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Computational Fluid Dynamics

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D. Leutloff R. C. Srivastava (Eds.)

Computational Fluid Dynamics Selected Topics

With 125 Figures

Springer

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Dr. Dieter Leutloff

Technische Hochschule Darmstadt, Fachbereich Mechanik, Hochschulstrasse 1

D-64289 Darmstadt, Germany

Dr. Ramesh C. Srivastava

Gorakhpur University, Department of Mathematics and Statistics 273009 Gorakhpur, India

ISBN-13:978-3-642-79442-1 e-ISBN-13:978-3-642-79440 -7 DOl: 10.1007/978-3-642-79440-7

Library of Congress Cataloging-in-Publication Data. Computational fluid dynamics: se­lected topics / [edited by) D. Leutloff, R. C. Srivastava. p. cm. Papers in honor of Prof. K. G. Roesner.Includes bibliographical references.ISBN-13<978-3-(j,p-79442-11.Fluiddynamics-Matlt­ematical models. 2. Turbulent boundary layer-Mathematical models. 3. Nonlinear waves-Mathematical models. 4. Tsunamis-Mathematical models.!. Leutloff, D. (Dieter), 1942- . II. Srivastava, R. C. (Ramesh C.), 1953- . III. Roesner, K. G. TA357.C5878. 1995 532' .05' 015194-dc20 94-48557

This work is subject to copyright. All rights are reserved, whetlter 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 thereofis permitted only under tlte provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under tlte German Copyright Law.

©Springer-Verlag Berlin Heidelberg 1995 Softcover reprint of tlte hardcover lSt edition 1995

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, tltat such names are exempt from the relevant protective laws and regulations and therefore free for general use.

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Preface

This special volume is dedicated to Professor Karl G. Roesner on the occa­sion of his 60th birthday on the 16th January 1995. Professor Roesner has a large family of friends all over the world. Some of them have expressed their best wishes and respects which are contained in this volume in the form of scientific contributions. The papers in this volume are mostly in the area of computational fluid dynamics (CFD), which is Professor Roesner's favorite field of research. These papers cover almost all aspects of CFD. They cover diverse topics such as, the tsunami problem, group invariant solution of hy­drodynamic equations, non-linear waves, and the modelling of the problem of evaporation-condensation. There is also a survey article on exact solutions for discrete models of the Boltzman equation by one of the foremost experts in this area. Articles are also devoted to turbulent boundary layer problems and quasi-geostrophic drag on a sphere in a rotating fluid. The editors would like to express their gratitude to all the contributors, especially to Soubbara­mayer who offered to write the memorial for this volume. We are also thankful to Professor Beiglbock and Mrs. Beiglbock and their colleagues at Springer­Verlag, Heidelberg, for their friendly cooperation in producing this volume.

Darmstadt, July 1994 R. C. Srivastava D. Leutloff

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K.G. Roesner (1994)

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Dedication

This volume contains several articles written in honour of Professor Karl G. Roesner, Darmstadt Institute of Technology, on the 60th anniversary of his birthday.

Professor Roesner was born on January 16th, 1935, in Beuthen/Ober­schlesien, an industrial town close to the border of Poland, which is now in Poland. There he lived untill January 1945, the end of World War II. One can imagine the situation after the war, where there was no definite tomorrow and no prospective view for the future. This was the time he started his school career. This situation had a great impact on his life.

He started his studies in Physics at the University of Munster, changed to the Georg-August-University of Gottingen and got his diploma in Physics there. Gottingen offered to the students at that time many possibilities to qualify by doing research work. Professor Roesner decided to join the research group of the Max-Planck-Institut fur Stromungsforschung, the famous insti­tute where Ludwig Prandtl started to build up the modern Fluid Dynamics. At that time the first digital computers where installed in Germany, and Professor Roesner was one of the first who got used to that type of research tool, especially useful for the numerical analysis of gasdynamics. This early contact with computers had a long-lasting impact on his scientific work. He himself now jokes about that period as the 'stone-age period of computation', where scientists had to perform real physical work to let their programs run on the machine by carrying big packages of punched cards to the card reader.

