum and total energy). AVBP is dedicated to LES and it has been widely used and validated in the past years in all kinds of conﬁgurations. The present simulations are based on the WALE sub-grid scale model [16]. The numerical scheme is the TTGC scheme [17] (third order in time and space): this scheme was speciﬁcally developed to handle unsteady turbulent ﬂows with unstructured meshes. The grid used for the calculations contains 1,500,000 tetrahedral cells. In this grid, ﬁfteen points describe the diameter of the hole and on average the ﬁrst oﬀ-wall point is situated at y+ ≈ 5. Typically the cells along the wall to be cooled and in the hole are sized to a height of 0.3 mm. This rather coarse grid is suﬃcient to show the ability of the method to reproduce the main features of the ﬂow. Another grid is used in this paper in section 4.2 to evaluate the eﬀect of the grid resolution: this is a very coarse grid with 150,000 tetrahedral cells. The results are discussed and compared to the results obtained with the ﬁrst grid. The grid with 1,500,000 cells will then be called FINE and the one with 150,000 cells will be called COARSE. All the results presented, except in section 4.2, are obtained with the ﬁner grid.

4 Results and Discussion 4.1 Operating Point Results are presented for a typical operating point of the experimental isothermal database provided by TURBOMECA, concerning the LARA test rig. Both the primary ﬂow which represents the burned gases ﬂow inside the combustion chamber, and the secondary ﬂow which represents the cold air coming from the compressor, are at the same temperature. The main aerodynamical parameters, given for the region up of the perforations, are summarized here:

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• The Reynolds number for the primary ﬂow (based on the duct centerline velocity and the half height of the rectangular duct) is Re=17750. • The Reynolds number for the secondary ﬂow (based on the duct centerline velocity and the half height of the rectangular duct) is Re=8900. • The pressure drop across the plate is 42 Pa. • The blowing ratio is τ1 = 1.78. In simulations, the ﬁrst three parameters are ﬁxed. The behavior of the ﬂow in the hole and related blowing ratio result from the calculations. Numerical results are compared with measurements performed at the ninth row of the LARA experimental test rig. Twelve rows of holes are studied in the experiment but the ninth row has been chosen to compare with numerical results because it is the location where measurements are most numerous. Further details about this experiment can be found in [1] and [18].

Fig. 3. Experimental measurements: mean streamwise velocity proﬁles evolution in the injection region.

Measurements show the formation of a ﬁlm created by eﬀusion through the plate. Jets interact together to form a ﬁlm that is able to protect the plate from the primary ﬂow (hot gases in real cases), even after the perforated zone. The ﬁlm needs several rows of holes to be formed, but only the structure of the ﬁlm far from the ﬁrst rows is interesting for comparisons with our numerical simulations. The primary ﬂow is disturbed in the neighborhood of the plate. Figure 3 shows the evolution of the streamwise velocity proﬁle, 3 diameters downstream of the center of the hole at rows 5, 7, 9 and 11. After a few rows, just downstream of the hole, mean streamwise velocity proﬁles

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are characterized by two peaks: the ﬁrst one, next to the wall, represents the jet core. It is the result of the interaction between the jet coming out of the hole and the ﬁlm. The second peak represents the ﬁlm core, which is the result of the interaction of all former jets with the main ﬂow. The value of the peaks behave diﬀerently: the peak related to the jet does not change a lot from one row to another, whereas the peak corresponding to the ﬁlm core is highly inﬂuenced by the number of rows upstream of the position of the proﬁle. The presence of a secondary velocity peak can be observed under the jet, for y/d ≈ 0.5. This is the sign of a well-known phenomenon for inclined jets in crossﬂow: the incident ﬂow is deﬂected by the jet and a part of the ﬂow is entrained towards the wall by the eﬀect of the rotating structure of the jet. The entrainment process has been widely discussed, and further details can be found in Tyagi and Acharya [10] and Yavuzkurt et al. [5]. 4.2 Validation of the Results by Comparison with Experimental Data Averages are performed over a period that corresponds to twenty characteristic convection times for the primary ﬂow. Comparisons are made with experimental proﬁles at locations shown in Fig. 4. Figure 5 shows the evolution of the mean streamwise velocity proﬁle upstream, above and downstream of the hole. Symbols correspond to measurements, lines to AVBP calculations: dotted lines are the results from the coarse grid and continuous lines are the results from the ﬁner grid. Figure 5.a shows the ﬂow upstream of the hole. Under y/d=6, the proﬁle shows the presence of the cooling ﬁlm, formed by the former injections. This proﬁle interacts with the jet in Fig. 5.b. At y=0, high velocities are present at the outlet of the hole. The jet strongly modiﬁes the proﬁle downstream of the jet, as shown in Fig. 5.c. The characteristic form of eﬀusion proﬁle can be observed, with a

Fig. 4. Location of the proﬁles measured on the experimental data.

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Fig. 5. Mean streamwise velocity proﬁles at four locations: dotted lines correspond to LES results with the coarse grid, continuous line to LES with the ﬁner grid and symbols correspond to measurements.

peak marking the jet just upstream, and a second peak that represents the ﬁlm created by injection through the former holes. The second peak is more pronounced in the experimental and in the coarse grid results. More downstream (Fig. 5.d), the jet loses its strength and progressively coalesces with the ﬁlm to add its contribution. In Fig. 5.d, the velocity near the wall increases compared to Fig. 5.c. This is an eﬀect of the entrainment process. A general good agreement is obtained between the simulations and the experiment. The results from the coarse grid reproduce very well the form of the proﬁle. However, the near-wall region is not discretized ﬁnely enough: this leads to important errors on the velocity gradient. These errors are due to a bad description of the vortical structure of the ﬂow: the entrainment process, that is a major characteristic of the ﬂow, is not reproduced with the coarse grid. Figure 5 shows two diﬀerent trends for the ﬁner grid: ﬁrst of all, the behavior in the near-wall region seems to be quite well reproduced and the velocity peak due to the jet is situated as in the experiment. This ability to describe the near-wall region is crucial because the aim is to get information about what happens in this region, in which measurements are very diﬃcult to perform. This time the entrainment process is reproduced nicely.

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Larger diﬀerences can be found above the jet, in the ﬁlm core region. It is believed that it is mainly due to the diﬀerence between the conﬁgurations that are being studied. Simulations characterize the ﬂow around an inﬁnite perforated plate, while measurements are made at the ninth row. This eﬀect of accumulation for mean streamwise velocity is coherent with what is observed experimentally in Fig. 3: the number of rows has an eﬀect on the mean velocity proﬁle above the peak due to the jet. Then, a nine-row plate and an inﬁnite plate cannot provide the same results and diﬀerences are coherent with Fig. 3, in which it can be observed that the velocity of the ﬁlm core tends to increase with the number of rows. This accumulation eﬀect is not reproduced in the coarse grid simulation, as the accumulation of ﬂuid near the wall is a consequence of the entrainment process. For quantities that are not directly aﬀected by the eﬀect of accumulation due to the inﬁnite conﬁguration, comparisons between experiments and simulations show very good agreement for the ﬁne grid, as shown for the mean vertical velocity (top of Fig. 6) for stations c and d or for the root mean square velocities (bottom of Fig. 6). All the proﬁles for these quantities show the same agreement for the ﬁne grid simulations. On the contrary the results from the coarse grid are clearly worse.

Fig. 6. Validation of the calculations on diﬀerent quantities: dotted lines correspond to LES results with the coarse grid, continuous line to LES with the ﬁner grid and symbols correspond to measurements. TOP: Mean vertical velocity proﬁles at two locations downstream of the jet. BOTTOM: Root Mean Square streamwise and vertical velocity at station c.

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This conﬁrms that the eﬀusion process is well reproduced with the ﬁne grid and that the diﬀerences between experimental and numerical results are mainly due to the diﬀerence between a ﬁnite and an inﬁnite perforated plate. 4.3 General Features of the Flow Numerical simulations are also able to provide information that is diﬃcult to obtain from experimental measurements, like the topology of the ﬂow in the hole or wall shear stress maps over the cooled wall. Behavior in the Hole The topology of the ﬂow in the hole starts to be known thanks to numerical studies that have conﬁrmed assumptions made by experimenters. Figure 7 shows ﬁelds of mean streamwise velocity in the hole.

a) Centerline plane

b) Plane perpendicular to the jet ﬂow

Fig. 7. Computed mean streamwise velocity (in m/s) and velocity vectors in the hole.

The ﬂow computed in the case of short holes with high blowing ratios is complex. Figure 7 allows to observe trends that have already been detected by Leylek and Zerkle [19] and by Walters and Leylek [20, 21] on a similar conﬁguration thanks to RANS calculations: the jet separates at the entry of the hole (Fig. 7.a) and two diﬀerent regions can be deﬁned: the jetting region, near the upstream wall of the hole, where the jet shows high velocities, and the low-momentum region. As the length of the holes is small, the jet at the exit of the hole is still highly inﬂuenced by what happens at the entry of the hole. Figure 7.b also shows that two counter rotating vortices are present in the hole itself. The sense of rotation of these vortices is the same as for the classical counter rotating vortex pair that is observed in jets in crossﬂow studies (see [22]). Here it can be noticed again that, as suggested earlier by

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many authors, a strong interaction between the secondary ﬂow, the ﬁlm-hole and the primary ﬂow exists and has to be reproduced to be representative for gas turbines cooling applications. Wall Shear Stress Numerical studies can easily provide information about the wall shear stress. Even if the reﬁnement at the wall is not optimal (the ﬁrst oﬀ-wall point is situated at y+ ≈ 5), reasonable assessment of the shear stress at the wall is expected from the LES. Figure 8 shows the mean shear stress at the wall that has to be cooled. It shows the main aspects of the near-wall ﬂow: • High values of wall shear stress (approximately 0.2 N.m−2 ) can be observed on the sides of the hole and just downstream. They are signs of the entrainment process. Here, the ﬂow is deﬂected towards the wall and particles with high velocities go close to the wall, generating high shear. • Downstream of the hole, a zone of low values (approximately 0.05 N.m−2 ) shows the presence of the jet lift-oﬀ just after the hole. Moreover, the jet plays the role of an obstacle for the incident ﬂow. • On the remaining of the plate, the wall shear stress appears to be quite homogeneous. Figure 8 also allows to understand the contribution of the jet to the wall shear stress: even if the contribution of the hole region on this plane is not very

Fig. 8. Wall shear stress (in N.m−2 ) map on the ”combustion chamber” side of the wall.

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high, disturbances of the near-wall region due the jet are very important. The shear stress at the wall is not constant any more, like over an impermeable plate: the jet has a strong eﬀect on the topology of the ﬂow, noticeably by creating locally high and low shear stress regions. The ﬂow in the near-wall region appears to be far too complicated to be modeled by a classical log-law or by a law that takes into account the porosity of the plate by introducing a scaling factor in the classical log-law.

5 Conclusions To compute realistic cases of eﬀusion cooling for combustion chamber liners, the generic conﬁguration of an inﬁnite perforated plate has been chosen. The method proposed to generate the ﬂow in a periodic conﬁguration (inﬁnite plate reduced to a domain containing one single hole with periodic boundary conditions in the tangential directions) provides interesting results. They are compared to an experimental database on a large-scale isothermal conﬁguration. Several comments can be made: • The velocity ﬁeld in the jet shows a realistic form: separation of the jet due to the sharp-edged, inclined inlet along the downstream portion of the hole is reproduced. • At the outlet of the hole, the jet lift-oﬀ and the entrainment process are observed. • Good general agreement with experimental measurements is observed. Velocity gradients at the wall are correctly predicted. The mean streamwise velocity shows an eﬀect of the inﬁnite plate, with diﬀerences on the ﬁlm core prediction, compared to the experimental data. • RMS velocity levels show a good general behavior, with good levels of turbulent intensities near the wall (y/d < 4). The isothermal conﬁguration is the ﬁrst step of the work on eﬀusion cooling modeling. In order to study the heat transfer on the multi-perforated plate, the method presented in this article should be adapted to compute Large-Eddy Simulations of a non-isothermal ﬂow.

Acknowledgments The authors are grateful to the European Community for funding this work under the project INTELLECT-DM (Contract No. FP6 - AST3 - CT 2003 - 502961), and to the CINES (Centre Informatique National pour l’Enseignement Sup´erieur) for the access to supercomputer facilities. The authors also acknowledge TURBOMECA for allowing the use of the two snapshots displayed in Fig. 1.

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References ´ [1] P. Miron. Etude exp´erimentale des lois de parois et du ﬁlm de refroidissement produit par une zone multiperfor´ee sur une paroi plane. PhD thesis, Universit´e de Pau et des Pays de l’Adour, 2005. [2] A. H. Lefebvre. Gas Turbines Combustion. Taylor & Francis, 1999. [3] R.J. Goldstein. Advances in Heat Transfer. Academic Press, New-York and London, 1971. [4] K. M. Bernhard Gustafsson. Experimental Studies of Eﬀusion Cooling. PhD thesis, Chalmers University of Technology. Goteborg, 2001. [5] S. Yavuzkurt, R.J. Moﬀat, and W.M. Kays. Full coverage ﬁlm cooling. Part 1. Three-dimensional measurements of turbulence structure. J. Fluid Mech., 101:129–158, 1980. [6] F. Bazdidi-Tehrani and G.E. Andrews. Full-coverage discrete hole ﬁlm cooling: investigation of the eﬀect of variable density ratio. Journal of Engineering for Gas Turbines and Power, 116:587–596, 1994. ´ [7] S. Rouvreau. Etude exp´erimentale de la structure moyenne et instantan´ee d’un ﬁlm produit par une zone multiperfor´ee sur une paroi plane. Application au refroidissement des chambres de combustion des moteurs aeronautiques. PhD thesis, E.N.S.M.A. et Facult´e des Sciences Fondamentales et Appliqu´ees, 2001. [8] M. K Harrington, M. A. McWaters, D. G. Bogard, Lemmon C. A., and K. A. Thole. Full-coverage ﬁlm cooling with short normal injection holes. ASME TURBOEXPO 2001. 2001-GT-0130, 2001. [9] E. Papanicolaou, D. Giebert, R. Koch, and A. Schultz. A conservationbased discretization approach for conjugate heat transfer calculations in hot-gas ducting turbomachinery components. International Journal of Heat and Mass Transfer, 44:3413–3429, 2001. [10] M. Tyagi and S. Acharya. Large eddy simulation of ﬁlm cooling ﬂow from an inclined cylindrical jet. Journal of Turbomachinery, 125:734– 742, 2003. [11] L. L. Yuan, R. L. Street, and J. H. Ferziger. Large-eddy simulations of a round jet in crossﬂow. J. Fluid Mech., 379:71–104, 1999. [12] P. Moin and K. Mahesh. Direct numerical simulation: A tool in turbulence research. Annu. Rev. Fluid Mech., 30(539-578), 1998. [13] S. Mendez, F. Nicoud, and P. Miron. Direct and large-eddy simulations of a turbulent ﬂow with eﬀusion. In ERCOFTAC WORKSHOP. Direct and Large-Eddy Simulations 6. Poitiers FRANCE, 2005. [14] T. Poinsot and S. Lele. Boundary conditions for direct simulations of compressible viscous ﬂows. J. Comp. Physics, vol.101(1):104–129, 1992. [15] V. Moureau, G. Lartigue, Y. Sommerer, C. Angelberger, O. Colin, and T. Poinsot. Numerical methods for unsteady compressible multicomponent reacting ﬂows on ﬁxed and moving grids. J. Comp. Physics, 202(2):710–736, 2005.

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[16] F. Nicoud and F. Ducros. Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion, 62(3):183–200, 1999. [17] O. Colin and M. Rudgyard. Development of high-order Taylor-Galerkin schemes for unsteady calculations. J. Comp. Physics, 162(2):338–371, 2000. [18] P. Miron, C. Berat, and V. Sabelnikov. Eﬀect of blowing rate on the ﬁlm cooling coverage on a multi-holed plate: application on combustor walls. In Eighth International Conference on Heat Transfer. Lisbon, Portugal, 2004. [19] J.H. Leylek and R. D. Zerkle. Discrete-jet dilm cooling: A comparison of computational results with experiments. Journal of Turbomachinery, 116:358–368, 1994. [20] D.K. Walters and J.H. Leylek. A systematic methodology applied to a three-dimensional ﬁlm-cooling ﬂowﬁeld. Journal of Turbomachinery, 119:777–785, 1997. [21] D.K. Walters and J.H. Leylek. A detailed analysis of ﬁlm-cooling physics: Part 1- Streamwise injection with cylindrical holes. Journal of Turbomachinery, 122:102–112, 2000. [22] J. Andreopoulos and W. Rodi. Experimental investigation of jets in a crossﬂow. J. Fluid Mech., 138:93–127, 1984.

Simulation of Separation from Curved Surfaces with Combined LES and RANS Schemes F. Tessicini, N. Li, and M. A. Leschziner Department of Aeronautics, Imperial College, London, UK [email protected]

Summary. The focus of the paper is on the performance of approximate RANStype near-wall treatments applied within LES strategies to the simulation of ﬂow separation from curved surfaces at high Reynolds numbers. Two types of combination are considered: a hybrid RANS-LES scheme in which the LES ﬁeld is interfaced, dynamically, with a full RANS solution in the near-wall layer; and a zonal scheme in which the state of the near-wall layer is described by parabolized Navier-Stokes equations which only return the wall shear stress to the LES domain as a wall boundary condition. In both cases, the location of the interface can be chosen freely. The two methods are applied to a ﬂow separating from the trailing edge of a hydrofoil. A second ﬂow considered is one separating from a three-dimensional hill, for which the performance of the zonal method is contrasted with a ﬁne-grid LES and simulations in which the near-wall layer is treated with log-law-based wall functions.

1 Introduction Separation from curved surfaces is characterized by intermittent, rapidly varying patches of reverse ﬂow and the ejection of large-scale vortices over an area that can extend to several boundary-layer thicknesses in the streamwise direction. The dynamics of this process are extremely complex, and attempts to represent it with RANS schemes, even those based on elaborate secondmoment closure, have been found to almost always result in predictive failure [1]. While LES naturally captures the dynamics of the separation process, in principle, its predictive accuracy greatly depends, in practice, on the details of the numerical mesh, in general, and the near-wall resolution, in particular. As found in [2], for the case of separation from a ducted two-dimensional hill, a 1% error in the prediction of the time-mean separation line results, approximately, in a 7% error in the length of the recirculation region. This sensitivity, associated with the representation of the near-wall physics, is a serious obstacle to the eﬀective utilization of LES in the prediction of separation from curved surfaces, because the demands of near-wall resolution rise roughly in proportion to Re2 .

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Approaches that aim to bypass the above exorbitant requirements are based on wall functions and hybrid or zonal RANS-LES schemes. The use of equilibrium-ﬂow wall functions goes back to early proposals of Deardorﬀ [3] and Schumann [4], and a number of versions have subsequently been investigated, which are either designed to satisfy the log-law in the time-averaged ﬁeld or, more frequently, involve an explicit log-law or closely related powerlaw prescription of the instantaneous near-wall velocity (e.g. [5–7]). These can provide useful approximations in conditions not far from equilibrium, but cannot be expected to give a faithful representation of the near-wall layer in separated ﬂow. The alternative of adopting a RANS-type turbulence-model solution for the inner near-wall layer is held to oﬀer a more realistic representation of the near-wall ﬂow in complex ﬂow conditions at cell-aspect ratios much higher than those demanded by wall-resolved simulations. The best-known realization of the combined RANS-LES concept is Spalart et al.’s DES method [8], in which the interface location, yint , is dictated by the grid parameters, through the switching condition yint = min(ywall , CDES × max(∆x∆y∆z)). A single turbulence model - the one-equation SpalartAllmaras model - is used, both as a RANS model in the inner region and a subgrid-scale model in the outer LES region. This is thus a ‘seemless’ approach. A feature of the method is that it enforces the location of the RANSLES switch at the y-location dictated by the streamwise and spanwise grid density. In general ﬂows, this density often needs to be high to achieve adequate resolution of complex geometric and ﬂow-dynamic features, both close to the wall (e.g. separation and reattachment) and away from the wall. Thus, the interface can be forced to be close to the wall, often as near as y + = O(20), defeating the rationale and objective of DES. Also, it has been repeatedly observed, especially at high Reynolds numbers and coarse grids and with the interface location being around y + = O(100 − 200), that the high turbulent viscosity generated by the turbulence model in the inner region extends, as subgrid-scale viscosity, deeply into the outer LES region, causing severe damping in the resolved motion and a misrepresentation of the resolved structure as well as the time-mean properties. A hybrid method allowing the RANS near-wall layer to be pre-deﬁned and to be interfaced with the LES ﬁeld across a prescribed boundary has recently been proposed by Temmerman et al. [9]. With any such method, one important issue is compatibility across the interface; another (related one) is the avoidance of ‘double-counting’ of turbulence eﬀects - that is, the overestimation of turbulence activity due to the combined eﬀects of modelled and resolved turbulence. This approach, outlined in Section 2, is one of those used below for simulating separation from curved surfaces. Zonal schemes are similar to hybrid strategies, but involve a more distinct division, both in terms of modelling and numerical treatment, between the near-wall layer and the outer LES region. Such schemes have been proposed and/or investigated by Balaras and Benocci [10], Balaras et al. [11], Cabot and Moin [12] and Wang & Moin [13]. In all these, unsteady forms of the

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Fig. 1. Two types of near-wall treatment: (a)The hybrid LES-RANS scheme; (b)The two-layer zonal scheme.

boundary-layer (or thin shear-ﬂow) equations are solved across an inner-layer of prescribed thickness, which is covered with a ﬁne wall-normal mesh, with a mixing-length-type algebraic model providing the eddy viscosity. Computationally, this layer is partially decoupled from the LES region, in so far as the pressure ﬁeld just outside the inner layer is imposed across the layer, i.e. the pressure is not computed in the layer. The wall-normal velocity is then determined from an explicit application of the mass-continuity constraint, one consequence being a discontinuity in this velocity at the interface. The principal information extracted from the RANS computation is the wall shear stress, which is fed into the LES solution as an unsteady boundary condition. In this paper, the eﬀectiveness of the hybrid LES-RANS scheme of Temmerman et al. [9] and the zonal scheme of Balaras et al. [11] is investigated in the context of simulating separation from curved surfaces at high Reynolds number. The emphasis is on two particular conﬁgurations: a statistically spanwise-homogeneous ﬂow over the rear portion of a hydrofoil, separating from the upper suction surface, and a three-dimensional ﬂow, separating from the leeward side of a circular hill in a duct.

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2 Outline of Methods 2.1 The Hybrid Scheme The principles of the hybrid scheme are conveyed in Fig. 1(a). The thickness of the near-wall layer may be chosen freely, although in applications to follow, the layer is simply bounded by a particular wall-parallel grid surface. The LES and RANS regions are bridged at the interface by interchanging velocities, modelled turbulence energy and turbulent viscosity, the last being subject to the continuity constraint across the interface: LES νSGS = νtRAN S .

(1)

The turbulent viscosity can be determined, in principle, from any turbulence model. In the case of a two-equation model, νtRAN S = Cµ fµ

k2 , ǫ

(2)

and matching the subgrid-scale viscosity to the RANS viscosity at the interface is eﬀected by: < fµ (k 2 /ǫ)νSGS > , (3) Cµ = < (fµ (k 2 /ǫ))2 > where < ... > denotes averaging across any homogeneous direction, or over a predeﬁned patch, in case no such direction exists. An analogous approach may be taken with any other eddy-viscosity model. With the interface Cµ determined, the distribution across the RANS layer is needed. Temmerman et al. [9] investigate several sensible possibilities, and the one adopted here, based on arguments provided in the aforementioned study, is: Cµ (d) = 0.09 + (Cµ,int − 0.09)

(1 − e−d/∆ ) . (1 − e−dint /∆int )

(4)

The numerical implementation of the coupling is straightforward and indicated in Fig. 1. The numerical solution within the near-wall layer is identical to that of the outer LES domain. The LES ﬁeld at nodes nearest to the interface provides the ‘boundary conditions’ for the inner layer. Also, at these nodes, Cµ is computed from (3). ‘Boundary conditions’ at the interface needed for solving the k-equation in the RANS layer are provided by the subgrid-scale energy in the LES domain, while the interface dissipation rate is evaluated from the subgrid-scale energy as k 1.5 /(C∆), where ∆ = (∆x∆y∆z)1/3 . In the case of two-equation RANS modelling, the turbulence equations in the sublayer are solved by a coupled, implicit strategy, replacing an earlier sequential, explicit solution applied to one-equation models, which was found to cause stability problems with two-equation models.

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2.2 The Zonal Two-Layer Strategy The objective of the zonal strategy is to provide the LES region with the wall-shear stress, extracted from a separate modelling process applied to the near-wall layer. The wall-shear stress can be determined from an algebraic law-of-the-wall model or from diﬀerential equations solved on a near-walllayer grid reﬁned in the wall-normal direction - an approach referred to as “two-layer wall modelling”. The method, shown schematically in Fig. 1(b), was originally proposed by Balaras and Benocci [10] and tested by Balaras et al. [11] and Wang and Moin [13] to calculate the ﬂow over the trailing edge of a hydrofoil. At solid boundaries, the LES equations are solved up to a near-wall node which is located, typically, at y + = 50. From this node to the wall, a reﬁned mesh is embedded into the main ﬂow, and the following simpliﬁed turbulent boundary-layer equations are solved: ˜j ˜i ˜i U ˜i ∂ρU ∂ρU ∂ P˜ ∂U ∂ + + [(µ + µt ) ] i = 1, 3 = ∂t ∂xj ∂xi ∂y ∂y ( )* +

(5)

Fi

where y denotes the direction normal to the wall and i identify the wallparallel directions (1 and 3). The left-hand-side terms are collectively referred to as Fi . In the present study, either none of the terms or only the pressuregradient term has been included in the near-wall approximation. The eﬀects of including the remaining terms are being investigated and will be reported in future accounts. The eddy viscosity µt is here obtained from a mixing-length model with near-wall damping, as done by Wang and Moin [13]: + µt + = κyw (1 − e−yw /A )2 µ

(6)

The boundary conditions for equation (5) are given by the unsteady outerlayer solution at the ﬁrst grid node outside the wall layer and the no-slip condition at y = 0. Since the friction velocity is required in equation (6) to determine y + (which depends, in turn, on the wall-shear stress given by equation (5)), an iterative procedure had to be implemented wherein µt is calculated from equation (6), followed by an integration of equation (5).

3 The Computational LES Framework The computational method rests on a general multiblock ﬁnite-volume scheme with non-orthogonal-mesh capabilities allowing the mesh to be body-ﬁtted. The scheme is second-order accurate in space, using central diﬀerencing for advection and diﬀusion. Time-marching is based on a fractional-step method, with the time derivative being discretized by a second-order backward-biased

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approximation. The ﬂux terms are advanced explicitly using the AdamsBashforth method. The provisional velocity ﬁeld is then corrected via the pressure gradient by a projection onto a divergence-free velocity ﬁeld. To this end, the pressure is computed as a solution to the pressure-Poisson problem by means of a partial-diagonalisation technique and a V-cycle multigrid algorithm operating in conjunction with a successive line over-relaxation scheme. The code is fully parallelised and was run on several multi-processor computers with up to 128 processors.

