Aerodynamic Drag Reduction Technologies: Proceedings of the CEAS/DragNet European Drag Reduction Conference, 19–21 June 2000, Potsdam, Germany (Notes ... Mechanics and Multidisciplinary Design, 76) 354041911X, 9783540419112

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90

Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM)

Editors E. H. Hirschel/München K. Fujii/Kanagawa W. Haase/München B. van Leer/Ann Arbor M. A. Leschziner/London M. Pandolfi/Torino J. Periaux/Paris A. Rizzi/Stockholm B. Roux/Marseille Y. I. Shokin/Novosibirsk

New Developments in Computational Fluid Dynamics Proceedings of the Sixth International Nobeyama Workshop on the New Century of Computational Fluid Dynamics, Nobeyama, Japan, April 21 to 24, 2003

Kozo Fujii Kazuhiro Nakahashi Shigeru Obayashi Satoko Komurasaki (Editors)

ABC

Professor Dr. Kozo Fujii

Professor Dr. Shigeru Obayashi

Institute of Space and Astronautical Science (ISAS) Yoshinodai, Sagamihara 3-1-1, 229-8510 Kanagawa Japan

Institute of Fluid Science Tohoku University Katahira 2-1-1, 980 Sendai Japan

Professor Dr. Kazuhiro Nakahashi

Professor Dr. Satoko Komurasaki

Department of Aeronautics/ Space Engineering Tohoku University Katahira 1-1-2, 980-8577 Sendai Japan

College of Science & Technology Department of Mathematics Nihon University Kanda-Surugadai 1-8 101-0062 Tokyo Japan

Library of Congress Control Number: 2005930441 ISBN-10 3-540-27407-3 Springer Berlin Heidelberg New York ISBN-13 978-3-540-27407-0 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005
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NNFM Editor Addresses

Prof. Dr. Ernst Heinrich Hirschel (General editor) Herzog-Heinrich-Weg 6 D-85604 Zorneding Germany E-mail: [email protected]

Prof. Dr. Kozo Fujii Space Transportation Research Division The Institute of Space and Astronautical Science 3-1-1, Yoshinodai, Sagamihara, Kanagawa, 229-8510 Japan E-mail: fujii@flab.eng.isas.jaxa.jp

Dr. Werner Haase Höhenkirchener Str. 19d D-85662 Hohenbrunn Germany E-mail: [email protected]

Prof. Dr. Bram van Leer Department of Aerospace Engineering The University of Michigan Ann Arbor, MI 48109-2140 USA E-mail: [email protected]

Prof. Dr. Michael A. Leschziner Imperial College of Science, Technology and Medicine Aeronautics Department Prince Consort Road London SW7 2BY U. K. E-mail: [email protected]

Prof. Dr. Maurizio Pandolfi Politecnico di Torino Dipartimento di Ingegneria Aeronautica e Spaziale Corso Duca degli Abruzzi, 24 I-10129 Torino Italy E-mail: pandolfi@polito.it Prof. Dr. Jacques Periaux Dassault Aviation 78, Quai Marcel Dassault F-92552 St. Cloud Cedex France E-mail: [email protected] Prof. Dr. Arthur Rizzi Department of Aeronautics KTH Royal Institute of Technology Teknikringen 8 S-10044 Stockholm Sweden E-mail: [email protected] Dr. Bernard Roux L3M – IMT La Jetée Technopole de Chateau-Gombert F-13451 Marseille Cedex 20 France E-mail: [email protected] Prof. Dr. Yurii I. Shokin Siberian Branch of the Russian Academy of Sciences Institute of Computational Technologies Ac. Lavrentyeva Ave. 6 630090 Novosibirsk Russia E-mail: [email protected]

Preface

It is a joyful and fitting moment that we, the friends, colleagues and supporters of Prof. Kunio Kuwahara, dedicate this Workshop to Prof. Kuwahara. We gathered in late April of 2003 in the tranquility of Nobeyama mountain resort to commemorate the 60th birthday of Prof. Kuwahara which had fallen in November, 2002. In the cultural backdrop of East Asia, the 60th birthday carries additional significance. Looking back on the occasion of Kan-re-ki (the 60th birthday), a man is supposed to have accomplished something of meaningfulness and value. With these undertones, it will be a useful exercise to recount the splendid accomplishments of Prof. Kuwahara. The major professional achievements of Prof. Kuwahara may be compressed into two main categories. First and foremost, Prof. Kuwahara will long be recorded as the front-line pioneer in using numerical computations to tackle complex problems in fluid mechanics. His unquenching zeal in computation and utilization of computers is unmatched throughout the globe. His infatuation with the Supercomputers of 1980’s and 1990’s is now a legend in the fluid dynamics communities. He continues to stand tall on the leading edge of computational fluid mechanics research and industrial applications. In short, Prof. Kuwahara has filled in a chapter in the history of modern fluid dynamics research. Another important contribution of Prof. Kuwahara has been made in the training and fostering of talented manpower of computational mechanics research. The early-time members of Prof. Kuwahara’s private research institute in Tokyo were more of recipients of non-profit research fellowships rather than employees of a commercial establishment. These then-young technical staff of Prof. Kuwahara grew up to dominate the computational sectors of industries and research circles of Japan. Prof. Kuwahara has had the vision and energy to nurture the next-generation developers and practitioners of computational fluid dynamics. We have no doubt that Prof. Kuwahara will exert ceaseless effort and enthusiasm to uplift the status of computational fluid dynamics to the next stratum. We expect to see Prof. Kuwahara in good health and high spirit in the years to come. A happy and prosperous 60th birthday of Prof. Kuwahara. Prof. Jae Min Hyun, Friends of Prof. Kuwahara

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Biography

Kunio Kuwahara Prof. Kunio Kuwahara was born on November 30, 1942 in Tokyo, Japan. He received his undergraduate and graduate education at the University of Tokyo, majoring in physics. In 1975, he obtained a doctor’s degree of Science from the University of Tokyo. He started his academic career in 1970 as Assistant Professor in the Department of Applied Physics at the University of Tokyo. After one year at the NASA Ames Research Center as a NRC Senior Research Associate, he joined the Institute of Space and Astronautical Science in 1981. Prof. Kuwahara has made significant and seminal contributions to the computational fluid dynamics since the infancy of the field. He pioneered the vortex method in his doctor’s thesis. This simple, but physically insightful approach, has been continuously improved by many researchers since then. From seventies to eighties, he and co-workers developed the basic numerical methods for solving the equations of incompressible and compressible fluid flows. The algorithms have been used for simulations of the three-dimensional, unsteady flows dominated by dynamic vortices. These successes opened a new frontier in CFD for both flow physics and flow engineering. Prof. Kuwahara’s contributions to CFD have been recognized by a number of major awards. He received the Awards from the Japan Society of Automobile Engineers and the Japan Society of Mechanical Engineers in 1992, the Computational Mechanics Achievement Award from the Japan Society

Biography

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of Mechanical Engineers in 1993. He also received the Max Planck Research Award in 1993. The editors are pleased to bring together the researchers who have contributed to this volume to express our thanks to Prof. Kuwahara. We wish Prof. Kuwahara continued success for many years to come. Kozo Fujii Kazuhiro Nakahashi Shigeru Obayashi Satoko Komurasaki

Contributions of Kunio Kuwahara to Computational Fluid Dynamics

K. Kuwahara and I. Imai, “Steady, Viscous Flow within a Circular Boundary”, The Physics of Fluids Supplement, Vol.12, 1969. K. Kuwahara and H. Takami, “Numerical Studies of Two-Dimensional Vortex Motion by a System of Point Vortices”, Journal of The Physical Society of Japan, Vol.34, No.1, 1973. K. Kuwahara, “Numerical Study of Flow past an Inclined Flat Plate by an Inviscid Model”, Journal of the Physical Society of Japan, Vol.34, No.1, 1973. H. Takami and K. Kuwahara, “Numerical Study of Three-Dimensional Flow within a Cubic Cavity”, Journal of Physical Society of Japan, Vol.37, No.6, 1974. K. Kuwahara, “Study of Flow past a Circular Cylinder by an Inviscid Model”, Journal of the Physical Society of Japan, Vol.45, No.1, 1978. K. Kuwahara and Y. Oshima, “Thermal Convection Caused by Ring-Type Heat Source”, Journal of Physical Society of Japan, Vol.51, No.11, pp3711-3719, 1982. S. Obayashi and K. Kuwahara, “An Approximate LU Factorization Method for the Compressible Navier-Stokes Equations”, Journal of Computational Physics, Vol.63, No.1, pp157-167, 1986. T. Kawamura, H. Takami and K. Kuwahara, “Computation of High-ReynoldsNumber Flow around a Circular Cylinder with Surface Roughness”, Fluid Dynamics Research, pp145-162, 1986. Y. Shida, K. Kuwahara, K. Ono and H. Takami, “Computation of Dynamic Stall of a NACA-0012 Airfoil”, AIAA Journal, Vol.25, No.3, 1987. K. Naitoh and K. Kuwahara, “Large Eddy Simulation and Direct Simulation of Compressible Turbulence and Combusting Flows in Engines Based on the BISCALES Method”, Fluid Dynamics Research, Vol.10, pp299-325, 1992. H. Suito, K. Ishii and K. Kuwahara, “Simulation of Dynamic Stall by MultiDirectional Finite-Difference Method”, AIAA Paper 95-2264, 1995. K. Kuwahara, “Unsteady Flow Simulation and Its Visualization”, AIAA Paper 99-3405 (Invited), 1999. K. Kuwahara and S. Komurasaki, “Semi-Direct Simulation of a Flow Around a Subsonic Airfoil”, AIAA Paper 2000-2656, 2000. K. Kuwahara and S. Komurasaki, “Direct Simulation of a Flow around a Subsonic Airfoil”, AIAA Paper 2001-2545, 2001. K. Kuwahara and S. Komurasaki, “Simulation of High Reynolds Number Flows Using Multidirectional Upwind Scheme”, AIAA Paper 2002-0133, 2002. A. Bethancourt, K. Kuwahara and S. Komurasaki, “Grid Generation and Unsteady Flow Simulation around Bluff Bodies”, AIAA Paper 2003-1129, 2003. J. Ooida and K. Kuwahara, “Implicit LES of Turbulence Generated by a Lattice”, AIAA Paper 2003-4097, 2003. S. Komurasaki and K. Kuwahara, “Implicit Large Eddy Simulation of a Subsonic Flow around NACA0012 Airfoil” AIAA Paper 2004-594, 2004.

Table of Contents

A Multidomain Spectral Collocation Method for Computational Electromagnetics with Application to Optical Waveguides . . . . . . . . . . C.C. Huang and J.Y. Yang 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Formulations of Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Multidomain Spectral Collocation Method . . . . . . . . . . . . . . . . . 3.2 Interfacial Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Cardinal Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Determination of Scaling Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Numerical Results and Disscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Fundamental Mode of Symmetric Three-Layer Step-Index Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Asymmetric Planar Waveguide with Exponential RIP . . . . . . . 5.3 Semiconductor Rib Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermonuclear Supernovae: Combining Astrophysical and Terrestrial Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. S. Oran 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Thermonuclear Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Transition from a Deflagration to a Detonation . . . . . . . . . . . . . . . . . 4 Conclusions to Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypersonic Magneto-Aerodynamic Interaction . . . . . . . . . . . . . . . . . . . . . J. S. Shang 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Experimental Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Non-Equilibrium Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Electromagnetic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Remarks on the CFD Research for Space Transportation System Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Fujii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 RLV Aerodynamics – CFD Capability for the Estimation of RLV Aerodynamics- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 3 4 4 5 5 5 6 7 9 11 11 12 15 19 23 23 24 24 27 31 31 34 34 36

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1.2 Supersonic Base Flows -Finding a New Efficient Tools- . . . . . . 2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40 44

Development of Implicit Large Eddy Simulation . . . . . . . . . . . . . . . . . . . K. Kuwahara, S. Komurasaki, J. Ooida, A. Betancourt 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Rearrangement of Karman Vortex Street . . . . . . . . . . . . . . . . . . 3.2 Transition to Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Turbulence Generated by a Lattice . . . . . . . . . . . . . . . . . . . . . . . 3.4 Circular Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Subsonic Flow Around an Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Simulation with Using a Body-Fitted Grid . . . . . . . . . . . . . . . . .

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DNA Computing Based on Actual Biological Sequences and Accurate Reaction Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Naitoh 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 DNA Computing as Nano-Computing . . . . . . . . . . . . . . . . . . . . . . . . . 3 DNA Computing Using Actual Living Organisms . . . . . . . . . . . . . . . 3.1 Problem Solved in this Session . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Encoding Based on the Segments of Biological DNA Cut by Restriction Enzymes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Library of the Candidates Obtained with Ligation Reactions and Analysis of the Electrophoresis Photographs . . 4 Strategy for Controlling Polymerase Chain Reaction (PCR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 PCR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Control of PCR for DNA Computing . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From One-Month CFD to One-Day CFD – Efforts for Reducing Time and Cost of CFD – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Nakahashi, T. Fujita and Y. Ito 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Mesh Generation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Surface Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Parallel Flow Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Basic Flow Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Parallel Flow Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 JAXA Supersonic Airplane in Ascending Flight . . . . . . . . . . . . 4.2 Flow around a Hornet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 48 52 52 53 54 55 55 57 60 60 61 61 61 61 62 64 64 64 66 68 68 69 69 69 72 72 73 74 74 77

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5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Numerical Attempts of Capturing Contact Surface . . . . . . . . . . . . . . . . . K. Sawada, N. Ohnishi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Scheme-A (Lagrangian Based Approach) . . . . . . . . . . . . . . . . . . 2.2 Scheme-B (Eulerian Based Approach) . . . . . . . . . . . . . . . . . . . . . 3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 82 82 83 86 89

Parallel Visualization Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 H. Miyachi, S. Hayashi, Y. Nakai, Y. Itoh, M. Shirazaki and R. Himeno 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2 Visualization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3 Various Parallel Visualization Systems . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.1 Parallel Volume Rendering System . . . . . . . . . . . . . . . . . . . . . . . 92 3.2 On-Demand Parallel Rendering System[1] . . . . . . . . . . . . . . . . . 93 3.3 Parallel Visualization System for Cluster Machine . . . . . . . . . . 95 4 Performance of the Parallel Visualization Systems . . . . . . . . . . . . . . . 96 4.1 Performance of On-Demand Rendering System . . . . . . . . . . . . . 96 4.2 Performance of Parallel Visualization System for Cluster Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Numerical Simulation of Time-Dependent Buoyancy-Driven Convection in an Enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.S. Kwak and J.M. Hyun 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Advances in Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Resonant Convection Responding to Periodic Excitations . . . . . . . . 4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Automatic Topology Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P.R. Eiseman and K. Rajagopalan 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 GridPro’s Topological Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Cartesian Approach to Automatic Topology Generation . . . . . . . . . . 3.1 Cartesian Topology Generation in Three Dimensions . . . . . . . . 3.2 Improvements to the Cartesian Topology Generation Technique in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Automatic Nested Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Application of Nest to a Thin Curved Wavy Tube in a Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102 102 103 106 109 112 112 113 114 116 119 119 122

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Computational Study of Influences of a Seam Line of a Baseball on Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Himeno 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Computation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 MAC Based Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Third-Order Upwind-Difference Scheme . . . . . . . . . . . . . . . . . . . 3 Grid System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Shape of the Seam on the Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Computed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Four-Seam Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Two-Seam Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vortex Breakdown Flows in Cylindrical Geometry . . . . . . . . . . . . . . . . . . R. Iwatsu 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Top Rigid-Cover (Case A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Top Free-Slip-Condition (Case B) . . . . . . . . . . . . . . . . . . . . . . . . Aerodynamic and Structural Analyses of Joined Wings of Hale Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Sivaji, S. Marisarla, V. Narayanan, V. Kaloyanova, U. Ghia and K. N. Ghia 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Aerodynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 1-D Approximation of 3-D Joined-Wing Structure: Aerodynamics-Structure Coupling. . . . . . . . . . . . . . . . . . . . . . . . 3.2 Box-Wing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Reinforced Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Aerodynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Simulation of Formation and Movement of Various Sand Dunes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Zhang, T. Kawamura and M. Kan 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Calculation of the Air Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 125 126 126 126 127 127 128 130 130 135 136 141 141 143 144 144 147 152

152 153 154 154 155 155 156 156 158 162 165 165 166 167

Table of Contents

XV

2.2 Estimation of Sand Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Determination of the Shape of the Sand Dunes . . . . . . . . . . . . . 3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Barchan Dunes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Transverse Dunes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Linear Dunes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Chinese Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167 167 168 168 169 169 172 173

Evolutionary Multi-Objective Optimization and Visualization . . . . . . . S. Obayashi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Evolutionary Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . . 2.1 MOGAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 CFD Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Neural Network and SOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Cluster Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Four-Objective Optimization for Supersonic Wing Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Formulation of Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Visualization of Design Tradeoffs: SOM of Tradeoffs . . . . . . . . 3.3 Data Mining of Design Space: SOM of Design Variables . . . . . 4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175

The Framework of a System for Recommending Computational Parameter Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Shirayama 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Parameter Recommendation System . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Parameter Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Framework of the Recommendation System . . . . . . . . . . . . . . . . 3 Components of the Parameter Recommendation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Parameter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Search Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Optimization by GA and Rule Extraction by Schema Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 System Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Framework Verification by Numerical Experiments . . . . . . . . . . . . . . 5.1 Experimental System 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Numerical Experiments Using System 1 and Discussion . . . . . 5.3 Experimental System 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 177 177 177 177 178 179 179 180 181 183 186 186 187 187 188 189 189 190 190 191 192 192 193 194 195 196

XVI

Table of Contents

Gas flow in Close Binary Star Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Matsuda, K. Oka, I.. Hachisu, H.M.J. Boffin 1 Accretion Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Numerical Simulations of Gas Flow in a Close Binary System 1.2 Modern Calculation of Accretion Flow . . . . . . . . . . . . . . . . . . . . 1.3 Discovery of Spiral Shocks by Observation . . . . . . . . . . . . . . . . . 2 Flow on a Companion Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Flow Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Doppler Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Simulation using Combined Compact Difference Scheme with Spectral-like Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tomonori Nihei (CSE, Nagoya University), Katsuya Ishii (ITC, Nagoya University) 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Combined Compact Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 3 Simulation of Incompressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Poisson Equation Solver for Two-Dimensional Problems . 3.2 The Poisson Equation Solver for Three-Dimensional Problems 3.3 Three-dimensional incompressible lid-driven cavity flow . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A New High Order Finite Volume Method for the Euler Equations on Unstructured Grids . . . . . . . . . . . . . . . . . . . . Z.J. Wang 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Idea of the Spectral Volume Method . . . . . . . . . . . . . . . . . . . . . 3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Accuracy Study with Vortex Evolution Problem . . . . . . . . . . . . 3.2 Scattering Of Periodic Acoustic Source from Two Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

198 198 198 200 202 202 202 203 206

206 207 209 209 211 212 213 215 215 216 217 217 220 222

Participants Photograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Participants List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 The Sixth International Nobeyama Workshop on the New Century of Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

A Multidomain Spectral Collocation Method for Computational Electromagnetics with Application to Optical Waveguides C.C. Huang and J.Y. Yang Computational Electromagnetics and Plasma Lab., Institute of Applied Mechanics, National Taiwan University, Taipei 10764, Taiwan, ROC

Summary A novel solution method using multidomain spectral collocation method for computational electromagnetics with application to the modal analysis of optical waveguides is proposed. The use of domain decomposition divides the computational domain into a few subdomains at the interfaces of different materials. The fields in each subdomain can be expanded by a set of orthogonal basis functions and patched by applying the electromagnetic boundary conditions. A new technique for the a priori determination of the scaling factor in Gauss-type basis functions is introduced to reduce the computational effort choosing the optimum value. The high accuracy of spectral collocation method is kept by the present method but the poor convergence is improved. Computations of several 2-D and 3-D waveguide structures have been tested for the accuracy and efficiency of the present method. Detailed comparisons of the present results with exact solutions or previously published data based on other methods are included.