At the Max-Planck-Institut he finished his dissertation devoted to gasdy­namics and obtained his doctorate in Physics in 1967. Some years later he went with his family to Freiburg and became an assistent at the Institute of Applied Mathematics of the Albert-Ludwigs-University, where Henry Gortler encouraged him to qualify and think about an academic career on the basis of a habilitation. In 1976 Professor Roesner finished his habilitation work and moved to Karlsruhe where he was incorporated into the Faculty of Mechani­cal Engineering as 'Privatdozent'. At the Institute of Fluid Flow and Fluid Machinery he gave lectures in the field of Numerical Fluid Mechanics with the aim to transfer his enthusiasm in numerical simulations of fluid flows to his students and to teach them how to apply this tool carefully and critically. As a physicist he looks at computers as an experimental device with all its preferences and drawbacks which the user has to take into account to get insight into the Physics he wants to investigate.

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His early numerical investigation on the method of characteristics was the subject about which he reported on the occasion of the 8th Biennial Sympo­sium on Advanced Problems and Methods in Fluid Dynamics 1967 in Poland. This was at that time the only chance to have contact with colleagues and sci­entists of the Eastern countries. One of them was the late Professor Yanenko from the Sibirian Branch of the Academy of Sciences of the Soviet Union. This encounter with the Russian mathematicians influenced very much Pro­fessor Roesners' further work. In 1970 he joined the Second International Conference on Numerical Methods in Fluid Dynamics at Berkeley, hosted by Maurice Holt. Here he came in contact with Henry Cabannes, Maurice Holt, and many other eminent personalities in Fluid Dynamics. He was included as one of the members of the International Organizing Committee for Compu­tational Fluid Dynamics Conferences. Since that time he has been promoting the idea to make the international family of numerical analysts as large as possible. Professor Roesner is an active member of that committee, repre­senting admirably the European computational fluid dynamicists. He plays an important role in the rigorous selection of papers to be presented at each conference. Having a paper selected by Professor Roesner is a guarantee of scientific high quality.

In 1980 he changed his place of work when he followed an offer to continue teaching and research work at the Darmstadt Institute of Technology in the Department of Mechanics. From this time on he began to design experiments for fluid flows in the laboratory parallel to their numerical simulation. The Workshops on Gases in Strong Rotation, which existed since 1975 and took place every two years, were extended on his instigation to include Seperation Phenomena in Liquid and Gases. Professor Roesner has generously hosted in Darmstadt the first of the new-version workshops. Let me tell you that the 'baby' of Professor Roesner, born in Darmstadt in 1987, has grown and is developing perfectly well. The last of the new series workshops was held in China 1994, and the next one will take place in Brasil in 1996. Hydrody­namic instability phenomena of rotating fluids especially attracted him from the point of view of how to visualize the velocity field without disturbing the flow field. He applied a visual technique on the basis of chemical reaction with photochromic compounds which made it possible to look at the microstruc­ture of the velocity field in totally closed cavities filled with liquid, whilst working on other topics such as boundary layer effects, numerical simula­tion of shock problems and separation phenomena in two phase flows, which are of great practical interest in the field of environmental problems. He has international scientific collaboration, and scientists from other countries are joining his group from time to time.

He is on the editorial board of the Computational Fluid Dynamic Jour­nal, and Computers and Fluids. He has translated many important Russian articles into English. He has made available to the West the knowledge of numerics and many branches of fluid dynamics through his translations. An important work by Yu.1. Shokin in Russian, 'The Method of First Differen-

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tial Approximation' was translated by Professor Roesner and published by Springer-Verlag in 1984.

A few comments about Roesners' human qualities. Very early in his life he was influenced by the destructive effects of war. During his student life in Gottingen he worked as an adviser to the foreign students, helping both those who asked for it and those whom he felt needed it. This idea has developed in him stronger with time. He does not consider lecturing as a mere obligation of an academic teacher but he enjoys teaching, always aiming to transfer some of his own enthusiasm for fluid dynamics to his students. In his every-day relationeships with students and co-workers he shows patience, compassion for personal problems and benevolence for their advancement. He prefers influencing his co-workers by his personal example, rather than by instruction and control. Roesner speaks, in addition to other languages, English, French, and Russian. In this way he has a strong medium of communication and can reach the people of different nations and different cultures.