4 Application 4.1 The Conﬁgurations Simulated In earlier papers of Temmerman et al. [7, 9]) and Tessicini et al. [14], results obtained with wall functions, hybrid RANS-LES and zonal schemes are presented for channel ﬂow, at a Reynolds number of up to 42000, and separated ﬂow from a two-dimensional hill, at the relatively low Reynolds number of 21000. In both ﬂows, the hybrid scheme was shown to perform well, and, importantly, to display only weak dependence to the interface location. Here, attention focuses on the ﬂow around a hydrofoil with separation from its trailing edge, and on a ﬂow separating from a three-dimensional hill. The conﬁguration of the hydrofoil ﬂow is shown in Fig. 2.

Fig. 2. One instantaneous realisation of ﬂow over the trailing edge of a hydrofoil. The ﬁve marked streamwise locations B to F are at x = −3.125, −2.125, −1.625, −1.125 and −0.625 upstream of the trailing edge, where ﬂow statistics are examined.

The ﬂow separates from the upper side of the asymmetric trailing edge. The Reynolds number, based on free stream velocity U∞ and the hydrofoil chord, is 2.15 × 106 . The corresponding Reynolds number, based on hydrofoil thickness, is 1.01×105 . Simulations were performed over the rear-most 38% of the hydrofoil chord. The ﬂow was previously investigated experimentally by

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Blake [15] and numerically by Wang and Moin [13]. The computational domain is 16.5H ×41H ×0.5H, where H denotes the hydrofoil thickness. Table 1 lists the simulations performed, including the grids and the interface locations, in terms of y + , in the boundary layer upstream of the curved trailing edge. The present ‘coarse-grid’ results are compared to those obtained by Wang and Moin [13] on a C-grid of 1536×96×48 nodes1 , claimed to be well-resolved. The inter-nodal distance in both streamwise and spanwise directions is ∆+ x = 120 for both wall-approximate methods. However, the wall-normal distributions are diﬀerent, the mesh used for the hybrid scheme being considerably ﬁner close to the interface (see later discussion) than that for the two-layer scheme. It is noted that the grid for both approximate methods contains only one quarter of the number of nodes used for the reference simulation. Inﬂow conditions were taken from Wang and Moin [13]. These had been generated in two parts: ﬁrst, an auxiliary RANS calculation was performed over the full hydrofoil, using the v2f turbulence model by Durbin [16]; the unsteady inﬂow data were then generated from two separate LES computations for ﬂat-plate boundary layers at zero pressure gradient. Discrete time-series of the three velocity components at an appropriate spanwise (y-z) plane were then saved. These data, appropriately interpolated, were fed into the inﬂow boundaries of the present simulations. The upper and lower boundaries are located at 20 hydrofoil thicknesses away from the wall, to minimize numerical blocking eﬀects. At the downstream boundary, convective outﬂow conditions are applied. Table 1. Grids, modelling practices and interface locations for hydrofoil simulations. Case Reference LES ∂p Two-layer Fi = ∂x i Two-layer Fi = 0 Log-law WF Hybrid (2EQ-j12) Hybrid (2EQ-j19)

Grid SGS Model Interface y + 768 × 192 × 48 Dynamic 512 × 128 × 24 Dynamic 40 512 × 128 × 24 Dynamic 40 512 × 128 × 24 Dynamic 40 512 × 128 × 24 Yoshizawa [17] 60 512 × 128 × 24 Yoshizawa [17] 120

The three-dimensional circular hill, of height-to-base ratio of 4, is located on one wall of a duct, as shown in Fig. 3. This ﬂow, at a Reynolds number of 130000 (based on hill height and duct velocity) has been the subject of extensive experimental studies by Simpson et al. [18] and Byun & Simpson [19]. The size of the computational domain is 16H × 3.205H × 11.67H, with H being the hill height. The hill crest is 4H downstream of the inlet plane. The inlet conditions required particularly careful attention in this ﬂow, because the inlet boundary layer is thick, roughly 50% of the hill height. As indicated 1

The reference LES was performed on a C-type mesh. It can be approximately translated into a 768 × 192 × 48 H-type mesh, which is directly comparable to the wall-model meshes.

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in Fig. 4, the mean ﬂow was taken from a RANS simulation that accurately matched the experimental conditions (Wang, et al. [1]). The spectral content was then generated separately by superposing onto the mean proﬁle ﬂuctuations taken from a separate precursor boundary-layer simulation performed with a quasi-periodic recycling method and rescaling the ﬂuctuations by reference to the ratio of the friction velocity values of the simulated boundary layer, at Reθ = 1700, and the actual boundary layer ahead of the hill, at Reθ = 7000. Although the ﬂuctuations only roughly match the experimental conditions at the inlet - as can be seen from the turbulence-energy proﬁles in Fig. 4 - specifying this reasonably realistic spectral representation proved to be decisively superior to simply using uncorrelated ﬂuctuations, even if the latter could be matched better to the experimental proﬁle of the turbulence energy. Because the upper and side walls of the domain were far away from the hill, the spectral state of the boundary layers along these walls was ignored.

Y

Z

X

-4 -2

1 0

x/H 2

0 0

4

2

6 8

z /H

4 6

Fig. 3. Flow over a three-dimensional hill.

1.2 RANS Exp. LES

0.009

Simulation Exp.

1

0.006

k/U 2ref

U/Uref

0.8 0.6 0.4

0.003

0.2 0

10-3

10-2

10-1 y/H

100

0

10-3

10-2

y/H

10-1

100

Fig. 4. The mean-velocity and turbulence-energy proﬁles at the inlet plane of the three-dimensional hill domain.

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Table 2 summarizes the simulations performed for the three-dimensional hill. Only log-law-based wall functions and the zonal two-layer method will be reported in this paper. To provide a reference point or yard-stick, pure LES computations, without wall modelling, were performed on a mesh of 9.6 million nodes using both the constant-coeﬃcient, van-Driest damped Smagorinsky model and its dynamic variant. Despite this rather ﬁne resolution, this simulation cannot be claimed to be fully wall-resolving, as the y + values at the wall-nearest nodes upstream of the hill were of order 5. The two simulations also display a non-negligible sensitivity to subgrid-scale modelling, which reinforces the observation that resolution is insuﬃcient. With wall models, the aspect ratio of the near-wall grid is (supposedly) no longer a crucial constraint for LES, and major savings in computational costs can be achieved by reducing the grid resolution in the streamwise and spanwise directions. This resulted in a much coarser 3.5-million-point mesh for the wall model simulations. Two computations for an even coarser grid of 1.5 million nodes are also included, one a pure LES with no-slip wall conditions imposed and another performed with the two-layer near-wall scheme. Attention is drawn to the fact that the grid disposition in the wall-normal direction depends greatly on the wall model used. In the case of simple wall laws or the zonal method, the wall-distance of the ﬁrst grid point (of the LES mesh) away from the wall can be placed at around y + = 40 - although the separate 1d sub-layer grid over which the parabolized RANS equations are solved is much ﬁner, of course. In contrast, the hybrid RANS/LES approach requires a high wall-normal reﬁnement to be maintained, with the wall-closest node being at y + = 1. Table 2. Grids, modelling practices and interface locations for three-dimensionalhill simulations. Case Fine-grid LES Fine-grid LES Log-law WF Two-layer scheme Coarse-grid LES Two-layer scheme Coarsest-grid LES

y + of interface or near-wall-node location 448 × 112 × 192 Dynamic 5-10 448 × 112 × 192 Smagorinsky 5-10 192 × 96 × 192 Dynamic 20-40 192 × 96 × 192 Dynamic 20-40 192 × 96 × 192 Dynamic 20-40 192 × 64 × 128 Dynamic 40-60 192 × 64 × 128 Dynamic 40-60 Grid

SGS Model

4.2 Results for the Hydrofoil Results for the hydrofoil are presented in Figs. 5 to 10. First, solutions obtained with the two-layer strategy and the wall-function method are contrasted against the highly-resolved LES in Figs. 5 and 6. The ﬁgures contain

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proﬁles of streamwise velocity and turbulence intensity at the ﬁve streamwise sections along the upper hydrofoil surface, denoted B to F, respectively, and velocity proﬁles at a further ﬁve sections in the wake region. Fig. 6 also contains distributions of skin friction along the upper surface. 0.6 B C D EF

two layer Fi=0 two layer Fi=dp/dxi wall function full LES

0.5

(y−yw )/h

(y−yw )/h

0.6

0.4 0.3 C

B

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two layer Fi =0 two layer Fi =dp/dxi wall function full LES

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5.5

0

0.3

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0.6

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0.75

′

(U 2 +V 2 )1/2 /Ue

rms(u )/U∞

Fig. 5. Mean magnitude velocity (l.h.s.), rms streamwise velocity (r.h.s.) at x/H = −3.125 (B),−2.125 (C),−1.625 (D),−1.125 (E) ,−0.625 (F). 0.006

1

two layer Fi =0 two layer Fi =dp/dxi wall function full LES

0.005 0.004

Cf

y/H

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0.003 0.002 0.001

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two layer Fi=0 two layer Fi =dp/dxi wall function full les

0 -1

0

1

2

3

U/U∞

4

5

6

-8

-6

-4

-2

0

x/H

Fig. 6. Proﬁles of mean streamwise velocity in the wake at x/H = 0, 0.5, 1.0, 2.0 and 4.0 (l.h.s) and mean skin-friction coeﬃcient Cf (r.h.s.)

With respect to the skin-friction data, it is noted ﬁrst that the abrupt change in the distribution of the reference LES solution reﬂects the sudden change in surface curvature at the junction of the plane and the curved parts of the surface. At this junction, there is also a fairly abrupt change in the pressure gradient, and this explains the sensitivity of the results to the inclusion or exclusion of the gradient in the two-layer formulation: as seen, the inclusion of the pressure gradient leads to the peak in Cf being reproduced. With this diﬀerence aside, the overall agreement achieved between all the approximations, particularly that including the pressure gradient, and the LES solution is encouraging - indeed, surprisingly good, considering the relative simplicity of the methods. The simplest two-layer variant, from which the pressure-gradient term is omitted, gives distributions for all quantities, which are close to those derived from the wall-function approach. This is the expected behaviour, for the former is constrained to return a solution in the

2EQ-j12 2EQ-j19 full LES

0.5

(y−yw )/h

(y−yw )/h

LES/RANS Modelling of Separated Flows

0.4 0.3

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0.2

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315

0

0.2

0.4

(U 2 +V 2 )1/2 /Ue

0.8

0.6 ′

rms(u )/U∞

Fig. 7. Mean magnitude velocity (l.h.s.), rms streamwise velocity (r.h.s.) at x/H = −3.125,−2.125,−1.625,−1.125 and −0.625.

1

0.006

2EQ-j12 2EQ-j19 full-LES

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Fig. 8. Proﬁles of mean streamwise velocity in the wake at x/H = 0, 0.5, 1.0, 2.0 and 4.0 (l.h.s) and mean skin-friction coeﬃcient Cf (r.h.s.)

near-wall layer that complies with the log law, which is imposed explicitly within the wall-function formulation. The inclusion of the pressure gradient causes early separation relative to the LES result, as observed by reference to the skin-friction distributions. However, the velocity proﬁles at E and F agree closely with the reference LES. Hence, there is a degree of inconsistency in the predictive quality of the skin friction and the ﬂow ﬁeld, implying that the recirculation zone predicted by the approximate schemes develops at a diﬀerent rate than that predicted by the LES and recovers more quickly. As a consequence, when the ﬂow reaches location E, the premature separation predicted with the pressure gradient included is not detectable and agreement is close. Conversely, when a broadly correct separation location is predicted, the recirculation zone is thicker than it should be at locations E and F. Fig. 7 and 8 present results for the hybrid LES-RANS scheme, corresponding to Figs. 5 and 6, again in comparison with the highly-resolved LES solution. Results are here included for two hybrid simulations, both obtained with a two-equation eddy-viscosity model applied in the near-wall layer. The simulations only diﬀer in respect of the location of the interface. Fig. 9 contains, in the upper l.h.s. plot, a magniﬁed view of the hybrid RANS-LES grid above the ﬂat portion of the upper surface preceding the curved trailing edge.

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This identiﬁes the two prescribed interface locations, denoted by ‘j = 12’ and ‘j = 19’, respectively. The other three plots in the ﬁgure show variations of the turbulent viscosity at the three streamwise locations identiﬁed by B, C and D in Fig. 10, and these illustrate, as expected, that shifting the interface outwards increases the proportion of turbulence that is represented by the RANS model within the thickening near-wall layer. However, the comparisons given in Figs. 7 and 8 convey a rather disconcerting sensitivity to the interface location, not seen previously in channel-ﬂow investigations [9]. Reference to the skin-friction distributions reveals that, while the shift in the interface location does not aﬀect the result upstream of the separation point or the ability of the scheme to resolve the sharp peak in Cf , it has a marked eﬀect on the structure within the separation zone. The lack of sensitivity upstream of separation is consistent with the behaviour observed in earlier channel-ﬂow studies already referred to above. With the interface located at j = 12, it is observed that the skin friction initially drops below zero at x/H = −1.5, indicating premature separation at the start of the curved surface, but then recovers to a positive or near-zero level. In contrast, with the interface placed at j = 19, the skinfriction maintains its negative value following the initial separation, indicating

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a fully-separated ﬂow. Corresponding to the above diﬀerences in skin-friction behaviour, there arise signiﬁcant structural diﬀerences in the ﬂow, as is shown in Fig. 10. Thus, with the interface at j = 12, no distinct recirculation zone is observed, while for j = 19, a well-formed recirculation zone covers the rear portion of the suction side - although, as noted earlier, separation occurs too early.

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A possible source of the sensitivity observed above is the grid disposition close to the interface. Fig. 9 shows that, at j = 12, the cell-aspect ratio is around 10. While this is an entirely appropriate level for LES in an attached boundary layer or a channel ﬂow, the present ﬂow in the trailing-edge region is much more complex, and this aspect ratio may well be inappropriately high for the LES solution just above the interface and the near-LES (URANS) solution just below it. With the interface shifted outwards to the location j = 19, the aspect ratio drops to around 5, and this is a much more benign level. Hence, a tentative conclusion emerging from this application is that caution is required when placing the interface in a region in which wall-normal reﬁnement, eﬀected to ensure appropriate RANS resolution, coincides with a coarse streamwise distribution. In practice, this constraint can be adhered to by sensitising the interface location, locally, to the cell aspect ration, thus ensuring that the LES solution above the interface is supported by a mesh of low aspect ratio. It is noted here that this is not an issue in applying the zonal two-layer scheme, as the LES mesh is divorced from the wall-layer mesh and has a low aspect ratio even in the cells closest to the wall. As seen from the velocity and turbulence-intensity proﬁles in Fig. 7, a disadvantageous

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consequence of the outward shift in the interface location is, unfortunately, an excessive outward displacement of the shear layer above the separation zone. This is, essentially, a consequence of the premature separation, already observed by reference to the skin-friction distributions, and the elevation of the turbulence that is associated with the longer stretch of separated shear layer from the point of separation to the streamwise locations ‘E’ and ‘F’. Whether this defect is directly linked to the near-wall approximation is not clear at this stage. As noted in Table 1, the subgrid-scale model used in the hybrid RANS-LES simulation is the one-equation formulation of Yoshizawa. Experience suggests that this model tends to be excessively dissipative, even in channel ﬂow and box turbulence. Hence, further simulations, with the dynamic model, need to be performed in an eﬀort to separate eﬀects arising from nearwall and subgrid-scale modelling. Additional uncertainties arise from the wellknown weaknesses of the two-equation (k − ǫ) turbulence model used in the near-wall region, especially in separated-ﬂow conditions, and this may well be a contributory factor in the observed sensitivity. Finally, the manner in which the dissipation rate is prescribed at the interface may be an inﬂuential issue. The present practice is to extract the interface dissipation rate from the length scale Cµ −3/4 κy, while a dynamic process may be more appropriate, based on the subgrid-dissipation rate. At this stage, all that can be said is that the insensitivity to the interface location, emerging as an advantageous property of the hybrid method in channel-ﬂow investigations, may not carry over to separated ﬂow conditions. When the main ﬂow feature to be resolved is ‘weak’ – in this case, incipient separation over a gently curved surface – the grid characteristics may play an important role, and this needs to be quantiﬁed. 4.3 Results for the Three-dimensional Hill Prior to a consideration of results obtained with the near-wall approximations, attention is directed brieﬂy to the pure LES solution on the 9.6-million-node mesh, some of which have already been reported by Tessicini et al. [20]. Although this mesh may be regarded as ﬁne for the Reynolds number in question, it is, in fact, too coarse and one that compromises the accuracy of the simulation. As noted previously, the nodal plane closest to the wall is at a distance of y + = 5 − 10, while the streamwise-to-wall-normal cell-aspect ratio is of order 80 near the wall. This grid is thus found to render the simulation sensitive to sub-grid-scale modelling, especially very close to the wall, where the asymptotic variation of the subgrid-scale viscosity and stresses can be very important. This sensitivity is illustrated in Figs. 11, which shows the mean velocity-vector ﬁelds across the hill centre-plane. With the Smagorinsky model, separation occurs too early and gives rise to a more extensive recirculation zone, which results in a slower pressure recovery in the wake following the separation and consequent diﬀerences in the ﬂow ﬁelds downstream of

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reattachment. The dynamic model gives a shorter and thinner recirculation zone, in better agreement with the experimental observations. Despite the broadly satisfactory results derived with the dynamic model, some caution is called for when assessing the physical ﬁdelity of the results. The use of the dynamic model poses uncertainties when it is applied on an under-resolving grid, because the near-wall variation of the Smagorinsky constant, following spatial averaging, is quite sensitive to the near-wall grid, and that grid is too coarse in the present LES computation. The fact that the dynamic model nevertheless performs better than the constant-coeﬃcient variant is due to the former returning a better representation of the required wallasymptotic variation of the Smagorinsky viscosity (O(y 3 )). An estimate of the grid density required to yield a suﬃciently well wall-resolved near-hill representation suggests the need for a grid of 30 − 50 million nodes, an extremely expensive proposition in view of the modest Reynolds number.

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Results obtained with the wall-functions and zonal two-layer near-wall approximations are given in Figs. 12 to 15. These show, respectively, distributions of the pressure coeﬃcient on the hill surface at the centre-plane, velocityvector ﬁelds across the centre-plane, ﬂow topology maps on the leeward side of the hill and velocity proﬁles in a cross-ﬂow plane at the downstream location,

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x/H = 3.63, for which experimental results are available. As seen, the wallmodel solutions all give quite encouraging agreement with the experiment. Fig. 12(a) shows that, except for the coarsest-mesh (1.5 million nodes) pure LES, all simulations predict the pressure-coeﬃcient distribution reasonably well. The magniﬁed views provided in Fig. 12(b) and (c) reveal, in particular, that the inﬂexion in the Cp curves, associated with the weak separation on the leeward side of the hill, is well captured. The beneﬁt of using wall models becomes especially evident in the case of extremely poor spatial resolution, on the 1.5 million-node mesh, where the use of no-slip conditions results in a grossly erroneous prediction of the separation process. In fact, as will be shown below, an attached ﬂow is predicted, and an excessively fast pressure recovery after the hill crest is returned. In contrast, applying a wall model on the coarsest grid results in the resolution of the separation process and thus a better representation of the pressure-recovery process. In the ﬁne-grid simulations and all cases using wall models, the size and extent of the recirculation zone on the leeward side of the hill agree fairly well with the experimental results, as shown in Fig. 13. With poor spatial resolution and no-slip conditions, the recirculation zone predicted is either too small - as is the case for the coarse-grid LES on the 3.5 million-node mesh - or entirely absent - as is the case for the 1.5-million-node mesh. Fig. 14 demonstrates that both wall approximations also give a broadly faithful representation of the ﬂow topology, reﬂecting the presence of a single pair of vortices detaching from the surface. With the combination of a very coarse grid and no-slip conditions, the topology is seen to be characterised by smooth attached-ﬂow streaklines. In contrast, use of the two-layer model allows the correct vortical structures to be recovered. Finally, the downstream velocity proﬁles at x/H = 3.63, included in Fig. 15, are generally in fairly close agreement with the experimental data, although the computed ﬂow appears

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Fig. 14. Topology maps predicted by the wall models, relative to the pure LES and experiment.

Fig. 15. Mean streamwise- and spanwise-velocity proﬁles at various spanwise locations on the downstream plane x/H = 3.63.

to recover slightly more quickly in the wake than suggested by the experiment, because all models predict slightly early reattachment of the ﬂow at the foot of the hill. In the region far away from the wall, beyond y/H = 0.4, the predicted velocity proﬁles are noticeable diﬀerent from the experimental LDA data [18]. However, they agree very well with hotwire measurements [21] made in the same ﬂow facility, and this discrepancy remains to be resolved. Overall, the level of agreement with the experimental observations achieved with the wall-function and zonal schemes is far closer than that reported in Wang et al. [1] for RANS closures, and also closer than for the pure LES on the same coarse grids for which the wall approximations were used. The fact

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that both approximate schemes yield very similar solutions is not surprising, for if neither the pressure gradient nor advection is included in the solution of the RANS equations (5) in the near-wall layer, the zonal scheme inevitably returns instantaneous representations of the log law as the solution to the one-dimensional RANS equations in the near-wall layer. What is surprising, however, is how good the ﬁdelity of the solution is with experiment, in view of the exceedingly simple approximations employed.

5 Conclusions Two drastically diﬀerent approaches to approximating the near-wall region within a combined LES-RANS strategy have been examined by reference to high-Reynolds-number ﬂows involving separation from curved surfaces. The hybrid RANS-LES scheme, applied only to the separated hydrofoil ﬂow, has been found to display some weaknesses not encountered previously in channelﬂow computations. Thus, the solutions in the separated zone have been found to be sensitive to the grid disposition at the interface – speciﬁcally, the cellaspect ratio at the interface – although other factors may play a contributory role, including the subgrid-scale model, the quality of the turbulence model in the near-wall layer and the manner in which the dissipation rate is prescribed at the LES-RANS interface, so as to serve as a ‘boundary condition’ for the dissipation-rate equation solved in the near-wall layer. The nature of the ﬂow may also be an issue: in this case, the separation is a relatively weak feature ensuing from a gently-curved surface, and this can therefore be expected to be quite sensitive to near-wall modelling. Two forms of the zonal two-layer model have been tested for the hydrofoil ﬂow, one excluding and the other including the pressure-gradient term in the solutions of the thin-shear-ﬂow equations in the near-wall layer. The simplest form, with the pressure gradient and advection ignored, returns a solution that is, in eﬀect, an instantaneous representation of the log law. Thus, as expected, the simulations display a behaviour very close to one in which loglaw-based wall functions are imposed explicitly in the course of the simulation. Both the two-layer formulations and the log-law-based wall-function approach have been found to return surprisingly good results for the hydrofoil ﬂow. In particular, the separation behaviour is broadly correctly predicted, and the consequence is a generally realistic representation of other ﬂow features. The simple zonal scheme and the wall-function approach have also yielded pleasingly realistic solutions for the ﬂow around the three-dimensional hill. On coarse grids, the solutions returned by both approximate methods have been observed to be deﬁnitively superior to the pure LES solutions on the same grids. Most dramatically, no separation could be resolved on the coarsest grid by the LES, while the zonal scheme still gave a credible representation of the major features associated with the separation process. A task that

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remains is to determine whether the inclusion of advection leads to a further improvement of the performance of the zonal scheme.

Acknowledgments This work was undertaken, in part, within the DESider project (Detached Eddy Simulation for Industrial Aerodynamics). The project is funded by the European Union and administrated by the CEC, Research DirectorateGeneral, Growth Programme, under Contract No. AST3-CT-2003-502842. N. Li and M.A. Leschziner gratefully acknowledge the ﬁnancial support provided by BAE Systems and EPSRC through the DARP project “Highly Swept Leading Edge Separation”.

References [1] C. Wang, Y.J. Jang, and M.A. Leschziner. Modelling 2 and 3-dimensional separation from curved surfaces with anisotropy-resolving turbulence closures. Int. J. Heat and Fluid Flow. 25, pp. 499–512, 2004. [2] L. Temmerman, M.A. Leschziner, C.P. Mellen, and J. Frohlich. Investigation of wall-function approximations and subgrid-scale models in large eddy simulation of separated ﬂow in a channel with streamwise periodic constrictions. Int. J. Heat and Fluid Flow. 24(2), pp 157–180, 2003. [3] J.W. Deardorﬀ. A numerical study of three-dimensional turbulent channel ﬂow at large Reynolds numbers. Journal of Fluid Mechanics, 41, pp. 453–480, 1970. [4] U. Schumann. Subgrid scale model for ﬁnite diﬀerence simulations of turbulent ﬂows in plane channels and annuli. J. Comp. Phys., 18, pp. 376–404, 1975. [5] H. Werner and H. Wengle. Large-eddy simulation of turbulent ﬂow over and around a cube in a plate channel. 8th Symposium on Turbulent Shear Flows. pp. 155–168, 1991. [6] G. Hoﬀmann and C. Benocci. Approximate wall boundary conditions for large-eddy simulations. Advances in Turbulence V, Benzi, R. (Ed.), pp. 222–228, 1995. [7] L. Temmerman, M.A. Leschziner, and K. Hanjali´c. A-priori studies of a near-wall RANS model within a hybrid LES/RANS scheme. Engineering Turbulence Modelling and Experiments. Rodi, W. and Fueyo, N. (Eds.), Elsevier, pp 317–326, 2002. [8] P.R. Spalart, W.-H. Jou, M. Strelets, and S.R. Allmaras. Comments on the feasibility of LES for wings and on the hybrid RANS/LES approach. Advances in DNS/LES, 1st AFOSR International Conference on DNS/LES , Greyden Press, pp. 137–148, 1997.