1

Introduction

Accurate analysis and simulation of dielectric waveguides is essential for designing and fabricating photonics integrated circuits. Many devices are concerned with the interactions between modes of waveguides, hence modal analysis is important and necessary to realize performances of these components. In the past decade, the powerful series expansion method based on Galerkin algorithm [1–3] with distinct basis functions and mathematical formulations has been applied to optical waveguides and optical fiber. A novel Hermite-Gauss (HG) orthogonal collocation method with single domain based on scalar wave equation is proposed by Sharma and Banerjee [4,5] for solving propagation characteristics of optical waveguides. The orthogonal collocation method not only avoids the evaluations of laborious integral elements as undertaken by a Ritz-Galerkin method for graded refractive index profile (RIP) problems but also attained the same convergence rate. However, the results obtained by using the orthogonal collocation method with

2

C.C. Huang and J.Y. Yang

usual single domain usually oscillate about the exact values and show relatively poor convergence for discontinuous RIP structures [5]. Consequently, the advantages of rapid convergence and high accuracy of orthogonal collocation method are lost. Besides, the scaling factor used by [5] was adopted by additional trial and error to determine it. Hence, the additional computational time is needed. As a result, we have proposed an efficient method which employs multidomain SCM to tackle these oscillatory solutions and poor convergence problems associated with single domain SCM and utilize a new technique to the a priori determination of scaling factor to improve computational efficiency.

2

Formulations of Wave Equations

Assuming a wave propagates along longitudinally (i.e., z-direction) invariant structure with the form exp(j(ωt − βz)), where j 2 = −1, and ω and β are the angular frequency and propagation constant along z-direction, respectively. Maxwell’s two curl equations can be written as follows: →

→

∇× E = −jωµ0 H →

(1) →

∇× H = jωε0 n2 (x, y) E

(2)

where µ0 and ε0 denote the permeability and permittivity in vacuum, respectively, and n(x, y) denotes the transverse RIP of waveguide. The vector wave equation based on the magnetic field vector is thus given by
∇n2 × (∇ × H) = 0 (3) n2 The desired coupled equations in terms of the transverse magnetic field components Hx and Hy are obtained as follows:





∇2 H + k02 n2 H +

∂ 2 Hx 1 ∂n2 ∂Hy ∂ 2 Hx 1 ∂n2 ∂Hx + 2 =0 + + k02 (n2 − n2e )Hx − 2 2 2 ∂x ∂y n ∂y ∂y n ∂y ∂x

(4)

∂ 2 Hy 1 ∂n2 ∂Hx ∂ 2 Hy 1 ∂n2 ∂Hy 2 2 2 + =0 (5) + + k (n − n )H − y 0 e ∂x2 ∂y 2 n2 ∂x ∂x n2 ∂x ∂y where k0 is the wavenumber in free space and ne = β/k0 is the mode effective index. For 2-D waveguide structures, if the light is only confined along ydirection, then we have the formulas 1 ∂n2 ∂Hx ∂ 2 Hx + k02 (n2 (y) − n2e )Hx = 0 − 2 2 ∂y n ∂y ∂y for TM modes and ∂ 2 Hy + k02 (n2 (y) − n2e )Hy = 0 ∂y 2 for TE modes.

(6)

(7)

Collocation Method for Computational Electromagnetics

3 3.1

3

Numerical Techniques

Multidomain Spectral Collocation Method

For the structures which involve discontinuous RIP, we divide the single interest domain at interfaces between different mediums into a number of subdomains. The unknown fields in each subdomain can be expanded by a distinct cardinal basis function. Subsequently, these subdomains are patched by physical boundary conditions of electromagnetic wave at the interfaces. The unknown fields Hx and Hy in each subdomain can be expanded by two sets of orthogonal basis functions which is called cardinal basis functions [6] and grid point values of Hx and Hy fields at (Nx + 1) × (Ny + 1) collocation x y and Hp,q as follows: points denoted as Hp,q Hx (x, y) =

Ny Nx



x Ap (x)Bq (y)Hp,q

(8)

y Ap (x)Bq (y)Hp,q

(9)

p=0 q=0

Hy (x, y) =

Ny Nx

p=0 q=0

where Aj (xi ) = δij and Bj (yi ) = δij and δij is the Kronecker delta. Substituting (8) and (9) into the equations (4) and (5), and requiring that (4) and (5) are perfectly satisfied at these (Nx + 1) × (Ny + 1) collocation points, we convert the two partial differential equations into an eigenvalue equation ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ S + P H H C x xy x x ⎣ ⎦ ⎣ ⎦ = (β)2 ⎣ ⎦, (10) Cyx S + Py Hy Hy where S,Px ,Py ,Cxy , and Cyx represent operators and the elements SHr (r = x, y), are respectively given by  2  ∂ Hr ∂ 2 Hr 2 2 2 + + k0 (n − ne )Hr |x=xi ,y=yj , (r = x, y) SHr = ∂x2 ∂y 2 Ny  Nx

(2) = A(2) p (x)Bq (y) + Ap (x)Bq (y) p=0 q=0

+ k02 (n2 (x, y) − n2e )Ap (x)Bq (y) |x=xi ,y=yj · [Hr ] (11)   2 1 ∂n ∂Hx P x Hx = − 2 |x=xi ,y=yj n ∂y ∂y ⎡ ⎤ Ny Nx
2
∂n (x, y) 1 (1) = ⎣− 2 Ai (x)Bj (y)⎦ |x=xi ,y=yj .[Hx ] (12) n (x, y) ∂y i=0 j=0 (h)

(h)

In (11)–(12), Ap (x) and Bq (y) indicate the h-th order derivatives of cardinal basis functions Ap (x) and Bq (y), respectively.

4

3.2

C.C. Huang and J.Y. Yang

Interfacial Boundary Conditions

Besides that the normal and tangential components of magnetic fields Hx and Hy at each intra-element boundary are continuous for both horizontal and vertical interfaces, another set of continuity conditions are Hz and Ez at → → → the interfaces through ∇ × H = jωε0 n2 (x, y) E and ∇· H = 0. First, we consider a horizontal interface ay y=0. According to the continuities of both Ez and Hz at the interface, we obtain the follows:



∂Hx

∂Hy 2 ∂Hx

n2y+ (13) − n = (n2y+ − n2y− ) y− y−

∂y ∂y y+ ∂x ∂Hy ∂Hy |y+ = |y− ∂y ∂y

(14)

where y+ and y− are referred to the locations at the infinitesimally upper and lower sides of the horizontal interface, respectively.

3.3

Cardinal Basis Functions

The cardinal basis function Cj (y) satisfying Cj (yi ) = δij of Chebyshev polynomials [11] for the independent variable x can be expressed as (−1)j+1 (1 − y 2 )TN (y) , cj N 2 (y − yj )

Cj (y) =

y = yj

(15)

where TN (y) denotes Chebyshev polynomial of order N , c0 = cN = 2 and cj = 1 (1 ≤ j ≤ N −1) and the prime denotes the first derivative with respect to y. The LG functions for semi-infinite interval subdomains are given by Cj (αy) =

αyLN (αy) e−αy/2 , e−αyj /2 (αyLN ) (αyj )(αy − αyj )

y = yj

(16)

where LN (αy) denotes Laguerre polynomials of order N and the roots of Laguerre polynomial LN (αy) of order N plus y0 = 0 are used to incorporate the boundary conditions as collocation points. Finally, using HG as basis functions and the roots of Hermite polynomials HN +1 (αy) of order N + 1 as collocation points, we have 2 2

Cj (αy) =

e−α

y /2

−α2 yj2 /2

e

HN +1 (αy) ,  HN +1 (αyj )(αy − αyj )

In (16) and (17), α is a scaling factor to be determined.

y = yj

(17)

Collocation Method for Computational Electromagnetics

4

5

Determination of Scaling Factor

Any orthogonal basis functions defined over an infinite interval, for instance, HG or LG implicitly contains a scaling factor α cited earlier to adjust the extension of the mapping. In general for a given N , the scaling factor α is usually picked up through a few trial and errors with additional expense of a considerable amount of computational effort [5]. Here, we combine two techniques, one from Tang [7] and the other, effective index method (EIM) [8] to provide an effective method to determine α for optical waveguide problems. In [7], Tang proposed a definite procedure to decide α for Gaussian-type functions that is expanded by the first N + 1 HG of order N . For a given number of N basis functions, the optimum α value can be defined via the following relationship: {xj } . (18) α = max 0≤j≤N M where {xj }N j=0 are the collocation points selected from the root of HG or LG basis function and M denotes the finite support. The finite support hasn’t yet defined for different physical problems in [12]. Hence, the finite support M remains uncertain and needs to be determined for unknown functions of differential equations. The finite support M is related by optical field profiles of guided modes one interests in. Moreover, we have to set a criterion to take κ as the ratio of the field strength at the position x = M relative to the field strength at turning point x = xt from [9]. Ramanujam et al. [9] took κ = 0.01 for sine functions applied in Galerkin method. The value κ = 0.01 is also taken in our examples. The detail derivation to determine finite support can refer to our previous work [10].

5

Numerical Results and Disscussion

Several numerical examples are checked to show the efficiency and accurate of the present method in this section. 5.1

Fundamental Mode of Symmetric Three-Layer Step-Index Waveguides

Figure 1 shows the geometry of the three-layer step-index waveguide, which supports single TE mode. The refractive indices of core and claddings are nc = 1.45 and ncl = 1.447636, respectively. The half width of the core is W = 3 µm and the wavelength is λ = 1.3 µm. The whole domain is divided into three subdomains. The Hy field in region I is expanded by Chebyshev polynomials, region II and region III are both expanded by LG. In addition, the finite support are calculated from [10] as MII = 15.77 µm and MIII = 15.77 µm for the two outer regions II and III, respectively.NI , NII and NIII

6

C.C. Huang and J.Y. Yang n(y)

II

I

III

nc

ncl

2W

y Figure 1 Refractive index profile for a three-layer step-index waveguide divided into three subdomains by the vertical dotted lines Table 1

Convergence of effective indices ne of a three-layer step index structure

Exact

Table 4 in [5]

This Work (MII = MIII = 15.77 µm)

ne

N

ne

NII

NI

NIII

α

ne

1.44890

40

1.44886

3

3

3

0.217

1.449005

50

1.44875

5

3

5

0.596

1.448978

60

1.44895

6

3

6

0.802

1.448978

68

1.44888

5

5

5

0.596

1.448898

6

5

6

0.802

1.448898

represent the number of terms of orthogonal basis functions for each region in Table 1 and αII , αIII are scaling factors evaluated by (18). In [5], N = 40−68 terms of HG in single domain were used, however the convergent results can’t be achieved yet. Using the present method, our solutions achieve the exact solution using merely 15 terms of orthogonal basis functions. 5.2

Asymmetric Planar Waveguide with Exponential RIP

The second example is asymmetric planar diffused waveguides with Exponential RIP. The exact solutions have been obtained in [8]. The RIP is described as follows: ⎧   ⎪ ⎨ n2s + 2ns ∆n exp − y , y ≥ 0 d (19) n2 (y) = ⎪ ⎩ 2 y 108 . With the advances in computational capability, numerical works have aimed at more computationally-intensive problems. The numerical studies today extend the coverage of Ra over 1010 and conduct three-dimensional computations with more than millions of grid points [6–9]. Another serious stream has been toward buoyant convection in more complex and realistic situations. Extensions have been attempted to deal with complicated geometry, realistic working fluids, practical thermal boundary conditions, etc.

3

Resonant Convection Responding to Periodic Excitations

Recently, increasing attention has been given to a separate class of unsteady problem, buoyant convection responding to time-periodic thermal boundary conditions [5,11–23]. The interest stems from the frequent occurrence of such situations in practical thermo-fluid systems, e.g., solar heating varying on a daily basis, and periodic energizing or cooling of an electronic device. The problems involve an important physical feature, i.e., the possibility of resonance; the external excitations with correct natural frequency may lead to substantial amplifications of eigenmodes of the system. The numerical work of Lage and Bejan [11] was the first to discuss the occurrence of resonance in the context of natural convection in an enclosure. They suggested a new conceptual model for time-dependent natural convection affected by an on-an-off mode of heating. The flow configuration was the same as shown in Fig. 1, and a periodic heat flux fluctuating in a square wave fashion was applied to the heating sidewall. It was disclosed that the fluctuation of the instantaneous Nusselt number in the cavity was substantially amplified at certain frequencies. Resonance was defined as a maximal amplification of fluctuation of heat transfer in the interior. This is related to the occurrence of maximum fluctuations of local velocity and temperature. The presence of resonance was reconfirmed by the numerical study of Kwak and Hyun [12] for a different type of thermal boundary condition. For a square cavity depicted in Fig. 1, the temperature at the heating sidewall was given by a sinusoidal function of time Th (t) = To + ∆T (1 + εT sin f t

(5)

where εT and f represent the amplitude and frequency of periodic thermal forcing. They monitored the temporal variation of the Nusselt number

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Fluctuating Amplitude of the Nusselt Number

0.8 0.6 0.4 0.2 0.0

0.1

1

10

Frequency, f / N Figure 2 Variation of the fluctuating amplitude of the Nusselt number at the central vertical plane vs. frequency. Ra = 107 , P r = 0.7, A = 1, εT = 0.1. The amplitude and frequency are normalized by the Nusselt number for the corresponding steady heating case and the Brunt-Vaisala frequency, N = (βg∆T /H)1/2 , respectively

averaged over the center vertical plane of the cavity. The key findings are reproduced in Fig. 2. The sharp peaks demonstrate the occurrence of resonance; the fluctuating amplitude of the Nusselt number is maximized at certain intermediate frequency. This feature becomes more distinct as the Rayleigh number increases for P r ∼ O(1). The task is to describe the nature of resonance. Among several attempts, the explanation of Kwak and Hyun [12] appears useful. There are several oscillatory modes found in a cavity with lateral heating (see, e.g., [4–9]. Their argument was that resonance was associated with one of these eigenmodes. In order to identify the resonating mode, time-dependent flow and temperature fields were analyzed by using numerical animations. It was seen that the isotherms in the interior exhibit a periodic tilting, and the flow fluctuations are not confined to the boundary layers; rather, they are most vigorous in the interior. These are unique only for the resonance cases. All these support the argument that the flow resonates with internal gravity waves in the stablystratified interior. The predictions based on the principal modes of internal gravity waves were in good agreement with the numerical results [12, 13]. The studies of Lage and Bejan [11] and Kwak and Hyun [12] were followed by the subsequent numerical studies. The presence of resonance was extended to various types of working fluids; e.g., a saturated porous medium with and without heat generation, and a non-Boussinesq fluid [14–17]. The physical aspects of resonant convection were scrutinized numerically: independent influences of amplitude of thermal forcing and the Prandtl number [14, 15, 17, 18], the effects of finite thickness and conductivity of the heating wall [19], and the effective way to suppress resonant fluctuations due to internal gravity waves [20].

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Advances have also been made on the experimental front. Antohe and Lage [21] conducted an experimental study for the problem of Lage and Bejan [11], which confirmed the presence of resonance of natural convection. More recently, Kim et al. [22] carried out precision experiments by devising an apparatus to enforce time-periodic mechanical disturbances to the flow. The experimental results demonstrated that the flow resonates with internal gravity wave oscillations. Considerable efforts have been devoted for practical utilization of resonant buoyant convection [18, 21, 23]. An expectation here is that the oscillatory heating may give rise to an augmentation of the time-mean heat transfer rate. This idea was previously discussed by Ho and Chu [23]; in the melting process in a sidewall-heated cavity, an unsteady boundary condition of a sinusoidally-varying wall temperature gave a faster melting than the steady heating condition. The numerical study of Kwak et al. [18] also showed that the sidewall temperature oscillation led to a measurable enhancement of the time-mean Nusselt number. An important notion here is that the maximum gain of the time-mean heat transfer was found near the resonance frequency at which maximal fluctuations of flow and heat transfer occur. The evidence was also found experimentally by Antohe and Lage [21]; the oscillatory heating fluctuating in a square wave fashion resulted in a 20% augmentation of the time-mean heat transfer rate in comparison to the corresponding steady heating with the same time-mean heat flux. A practical issue for utilization of resonant convection is how to control the flow. The problem is that it is difficult to regulate the frequency and amplitude of time-periodic thermal condition precisely. An important clue here is that the occurrence of resonance is independent of the type of external forcing, and dependent on the forcing frequency. This suggests an alternative way to apply time-periodic mechanical forcing, which is easy-to-control, instead of the periodic thermal condition. Such a possibility was explored previously and the presence of resonance was seen in the context of mixed convection [24–26]. Fu and Shieh [24] conducted a numerical investigation of thermal convection in a sidewall-heated cavity vibrating vertically in a sinusoidal manner, which showed the presence of resonant convection regime. More recently, Kim et al. [26] performed a similar exercise by considering three different types of external forcing; a time-periodic sidewall temperature variation as in (5), a horizontal vibration and a vertical vibration of the full system. The vibration was modelled by the modulation of gravity vector g (6) g = g(1 + εY sin f t) j + gεX sin f tI where i and j denote the unit vectors in the horizontal and vertical direction, respectively. εX and εY are the amplitudes of vibration in the i and j directions. The results are summarized in Fig. 3. Despite the differences in the type of external forcing, the characteristic features of resonant convection were essentially the same. There were similarities and analogies, both

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Fluctuating Amplitude of the Nusselt Number

8 6 4 2 0

0.0

0.5

1.0

1.5

2.0

Frequency, f /N Figure 3 Legends are the same as in Fig. 1. The amplitude is multiplied by the multiplication factor λ/ε · •, periodic thermal forcing (ε = εT = 0.1, εX = εY = 0, λ = 1); ◦, horizontal vibration (ε = εX = 0.01, εT = εY = 0, λ = 0.113)
, vertical vibration (ε = εY = 0.01, εT = εX = 0, λ = 4.14)

in qualitative and quantitative senses, among resonant convections responding to the above three types of external excitations. The implication is that oscillatory thermal forcing can be replaced by mechanical vibration with a proper amplitude. The experimental work by Kim et al. [22] confirmed this argument.