Finally, I will give a brief glimpse of Professor Roesner's private life. In 1962 Roesner was married to Lotte. A daughter and two sons were born of this very harmonious marriage. Now the children have grown up and are pursuing their professional training. On the many occasions when I was a guest of the Roesners', I admired his great understanding and broad minded view towards his developing children.

During my stay in Darmstadt Mrs. Petra Leutloff invited me one evening to her home together with Professor Roesner and his wife. It was a really nice evening and we freely discussed different problems. Mrs. Petra was serving soft drinks when Mrs. Roesner made a very nice comment about Karl Roes­ner, my husband is like a 'steam engine'. I am sure this is a most suitable remark about Karl Roesner. Work is steam, work is spirit for him. But Mrs. Roesner is also a strong 'wheel' for this running steam engine.

I express my best wishes to Karl G. Roesner on his 60th birthday, also on behalf of all the authors who contributed to this volume. May he have the same joy in his scientific work in the years ahead. We wish Karl and Lotte many years of health and happiness.

Darmstadt, July 1994 R. C. Srivastava

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Table of Contents

Continuum Hypothesis in the Computation of Gas-Solid Flows J.Y. Tu and C.A.J. Fletcher. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. Introduction................................................. 1 2. Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Gas Phase .............................................. 2 2.2 Particulate Phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3. Numerical Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4. Numerical Results ........................................... 6 5. Conclusions................................................. 10

Numerical Modelling of Two- and Three-Dimensional External and Internal Unsteady Incompressible Flow Problems Gunter K.F. Barwolff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13

1. Introduction................................................. 13 2. The Mathematical Model - The Governing Equations ............ 13 3. The Numerical Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15 4. The Solution of the Equation Systems (3.9) and (3.10) - Solvability

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 5. The Application of the Solution Methods to Crystal Melt Flow .... 19

5.1 Wheeler's Benchmark Problem ............................ 19 5.2 The Numerical Simulation of Vertically Situated (Bio.5Sb1.5Te3)

Melt Zones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22 5.3 Further Investigations .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25

6. The External Flow Around a Cylinder ........................ " 25

Numerical Experiments in Double-Diffusive Convection R. Peyret and J .M. Vanel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33

1. Introduction................................................. 33 2. The Motion Equations and Their Numerical Solution. . . . . . . . . . . .. 37 3. Lateral Heating of a Stratified Fluid. . . . . . . . . . . . . . . . . . . . . . . . . . .. 40 4. Conclusion.................................................. 49

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New Potentialities of Computational Experiment in Tsunami Problem Yu.I. Shokin, G.S. Khakimsyanov, and L.B. Chubarov . . . . . . . . . . . . . .. 53

1. Introduction................................................. 53 2. Basic Mathematical Models ................................... 53 3. Methodological Principles of Some Applied Problems of Tsunami

by Computational Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58 3.1 Apriori Zoning of the Coast ...... . . . . . . . . . . . . . . . . . . . . . . . .. 58 3.2 On-Line Assessment of the Hazardous Wave Parameters. . . . .. 58

4. Computational Experiment Facilities Employed for the Training of the Tsunami Warning Service Personnel and of the Population in the Threatened Zones ........................................ 59

General Balance Equations for a Fluid-Fluid Interface in Magnetofluiddynamics Soubbaramayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63

1. Introduction................................................. 63 2. MFD Model: Integral Conservative Form. . . . . . . . . . . . . . . . . . . . . . .. 63 3. Balance Equations for the Interface ............................ 66 4. Application to the Calculation of the Surface Depression in a High­

Current Arc Weld Pool . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 66 5. Conclusion.................................................. 68

Group-Invariant Solutions of Hydrodynamics S.V. Coggeshall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71

1. Introduction................................................. 71 2. Lie Groups Applied to Differential Equations. . . . . . . . . . . . . . . . . . .. 72 3. Hydrodynamics Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75 4. One-Dimensional Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 78