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[9] L. Temmerman, M. Hadziabli´c, M.A. Leschziner, and K. Hanjali´c. A hybrid two-layer URANS-LES approach for large eddy simulation at high Re. Int. J. Heat and Fluid Flow. 26(2), pp 173–190, 2005. [10] E. Balaras and C. Benocci. Subgrid-scale models in ﬁnite diﬀerence simulations of complex wall bounded ﬂows. Applications of Direct and Large Eddy Simulation, AGARD. pp. 2-1–2-6, 1994. [11] E. Balaras, C. Benocci, and U. Piomelli. Two-layer approximate boundary conditions for large-eddy simulations. AIAA Journal, 34(6), pp. 1111– 1119, 1996. [12] W. Cabot and P. Moin. Approximate wall boundary conditions in the large eddy simulation of high Reynolds number ﬂow. Flow, Turbulence and Combustion, 63, pp. 269–291, 1999. [13] M. Wang and P. Moin. Dynamic wall modelling for large-eddy simulation of complex turbulent ﬂows. Physics of Fluids. 14(7) pp. 2043–2051, 2002. [14] F. Tessicini, L. Temmerman, and M.A. Leschziner. Approximate nearwall treatments based on zonal and hybrid RANS-LES methods for LES at high Reynolds numbers. Proceedings of the Engineering Turbulence Modelling and Measurements (ETMM6), Sardinia, 2005. [15] W.K. Blake. A statistical description of pressure and velocity ﬁelds at the trailing edge of a ﬂat strut. David Taylor Naval Ship R&D Center Report 4241, Bethesda, Maryland, 1975. [16] P.A. Durbin. Separated ﬂow computations with the k-e-v2 model. AIAA Journal. 55, pp. 659–664, 1995. [17] A. Yoshizawa and K. Horiuti. A statistically-derived subgrid-scale kinetic energy model for the large-eddy simulation of turbulent ﬂows. J. Phys. Soc. of Japan. 54, pp. 2834–2839, 1985. [18] R.L. Simpson, C.H. Long, and G. Byun. Study of vortical separation from an axisymmetric hill. Int. J. Heat and Fluid Flow, 140 (2), pp. 233–258, 2002. [19] G. Byun and R.L. Simpson. Structure of three-dimensional separated ﬂow on an axisymmetric bump. AIAA Paper, 2005-0113, 2005. [20] F. Tessicini, N. Li, and M.A. Leschziner. Zonal LES/RANS modelling of separated ﬂow around a three-dimensional hill. ERCOFTAC Workshop Direct and Large-Eddy Simulation-6, Poitiers, 2005. [21] R. Ma and R.L. Simpson. Characterization of Turbulent Flow Downstream of a Three-Dimensional Axisymmetric Bump. 4th Int. Symp. on Turbulence and Shear Flow Phenomena (TSFP4), Williamsburg, pp. 1171–1176, 2005.

Highly Parallel Large Eddy Simulations of Multiburner Conﬁgurations in Industrial Gas Turbines G. Staﬀelbach1 , L. Y. M. Gicquel1 , and T. Poinsot2 1

2

CERFACS, 42 Av. G. Coriolis, 31057 Toulouse, France [email protected] IMFT, All´ee du Professeur C. Soula, 31400 Toulouse, France

Summary. Recent advances in computer science and highly parallel algorithms make Large Eddy Simulation (LES) an eﬃcient tool for the study of complex ﬂows. The available resources allow today to tackle full complex geometries that can not be installed in laboratory facilities. The present paper demonstrates that the state of the art in LES and computer science allows simulations of combustion chambers with one, three or all burners and that results may diﬀer considerably from one conﬁguration to the other. Computational needs and issues for such simulations are discussed. A single burner periodic sector and a triple burner sector of an annular combustion chamber of a gas turbine are investigated to assess the impact of the periodicity simpliﬁcation. Cold ﬂow results validate this approach while reacting simulations underline diﬀerences in the results. The acoustic response of the set-up is totally diﬀerent in both cases so that full geometry simulations seem a requirement for combustion instability studies.

1 Introduction In the highly competitive ﬁeld of power generation, gas turbines have gained an increasing role over the years. New emission regulations and growing energy demand increase the weight on the research and development of the manufacturers of such machinery. Outstanding advances have been made and ever more complex designs have been developed to meet the increasingly stringent needs. Unfortunately sometimes the new designs are subject to combustion instabilities [1–3]. Experimental tests must be conducted to evaluate the risk and impact of such phenomena on the machines. Building a full burner for each test is unpractical and geometric simpliﬁcations must be made leading to approximations in the results. Numerical tools, which do not have this limitation, are an attractive alternative to experimental set-ups especially Large Eddy Simulation (LES). Recent studies using Large Eddy Simulation (LES) have shown the accuracy of this approach in comparison with experimental data. LES is able to

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predict mixing [4], stable ﬂame behaviour [5] and ﬂame acoustic interaction [6]. It is also used for ﬂame transfer function evaluation[7]. The objective of the present paper is to demonstrate the necessity to investigate full burner conﬁgurations for instability studies. Technical diﬃculties encountered for full burner LES are discussed. The interest of such simulations is shown by an a posteriori study of the ﬂow for a single and a triple burner conﬁgurations. Note that the current work is part of the European project DESIRE with the support of SIEMENS PG.

2 Target Conﬁguration and LES Models To demonstrate the feasibility and usefulness of multi-burner and full burner simulations, LES of an annular combustion chamber are performed. The injection system consists of two co-rotating partially premixed swirlers. The swirler vanes are not simulated and appropriate boundary conditions are set to mimic the vane eﬀects on the ﬂow. LES is carried out with a parallel solver called AVBP, simulating the full compressible Navier Stokes equations on structured, unstructured or hybrid grids. The sub-grid scale inﬂuence is modeled with the standard Smagorinsky model[8]. A one step chemical scheme for methane matching the behaviour of the GRI-mech 3.0 scheme [9] at the target conditions is employed to represent the chemistry. The Thickened Flame Model (TFLES) ensures that the ﬂame is properly solved on the grid [10, 11]. Finally all simulations employ the Lax-Wendroﬀ numerical scheme.

3 Computational Issues When dealing with very large conﬁgurations, computer related issues rapidly arise. In order to perform a full burner LES, one needs to adapt the available tools. The potential diﬃculties are divided in three themes: • Mesh generation • Fast and eﬃcient LES • Post-processing of the results Generating a mesh for the calculation is by far the most time consuming and remains a critical point in numerical simulations. The diﬃculty is greatly increased when trying to build a mesh for a full set-up. The memory requirements become important (over ﬁve gigabytes of RAM for a 5.106 cells mesh) and powerful computers are needed. Most designs have a natural geometric periodicity and the most practical solution to generate the mesh is to create a single periodic mesh and then duplicate it as many times as needed to build the full model.

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Fig. 1. Full burner set-up: 40 million tetrahedra

A full twenty four burner set-up was meshed using this procedure (cf. Figure 1). The resulting computational domain has over forty million cells and requires three hundred Megabytes. In order to perform a fast and eﬃcient LES for such a large mesh, a large number of processors is required. However using a large number of processors means decomposing the computational domain into a large number of parts. Since message passing communications are proportional to the quantity of domains, the code must not suﬀer any degradation in performance by increasing the number of processors. Figure 2 shows the ideal behaviour (black line) compared to the behaviour of AVBP for a ﬁve million cells LES (squares) and for a forty million cells case (circles). A close to optimal result is observed even for up to 5000 processors3 . Once LES is performed, a lot of data has been generated (over 20 gigabytes for the ﬁve million cells case). Memory limitations encountered during mesh generation are also present during the post-processing step. Retrieving such large quantities of data from a computer center to a visualization post can also be troublesome. Remote visualization applications are a potential solution and oﬀer an interesting way to reduce post-processing time.

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Fig. 2. Behaviour of AVBP for a large number of processors (BlueGene conﬁguration).

For the present study and especially the 40.106 cells case, a computer with over 16 gigabytes of memory was necessary to create the mesh (using the concatenation method described above), and to post-process the data. Remote visualization software was also used to accelerate the analysis phase.

4 Results and Discussion To evaluate the possible beneﬁts from multi-burner simulations compared to a simulation using a simpliﬁed geometry, a single burner (cf. Figure 3a) and a triple burner LES (cf. Figure 3b) are presented. The single burner side

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boundary conditions are periodic whereas the triple burner’s are simple walls. Details of the computational domain are given on Figure 4.

Fig. 4. Single burner conﬁguration. (See Plate 40 on page 434)

In the following the unsteady and averaged behaviours of the conﬁgurations are discussed. The single periodic and the central burner of the triple set-up are compared to assess the impact of the periodicity. Results show that the main hydrodynamic features remain the same but some substantial diﬀerences are observed in the reacting cases. 4.1 Cold Flow Results Swirled ﬂows have been used for a long time in gas turbines [5, 12]. Their main objective is to create a central recirculation zone to anchor the ﬂame without ﬂame holders. Precessing structures, also called precessing vortex cores (PVC), are commonly observed for this type of ﬂows and are usually located right at the outlet of the injector system. Figure 5 shows instantaneous views of the LES results for the single periodic burner (a) and for the triple burner (b). On the top part of each ﬁgure, the diﬀerent velocity components are displayed. The central recirculation zone is visible in the axial velocity component snapshot. For the periodic sector, a precessing vortex core rotating in the same direction as the imposed swirl is evidenced using a low pressure

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b)

Fig. 5. Precessing structure and velocity components in the mid clip plane: a) single burner, b) triple burner.

isosurface near the axial hub. Spectral analysis in the axial swirler region of the radial velocity component reveals that the PVC revolves at 300Hz (cf. Figure 6a). In the triple burner case, pressure isosurfaces reveal the existence of a PVC for each burner. They are located at the exit of the corresponding burner as observed in the single sector conﬁguration. Spectral analysis of the central burner PVC conﬁrms that the precession motion is also at 300Hz (cf. Figure 6b).

a)

b)

Fig. 6. Radial velocity spectra in the axial region for the single burner (a) and triple conﬁguration central burner (b).

From the instantaneous behaviour we can conclude that the periodicity simpliﬁcation seems to have no impact on LES results. The averaged quantities must be checked to enforce this conclusion: Figures 7a and 7b show the averaged axial velocity and the pressure ﬂuctuations for the single burner (black line) and the central burner of the triple case (dashed line) for the non-reacting case. Proﬁles are extracted at diﬀerent locations on a horizon-

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tal plane along the axis of the burner. The results match quite well even for pressure ﬂuctuations. The main hydrodynamic features of the ﬂow seem to be identical in the non-reacting cases for the single and triple sector.

a)

b)

Fig. 7. a) Mean axial velocity and b) pressure ﬂuctuations. Single burner (black line) vs central burner of the triple set-up (dashed line).

4.2 Reacting Flow The cold ﬂow observations suggest that the triple burner LES oﬀers no additional information compared to the periodic LES. The situation is diﬀerent for reacting cases as shown in the following section.

a)

b)

Fig. 8. Flame (1000K isosurface) and velocity components (reaction rate, black contours): a) periodic burner b) triple burner.

The recirculation zones observed in the cold ﬂow simulations suggested that the ﬂame should attach near the axial hub and at the burner outlet. Figures 8a) and 8b) show the ﬂame zone represented by a one thousand Kelvin

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temperature isosurface. The diﬀerent velocity components are shown at the top of the Figures in addition to a reaction rate iso-contour. The ﬂame is anchored in both conﬁgurations and for all burners by the central re-circulation zone as expected. The position of the central recirculation zone seems unaffected by the presence of the ﬂame. Using the coherent structure detection criterion from Hussain [13], a PVC is evidenced in the reacting case for the single burner (Fig. 9a). Its revolving movement matches the swirl’s. Spectral analysis of the radial velocity component reveals that the precessing structure’s frequency is 780Hz (cf. Figure 9b). Which diﬀers from the one observed in the cold ﬂow results. Since the presence of the ﬂame disrupts considerably the near axial hub region this is expected. The structure is also present in the triple sector case .

a)

b)

Fig. 9. Single periodic burner: a) ﬂame (1000K transparent isosurface ) and Q criterion (solid isosurface) b) power spectrum in the axial region.

The observations of the unsteady solutions highlight no clear diﬀerence between the two conﬁgurations. Figure 10a) shows that the mean axial velocity matches well for both simulations. As was the case in the non-reacting ﬂow (cf. Figure 7a), the match is quite good. The mean temperature proﬁles also match reasonably well (cf. Figure 10b). However, pressure and temperature ﬂuctuations diﬀer greatly (cf. Figure 11). Predicted pressure ﬂuctuations are two times higher in the three sector case than in the single sector case. The presence of the side burners modiﬁes temperature ﬂuctuations. Since pressure and heat release (therefore temperature ﬂuctuations) are closely linked to the acoustic behavior of the set-up, acoustic analysis is required to evaluate the impact of the diﬀerences and their origin.

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a)

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b)

Fig. 10. a) Mean axial velocity b) mean temperature ﬁeld: single burner (black line) vs central burner of the triple set-up (dashed line).

b)

a)

Fig. 11. a) Pressure ﬂuctuations b) temperature ﬂuctuations: single burner (black line) vs central burner of the triple set-up (dashed line).

5 Acoustic Description Eigen-value solvers are useful tools for visualizing and analyzing the possible acoustic modes present in a set-up. To conduct the acoustic analysis a Helmholtz solver [14, 15] using the averaged speed of sound distribution from the reacting LES is used. The lowest eigen-frequencies obtained for both cases are displayed in Table 1. Table 1. Eigen-frequencies for one and three sectors Eigen Value 1 2 3 4 One sector (Hz) 129 560 679 Three sectors (Hz) 128 371 561 680

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Some eigen values match in both conﬁgurations (cf. table 2 ). Since the triple burner contains three single burners side by side this is no surprise. However the triple burner has a 370Hz eigen-value which is not present in the one sector conﬁguration (cf. Table 2 eigen-frequencies number 2 ). Table 2. Eigen-modes for both conﬁgurations (Helmholtz analysis). (See Plate 41 on page 434)

1

2

3

4

The excitation of the 370Hz eigen mode explains the diﬀerences observed in the reacting LES. Spectral analysis of the LES pressure ﬁeld in the middle of each sector composing the triple burner yields a dominant harmonic at 370Hz for the side burners (cf. Figure 12). This frequency matches very well mode 2 as predicted by the Helmholtz solver (cf. table 2) . The single burner simulation is unable to reproduce the right behavior for the reacting case since 370Hz is not an eigen-frequency for the one sector conﬁguration. Therefore for combustion instability studies, considering a full burner LES may be necessary.

6 Conclusions The feasibility of full burner LES is demonstrated. The methodology and challenges behind such simulations are enumerated and possible solutions are given. To assess the beneﬁts from full burner simulations compared to simpliﬁed conﬁgurations, a single periodic burner LES and a triple burner LES were performed. The periodic simpliﬁcation seems to be adapted to cold ﬂow studies. However, to retrieve the correct acoustic behaviour of the set-up for combustion instability studies, considering the full geometry seems paramount. The acoustic properties of the full chamber are too diﬀerent from the single

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Fig. 12. Pressure spectra in the center of each sector of the triple burner case (LES result)

or even the triple burner case. Addition of more burners in the computation induces new ﬂame interactions as well as thermal perturbations which coupled with the new acoustic ﬁeld of the new domain result in totally diﬀerent ﬂow ﬁelds. Clear identiﬁcation of the leading mechanisms yielding these diﬀerences is not established yet. However, preliminary investigations presented in this work indicate acoustics as being paramount if proper gas turbine numerical simulations are to be trusted. Finally and for reacting LES ﬂow simulations, the full turbine behavior can not be extrapolated from the simpliﬁed case.

Acknowledgments This work is funded by the European Community through project DESIRE (Design and Demonstration of Highly Reliable Low NOx Combustion Systems for Gas Turbines, contract no NNE5/388/2001). The authors would like to thank CINES (Centre Informatique National de l’Enseignement Sup´erieur), French national computing center, for their help and support. They are also indebted to IBM for the opportunity to use their latest BlueGene machine.

References [1] S. Candel. Combustion instabilities coupled by pressure waves and their active control. Proc. Combust. Inst., 24:1277–1296, 1992.

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[2] D.G. Crighton, A. Dowling, J. Ffowcs Williams, M. Heckl, and F. Leppington. Modern methods in analytical acoustics. Lecture Notes. Springer Verlag, New-York, 1992. [3] T. Poinsot and D. Veynante. Theoretical and numerical combustion 2nd Edition. R.T. Edwards, 2005. [4] C. Pri`ere, L.Y.M. Gicquel, A. Kaufmann, W. Krebs, and T. Poinsot. LES of mixing enhancement: LES predictions of mixing enhancement for jets in cross-ﬂows. J. Turb., 5:1–30, 2004. [5] S. Roux, G. Lartigue, T. Poinsot, U. Meier, and C. B´erat. Studies of mean and unsteady ﬂow in a swirled combustor using experiments, acoustic analysis and large eddy simulations. Combust. Flame, 141:40–54, 2005. [6] L. Selle, G. Lartigue, T. Poinsot, R. Koch, K.-U. Schildmacher, W. Krebs, B. Prade, P. Kaufmann, and D. Veynante. Compressible large-eddy simulation of turbulent combustion in complex geometry on unstructured meshes. Combust. Flame, 137(4):489–505, 2004. [7] A. Kaufmann, F. Nicoud, and T. Poinsot. Flow forcing techniques for numerical simulation of combustion instabilities. Combust. Flame, 131:371– 385, 2002. [8] J. Smagorinsky. General circulation experiments with the primitive equations: 1. The basic experiment. Monthly Weather Review, 91(3):99–164, 1963. [9] M. Frenklach, H. Wang, M. Goldenberg, G.P. Smith, D.M. Golden, C.T. Bowman, R.K. Hanson, W.C. Gardiner, and V. Lissianki. Gri-mech: an optimized detailed chemical reaction mechanism for methane combustion. Technical Report GRI-Report GRI-95/0058, Gas Research Institute, 1995. [10] O. Colin, F. Ducros, D. Veynante, and T. Poinsot. A thickened ﬂame model for large eddy simulations of turbulent premixed combustion. Phys. Fluids, 12(7):1843–1863, 2000. [11] J.-Ph. L´egier, T. Poinsot, and D. Veynante. Dynamically thickened ﬂame large eddy simulation model for premixed and non-premixed turbulent combustion. In Summer Program 2000, pages 157–168, Center for Turbulence Research, Stanford, USA, 2000. [12] A.K. Gupta, D.G. Lilley, and N. Syred. Swirl ﬂows. Abacus Press, 1984. [13] F. Hussain and J. Jeong. On the identiﬁcation of a vortex. J. Fluid Mech., 285:69–94, 1995. [14] F. Nicoud and L. Benoit. Global tools for thermo-acoustic instabilities in gas turbines. In APS/DFD meeting, Bull. Amer. Phys. Soc., New York, 2003. [15] L. Benoit and F. Nicoud. Numerical assessment of thermo-acoustic instabilities in gas turbines. Int. J. Numer. Meth. Fluids, 47:849–855, 2005.

Response of a Swirled Non-Premixed Burner to Fuel Flow Rate Modulation A. X. Sengissen1 , T. J. Poinsot2 , J. F. Van Kampen3 , and J. B. W. Kok3 1

2

3

CERFACS, 42 Av. G. Coriolis, 31057 Toulouse cedex, France [email protected] IMFT, All´ee du Professeur C. Soula, 31400 Toulouse, France [email protected] University of Twente, 7500 AE Enschede, The Netherlands [email protected]

Summary. Combustion instability studies require the identiﬁcation of the combustion chamber response. In non-premixed devices, the combustion processes are inﬂuenced by oscillations of the air ﬂow rate but may also be sensitive to ﬂuctuations of the fuel ﬂow rate entering the chamber. This paper describes a numerical study of the mechanisms controlling the response of a swirled non-premixed combustor burning natural gas and air. The ﬂow is ﬁrst characterized without combustion and LDV results are compared to Large Eddy Simulation (LES) data. The non-pulsated reacting regime is then studied and characterized in terms of ﬁelds of heat release and equivalence ratio. Finally the combustor fuel ﬂow rate is pulsated at several amplitudes and the response of the chamber is analyzed using phase-locked averaging and ﬁrst order acoustic analysis.

1 Introduction Many combustors exhibit combustion instabilities which constitute signiﬁcant risks for project developments. Being able to predict the stability of a given burner is the center of many present research programs: these eﬀorts can be experimental [1–10] or numerical [11–14]. A speciﬁcity of modern gas turbines is that these systems operate in very lean regimes to satisfy emission regulations. The resulting ﬂames are extremely sensitive to combustion oscillations but the exact phenomena leading to instabilities are still a matter of discussions. A central question for modeling approaches is to know what induces an unsteady reaction rate (necessary to sustain the oscillations) when an acoustic wave enters the combustion chamber. This mechanism described in ﬁgure 1 may be due to (at least) two main eﬀects: 1. The formation of vortices in the combustion chamber (Fig. 1-a): These vortices are usually triggered by strong acoustic waves propagating in the

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air passages. These structures capture large pockets of fresh gases which burn only later in a violent process leading to small scale turbulence and high reaction rates [15, 16]. 2. The modiﬁcation of the fuel and oxidizer ﬂow rates when the acoustic wave propagates into the fuel and air feeding lines (Fig. 1-b). This can lead to a local change of the equivalence ratio (rich and poor pockets) and therefore to a modiﬁcation of the burning rate when these pockets enter the chamber. If the burner operates in a very lean mode, this eﬀect may be important since variations of inlet ratio may trigger strong combustion oscillations [17].

(a)

(b)

Fig. 1. Mechanism of the ﬂame response to (a) velocity perturbations and (b) equivalence ratio perturbations.

In premixed combustors, the second mechanism has been identiﬁed as a key element controlling combustor stability [6, 17] but its eﬀect on non-premixed devices remains unclear. According to Lieuwen, the mechanism is the following: even away from Lean or Rich Blow Oﬀ (LBO or RBO), equivalence ratio ﬂuctuations produce very large heat-release oscillations which trigger combustion instabilities through pressure oscillations feedback. A direct proof of the importance of fuel injection on stability is that the location of fuel injectors often alters the stability of the system. The crucial role of fuel modulation can also be readily identiﬁed by considering active control examples in which a small modulation of the fuel lines feeding a combustor can be suﬃcient to alter the stability of the combustor [18–20]. Even though the general idea of the mechanism proposed by Lieuwen is fairly clear, the details of the coupling phenomenon are unknown. For instance, real instability mechanisms are often a mixture of mechanism 1 and 2 and not of only one of them. A proposed approach to gain more insights into this instability mechanism is to pulsate the fuel ﬂow rate in a non-premixed

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combustor. Multiple studies have examined the behavior of combustors submitted to a pulsation of the air stream to measure their transfer function [15, 21, 22]. Much less data is available for fuel pulsation in non-premixed devices [23]. The objective of this paper is to analyze the response of a swirled nonpremixed combustor to a pulsation of the fuel ﬂow rate. The work is performed using Large Eddy Simulation (section 2). The corresponding experimental setup is a 125 kW burner installed at University of Twente (The Netherlands) and described in Section 3. LES and experimental results are ﬁrst compared for the non-reacting ﬂow (section 4.1). Reacting unforced results are detailed in section 4.2 before presenting results with fuel ﬂow rate pulsation (section 5).

2 Numerical Approach The LES solver AVBP (see www.cerfacs.fr/cfd) simulates the full compressible multi-species (variable heat capacities) Navier Stokes equations on hybrid grids. Subgrid stresses are described by the classical Smagorinsky model [24]. A two-step chemical scheme is ﬁtted for lean regimes on the GRI-Mech V3 reference [14]. The ﬂame / turbulence interaction is modeled by the Dynamic Thickened Flame (DTFLES) model [25] and allows to handle both mixing and combustion (crucial in partially premixed ﬂames). What the DTFLES model guarantees is that each ﬂame element moves locally at the ﬂame speed corresponding to the equivalence ratio and turbulence level. The ﬂame structure itself obtained ultimately with the DTFLES model has no physical meaning, but the response to acoustic waves or to equivalence ratio perturbations is captured adequately [26]. The numerical scheme uses third-order spatial accuracy and third-order explicit time accuracy [27]. The boundary condition treatment is based on a multi-species extension [28] of the NSCBC method [29], for which the acoustic impedance is fully controlled [30]. The walls are treated as adiabatic, with a law-of-the-wall formulation. Typical runs are performed on grids between 600, 000 and 2.7 million tetrahedra on several parallel architectures (SGI origin 3800, Compaq alpha server, Cray XD1 ).

3 Computed Conﬁguration 3.1 Target Geometry The test rig is a 125 kW lab-scale burner developed by University of Twente and Siemens PG in the context of the European Community project DESIRE (Design and Demonstration of Highly Reliable Low Nox Combustion Systems for Gas Turbines).

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Figure 2 presents the whole geometry and summarizes the ﬂow path. Figure 3 shows closer views of the various ﬂow passages. The preheated air comes out of the compressor into the air supply room. Then it ﬂows into the plenum through the acoustic decoupling system pipes (Fig. 3-b). After the swirler (Fig. 3-a), the methane is injected in the air cross ﬂow through four small holes to ensure suﬃcient mixing. Then, the mixture reaches the combustion chamber where the ﬂame is stabilized and burnt gases leave the chamber through the outlet ﬂange (Fig. 3-c).

Fig. 2. Full LES computational domain: red, fuel inlet; blue, air inlet; purple, acoustic decoupling system; yellow, swirler vanes; green, outlet ﬂange. (See Plate 42 on page 435)

The LES computational domain (Fig. 2) includes all these upstream parts from the air supply room to the outlet ﬂange. This is necessary to have the right acoustic impedance, to predict accurately the chamber acoustic modes and to minimize the uncertainties on boundary conditions.

(a)

(b)

(c)

Fig. 3. Details of the computational domain: (a) swirler vanes, (b) acoustic decoupling system and (c) outlet ﬂange. (See Plate 43 on page 435)

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3.2 Operating and Boundary Conditions The operating point simulated is the same for cold, reacting and pulsated ﬂows: • The air supply room feeds the chamber with 72.4g/s of air, preheated at 573K. This leads to a Reynolds number of 22000 (based on the bulk velocity at the burner mouth and its diameter), and a swirl number [31] of 0.7 (at the same location). • 3.06 g/s of the natural gas used in industrial gas turbines at ambient temperature (298K) is provided to the fuel line. Note that the so-called “Groningen” natural gas is modeled by a mixture of methane (76.7%) and nitrogen (23.3%) in order to have equivalent thermal properties, so that the global equivalence ratio of the setup is 0.55. • The mean pressure of the test rig is 1.5 bar. In the computation, the acoustic behavior upstream the combustion chamber is ensured by the fully reﬂecting acoustic decoupling system (Fig. 3-b) and downstream the combustion chamber, the impedance at the outlet (Fig. 3-c) is controlled through the NSCBC linear relaxation method [30].

4 Non-Pulsated Cold & Reacting Cases 4.1 Cold Flow Results The cold ﬂow experiment has been done by University of Twente on a watertunnel based on the Reynolds number similarity. The level of agreement between LES and the experiment is assessed by comparing one dimensional velocity proﬁles on the central plane at several locations from the burner exit (Plane A : 5mm, Plane B : 15mm, Plane C : 25mm, Plane D : 45mm and Plane E : 65mm). Note that the scale employed on all the proﬁles of a ﬁgure is the same. Figure 4 shows the good agreement between experimental data and LES results in both shape and amplitude of the mean velocity components and even on its RMS ﬂuctuations. In particular, the opening angle of the swirled jet, the intensity of central recirculation zone and the re-attachment of the top/bottom recirculation zones are predicted correctly. 4.2 Reacting Flow Results The steady-state reacting ﬂow and the cold ﬂow dynamics are very similar. The only noticeable diﬀerence is the larger opening angle of the swirled jet. Figure 5-(a) exhibits the instantaneous three-dimensional ﬂame structure, materialized by an isosurface of temperature at 1200K. Even though the ﬂame is compact, it is strongly wrinkled by the turbulence. The corresponding combustion regime is characterized on ﬁgure 6 by scatter plots of reaction rate

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(a)

(a’)

(b)

(b’)

(c)

(c’)

Fig. 4. Comparison of statistical proﬁles: (a) axial, (b) radial and (c) swirling mean velocity; (a’) axial, (b’) radial and (c’) swirling RMS velocity; Symbols : Experiment; Solid line: LES; dashed line: zero line.

and temperature versus local equivalence ratio (for the whole computational domain). It assesses the quality of the mixing, since very few points burn at equivalence ratio below 0.4 or above 0.7.