4

Concluding Remarks

With remarkable advances in computational capability, numerical simulations have become a very useful tool for investigation of unsteady thermal convection in a confined space. Today, numerical studies accommodate twodimensional computations up to Ra = 1010 and are challenging the threedimensional computations. Another stream is to explore a new conceptual idea by taking advantage of the benefits of simulation. An example is the illustration of resonant buoyant convection of an enclosed fluid responding to oscillatory heating. The physical mechanisms and practical utilizations have been examined. As a subsequent task, design of a practical device utilizing resonance phenomena is needed. More efforts are called for experimental verifications.

Acknowledgements Support was provided in part by “The National Program for Development of Advanced Machinery and Parts”.

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References [1] S. Ostrach, “Natural Convection Heat Transfer in Cavities and Cells”. Proceedings of the 7th International Heat Transfer Conference, Volume 6, 1982, pp. 365–379. [2] C.J. Hoogendoorn, “Natural Convection in Enclosures”. Proceedings of the 8th International Heat Transfer Conference, Volume 1, 1986, pp. 111–120. [3] S. Ostrach, “Natural Convection in Enclosures”. Trans. ASME Journal of Heat Transfer, Volume 110, 1988, pp. 1175–1190. [4] J.M. Hyun, “Unsteady Buoyant Convection in an Enclosure”. Advances in Heat Transfer, Volume 34, 1994, pp. 277–320. [5] H.S. Kwak and J.M. Hyun, “Unsteady Natural Convection in an Enclosure”. Proceedings of the 11th International Heat Transfer Conference, Volume 1, 1998, pp. 341–356. [6] J. C. Patterson and J. Imberger, “Unsteady Natural Convection in a Rectangular Cavity”. Journal of Fluid Mechanics, Volume 100, 1980, pp. 65–86. [7] R.J.A. Janssen and R.A.W.M. Henkes, “Influence of Prandtl Number on Instability Mechanisms and Transition in a Differentially Heated Square Cavity”. Journal of Fluid Mechanics, Volume 290, 1995, pp. 319–344. [8] P. Le Qu´er´e and M. Behina, “From Onset of Unsteadiness to Chaos in a Differentially Heated Square Cavity”. Journal Fluid Mechanics, Volume 359, 1998, pp. 81–107. [9] T. Fusegi, J.M. Hyun and K. Kuwahara, “Transient Three-Dimensional Natural Convection in a Differentially Heated Cubical Enclosure”. International Journal of Heat & Mass Transfer, Volume 34, 1992, pp. 1559–1564. [10] R.A.W.N. Henkes and P. Le Qu´er´e, “Three- Dimensional Transition of Natural Convection Flows” Journal of Fluid Mechanics, Volume 319, 1996, pp. 281–303. [11] J. L. Lage and A. Bejan, “The Resonance of Natural Convection in an Enclosure Heated Periodically from the Side”. International Journal of Heat & Mass Transfer, Volume 36, 1993, pp. 2027–2038. [12] H.S. Kwak and J.M. Hyun, “Natural Convection in an Enclosure Having a Vertical Sidewall with Time-Varying Temperature”. Journal of Fluid Mechanics, Volume 329, 1996, pp. 65–88. [13] H.S. Kwak, K. Kuwahara and J.M. Hyun, “Prediction of Resonance Frequency of Natural Convection in an Enclosure with Time-Periodic Heating Imposed on One Sidewall”. International Journal of Heat and Mass Transfer, Volume 41, 1998, pp. 3157–3160. [14] B.V. Antohe and J.L. Lage, “Amplitude Effect on Convection Induced by Time-Periodic Horizontal Heating”. International Journal of Heat and Mass Transfer, Volume 38, 1996, pp. 1121–1133. [15] B.V. Antohe and J.L. Lage, “The Prandtl Number Effect on the Optimum Heating Frequency of an Enclosure Filled with Fluid or with a Saturated Porous Medium,” International Journal of Heat and Mass Transfer, Volume 40, 1997, pp. 1313–1323. [16] C.H. Lee, J.M. Hyun and H.S. Kwak, “Oscillatory Buoyant Convection in an Enclosure of a Fluid with Density Inversion”. International Journal of Heat and Mass Transfer, Volume 43, 2000, pp. 3747–3751. [17] G.B. Kim, J.M. Hyun and H.S. Kwak, “Buoyant Convection in a Square Cavity Partially Filled With a Heat-Generating Porous Medium”. Numerical Heat Transfer A, Volume 40, 2001, pp. 601–618.

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[18] H.S. Kwak, K. Kuwahara and J.M. Hyun, “Resonant Enhancement of Natural Convection Heat Transfer in a Square Enclosure”. International Journal of Heat and Mass Transfer, Volume 41, 1998, pp. 2837–2846. [19] K.H. Chung, H.S. Kwak and J.M. Hyun, “Finite-Wall Effect on Buoyant Convection in an Enclosure with Pulsating Exterior Temperature”. International Journal of Heat and Mass Transfer, Volume 44, 2001, pp. 721–732. [20] J.H. Bae, J.M. Hyun and H.S. Kwak, “Buoyant Convection in a Cavity with a Baffle under Time-Periodic Wall Temperature”. Numerical Heat Transfer A, Volume 39, 2001, pp. 723–736. [21] B.V. Antohe and J.L. Lage, “Experimental Investigation on Pulsating Horizontal Heating of an Enclosure Filled with Water”. Transactions of ASME Journal of Heat Transfer, Volume 118, 1996, pp. 889–896. [22] S.K. Kim, S.Y. Kim and Y.D. Choi, “Resonance of Natural Convection in a Side Heated Enclosure with a Mechanically Oscillating Bottom Wall”. International Journal of Heat and Mass Transfer, Volume 45, 2002, pp. 3155–3162. [23] C.J. Ho and C.H. Chu, “Periodic Melting within a Square Enclosure with an Oscillatory Surface Temperature”. International Journal of Heat and Mass Transfer, Volume 36, 1993, pp. 725–733. [24] L. Iwatsu, J.M. Hyun and K. Kuwahara, “Convection in a Differentially-Heated Square Cavity with a Torsionally-Oscillating Lid”. International Journal of Heat and Mass Transfer, Volume 32, 1992, pp. 1069–1076. [25] W.S. Fu, and W.J. Shieh, A Study of Thermal Convection in an Enclosure Induced Simultaneously by Gravity and Vibration”. International Journal of Heat and Mass Transfer, Volume 35, 1992, pp. 1695–1710. [26] K.H. Kim, J.M. Hyun and H.S. Kwak, “Buoyant Convection in a Side-Heated Cavity under Gravity and Vibration”. International Journal of Heat and Mass Transfer, Volume 44, 2001, pp. 857–862.

Automatic Topology Generation P.R. Eiseman and K. Rajagopalan Program Development Company, 300, Hamilton Ave, #409, White Plains, NY-10601, U.S.A.

Summary Some techniques for automatically generating topology to generate multiblock structured grids are discussed. The basic topology paradigm of GridPro’s robust grid generation engine and its ability to generate high quality multi-block structured grids from a crudely defined set of topology corners and their connectivity is shown. The automatic topology techniques are applied in tandem with GridPro’s grid generation engine to get high quality block structured grids automatically. The Cartesian trimming approach is shown to be a good means of generating topology automatically. Examples and illustrations of the application of the Cartesian method to some problems are given. Finally, examples of the application of an automatic recursive adaptive technique called nested refinement, which is applied in the topological sense, is discussed.

1

Introduction

A reliable and automatic method to generate high quality hexahedral grids for CFD and structural simulations remains to be an elusive and difficult problem. While the challenge of generating high quality hex meshes remains a difficult one, the need for such meshes for accurate and reliable analysis is acute. There exist many tools and techniques to build such meshes. Often, these tools require the user to craft the mesh block by block, and build the mesh interactively in a step by step procedure. This kind of procedure can take days to weeks for complex geometries. Of the tools and techniques available, the software package GridPro is unique for its topology paradigm. GridPro’s grid generation engine requires only the specification of a crudely defined set of corners and their connectivity. The optimization procedures in the grid generation engine are very robust and give the highest quality grid for the particular topology. With GridPro, the generation of a high quality block structured grid is reduced to the specification of a loose set of corners and their connectivity – the topology. Since the path from topology to grid has been automated, the automatic creation of topology will complete the path to get from geometry to grid automatically. This paper discusses and reviews some techniques for automating

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the process of building topology for complex geometries. In particular, the Cartesian approach, which has been successfully used to produce grids for inviscid analysis, is shown to be a promising method for generating topology automatically.

2

GridPro’s Topological Paradigm

GridPro’s topology paradigm is a very powerful and unique technique. This paradigm reduces the problem of generating multi-block grids to that of generating a set of loosely positioned topology corners and their connectivity. The paradigm provides for ease of use and automation – because one does not have to worry about positioning the corners exactly. Also, topology can be used as a template and can be quite independent of the surfaces themselves. This gives rise to an important advantage that topology need be built only once for a grid generation problem. Figure 1 shows a simple example to illustrate how the topology paradigm works in GridPro. Both topology 1 and topology 2 produces almost identical grid point positions. The iterative grid optimization algorithms in GridPro are very robust and ensure the highest quality of the grid for a particular topology. The robust optimization algorithms can untwist a folded block as shown in Fig. 2 Figure 3 illustrates the grid point movement algorithms by showing pictures of a grid at different iteration steps for the grid generation engine. Again, we see that the folded grid cells in (a) have been unfolded and neatly placed in (c). Figure 3 also serves to illustrate the curvature clustering done by GridPro. Now that we have a robust topology engine behind us, we can use

Figure 1

Loose Positioning of topology corners

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Figure 2

3 (a)

Twisted cubical topology and its corresponding GridPro grid

3 (b)

3 (c)

Figure 3 Grid Optimization in GridPro. Grid for iteration step -0 (a) 100 (b) 500 (c) 3000

this for generating multi-block structured grids for complex geometries. One option is to develop schemes to automatically generate topology. Another is to develop an interactive topology generator. An interactive topology generator called the AZ manager comes with GridPro which has numerous features for automating parts of topology creation. This provides a kind of semiautomatic approach to topology creation. The fully automatic approach to topology creation is a present research topic in Program Development Company and is discussed in the coming sections. For more details on topology creation and other aspects of grid generation in GridPro, please refer to the GridPro TIL manual [1] and the GridPro tutorials [2].

3

Cartesian Approach to Automatic Topology Generation

There are many techniques which can help in generating topology automatically from the surface geometry. The medial axis based approach [6], Cartesian methods, fuzzy logic based techniques [7] are some examples. The Cartesian method has been very successfully applied in generating meshes for inviscid flows [5]. Cartesian methods are highly reliable, but when applied to generate viscous meshes, they suffer from the disadvantage of having a large number of blocks and a relatively poor alignment with the flow of

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1:BoundingBox

4: Wrapping

Figure 4

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2: Cartesian Blocks

3: Trimming, Adaptation

Generation of a Cartesian topology for a two dimensional geometry

the geometric boundaries. Boundary alignment is particularly important for generating grids for viscous flows. It is shown here that the Cartesian trimmed approach is an attractive technique to generate topology for viscous meshes. The reliability of the Cartesian method translates to a high success rate of the automatic topology scheme. Also, a number of topology techniques can be used to ensure that the grid is boundary conforming and has the required qualities. Figure 4 illustrates the use of such a technique on a reasonably complex two dimensional geometry. The first step is to construct a generalized bounding box. The next step divides the bounding box into a number of Cartesian blocks. The next step is to trim and adapt these blocks to the boundary. The final step is to construct an o-grid around the boundary of this trimmed topology to make it boundary conforming. Figures 5 and 6 shows the grid and topology generated by GridPro for the topology generated in step 4 of Fig. 4. The topology corners with the white dots have been assigned to the surface geometry, associating topology corners with the actual geometry in GridPro. Each block has been colored differently in the grid. Most of the steps shown in Fig. 4, creating the Cartesian blocks, trimming and creating the O-grid etc, are pretty straight forward to implement. But choosing a good technique for adaptation and implementing it can be tricky in three dimensions. The adaptation shown in Fig. 4 uses quadrilateral adaptation templates [8].

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Figure 5

Figure 6

3.1

Grid and corresponding topology

Zooming into the section in the box in figure 5

Cartesian Topology Generation in Three Dimensions

The techniques illustrated in Figs. 4 to 8 illustrate the successful application of automatic topology techniques for two dimensional objects. The same techniques have been extended to three dimensional objects. A C++ program automatically generates a Cartesian trimmed, wrapped (with O-grid) topology from a given geometry. Details of such Cartesian trimming methods can be obtained from [4]. This program was used to generate a high quality boundary conforming grid for a number of three dimensional geometries.

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Figure 7

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A three dimensional blob and the automatically generated grid

Figure 8

Two grid sheets through the blob grid

Figure 7 shows the picture of the three dimensional blob geometry and the corresponding 1338 block grid generated by the program. This topology is fed straight into the GridPro’s grid generation engine to obtain a smooth boundary conforming grid. Figure 8 shows two grid sheets for this grid. The clustering of the grid cells near the boundary was done using GridPro’s cluster utility. Recently, this program was used to generate a high quality block-structured grid for an actual carotid artery geometry provided by the University of California at Berkeley. The resulting grid from the Cartesian program applied in tandem with GridPro’s grid generation engine is shown in the

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Figure 9

Figure 10

Carotid Artery and the automatically generated grid

Cross Sectional Grid Sheets for the Carotid Artery

Figs. 9 and 10. This grid is also featured in the GridPro grid generation gallery website [9]. Readers can download the grid in a number of popular formats from this site. For the geometry shown in Fig. 9, the Cartesian topology generation program was applied only to the middle section of the geometry. Since the beginning and the end sections of the geometry were tubular, a simple topology

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extrusion was used to extend the ends of this Cartesian topology to the ends of the geometry. The resulting topology had 13488 blocks. This is a large number of blocks, and the grid is best treated as an unstructured hexahedral grid and solved by modern finite volume CFD codes. 3.2

Improvements to the Cartesian Topology Generation Technique in 3D

The results generated by the automatic topology generation program and the grids indicate that the Cartesian method can be effectively used to generate topology for complex three dimensional objects. The boundary wrapping technique is very effective in creating boundary conforming grids for viscous cases. The Cartesian technique produces hexahedral cells which are almost cubes in the volume of the grid. GridPro’s grid optimization algorithms takes care that grid cells near the boundary are orthogonal to the boundary and the O-grid layer transitions smoothly into the volume of the grid. The main disadvantage of the Cartesian topology generation method is the huge number of blocks it produces. Geometries may have a wide variety of scale variations and it is clear that the method will need to have less number of blocks if it can be applied successfully to more cases. In general, it will require some kind of conformal adaptive Cartesian topology generation technique to reduce the total number of blocks in the geometry. The two dimensional Cartesian topologies shown in Figs. 4, 5 and 6 illustrate some possible ways to achieve such a technique in two dimensions. Extension of these techniques to three dimensions is a present research topic in PDC. In the next section, we will talk about a technique for handling large scale differences in geometries by an automatic topology generation technique called nested refinement.

4

Automatic Nested Refinement

The basic idea behind nested refinement is simple and has been around for sometime. There are many ways of generating topology to get such grids in GridPro. Some references can be quoted from the GridPro TIL manual [1], where a nested refinement structure has been used to illustrate the idea of components in the topology input language (TIL). Nested refinement can be looked at as a generalization of a structure called the clamp [10]. Figures 11, 12, 13 and 14 illustrate the idea of a self-similar topological structure which has a high resolution in the bottom and less in the top. A self-similar topological template is chosen and recursively stacked to make such a structure. This process is completely automatic once the top and the bottom surfaces have been specified. Figure 15 shows flow field solution for the wavy wall problem computed R . Comparisons between flow field solutions using the nested using Fluent


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Figure 11

Nested Topology and Grid

grid (shown in Fig. 11 through 15) and a regular structured grid on the same geometry shows that the nested grid runs much faster but gives the same flow results. For more details on this comparative simulation, refer to [3]. This concept has been extended into three dimensions. It can be accessed as from the GUI of GridPro.

Figure 12

zoom the box in Fig. 11

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Figure 13

zoom the box in Fig. 12

Figure 14

zoom the box in Fig. 13

Figure 15

Flow field solutions

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Figure 16

Figure 17

5

Curved wavy tube in a box

A Cross-sectional sheet

Application of Nest to a Thin Curved Wavy Tube in a Box

A thin curved tube in a box whose radius fluctuates by a small amount is another example of a multi-scale problem. Figure 30 illustrates the geometry. The arrows indicate a flow direction, and the problem posed can be that of calculating the drag experienced by the tiny tube by such a flow. Figures 31 and 32 show a cross-sectional grid sheet for the grid generated on this geometry using nest. Figure 33 shows two cross-sectional sheets perpendicular to each other. Figure 33 illustrates the resolution of the grid near the wavy tube. Nest achieves the high resolution near the tube and the gentle transition to a lower resolution away from the tube.

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Figure 18

Figure 19

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Zooming Fig. 31

Two cross-sectional sheets near the tube

Acknowledgement(s) The authors would like to Prof Stanley Burger at the University of California at Berkeley for providing the geometry of the carotid artery. The authors would also like to thank Prof Houston Wood at the University of Virginia for his help and support in developing the automatic nested refinement utility.

References [1] “GridPro TIL Manual”. Distributed by Program Development Company, White Plains, NY, USA. [2] “GridPro Tutorials – Basic”, Tutorials 1 to 11. Distributed by Program Development Company, White Plains, NY, USA. [3] Krishnakumar Rajagopalan. “Automatic Nested Refinement – A Technique for the generation of high quality Multi-Block structured grids for Multi-Scale R ” - Master’s Thesis in Mechanical and Aerospace problems using GridPro
Engineering presented to the University of Virginia, May 2003. [4] M.J. Aftosmis. “Solution Adaptive Cartesian grid methods for aerodynamic flows with complex geometries”. In Algorithms and data structures for grid

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[5] [6]

[7]

[8]

[9] [10]

[11]

P.R. Eiseman and K. Rajagopalan generation, selected special topics from previous VKI lecture series, Von Karman institute for Fluid Mechanics. M.J. Aftmosis, M.J. Berger, J.E. Melton, “Robust and efficient Cartesian mesh generation for component based geometry”, AIAA Paper 97–0196, Jan 1997. CG Armstrong, DJ Robinson, RM McKeag, TS Li, SJ Bridgett, RJ Donaghy & CA McGleenan. “Medials for Meshing and More”. Proceedings, 4th International Meshing Roundtable, Sandia National Laboratories, pp.277–288, October 1995. Reza Taghavi. “Automatic Block Decomposition Using Fuzzy Logic Analysis”. In the proceedings of the 9th International Meshing Roundtable, pp.187–192, October 2000. Robert Schneiders. “Quadrilateral and Hexahedral Element meshes”. Chapter 21 in the handbook of grid generation, edited by Joe. F. Thompson, Bharat. K. Soni, Nigel. P. Weatherill, CRC Press, 1999. The grid gallery website at www.gridpro.com/gridgallery/index.html Jochem Hauser, Peter Eiseman, Yang Xia, Zheming Cheng. “Parallel MultiBlock Structured Grids”. Chapter 12 in the handbook of grid generation, edited by Joe. F. Thompson, Bharat. K. Soni, Nigel. P. Weatherill, CRC Press, 1999. Krishnakumar Rajagopalan, Peter Eiseman. “Automatic Nested Refinement – A Technique for the generation of high quality Multi-Block structured grids for R ” – Proceedings of the 12th international Multi-Scale problems using GridPro
Meshing roundtable, September 2003.