4.1 Traditional Similarity Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . .. 78 4.2 Exponential Solutions .................................... 79 4.3 Projective Group Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80 4.4 Solutions Including Conduction. . . . . . . . . . . . . . . . . . . . . . . . . . .. 83 4.5 Solutions with Shocks .................................... 84 4.6 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86

5. Two-Dimensional Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86 5.1 Multiple Reductions Using Lie Groups. . . . . . . . . . . . . . . .. . . ... 86 5.2 Reductions to ODE's. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88

6. 3-D Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92 7. Analytic Solutions ........................................... 94

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Survey on Exact Solutions for Discrete Models of the Boltzmann Equation Henri Cabannes ................................................. 103

1. Introduction ................................................. 103 2. Broadwell Equations ......................................... 104

2.1 Self-Similar Solutions ..................................... 104 2.2 Cornille's Solutions ....................................... 106

3. Models with 14 Velocities ..................................... 107 4. Models with Triple Collisions .................................. 109 5. Two-Dimensional Semi-Continuous Model ....................... 111 6. Conclusion .................................................. 113

Boundary Conditions for Discrete Models of Gases and Applications to Couette Flows Amah D'Almeida and Renee Gatignol ............................. 115

1. Introduction ................................................. 115 2. Discrete Kinetic Theory ...................................... 116 3. Boundary Conditions ......................................... 118 4. Couette Flows ............................................... 122 5. Results ..................................................... 125

5.1 Study of the Tangential Velocity ........................... 125 5.2 Study of the Temperature ................................. 128

6. Conclusion .................................................. 129

Computation of Viscous Transonic Flow Around the F5 Wing Arthur Rizzi ................................................... 131

1. Introduction ................................................. 131 2. Governing Equations and Boundary Conditions .................. 132

2.1 Navier-Stokes Equations .................................. 132 2.2 Thrbulence Model ........................................ 134 2.3 Boundary Conditions ..................................... 135

3. Numerical Method ............................ " ............. 136 3.1 Spatial Discretization ..................................... 136 3.2 Numerical Damping ...................................... 138 3.3 Time Integration ......................................... 139 3.4 Initialization and Boundary Treatment ..................... 139 3.5 Stability ................................................ 140

4. Results ..................................................... 143 4.1 0-0 Mesh .............................................. 143 4.2 Case 1: a = 0° ........................................... 145 4.3 Case 2: a = 2° ........................................... 146

5. Conclusions ................................................. 147

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A Time-Dependent Space Marching Algorithm for Three-Dimensional PNS Equations Wang Ru-quan and Xue Ju-kui ................................. , . 151

1. Introduction ................................................. 151 2. Governing Equations ......................................... 152 3. The Flux-Difference Splitting .................................. 152 4. Explicit-Implicit Difference Scheme ............................. 154

4.1 Multi-Sweep Technique ................................... 156 4.2 Single-Sweep Technique ................................... 156

5. Numerical Tests ............................................. 157 5.1 The Hypersonic Flow Past the Sphere-Cone at High Angle of

Attack .................................................. 157 5.2 Hypersonic Flow Around the Simplified Space-Shuttle Orbiter. 157

6. Conclusions ................................................. 159

New Potential-Field Properties of General Laminar and Turbulent Motions of Newtonian Fluids H. Bischoff and E. Kaucher ...................................... 163

1. Introduction ................................................. 163 2. Physical and Mathematical Preliminaries ....................... 164

2.1 Mathematical Laws ...................................... 164 2.2 Physical Terminologies and Fundamental Laws .............. 169

3. Derivation of Some Field Properties of NSE ..................... 172 3.1 Relation Between Mechanical and Internal Energy ........... 172 3.2 The Dissipation Minimum Principle in the Stationary Case . ... 173 3.3 Potential Field Properties in the Stationary Case ............ 175

4. Thermodynamical Stability Criterion for Instationary Flows ....... 177 5. The Instationary Turbulent Flow in Temporal Mean .............. 182 6. Conclusion .................................................. 185