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The mean location of the ﬂame is shown on ﬁgure 5-(b): heat release is exhibited on a longitudinal cut in the central plane. The grey isosurface surrounds the part of the domain where the equivalence ratio φ is higher than 0.6. The mean path of the fuel jets in the swirled cross ﬂow is clearly visible. Note that φ is computed using passive scalar “z” following [32]. Therefore, it is still valid in burnt gases.

(a)

(b)

Fig. 5. (a) Instantaneous view of the ﬂame (isosurface of temperature at 1200K) and of the methane jets (isosurface of fuel mass fraction at 0.1); (b) Mean heat release in central plane and isosurface of equivalence ratio φ = 0.6. (See Plate 44 on page 436)

(a)

(b)

Fig. 6. Combustion regime corresponding to instantaneous solution presented on Fig. 5-a: scatter plots of (a) reaction rate and (b) temperature versus equivalence ratio.

5 Pulsated Reacting Cases 5.1 Forcing Method and Phenomenology Forcing the reacting ﬂow is achieved by pulsating the mass ﬂow rate of methane in the four fuel pipes (Fig. 3-a). Forcing is performed at 300Hz for

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several amplitudes: 5, 10, 15, 30, 50, and 80 percent of the unpulsated mean mass ﬂow rate. For all these amplitudes, the fuel pipes remain subsonic. The mean temporal air ﬂow rate provided by the “air supply room” remains constant, and oscillates instantaneously due to the ﬂow modulations inferred by acoustic waves propagation. Section 5.4 will show that these modulations are not negligible. Since the momentum of the fuel jets is very small compared to the momentum of the air ﬂow, only the second mechanism described in section 1 is involved. Pulsating the fuel lines should not create ring vortices that could wrinkle the ﬂame and capture pockets of fresh gases. The modulation of fuel ﬂow rate should create alternatively pockets of rich and poor mixture, which after a given convective time will excite the ﬂame. Note that results shown in sections 5.2, 5.3 and 5.4 are related to a pulsation amplitude of 15 percent. 5.2 Acoustic Modes Analysis A ﬁrst order acoustic analysis is a simple and powerful tool to understand the mechanisms leading to combustion instabilities. The acoustic eigenmodes of the setup can easily be computed using a 3D Helmholtz code. The ﬁeld required for this analysis is the local speed of sound and is provided by a temporally averaged solution of the LES. Table 1 shows the lowest frequencies found numerically, and the corresponding ones measured in the experiment. The agreement between the predicted eigenmodes frequencies and the measured results is reasonable : the Helmholtz code provides higher frequencies for most modes because the mean temperature ﬁeld was obtained from a LES using adiabatic walls, yielding higher temperature and higher sound speeds. Table 1. Frequencies computed by the Helmholtz code and measured in the experiment. Physical description of these modes: 1/4 means quarterwave mode. “S” for full Setup, “C” for Chamber and “P” for Plenum. Frequencies computed (Hz) 72 272 298 487 Frequencies measured (Hz) 62 192 239 445 Mode description 1/4 5/4 7/4 5/4 Mode related to S P P C

705 630 7/4 C

926 1046 846 983 9/4 1/2 C P

Coming back to LES, ﬁgure 7-(a) now demonstrates that the main peak around 450Hz is clearly captured by LES. This eigenmode of the setup seems to be fed by the ﬂame, especially when the ﬂame is excited by the 300Hz pulsation which is also visible on the LES spectrum (Fig. 7-b). From a physical point of view, it is expected that only the eigenmodes related to the combustion chamber itself may be excited by the ﬂame response. Among these modes, the most sensitive is the closest to the excitation frequency. This is

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(b)

Fig. 7. Pressure spectra measured in the experiment (thin line) and computed with LES (thick line) for non pulsated (a) and pulsated (b) cases.

why the 450Hz is especially ampliﬁed. The other modes near 300Hz (at 72, 272 and 298 Hz) are indeed either eigenmodes of the plenum, or eigenmodes of the full setup. Therefore, they are not sensitive to ﬂame response. 5.3 Phase-Locked Averaged Analysis Due to the high levels of turbulence which strongly aﬀect the ﬂame (as shown in section 4.2 and ﬁgure 5), diagnostic tools are necessary to ﬁlter the results. The most obvious one to study harmonic phenomena is the phase-locked averaging procedure. Figure 8 evidences the shape and intensity of the ﬂame at eight phases of the cycle. It also shows the evolution of rich pockets along this cycle, materialized by an isosurface of equivalence ratio at φ = 0.6 (slightly richer than the mean φ = 0.55). After a certain time lag, these pockets reach the reacting zone. The ﬂame does not move signiﬁcantly when it is reached by these pockets but the local heat release oscillates and triggers the pressure ﬂuctuations feeding the 450Hz acoustic mode (as described in section 5.2). 5.4 Ampliﬁcation Eﬀect A controversial question to understand the excitation mechanism is the following: is the air ﬂow rate remaining constant during fuel ﬂow rate pulsation? This can be checked in the LES by evaluating the fuel and air mass ﬂow rates at the mouth of the burner. Using a simple derivation, the variation of the equivalence ratio can be easily split in two parts (Eq. 1): The contribution of instantaneous fuel ﬂow rate to equivalence ratio ﬂuctuations φ′F and the

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Fig. 8. Phase locked heat release in the central plane and isosurface of equivalence ratio φ = 0.6. (See Plate 45 on page 436)

contribution of instantaneous air ﬂow rate to equivalence ratio ﬂuctuations φ′A . φ′ =

m˙F ′ φ m˙ ( )*F+

Fuel contribution

φ′F

m˙A ′ − φ m˙ ( )* A+

Air contribution

(1) φ′A

Figure 9 presents the two contributions φ′F and φ′A for two pulsation amplitudes: 15% and 30%. After a time delay of two cycles, the acoustic waves produced by the ﬂame and partially reﬂected at the end of the chamber clearly perturb the air ﬂow rate. In other words, the X % pulsation of the fuel line is seen by the ﬂame as a 1.2 · X % equivalence ratio excitation. In the present situation and for a forcing frequency of 300Hz, the air ﬂow is also aﬀected by the fuel ﬂow modulation and ampliﬁes its impact on the ﬂuctuations of equivalence ratio at the burner inlet. This conclusion is not general but shows that this eﬀect should be taken into account for modeling.

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(b)

Fig. 9. Contribution of the fuel (thin line) and the air (thick line) ﬂuctuations to equivalence ratio oscillations at the mouth of the burner. The amplitudes of pulsations are: (a) 15%; (b) 30%.

Figure 10 presents scatter plots of local reaction rate versus equivalence ratio at the extrema of the cycle (respectively maximum and minimum global reaction rate). Compared to Fig. 6-(a) (unforced case), the width of the scatter plot is approximately the same which indicates that the eﬀect of forcing on mixing eﬃciency is not obvious. The distribution of points is just moved alternatively up-right and down-left in the reaction rate - equivalence ratio space.

(a)

(b)

Fig. 10. Combustion regime during the 15% forcing cycle corresponding to (a) maximum reaction rate and (b) minimum reaction rate versus equivalence ratio.

5.5 Saturation Eﬀect The last point of this paper is the linearity of the ﬂame response to the equivalence ratio pulsation. Recent experimental studies [33, 34] show that beyond a certain pulsation amplitude, a saturation eﬀect is observed. Figure 11 presents the reaction rate ﬂuctuation level for several pulsation amplitudes (Fig. 11-a), up to 80%, and its evolution along the cycle (Fig. 11-b). No saturation eﬀect is noticed here: the ﬂame really behaves linearly within the range considered. This major diﬀerence may be due to the way the equivalence ratio is pulsated: as explained in section 5.1, the pulsation method should not create

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(a)

(b)

Fig. 11. Dependence of normalized global heat release ﬂuctuation upon the forcing amplitude (a) and its evolution along the cycle (b).

any ring vortices, so that only the second mechanism described in section 1 is concerned. In Balachandran et al., the fuel ﬂow rate is constant (fuel line choked) and they pulsate the air ﬂow. Therefore, both mechanisms 1 and 2 are involved.

6 Conclusion Computations of a partially premixed lab-scale burner are carried out using LES for both non-pulsated and pulsated cases. LES results are validated from velocity measurements performed at the University of Twente. The overall agreement with experiment is very good for both mean and RMS values. The mechanism leading to heat release oscillations is characterized using phaselocked analysis and a simple 3D Helmholtz solver. An ampliﬁcation eﬀect of the equivalence ratio excitations due to reﬂected acoustic waves perturbing the air ﬂow rate is observed. Note that no saturation eﬀect is noticeable for the range of observation considered, such as presented in Balachandran et al. [33, 34]. More generally, this study demonstrated that LES is able to capture the speciﬁc role of equivalence ratio ﬂuctuations in phenomena leading to combustion instabilities.

Acknowledgments Most numerical simulations have been conducted on the computers of CINES, French national computing center, consuming about 40, 000 hours on a SGI origin 3800. This work was carried out in the framework of the EC project DESIRE with Siemens PG.

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References [1] S. Candel, C. Huynh, and T. Poinsot. Some modeling methods of combustion instabilities. In Unsteady combustion, pages 83–112. Nato ASI Series, Kluwer Academic Publishers, Dordrecht, 1996. [2] F.E.C Culick. Some recent results for nonlinear acoustic in combustion chambers. AIAA Journal, 32(1):146–169, 1994. [3] A.P. Dowling. The calculation of thermoacoustic oscillations. J. Sound Vib., 180(4):557–581, 1995. [4] A.P. Dowling. A kinematic model of ducted ﬂame. J. Comput. Phys., 394:51–72, 1999. [5] W. Krebs, P. Flohr, B. Prade, and S. Hoﬀmann. Thermoacoustic stability chart for high intense gas turbine combustion systems. Combust. Sci. Tech., 174:99–128, 2002. [6] T. Lieuwen and B.T. Zinn. The role of equivalence ratio oscillations in driving combustion instabilities in low NOx gas turbines. Proc. of the Combustion Institute, 27:1809–1816, 1998. [7] T. Poinsot and D. Veynante. Theoretical and numerical combustion, second edition. R.T. Edwards, 2005. [8] W. Polifke, A. Fischer, and T. Sattelmayer. Instability of a premix burner with nonmonotonic pressure drop characteristic. J. Engng for Gas Turb. and Power, 125:20–27, 2003. [9] T. Schuller, D. Durox, and S. Candel. Self-induced combustion oscillations of laminar premixed ﬂames stabilized on annular burners. Combust. Flame, 135:525–537, 2003. [10] T. Schuller, D. Durox, and S. Candel. A uniﬁed model for the prediction of laminar ﬂame transfer functions: comparisons between conical and Vﬂames dynamics. Combust. Flame, 134:21–34, 2003. [11] H. Pitsch and L. Duchamp de la Geneste. Large eddy simulation of premixed turbulent combustion using a level-set approach. Proceedings of the Combustion Institute, 29:in press, 2002. [12] S. Roux, G. Lartigue, T. Poinsot, U. Meier, and C. B´erat. Studies of mean and unsteady ﬂow in a swirled combustor using experiments, acoustic analysis and large eddy simulations. Combust. Flame, 154:40–54, 2005. [13] V. Sankaran and S. Menon. LES of spray combustion in swirling ﬂows. J. Turb., 3:011, 2002. [14] L. Selle, G. Lartigue, T. Poinsot, R. Koch, K.-U. Schildmacher, W. Krebs, B. Prade, P. Kaufmann, and D. Veynante. Compressible large-eddy simulation of turbulent combustion in complex geometry on unstructured meshes. Combust. Flame, 137(4):489–505, 2004. [15] A. Giauque, L. Selle, T. Poinsot, H. Buchner, P. Kaufmann, and W. Krebs. System identiﬁcation of a large-scale swirled partially premixed combustor using LES and measurements. J. Turb., 6(21):1–20, 2005.

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[16] T. Poinsot, A. Trouv´e, D. Veynante, S. Candel, and E. Esposito. Vortex driven acoustically coupled combustion instabilities. J. Comput. Phys., 177:265–292, 1987. [17] J. H. Cho and T. Lieuwen. Laminar premixed ﬂame response to equivalence ratio oscillations. Combust. Flame, 140:116–129, 2005. [18] C. Hantschk, J. Hermann, and D. Vortmeyer. Active control with direct drive servo valves in liquid-ﬁlled combustion systems. Proc. Combust. Institute, 26:2835–2841, 1996. [19] K. McManus, T. Poinsot, and S. Candel. A review of active control of combustion instabilities. Prog. Energy Comb. Sci., 19:1–29, 1993. [20] T. Poinsot, D. Veynante, B. Yip, A. Trouv´e, J-M. Samaniego, and S. Candel. Active control methods and applications to combustion instabilities. J. Phys. III, July:1331–1357, 1992. [21] W.S. Cheung, G.J.M. Sims, R.W. Copplestone, J.R. Tilston, C.W. Wilson, S.R. Stow, and A.P. Dowling. Measurement and analysis of ﬂame transfer function in a sector combustor under high pressure conditions. In ASME Paper, Atlanta, Georgia, USA, 2003. [22] A. Kaufmann, F. Nicoud, and T. Poinsot. Flow forcing techniques for numerical simulation of combustion instabilities. Combust. Flame, 131:371– 385, 2002. [23] D. Bernier, S. Ducruix, F. Lacas, S. Candel, N. Robart, and T. Poinsot. Transfer function measurements in a model combustor: application to adaptative instability control. Combust. Sci. Tech., 175:993–1013, 2003. [24] J. Smagorinsky. General circulation experiments with the primitive equations: 1. The basic experiment. Mon. Weather Review, 91:99–164, 1963. [25] J.-Ph. L´egier, T. Poinsot, and D. Veynante. Dynamically thickened ﬂame large eddy simulation model for premixed and non-premixed turbulent combustion. In Summer Program 2000, pages 157–168, Center for Turbulence Research, Stanford, USA, 2000. [26] P. Schmitt, T.J. Poinsot, B. Schuermans, and K. Geigle. Large-eddy simulation and experimental study of heat transfer, nitric oxide emissions and combustion instability in a swirled turbulent high pressure burner. J. Comput. Phys., in print, 2006. [27] O. Colin and M. Rudgyard. Development of high-order Taylor-Galerkin schemes for unsteady calculations. J. Comput. Phys., 162(2):338–371, 2000. [28] V. Moureau, G. Lartigue, Y. Sommerer, C. Angelberger, O. Colin, and T. Poinsot. Numerical methods for unsteady compressible multi component reacting ﬂows on ﬁxed and moving grids. J. Comput. Phys., 202:710–736, 2005. [29] T. Poinsot and S. Lele. Boundary conditions for direct simulations of compressible viscous ﬂows. J. Comput. Phys., 101(1):104–129, 1992. [30] L. Selle, F. Nicoud, and T. Poinsot. The actual impedance of nonreﬂecting boundary conditions: implications for the computation of resonators. AIAA Journal, 42(5):958–964, 2004.

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[31] A.K. Gupta, D.G. Lilley, and N. Syred. Swirl ﬂows. Abacus Press, 1984. [32] J.-Ph. L´egier. Simulations num´eriques des instabilit´es de combustion dans les foyers a´eronautiques. PhD Thesis, INP Toulouse, 2001. [33] R. Balachandran, B. O. Ayoola, A. P. Kaminski, A. Dowling, and E. Mastorakos. Experimental investigation of the non-linear response of turbulent premixed ﬂames to imposed inlet velocity oscillations. Combust. Flame, 143:37–55, 2005. [34] R. Balachandran, A. Dowling, and E. Mastorakos. Response of turbulent premixed ﬂames to inlet velocity and equivalence ratio perturbations. In European Combustion Meeting, 2005.

Investigation of Subgrid Scale Wrinkling Models and Their Impact on the Artiﬁcially Thickened Flame Model in Large Eddy Simulations Tim Broeckhoven1 , Martin Freitag2 , Chris Lacor1 , Amsini Sadiki2 , and Johannes Janicka2 1

2

Department of Mechanical Engineering Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussel, Belgium [email protected] Energy and Powerplant Technology TU-Darmstadt Petersenstr. 30, 64287 Darmstadt, Germany

Summary. In this contribution a Large Eddy Simulation together with the Artiﬁcially Thickened Flame approach is used to study a well known experimental set-up consisting of a rectangular dump combustor - ORACLES. The major drawback of artiﬁcially thickening the ﬂame is that the interaction between turbulence and ﬂame is altered. To compensate for the inability of small vortices to wrinkle the ﬂame a subgrid scale wrinkling model has to be introduced. In this contribution the inﬂuence of the subgrid scale wrinkling on the ﬂame front in a high Reynolds number ﬂow is investigated. Moreover the inﬂuence of diﬀerent approximations for the subgrid scale velocity on the prediction of the ﬂow ﬁeld and ﬂame structure is studied.

1 Introduction A world wide increasing demand for energy, a decline of available resources and compelling environmental requirements with regard to combustion based energy conversion processes make the development of reliable predicting tools for combustion processes a must. The desire in terms of combustion engines and gas turbines is to enhance the thermal output and, at the same time, reduce speciﬁc fuel consumption and thereby also the amount of pollutants. Switching from diﬀusion ﬂames to premixed combustion, or at least to partially premixed combustion, might be one of the key-factors to reduce pollutants drastically - however the control and stabilization of premixed ﬂames appears to be a challenging task. Beside experimental investigations, which

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are expensive and time consuming, the numerical simulation of turbulent premixed and non-premixed ﬂames has made considerable progress in the last years as reported in diﬀerent contributions [1–3]. The Direct Numerical Simulation (DNS), resolving all ﬂow scales up to the Kolmogorov scale, of a turbulent reacting ﬂow in a geometry similar to those encountered in industrial applications, will remain diﬃcult in the near future due to its immense requirement of computer resources. In the past decade Large Eddy Simulation (LES) has become a mature approach for non-reacting ﬂows. The capabilities of LES to predict unsteady turbulent ﬂows make the approach very well suited for reactive ﬂows [1, 2]. The classical concept of LES is to solve spatially ﬁltered transport equations, by explicitly computing the large structures of the ﬂow ﬁeld, whereas the eﬀect of the smaller ones, the sub grid scales, is modeled. This idea makes the computational cost acceptable for high Reynolds number ﬂows compared to DNS. Improved modeling concepts have made it possible to perform Large Eddy Simulations of real combustion systems recently [4–7]. It turns out that LES provides an adequate description of turbulence-combustion interaction because the large structures are explicitly computed and instantaneous fresh and burnt gas zones, where turbulence characteristics are quite diﬀerent, are clearly identiﬁed. A diﬃcult problem encountered in LES of premixed ﬂames is that the 0 is generally a lot smaller than the mesh size ∆. typical ﬂame thickness δL Thus the ﬂame front can not be resolved on the computational mesh leading to numerical problems. To overcome this, several approaches have been described in the literature: the ﬂame front tracking technique (G-equation) [8], the use of a ﬁlter larger than the grid resolution [9] and an artiﬁcial thickening of the ﬂame [4, 10, 11]. The artiﬁcially thickened ﬂame (ATF) model, ﬁrst introduced by Butler et al [12] is very attractive due to its inherent numerical stabilizing features. The basic idea of the approach is to consider a ﬂame thicker than the actual one, with the same laminar ﬂame speed, so that the thickened ﬂame front can be resolved on the computational mesh. The major drawback of this approach is the reduction of the Damk¨ ohler number, altering the interaction between the turbulence and the ﬂame. To counteract for the reduction of the ﬂame surface wrinkling a Subgrid Scale Wrinkling Model is introduced. To this day diﬀerent model approximations have been proposed in the literature. A large number of reported works concentrate rather on a priori and a posteriori tests of the ATF model using DNS data [10, 13, 14] of relative low Reynolds numbers. Therefore the main objective of this paper is to investigate how the subgrid scale wrinkling model will aﬀect the ﬂame structure and ﬂow ﬁeld in a high Reynolds number conﬁguration. In this contribution a LES-ATF approach is used to study a well known experimental set-up consisting of a rectangular dump combustor (ORACLES [13, 15–18]). The same conﬁguration has been investigated by Pitsch et al [19], under partially premixed conditions, by Domingo et al [20] , testing a new model for Σ-PDF-SGS-closure and by D¨ using [21], who used the Gequation approach to predict the ﬂame and ﬂow ﬁeld and also by Fureby [22],

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who tested a recently proposed ﬂame-wrinkling model. In this contribution the inﬂuence of diﬀerent approximations for the subgrid scale velocity, used to locally estimate the subgrid scale wrinkling and thus the turbulent burning velocity, on the prediction of the ﬂow and combustion ﬁeld, is also studied. The structure of the paper can be outlined as follows: Section 2 introduces the used model and its dependencies and in section 3 details of the investigated conﬁguration are given. After describing the numerical technique in section 4 the results are presented and discussed in the next section. Finally a conclusion will close the paper.

2 Modeling and Governing Equations The governing equations for the investigated problem are the ﬁltered conservation equations of mass and momentum for an incompressible Newtonian ﬂuid in their instantaneous, local form: ∂ ρ¯ ∂ + (¯ ρui ) = 0 ∂t ∂xi

∂ ρ¯ui ∂ ∂ p¯ ∂ ′ u′ + τij − ρ¯u5 (¯ ρui uj ) = − + i j ∂t ∂xj ∂xi ∂xj

(1) (2)

where ¯ denotes ﬁltered quantities and denotes mass-weighted ﬁltering i.e. φ = ρφ/¯ ρ. A Boussinesq assumption is used to model the unresolved ﬂuxes: ′ u′ = −2ν S ˜ ˜ u5 t ij + 2/3 νt Skk δij and the Smagorinsky model is used to express i j the subgrid scale viscosity νt : 2 2S˜ij S˜ij (3) νt = (CS ∆)

where the Smagorinsky coeﬃcient CS is calculated with Germano’s dynamic model based on a test-ﬁlter ∆ˆ = 2∆. To describe the turbulent premixed combustion the ATF ﬂame model is chosen with a progress variable representing the non-dimensional temperature, u Θ = TTb−T −Tu , where subscripts u and b respectively denote burnt and unburnt quantities. Note that for laminar premixed ﬂames the ﬂame speed s0L and 0 may be expressed as ﬂame thickness δL s0L ∝

√

DW

0 δL ∝

D s0L

(4)

where D is the molecular diﬀusivity and W the reaction rate. In the frame of the ATF model, the thickening of the ﬂame is achieved by decreasing the reaction rate with a factor F whereas the diﬀusion is increased by the same factor F . This operation leaves the laminar ﬂame speed s0L unchanged but 1 0 increases the ﬂame thickness i.e. δL = F δL so it can be resolved on the LES computational mesh. The progress variable can then be described by [10]

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˜ ˜ ∂ ρ¯Θ ∂ ρ¯u˜i Θ ∂ + = ∂t ∂xi ∂xi

˜ ∂Θ ρ¯F D ∂xi

+

ω7 ˙Θ . F

(5)

The source term for the progress variable is modeled using Arrhenius law so various phenomena like ignition, ﬂame-stabilization and ﬂame wall interaction are taken into account Ta ˜ exp − ω7 ˙ Θ = Aρ 1 − Θ . (6) ˜ b − Tu ) Tu + Θ(T

In (6), A is the pre-exponential constant (taken independent of the temperature) and Ta the activation temperature. Thickening the ﬂame modiﬁes the interaction between turbulence and chemistry because the Damk¨ ohler number Da which describes the relation between turbulent and chemical time scales Da =

τt lt s0 = ′ L 0 τc u δL

(7)

is reduced by a factor F . To compensate for this reduction of ﬂame surface wrinkling due to the thickening, an eﬃciency function E is introduced [10, 23]. This eﬃciency function measures the subgrid scale wrinkling as a function of the local subgrid turbulent velocity u′∆e and a length scale ∆e . The diﬀusion coeﬃcient in (5) is replaced by the product EDF while the preexponential constant is replaced by AE/F so that the conservation equation for the progress variable reads now ˜ ˜ ∂ ρ¯u˜i Θ ˜ ∂ ρ¯Θ Ta ∂Θ AE ∂ ˜ exp − + ρ 1−Θ ρ¯EF D + = ˜ b − Tu ) ∂t ∂xi ∂xi ∂xi F Tu + Θ(T (8) Thus the turbulent ﬂame propagates at a turbulent ﬂame speed sT = Es0L (where E goes to unity in laminar regions) and keeps a thickness of the order 1 0 = F δL . Since the eﬃciency function describes the underestimation of of δL the actual wrinkling of the ﬂame front one can express E as the ratio between the subgrid wrinkling factors of the actual ﬂame and the thickened ﬂame. To estimate the subgrid wrinkling as a function of the local subgrid turbulent velocity u′∆e and local ﬁlter size ∆e the model proposed in Colin et al [10] can be used: 0 0 1 + αΓ ∆e /δL , u′∆e /s0L u′∆e /s0L Ξ δL (9) E= 1) = 1 , u′ /s0 u′ /s0 Ξ (δL 1 + αΓ ∆e /δL L ∆e L ∆e where the Γ function corresponds to the integration of the eﬀective strain rate induced by all scales aﬀected by the artiﬁcial thickening and α is a model −1/2 parameter that scales as α ∝ Ret . The function Γ obtained from DNS results [10, 24] is written as 2/3 ∆e 1.2 0 0 ′ (10) Γ ∆e /δL , u∆e /sL = 0.75 exp − 0.3 0 δL u′∆e /s0L

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The subgrid scale velocity u′∆e in (9) remains to be estimated. This can be achieved through the subgrid scale turbulent viscosity provided by the ﬂow-ﬁeld SGS-model u′∆e = CS ∆e 2S˜ij S˜ij (11) However, the problem with this formulation is that in the local absence of tur0 bulence the laminar ﬂame speed s is not recovered since the strain 2S˜ij S˜ij L

is inﬂuenced by the thermal expansion. In Colin et al [10] it is therefore proposed to use an operator which naturally subtracts the dilatational part of the velocity ﬁeld 8 9 (12) u′∆e = 2∆3x ∇2 (∇ × u) In this work the expressions given in (11) and (12) will be used and their inﬂuence on the prediction of the ﬂow and ﬁeld and the ﬂame structure will be examined.