Computational Study of Influences of a Seam Line of a Baseball on Flows R. Himeno RIKEN (The Institute of the Physical and Chemical Research), 2-1 Hirosawa Wako-shi Saitama, 351-0198 Japan

Summary Flows around a baseball without rotation are calculated using third-order upwind-difference method with various seam positions determined by two angles in orthogonal two directions. Those are four-seam rotation direction with an angle: α and two-seam rotation direction with an angle: β. The computed results in the four-seam rotation cases are compared with experimental data measured in a wind tunnel and computed drag coefficients qualitatively agree well with experiments. However, lift coefficients do not agree well. The computed results and geometrical symmetry suggest that a supporting rod in the wind tunnel would have strong influence on the accuracy of the measurement. Flow changes in two-seam rotation direction are also simulated. It is found that the lowest drag force is observed at α = 90 and that the value is less than half of the largest drag force at α = 30 and 60 degrees. The largest lift force is observed at α = 20 degree. In this case, the seam line on the top causes a large separation while smooth surface without the seam at the bottom dose not separate the flow. A pair of longitudinal vortices are found in the wake, which make wake slant and generate large lift force.

1

Introduction

A ball used in baseball games has a peculiar seam line on its surface. The seam line makes a groove and rib, whose height is about 1 mm and very small to the diameter of the ball: about 70 mm. It is well known that the unique curved seam line gives large influence on flows around the ball. Reference [9] reported that the seam line works as surface roughness on a sphere and makes Magnus force larger than a ball without seam. Reference [10] reported that asymmetric aerodynamic force acts on the ball and changes its direction when the ball is slowly rotating, which is the reason why knuckle ball has curious trajectory. Reference [5] performed wind tunnel experiments and measured aerodynamic forces of the ball at various seam position. They reported the seam makes the boundary layer turbulent and the asymmetric position of the

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seam lets wake asymmetric, resulting asymmetric aerodynamic force. However, balls were supported by rods in all these wind tunnel experiments and the influence of the rod is not clear. Reference [5] showed the asymmetric position of the seam makes asymmetric wake but they only measured the influence in two-dimensional section. Three-dimensional structure of the wake is not clear now. Recently, CFD has been rapidly developed and is widely applied now. The author calculated flows around the ball with the seam line but no rotation. In this paper, the computed results will be shown comparing experimental results with the same conditions by [5] and the accuracy of both computation and experiments will be discussed. Three dimensional wake structure of the ball will be also investigated here.

2

Computation Method

The fastest ball speed in baseball games is about 45 m/s (162 km/h). Therefore the flow around the ball is incompressible. Usually, ball speed in baseball games is in between 30 m/s (108 km/h) and 45 m/s (162 km/h). Reynolds number based on the ball diameter d: 0.0715 m is in between 1.3 × 105 and 2.1×105 . Flow fields would be consist of laminar and turbulent regions. In this paper, whole flow field is simulated by using third-order upwind-difference scheme without any turbulence model. This approach was successfully applied to a turbulent inner flow [2] and is regarded as a quasi-DNS. 2.1

Basic Equations

Basic equations are unsteady incompressible Navier-Stokes equations and the equation of continuity. These equations are expressed on a coordinate system → fixed on the ball moving with the velocity v 0 through stationary fluid as follows. ∇·v=0, (1) 1 ∂v + {(v − v0 ) · ∇} v = −∇p + ∆v , (2) ∂t Re where v is flow velocity, p is pressure and Re is Reynolds number. All values are non-dimensional ones. 2.2

MAC Based Scheme

In order to couple the equation of continuity and the Navier-Stokes equation, MAC method [8] is used here. Poisson equation for pressure in high Reynolds number regime is as follow. ∆p = −∇ · {(v − v0 ) · ∇} v −

∂D , ∂t

(3)

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where D = ∇ · v. When the second term of the right hand side of this equation is expressed by a finite difference approximation, the next equation is obtained. Dn+1 − Dn , (4) ∆p = −∇ · {(v − v0 ) · ∇} v − ∆t where superscript n denotes the n-th time step and ∆t is a time increment. Although the second term of the right hand side of this equation is zero due to the equation of continuity, Dn is not omitted so that any error in satisfying the equation of continuity at the present time step can be compensated as follows; Dn . (5) ∆p = −∇ · {(v − v0 ) · ∇} v + ∆t The equation of continuity is not directly satisfied but implicitly satisfied. This derivation is based on a MAC method, which may cause the error in the flow quantity in case of internal flows but may not cause significant errors in case of external flows. 2.3

Third-Order Upwind-Difference Scheme

Equations (2) and (5) are transformed into a generalized coordinate system and discritized. A first-order Euler implicit method is used for time derivatives, a third-order upwind-difference scheme [3] for convective terms, and second-order central-difference for the other spatial derivative terms. This scheme is applied to various three-dimensional high Reynolds number flows and shows very good results [1, 2, 7]. The third-order upwind-difference scheme is as follows.

ui ∂u

{−ui+2 + 8 (ui+1 − ui−1 ) − ui−2 } u x=xi = ∂x

12∆x |ui | (ui+2 − 4ui+1 + 6ui − 4ui−1 + ui−2 ) (6) + 12∆x where subscript i denotes i-th grid number, u is velocity component in xdirection and ∆x is an finite difference increment in x-direction.

3

Grid System

The simplest grid system may be an O-type system which has a singular axis like the longitude and latitude system on the globe. However, this causes unnecessary grid concentration near the axis. Figure 1 shows grid system used in the computation. Positions and heights of the seam line are obtained by measuring an official ball for a baseball game. Grid lines near the seam line are generated to be aligned along the seam line. Two kind of grid system are used in the computation, a) coarse grid system with 169 × 92 × 101, about

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Fine grid system (337 × 181 × 101 grid points)

Figure 1

1.6 million grid points and b) fine grid system with 337 × 181 × 101, about 6.2 million grid points, which uses four times finer grids of the coarse grids in each grid plane parallel to the ball surface.

4

Shape of the Seam on the Ball

In the all computations, the ball has no angular velocity but has some angles in two kinds of axes of the rotation direction. Before presenting the results, changes of the seam in the two kinds of rotation direction are shown here to see its geometrical symmetry. “Two-seam rotation” and “four-seam rotation” are defined as the rotation around the Z axis and the X-axis in Fig. 2, respectively. As Fig. 3 shows, the seam appears four times when the ball turns around in α-direction while the seam appears two times in β-direction. Geometrical symmetry of the seam shape is very important to investigate the

Z

β Y X

Figure 2

α

Definition of coordinates and angle α, β

Computational Study of Influences of a Seam Line of a Baseball on Flows

(a) four-seam rotation Figure 3

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(b) two-seam rotation

Four-seam and two-seam rotation directions

influence of the seam line on the flow. Figures 4 and 5 show seam line shape during the two kinds of rotations. In the case of four-seam rotation, the shape change has an apparent period of α : 180 degrees. Watching Fig. 4 carefully, it is found that the shape at α = 90 degrees is the inverted shape at α = 0 or 180 degrees and that the shape at α = 135 degrees is the reversed shape at α = 45 degrees. To get the variation of aerodynamic forces in this rotation, it is not necessary to compute flows from α = 0 degree to 360 degrees but it is sufficient to compute flows

α=0,

α=45,

α=90, (a) front view

α=135,

α=180

α=0,

α=45,

α=90, (b) side view

α=135,

α=180

α=0,

α=45,

α=90, (c) top view

α=135,

α=180

Figure 4

Shape change of the seam (four-seam rotation)

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β=−180,

β=0, Figure 5

β=−135,

β=−90,

β=45,

β=90,

β=−45,

β=135,

β=0

β=180

Shape change of the seam (two-seam rotation, front view)

only from α = 0 to 45 degrees for the sake of these geometrical symmetries. However, to see the symmetry effect not only the cases of α = 0 to 45 degrees but also the cases of α = 45 to 90 degrees are calculated. In the case of the two-seam rotation, the shape change has a period of β : 360 degrees. However, watching Fig. 5 carefully, it is found that the shape at β = 180 and −180 degrees is the inverted shape at β = 0 degree, and that the shape at β = 135 degrees is also the inverted shape at α = 45 degrees. Similarly, it is found that the shape at β = −135 degree is the inverted shape at β = −45 degrees. To get variation of aerodynamic forces in this rotation, it is sufficient to compute flows only from β = −90 to 90 degrees.

5

Computed Results

Two series of computations are performed, which are 1) four-seam case and two-seam case. Reference [5] performed wind tunnel measurements in the four-seam case at the Reynolds number 1.0 × 105 and reported the aerodynamic coefficients at various angle α. Computed results are firstly compared with their measurements. All computations were done on a Fujitsu VPP500 with 28 processor elements at RIKEN, which is a vector-parallel computer whose theoretical peak performance is 44.8 GFLOPS. It took about 8 to 10 hours to calculate flows up to non-dimensional time 100 in the coarse grid case and 100 hours in the fine grid case. All aerodynamic forces and flow fields are taken averaged from non-dimensional time 20.0 up to 100.0. The code was parallelized and executed on 28 processor elements on the VPP500. 5.1

Four-Seam Case

(1) Aerodynamic coefficients

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B

A Calculations

Experiments

α Figure 6

Comparison of drag coefficients

The shape of the seam has a period of 90 degrees in α-direction because of the geometrical symmetry. Calculated drag coefficients using the coarse grid system are plotted in Fig. 6 as the solid line, compared with the measurements (dotted lines). The experimental data were originally measured from α = 0 to 360 degrees and are plotted as in between 0 and 90 degree. Computed results qualitatively show good agreement with the experiments although the values are about 20 percent lower than the measured values. Points “A” and “B” in the Fig. 6 show the computed drag coefficients using the fine grid system. Although the grid density increases from the coarse grid system, it dose not improve the gap. Figure 7 shows the calculated side force coefficients compared with the experimental data. The side force measured in experiments has sharp transition near α = 45 degrees although the calculated force have only gradual change there. The calculated forces are about half in value. However, the experiments show that the transition point is not fixed and varies from 30 to 60 degrees. It becomes slower transition when those values are taken averaged. The results for the finer grid calculations are shown in Fig. 7 as point “A” and “B”. In this case, discrepancy decreases at B as the grid density goes up. Figure 8 shows the calculated lift coefficients as the solid lines compared with the experimental data plotted as the dotted lines. The calculated values themselves are not much different but the characteristics of the curve are

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Experiments B Calculations

A

α Figure 7

Comparison of side coefficients

Experiments

A Calculations

B

α Figure 8

Comparison of lift coefficients

totally different. Because of the geometric symmetry of the shape shown in Fig. 4, the curve of lift coefficient variation should be the point symmetric with respect to α = 45 degrees. The calculated results clearly show point symmetry but the measured values are not. This would be caused by a supporting rod that suspends the ball at the bottom in the wind tunnel.

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(a) Front view Figure 9

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(b) Rear view

Surface static pressure distribution: α = 45 degrees

Finer grid results are also plotted in Fig. 8 as points “A” and “B”, and “B” shows larger negative values than that of the coarse grids. (2) Flow fields As shown in Fig. 7, large asymmetric force acts on the ball around α = 60 degrees. The largest side force is observed at α = 65 degrees and nearly no side force acts at α = 45 degrees. Figures 9 and 10 show the front and rear views of time-averaged static pressure distributions on the ball surface at α = 45 and 65 degrees, respectively. The pressure distribution at α = 45 degrees in Fig. 9 (a) is almost symmetric and Fig. 9 (b) shows that there is little pressure difference on the rear half surface. The slight asymmetrical pressure distributions in the front half surface generates Cs: 0.006 in this case. On the other hand, pressure distributions at α = 65 degrees in Fig. 10 are much more asymmetrical than the case of α = 45 and the peak negative pressure is much higher. Cs in this case is 0.192.

(a) Front view Figure 10

(b) Rear view

Surface static pressure distribution: α = 65 degrees

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Reference [5] reported that wake of a ball is slanted when side force is acting and that the wake is symmetric at α = 45 degrees. To validate the computed results, time-averaged total pressure distributions at α = 45 and α = 65 degrees are compared with each other in Fig. 11 The horizontal red lines in Fig. 11 show lines of symmetry to see how much the wake is slanted. Computed wake at α = 45 degrees is little slanted downward but the wake at α = 65 apparently shifts downward.

Figure 11

(a)

=45 degrees

(b)

=65 degrees

Time-averaged total pressure distributions

Computational Study of Influences of a Seam Line of a Baseball on Flows

(a) side view Figure 12

(a)side view Figure 13

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(b) rear view

Streamline in the wake at α = 45 degrees

(b) rear view Streamline in the wake at α = 65 degrees

Figures 12 and 13 show streamline in the wake at α = 45 and α = 65 degrees, respectively. A vortex ring and a pair of weak longitudinal vortices are found in the wake at α = 45 degrees. At α = 65 degrees, the ring vortex disappears and the longitudinal vortices increase their strength. The slanted wake and the side force are caused by these longitudinal vortices. 5.2

Two-Seam Case

(1) Aerodynamic coefficients Aerodynamic coefficients are calculated and plotted in Fig. 12 from β = −75 to 90 degrees. The drag coefficient at β = 90 degrees is 0.243 and is the lowest not only in β variation but also in α variation. The largest drag coefficient is observed at β = 30, 60 degrees and is 0.514 which is more than twice of the lowest value: 0.243 at β = 90 degrees. The largest lift force is observed at β = 20 degrees and its value is −0.316. This absolute value is the largest asymmetric force among Figs. 7, 8 and 14. In other words, this is

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β

Figure 14

Aerodynamic coefficient variations in two-seam cases

the largest force perpendicular to the uniform flow among the all calculated cases. (2) Flow fields In order to investigate why the lift coefficient at β = 20 degrees is so high, the flow fields at β = 20 degrees are compared with other cases. Figure 15 shows the side view of the time-averaged surface pressure distributions at β = 20 degrees compared with β = 0 degree. Main difference appears at the top of the ball. The seam at β = 20 degrees causes large separation at the top and makes the pressure higher. This is more clearly shown in Fig. 16 that shows the total pressure distributions at β = 20 and 0 degree. At β = 20 degrees, the wake is strongly shifted upward. Figure 17 shows streamlines at β = 20 degrees, which clearly shows a pair of longitudinal vortices in the wake. The longitudinal vortices have counter rotation with each other and generate lift force on the ball like an airplane. At the same time, the vortices make the wake slanted. When the separation at the top is not so large, a vortex ring appears instead of longitudinal vortices as shown in Fig. 18.

6

Conclusions

Flows around a baseball with a seam line and no rotation are numerically simulated in two kinds of rotation directions. Those are four-seam rotation direction with an angle: α and two-seam rotation direction with an angle : β. The computed results of the four-seam rotation cases are compared with experimental data measured in a wind tunnel. The computed drag coefficients qualitatively agree well with experiments. However, lift coefficients do not agree well. The computed results and geometrical symmetry suggest that a

Computational Study of Influences of a Seam Line of a Baseball on Flows

(a)

=20 degrees

Figure 15

(b)

=0 degree

Surface pressure distributions at β = 20 and 0 degree

(a) β =20 degrees

(b) Figure 16

=0 degree

Total pressure distributions at β = 20 and 0 degree

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(a) side view

(b) rear view

(c) top view

Figure 17

Streamline in the wake at β = 20 degrees

Figure 18

Streamline in the wake at β = 5 degrees

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supporting rod at the bottom of the ball in the wind tunnel causes errors in measuring lift force. It is also observed in the computed results that wake is periodically changing from symmetric shape to slant shape as α increases. Present computation method is validated by these facts. When the wake is slanted, a pair of strong vortices are observed in the wake. Flow changes in two-seam rotation direction are also simulated. It is found that the lowest drag force is observed at β = 90 and the value is less than half of the largest drag force at α = 30 and 60 degrees. The largest lift force is observed at β = 20 degrees. In this case, the seam line on the top causes a large separation while smooth surface without the seam at the bottom dose not separate the flow. A pair of longitudinal vortices are found in the wake, which make wake slant and generate large lift force. When wake is nearly symmetric, a ring vortex is observed instead of a pair of longitudinal vortices.

Acknowledgement The author would like to sincerely thank Mrs. Sanae Sato for her efforts on generating the grid system used in this study. The author also would like to express his thanks to Prof. Mizota for his kindness of offering his experimental data.

References [1] Himeno, R., Takagi, M., Fujitani, K. and Tanaka, H., “Numerical Analysis of the Airflow around Automobiles Using Multi-block Structured Grids”, SAE Paper 900319 (1990), The Society of Automotive Engineers. [2] Kawamura, T. and Kuwahara, K., “Direct simulation of a turbulent inner flow by finite difference method”, AIAA paper, AIAA-85–0376 (1985). [3] Kawamura, T., Takami, H. and Kuwahara, K., “Computation of high Reynolds number flow around a circular cylinder with surface roughness”, Fluid Dynamic Research 1 (1986), 145–162. [4] Mehta, R.D., “Aerodynamics of sports ball”, Annual Review of Fluid Mechanics, Vol.17 (1989), 151–189. [5] Mizota, T., Kuba, H. and Okajima, A., “Erratic behavior of knuckle ball (1) Quasi-steady flutter analysis and experiment”, Journal of Wind Engineering, No.62, January (1995), 3–13. [6] Mizota, T., Kuba, H. and Okajima, A.,” Erratic behavior of knuckle ball (2) Wake field and aerodynamic forces”, Journal of Wind Engineering, No.62, January (1995), 15–22. [7] Ono, K., Himeno, R., Fujitani, K. and Uematsu, U., “Simultaneous computation of the external flow around a car body and the internal flow through its engine compartment”, SAE Paper 920342 (1992), The Society of Automotive Engineers. [8] Roache, P.J.,” Computational Fluid Dynamics”, (1976), Hermora Publishers, New York.

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[9] Tani, I., “Curve in a baseball game”, Science Journal of Kagaku, 20–9 (1950), 405–409 (in Japanese). [10] Watts, R.G. and Sawyer, E., “Aerodynamics of knuckleball”, American Journal of Physics, 43-11 (1975), 961–963.