Boundary Layer Turbulence and the Control by Suction Y. Aihara ...................................................... 187

1. Introduction ................................................. 187 2. Flow Change with the Development of Boundary Layer ........... 188 3. Discussion on Energy ......................................... 190 4. Self-Organization of Turbulent Boundary Layer .................. 191 5. Experiment on Control of Turbulent Boundary Layer ............. 192 6. Conclusions ................................................. 195

On the Quasi-Geostrophic Drag on a Rising Sphere in a Rotating Fluid M. Ungarish .................................................... 197

1. Introduction ................................................. 197

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2. Solution .................................................... 198 3. Results ..................................................... 199

Two-Dimensional Nonlinear Saturation Behaviour of Instability Waves in a Boundary Layer at Mach 5 N .A. Adams and 1. Kleiser ....................................... 203

1. Introduction ................................................. 203 2. Governing Equations ......................................... 204 3. Numerical Method ........................................... 206 4. Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 5. Perturbation Evolution ....................................... 207 6. Evolution of the Flow Field ................................... 211 7. Shock Identification .......................................... 213 8. Conclusions ................................................. 216

Inertial Convection in Turbulent Rayleigh-Benard Convection at Small Prandtl Numbers G. Grotzbach and M. Worner ..................................... 219

1. Introduction ................................................. 219 2. Simulation Method ........................................... 221 3. Case Specifications and Initial Data ............................ 222 4. Results..................................................... 225

4.1 Verification ............................................. 225 4.2 Flow Mechanisms and Dynamics ........................... 226 4.3 Heat Transfer Statistics ................................... 229

5. Conclusions ................................................. 229

The GRP Treatment of Flow Singularities M. Ben-Artzi, A. Birman, and J. Falcovitz ......................... 233

1. Tracking of Singularities ...................................... 236 2. Geometrical Singularities ..................................... 238 3. Narrow Reaction Zones in Combustion Calculations .............. 240

Application of the Multidomain Local Fourier Method for CFD in Complex Geometries 1. Vozovoi, M. Israeli, and A. Averbuch ............................ 245

1. Introduction ................................................. 245 2. Multidomain Local Fourier Method (MDLF) .................... 247 3. Problems in Complex Geometries .............................. 249 4. Preconditioned Iteration Method with Spectral Preconditioner ..... 250 5. Demonstration for the N avier-Stokes System .................... 251

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Confined Swirling Flows - A Continuing Challenge Pinhas Bar-Yoseph .............................................. 257

1. Introduction ................................................. 257 2. Statement of the Problem ..................................... 257

2.1 Disk-Cylinder System .................................... 259 2.2 Spherical Annulus ........................................ 259 2.3 Cylinderical Annulus ..................................... 260 2.4 Sphere-Capsule System .................................. 260

3. Numerical Formulation ....................................... 260 4. Results and Discussion ....................................... 261

4.1 Disk-Cylinder System .................................... 261 4.2 Spherical Annulus ........................................ 262 4.3 Cylinderical Annulus ..................................... 262 4.4 Sphere-Capsule System ................................... 264

5. Conclusions ................................................. 266 5.1 Present Study ........................................... 266 5.2 Future Trends ........................................... 266

Eshelbian Continuum Mechanics and Nonlinear Waves G.A. Maugin ................................................... 269

1. Introduction ................................................. 269 2. Momentum and Pseudo-Momentum; Force and Pseudo-Force ...... 270 3. Field Formulation ............................................ 273 4. Non-Equivalence Between Global Formulations .................. 275 5. The Role of Pseudomomentum and Energy in Nonlinear-Wave

Propagation ................................................. 277 5.1 General Features ......................................... 277 5.2 First Example: The "Good" Boussinesq (GoB) Equation ...... 279 5.3 Second Example: The "Generalized" Boussinesq Equation (GB) 279 5.4 Material Global Forces as Perturbations .................... 282

6. A Newtonian Mechanics for Global Material Forces? ............. 284

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List of Contributors

N.A.Adams DLR, Institute for Theoretical Fluid Mechanics Bunsenstraf3e 10 D-37073 Gottingen, Germany