3 Oracles Conﬁguration The conﬁguration investigated is documented in the literature as ORACLES (One Rig for Accurate Comparisons with Large Eddy Simulations) burner [13, 15–18]. It consists of two separate mixing chambers, from where a propane-air mixture enters the combustion chamber as two fully developed turbulent channel ﬂows. Behind the two meter long separated channels the splitter plate narrows with an angle of 14◦ . Further downstream the ﬂow is exposed to a sudden expansion, with a step height of D= 29.9 mm. For a better understanding, a sketch of the ORACLES combustion chamber is presented in Fig. 1. For the isothermal and for the reacting case stream-wise and transverse velocity components have been measured using LDV. At position −5D and −5.8D upstream of the backward facing step the data set of the upper and lower channel exhibit a slight diﬀerence, which has to be considered when generating inﬂow data for LES. For the combustion chamber mean velocities and ﬂuctuations have been reported between 1D and 10D downstream from the step [13, 17]. Commercial propane was used as fuel. The global, average composition can be expressed by C3.01 H7.94 . A more detailed itemization including the exact species concentration is given in [18]. The equivalence ratio in the two channels can be varied independently within the limits of Φ = 0.0 and Φ = 0.85. A variation of the equivalence ratio allows to study combustion under partially premixed conditions and the occurrence of extinction. The present simulation deals with a constant equivalence ratio of Φ = 0.75 for both inﬂow channels. In this case the ﬂame will anchor in the shear layer at the edge of the expansion. The Reynolds number is calculated based on the ﬂow properties of the unburnt mixture, the bulk velocity of the channel

T. Broeckhoven, M. Freitag, C. Lacor, A. Sadiki, and J. Janicka

70, 4 130, 6

39, 7

z

y

29, 9

14

30, 4

◦

30, 4

29, 9

358

x

Fig. 1. 2D sketch of the ORACLES burner, the channel width in y direction is 150, 5mm (all dimensions are in mm) Table 1. Relevant physical properties Properties Re Ubulk ρu ρb Tu Tb νu νb s0L

Value 20000 11.0 1.296 0.166 276 1980 1.66 E-5 6.40 E-5 0.27

Unit m/s kg/m3 kg/m3 K K m2 /s m2 /s m/s

and the height of the sudden expansion (29.9 mm). In the investigated case the Reynolds number is set to be equal 20, 000, assuming a bulk velocity of 11 m/s for both channels. More details of the experimental set-up can be found in Besson et al [13] and Bruel et al [15]. Using CHEM1D [25] the physical properties of a 1D laminar ﬂame were calculated with the GRI3.0 mechanism. All relevant physical properties, used in the simulation are summarized in table 1. To classify the quality of the combustion LES the computational ﬁlter size has to be compared with ﬂame and turbulence properties, as recommended by the TNF7 workshop [26]. To satisfy this requirement we make use of the experimental classiﬁcation of the burner in the Borghi-Peters [8, 27] Diagram presented in the report of Bruel et al [17]. The ratio of the integral length scale lt over the ﬂame thickness lf is around 500, and the turbulence level u′ over laminar ﬂame speed sL is given to be approximately 9. Therefore the region of interest is located close to a Karlovitz number of unity, which describes the border between the corrugated ﬂamelets and the thin reaction zone. The fundamental requirement for LES is that the ﬁlter length is larger than the Kolmogorov scale. A further requirement of Pitsch et al [28] is that the ﬁlter

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size is larger than the laminar ﬂame thickness. Taking the Karlovitz number of one, and assuming the ﬁlter size to be in the order of 1 mm, the ratio of ﬁlter size over laminar ﬂame thickness is calculated to be approximately 7. A classiﬁcation into the Regime Diagram of Pitsch et al [28] or D¨ using et al [29], extended to numerical requirements, shows that the chosen ﬁlter size is adequate. For more details about the method and theory of the regime diagram the following literature is recommended [28, 29].

4 Numerical Technique The governing equations for the investigated problem are solved using a ﬁnite volume technique on a Cartesian mesh with velocities located on a staggered grid whereas the progress variable as well as the density are stored in the cell center. The molecular diﬀusion coeﬃcient of the scalar transport equation, is expressed by the ratio of kinematic viscosity and the Schmidt number, which was set to a constant value of 0.7 in the simulation. For spatial discretization second order central diﬀerences are used for the momentum equation. The convective ﬂuxes of the scalar equation are discretized using a nonlinear TVDscheme (CHARM-limiter) [30], to keep the solution bounded, and to avoid unphysical oscillations in the progress variable ﬁeld. The overall temporal discretization is an explicit third order Runge-Kutta method and the Poisson equation for the pressure is inverted using a direct fast elliptic solver. Upstream the combustion chamber the backward facing step is included in the computational domain using the immersed boundary technique [31]. Hereby, the walls of the step and the splitter plate are expressed by an artiﬁcial force added to the momentum equation. This force has been added to the velocities after the pressure correction at every Runge-Kutta time step. The splitter plate between the two separated channels, in the upstream direction, induces a periodic instability of Helmholtz type, which can not be captured if the computational domain starts at the backward facing step. This instability is expected to inﬂuence the behavior of the premixed ﬂame and thus this part should be included in the computational domain. The total extension of the computational domain in axial (x), transverse (z) and the periodic (y) direction is 23.7D × 4.35D × 2.34D where D represents the height of the backward facing step. It has to be mentioned that in stream-wise direction 5D of the domain are located upstream of the sudden expansion. The computational domain is resolved with 712× 130 × 64 ≈ 6 · 106 grid points. Setting the velocities to zero at the upper and lower wall of the combustion chamber and using Neumann boundary conditions for the pressure mimics the walls of the combustion chamber. At the outﬂow Neumann boundary conditions for the velocity and the pressure are prescribed, negative velocities are clipped. The inlet boundary conditions for the two channels are fully developed channel mean velocity proﬁles with superimposed ﬂuctuations, generated by a method presented by Klein et al [32]. The method uses

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T. Broeckhoven, M. Freitag, C. Lacor, A. Sadiki, and J. Janicka

digital ﬁltering of random data to achieve prescribed two point correlations together with components of the Reynolds stress tensor. Experimental data [13] has been used to generate the inﬂow data 5D upstream of the step. These measurements contain axial and transverse ﬂuctuations and cross-correlations. The length scales at the inﬂow for both channels have been set to 0.4 times the width of the supplying channels in streamwise direction and 0.125 times in the two other directions. Since the computational domain starts more than 150 mm before the area of interest, it is expected that the inﬂow conditions can be considered as reasonable. All simulations run 60,000 time steps which is equivalent to approximately ﬁve ﬂow through times, based on the bulk velocity, so the number of collected samples can been considered as suﬃcient. The computations run parallel for approximately 80h on four CPUs of a Transtec 8000L Itanium2 1.5 Ghz cluster.

5 Results and Discussion In the following sections the results of the investigation of the ORACLES conﬁguration will be presented. Initially results of the isothermal ﬂow case will be presented and discussed. Furthermore three simulations of the reacting ﬂow ﬁeld have been conducted with diﬀerent approximations for the SGS-velocity, which has been used to model the eﬃciency function. Results will be compared to experimental data and discussed. 5.1 Non-Reacting Flow Initially the non-reacting ﬂow is simulated to ensure the capability of the chosen grid resolution. The agreement between the results of the simulation and the experimental data can be considered to be very good in terms of the mean streamwise velocity component. The recirculation zone attached to the lower wall of the channel is predicted very well, however at the upper wall the numerical simulation over-predicts the length of recirculating ﬂuid slightly. The corresponding ﬂuctuations yield, although not perfect an encouraging agreement, as can be seen in Fig. 2. Interesting to note, both experimental and former numerical investigations [19–22] have shown a non-symmetric behavior of the isothermal ﬂow ﬁeld, which is also identiﬁed in the current simulation. Abbot and Kline investigated a large series of single and double backward facing steps, where several inﬂuencing variables were investigated independently [33]. They found that an asymmetry of the ﬂow pattern can be observed for expansion ratios bigger then 1.5 which is evidently the case in the current conﬁguration. Another conclusion of their study was that the re-attachment length is less inﬂuenced by the level of ﬂuctuations or the mean upstream velocity proﬁles, but strongly depends on the expansion ratio of the channel.

12.0 8.0 4.0 0.0 -4.0

x = 8D

12.0 8.0 4.0 0.0 -4.0

x = 7D

12.0 8.0 4.0 0.0 -4.0

x = 4D

12.0 8.0 4.0 0.0 -4.0

x = 3D

12.0 8.0 4.0 0.0 -4.0

x = 2D

12.0 8.0 4.0 0.0 -4.0

x = 1D

4.0

361

x = 8D

2.0 0.0 4.0

x = 7D

2.0 0.0 4.0

uax' [m/s]

〈uax〉 [m/s]

Impact of Subgrid Scale Wrinkling Models on the ATF Model in LES

x = 4D

2.0 0.0 4.0

x = 3D

2.0 0.0 4.0

x = 2D

2.0 0.0

Exp LES

4.0

Exp LES

x = 1D

2.0 0.0 -50

-25

0

z [mm]

25

50

-50

-25

0

25

50

z [mm]

Fig. 2. Transverse proﬁles of time averaged stream-wise velocities and ﬂuctuations at diﬀerent axial positions (non-reacting case): comparison of LES results (solid) and experimental data (points)

Two shear layers 35 mm away from the centerline of the combustion chamber are clearly visible in the proﬁles of the velocity ﬂuctuations, together with a mixing layer of lower intensity level at the centerline. Starting the computational domain at the backward facing step, the simulation failed to predict the level of the turbulent ﬂuctuations adequately. 5.2 Reacting Flow To resolve the density jump across the ﬂame front, the product of the thickening factor and the laminar ﬂame thickness must become resolvable with suﬃcient grid points. In the present case the thickened ﬂame front, 1 0 δL = F δL = 20 · 0.2mm ≈ 4mm, is resolved using roughly about ﬁve grid points. To guarantee the ability of the thickened ﬂame to interact with the integral turbulent scales, the thickened ﬂame must be considerably smaller than the integral length scales. Considering the integral scale of the streamwise velocity component to be about 0.4 times the width of the supplying channels, one ends up with a dominant length scale of approximately 12mm. Hence, the ability of large scales, to interact with the thickened ﬂame front is conserved. Therefore the chosen thickening factor can be addressed as a reasonable choice.

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Fig. 3. Transverse proﬁles of time averaged mean stream-wise and transverse velocities, using a constant eﬃciency function E = 1.5 (dotted), using Colin’s Model for u′∆e (dashed-dotted) and using the ﬁltering technique for u′∆e (solid) compared to experimental data (points)

The subgrid scale velocity u′∆e needed to model the subgrid scale wrinkling, was calculated with two diﬀerent approaches, expressed by (11) and (12). These two simulations, together with an additional third calculation, where the eﬃciency function E was set to a constant value of 1.5, which is approximately the average value of E in the entire un-burnt region of the domain, are compared to experimental ﬁndings [13]. To simplify the discussion of the results the following abbreviations will be used: using the model presented by Colin et al [10] for u′∆e is named Ecol, using the ﬁltering technique for u′∆e is named Ef il and using a constant value for the mean eﬃciency function is denoted Econ. Under reacting conditions the average ﬂow ﬁeld becomes much more symmetric, which can be seen in the comparison between the experiments and the numerical simulations, plotted in Figs. 3 and 4. Also the time averaged ﬂame front position is found to be symmetric with respect to the center of

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Fig. 4. Transverse proﬁles of the ﬂuctuations of the stream-wise and the transverse velocities, using a constant eﬃciency function E = 1.5 (dotted), using Colin’s Model for u′∆e (dashed-dotted) and using the ﬁltering technique for u′∆e (solid) compared to experimental data (points)

the channel, in all three simulations, Ecol, Econ and Ef il. In the central region of the channel, where the ﬂuid can be assumed to be generally unburnt, the numerical and experimental data of the mean velocities (Fig. 3) are found to be in very good agreement, but closer to the wall, where the ﬂuid is sometimes burnt and sometimes un-burnt some discrepancies between experimental results and the simulations can be identiﬁed. All three simulations fail to predict the recirculation zone, at x = 1D, independently of the eﬃciency function modeling. Further downstream the channel, the Econ simulation over-predicts the streamwise velocity component, in the same amount as the other two, Ef il and Ecol, under-predict the streamwise velocity of the experimental investigation. For the transverse velocity component the diﬀerence between the simulations is more evident. The large discrepancy at x = 1D, z = −50mm can be explained with the inability to capture the recirculation zone properly. Taking

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into account that adiabatic boundary conditions have been used for the walls of the channel, the ﬂame will always be attached to the edge of the sudden expansion. The experimental data-set gives no further insight if the ﬂame is attached to the wall, or burns slightly lifted. This might explain the disagreement between experimental and numerical results. Comparing the runs with diﬀerent approximations for the eﬃciency functions one can clearly see that Ef il and Ecol behave similarly well, while Econ shows a strong deviation, especially visible at z = ±50mm in the proﬁles of the transverse velocity distribution. The time averaged ﬂuctuations of the stream-wise velocity component, plotted in Fig. 4 seem to be almost identical for all three calculations. Comparing the results to the experimental ﬁndings, yields again relatively good agreement in the central region of the channel, but especially at x = 2D and at x = 8D the absolute level of ﬂuctuations is not predicted properly in the close to wall regions. Beside the fact that the resolution is insuﬃcient to capture the viscous sub-layer of the channel, another possible explanation for the under-prediction is given in the following. It is reported in the literature that under reacting conditions an oscillatory motion can be observed in the combustion chamber. The impact of this oscillations can be identiﬁed up to −5D upstream of the splitter plate. The corresponding acoustic frequency, about 50Hz might result in a temporarily uneven mass ﬂux, approaching the combustion chamber. Clearly, this deviates from the constant incoming mass ﬂux, used in the current simulation. Unfortunately no detailed information about the amplitude of this oscillation can be found in the literature. In the work of Domingo et al the eﬀort was made to include this frequency in the simulation, by adding a 50Hz oscillation to the incoming mass ﬂux [34]. For the ﬂuctuations of the transverse velocity component (Fig. 4) an overall encouraging agreement is found at all stream-wise positions presented. But again, all simulations fail to predict the shear layers at z = ±50 (x = 1D), which might be contributed to the fact that the recirculation zone was not captured precisely. Comparing the three calculations, it can be concluded that the inability of the ﬁltering method to correct for the dilatational part of the SGS-ﬂuctuations is less important in high Reynolds number ﬂows, but remains signiﬁcant for moderate and low Reynolds number ﬂows. Removing the local information of the SGS-velocities and taking the eﬃciency function as constants leads to results, which are less precise, but also end up in a similar level of prediction quality. Although one expects the SGS-ﬂuctuations to be more or less uniform in a channel, apart from the wall, this seems not to be the case under reacting conditions in the investigated conﬁguration. 5.3 Flame Front Wrinkling In Fig. 5 instantaneous snapshots of the ﬂames together with the approximated eﬃciency function mapped on a plane are presented. The top ﬁgure

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Fig. 5. Instantaneous snapshot of the ﬂame front (black) with the level of the eﬃciency function, mapped on a plane (for comparison step height=29.9 mm). Light colors mean small ﬂuctuation level and dark color high ﬂuctuations. A constant eﬃciency function E = 1.5 is used in the upper plot, Colin’s Model for u′∆e in the middle plot and the ﬁltering technique for u′∆e in the lower plot

(Econ), with a constant eﬃciency function of 1.5 in the entire domain, shows a shorter ﬂame compared to the other two ﬁgures (Ecol, Ef il) and additionally the ﬂame is located further from the bottom and top wall of the combustion chamber. Introducing the eﬃciency function locally increases the ﬂame speed from s0L to sT = Es0L which shortens the ﬂame considerably. The ﬂame fronts are clearly attached to the corners of the sudden expansion due to the assumption of adiabatic walls, and show especially in the Ecol and Ef il simulations large scale wrinkles caused by the separation of the ﬂow from the steps. Assuming a large wrinkling at the subgrid scale level, the maximum value for the eﬃciency function can be estimated to be E max = F 2/3 ≈ 7 with the used thickening factor. Locally the turbulent ﬂame speed can thus become signiﬁcantly higher than the laminar ﬂame speed in zones of high subgrid scale velocities u′∆e . The mean eﬃciency function along the ﬂame front is a lot lower than this maximum. From Fig. 5 it can be seen that E is locally high in the shear layers and in the central zone of the combustion chamber due to the splitter plate.

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Comparing the snapshots between the two diﬀerent approximated eﬃciency functions the ﬂuctuations seem to look considerably diﬀerent, but the impact on the ﬂame front seems to be rather weak, which might come from the relatively high Reynolds number of the investigated problem. The statistical behavior of both approximations (Fig. 3 and Fig. 4) with respect to the mean velocity components is almost identical. Downstream the channel the mean streamwise velocities, using Colin’s model for u′∆e , are slightly higher than with the ﬁltering approach. Although locally the eﬃciency function in the Ecol and Ef il simulations is a lot larger than the constant value of 1.5 used in the Econ simulation the ﬂames are considerably longer than in the latter case.

6 Conclusions A code for large eddy simulation of turbulent premixed combustion based on the concept of the artiﬁcially thickened ﬂame has been developed and tested on the ORACLES burner. The test case has been investigated in ﬁrst instance on the inert case. The results demonstrate good agreement with the experimental data [13]. To evaluate several ways of modeling the subgrid scale velocity u′∆e for the calculation of the eﬃciency function, a systematic comparison in three dimensional LES calculations is performed. The results show that these approaches have a relatively small impact on the behavior of the ﬂame front, i.e. the ﬁltering approach and the approach given in (12) behave equally well. But it must be noted that in terms of computational costs, the LES using the ﬁltering approach, was about 30% less expensive. Results, using an average eﬃciency function of constant value, showed that the local level of ﬂuctuations is required to gain better results, even though this speciﬁc simulation was also in relatively good agreement to the experimental data. In contrast to DNS investigations done at lower Reynolds numbers it can be concluded that for relatively high Reynolds numbers, the absolute level of the ﬂuctuation is the dominant factor towards ﬂame wrinkling and ﬂame length.

Acknowledgments The authors gratefully acknowledge the ﬁnancial support by the Flemish Research Foundation (FWO-Flanders) and Deutsche Forschungs Gesellschaft (DFG) SFB 568 (project B3 and D3).

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References [1] J. Janicka and A. Sadiki. Large eddy simulation of turbulent combustion systems. In 30th Int. Symp. on Combustion, pages 537–548, Chicago, 2004. [2] D. Veynante. Large eddy simulation for turbulent combustion. In Proc. European Combustion Meeting, pages 311–332, Louvain-la-Neuve, 2005. [3] T. Poinsot and D. Veynante. Theoretical and Numerical Combustion. Edwards, 2001. [4] L. Selle, G. Lartigue, T. Poinsot, R. Koch, K.U. Schildmacher, W. Krebs, B. Prade, P. Kaufmann, and D. Veynante. Compressible large-eddy simulation of turbulent combustion in complex geometry on unstructured meshes. Combustion and Flame, 137:489–505, 2004. [5] P. Moin. Advances in large eddy simulation methodology for complex ﬂows. Heat and Fluid Flow, 23:710–720, 2002. [6] P. Moin and S. Apte. Large-eddy simulation of realistic gas turbines combustors. In 42nd AIAA Aerospace Sciences Meeting 2004-0330, pages 1–10, Reno, 2004. [7] S. Menon, C. Stone, and N. Patel. Multi scale modeling for LES of engineering designs of large-scale combustors. In 42nd AIAA Aerospace Sciences Meeting 2004-0157, pages 1–15, Reno, 2004. [8] N. Peters. Turbulent Combustion. Cambridge University Press, 2000. [9] M. Boger, D. Veynante, H. Boughanem, and A. Trouve. Direct numerical simulation analysis of ﬂame surface density concept for large eddy simulation of turbulent premixed combustion. In 27th Symposium on Combustion, pages 917–925, 1998. [10] O. Colin, F. Ducros, D. Veynante, and T. Poinsot. A thickened ﬂame model for large eddy simulations of turbulent premixed combustion. Physics of Fluids, 12(7):1843–1863, 2000. [11] C. Nottin, R. Knikker, M. Boger, and D. Veynante. Large eddy simulation of an acoustically excited turbulent premixed ﬂame. In 28th Int. Symp. on Combustion, pages 67–73, Edinburgh, 2000. [12] T.D. Buttler and P.J. O’Rourke. A numerical method for two dimensional unsteady reacting ﬂows. In 16th Int. Symp. on Combustion, pages 1503– 1515, 1977. [13] M. Besson, P. Bruel, L. Champion, and B. Deshaies. Experimental analysis of combustion ﬂows developing over a plane-symmetric expansion. Journal of Thermophysics and Heat Transfer, 14(1):59–68, 2000. [14] F. Charlette, C. Meneveau, and D. Veynante. A power-law ﬂame wrinkling model for LES of premixed turbulent combustion part 1: Nondynamic formulation and initial tests. Combustion and Flame, 131:155– 180, 2002. [15] P. Bruel and P.D. Nguyen. Generic 2-d combustor: Determination of lean extinction limits (1/2). WP3 6M, Modeling of Low Emmisions Combus-

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[29] M. D¨ using, A. Sadiki, and J. Janicka. Towards a classiﬁcation of models for the numerical simulation of premixed combustion based on a generalised regime diagram. Combust. Theory Modeling, 10(1):105–132, 2006. [30] PHOENICS. www.cham.co.uk/phoenics/d polis/d lecs/numerics /scheme.htm, 2002. [31] E.M. Saiki and S. Biringen. Numerical simulation of a cylinder in uniform ﬂow: Application of a virtual boundary method. J. Comp. Phys, 123(2):450–465, 1996. [32] M. Klein, A. Sadiki, and J. Janicka. A digital ﬁlter based generation of inﬂow data for spatially developing direct numerical or large eddy simulations. J. Comp. Physics, 186:652–665, 2003. [33] D.E. Abbot and S.J. Kline. Experimental investigation of a subsonic turbulent ﬂow over single and double backward-facing steps. Journal of basic engineering, 84:317–325, 1962. [34] P. Domingo, L. Vervish, S. Payet, and R. Hauguel. DNS of a premixed turbulent V ﬂame and LES of a ducted ﬂame using a FSD-PDF subgrid scale closure with FPI-tabulated chemistry. Combustion and Flame, 143(4):566–586, 2005.

Analysis of Premixed Turbulent Spherical Flame Kernels Rob J.M. Bastiaans, Joost A.M. de Swart, Jeroen A. van Oijen, and L. Philip H. de Goey Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands [email protected] Summary. The present study discusses results from a number of DNS simulations of turbulent ﬂame kernels. The description of the chemistry in these calculations was based on the Flamelet Generated Manifolds (FGM) technique. The diﬀerences are imposed by varying the turbulence intensity and length scale within the thin reaction zones regime. This results in changes in the ﬂame-turbulence interaction. The goal of the study is to see if the presently used reduced chemistry is able to properly deal with the turbulent modulations deﬁned by stretch and curvature of the local ﬂamelets. Especially the inﬂuences of the given turbulence eﬀects to the local mass burning rate is investigated. Also global ﬂame dynamics are described and an interpretation of the latter is given in terms of local quantities.

1 Introduction Combustion in lean premixed gas-turbines predominantly takes place in the ﬂamelet regime and in the thin-reaction zones combustion regime. In the past a lot of so-called ﬂamelet models have been developed for premixed combustion processes in the ﬂamelet regime. The ﬁrst and most well-known models are derived by assuming that the ﬂame-front is inﬁnitely thin. Flame stretch is an important parameter that is recognized to have a determining eﬀect on the burning velocity in premixed ﬂames. Up to now this eﬀect has not been taken into account in the ﬂamelet approach for turbulent combustion in a satisfying way. The laminar burning velocity, which is largely aﬀected by stretch, is an important parameter for modeling turbulent combustion. In the turbulent case, stretch rates vary signiﬁcantly in space and time. An expression for the stretch rate is derived directly from its mass-based deﬁnition as in [1]. On the basis of this deﬁnition a model for the inﬂuence of stretch and curvature on the mass burning rate has been developed. The predictive power of this theory will be evaluated in cases of turbulent combustion in the thin reaction zones regime. In this regime it is found by [2] that the

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preheat is modulated by turbulence but that the reaction zones stay intact and therefore it is considered to be part of the class of ﬂamelet regimes. The study of statistically spherical ﬂame kernels is quite attractive from a physical point of view. There are no complex interactions with walls and these ﬂames can be investigated experimentally as well. In fact the ignition phase of premixed combustion in an Otto engine is a good example of these kind of structures in practice and this could on itself already be a very good reason for studying turbulent ﬂame kernels. From a numerical viewpoint, the setup of a ﬂame kernel is relatively straight forward. There are no walls and due to the expansion of the kernel itself, all boundaries are considered as outﬂow boundaries. The initial turbulence is decaying, as is the case in many practical combustion applications (discontinuous-ﬂux devices, but not in continuousﬂux applications like in jet engines). A disadvantage of studying ﬂame kernels is that only short time integrations can be performed since ﬂuctuations appear fast and reach the boundaries of the domain early. Especially if the ﬂuctuations and turbulent length scale are large. Because of the advantage of the simplicity of boundary conditions while maintaining technical meaning we chose to perform the study of ﬂame kernels, later we will consider the study of statistically ﬂat ﬂames. Nevertheless, with the present simulations we ﬁnd that we are able to reach the edge of the ﬂame regime for which the present method works, which is located quite deep into the thin reaction zones regime. As a consequence of the previously mentioned facts many researchers have used ﬂame kernels for the study of premixed turbulent combustion. Of particular interest is the research in [3–5]. The last study is associated with rich ﬂames, which are not addressed here. In [3] the evolution of shape parameters were studied in terms of ﬂame normals and curvatures by means of DNS combined with single step chemistry. One of their conclusions is that at low turbulence intensities there is a tendency to favor spherical over cylindrical curvature. In [4], the numerical simulations of [3] were extended and supplemented by experimental PLIF observations for both methane and hydrogen combustion at stoichiometric and lean conditions. The result of the study is a qualitatively good agreement between the simulations and experiments. The goal of the present study is to see if a ﬂamelet method is able to qualitatively predict a correct turbulence ﬂame interactions. Another study, which also gives an overview of many related papers, is given in [6]. In this study k − ǫ simulations have been performed and compared to measurements in terms of ﬂame speed. Here the turbulent length scale is taken as an important parameter and relatively long time scales are considered, starting at a time where the present simulations have already ﬁnished.