Vortex Breakdown Flows in Cylindrical Geometry R. Iwatsu Tokyo Denki University, Chiyoda-ku, Tokyo 101-8457, Japan

Summary This paper reports the numerical results of swirling flows driven in cylindrical container by constantly rotating the bottom disc while either no-slip or free-slip boundary condition is imposed on the top lid. For the closed cylinder case, axisymmetric steady state solutions exhibit good agreement with available experimental measurements on the parameter dependency of the flow pattern as well as the position of stagnation points on the axis of rotation when the values of parameters are within the range of axisymmetric flows. Based on the numerical solutions, the idea in [1] is pursued under an effort to predict the position of stagnation points by means of using simple criteria. A reduced equation is derived by post-processing the momentum equations which successfully approximates the Re dependency of the axial position of the stagnation points. For the cylinder with top free-slip case, extensive parametric study was performed over a wide parameter space which includes parameter regions not fully explored by the previous studies. Momentum balance, whose information is ignored in the previous studies is examined in detail and it is shown that the nature of the flow is different between the flows with small h and these with large h for this flow configuration. While simulated flow state diagram plotted on the (h, Re) plane for the top no-slip condition case exhibits good agreement with the previous simulation by Br/ ons et al. [2], the counterpart for the top free-slip case does not coincide qualitatively with their numerical simulation [3] on some of the small parameter ranges. It is suggested furthermore that there is a possibility that many such small ranges exist which were not noticed in the previous numerical studies. These numerically predicted flow states are not yet confirmed by the experimental visualization nor by the full threedimensional numerical computation and the results are open to the future studies.

1

Introduction

Confined swirling flows of viscous incompressible fluid driven by steady rotation of end wall(s) within cylindrical containers had been extensively studied

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in the past [4–9]. The flows driven by the constant rotation of one of the top or bottom end walls is known to exhibit axisymmetric bubble type vortex breakdown phenomenon for certain range of parameter values [4,9]. The agreement between the experimental visualization and the numerical simulation under the assumption of the axisymmetry for this flow configuration is good and the cross comparison of the experimental and numerical results enables in depth investigation of the steady/unsteady vortex breakdown bubbles appearing in the confined swirling flows (see e.g. [6]). A similar flows driven in open cylindrical containers by the rotation of bottom end wall with top free surface have not been studied in the past to the same extent as those in the sealed cylinders. Experimental study on this flow configuration was conducted by Spohn et al. by utilizing various experimental visualization techniques [10–12]. The existence of deformable top free surface bestow an additional complexity to this flow configuration [13], though it has a close relationship to the flows in cylinders driven by the co-rotation of the both end walls. Spohn et al. found out that replacement of the top rigid cover of the sealed cylinder problem to the free surface has large influence over the whole flow field. They observed recirculating zones displaying different appearances from the recirculating bubbles observed in the sealed cylinder and lower stability limit of the steady swirling flows compared with the counterpart in the sealed cylinder. creation of off-axis bubble and the axis bubble is numerically observed for the flows in cylindrical containers driven by the co-rotation of top and bottom end walls [14, 15] and for the flows when inner cylinder with small radius is inserted in the cylindrical container [16]. Stability and unsteady flows are studied in [17–20]. Recently Br/ons et al. [2] applied the theory of two-dimensional dynamical systems to the velocity vector field on the meridional plane (r, z) and derived a list of possible bifurcations of streamline structures on varying two governing parameters. By comparing their results with the experiments by Spohn [11], they found both similarities and differences. The swirling flows in the above cylindrical configuration poses some unsolved questions e.g., upstream closed but downstream open bubble experimentally observed by Spohn et al. [12], the relationship between the flow asymmetry and the boundary layer instability and the steady to unsteady flow transition. These fundamental fluid physics are tackled in recent studies (see eg. [21–24]). In engineering applications on the other hand, it is much of practical use if we are able to predict the occurrence of the vortex breakdowns based on some concise criterion without solving the whole flow field. The swirl angle criterion had been known for long time, however it is believed that the breakdown is influenced by at least three or more parameters (see e.g. [25–27]). Criterion proposed by Brown and Lopez [28] was later reported to be insufficient as a predicting tool [29]. The present authors are motivated by deriving a simple criterion and with this goal in mind, the momentum transport is analyzed based on the numerical solutions computed for both flow configurations. The results will be reported in the following.

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2

143

Numerical Model

The governing equations are solved under the assumptioin of axisymmetry for the Stokes streamfunction ψ, the azimuthal vorticity component ω ≡ ωϕ and the azimuthal velocity component v on the cylindrical coordinate system (r, ϕ, z) with velocity components (u, v, w).     ∂ ∂ 1 ∂v 2 1 ∂ 1 ∂rω ∂ω ∂2ω + (uω) + (wω) − = , (1) + ∂t ∂r ∂z r ∂z Re ∂r r ∂r ∂z 2     ∂ ∂ 2uv 1 ∂ 1 ∂rv ∂v ∂2v + (uv) + (wv) − = (2) + 2 . ∂t ∂r ∂z r Re ∂r r ∂r ∂z   ∂ 1 ∂ψ 1 ∂2ψ + = −ω (3) r ∂z 2 ∂r r ∂r where the streamfunction and the azimuthal vorticity component are defined respectively as u=−

1 ∂ψ , r ∂z

w=

1 ∂ψ r ∂r

and ω ≡ ωϕ =

∂u ∂w − . ∂z ∂r

Reference scales for the length, time, velocity, vorticity and stream function are R, Ω −1 , RΩ, Ω and R3 Ω respectively, where R is the radius of the cylindrical container, Ω the constant angular velocity of the rotating bottom wall, and ρ the density of the fluid. Non-dimensional physical parameters which appear in the basic equations are the rotational Reynolds number Re = R2 Ω/ν where ν is the kinematic viscosity of the fluid, and the radius to height cylinder aspect ratio h = H/R where H is the height of the cylindrical container. The boundary conditions for the top rigid-cover case (case A) are as follows ψ = 0, ω = 0, v = 0 (r = 0, 0 ≤ z ≤ h) , (4) 1 ∂2ψ ∂ψ = 0, ω = − , v = 0 (r = 1, 0 ≤ z ≤ h) , ∂r r ∂r2 ∂ψ 1 ∂2ψ ψ = 0, = 0, ω = − , v = −r (z = 0, 0 ≤ r ≤ 1) , ∂z r ∂z 2 ∂ψ 1 ∂2ψ ψ = 0, = 0, ω = , v = 0 (z = h, 0 ≤ r ≤ 1) . ∂z r ∂z 2 For the top free-slip case (case B), condition (7) is replaced with ψ = 0,

ψ = 0,

∂v ∂2ψ = 0 (z = h, 0 ≤ r ≤ 1) . = 0, ω = 0, ∂z 2 ∂z

(5) (6) (7)

(8)

The steady solutions are obtained as converged solutions of above system of equations. In the case B flows, Grid dependency check was performed for both cases and it was shown that the number of grid points utilized in the present computation is sufficiently large to assure sufficient numerical accuracy.

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3 3.1

Results

Top Rigid-Cover (Case A)

Numerically constructed flow state diagram is shown in Fig. 1 and compared with the previous experiments in Fig. 2. Agreement on the values of the critical Re for the breakdown bubble creation, between the present computation and the previous experiments is within 2% when h < 2.8 and Re is below the linear stability limit predicted by Gelfgat et al. [9]. It is also noted that the agreement is quite good for the curve 1 (which separates the flow state (a) and (b)) well into the parameter region where axisymmetric flows are considered to be unstable to the asymmetric disturbances (h > 2.8). The location of the stagnation point on the axis of rotation is shown in Fig. 3 for selected values of h and compared with available previous experimental measurement (for definition of the stagnation point, see the figure caption). The agreement is again very good and it is seen that the location as well as the Re dependency of the breakdown bubbles computed by axsymmetric governing equations appear to faithfully reproduce the experimental measurements. The same is true for flows at h = 3.0 in Fig. 3(c). Based on these observations, it appears that the influence of the flow asymmetry on the occurence and the axial position of the vortex breakdown is slight for this flow configuration.

Figure 1 Computed flow state diagram for closed cylindrical container. S(0) stands for flows with no bubble, S(1) one bubble, S(2) two bubbles and S(2m) two bubbles merged

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Figure 2 Comparison with the experimental data by Escudier (1984). Red solid lines: experimental data, red dashed line: experimental stability limit. Blue markers and lines: present numerical results

The flow data are then analyzed to evaluate the magnitude of each terms contained in the momentum equations [31]. Based on the analysis of momentum balance in the vicinity of the rotating axis by neglecting terms of minor magnitude, after some manipulations, a reduced equation is derived for the axial velocity component w at appropriately defined edge of the vortex core at r = r¯:  r¯ 1 d2 w 1 u ∂ dw − (rv)2 dr = w 3 dz Re dz 2 w ∂r 0 r    r¯   1 v ∂ 1 ∂ 2 (rv) dr ≡ A (9) − Re 0 r w ∂r r ∂r Berger and Erebacher [1] in an effort to eliminate the parabolicity of the second-order quasi-linear differential equations, included the radial diffusion term in the u equation. The present analysis, based on the assessment of numerical flow data indicates that radial balance is dominantly between the centrifugal and pressure force. The diffusion term survives in the final equation from the w equation instead of u equation (see the details in [25]). It should be noted that w is defined in the above equation at r = r¯ instead of at r = 0 and d2 w/dz 2 |r=0 ∼ d2 w/dz 2 |r=¯r is assumed. It is anticipated that ∂v/∂r is a small quantity and do not vary with Re for sufficiently large Re from the classical interpretation of flow between two discs that the fluid rotates with a constant vorticity with a slow drift from stationary top disc to the rotating

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Figure 3 Location of the stagnation point z on the axis of rotation as a function of Re. z is defined as the upstream stagnation point on the axis of each bubble. (a): h = 1.5, (b): h = 2.0, and (c) h = 3.0

Figure 4 A|r=0.0194 dependency on Re. h = 2.0. The value of A is evaluated at an axial position where u ∼ 0 holds

bottom disc in the bluk of the fluid. Figure 4 exhibits that the quantity A retains a fair degree of constancy over the range of Re where breakdown occurs and thus above conjecture is justified to some extent. Although accurate

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estimation of the value of A at upstream location is not given, the reduced ordinary differential equation might be useful in predicting the occurence and the location of the stagnation points on the axis of rotation. 3.2

Top Free-Slip-Condition (Case B)

The steady state flows are classified according to the meridional flow pattern (Fig. 5) and mapped on the (h, Re) plane in Fig. 6. Comparison with experimental visualization by Spohn et al. [10, 11] shown in Fig. 7 confirmed that the agreement between the present computation and the experiment appears to be good for h ≤ 1.0. Present result is compared also with the numerical studies [2, 13, 15] and overall similarity was observed [32]. Post processing analysis similar to the case A flows is carried out for case B flows in order to evaluate the relative magnitude of terms in momentum equations. According to the estimation of the momentum balance, it is shown

Figure 5 Meridional flow patterns classified in cylindrical container with top free-slip boundary condition

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Figure 6 Computed flow state diagram for flows in cylindrical container with top free-slip boundary condition

Figure 7 Comparison with the experimental data by Spohn et al. [10, 11]. Signs are experimental data reconstructed from [10, 11]. Triangle: flow state (a), solid triangle: (b), diamond: (c) or (d), circle: (f ) or (g) and square: (h) or (l). Solid lines are the present numerical result. Color of the lines corresponds to that of the signs

that the flow behaviour is different for flows with h > < O(1). When h ≤ 1, a considerable portion of the fluid inside the cylindrical container undergoes a quasi-rigid rotation with its rotation rate equal to the rotation rate of the bottom disc Ω. In the flow state diagram in Fig. 6 a variety of the corner/surface bubbles are seen to appear in this parameter region, however they are caused by relatively weak secondary motion on the (r, z) plane. If the absolute magnitude of the velocity components are taken into account, the Taylor-Proudman theorem is considered to hold as a good approximation in

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the fluid motion around the axis of rotation. When h > 1 on the other hand, the momentum balance reveals that quasi-cylindrical equations hold as an approximation in the vicinity of the axis of rotation. The result suggests that the utilization of these reduced equations might provide a useful information on the prediction of the vortex breakdown in engineering situations for this parameter range. Enlarged diagrams in Fig. 8 for case B flows contain very narrow parameter regions where flow pattern changes on varying h and Re. Some of the results shown in Fig. 8 coinside and some do not agree with [3]. Br/ ons et al. [2, 3] utulized the dynamical systems theory to account for the topological change of two dimensional streamlines as two control parameters are varied. They combined the theory with the numerical simulation and constructed a flow pattern diagram. However, computations were performed for not many parametric points. In the present study, contrary to the previous attempts, parametric computation was carried out for a very large number of parameter points. The results indicated that for the parameter region shown in Fig. 8(a), the arrangement of border curves which separate different flow

(a)

(b)

h

(c)

(d)

Figure 8 Enlargement of figures. (a): (h,Re)=(2.2,1200), (b): (h,Re)=(1.4,1800), (c): (h,Re)=(0.4,1600), and (d): (h,Re)=(0.3,1200)

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patterns is same with that derived by Br/ons et al. [3]. However for the parameter region plotted in Fig. 8(b), the present diagram exhibited qualitatively different behavior from the one presented in [3]. Br/ ons et al. mapped the diagram in this region involving flow patterns (h) (j) (k) and (l). In the present computation, a flow pattern (type i in Fig. 5) not recognized in the previous study was found on a very small parameter plane. Because of the existence of flow pattern (i), the type of bifurcation presumed in [3] is impossible and possible bifurcation diagram might be the one shown in Fig. 8(b) which involves five flow patterns (h) (i) (j) (k) and (l). It was found furthermore that Fig. 8(c) was not noticed in the previous studies. The diagram in Fig. 8(d) was also previously not reported. In the present study, it was not attempted to map flow patterns for high Re(Re > 3000) small h(h ≤ 0.3) region. Flow types other than shown in Fig. 5 was noted in this region and it might be natural to anticipate that this region contains also narrow parametric ranges where topologic change of flow patterns occur. All of these parameter dependencis reported in the present as well as previous studies are open to the future studies.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

S.A. Berger and G. Erlebacher Phys. Fluids 7 (5), 1955 972–982. Br/ ons, M., L.K. Voigt and S/ orensen, J.N. (1999) J. Fluid Mech. 401 275–292. Br/ ons, M., L.K. Voigt and S/ orensen, J.N. (2001) J. Fluid Mech. 428, 133–148. Escudier, M.P (1984) Exp. Fluids 2, 189–196. Lugt, H.J and Abboud, M. (1987) J. Fluid Mech. 179, 179–200. Lopez, J.M (1990) J. Fluid Mech. 221, 533–552. Tsitverblit, N. (1993) Fluid Dyn. Res. 11, 19–35. Gelfgat, A.Y, P.Z.B-Yoseph and A. Solan (1996) J. Fluid Mech. 311, 1–36. Gelfgat, A.Y, P.Z.B-Yoseph and A. Solan (2001) J. Fluid Mech. 438, 363–377. Spohn, A. (1991) Th` ese de Doctorat de l’Universit´e Joseph Fourier-Grenoble I. Spohn, A., M. Mory and E.J. Hopfinger (1993) Exp. Fluids 14, 70–77. Spohn, A., M. Mory and E.J. Hopfinger (1998) J. Fluid Mech. 370, 73–99. Lopez, J.M and Chen, J. (1998) Trans. ASME: J. Fluids Engng 120, 655–661. Valentine, D.T and C.C. Jahnke (1994) Phys. Fluids 6 (8), 2702–2710. Jahnke, C.C and D.T. Valentine (1998) Trans. ASME: J. Fluids Engng 120, 680–684. Mullin, T., S.J. Tavener and K.A. Cliffe (1998) Trans. ASME: J. Fluids Engng 120, 685–689. Gelfgat, A.Y, P.Z. Bar-Yoseph and A. Solan (1996) FED-Vol.238, Proc. ASME Fluids Engineering Summer Meeting 105–112. Gelfgat, A.Y, P.Z.B-Yoseph and A. Solan (1996) Phys. Fluids 8 (10), 2614– 2625. Lopez, J.M (1995) Phys. Fluids 7 (11), 2700–2714. Sorensen, J.N and E.A. Christensen (1995) Phys. Fluids 7 (4), 764–778. Sotiropoulus, F., Ventikos, Y. & Lackey, T.C (2001) J. Fluid Mech. 444, 257– 297.

Vortex Breakdown Flows in Cylindrical Geometry [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

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Serre, E. and bontoux, P. (2002) J. Fluid Mech. 459, 347–370. Blackburn, H.M and Lopez, J.M (2000) Phys. Fluids 12 (11), 2698–2701. Marques, F. and Lopez, J.M (2001) Phys. Fluids 13 (6), 1679–1682. Hall, M.G (1972) Ann. Rev Fluid Mech. 4, 195–218. Delery, J.M. (1994) Prog. Aerospace Sci. 30, 1–59. Escudier, M. (1988) Prog. Aerospace Sci. 25, 189–229. Brown, G.L. and Lopez, J.M (1990) J. Fluid Mech. 221, 553–576. Watson, J.P and Neitzel, G.P (1996) Phys. Fluids 8 (11), 3063–3071. Fujimura, K., Koyama, H. and Hyun. J.M. (1997) J. Fluids Eng. 119, 450–453. Iwatsu, R. and Koyama, H.S in preparation. Iwatsu, R. (2005) J. Phys. Soc. Jpn. 74 (1), 333–344.

Aerodynamic and Structural Analyses of Joined Wings of Hale Aircraft R. Sivaji1 , S. Marisarla1 , V. Narayanan1 , V. Kaloyanova2 , U. Ghia1 and K. N. Ghia2 1

Department of Mechanical, Industrial and Nuclear Engineering Department of Aerospace Engineering and Engineering Mechanics, Computational Fluid Dynamics Research Laboratory, University of Cincinnati, Cincinnati, OH 45221-0072, USA 2

Summary High Altitude Long Endurance (HALE) aircraft high-aspect ratio wings undergo significant deflections that necessitate consideration of structural deformations for accurate prediction of the flow behavior. The objective of this research is to simulate the complex, three-dimensional flow past the joined wing of a HALE aircraft, and to predict its structural behavior based on three different structural models. A Reynolds-Averaged Navier-Stokes (RANS) based flow solver, COBALT, is used for determining the aerodynamic loads on the structure. The structural models considered include a 1-D approximation of the 3-D structure, a twin-fuselage equivalent box-wing model, and a reinforced shell model. Linear static and modal analyses are performed using ANSYS, a finite-element analysis software, to determine the deformation and mode shapes of the structure. The resulting structural deformations in turn affect the flow domain, which has to be re-meshed in a grid generator software and the flow analysis performed again on the deformed shape.

1

Introduction

Recent research trends have indicated an interest in High-Altitude, LongEndurance (HALE) aircraft as a low-cost alternative to space missions. For example, the HALE aircraft based on the Intelligence, Surveillance and Reconnaissance (ISR) or Sensorcraft configuration finds application in fields such as telecommunication relay, environmental sensing, and military reconnaissance. In order to effectively utilize HALE flight vehicles for such missions, the aircraft must be capable of operating at extremely high altitudes to obtain maximum coverage. In addition, the aircraft must also be capable of extended mission duration. At these altitudes, atmospheric density is significantly reduced, and low-Reynolds number (Re) flight conditions exist. Due to the long-endurance requirement, a light vehicle operating at high lift and minimal total drag, and hence, with high-aspect ratio wings, is necessary.