Y.Aihara Department of Aeronautics and Astronautics The University of Tokyo Tokyo, Japan

A.Averbuch School of Mathematical Sciences Tel Aviv University Tel Aviv 69978, Israel

P. Bar-Yoseph Computational Mechanics Laboratory Faculty of Mechanical Engineering Technion, Haifa 32000, Israel

G.K.F. Barwolff Technische Universitiit Berlin Hermann-Fottinger-Institut Rudower Chaussee 6 D-12484 Berlin, Germany

M. Ben-Artzi Institute of Mathematics Hebrew University Jerusalem 91904, Israel

A.Birman Dept. of Physics Technion-Israel Haifa 32000, Israel

H. Bischoff Institut fur Wasserbau Technische Hochschule Darmstadt D-64287 Darmstadt, Germany

H. Cabannes Lab. Modelisation en Mecanique, Associe au CNRS Universite Pierre et Marie Curie 4 Place J ussieu F-75005 Paris, France

L.B. Chubarov Institute of Computational Technologies, Siberian Branch of the Russian Academy of Science pr. Lavrentyeva 6 Novosibirsk 630090, Russia

S.V. Coggeshall Los Alamos National Laboratory, Los Alamos, NM 87545, USA

A. D'Almeida Lab. Modelisation en Mecanique, Associe au CNRS Universite Pierre et Marie Curie 4 Place J ussieu F -75252 Paris Cedex 05, France

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J. Falcovitz Dept. of Aerospace Engineering Technion-Israel Haifa 32000, Israel

C.A.J. Fletcher CANCES The University of New South Wales Sydney 2052, Australia

R. Gatignol Lab. Modelisation en Mecanique, Associe au CNRS Universite Pierre et Marie Curie 4 Place Jussieu F -75252 Paris Cedex 05, France

G. Grotzbach KfK, Institut fur Reaktorsicherheit, Postfach 3640 D-76021 Karlsruhe, Germany

M. Israeli Faculty of Computer Science Technion, Haifa 32000, Israel

E.Kaucher Institut fur Angewandte Mathematik U niversitiit Karlsruhe D-76128 Karlsruhe, Germany

G.S. Khakimsyanov Institute of Computational Technologies Siberian Branch of the Russian Academy of Science pro Lavrentyeva 6 Novosibirsk 630090, Russia

L. Kleiser DLR, Institute for Theoretical Fluid Mechanics BunsenstraBe 10 D-37073 Gottingen, Germany

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G.A.Maugin CNRS-Laboratoire de Modelisation en Mecanique Universite Pierre et Marie Curie Paris, France

R. Peyret Laboratoire de Mathematiques, CNRS URA 168 Universite de Nice-Sophia Antipolis Parc Valrose F-06108 Nice Cedex 2, France

A. Rizzi The Royal Institute of Technology KTH, S-10044 Stockholm, Sweden

Yu.I. Shokin Institute of Computational Technologies,Siberian Branch of the Russian Academy of Science pr. Lavrentyeva 6 Novosibirsk 630090, Russia

Soubbaramayer Departement des Pro cedes d 'Enrichissement CEA-Saclay F-91191 Gif sur Yvette Cedex, France

J.Y.Tu CANCES The University of New South Wales Sydney 2052, Australia

M. Ungarish Department of Computer Science Technion, Haifa 32000, Israel

J.M. Vanel Laboratoire de Mathematiques, CNRS URA 168 Universite de Nice-Sophia Antipolis Parc Valrose F-06108 Nice Cedex 2, France

Page 17: Computational Fluid Dynamics - Home - Springer978-3-642-79440-7/1.pdf · computational fluid dynamics (CFD), which is Professor Roesner's favorite ... senting admirably the European

L. Vozovoi Faculty of Computer Science Technion, Haifa 32000, Israel

R. Wang The Computing Center Academia Sinia Beijing, China

M.Worner KfK, Institut fur Reaktorsicherheit, Postfach 3640 D-76021 Karlsruhe, Germany

J.Xue Northwestern Normal University Lanzhou

XIX


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