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2 Numerical Approach and Physical Conditions In the present study simulations will be analyzed with respect to subgrid scale and RANS-model analysis of the turbulent combustion ﬁeld. Fully compressible DNS simulations are carried out using FGM, based on a single progress variable, for a turbulent spherical premixed ﬂame on a 2543 grid in a 1.23 cm3 box. Details of the code are described in [7]. The fuel consists of methane, premixed with air at an equivalence ratio of 0.7. An initially laminar ﬂame kernel was obtained from a time dependent, 1D spherical calculation with detailed chemistry (with the code of [8]) up to an inner layer radius of r = 2.9 mm. This laminar ﬂame kernel was subjected to a turbulent ﬂow environment. The turbulence was generated by prescribing random numbers to a stream function. This stream function was spatially ﬁltered, converted to a velocity ﬁeld and rescaled to obtain the proper length and velocity scales for an incompressible ﬂow. This procedure gives a quite realistic turbulent ﬂow and it was found in non-combusting cases of decaying isotropic turbulence that it relaxes quite fast to simulations with an initial realistic spectrum as in [9]. The relative turbulence intensities are u′ /sL = 4 and u′ /sL = 8 and the the length scale ratio was varied from lT /δf = 1.87 to lT /δf = 3.74. Here the relevant turbulent length scale is deﬁned as the longitudinal Taylor scale λ. In table 2 the considered cases are given. Table 1. Physical properties corresponding to the diﬀerent simulations. Case u′ /sL lT /δf Re Da Ka C1 C2 C3

4.0 8.0 8.0

1.87 7.5 0.47 5.8 1.87 15.0 0.23 16.5 3.74 30.0 0.47 11.7

The chemistry is chosen due to the large interest in the power industry in lean premixed combustion engines and there is detailed knowledge of its chemical kinetics. Therefore premixed combustion of a methane/air mixture is used, with an equivalence ratio of φ = 0.7 for all cases. Moreover preferential diﬀusion of all the chemical species i is not taken into account, or Lei = 1. In principle this is not a restriction of the method, but it will simplify the analysis considerably, since turbulent stretch and curvature eﬀects will not be obscured by it. Starting conditions are assumed to be atmospheric, i.e. a pressure of 1 atm and temperature of 300 K. With the speciﬁcation of these conditions and specifying a kinetic scheme for the chemistry the burning velocity of a ﬂat unstretched ﬂame with respect to the unburnt mixture, sL , is a ﬁxed physical quantity. In the present study the chemical kinetics is based on the GRI3.0 scheme [10]. Then the numerical value of the laminar burning velocity is given

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by s0L = 18.75 cm/s and the corresponding mass burning rate is m0 = 0.213 kg/m2 s. It must be remarked that the simulations are direct in the sense that the smallest scales of motion are fully resolved, while the chemical kinetics are solved in advance and parameterized in a table by the method of the ﬂamelet generated manifolds (FGM), [11]. The state of the reactions are assumed to be directly linked to a set of progress variables. The conservation equations for these progress variables are solved using DNS, with the unclosed terms coming from the table. This allows the use of detailed chemical kinetics without having to solve the individual species conservation equations. Here we will apply only a single progress variable, which is of course a rather crude approach. The use of more progress variables will increase the accuracy of the method. However it will be shown that by using one progress variable already most of the interesting ﬂame turbulence interactions are resolved quite satisfactory. One could argue that using a single progress variable is essentially the same as using one step chemistry. However in the present case detailed kinetic schemes can be used without the need for tuning of the rate parameters. The reaction rate and other lookup variables are functionally related to detailed and distributed chemical events through the reaction zone. Furthermore an arbitrary number of progress variables can be added up to a description at which suﬃcient accuracy is attained. As a ﬁrst prerequisite for the simulations, but also for the subsequent analysis, the ﬂame should be well resolved, which is the case with the chosen parameters (grid size of 47.5 µm, which is a little bit more then 10 points through the ﬂame). Moreover the eﬀects of ﬂame stretch and curvature on the local and turbulent burning velocity should be well recovered with the present FGM. In a previous study, [12], it was found that this is the case, good correlations were found with the model based on [1]. Also back-substitution of strain in a detailed chemistry calculations by extracting ﬂamelets from the DNS, gives consistent results, [13]. Moreover, this study revealed that the addition of an extra dimension by using two progress variables is shown to give a very good convergence. The mass fraction of carbon dioxide, which is monotonously increasing, burnt . Since pressure, enis used as single controlling variable Ycv = YCO2 /YCO 2 thalpy and element mass fractions are constant in the present ﬂames, they are not needed as additional controlling variables. When FGM is applied, transport equations for all species mass fractions do not have to be solved. Instead a diﬀerential equation is solved for the controlling variable only. Since the reaction layer for this ”slowly changing” variable is thicker than for radicals such as CH, a relatively coarse grid is suﬃcient to resolve the structure of the ﬂame. The entire set of equations that are solved are given in [12]. Also initial and boundary conditions are treated in more detail.

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Fig. 1. Surface area of the inner layer value of the progress variable as function of time. Left: averaged surfaces (lower 3 lines) and turbulent surfaces (upper 3 lines) for case C1, drawn, C2, dashed and C3, dash-dotted. Right: actual turbulent surface divided by the averaged surface, same line types. Thick lines denote mean values, thin lines are associated with ﬂuctuations.

3 Results From the cases considered, ﬁrst the ﬂame surface evolution was analyzed. For the ﬂame surface we take the so called ”inner layer” of the ﬂame, which is the position of maximum heat release. This position is related to the progress variable (locally monotonically increasing in space) for the laminar case on which the ﬂamelet approach is based. Thus this value can be computed in advance. For the present lean methane air ﬂame without preferential diﬀusion il = 0.605. Furthermore a distinction is made between the averaged it is Ycv ﬂame surface and the turbulent ﬂame surface. The latter is just the actual surface deﬁned by the inner layer progress variable and it is evaluated by a method described in [14]. The averaged ﬂame surface is deﬁned as the mean radius corresponding to all observations of the progress variable that have a value within a range of il il < Ycv < 1.02Ycv . Here the radius is corrected for a movement of the 0.98Ycv entire ﬂame kernel by subtracting the center of mass of the progress variable itself. Note that, since the progress variable is deﬁned to be the normalized mass fraction of carbon dioxide, this gives the mass weighted center of the burnt region. With the given deﬁnition of the averaged surface it can be interpreted as an ensemble average and thus a RANS of a spherical ﬂame, which of course is a one-dimensional phenomenon. Note however that the ensemble is not obtained by independent observations, depending on the case as deﬁned in table 2 there is more or less correlation or coherence. Results of the surface evaluation as function of time are given in ﬁgure 1. The end of the lines are deﬁned by the end of the simulations which are deﬁned by the moment at which at some point the ﬂame structure reaches

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Fig. 2. Contour plots of the progress variable in a central cross section, left to right: cases C1, C2, C3 respectively, top: time T1, bottom: time T2. The values of the contours are given by, drawn: 0.2, 0.3, 0.4, 0.5, 0.7, 0.8, 0.9; dashed: 0.605 (inner layer).

the boundaries of the simulation domain. From the ﬁgure it can be observed clearly that the increase of turbulent ﬂame surface area depends on lT /u′ . The cases C1 and C3 have the same value of the turbulent time scale, lT /u′ , and at least the initial part of the surface increase is similar. In case C2 the turbulent time scale is two times as large and consequently in the beginning the surface increases two times as fast. This all would indicate a more or less passive wrinkling of the surface by the turbulent velocity ﬁeld. However there is also an increase in the averaged ﬂame surface. This should be attributed to two eﬀects, being the reactive consumption of mass and the expansion of the hot ﬂame kernel. Furthermore it can be observed that the mean surface increase is almost independent of the turbulent parameters. It must be recognized that turbulent scales are relatively fast compared to the reactive front propagation, which is obvious from the speciﬁcation of the ratio of turbulent velocity to the burning velocity. However with higher ratios the most interesting region is entered, especially also for assessing the performance of the ﬂamelet method. For higher velocity ratios the combustion regime changes gradually from a ﬂamelet type of reaction zones to a distributed type of chemical conversion. It is at this edge where the ﬂamelet approach is presumed to lose its validity. This will be given attention to in ﬂamelet analysis, later on in the paper. The evolution of the ﬂame surface at later times is not so easily explained. Case C1 behaves not very special but the other cases do. In case C2 at a certain point in time the turbulent surface increase drops, which causes the ratio of turbulent and averaged surface to tend to an almost constant value. For case

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C3 ﬁrst a more or less sudden increase in turbulent surface can be observed after which, at about t = 0.5ms a change is observed in both turbulent and averaged surface causing a sharp transient in the ratio. In order to investigate these changes we will have a look at the obtained turbulent ﬂame structures at two instants in time, T1 : t = 0.3ms and T2 : t = 0.5ms. These times correspond to the time at which, for T1 , case C1 and C3 have still a common result in terms of ﬂame surface, quite diﬀerent from C2, and for T2 which marks the point just after the beginning of the transient in C3 and it coincides with the end of simulation C2. 0.045 0.04 0.035

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In ﬁgure 2 contour plots are given for the progress variable at the given two incidents in time. The ﬁgure shows a more or less expected behavior. From the beginning the ﬂame structure starts to deform from an initially perfect sphere. This progresses in time as can be observed from the plots. Additionally it can be seen that in case C2 the layer at the unburnt side of the inner layer, also known as the preheat zone, is disrupted. Here the preheat zone deviates from a neatly layered structure. This is caused by the more vigorous turbulence in combination with having turbulent kinetic energy in relative small length scales; it is not observed anymore in case C3. Here, with case C2, we have clearly entered the ”thin reaction zones” regime in which the preheat zone is perturbed but the reaction layer is still intact. Note that the initial generation of turbulent ﬂuctuations was the same for all three cases; case C2 shows more ampliﬁed deviations and case C3 shows low pass equivalents of the phenomena in cases C1 and C2. Moreover from inspecting the plots from case C3 it can be observed that large coherent bulges and crevices appear. At time T2 the bulge at the left side gets chemically converted, which has a big impact on the amount of the turbulent surface. This has to be considered in view of the relatively low

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amount of turbulent length scales that ﬁt on the sphere, which has the same size for all the cases. Therefore it can be concluded that there is a lack of statistics associated with case C3. This shows itself also in the movement of the ﬂame structure as a whole. In ﬁgure 3 the change of the center of mass of the progress variable is displayed as function of time. Clearly the entire ﬂame kernel of case C3 starts to move around. This is much less the case for C1 and C2, the fact that the amount of C2 is twice that of C1 is obvious. −3

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In ﬁgure 4 some more statistics of the position of the inner layer and the mean inner layer ﬂuctuations of the progress variable are given. The mean inner layer radius is also presented in ﬁgure 1, converted to an averaged surface, the ﬂuctuations in radius can also be associated with the increase in turbulent surface, though much less pronounced. Given the mean inner layer position as function of time, at each incident the statistics of the progress variable is given in the right side of ﬁgure 4. Thus these are statistics for all observations for which holds that 0.98ril < r < 1.02ril . For case C3 the ﬂuctuations of the progress variable near the inner layer are largest at later times. However still a ﬂamelet behavior is expected since this is non-local statistics. The Karlovitz-number is an indicator for ﬂamelet behavior to be valid or not, [15]. It is given by Ka =5.8, 16.5, 11.7 for the cases C1, C2 and C3 respectively. For higher Karlovitz numbers the combustion starts to deviate from a ﬂamelet. Therefore it is expected that for case C2, the ﬂamelet generated manifolds method would be worst. Also from the contour plots it looks like the structure is most disrupted for case C2. In the following section the ﬂamelet behavior is further investigated.

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4 Flamelet Analysis In this section an analysis of ﬂamelet behavior is given based on the theory [1]. Based on a ﬂamelet concept and integral analysis ([17]) an expression for the eﬀect of stretch and curvature on the mass burning rate can be derived. For the special case of unity Lewis numbers, applied here, the parts related to preferential diﬀusion in the expression for the mass burning rate at the burnt side of the ﬂame mb vanish. Later this expression was improved, [16], by relating it to the inner layer, yielding mil = 1 − Kail , m0il

(1)

with the inner layer Karlovitz integral s  b sb 1  σρKY ds − σρKds Kail := σil m0il su

(2)

sil

with K the ﬂame stretch. The integrals have to be taken over paths normal to the ﬂame and su , sb and sil are the positions at the unburned side, the burned side and the inner layer, respectively. The ﬂame surface area, σ, is related to the ﬂame curvature, κ, which is related to the ﬂame normals, ni on the basis of the progress variable, Y , κ=

∂ni 1 ∂σ . =− ∂xi σ ∂s

(3)

The local mass burning rate, m = ρsL , can be evaluated in postprocessing by following ﬂamelets, tracks in normal direction to the progress variable. Thus the local burning velocity sL , ∂ λ ∂Y + ρ ˙ ∂xi Le¯ c ∂x p i , (4) sL = ∂Y ∂xi

can be evaluated based on a consequence of the combination of the conservation equation for Y with the kinematic equation for Y . More details on the ﬂamelet tracking can be found in [13]. In the ﬂamelet analysis the result of the actual burning velocity, equation 4, will be confronted to the model, equation 1. This was done before, [12, 13, 18– 20], and proven to be successful. Of course this depends on the assumptions in the model compared to the actual phenomena occurring in the simulations. A loss of correlation, therefore, does not imply a priori falsiﬁcation of the model on the one hand or poor numerics or physics in the simulations on the other hand. On the basis of the earlier results, this option is proven to be false. A mismatch between observations from simulations compared to

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model prediction is therefore caused by phenomena in the simulations that do no longer belong to the ﬂamelet regime. With this in mind, the test is a validation method for the ﬂamelet generated methods (FGM) technique for dealing with the chemical kinetics, since this technique is based as well on ﬂamelet assumptions. The ﬂamelet assumptions, connected to locally one-dimensional ﬂames, neatly layered without large changes along the ﬂame, will be violated at some point if the length scales of turbulent ﬂuctuations decrease. At that point these ﬂuctuations are able to penetrate the ﬂame front. To the opinion of the authors the ﬂamelet assumptions need only to be valid near the inner layer since all the chemical source terms are concentrated in this region. This extends the ﬂamelet regime in which FGM can be applied to the thin reaction zones regime by deﬁnition. These regimes are indicated by the Karlovitz number, Ka, for Ka > 1 the thin reaction zones regime is already entered. This is the case for all three simulations as given by the estimations of Ka at the end of the previous section. It must be remarked that the present Karlovitz numbers are based on a Taylor scale whereas in the standard deﬁnition the integral turbulent length scale is used. In the present simulations, however, a considerable part of the numerical resolving power is invested in the chemistry resulting in a limited turbulent spectrum. This causes the diﬀerence in Ka deﬁnitions to be negligible. In ﬁgure 5 correlations are given for actual mass burning rates and the model, again at the two times, T1 and T2. The ﬁgure shows correlations in qualitative agreement with expectations from both the contour plots of ﬁgure 2 and the Ka numbers. Case C2 gives a much wider scatter, i.e. local deviations from the model, then cases C1 and C3, which look almost the same. On the one hand this is peculiar since Ka changes gradually, on the other hand it is consistent with the contour plots in which case C2 gives a clear indication of turbulent mixing in the preheat zone, in contrast to both case C1 and C2. Looking at the diﬀerence between the two incidents in time it seems that the mass burning rate increases. This is consistent with the notion of the decaying turbulence, resulting in contraction of the distributions of curvature and stretch to their mean values. The present evaluations of the local mass burning rates show trends and it appears that for all the cases considered the ﬂamelet assumptions are not really violated so that the simulations are reasonably valid. Numerical errors in the evaluation of equations 4 (with singular behavior at non-interesting points) and 2 should be assessed in order to obtain good ﬂamelet statistics. At the moment this prevents us from giving statistics on the local mass burning rates. From the results in this section and the previous section the question arises how the statistics of fuel conversion (of ﬁgure 5) and local turbulent ﬂame surface (ﬁgure 1) determines the total fuel conversion and how this aﬀects the total displacement (ﬁgure 1 and 4) and how to discern between turbulent consumption eﬀects and total expansion. The research will continue

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Analysis of Premixed Turbulent Spherical Flame Kernels

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Fig. 5. Scatter plots of actual mass burning rates versus model values. Top to bottom: C1, C2 and C3 respectively, left: time T1, right: time T2. Results are obtained by entering the ﬂame zones at regular intervals, to cover a representative set of realizations, resulting in 3500 ﬂamelets, typically.

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to answer these questions, especially in a framework of LES modelling in the context of coherent ﬂamelet models in which the local ﬂame surface density is a key parameter. In this way the advantages of FGM can play an important role.

5 Conclusions Turbulent spherical ﬂame kernels under diﬀerent turbulent conditions have been simulated with a DNS-FGM method. It appears that initially ﬂame surface generation is governed by lT /u′ . However no real increase in averaged ﬂame expansion can be detected. Detailed inspection shows that in one case the preheat zone is clearly penetrated by turbulence. A ﬂamelet analysis shows that for this case the correlation with a model for the mass burning rate based on ﬂamelet assumptions drops. However the drop is only moderate. With the present study only short time integrations can be performed since ﬂuctuations appear fast and with large turbulent length scales they reach the boundaries of the domain early. However, simulations were performed in the thin reaction zones up to a level where eddies penetrate clearly in the preheat zone and it is shown that even at this point the method still correlates well with the model. For further research the study of statistically ﬂat ﬂames will be considered.

Acknowledgments The authors would like to thank the Dutch Technology Foundation (STW) under grant no. EWO.5874 and the support of NCF, the Netherlands Computing Facilities Foundation.

References [1] L.P.H. de Goey and J.H.M. ten Thije Boonkkamp. A Flamelet description of premixed laminar ﬂames and the relation with ﬂame stretch. Combust. Flame, 119:253–271, 1999. [2] L.P.H. de Goey, T. Plessing, R.T.E. Hermanns, and N. Peters. Analysis of the ﬂame thickness of turbulent ﬂamelets in the thin reaction zones regime. Proc. Combust. Inst., 30:859–866, 2005. [3] K.W. Jenkins and R.S. Cant. Curvature eﬀects on ﬂame kernels in a turbulent environment. Proc. Comb. Inst., 29:2023–2029, 2002. [4] S. Gashi, J. Hult, K.W. Jenkins, N. Chakraborty, R.S. Cant, and C.F. Kaminski. Curvature and wrinkling of premixed ﬂame kernelscomparisons of OH PLIF and DNS data. Proc. Combust. Inst., 30:809– 817, 2003.

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[5] D. Th´evenin. Three-Dimensional direct simulation and structure of expanding turbulent methane ﬂames. Proc. Combust. Inst., 30:2005. [6] A.N. Lipatnikov and J. Chomiak. Transient geometrical eﬀects in turbulent ﬂames. Comb. Sci. Techn., 154:75–117, 2000. [7] R.J.M. Bastiaans, L.M.T. Somers, and H.C. de Lange. DNS of nonpremixed combustion in a compressible mixing layer, In: Geurts, B.J. (ed) Modern Simulation Strategies for Turbulent Flow, 247–262. R.T. Edwards Publishers, Philadelphia, USA, Book Chapter ISBN 1-93021704-8, 2001. [8] CHEM1D. A one-dimensional laminar ﬂame code. Eindhoven University of Technology. http://www.combustion.tue.nl/chem1d/ [9] J. Meyers, B.J. Geurts, and M. Baelmans. Database analysis of errors in large-eddy simulations. Phys. Fluids, 15(9):2740–2755, 2003. [10] G.P. Smith , D.M. Golden, M. Frenklach, N.W. Moriarty, B. Eiteneer, M. Goldenberg, C.T. Bowman, R.K. Hanson, S. Song, W.C. Gardiner Jr., V.V. Lissianski, and Z. Qin. http://www.me.berkeley.edu/gri mech/ [11] J.A. van Oijen. Flamelet-generated manifolds: development and application to premixed laminar ﬂames. Ph.D. thesis, Eindhoven University of Technology, The Netherlands, 2002. [12] R.J.M. Bastiaans, J.A. van Oijen, S.M. Martin, L.P.H. de Goey, and H. Pitsch. DNS of lean premixed turbulent spherical ﬂames with a ﬂamelet generated manifold, CTR Annual Research Briefs, 257–268, 2004. [13] J.A. van Oijen, R.J.M. Bastiaans, G.R.A. Groot, and L.P.H. de Goey. Direct numerical simulations of premixed turbulent ﬂames with reduced chemistry: Validation and ﬂamelet analysis. Flow, Turbulence and Combustion, 75:67–84, 2005. [14] B.J. Geurts. Mixing eﬃciency in turbulent shear layers. J. Turbulence, 2(1):17, 2001. [15] N. Peters. Turbulent Combustion. Cambridge University Press, 2000. [16] G.R.A. Groot and L.P.H. de Goey. A computational study on propagating spherical and cylindrical premixed ﬂames. Proc. Combust. Inst., 29:1445–1451, 2002. [17] S.H. Chung and C.K. Law. An Integral Analysis of the structure and propagation of stretched premixed ﬂames. Combust. Flame, 72:325–336, 1988. [18] R.J.M. Bastiaans, S.M. Martin, H. Pitsch, J.A. van Oijen, and L.P.H. de Goey. Flamelet analysis of turbulent combustion. Lecture notes in computer science, 3516:64–71, 2005. [19] J.A. van Oijen, G.R.A. Groot, R.J.M. Bastiaans, and L.P.H. de Goey. A ﬂamelet analysis of the burning velocity of premixed turbulent expanding ﬂames. Proc. Combust. Inst., 30:657–664, 2005. [20] J.A. van Oijen, R.J.M. Bastiaans, and L.P.H. de Goey. Flamelet analysis of direct numerical simulations of premixed turbulent ﬂames. ECCOMAS, Lisbon, 2005.

Large Eddy Simulation of a Turbulent Ethylene/Air Diﬀusion Flame D. Cecere1 , G. Gaudiuso2 , A. D’Anna1 , and R. Verzicco2 1

2

Department of Chemical Engineering Universit` a degli Studi di Napoli Federico II P.le Tecchio 80, 80125 Napoli, Italia [email protected] DIMeG and CEMeC, Politecnico di Bari, Via Re David 200, 70125 Bari, Italia [email protected]

Summary. As combustion generated nano-organic particles (NOC) may pose signiﬁcant health and environmental problems, there is great scientiﬁc interest in studying their formation and evolution in turbulent combustion systems. Traditional approaches to turbulent combustion numerical modeling apply Reynolds averaging techniques (RANS) to predict the behavior of the mean values of the reacting ﬂow properties. In this way, unsteady eﬀects are not taken into account in the formation of nanoparticles. Large Eddy Simulation represents an attractive methodology for studying turbulent reacting ﬂows and this approach is becoming possible as computational resources are increasing. A LES approach involves the direct numerical resolution of the large turbulence scales while the small ones and their interaction with the large-scale ﬂow is modeled. The chemistry model used here is based on the mixture fraction transport so that turbulent combustion is modeled as a simple laminar diﬀusion reactor in an unsteady straining environment created by turbulent advection. In ﬂamelet models the explicit dependence on velocity is removed, from the species concentration transport equation, by relating scalars to the mixture fraction which, in turn, is related to the velocity ﬁeld. A reaction progress variable is introduced to take into account the transition from premixed combustion occurring near the ﬂame base to non-premixed combustion occurring downstream in the jet ﬂame. In the present study, LES of a turbulent ethylene jet diﬀusion ﬂame are performed with the aim of computing the formation of nano-organic particles; the results are then compared with an ad hoc experiment.

1 Introduction Combustion processes are essential for power generation, since an overwhelming majority of energy-producing devices rely on the combustion of fossil or renewable fuels. In most applications, the ﬂow ﬁeld in which chemical reactions take place and release heat is turbulent. Due to a growing awareness

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concerning combustion related pollutant emissions and global warming, there is an increasing need towards optimization of these systems. Owing to the high cost of experimental testing and prototyping, numerical simulations become more and more important in the investigation of turbulent combustion. Traditional Reynolds-Averaged approaches are presently the only ones really suited for the simulation of industrial applications, but on the other hand, using RANS simulations only the mean ﬁelds are computed, and this allows only the identiﬁcation of the major trends. Large Eddy Simulation, in contrast, can be seen as intermediate between Direct Numerical Simulation (DNS) and RANS, and it represents a powerful and promising method for studying combustion processes. In LES only the larger ﬂow structures (typically those larger than computational mesh size) are computed, whereas the eﬀect of smaller structures is modeled. Large structures in turbulent ﬂows generally depend on the geometry of the system, while smaller scales feature more universal properties. So turbulence models may be more eﬃcient when they have to describe only the smallest structures. Turbulent mixing controls most of the global ﬂame properties. A key role in the modeling of turbulent combustion systems is played by the interaction of turbulent structures and chemical reactions. In LES, unsteady large scale mixing (between fuel and oxidizer in non-premixed burners) is simulated, instead of being averaged, and instantaneous fresh and burnt gases zones, with diﬀerent turbulence characteristics, are clearly identiﬁed. One method for modelling turbulent combustion with LES is a simple ﬂamelet conserved scalar approach, since it provides a reasonable compromise between implementation costs and accuracy. When chemical reactions are conﬁned to very thin laminar ﬂame-like regions, small compared to the scale of turbulence, and chemical time scales are short compared with the smallest convection scales, it is possible to deﬁne a ﬂame structure that is convected by the ﬂow structures[1]. In this case a single strictly conserved scalar, the mixture fraction Z, is suﬃcient to describe the thermo-chemical state of the ﬂow and to decouple the modeling of turbulence from that of reactive phenomena. The functional dependence of composition, temperature and density on Z can be obtained from a priori laminar ﬂamelet calculations. Unfortunately the chemical time-scales of pollutant formation such as soot and NOx are comparable or even much larger than turbulence time-scales in practical combustion systems so that a simple steady ﬂamelet approach cannot be used for the modeling of such pollutants. Measurements and modeling performed in rich-premixed ﬂames across the soot threshold limit demonstrated that soot precursor particles, named Nanosized Organic Carbon (NOC), have a fast formation rate even comparable with the rate of oxidation and hence suitable to be modeled with a ﬂamelet assumption [2]. Moreover, most of the particulate formed in rich combustion, including PAH and NOC, seems to be originated by the fast rearrangement of aromatic radicals more than by the slow process of mass addiction from the gas phase. It means that prediction of gas phase aromatics corresponds to the prediction of nanosized soot precursor. In this paper, a detailed kinetic mechanism previously developed and

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tested in laminar diﬀusion ﬂame conditions is introduced and coupled with the turbulent ﬂamelet-progress variable approach, to model the formation of aromatics in a turbulent jet diﬀusion ﬂame of ethylene-air, and then compared with experimental results in terms of mean NOC volume fraction proﬁle.