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Figure 1

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Sesorcraft Configuration

Current state-of-the-art HALE aircraft, such as the Global Hawk, is designed to loiter on station for durations of over 24 hours, and fly at high altitudes and low speeds. The proposed model is designed to be in flight for as long as 8 days continuously at altitudes of around 60,000 feet. The present paper examines the aerodynamic performance of a Sensorcraft, a current generation HALE aircraft [1], as shown in Fig. 1. Its highaspect ratio wings undergo significant deflections as a result of the fluid loads acting on them, and require the consideration of aeroelastic effects. The objective of the current work is to analyze the interaction between the flow and the joined-wing [2] structure of HALE Sensorcraft.

2

Aerodynamic Analysis

The flow solver used for the aerodynamics analysis is COBALT60 [3]. It is a code developed by the Air Force Research Laboratories at Wright Patterson Air Force Base in Dayton, Ohio. The fundamental algorithm of COBALT60 is a finite-volume, cell-centered scheme, which is second-order accurate in time and space. COBALT is capable of employing the Spalart-Allmaras model, Detached Eddy Simulation (DES), Menter Baseline model with/without Shear Stress Transport (SST), and the Wilcox k − ω models to model the fine-scale effects of turbulence. COBALT solves the flow on unstructured grids. The unstructured grid generated for the joined wing geometry contains tetrahedral, prism and hexahedral elements. The unstructured grids used were generated using the GRIDGEN software. Viscous flow simulations were carried out using RANS equations with a one-equation Spalart-Allmaras turbulence model for a steady-state simulation, whereas the time-accurate simulations were performed using DES [4, 5]. The boundary conditions, as implemented in the flow solver, are shown graphically in Fig. 2. The farfield boundary is described by the reference flow-field with a specified free-stream Mach number, static pressure and temperature, and angle of attack. The surface of the wing is modeled as an adiabatic, no-slip, solid wall. The “fixed” boundary, or the mirror plane, of the computational domain is modeled by a “symmetry” condition with the use of a slip wall allowing all flow variables to extend freely beyond the computational domain without the end effects of the fixed end of the main and aft

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Farfield

Farfield Farfield

Symmetry

Figure 2

Aerodynamic Analysis Boundary Conditions

wings. Also, Fig. 2 shows that the body occupies about 15% of the streamwise extent and about 50% of the spanwise extent of the computational domain. The block arrow in the figure indicates the flow direction.

3

Structural Analysis

The structural response of the joined-wing model is examined by three different models – 1-D approximation of the 3-D joined wing structure, box-wing model for a twin-fuselage configuration, and reinforced shell model. 3.1

1-D Approximation of 3-D Joined-Wing Structure: Aerodynamics-Structure Coupling.

The 1-D approximation is a simple structural model, created to present a possible approach for the aerodynamics-structure coupling. The pressure loads from the aerodynamic analysis are integrated numerically to obtain the resultant aerodynamic forces and moments (spanwise lift and pitching moment distributions, acting at the aerodynamic center). These are applied on the 1-D structural model. A linear static analysis is performed under this equivalent load, and the deformed shape of the 1-D model is used to obtain the deformed shape of the actual 3-D joined wing. The 1-D model is built for a preliminary structural analysis, and is based on the joined-wing geometry of the aerodynamic model. It represents the original 3-D joined wing as regards its material and structural properties, chosen to be identical to those of the equivalent twin-fuselage box-wing model, provided by AFRL, even though the fuselage is not presently considered. The 1-D beam structure is discretized as a finite element model in ANSYS using BEAM4 elements where the beam axis coincides with the aerodynamic axis of the actual 3-D wing. The relationship between the 1-D approximation and the deformed 3-D geometry is based on solid mechanics assumptions as well as geometrical considerations. A MATLAB code is developed, for which the input consists

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of the (x, y, z) coordinates of the surface grid points of the undeformed 3-D joined wing as well as the ANSYS output for the deformed shape of the 1-D model; the code provides the new (x, y, z) coordinates of the same grid points after deformation. 3.2

Box-Wing Model

The Sensorcraft box-wing model was developed for multi-control surface trim optimization, based on the AFRL/VA In-House Technical Assessment [6] of a vehicle to meet the UAV/ISR mission. This model represents an “equivalent”, twin-fuselage, twin-tail model of the joined-wing, and consists of shear panels to represent the ribs and spars which carry the shear loads, rods to stiffen the structure, and wing skins on which the bending loads act. Masses are attached at specified locations in the box-wing model to account for the weight added by signal processing equipment, VHF antennae, X-Band, fuel, etc. The fuselage and the tail are modeled at about 30 feet from the centerline of the aircraft. The symmetry of the aircraft permits modeling only one half of the total aircraft. The span of this symmetric thin-walled model is 122.86 ft and the chord is 3.35 ft. The wing material is generic aircraft type aluminum with Young’s Modulus 144e7 lbf/sq.ft, Poisson’s ratio 0.28 and density 5.37 slug/cu.ft. Wing skins are considerably thicker at the joint and at the roots to accommodate high stress concentrations. Similarly, the spars at the roots of the wing and at the joint are thicker compared to those in other regions. 3.3

Reinforced Shell Model

In the reinforced shell model, shown in Fig. 3, the surface mesh used for structural analysis is the same as the CFD grid. Even though, in general, the fluid grid required for flow analysis is finer than the structure mesh, retaining the same grid permits the pressure distribution obtained from the

Figure 3

Reinforced Shell Model

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CFD analysis to be accurately transferred to the structure without the need for any interpolation. When translated into ANSYS, this grid defines the outer skin of the wing, and is modeled with shell elements. The reinforced shell model has a span of 38.5 m and a chord length of 2.5 m. The hollow wing is reinforced with ribs, spars and rods that provide adequate stiffness to withstand the pressure loads. The main and the aft wings are reinforced with 13 and 7 ribs, respectively, equally spaced along the span. Both the main and aft wings have eight spars located between 15% and 65% of the chord. The thickness of the reinforcements and the cross section of the rods are identical to those of the box wing, even though the latter is a twin-fuselage model. The mass of the wing is maintained equal to that of the box-wing, mass, i.e., approximately 3975 kg. The ribs and spars are modeled with shear panel elements, while the rods are represented by link elements in ANSYS. Structural masses are added to the wing to represent the weight of the sensor equipment and fuel. This model meets the requirements of a lightweight structure, and represents a realistic wing by retaining the aerodynamic shape of the joined wing. A linear static analysis is performed to determine the deflections and stresses in the model.

4 4.1

Results and Discussion

Aerodynamic Analysis

After suitable test simulations [7] performed on the ONERA M6 Wing and also on the Sensorcraft geometry, viscous simulations were carried out on the joined wing for two different angles of attack, namely, 0◦ and 12◦ , and two Mach numbers. The airfoil cross-section is a standard NACA 4421 airfoil. At the high angle of attack, the airfoil is expected to have maximum lift, and at 12◦ , flow separation is expected. The Mach numbers of 0.4 and 0.6 form the lower and upper limits of the design operating regime of this aircraft. The viscous simulations for M = 0.4, α = 0◦ , and M = 0.4, α = 12◦ were performed on a grid containing 2,091,000 cells. This grid was coarse (y+ = 7) compared to the subsequent grids used. The M = 0.6, α = 0◦ case was computed on a dense volume grid with 6,205,760 cells (y+ = 0.7). This grid is considered adequately clustered for capturing the boundary layer. Currently, only the pressure loads are considered in the procedure of load transfer onto the structure. Also, the computational time taken for obtaining a converged viscous solution even for the coarse grid was about 8 days on a 64-processor PC cluster at UC. Hence, it was deemed necessary to explore inviscid flow simulations as they may provide a reasonable estimate of the fluid pressure loads, at least for lower angles of attack. To confirm this, both inviscid and viscous simulations were performed for M = 0.6, α = 0◦ , and the results compared. The inviscid grid contained 1,732,528 cells, whereas the viscous grid contained over 6 million cells (dense grid). The coefficient of pressure, Cp, compares reasonably for this low angle of attack. Figure 4 shows

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Slice 5 Slice 4

V-section Slice 3

Slice 9 Slice 7 Slice 6

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Figure 4

Airfoil Slice Locations

Mach = 0.6,

= 0°, Slice 1

+ Inviscid

* Viscous

7.0

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0.5 0.0 6.0

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Viscous and Inviscid Cp − Mach = 0.6, α = 0◦

the locations of the various airfoil cross-sections over which the comparisons of the surface Cp were performed. Figure 5 compares the Cp at a particular airfoil cross-section on the main wing (Slice 1 from Fig. 4. At higher angles of attack, viscous effects are expected to cause major deviation from the inviscid behavior. Consequently, M = 0.6, α = 12◦ inviscid and viscous results were also compared. The viscous results for this case were obtained on a grid containing 2,761,440 cells. This “intermediate” grid was obtained by coarsening the dense grid so as to reduce the computational time considerably. Again, surface Cp is compared at the same airfoil section (Slice 1) on the main wing (Fig. 6). As expected, the inviscid and viscous results differ significantly for this case. The viscous results exhibit flow separation at the trailing edge of the main wing in the regions of the joint (Fig. 7). The velocity vectors are colored by the streamwise component of velocity, and the blue region at the trailing edge of the main wing is the region of separated flow. Also, a weak oblique shock is observed at the trailing edge of the aft wing (Fig. 8) as seen from the Mach field at a spanwise section in the joint region. The supersonic region encompasses 255,000 cells.

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Mach = 0.6,

= 12°, Slice 1

+ Inviscid

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x Figure 6

Figure 7

4.2

Viscous and Inviscid Cp − Mach = 0.6, α = 12◦

Velocity Vectors at Joint − March = 0.6, α = 12◦

Structural Analysis

1-D Approximation of 3-D Joined-Wing Structure: AerodynamicsStructure Coupling. The applied integrated aerodynamic load corresponds to the flow conditions M = 0.6, α = 0◦ , altitude 30,000 ft. A significant drop in the computed lift is observed in the region of the joint (Fig. 9a). For the structural analysis, wing roots are constrained in all six degrees of freedom. The maximum displacement is at the tip, and is equal to 20.055 ft (Fig. 9b), so that δ/L is 16.3 % – large deflection, indicating the wing’s flexibility. The ANSYS output for the deformation of the 1-D model is used to obtain the deformed surface grid of the actual 3-D wing, which is supplied to the grid generation software, so an aerodynamic analysis can be performed on the deformed shape.

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Figure 8

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March Contours − March = 0.6, α = 12◦

(a) (b)

Figure 9

1-D Approximation. (a) Lift Distribution. (b) Deformed Shape

Box-Wing Model The results obtained from evaluating the structural behavior of the box-wing model by performing modal, linear, eigenvalue buckling and dynamic analyses are presented. It should be noted that the rotational degrees of freedom are arrested throughout the model in all these analyses. Modal analysis with clamped boundary conditions is performed using Block Lanczos eigenvalue extraction method to extract the first five modes shown in Fig. 10. For the linear static and eigenvalue buckling analyses, the aerodynamic loads from the flow analysis results corresponding to the case M = 0.4, α = 12◦ , and altitude = 60,000 feet are applied on the top (104 psf) and bottom (125 psf) surfaces of the box-wing as averaged uniform pressure distribution accounting for the lift forces acting on the wing. The main and aft wing roots are clamped in all degrees of freedom. Linear static structural analysis is performed in order to study the response of the wing in terms of deflection and stress. Figure 11 shows the displacement and stress contours. The stresses are maximum at the constrained end and in the midsection in

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Mode 1 0.17143 Hz

Mode 3 1.6732 Hz

Mode 5 3.5718 Hz

Mode 2 0.9186 Hz Mode 4 2.6519 Hz

Figure 10

Figure 11

Mode Shapes and Frequencies of Box-Wing

von Mises Stress and Displacement Contours for Box-Wing

the aft wing. The maximum tip deflection obtained is δmax = 6.562 ft, and the maximum von Mises stress is 136.45 MPa. Figure 12 shows the deflection versus distance along the span at the leading edge of the wing. The deflection is seen to vary smoothly along the span for the optimized box-wing structure. Linear eigenvalue buckling analysis is performed in order to calculate the eigenvalues and buckling mode shapes (Fig. 13) of the structure. This is done by first performing linear static analysis activating the pre-stress effects, and subsequently the eigenvalue buckling analysis. The buckling eigenvalues represent buckling load factors which are useful in calculating the critical buckling loads. The response of the wing under the action of time varying aerodynamic loading is also studied. Time history data of integrated forces obtained from the flow analysis is applied as uniform distribution of forces over the upper and lower surfaces of the box wing model. Figure 14 shows the tip nodal z-displacement and von Mises stress versus time. As expected, the deflection and the stress curves follow the load time history. Reinforced Shell Model Pressure loads corresponding to α = 0◦ , M = 0.4 and altitude = 60,000 ft are applied to this structural model. The main and the aft wing roots are constrained in all degrees of freedom to represent their connection with the fuselage.

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Deflection vs Distance Along Span 7.00 6.50 6.00 5.50

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Figure 12

Box-Wing Diflection vs Distance along Span

Mode 1 9.536

Mode 2 13.641

Mode 3 13.956

Mode 4 16.222

Mode 5 20.821

Figure 13

Box-Wing Eigenvalues and Buckling Modes

The maximum deflection, δmax , occurs at the tip and is equal to 0.4 m (Fig. 15). The ratio of tip defection to span length is only 1%, indicating that the model is stiffer than necessary. Moreover, the wing skin “wrinkles” under the action of the pressure loads, leading to stress concentration in the regions directly above the reinforcements, as shown in Fig. 16. The plot of deflection versus distance along the span shows that the deflection is zero at the root, and maximum at the tip. There is noticeable retardation in the slope of the curve near the joint, indicating the drop in the lift force distribution and the role of the aft wing in reducing the deflections at the joint for this un-optimized structure. Further optimization of the wing skin and reinforcements is needed to obtain the appropriate stiffness and also prevent the skin from wrinkling.

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2.75 2.2 1.75 1.2 0

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Figure 14 Time History showing Pressure Loading, z-Displacement and von Mises Stress for Box-Wing Model

Deflection

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Figure 15

Reinforced Shell Model Displacement Contours

5

Conclusion

As a precursor to fluid pressure loads transfer to the structural model, viscous and inviscid results are compared for M = 0.6; the corresponding surface Cp distributions match satisfactorily for low angles of attack. This agreement is lost for α = 12◦ for which the flow separates at the joint. Structural analysis is performed for three different models of the joined wing. The modal

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Maximum stress concentration in the joint

Figure 16

Reinforced Shell Model von Mises Stress Contours

characteristics of the box wing were determined. The load factors determined from the linear buckling analysis provide information about the critical buckling loads for the structure. The linear static analysis for the reinforced shell model indicates that the wing skin “wrinkles” and causes stress concentration in regions directly above the reinforcements. Further optimization of this model is required to obtain the appropriate stiffness of the structure and prevent stress concentration on the wing skin. Linear static analysis performed for the 1-D approximation of the 3-D structure shows large deformations which will affect the flow domain, and a subsequent aerodynamic analysis is needed on the deformed shape. The aforementioned study is a step towards the future goal of transferring loads and deflections back and forth between the fluids and structures disciplines.

Acknowledgements This research was supported, in part, by the Dayton Area Graduate Studies Institute under Task VA-00-UC-03, and supercomputer time was provided by the Ohio Supercomputer Center. The authors would like to Drs. Philip Beran, Greg Reich, Matthew Grismer and Larry Huttsell of Air Force Research Laboratory for their assistance with various aspects of this research.

References [1] Blair M., Moorhouse D., and Weisshaar T., (2000), “System Design Innovation using Multi-Disciplinary Optimization and Simulation,” AIAA-2000-4705. [2] Wolkovitch, J., (1986), “The Joined Wing: An Overview,” Journal of Aircraft, Vol. 23, pp. 161–178. [3] Grismer, M. J., Strang, W. Z., Tomaro, R. F. and Witzeman, F. C., (1998), “Cobalt: A parallel, implicit, unstructured Euler/Navier-Stokers solver”, Advances in Engineering Software, Vol. 29, No. 3–6, pp. 365–373.

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[4] Morton, S. A., Forsythe, J. R., Squires K. D. and Wurtzler, K. E., (2002), “Assessment of Unstructured Grids for Detached-Eddy Simulation of High Reynolds Number Separated Flows,” Proceedings of the Eighth International Conference on Numerical Grid Generation in Computational Field Simulations. [5] Squires, K. D., Forsythe, J. R., Morton, S. A., Strang, W. Z., Wurtzler, K. E., Tomaro, R. F., Grismer, M. J. and Spalart, P. R., (2002), “Progress on Detached-Eddy Simulation of Massively Separated Flows,” 40th AIAA Aerospace Science Meeting & Exhibit, January, Reno, NV, AIAA-2002-1021. [6] Reich, G., (2002), “VA In-house Sensorcraft Box-Wing Configuration,” Private Communication. [7] Sivaji. R, Ghia, U. and Ghia, K., Thornburg, H., (2003), “Aerodynamic Analysis of a Joined-Wing Configuration of a HALE Aircraft,” 41st AIAA Aerospace Science Meeting & Exhibit, January, Reno, NV, AIAA-2003-0606.

Numerical Simulation of Formation and Movement of Various Sand Dunes R. Zhang, T. Kawamura and M. Kan Ootsuka 2-1-1, Bunkyo-ku, Ochanomizu University, Tokyo, Japan

Summary In this paper, four types of sand dunes are simulated by using Large-Eddy Simulation (LES) method. In the flow direction and span direction cyclic boundary conditions are imposed for velocity and pressure. The movement of the sand dunes which are well know as barchan dunes, transverse dunes, linear dunes and Chinese wall have been investigated. The numerical method employed in this study can be divided into three parts: (i) calculation of the air flow above the sand dunes using standard MAC method with a generalized coordinate system, (ii) estimation of the sand transfer caused by the flow through the friction, (iii) determination of the shape of the ground. Because the computational area has been changed due to step (iii), steps (i)–(iii) are repeated until prescribed times.

1

Introduction

We can find various sand dunes in the desert. Most dunes can be classified into barchan dunes, transverse dunes, linear dunes and star dunes Fig. 1 [1]. It is considered that the formation of them depends on the mass of the sand supply, the strength of the wind and the steadiness of the wind direction [2]. However, it is difficult to find out these effects on the formation and the movement of these typical sand dunes by observation or experiment because of the large spatial scale of the sand dunes and the long time scale of these formations. Barchan and star dunes are associated with limited sand supply and the growth of the dunes leads to the uncovering of bare desert floor. When the desert floor is uncovered, sand coalesces into barchan dunes and star dunes even faster because the probability of deposition when a sand grain impacts a dune is much greater than when a sand grain impacts a hard desert floor (since it rebounds with much more energy). Although barchan dunes and star dunes both form with limited sand supply, the difference is in the wind direction variability: bachan dunes form in unidirectional wind and star dunes form in regions with wind from multiple directions. Transverse dunes form in unidirectional wind with large sand supply than barchan dunes. Linear

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Figure 1

Various sand dunes (arrows show wind directions)

dunes form from barchan dunes by two directional winds and extends at the converging direction. We are interested in estimating the movement of the sand dunes quantitatively by means of the numerical simulations and applying the results to the environmental problems which might occur in arid lands. In this study, we are trying to simulate the formation of the typical sand dunes and to make clear the mechanism of the shape of the sand dunes.