2 Large Eddy Simulation The simulation is performed with an upgraded version of the CDP code, developed at Stanford University[3], in which, in addition to mass and momentum equations, scalar transport equations for mixture fraction and progress variable are solved. The enthalpy equation is here introduced to take into account heat losses; this allows better description of temperature proﬁles. 2.1 Mathematical Formulation In LES each ﬁeld variable is decomposed into a resolved and a subgrid-scale part. In this work, the ﬁltering operation is implicitly deﬁned by the computational mesh used for the large scale equations. Quantities per unit volume are treated using a Reynolds decomposition, while a Favre (density weighted) decomposition is used to describe quantities per unit mass. The instantaneous small-scale ﬂuctuations are removed by the ﬁlter, and they have to be modeled with a so called sub-grid model, while their statistical eﬀects remain in unclosed terms representing the inﬂuence of the subgrid scales on the resolved ones. The LES equations are formally derived substituting each variable appearing in the variable–density low–Mach–number form of the Navier-Stokes equations with its ﬁltered value, so that the following set of transport equations for mass, momentum and energy results: uj ) ∂ρ ∂(ρ + =0 ∂t ∂xj ∂τui uj ∂(ρ uj ) ∂(ρ ui u j + pδij ) ∂ τij + = + ∂t ∂xi ∂xi ∂xi ∂ 2 ∂ uk ui ∂ uj − µ τij = µ + δij ∂xj ∂xi 3 ∂xk

(1) (2)

∂(ρ hT ) ∂(ρ ui hT ) ∂p ∂τui h ∂τui uk uk ∂(uk τik ) ∂q + + = + + − i − ψr (3) ∂t ∂xi ∂t ∂xi ∂xi ∂xi ∂xi q i = −k

∂ T ∂xi

where ρ is the mixture density, u is the velocity vector, p is the pressure and hT is the total enthalpy, including the resolved sensible enthalpy hS , chemical and kinetic energy, and ψr is the net radiative heat ﬂux, related to

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CO, CH4 , H2 O and CO2 species[4]. The molecular viscosity of the mixture, µ, is assumed to be a mean of the species viscosities weighted with their mass fractions. The subgrid correlations associated with terms uk τik are negligible with respect to other terms in enthalpy equation and then these are treated as u k τ ik . The quantities τui uj and τui h represent the unknown subgrid scale correlation between the subscript variables and τui uk uk represents interactions between the resolved velocity ﬁeld and the subgrid scale kinetic energy. A set of two scalar transport equations is also needed in order to track the chemical properties of the ﬂow ﬁeld. Scalar transport equations are implicitly ﬁltered, as described above. The chemical model is discussed in detail in Sect. 2.3. 2.2 Subgrid Scale Models Unclosed terms that appear in (1)-(3) are modeled using a dynamic procedure. The residual stresses are modeled as subgrid turbulent stresses with an eddy viscosity assumption[5]: 1 1 i u j ≃ 2CR ρ∆2f Πf2 Sij − ρq 2 δij τui uj = −ρ u5 i uj − u 3

(4)

where 31 ρq 2 is the subgrid kinetic energy and the ﬁltered strain rate tensor, Sij , is deﬁned as: ui uk 1 ∂ ∂ uj 1 ∂ Sij = + − δij , 2 ∂xj ∂xi 3 ∂xk 1 Πf2 = Sij Sij , and ∆f is the grid ﬁlter width. A Smagorinsky model, with a gradient diﬀusion assumption is used to model the quantities: 1 ∂ h 2 2 5 , (5) τui h = −ρ ui h − u i h ≃ 2Ch ρ∆f Πf ∂xi while τui uk uk is ﬁrst decomposed as: τui uk uk = − ρ u i u k u k /2 i uk uk − u ′′ ′′ ′′ ′′ =−ρ u k u i − u k u k u i − ρ u k − ρ u i /2 ku i uk u k uk u ′′ ′′ ′′ ′′ ′′ − ρ ui u k u k + 2u k + u i uk u i uk uk /2

(6)

and then the ﬁrst, the third and the fourth terms are neglected, while the second is approximated using a Smagorinsky model[6]: 0 1 ′′ ′′ 1 1 ′′ ′′ k ≃ −ρ uk u k 2CR ρ∆2f Πf2 Sik − Smm δik . (7) −ρ u i uk u i uk = u 3

The modeling “constants” CR and Ch are space functions of the ﬁltered stresses and ﬂuxes when, as in this work, a dynamic procedure is adopted.

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As already mentioned, in the present study ﬁltering is implicitly deﬁned by the computational grid. For the dynamic procedure to be applicable, the quantities to be modeled must vary substantially between the grid scale and the test ﬁlter scale, that is assumed to be twice the grid scale. For the subgrid turbulent stress model the dynamic procedure gives: Mui uj Lui uj , Muk ul Muk ul ˘ ˘ = − ρ ui u j − ρu i u j ,

CR = Lui uj

(8)

1 1 2 2 ˘ 2 ˘ 2 Π Mui uj = 2 ρ∆ f f S ij − 2ρ∆f Πf Sij ,

˘ 2 f stands to indicate the test ﬁlter width. For the where u i = ρ uj / ρ and ∆ modeling constant appearing in (5) applying the dynamic procedure one obtains: Luj h Muj h , Mul h Mul h ˘ ˘j h− h , uj ρu = − ρ

Ch = Luj h

1

˘2 2 Π ρ∆ Muj h = f f

(9)

˘ 1 h ∂ 2 ∂h . − ρ∆2f Π f ∂xi ∂xi

The quantities Mui uj and Muj h represent the diﬀerence between subgrid scale models expressed in terms of test ﬁlters quantities and the subgrid scale models test ﬁltered, and the quantities Lui uj and Luj h constitute the Germano identity and are the resolved subgrid scale correlations at the test ﬁlter level[7]. 2.3 Chemistry Model In order to incorporate chemistry into large eddy simulation, diﬀerent strategies can be taken into account. One of these would be to ﬁnd a chemical kinetic mechanism for the system under investigation, solve scalar transport equations for all the species of the mechanism, and attempt to model the ﬁltered non linear source term in each equation. A serious problem involved with this approach is that realistic kinetic mechanism can involve tens of species and hundreds of reaction steps. This is computationally expensive and conﬂicts with minimizing the number of transported scalar variables. For non premixed combustion, mixture fraction seems to oﬀer the most detailed description of species concentration, temperature and ﬂow properties. The mixture fraction can be deﬁned as a normalized chemical element mass fraction and it takes the value of unity in the fuel stream and zero in the oxidizer stream. The transport equation for Z is that of a conserved scalar,

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does not contain source terms and is aﬀected by chemical reactions through its inﬂuence on ﬂow density: ∂(ρZ) ∂(ρ ui Z) ∂[ρ(αt + αZ )∂(Z)/∂x i] + = , ∂t ∂xi ∂xi

(10)

where αt is the turbulent diﬀusivity, deﬁned as:

1/2 αt = Cφ ∆2f Πf Sij ,

Mi Li , Mj Mj ˘ ˘ Lφ = − ρ ui φ − ρu i φ ,

(11)

Cφ =

(12)

1 1 2 2 ˘ 2 ˘ 2 Π Mφ = 2 ρ∆ f f S ij − 2ρ∆f Πf Sij ,

and αZ is an average mixture fraction diﬀusivity. Combustion chemistry is incorporated in the form of premixed (in the zones where scalar dissipation is grater then the scalar dissipation rate of quenching of a diﬀusion ﬂame) [8] and diﬀusion steady-state ﬂamelet library. The premixed ﬂamelet equations at diﬀerent value of the equivalence ratio within the ﬂammability limits and at diﬀerent values of the fresh gases temperature, are solved by means of PREMIX code from Chemkin. The diﬀusion ﬂamelet equations are solved in the mixture fraction space, at diﬀerent values of the scalar dissipation rate χ [10] and at diﬀerent values of the maximum temperature and then reparameterized, obtaining a ﬂamelet library as a function of the mixture fraction, progress variable and sensible enthalpy Yi (Z, C, hs ): ∂χ ∂Yi 1 1 χ 1 ∂ 2 Yi ωi + = 0, (13) −1 + 4 Lei ∂Z ∂Z 2 Lei ∂Z 2 ρ where Lei = k/Di is the Lewis number for the i–th chemical specie, being k the thermal diﬀusivity and Di the mass diﬀusivity of the specie. Species diﬀusivities are taken as binary coeﬃcients in nitrogen. At a ﬁxed mixture fraction value the ﬂame properties are described by the progress variable that evolves between zero and its equilibrium value[9]. A look-up table is constructed to parameterize ﬂamelet evolution as function of mixture fraction and progress variable and enthalpy Yi (Z, C, hs ). In order to better evaluate the structure of the ﬂame where high scalar dissipation rates occur and the classical steady ﬂamelet approach describes a quenched ﬂame, a balance equation for the progress variable has to be introduced: ∂(ρ ui C) ∂[ρ(αt + αC )∂ C/∂x ∂(ρC) i] + = + ρ ωC , ∂t ∂xi ∂xi

(14)

where ω C is the precomputed ﬁltered chemical source term of the progress variable:

Large Eddy Simulation of a Turbulent Ethylene/Air Diﬀusion Flame

ω C =

C Z

ωC (Z, C) P(Z, C) dZ dC .

391

(15)

The mixture fraction is introduced to characterize, at a given location, the amount of the ﬂuid coming from the fuel that has been mixed with air. The progress variable, C(= YCO + YCO2 ), is used to characterize the amount of fuel that has been consumed, or the degree of progress in the reaction of any relevant quantity, as reactants, or even intermediate radicals. All mixture-weighted thermodynamic properties such as laminar viscosity, molecular and thermal diﬀusivity are determined using the polynomial curve ﬁts from CHEMKIN library. In a turbulent ﬂow all these quantities ﬂuctuate in space and time so that 7′′ ), is necessary in order to compute the mean 7′′ , C, C Z a joint PDF, P (Z, properties of the ﬂame, as species concentration, density and temperature, in 7′′ stands to indicate the subgrid scalar variance space and time. The quantity Z and is deﬁned as[11]: 2 ′′ 2 5 ρZ = CZ ρ∆2f ∇Z , (16)

where:

M L , M 2 ˘ ˘ Z − L = − ρZ ρZ Z , 2 ˘ 2 2 ∇Z ρ∆ . M = − ρ∆2f ∇Z f

CZ =

(17)

In this paper the progress of reaction and the mixture fraction are considered as two statistically independent variables and the enthalpy subgrid ﬂuctuation 7′′ ) can be chosen φ are not taken into account. The functional form of P (φ, as a β-function, deﬁned in the interval (0, 1)[12][13]. The gas-phase kinetic mechanism used to model ethylene pyrolysis and oxidation is built on the GRI mechanism for C1 and C2 species and Miller and Melius suggestions[14] for higher molecular mass species reactions. It includes hydrocarbon oxidation and pyrolysis up to the formation of benzene and 2−, 3−ring PAH such as naphthalene, acenapthylene and phenanthrene; details on the mechanisms for aromatic formation and molecular growth are reported in [15]. There are 60 species and 280 reactions for the formation of the gas-phase compounds. Each elementary reaction in the mechanism is reversible and the rate coeﬃcients of the forward reactions are either taken from literature or estimated on the basis of analogous reactions. The rate constants of reverse steps are calculated using appropriate equilibrium constants. The model has been validated by comparison of light species and radical concentrations in premixed and diﬀusion ﬂames against experimental data[2]. Figures 1 and 2 show the functional dependence of the density, temperature and principal products of a ﬂamelet at a ﬁxed value of χ.

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Fig. 1. Dependence on mixture fraction of temperature and density, at χ = 20s−1 .

Fig. 2. Dependence on mixture fraction of major species mass fraction concentration.

3 Numerical Simulation The ﬁltered mass, momentum, enthalpy and scalar equations mentioned above are discretized with a 2nd–order accurate central ﬁnite–diﬀerence scheme over a cylindrical non-uniform grid using a staggered approach, and then integrated in time, adopting a semi-implicit, iterative procedure[3]. In this study, the terms that have been treated implicitly are advection and diﬀusion in both radial and azimuthal direction and the pressure in all directions via a

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Poisson equation that is solved, in the iterative procedure, to determine the adjustment to the pressure ﬁeld required to assure that continuity equation is satisﬁed. The treatment of the convective terms in the scalar equations only is performed by a Leonard upstream (upwind) diﬀerence scheme (QUICK). This scheme, without adding a large amount of artiﬁcial viscosity, reduces dispersive errors that can lead to undesired, unphysical spatial oscillations of the scalars.

4 Experimental Setup and Results As a test case we show the simulation of a turbulent ethylene-air diﬀusion ﬂame. The free jet consists of a d = 3mm wide jet with a thinned rim at the exit. The jet composition is 100 % C2 H4 by volume. The cold jet velocity was ﬁxed to 2260cm/s resulting in a Reynolds number of 6000. The temperature of the fuel mixture and air are respectively 330K, and 293K, The ambient pressure is set to 1bar. Convective conditions are used for the boundaries. The computational grid cylindrical tube has a length of 70cm and a radius of 7.5cm. The mesh is reﬁned in the radial direction near the annulus, where a high gradient of mixture fraction is present, and along axial direction at the jet oriﬁce. The size of the grid is 192 × 128 × 48 points, respectively in the axial, radial and azimuthal direction as shown in ﬁg. 3.

Fig. 3. Computational grid used for the simulation.

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The simulation is performed with a maximum CFL number of 0.6 and a mean time step of 5−6 s, on 6 processors. The number of inter–iterations per time step, in the Newton-Raphson procedure was ﬁxed to 6. All the statistical values of the quantities of interest are obtained from diﬀerent azimuthal planes with a ﬁxed sampling interval, and then averaged on a meridional plane. The turbulent ethylene-air diﬀusion ﬂame was characterized by Commodo et al.[15] by using UV-induced emission spectroscopy for the measurements of the volume fractions of NOC particles. Comparison of experimental available data and numerical radial proﬁle of NOC volume fraction are depicted in ﬁgures 4–7.

Fig. 4. Comparison between modeled PAH volume fraction (Fv) and UV-ﬂuorescing species concentrations along the radii of the ﬂame at 3 cm on the axis.

Radial proﬁles show the typical wing shape with a maximum moving towards centerline for increasing distances from the nozzle. NOC particles are formed close to the ﬂame axis. The volume fraction of NOC particles, at z = 3cm reaches a maximum at r = 2.5mm, thereafter gradually decreases moving towards the outer ﬂame zone. The maximum volume fraction of organic carbon particles decreases at increasing distances in the ﬂame. Radial proﬁles of the NOC volume fraction change in shape becoming ﬂatter at increasing heights, showing a trend to merge in the central region, owing to turbulent mixing. In the comparison of PAH/NOC volume fraction it seems that at a distance above the nozzle of z = 3cm and z = 11.6cm the model under estimates the formation of PAH, whereas for the central region at z = 5cm and z = 7cm only the peak is adequately predicted, whereas the broadening of the distribution is greater than the experimental one due to the over es-

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Fig. 5. Comparison between modeled PAH volume fraction (Fv) and UV-ﬂuorescing species concentrations along the radii of the ﬂame at 5 cm on the axis.

Fig. 6. Comparison between modeled PAH volume fraction (Fv) and UV-ﬂuorescing species concentrations along the radii of the ﬂame at 7 cm on the axis.

timation in this regions of the model of turbulent diﬀusivity. In ﬁgures 8-9 the comparison of experimental and numerical temperature radial proﬁles is shown at two diﬀerent axial sections. The comparison of the temperature proﬁles at 7cm shows that the temperature proﬁle is over estimated. This may be due to the simple radiation model adopted that does not take into account the radiation related to soot particles. In ﬁg. 10 an instantaneous proﬁle of the

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Fig. 7. Comparison between modeled PAH volume fraction (Fv) and UV-ﬂuorescing species concentrations along the radii of the ﬂame at 11.6 cm on the axis.

contour of mass fraction of PAH sum in the near injection region is depicted.

Fig. 8. Comparison between modeled and experimental temperature along the radii of the ﬂame at 3 cm on the axis.

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Fig. 9. Comparison between modeled and experimental temperature along the radii of the ﬂame at 7 cm on the axis.

Fig. 10. Snapshot of the mass fraction of the sum of the PAH near the nozzle. (See Plate 46 on page 437)

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5 Conclusions The kinetic scheme used in the calculation of ﬂamelet library used for the simulation is able to model the formation of 2-, 3-ring PAH. The qualitative good agreement between experimental concentration of UV-ﬂuorescing compounds (not fully solid particles of few nanometers), and LES modeled sum PAH concentrations at diﬀerent heights above the nozzle, suggests that the hypothesis of fast formation rate of Nanosized Organic Carbon (NOC) particles is correct. Their formation is related to turbulence mixing and the prediction of gas phase aromatics corresponds to the prediction of nanosized soot precursor. The results obtained open a new perspective in the possibility of predicting polycyclic aromatic hydrocarbons in turbulent combustion. A major characteristic of the proposed approach is the possibility to describe aromatics formation in the framework of fast chemical reactions. The use of a sectional method to predict also the size distribution of nanometer-sized particles is in progress.

References [1] N. Peters. Laminar diﬀusion ﬂamelet models in non-premixed turbulent combustion. Proc. Energy Combustion Sci., 10, 319–339, 1984. [2] A. D’Anna, A. Violi, A. D’Alessio and A. Saroﬁm. A reaction pathway for nanoparticle formation in rich premixed ﬂames. Combust. Flame, 127, 1995–2003, 2001. [3] C.D. Pierce. Progress Variable Approach for Large Eddy Simulation of Turbulent Combustion. PhD Thesis, Stanford University, 2001. [4] R.S. Barlow, A.N. Karpetis, J.H. Frank and J.-Y. Chen. Scalar proﬁles and NO formation in laminar opposed-ﬂow partially premixed methaneair ﬂames. Combust. Flame 127, 2102–2118, 2001. [5] C.D. Pierce and P. Moin. Progress variable approach for large eddy simulation of non premixed turbulent combustion. J. Fluid Mech., 504, 72–97, 2004. [6] E.P. DesJardins and H.S. Frankel. Large Eddy Simulation of a nonpremixed reacting jet: Application and assessment of subgrid-scale combustion models. Phys. Fluids, 10, 2298–2314, 1998. [7] M. Germano, U. Piomelli, P. Moin and W.H. Cabot. A dynamic subgridscale eddy viscosity model. Phys. Fluids, A 3, 1760–1765, 1991. [8] M. Rullaud. Mod´elisation de la combustion turbulente via une m´ethode de tabulation de la cin´etique chimique d´etaill´ee coupl´ee `a des fonctions de densit´es de probabilit´e. Application aux foyers a´eronautiques. Th´eses de physique, INSA Rouen, 2004. [9] L. Vervisch, R. Hauguel, P. Domingo and M. Rullaud. Three facets of turbulent combustion modelling: DNS of premixed V-ﬂame, LES of lifted nonpremixed ﬂame and RANS of jet-ﬂame. J. of Turbulence, 5, 2004.

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[10] N. Peters. Turbulent Combustion. Cambridge University Press, Cambridge, 2000. [11] C.D. Pierce and P. Moin. A dynamic model for subgrid-scale variance and dissipation rate of a conserved scalar. Phys. Fluids, 10, 3041–3044, 1998. [12] J. Jimenez, A. Linan, M.M. Rogers and F.J. Higuera. A priori testing of subgrid models for chemically reacting non-premixed turbulent shear ﬂows. J. Fluid Mech., 349, 149–171, 1997. [13] A.W. Cook and J.J. Riley. Subgrid-scale modeling for turbulent reacting ﬂows. Combust. Flame, 112, 593–606, 1998. [14] J.A. Miller and C.F. Melius. Kinetic and thermodynamic issues in the formation of aromatic compounds in ﬂames of aliphatic fuels. Combust. Flame, 91, 21–39, 1992. [15] M. Commodo, A. Violi, A. D’Alessio, A. D’Anna, C. Allouis, F. Beretta and P. Minutolo. Soot and nanoparticle concentration in a turbulent nonpremixed ethylene/air ﬂame from laser induced emission measurement at 213 nm. 28th Meeting of the Italian Section of the Combustion Institute, Naples, 2005.

Energy Fluxes and Shell-to-Shell Transfers in MHD Turbulence Daniele Carati1 , Olivier Debliquy1 , Bernard Knaepen1 , Bogdan Teaca2 , and Mahendra Verma3 1

2 3

Statistical and Plasma Physics, Universit´e Libre de Bruxelles Campus Plaine, CP 231, B-1050 Brussels, Belgium Faculty of Physics, 13 A.I. Cuza Street, 200585 Craiova, Romania Department of Physics, I. I. T. Kanpur, Kanpur 208016, India

Summary. A spectral analysis of the energy cascade in magnetohydrodynamics (MHD) is presented using high resolution direct numerical simulations of both forced and decaying isotropic turbulence. The triad interactions between velocity and magnetic ﬁeld modes are averaged into shell interactions between similar length scales phenomena. This is achieved by combining all the velocity Fourier modes that correspond to wave vectors with similar amplitude into a shell velocity variable. The same procedure is adopted for the magnetic ﬁeld. The analysis of the interactions between these shell variables gives a global picture of the energy transfers between diﬀerent length scales, as well as between the velocity and the magnetic ﬁelds. Also, two diﬀerent attempts to separate the shell-to-shell interactions into forward and backward energy transfers are proposed. They provide diagnostics that can be used in order to assess subgrid-scale modelling in large-eddy simulation for turbulent MHD systems.

1 Introduction A good knowledge of the interactions between large resolved scales and small unresolved scales is required to design accurate subgrid-scale models for largeeddy simulations (LES). Although these interactions have been extensively studied for Navier-Stokes turbulence [1–3], much less eﬀorts have been devoted to characterize nonlinear interactions in turbulent electrically-conducting ﬂuids. The objective of this study is to perform a detailed analysis of the nonlinear triad interactions in magnetohydrodynamic (MHD) turbulence and more speciﬁcally of the shell-to-shell interactions. The concept of shell variables is naturally introduced when a spectral analysis of the ﬁelds is considered. A physical space version can however be deﬁned by considering the parts of the velocity and magnetic ﬁelds that correspond to structures with a given length scale. The physical space analysis of the shell-to-shell interactions is then better suited to the design of subgrid-scale models, especially in complex

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geometries for which a Fourier decomposition of the MHD variables is not always available. The extension of the deﬁnition of triad interactions and shell-to-shell energy transfers to the MHD equations is fairly simple, though shell-to-shell interactions in MHD are more complex [4–6]. Indeed, the total energy is still an ideal invariant, but it can be split into two contributions, the kinetic energy and the magnetic energy. The characterization of energy ﬂuxes thus requires a detailed analysis of the various transfers of energy, not only between different length scales but also between the velocity and the magnetic ﬁelds. Simulations of isotropic decaying and forced MHD turbulence using up to 5123 Fourier modes have been analyzed and the energy ﬂuxes as well as the shell-to-shell energy transfers between the velocity and the magnetic ﬁelds are computed from these simulations. As usually anticipated in the analysis of triad interactions in turbulence, the total energy transfers are very much local, i.e. the transfers are dominated by exchanges of energy between structures that have similar sizes. The knowledge of the detailed backscatter and forward energy transfers is often considered as an additional interesting guide to describe the physics of the nonlinear interactions in turbulence [7–9]. It is also a valuable diagnostics in the framework of subgrid-scale modelling. For this reason, shell-to-shell interactions have been decomposed into forward (from large to small scales) and backward (from small to large scales) energy transfers. It is pointed out that such a decomposition is not unique and two possible strategies to identify the energy backscatter are suggested. The ﬁrst one is based on the Fourier representation of the turbulent ﬁelds and, in this case, the forward and the backward transfers, like the total energy exchanges, are essentially local. On the contrary, the physical space decomposition shows very non-local exchanges of energy between the shells [10] (signiﬁcant forward and backward exchanges of energies between structures with characteristic size ratios larger than 10 are commonly observed).

2 Methodology 2.1 Equations for a Conducting Fluid When a conducting ﬂuid interacts with an electromagnetic ﬁeld, hydrodynamics and electromagnetisms must be coupled. First, the electromagnetic ﬁelds inﬂuence the momentum balance through the Lorentz force. The ﬂuid is usually assumed to be almost electro-neutral, so that the electric charge density is neglected. In this case, the Lorentz force reduces to its magnetic part. The equations for the electromagnetic ﬁeld are also inﬂuenced by the ﬂuid motion through the closing assumption made to relate the electric current to the electric ﬁeld in a moving ﬂuid. The most typical approximation is to consider a linear relation (Ohm’s law). Combining all these assumptions

Energy Fluxes and Shell-to-Shell Transfers in MHD Turbulence

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yields the coupled velocity and magnetic ﬁelds equations ∂t ui = −uj ∂j ui + bj ∂j bi − ∂i p + ν∇2 ui , 2

∂t bi = −uj ∂j bi + bj ∂j ui + η∇ bi .

(1) (2)

In these equations, ν is the kinematic viscosity, p is the total (hydrodynamic + magnetic) pressure divided by the ﬂuid density. The magnetic ﬁeld has been rescaled using Alfv´en’s units and has the dimensions of a velocity. In the examples treated below, the magnetic diﬀusivity η will be chosen to be equal to the viscosity, so that the magnetic Prandtl number is unity. The ﬂuid density is here assumed to be constant and the velocity is divergencefree (as well as the magnetic ﬁeld, as a consequence of Maxwell’s equations): ∂ l u l = 0 = ∂ l bl . 2.2 Energy Transfers and Shell Variable Deﬁnition All the simulations that have been post-processed in order to extract energy ﬂuxes and shell-to-shell energy transfers have been performed using a fully de-aliased pseudo-spectral code. The use of a Fourier representation for the variables allows a straightforward deﬁnition of the shell variables. Indeed, the Fourier space is sliced into shells deﬁned by: sn ≡ k such that kn ≤ ||k|| < kn+1 . (3)

The shell sn thus contains all the modes with a wave vector amplitude larger than kn and smaller than kn+1 . The kinetic and magnetic energies in a shell may be deﬁned by diﬀerent expressions: 1 1 1 n Eun = |u (k)|2 = dx un (x)2 , (4) |u(k)|2 = 2 2 2 k∈sn k 1 1 1 n n 2 2 Eb = (5) |b (k)| = |b(k)| = dx bn (x)2 , 2 2 2 k∈sn

k

where un (k) is the shell velocity, i.e. the velocity ﬁeld obtained by setting all the Fourier modes outside the shell sn to zero. Its inverse Fourier transform, un (x), can be used to give a physical space representation of the shell velocity. The same deﬁnitions also apply to the shell magnetic ﬁeld. The shell-to-shell a,b energy transfers Tn,m represented on Figure 1 correspond to the variation of the variable a (which could be either the velocity or the magnetic ﬁeld) energy contained in a shell of wave vectors with index n caused by the interactions with the variable b from the shell of wave vectors with index m. The explicit mathematical deﬁnition is rather intricate but can be derived straightforwardly from Eqs. (1-2). The explicit form of these shell-to-shell transfers will not be needed in the following. The interested reader may ﬁnd a complete discussion of the mathematical structure of these shell-to-shell transfers in reference [10].

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u,u

T n,m k0

k1

kn

k2

u k! b,u

T n,m

u,b

T n,m b k! k0

k1

kn

k2

T

b,b n,m

Fig. 1. Schematic representation of the shell-to-shell energy transfers.