2

Numerical Methods

The numerical method employed in this study consists of the following three parts. (i) Calculation of the air flow above the sand dunes. (ii) Estimation of the sand transfer caused by the flow through the friction. (iii) Determination of the shape of the sand dunes. Because the shape of the sand surface is changed due to step (iii). Steps (i)–(iii) are repeated until prescribed times. We will explain the procedures below.

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Calculation of the Air Flow

The Reynolds number of the flow over the sand surface is high enough that the flow is in turbulence regime, therefore we use LES method to compute turbulent flow over the complex geometry. Standard MAC method is employed to solve three-dimensional Navier-Stokes equations. The shape of the sand dune is rather complex and will change in time. In order to impose boundary condition accurately along the sand surface, the time dependent body fitted coordinate system ξ = ξ(x, y, z, t) ,

η = η(x, y, z, t) ,

ς = ς(x, y, z, t)

(1)

is used in this study. Then the computation can be done on the time independent rectangular grids [3]. 2.2

Estimation of Sand Transfer

According to the study of Bagnold, there are three types of sand transfer. They are surface creep, saltation and suspension. These processes are depending on the radius of the sand granularity and the power of the wind. It is considered by the observation in the actual sand dunes that saltation is dominant among three types. In this study, we assume that the sand is transferred only by saltation. Equation of sand transfer is giver by ρ q = c u3∗ , g

(2)

where u∗ is the friction velocity, ρ is the density of the air and q is the mass transfer of the sand, c is the constant which is determined by the experiments [4]. The friction velocity is given by  d |U | , (3) |u∗ | = υ dz where U is the velocity parallel to the sand surface and υ is the turbulence viscosity. 2.3

Determination of the Shape of the Sand Dunes

The sand dune will change its shape by the sand transfer estimated by (2). Considering the local coordinate system along the sand surface, continuity equation of sand becomes ρs

dq2 dh dq1 − , =− dth dX dY

(4)

where ρs is the density of the sand and h is the normal distance from the base plane parallel to the sand surface. q1 , q2 are theX, Y components of the

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vector q. X and Y are the axes determined by the base plane and the original x-z plane and y-z plane respectively. This equation means that the increment of h with time equals to the net influx of the sand into the small region. By discretizing this equation, the shape of the sand dunes are determined in every time step. Because the time scale of the change of the air flow (∆t) is quite different from that of the change of the sand surface (∆th ), the time increment to integrate the (4) is 600 times of that used to integrate the Navier-Stokes equation (∆th = 600∆t). It means that we estimate the sand transfer every 600∆t. If the slope of the sand exceeds the maximum angle which is 320 , the height of the sand at the grid point is changed artificially both to keep the maximum value and to satisfy the conservation of the sand.

3 3.1

Results and Discussion

Barchan Dunes

We calculate the flow field over the fixed sand dune without movement during the first 4000 steps in order to obtain the initial conditions. The time increment ∆t for the Navier-Stokes equation is set to 0.02s. By using these initial conditions, we repeat steps (i)–(iii) mentioned in Sect. 2 and compute the change of the shape of the sand dunes. In the flow direction and span direction cyclic boundary conditions are imposed for velocity and pressure. We also imposed uniform flow on all other boundaries except on the sand surface. Although the shape of the sand surface changes with time, the no-slip condition is imposed since the sand moves very slowly. Initially wind velocity in the flow direction u0 is 11m/s and is assumed to blow all over the region. As is shown in Fig. 2, the sand dune initially has elliptic cross section parallel to x-y plane and parabolic cross section parallel to both x-z and y-z planes. The height of the dune is 20 m and lengths of major and minor axes of the base are 40 m and 30 m. The number of grid points is 85 × 114 × 24. Figure 3 is the bird’s eye view of the surface contours at various times. Figure 3 (a) is the initial shape. Figure 3 (b) is the shape after 3 days. It is clear that the slope of leeward side of the sand becomes steeper and that of windward side becomes gentler than the original slope in only one day although the top of the dune moves very little distance. We can see the crescent shape in Fig. 3 (c) clearly. Figure 4 shows the velocity vectors in the cross section of y = 0 m. Figure 4 (a)–(c) are the results corresponding to Fig. 3 (a)–(c). We can see the large separation of the flow behind the ridge in each figure and an ascending flow along the rear side of the dune. This flow is considered to make the slope of this side steep. In Fig. 4 (b) and (c), the slope has its maximum value which

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is set to 32 degrees before computation. On the other hand, the flow ahead of the ridge is smooth and fast. This flow gradually erodes the dune and makes the dune lower. 3.2

Transverse Dunes

Initially uniform wind velocity of x-direction is 10m/s and is assumed to blow all over the region. As is shown in Fig. 5, the initial sand dunes are three hills which have circle cross sections parallel to x-y plane and parabolic cross sections parallel to both x-z plane and y-z plane. The height of the dunes are 7m and the radius of the circles are 15 m. The number of grid points is 41 × 91 × 24. The other condition is as the same as in Sect. 3.1. Figure 6 is the time development of surface contours (x-y plane). Figure 6 (a) shows the initial shape and Fig. 6 (b) and (c) show the shape after 0.5 day and 2 days. It is clear that the hills develop wings from the each end of them and connect with each other already. Figure 6 (d) shows the shape after 17 days. The simulated dunes, which have circle cross sections initially, become essentially parallel straight ridges which are known as transverse dunes. One slip face can be seen because of the wind from only one direction. Figure 7 shows the shape of the dunes and the velocity vectors in the cross section of y = 0 m. 3.3

Linear Dunes

Barchan dunes and transverse dunes are formed by wind having a single dominant direction and are oriented with their axes at right angles to wind direction and have only one slip face. Perhaps the most controversial type of dune is the linear dune commonly known as the seif dunes in Africa and Saudi Arabia and as the longitudinal

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dune in many other regions. It is, in general, a straight ridge with slip faces on both sides, because it is formed by the wind which have two dominant directions. Groups of these ridges, appear as parallel straight lines. Initially wind velocity of y-direction u1 is 10 m/s and after 17 hours the wind (direction) is changed with u2 = 11 m/s, the angle between wind direction and y-axis is 1300 . We assume that the wind blow all over the region. We calculate the flow field over the fixed sand dune without the movement during the first 2000 steps of every 17 hours in order to obtain the initial conditions of the flow. The other condition is as the same as in Sect. 3.1. As is shown in Fig. 8, the initial sand dune has elliptic cross section parallel to x-y plane and parabolic cross section parallel to both x-z and y-z

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planes. The height of the dune is 25 m and lengths of major and minor axes of the base are 60 m and 40 m. The number of grid points is 85 × 77 × 26. Figure 9 shows the velocity vectors in the plane of 1.4 m above the sand surface. Figure 9 (a) indicates the flow field around the initial sand dune without movement. Large separations of twin vortices are found in the rear region of the sand dune. The flow near the ridge of the dune is fast and this makes the active movement of the sand dune. It looks like a barchan dune in Fig. 9 (b). Figure 9 (c) is the velocity vectors after 16 days. Straight ridge with slip faces on both sides can be seen. Two slip faces are the result of winds from two directions. These changes of the shape can be seen more clearly from the bird’s eye view shown in Fig. 10. 3.4

Chinese Wall

Chinese wall is the special example of linear dunes. It is common for winds to come from the opposite direction. One reason the great sand dunes reach such great heights is due to reversing winds. Reversing dunes grow atop themselves rather than moving forward across the landscape. Initially wind velocity u is 10 m/s and after 14 hours the wind is changed on the opposite direction as is shown in Fig. 11. We assume that the wind blow all over the region. We calculate the flow field over the fixed sand dune without the movement during the first 2000 steps of every 14 hours in order to obtain the initial conditions of the flow. The other condition is as the same as in Sect. 3.1. In Fig. 11, the arrows show the wind direction. We can see in Fig. 11 (b) and (c), when the angle of repose is exceeded by the dune, a sand avalanche occurs on the lee side of the dune, and in the middle of the Fig. 11 (d) is the dune which is called Chinese wall.

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Concluding Remarks

In this study, we proposed one method to simulate the sand movement by the wind, investigate the flow over the typical sand dunes and make clear the mechanism of the formation of it. At first, the flow field is obtained without the movement of the dune to get the initial condition for computation. Using these results, the sand dunes are reformed by the change of height of the surface. The flow field and the

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transfer of the sand are coupled through the relation between the friction and the transportation. Remaining problems which are not solved in this study but are planed to treat in the next step are as follows: (1) To investigate the relational factors of the wavelength of the transverse dunes. (2) To make clear the mechanism of the formation of star dunes. (3) To make clear the effect of the vegetation on the sand dunes.

References [1] R.A. Wasson and R. Hyde, “Factors Determining Desert Dune Type”. Nature Vol. 304 (1983), pp. 337–339. [2] Nagashima, “Sand Transfer and Dunes in Deserts”. J. Japan Society of Fluid Mechanics, Vol. 10. No. 33 (1991), pp. 166-180 (in Japanese). (1983), pp. 337– 339. [3] J.F. Thompson, Z.U.A Warsi, C.W Mastin, “Numerical Grid Generation Foundations and Applications”. Elsevier Science Pubulishing Co. Inc. (1985). [4] R.A. Bagnold, “The Movement of Desert Sand”. Proc. Roy. Soc. A157 (1963). [5] R.A. Bagnold, “Physics of Blown Sand and Desert Dunes”. Pp. 259–275.

Evolutionary Multi-Objective Optimization and Visualization S. Obayashi Institute of Fluid Science, Tohoku University 2-1-1, Katahira, Aoba-ku, Sendai, 980-8577 Japan

Summary Self-Organizing Maps (SOMs) have been used to visualize tradeoffs of Pareto solutions in the objective function space for engineering design obtained by Evolutionary Computation. Furthermore, based on the codebook vectors of cluster-averaged values of respective design variables obtained from the SOM, the design variable space is mapped onto another SOM. The resulting SOM generates clusters of design variables, which indicate roles of the design variables for design improvements and tradeoffs. These processes can be considered as data mining of the engineering design. Data mining example will be given for supersonic wing design.

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Design optimization has become a major application area of Computational Fluid Dynamics (CFD). Among various optimization approaches, Multiobjective Evolutionary Algorithms (MOEAs) are getting popular because they will provide a unique opportunity to address global tradeoffs between multiple objectives by sampling a number of non-dominated solutions. To understand tradeoffs, visualization is essential. Although it is trivial to understand tradeoffs between two objectives, tradeoff analysis in more than three dimensions is not trivial as shown in Fig. 1. To visualize higher dimensions, Self-Organizing Map (SOM) by Kohonen [1, 2] is employed in this paper. SOM is one of neural network models. SOM algorithm is based on unsupervised, competitive learning. It provides a topology preserving mapping from the high dimensional space to map units. Map units, or neurons, usually form a two-dimensional lattice and thus SOM is a mapping from the high dimensions onto the two dimensions. The topology preserving mapping means that nearby points in the input space are mapped to nearby units in SOM. SOM can thus serve as a cluster analyzing tool for high-dimensional data. The cluster analysis of the objective function values will help to identify design tradeoffs. Design is a process to find a point in the design variable space that matches with the given point in the objective function space. This is, however, very

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2 objectives

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difficult. For example, the design variable space considered here has 72 dimensions. One way of overcoming high dimensionality is to group some of design variables together. To do so, the cluster analysis based on SOM can be applied again. Based on the codebook vectors of cluster-averaged values of respective design variables obtained from the SOM, the design variable space can be mapped onto another SOM. The resulting SOM generates clusters of design variables. Design variables in such a cluster behave similar to each other and thus a typical design variable in the cluster indicates the behaviour/role of the cluster. A designer may extract design information from this cluster analysis. These processes can be considered as data mining for the engineering design. In this paper, SOM is applied to map objective function values of nondominated solutions in four dimensions. This will reveal global tradeoffs between four design objectives. The multipoint aerodynamic optimization of a wing shape for supersonic transport (SST) at both supersonic and transonic cruise conditions has been performed by using MOEAs previously [3]. Both aerodynamic drags were to be minimized under lift constraints, and the bending and pitching moments of the wing were also minimized instead of imposing constraints on structure and stability. A high fidelity CFD code, a Navier-Stokes code, was used to evaluate the wing performance at both conditions. In this design optimization, planform shapes, camber, thickness distributions and twist distributions were parameterized in total of 72 design variables. To alleviate the required computational time, parallel computing was performed for function evaluations. The resulting 766 nondominated solutions are analyzed to reveal tradeoffs in this paper. The resulting SOM is also used to create a new SOM of the cluster-averaged design variables.

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MOGAs

The genetic operators used here are based on Multiobjective Genetic Algorithms (MOGAs) [4, 5]. Selection is based on the Pareto ranking method and fitness sharing. Each individual is assigned to its rank according to the number of individuals that dominate it. A fitness sharing function is used to maintain the diversity of the population. To find non-dominated solutions more effectively, the so-called best-N selection is employed. For real function optimizations like the present research it is more straightforward to use real numbers for encoding. Thus, the floating-point representation is used here. Accordingly, blended crossover (BLX − α) [6] is adopted at the crossover rate of 100%. This operator generates children on a segment defined by two parents and a user specified parameter α. The disturbance is added to new design variables within 10% of the given range of each design variable at a mutation rate of 20%. Crossover and mutation rates are kept high because the best-N selection gives a very strong elitism. Details for the present MOGA were given in [3, 5] 2.2

CFD Evaluation

To evaluate the design, a high fidelity Euler/Navier-Stokes code was used. Taking advantage of the characteristic of MOGAs, the present optimization is parallelized on SGI ORIGIN2000 at the Institute of Fluid Science, Tohoku University. The system has 640 Processing Elements (PEs) with peak performance of 384 GFLOPS and 640 GB of memory. A simple master-slave strategy was employed: The master PE manages the optimization process, while the slave PEs compute the Navier-Stokes code. The parallelization became almost 100% because almost all the CPU time was dominated by CFD computations. The population size used in this study was set to 64 so that the process was parallelized with 32-128 PEs depending on the availability of job classes. The present optimization requires about six hours per generation for the supersonic wing case when parallelized on 128 PEs. 2.3

Neural Network and SOM

SOM [1, 2] is a two-dimensional array of neurons: M = {m1 · · · mp×q }

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This has the same dimension as the input vectors (n-dimensional). The neurons are connected to adjacent neurons by a neighbourhood relation. This dictates the topology, or the structure, of the map. Usually, the neurons are connected to each other via rectangular or hexagonal topology. One can also define a distance between the map units according to their topology relations. The training consists of drawing sample vectors from the input data set and “teaching” them to SOM. The teaching consists of choosing a winner unit by means of a similarity measure and updating the values of codebook vectors in the neighbourhood of the winner unit. This process is repeated a number of times. In one training step, one sample vector is drawn randomly from the input data set. This vector is fed to all units in the network and a similarity measure is calculated between the input data sample and all the codebook vectors. The best-matching unit is chosen to be the codebook vector with greatest similarity with the input sample. The similarity is usually defined by means of a distance measure. For example in the case of Euclidean distance the best-matching unit is the closest neuron to the sample in the input space. The best-matching unit, usually noted as mc , is the codebook vector that matches a given input vector x best. It is defined formally as the neuron for which (3) x − mc = min [ x − mi ] i

After finding the best-matching unit, units in SOM are updated. During the update procedure, the best-matching unit is updated to be a little closer to the sample vector in the input space. The topological neighbours of the bestmatching unit are also similarly updated. This update procedure stretches the best-matching unit and its topological neighbours towards the sample vector. The neighbourhood function should be a decreasing function of time. In the following, SOMs were generated in the hexagonal topology by using R SOMine 4.0 Plus [7]. Viscovery
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Cluster Analysis

Once SOM projects input space on a low-dimensional regular grid, the map can be utilized to visualize and explore properties of the data. When the number of SOM units is large, to facilitate quantitative analysis of the map and the data, similar units need to be grouped, i.e., clustered. The twostage procedure – first using SOM to produce the prototypes which are then clustered in the second stage – was reported to perform well when compared to direct clustering of the data [8]. Hierarchical agglomerative algorithm is used for clustering here. The algorithm starts with a clustering where each node by itself forms a cluster. In each step of the algorithm two clusters are merged: those with minimal distance according to a special distance measure, the SOM-Ward distance [7]. This measure takes into account whether two clusters are adjacent in the map.

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This means that the process of merging clusters is restricted to topologically neighbored clusters. The number of clusters will be different according to the hierarchical sequence of clustering. A relatively small number will be chosen for visualization (Sect. 3.2), while a large number will be used for generation of codebook vectors for respective design variables (Sect. 3.3).

3 3.1

Four-Objective Optimization for Supersonic Wing Design

Formulation of Optimization

Four objective functions used here are 1. 2. 3. 4.

Drag coefficient at transonic cruise, CD,t Drag coefficient at supersonic cruise, CD,s Bending moment at the wing root at supersonic cruise condition, Mb Pitching moment at supersonic cruise condition, MP

In the present optimization, these objective functions are to be minimized. The transonic drag minimization corresponds to the cruise over land; the supersonic drag minimization corresponds to the cruise over sea. Lower bending moments allow less structural weight to support the wing. Lower pitching moments mean less trim drag. The present optimization is performed at two design points for the transonic and supersonic cruises. Corresponding flow conditions and the target lift coefficients are described as 1. 2. 3. 4. 5.