2.3 Forward and Backward Energy Transfers Although the energy exchanges between the velocity and magnetic shell variables may be deﬁned unambiguously, the situation is diﬀerent for the forward and the backward energy transfers. Indeed, the energy transfer between two shell variables, for instance un and bm , results from a large number of triad interactions. Some of the triad interactions increase the energy of un , while other interactions have the opposite eﬀect. As a consequence, the shell-to-shell u,b,(+) transfer is, in general, the sum of many positive Tn,m > 0 and negative u,b,(−) Tn,m < 0 contributions. When the positive contributions are larger than u,b,(+) u,b,(−) u,b the negative contributions (Tn,m = Tn,m + Tn,m > 0) and when n > m, the global interaction is referred to as a direct energy cascade. Independent of u,b,(±) the deﬁnition of Tn,m , it seems natural to impose that the forward energy transfer from bm to un does correspond to a backward energy transfer from un to bm . Hence, the following property, a,b,(+) b,a,(−) Tn,m = −Tm,n ,

(6)

will be imposed to the deﬁnition of forward and backward energy transfers. Deﬁning these forward and backward energy transfers using a separate counting of all the triad interactions that increase or decrease the energy of un could seem the most natural strategy. For practical reasons however, this deﬁnition has not been adopted here. Indeed, in terms of energy transfers, the triad interactions involve cubic terms with a ﬁrst mode u(k) in the shell sn , a second mode b(p) in the shell sm and a third mode b(q) that can be in any shell as long as k + p + q = 0. Because of this last condition on the wave vectors, the number of triad interactions grows like M 2 in a simulation with M modes (typically, in our runs M = 5123 ≈ 125 106 ). It is well known that, in

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spectral codes, the computation of the nonlinear terms can be done in M log M operations using fast Fourier transform (FFT) algorithms. However, if the forward and backward energy transfers are deﬁned by a separate counting of the mode-to-mode interactions that lead to increases or decreases of the shell energies, the evaluation of all the forward and backward energy transfers cannot make use of FFTs any more. As a consequence, such a procedure would require M 2 operations and would be prohibitively costly. The deﬁnitions that have been introduced in [10] are compatible with the evaluation of all the shell-to-shell energy transfers using FFTs. They require only M log M operations. In practice, using the Fourier representation of the velocity and magnetic ﬁelds, the modes that correspond to the variable un are separated into those that experience an increase and those that experience a decrease of energy through the interactions with all the modes from the variable bm . In terms of the energy evolution, this deﬁnition is straightforward. Indeed, by deﬁnition, in the ideal limit (ν = η = 0) the shell energies evolve according to: u,u u,b ∂t Eun = Tn,m + Tn,m , (7) m

∂t Ebn

=

m

m

b,u Tn,m

+

b,b Tn,m ,

(8)

m

and, considering the kinetic energy in shell sn , we have: 1 n n ∗ n ∂ t Eu = [u (k)] · ∂t u (k) + c.c. . 2

(9)

k

If the contribution to this sum from the interactions of one mode un (k) with all u,b,(+) the modes bm (k) in the shell sm is positive, it is added to Tn,m,(k) . Moreover, in order to satisfy the condition (6), a symmetrization of the separation into forward and backward transfers has been imposed. An alternative deﬁnition, using the real space representation of the shell variables, has also been proposed. It is based on the following equivalent expression of the shell energy evolution: (10) ∂t Eun = dx un (x) · ∂t un (x) . The forward and backward energy transfers are now deﬁned by a separate counting of the energy exchanges from positions that correspond to local increase or decrease of energy in un due to the total interactions with the shell variable bm . In practice, if the contribution to this integral from the interaction of the velocity un (x), with all the velocity bm (y) from the shell sm is posiu,u,(+) tive, it is added to Tn,m,(x) . Here again, in order to satisfy the condition (6), a symmetrization of the separation into forward and backward transfers has been imposed. It is important to note that with these deﬁnitions,

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D. Carati, O. Debliquy, B. Knaepen, B. Teaca, and M. Verma a,b,(+)

a,b,(+)

Tn,m,(x) = Tn,m,(k) .

(11)

The origins of this diﬀerence between the two deﬁnitions will be discussed after the presentation of the results for both decaying and forced turbulence.

3 Numerical Results In both decaying and forced turbulence, the initial conditions correspond to an equipartition of energy between the magnetic and the velocity ﬁelds. The magnetic Prandtl number Pm = ν/η is always unity. The geometry is a (2π)3 box. The time stepping is done using a third order Runge-Kutta method and the time-step is computed automatically using the CFL criterion. The dissipative linear terms in the equations for both the velocity and the magnetic ﬁelds are treated analytically using exponential factors. 3.1 Decaying Turbulence u,u,(±)

Both the shell-to-shell transfer rates in physical space Tn,m,(x) and in Fourier u,u,(±)

space Tn,m,(k) are computed straightforwardly from the DNS code [11, 12]. The decaying turbulence simulation has been performed using 5123 modes. The initial Reynolds number Rλ (based on Taylor’s micro-scale) for the initial ﬁeld is 159. Since the magnetic Prandtl number is 1, the magnetic and the kinetic Reynolds numbers coincide. In decaying turbulence, the initial conditions may aﬀect the shell-to-shell energy transfers through at least two mechanisms. First, at the fairly low Reynolds numbers achievable with DNS, the energy decay is not supposed to be perfectly self-similar and the shape of the energy spectrum is not preserved during the simulation. So, it is important to chose an initial energy spectrum that is not peaked too strongly in the low wave number range nor in the dissipation range. Indeed, such choices would lead to a rapid modiﬁcation of the energy spectrum in the course of the simulation and measurements of the shell-to-shell energy transfers would most probably strongly depend on both the initial spectrum and time. In our simulation, the initial spectrum is described in details in [10] and has been chosen to exhibit a limited k −5/3 range with the hope that it is reasonably close to the self-similar solution. The initial conditions may also strongly aﬀect the energy transfers through the velocity and magnetic phases. Indeed, the traditional procedure to build up an initial ﬁeld is to impose the velocity and magnetic mode amplitudes so that they match a prescribed spectrum and to pick up random phases with the constraint that both ﬁelds are divergence-free. However, random phases correspond to vanishing triple correlations of the ﬁelds and, as a consequence, to vanishing energy transfers. The usual strategy is to let the ﬁelds evolve according to the Navier-Stokes or the MHD equations so that reasonable phases can build-up and realistic energy transfers may develop. For this reason, the energy transfers are presented here after about two

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Fig. 2. Two-dimensional representation of the shell to shell energy transfers. The horizontal coordinate corresponds to the receiving shell and the vertical coordinate to the giving shell. The u-to-u, b-to-b, u-to-b and b-to-u energy transfers are represented respectively in the top-left, top-right, bottom-left and bottom-right frames. (See Plate 47 on page 437)

eddy turn-over times, when a signiﬁcant fraction of the initial energy has been dissipated. As expected from the phenomenology of turbulence, the energy transfers a,b Tn,m are essentially local and direct. This can be seen in Figure 2. Locality can be observed since all the signiﬁcant transfers are along the diagonal where n is close m. Only direct transfers are observed which is conﬁrmed by positive values below the diagonal and negative values above. Also, scale independent energy transfers can be observed since all the horizontal lines are very much a,b is essentially a function similar when properly shifted by m boxes, i.e. Tn,m of n − m and not of n and m separately. Interestingly, the Fourier space based forward shell-to-shell energy transa,b,(+) fers, Tn,m,(k) , are also very local. They appear to be larger in amplitude than the total energy transfers (Figure 3, left), which indicates that signiﬁcant cancellations have to take place between forward and backward energy transfers. These forward transfers are positively deﬁned. The backward transfers are not shown since they can be obtained using the symmetry property (6). Finally, the physical space based forward shell-to-shell energy transfers, a,b,(+) Tn,m,(k) , show a very diﬀerent picture. They also appear to be larger in amplitude than the total energy transfers (Figure 3, right). However, these transfers a,b,(+) are highly non-local: Non-vanishing values of Tn,m,(k) are observed for moderate to large values of m − n.

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Fig. 3. Two-dimensional representations of the Fourier (left) and physical space (+) (right) Tn,m (same interpretation as in Figure 2). (See Plate 48 on page 438)

3.2 Forced Turbulence Forced turbulence has been simulated using a force that injects a constant rate of energy ǫ in the ﬂow. In Fourier space, this force is limited to one shell of wave vector sf (typically sf = s4 , so that energy is injected only at large scales). All modes in the shell sf are submitted to a force deﬁned by: f (k) = α

u(k) , |u(k)|2

(12)

where α = ǫ/Nf is the ratio between the desired total energy injection rate ǫ and the number of modes Nf in the forcing shell sf . Another form of this

Fig. 4. Two-dimensional representation of the shell-to-shell energy transfers in forced turbulence simulation. The u-to-u (left) and the b-to-b (right) energy transfers are observed to be direct and local. (See Plate 49 on page 438)

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409

force that simultaneously injects a constant rate of energy and a constant rate of helicity has also been used but this presentation is limited to the non-helicoidal forcing. Because convergence of the statistics appears to be rather slow, runs of forced turbulence have been limited to 2563 modes. The u-to-u and b-to-b transfers (Figure 4) do not show major diﬀerences with the decaying turbulence case. Again, they appear to be local and direct. Also, in a certain range of shells (s6 − s17 ), the shell-to-shell energy transfers are almost self-similar. Remarkably, the u-to-b and b-to-u shell-to-shell energy transfers are much more inﬂuenced by the forcing. Indeed, the forced velocity shell transfers energy to the magnetic ﬁeld at almost all the scales. This conﬁrms recent results [13, 14] obtained using a large-scale constant forcing which, contrary to the forcing (12), generates an average velocity ﬁeld. Hence, this phenomenon of distant interactions between the velocity and the magnetic ﬁelds in forced turbulence appears to be independent of the type of forcing.

Fig. 5. Same as Figure 4 for the b-to-u (left) and the u-to-b (right) energy transfers. (See Plate 50 on page 439)

The forward and backward energy transfers deﬁned using the Fourier and the physical space representations have also been computed for forced turbulence. Like in decaying turbulence, the physical space representation exhibits more non-local energy exchanges, although non-local energy ﬂux from the forced shell are observed using both decompositions.

4 Conclusions The work reported here contributes to the extension of the analysis of turbulence in terms of the shell-to-shell energy transfers to MHD systems. Two major conclusions may be drawn from the present study. The ﬁrst one concerns the non-uniqueness of the deﬁnition of forward and backward energy transfers in turbulence. It is not typical of MHD ﬂow and directly applies to

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(+)

Fig. 6. Two-dimensional representations of the Fourier Tn,m,(k) (left) and physical (+)

space (right) Tn,m,(x) for forced turbulence (same interpretation as in Figure 2). (See Plate 51 on page 439)

Navier-Stokes turbulence. The second conclusion concerns the non-local energy transfers observed in forced turbulence. They mostly aﬀect the velocitymagnetic ﬁelds exchanges, and as such, are speciﬁcally an MHD eﬀect. First, this study shows that the analysis of turbulence in terms of the shell-to-shell energy transfers can be easily extended to MHD ﬂows. Although the total shell-to-shell energy transfers are unambiguously deﬁned, their decomposition into forward transfer and backscatter of energy is not unique and two deﬁnitions, using respectively the Fourier space and the physical space representations of the shell variables have been proposed. The results for both decaying and forced MHD turbulence show that these two deﬁnitions yield different conclusions on the property of the forward and backward energy transfers. Since the nature of the energy transfers is, of course, independent of the choice adopted for this decomposition, it must be concluded that the quan(±) (±) tity Tn,m,(k) and Tn,m,(x) characterize diﬀerent properties of MHD turbulence. Considering their mathematical deﬁnitions, both quantities can legitimately be interpreted as forward or backward energy transfers. The fact that these transfers appear to be much more local when based on the spectral analysis may be interpreted as follows. The splitting of the shell variables into energy increasing and energy decreasing modes may be used to deﬁned two velocity ﬁelds: m m (13) un (k) = u{n,b ,+} (k) + u{n,b ,−} (k) . It is important to realize that this decomposition of un depends on the other m m shell variable bm . The quantities u{n,b ,+} (k) and u{n,b ,−} (k) have to be interpreted as the collections of modes in the shell variables un (k) that experience respectively an energy increase or an energy decrease through the interactions with the shell variable bm (k). The important point here is that both these variables are still shell variables in the sense of deﬁnition (3). On

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411

the contrary, the splitting of the shell variables into energy increasing and energy decreasing positions is quite diﬀerent: un (x) = u{n,b m

m

,+}

(x) + u{n,b

m

,−}

(x) .

(14)

m

Indeed, the quantities u{n,b ,+} (x) and u{n,b ,−} (x) have now to be interpreted as the ﬁeld un (x) ﬁltered so that it is put to zero except at locations where the shell variables un (x) experience respectively an energy increase or an energy decrease through the interactions with the shell variable bm (x). m It must be noted that u{n,b ,±} (x) is not the inverse Fourier transform of m {n,bm ,±} u (k). In particular, the inverse Fourier transform of u{n,b ,±} (x) has most probably non zero modes in several shells and it is not a pure shell variable. In that sense, it is not a surprise that the forward and backward energy transfers based on the analysis of the real space representation of the shell variables appear much more non-local. It would thus be tempting to simply abandon this picture. It must be stressed however that, when no Fourier representation of the variables is available, the determination of backscatter will most probably be performed using a deﬁnition that should be very close to (14). The second major conclusion that can be drawn from this study is the presence of non-local interactions between the velocity and the magnetic ﬁelds in forced turbulence. These non-local interactions are clearly related to the forcing and only aﬀect the forced velocity shell which transfers energy to the magnetic ﬁeld at almost all scales. The peculiar behavior of the forced velocity shell can be understood by considering the shell energy evolution equations. These equations contain three types of terms: Non-linear shellto-shell interaction terms as reported in equations (7) and (8), dissipation terms and forcing terms. In the range where dissipation and forcing may be neglected, the shell variables are in a regime in which their energy level is due to the balance between the diﬀerent non-linear interactions. For the forced velocity shell, the non-linear interactions entering this balance are thus quite naturally aﬀected by the presence of the forcing term. The analysis of the shell-to-shell energy transfers indicates that the statistically stationary energy level in the forced shell can only be reached when a signiﬁcant amount of energy is transferred not only to the neighbor shells but also to distant shells. Considering similar recent independent results [13, 14] obtained using another type of forcing, it seems that this phenomenon is qualitatively independent of the nature of the forcing. Why this eﬀect only aﬀects the u-b interactions and not the u-u interactions remains an open question.

Acknowledgments This work has been supported by the Fonds National pour la Recherche Scientiﬁque (Belgium, FRFC 2.4542.05), the Fonds Defay, the Communaut´e

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Fran¸caise de Belgique (ARC 02/07-283) and the contract of association EURATOM - Belgian state. The content of the publication is the sole responsibility of the authors and it does not necessarily represent the views of the Commission or its services.

References [1] R. Kraichnan. Inertial-range transfer in two and three-dimensional turbulence, J. Fluid Mech. 47, pp. 525-535, 1971. [2] Y. Zhou. Degrees of locality of energy transfer in the inertial range, Phys. Fluids A 5, pp 1092-1094, 1993. [3] R.M. Kerr, J.A. Domaradzki, and G. Barbier. Small-scale properties of nonlinear interactions and subgrid-scale energy transfer in isotropic turbulence, Phys. Fluids 8, pp. 197-208, 1996. [4] A. Pouquet, U. Frisch, and J. L´eorat. Strong MHD helical turbulence and the nonlinear dynamo eﬀect, J. Fluid Mech. 77, pp. 321-354, 1976. [5] O. Schilling and Y. Zhou. Triadic energy transfers in non-helical magnetohydrodynamic turbulence, J. Plasma Phys. 68, pp 389-406, 2002. [6] M.K. Verma. Statistical theory of magnetohydrodynamic turbulence: recent results, Phys. Report 401, pp 229-380, 2004. [7] C.E. Leith. Stochastic backscatter in a subgrid-scale model: Plane shear mixing layer, Phys. Fluids A 2, pp 297-299, 1990. [8] P.J. Mason and D.J. Thomson. Stochastic backscatter in large-eddy simulations of boundary layers, J. Fluid Mech. 242, pp 51-78, 1992. [9] D. Carati, S. Ghosal, and P. Moin. On the representation of backscatter in dynamic localization models, Phys. Fluids 7, pp 606-616, 1995. [10] O. Debliquy, M.K. Verma, and D. Carati. Energy ﬂuxes and shell-toshell transfers in three-dimensional decaying magnetohydrodynamic turbulence, Phys. Plasmas 12, 042309, 2005. [11] J.A. Domaradzki and R.S. Rogallo. Local energy-transfer and nonlocal interactions in homogeneous, isotropic turbulence, Phys. Fluids A 2, pp 413-426, 1990. [12] G. Dar, M. Verma, and V. Eswaran. Energy transfer in two-dimensional magnetohydrodynamic turbulence: formalism and numerical results, Physica D 3, pp 207-225, 2001. [13] A. Alexakis, P.D. Mininni, and A. Pouquet. Shell to shell energy transfer in MHD, Part I: Steady state turbulence, Phys. Rev. E 72, p046301, 2005. [14] P.D. Mininni, A. Alexakis, and A. Pouquet. Shell to shell energy transfer in MHD, Part II: Kinematic dynamo, Phys. Rev. E 72, p046302, 2005.

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Plate 1: Computational grid and contours of u-velocity from the inviscid Taylor problem. Reconstruction of Mahesh et al. [1] (center) – Ham and Iaccarino’s reconstruction [2] (right). Results have been copied in periodic directions for clarity. (See Fig. 1 on page 4)

Plate 2: Load balancing for multiphase ﬂow calculations of a jet engine fuel injector. Sprays are highly localized in the vicinity of the injector. A gas-phase cell-based partition leads to high particle counts on only few processors (left); a dual partition constraint provides nearly optimal load balancing for both the gas and liquid-phase (right). Top ﬁgures show the grid partition boundaries, whereas the cell and particle counts per processor are reported in the histograms. (See Fig. 7 on page 7)

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Plate 3: Schematic representation of the multi-code coupling for the simulation of the ﬂow in a jet engine (left). Flow simulation of the high-spool of a Pratt & Whitney engine. Contours of entropy are reported in the compressor and turbine, isosurfaces of temperature in the combustor (right). (See Fig. 8 on page 8)

Plate 4: Temperature distribution in cross-sections of a Pratt & Whitney combustion chamber. (See Fig. 9 on page 9)

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Plate 5: Schematic of optical propagation through a turbulent wake behind a circular cylinder. (See Fig. 11 on page 10)

Plate 6: Instantaneous far-ﬁeld intensity patterns for an aberrated beam (top) and a non-aberrated beam (bottom) at diﬀerent distances of propagation. The optical wavelength is 2.5 × 10−6 D. The intensity levels are normalized by the peak intensity at the aperture where a Gaussian proﬁle is assumed. (See Fig. 12 on page 11)

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Plate 8: Dual solution of the circular advection problem at t = .45. Shown are a carpet plot of the dual solution φh using quadratic space-time elements (left) and a carpet plot of the dual solution defect φh − πh φh (right) with πh a projection to space-time linear functions. (See Fig. 3 on page 41)

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Plate 9: Navier-Stokes solution at the non-dimensional time t = 635 computed on the reference 40K element mesh using P2 space-time elements. Presented here are velocity contours (left) and logarithmically scaled vorticity magnitude contours (right). (See Fig. 4 on page 43)

Plate 10: Locally linearized dual Navier-Stokes solution corresponding to the drag coeﬃcient functional with local linearization about P1 primal space-time data on a 12K mesh. Presented here are dual solution contours at the same approximate non-dimensional time as Plate 9 showing the x-momentum component of the dual solution φ (left) and the x-momentum component of the dual solution defect φ−πh φ (right). (See Fig. 5 on page 44)

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Plate 11: Crow (left) and short-wave or elliptic instability (right). (See Fig. 1 on page 54)

Plate 12: Simulation of the evolution of an inviscid elliptical vortex using the AGM particle method: vorticity (left), particle sizes (middle, dark areas represent coarse particle sizes) and grid (right). (See Fig. 2 on page 56)

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Plate 16: Fine grid, Reτ = 395, streamwise vorticity, isosurfaces of ωx = ±45; LES (left) and LES with corrected near-wall eddy-viscosity (right). (See Fig. 4 on page 131)

Plate 17: Fine grid, Reτ = 395, streamwise vorticity, isosurfaces of ωx = ±45; LES with corrected near-wall eddy-viscosity (left) and LES with RANS eddy-viscosity in near-wall region (right). (See Fig. 5 on page 131)

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Plate 25: Reτ = 100, 000 and Reτ = 1, 000, 000, 96 x 97 x 96, streamwise vorticity, isosurfaces of ωx = ±55 (left) and ωx = ±5 (right). (See Fig. 17 on page 137)

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Plate 29: Computational grid in the symmetry plane for the hollow sphere. (See Fig. 10 on page 231)

Plate 30: Left: Streamwise velocity. Right: instantaneous spanwise vorticity; a) smooth sphere; b) dimpled sphere; c) hollow sphere. (See Fig. 11 on page 231)

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L2 error in u1 at t = 1

10-1

cv-based dual-based

10-2

0

0.2

0.4

0.6

0.8

1

10

-3

10

-4

10

-5

10-6

10

-1

10

0

t relative ∆x Plate 32: Comparison of cv (dashed) and dual (solid) formulations: inviscid Taylor problem on uniform Cartesian meshes with Dirichlet boundary conditions displaced by (∆x, ∆y) = (0.25, 0.31), σ = 1.0. Results shown are for same meshes as Plate 31. In this case, the Taylor vortices cut the domain, resulting in inﬂow or outﬂow at all angles to the boundary. Thin dashed lines are slope 1 and 2. (See Fig. 5 on page 245)

Color Plates

mesh

initial ui

0.5

0.5

y

1

y

1

0

-0.5

-1

x cv-based p at t = 3.0 -1

-0.5

0

0.5

1

1.5

-1.5

0.5

0.5

y

1

y

1

0

-0.5

-1

-1

x cv-based u1 at t = 3.0 -1

-0.5

0

0.5

1

1.5

-1.5

1

0.5

0.5

y

1

0

-0.5

-1

-1

-1

-0.5

x 0

0.5

1

1.5

-0.5

0

0.5

1

1.5

x dual-based u1 at t = 3.0 -1

-0.5

0

0.5

1

1.5

0

-0.5

-1.5

x dual-based p at t = 3.0 -1

0

-0.5

-1.5

y

0

-0.5

-1

-1.5

429

-1.5

-1

-0.5

x 0

0.5

1

1.5

Plate 33: Comparison of L2 u1 -error history for cv (dashed) and dual (solid) formulations on the same highly skewed unstructured mesh for the inviscid Taylor problem with Dirichlet boundary conditions. (See Fig. 6 on page 246)

Color Plates

L2 error in u1

430

10

-1

10

-2

10

-3

10

-4

cv: base grid cv: 1 level of reﬁnement cv: 2 levels of reﬁnement dual: base grid dual: 1 level of reﬁnement dual: 2 levels of reﬁnement dual: 3 levels of reﬁnement

L2 error in u1

0

10

-1

10

-2

10

-3

10

-4

20

0

40

2

t

t

60

4

80

100

6

Plate 34: Time history of L2 velocity error for cv (dashed) and dual (solid) simulations: inviscid Taylor problem with Dirichlet boundary conditions on the highly skewed unstructured grid shown in ﬁgure 6. (See Fig. 7 on page 247)

Color Plates

431

Plate 35: Example of a mesh hierarchy with three levels. (See Fig. 1 on page 254)

10

−1

10

−2

10

−3

10

−4

10

−5

a b c d

10

100 N

1000

Plate 36: Convergence of the density error as a function of coarse mesh resolution (L1 norm) for the two vortex periodic system in four diﬀerent scenarios. The thick line denotes second-order rate. (See Fig. 3 on page 256)

Color Plates

20 LES−SAMR Gutmark & Wygnanski (1976)

(Uj/UM)

2

15

10

5

0

0

20

40

60

80

100

x/d

(a) Turbulent planar jet

(b) Compensated mean center-plane jet velocity 0.12

0.1

Mixing Zone Width [m]

432

0.08

37.2 m s-1 0.06

0.04

4.2 m s-1 0.02 0

0.002

0.004

0.006

0.008

t [s]

(c) RMI Interface

(d) RMI Mixing zone as a function of time (red line) and experimental growth rate (experiment)

Plate 37: Some results for diﬀerent ﬂows. (See Fig. 5 on page 259)

Color Plates

433

(a)

(b) Plate 38: Iso-surfaces of temperature (a) and overlay of the mesh (b) for a turbulent hydrogen diﬀusion ﬂame. (See Fig. 6 on page 260)

a)

b)

Plate 39: a) Single periodic burner mesh (1.6 million cells) b) triple burner mesh (5 million cells). (See Fig. 3 on page 328)

434

Color Plates

Plate 40: Single burner conﬁguration. (See Fig. 4 on page 329)

1

2

3

4

Plate 41: Eigen-modes for both conﬁgurations (Helmholtz analysis). (See Table 2 on page 334)

Color Plates

435

Plate 42: Full LES computational domain: red, fuel inlet; blue, air inlet; purple, acoustic decoupling system; yellow, swirler vanes; green, outlet ﬂange. (See Fig. 2 on page 340)

(a)

(b)

(c)

Plate 43: Details of the computational domain: (a) swirler vanes, (b) acoustic decoupling system and (c) outlet ﬂange. (See Fig. 3 on page 340)

436

(a)

Color Plates

(b)

Plate 44: (a) Instantaneous view of the ﬂame (isosurface of temperature at 1200K) and of the methane jets (isosurface of fuel mass fraction at 0.1); (b) Mean heat release in central plane and isosurface of equivalence ratio φ = 0.6. (See Fig. 5 on page 343)

Plate 45: Phase locked heat release in the central plane and isosurface of equivalence ratio φ = 0.6. (See Fig. 8 on page 346)

Color Plates

437

Plate 46: Snapshot of the mass fraction of the sum of the PAH near the nozzle. (See Fig. 10 on page 397)

Plate 47: Two-dimensional representation of the shell to shell energy transfers. The horizontal coordinate corresponds to the receiving shell and the vertical coordinate to the giving shell. The u-to-u, b-to-b, u-to-b and b-to-u energy transfers are represented respectively in the top-left, top-right, bottom-left and bottom-right frames. (See Fig. 2 on page 407)

438

Color Plates

Plate 48: Two-dimensional representations of the Fourier (left) and physical space (+) (right) Tn,m (same interpretation as in Plate 47). (See Fig. 3 on page 408)

Plate 49: Two-dimensional representation of the shell-to-shell energy transfers in forced turbulence simulation. The u-to-u (left) and the b-to-b (right) energy transfers are observed to be direct and local. (See Fig. 4 on page 408)

Color Plates

439

Plate 50: Same as Plate 49 for the b-to-u (left) and the u-to-b (right) energy transfers. (See Fig. 5 on page 409)

(+)

Plate 51: Two-dimensional representations of the Fourier Tn,m,(k) (left) and physi(+) Tn,m,(x)

cal space (right) for forced turbulence (same interpretation as in Plate 47). (See Fig. 6 on page 410)

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