Transonic cruising Mach number, M∞,t = 0.9 Supersonic cruising Mach number, M∞,s = 2.0 Target lift coefficient at transonic cruising condition, CL,t = 0.15 Target lift coefficient at supersonic cruising condition, CL,s = 0.10 Reynolds number based on the root chord length at both conditions, Re = 1.0 × 107

Flight altitude is assumed at 10 km for the transonic cruise and at 15 km for the supersonic cruise. To maintain lift constraints, the angle of attack is computed for each configuration by using CLα obtained from the finite difference. Thus, three Navier-Stokes computations per evaluation are required. During the aerodynamic optimization, wing area is frozen at a constant value. Design variables are categorized to planform, airfoil shapes and the wing twist. Planform shape is defined by six design variables, allowing one kink in the spanwise direction. Airfoil shapes are composed of its thickness distribution and camber line. The thickness distribution is represented by a B´ezier curve defined by nine polygons. The wing thickness is constrained for structural strength. The thickness distributions are defined at the wing root, kink and tip, and then linearly interpolated in the spanwise direction. The camber

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Figure 3 CFD code

Wing grid in C-H topology

Pressure contours on the upper surface of a wing computed by the

surfaces composed of the airfoil camber lines are defined at the inboard and outboard of the wing separately. Each surface is represented by the B´ezier surface defined by four polygons in the chordwise direction and three in the spanwise direction. Finally, the wing twist is represented by a B-spline curve with six polygons. In total, 72 design variables are used to define a whole wing shape. A three-dimensional wing with computational structured grid and the corresponding CFD result are shown in Figs. 2 and 3. See [3] for more details for geometry definition and CFD information. 3.2

Visualization of Design Tradeoffs: SOM of Tradeoffs

The evolution was computed for 75 generations until all individuals become non-dominated. An archive of non-dominated solutions was also created along the evolution. After the computation, the 766 non-dominated solutions were obtained in the archive as a three-dimensional surface in the four-dimensional objective function space. By examining the extreme non-dominated solutions, the archive was found to represent the Pareto front qualitatively. The present non-dominated solutions of supersonic wing designs have four design objectives. First, let’s project the resulting non-dominated front onto the two-dimensional map. Figure 4 shows the resulting SOM with seven clusters. For better understanding, the typical planform shapes of wings are also plotted in the figure. Lower right corner of the map corresponds to highly swept, high aspect ratio wings good for supersonic aerodynamics. Lower left corner corresponds to moderate sweep angles good for reducing the pitching moment. Upper right corner corresponds to small aspect ratios good for reducing the bending moment. Upper left corner thus reduces both pitching and bending moments.

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SOM of the objective function values and typical wing planform shapes

Figure 5 shows the same SOM contoured by four design objective values. All the objective function values are scaled between 0 and 1. Low supersonic drag region corresponds to high pitching moment region. This is primarily because of high sweep angles. Low supersonic drag region also corresponds to high bending moment region because of high aspect ratios. Combination of high sweep angle and high aspect ratio confirm that supersonic wing design is highly constrained. 3.3

Data Mining of Design Space: SOM of Design Variables

The previous SOM provides clusters based on the similarity in the objective function values. The next step is to find similarity in the design variables that corresponds to the previous clusters. To visualize this, the previous SOM is first revised by using larger number of clusters of 49 as shown in Fig. 6. Then, all the design variables are averaged in each cluster, respectively. Now each design variable has a codebook vector of 49 cluster-averaged values. This codebook vector may be regarded to represent focal areas in the design variable space. Finally, a new SOM is generated from these codebook vectors as shown in Fig. 7. This process can be done for encoded design variables (genotype) and decoded design variables (phenotype). In the earlier study, the genotype was used for SOM. However, the genotype and phenotype generated completely different SOMs. A possible reason is because the various scaling appears in phenotype, for example, one design variable is between 0 and 1 and another is between 35 to 70. The difference of order of magnitude in design variables may lead to different clusters. To avoid such confusion, the genotype is used for SOM here.

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SOM of objective function values with 49 clusters

In Fig. 7, the labels indicate 72 design variables. DVs 00 to 05 correspond to the planform design variables. These variables have dominant influence on the wing performance. DVs 00 and 01 determine the span lengths of the inboard and outboard wing panels, respectively. DVs 02 and 03 correspond to leading-edge sweep angles. DVs 04 and 05 are root-side chord lengths. DVs

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06 to 25 define wing camber. DVs 26 to 32 determine wing twist. Figure 7 contains seven clusters and thus seven design variables are chosen from each cluster as indicated. Figure 8 shows SOM’s of Fig. 4 contoured by these design variables. The sweep angles, DVs 02 and 03, make a cluster in the lower left corner of the map in Fig. 7 and the corresponding plots in Fig. 8 confirm that the wing sweep has a large impact on the aerodynamic performance. DVs 11 and 51 in Fig. 8 do not appear influential to any particular objective. By comparing Figs. 8 and 5, DV 01 has similar distribution with the bending moment Mb, indicating that the wing outboard span has an impact on the wing bending moment. On the other hand, DV 00, the wing inboard span, has an impact on the pitching moment. DV 28 is related to transonic drag. DV 04 and 05 are in the same cluster. Both of them have an impact on the transonic drag because their reduction means the increase of aspect ratio. Several features of the wing planform design variables and the corresponding clusters are found out in the SOMs and they are consistent with the existing aerodynamic knowledge.

4

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Design tradeoffs have been investigated for a multiobjective aerodynamic design problem of supersonic transport by using visualization and cluster analysis of the non-dominated solutions based on SOMs. The present optimization problem is to design supersonic wings defined by 72 design variables with four objectives to be minimized. Design data were generated by MOGAs. SOM is first applied to visualize tradeoffs between design objectives. In the present design case, four objective functions were employed and 766 nondominated solutions were obtained. Three-dimensional non-dominated front

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in the objective function space has been mapped onto the two-dimensional SOM where global tradeoffs are successfully visualized. Furthermore, based on the codebook vectors of cluster-averaged values for respective design variables obtained from the SOMs, the design variable space is mapped onto another SOM. Design variables in the same cluster are considered to have similar influences in design tradeoffs. Therefore, by selecting a member (design variable) from a cluster, the original SOM in the objective function space is contoured by the particular design variable. It reveals correlation of the cluster of design variables with objective functions and their relative importance. Because each cluster of design variables can be identified influential or not to a particular design objective, the optimization problem may be divided into subproblems where the optimization will be easier to lead to better solutions. These processes may be considered as data mining of the engineering design. The present work demonstrates that MOGAs and SOMs are versatile design tools for engineering design.

References [1] T. Kohonen, Self-Organizing Maps. Springer, Berlin, Heidelberg (1995). [2] J. Hollmen, Self-Organizing Map, http://www.cis.hut.fi/∼jhollmen/dippa/ node7.html, last access on October 3, 2002.

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[3] D. Sasaki, S. Obayashi and K. Nakahashi, “Navier-Stokes Optimization of Supersonic Wings with Four Objectives Using Evolutionary Algorithm”. Journal of Aircraft, Vol. 39, No. 4, (2002), pp. 621–629. [4] C. M. Fonseca and P. J. Fleming, “Genetic Algorithms for Multiobjective Optimization: Formulation, Discussion and Generalization”. Proc. of the 5th ICGA, (1993), pp. 416–423. [5] S. Obayashi, S. Takahashi and Y. Takeguchi, “Niching and Elitist Models for MOGAs”. Parallel Problem Solving from Nature – PPSN V, Lecture Notes in Computer Science, Springer, Vol. 1498, Berlin Heidelberg New York, (1998), pp. 260–269. [6] L. J. Eshelman and J. D. Schaffer, Real-Coded Genetic Algorithms and Interval Schemata. Foundations of Genetic Algorithms 2, Morgan Kaufmann Publishers, Inc., San Mateo, (1993), pp. 187–202. [7] Eudaptics software gmbh. http://www.eudaptics.com/technology/somine4. html, last access on October 3, 2002. [8] J. Vesanto and E. Alhoniemi, “Clustering of the Self-Organizing Map”. IEEE Transactions on Neural Networks, Vol. 11, No. 3, (2000), pp. 586–600.

The Framework of a System for Recommending Computational Parameter Choices S. Shirayama RACE, University of Tokyo, 4-6-1, Komaba, Meguro-Ku, Tokyo,153-8904, Japan

Summary Recently, many kinds of computer aided engineering (CAE) codes have become available. However, selecting a suitable code for effective and efficient computation is not easy for users with no knowledge of CAE or no interest in the features of computer systems. Additionally, it is quite difficult to determine appropriate computational parameters even if a well-established CAE code is used. In this study, we propose a framework for selecting parameters appropriate to distributed resources such as computers, databases and knowledge. Two prototype systems are implemented on a PC cluster and a Grid. Numerical experiments show that our proposed framework will be able to recommend appropriate computational parameter to users, and to extract unknown rules in the computations.

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With the development of computer aided engineering (CAE), many computational programs are being produced. Computational programs feature simple control structures and many of them are structured based on previously established outline frameworks of analytical procedures. Many programs can be configured with linear solvers as the core [1, 2]. These characteristics also play an important role in the integration of a program suite. For example, abstracting the computing environment by handling a linear solver section as a library, or its meta-level abstraction [3], changes a program suite into an analytical system. For individual problems, however, some believe that this method is no more effective than conventional methods involving individual reprogramming. To couple structural computations, the work can be simplified by making array variables common. A functionally equivalent program can be created without using object-oriented approaches or other new methodologies. This is why the number of independent programs is still increasing continuously, despite the trend towards systemization of CAE tools. In fields where strong nonlinear behavior is important, corrections at the

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computational algorithm level are necessary for each problem. These are often built into a new code or analytical system as a module. Consequently, the following problems have arisen: • Difficulties in finding a program suited to the specific application • Computational parameters for solving problems are not easy to determine, even after a program has been selected Researchers are now searching for some mechanisms to reduce the user burden. Some software programs support knowledge-based parameter selection [4, 5]. For efficient and highly accurate computations, however, expertise in numeric computations is needed in addition to knowledge of the object of the computation. No practical knowledge bases are currently available as there are few rules that are amenable to mechine processing. This paper discusses support methods for helping a user select a computational program suited to a particular situation and for determining optimal computational parameters for solving the problem. A method for searching for the best parameters and a rule extraction method for studying physical parameters, design parameters and computational parameters are then proposed, and are used to create a recommendation system. Search efficiency is also discussed, as the parameter search space is combinatorially explosive.

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Parameter Recommendation System

The core of the proposed recommendation system is searching for the best parameters by automatically performing a parameter analysis. For this reason, the following two functions are added: • Recommendation of high efficiency and accuracy • Automatic knowledge extraction Although the main purpose of knowledge extraction is to gather knowledge for searching for the best parameters, it is also to improve extensibility by incorporating parameters from a group of evaluation values. 2.1

Parameter Study

The main CAE parameters are design parameters, physical parameters and computational parameters. Studying parameters generally involves performing computations while systematically changing design parameters and physical parameters to obtain a group of computational results. Issues of parameter study can then be summarized as lack of: (A) Search techniques for using existing computational results (data) directly,

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(B) Methods for utilizing knowledge from individual skilled persons and utilizing the skilled persons themselves (human resources), (B1) Formulating implicit knowledge, (Ex) Extracting mathematically unknown rules, (B2) Gathering knowledge for machine processing, (C) Use of historical information, (D) Methodologies and systems for realizing (A) to (C). The aim of the proposed system is to recommend an appropriate computational program and corresponding computational parameters for a given problem, and the system can be constructed by making up for those lack. 2.2

Parameter Mining

An effective method for gathering knowledge in a parameter study is data mining. In a general sense, data mining is the extraction of knowledge from data. However, this is not a unidirectional search, but includes a feedback process for selecting search data from obtained knowledge or rules (upper part of Fig. 1). This process can also increase the efficiency of the parameter study. The problem is that data mining extracts knowledge from a very wide search space, which is the data itself. One of the solutions to this problem is data structuring. If data can be sorted into a set of classes, hierarchical searching becomes available and search efficiency is improved. This also allows for the introduction of a domain model that uses the attributes of the target data of the search. More specifically, information from the data generation process is used. For sets of data between parameters, data can be classified by parameters into classes having relations to the data generation process. Thus, a parameter space is placed between data and knowledge extraction and data is mined through the parameter space (lower part of Fig. 1). This is called parameter mining. Parameter mining attempts to determine rules while searching for

Figure 1

Data mining (up) and parameter mining (down)

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the parameters suited to a particular application. Because of the relationship between parameters and data, the search space has a hierarchical structure. A domain model using information from the data generation process is also easy to implement. As mentioned above, however, some contrivances are needed if the parameters do not reflect the data. 2.3

Framework of the Recommendation System

The proposed recommendation system is a combination of parameter study and parameter mining. The implemented system needs to address points (A) to (D) in Sect. 2.1, and function using information from the data generation process. The keys to the system are: • Further hierarchicalization of the search space, • Utilization of human resources and machine processing, • Use of the domain model. The proposed system recommends parameters through a series of processes: • Setting a problem and determining computational requirements, • Generating a parameter space, • Setting evaluation functions, • Formulating an optimization problem, • Extracting rules, • Utilizing existing knowledge and storing and reusing new knowledge, • Analyzing the history information. First, the parameter space and evaluation functions are determined according to the problem and requirements. Evaluation functions are chosen according to specific requirements of a problem, such as minimizing error in a certain region, preventing numerical solution overflow, and short computational times. For evaluation functions, the relationship between parameters and evaluation values need not be expressed explicitly. A mechanism whereby evaluation values can be computed if only the parameters are known is required. Once the evaluation functions have been determined, a kind of optimization problem is solved. Rules are extracted during this process. To the existing series of processes, a feedback process is added to utilize existing knowledge for limiting the search space and to store and reuse new knowledge.

3

Components of the Parameter Recommendation System

Making as many tasks as possible machine processable is important to all of the components of the recommendation system. For example, tasks that are difficult to machine process are parameterized or given in terms of the Semantic Web or other machine processable data structures.

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Parameter Space

There are many computational programs applicable to the same type of computation. The set of functionality that was coded into a computational program forms the first parameter space. For example, the absence or presence of each piece of program functionality can be represented by 0 or 1 to express computational programs in terms of parameters. As Table 1 shows, the ABC-flow program is expressed as 111011010. The program that was used to create a specific set of data can then be identified by this number. Table 1

Parameterized CFD programs Purpose

Method

Advection Term

Program compressible incompressible temperature free-surface 2D 3D FDM FEM Name

2nd Central

ABC-flow

0

1

1

0

1

1

1

0

1

XYZ-flow

1

1

1

0

1

1

0

1

0

The computational parameters passed to a program for execution is defined as the second parameter space. The computing environment then forms the third parameter space, with the best combination of parameters searched over all parameter spaces. 3.2

Search Procedure

If the parameter space is large, systematically restricting the space to be explored is necessary. To efficiently find parameters suitable for a problem solution, it is preferable to accumulate restrictive conditions by numerical calculation theories, and other methods that reduce the search space. For example, CFD requires parameters to be limited to fluid theories such as turbulent models, computational schemes to be selected by increasing the precision and speed, and computations to be stabilized by mathematical arguments. These methods limit the possible parameter range and reduce the search space. The proposed system uses the best parameter search process for knowledge extraction. More specifically, knowledge extraction is divided into the following steps: (a) Clarifying constraint conditions (machine-processable expressions), (b) Identifying areas containing constraint conditions, (c) Extracting rules from the search history.

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Step (a) is for extracting rules from the parameters and evaluation values belonging to the weak mathematical constraint area of the search space. Step (b) can be compared to machine learning. Even if no rules are found, supervised learning with parameters for divergent solutions and parameters for convergent solutions may allow the area to which a parameters belongs to be suggested from forecast values from given parameters. Step (c) is for extracting strategies as rules by analyzing search history information.

3.3

Optimization by GA and Rule Extraction by Schema Analysis

Optimization problems can be solved formally using the framework given in Sect. 2.3. However, machine learning and rule extraction using information obtained from the search process is the important goal, not the optimization problems explained in the previous section. Independent global searches are therefore needed, and it is not a good idea to use general optimization techniques. In addition, the search space consists of computational techniques and other discrete parameters, as well as timestep and other continuous parameters. This space also contains areas present under CFL condition and other mathematical constraints. Since user-defined evaluation functions have strong nonlinearity even when simple, the search space may be very complicated, containing many crests. For this kind of optimization problem, probabilistic techniques are more advantageous than the gradient method. This paper discusses optimization using a genetic algorithm (GA). The reason for using GA for optimization problems is generally to search only for the optimum value. A group of prospective schemas for determining optimum values are obtained by stopping the evolution process at a certain generation and examining their properties. Even when several evaluation functions exist, these can be processed in the same framework by multipurpose optimization. Such continuous quantities as timestep, grid control parameters, and relaxation coefficients in the solution of a matrix are quantized into discrete parameters, or a real-number GA coding [6] is adopted. The parameter space is then expressed by an encoded gene and an object function is created from the evaluation functions to solve the optimization problem. The complicated nature of a parameter space makes it impossible to obtain the optimum solution, even with a GA. Therefore, the GA search is stopped at a certain generation and a quasi-optimum solution is selected from the remaining solutions. By changing the initial system, a series of quasi-optimum solutions are determined. This group also provides search history information. By schema analysis, rules are extracted from this information. If the extracted rules can be generalized, they are imposed as restrictive conditions on the search space. The rules thus obtained are then fed back to the search process as constraint conditions or individual generation rules.

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4

System Implementation

When implementing the parameter recommendation system, it is necessary to note that the proposed process is highly parallelizable, and exhibits the following features: • A computational environment is set up for hierarchical or redundant computations. • There are sufficient computational resources available to the user. • Knowledge is close to that of individuals and is widely distributed. A system implementation is proposed in terms of a specific procedure. The parameter recommendation system requires problem and computational requirements (precision priority, computation time priority, etc.) to be set first. The problem is then analyzed and the range of computational programs that match is determined from the first parameter group. When a computational program has been chosen, schemas are determined to limit the search area. Based on the functions available, reference problems corresponding to the set problem are examined. Several test or reference problems are constructed at this stage. For example, if an advection term is judged to be important in a fluid analysis problem, test problems with the advection equation altered according to the problem are constructed. If there are no appropriate reference problems, the problem to be solved is left as it is. The evaluation function and fitness are then defined, and optimum computational parameters are determined from test or reference problems, or from the given problem. Figure 2 shows this processing schematically. First, the available computers are reserved dynamically (Fig. 2(A)). The initial individuals are then generated with considering restrictive conditions (Fig. 2(B)). The existence of results is checked (Fig. 2(C)). If there are no existing results, calculations are conducted using the initial system state (Fig. 2(D)). Using the evaluation value of the initial system state, the next generation is selected (Fig. 2(E)). Steps A to E are repeated up to a particular generation, at which point the processing is stopped and the schema, quasi-optimum solutions, and oscillatory solutions are stored (Figs 2(F) and (G)). A new initial system state is then generated and steps A to G are repeated. When a certain number of schemas have been stored, the schemas are analyzed by parameter mining. Rules are stored if found.

5

Framework Verification by Numerical Experiments

The proposed framework is designed to recommend not only the best computational parameters but also the best design and physical parameters. Considering the problem diversity and scale, it is very difficult to verify the effectiveness of the framework for all parameters. Considering that model verification problems exist in CFD and various other fields, knowledge may be obtained

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Figure 2

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System Concept

from a subset of model problems of a practical problem by parameter mining and used for a parameter recommendation to the practical problem. Setting the recommendation of model problems aside as a future subject, this paper introduces experiments as the first stage of verification of other sections. If CFD is taken as a domain, two kinds of model problems, basic and practical, can be considered. The former may be related to phenomena that can be expressed by the linear advection equation, diffusion equation, Burger’s equation, or Poisson’s equation on a scalar field. The latter may be related to basic flows, such as cavity flow, flow around a cylinder, or pipe flow. When handling a high Reynolds number flow, for example, computational parameters for at least solving the linear advection equation with high accuracy are necessary. Based on the assumption that thus recommended functionality is available, this paper verifies that a parameter recommendation is effective for a basic model problem. 5.1

Experimental System 1

To verify the effectiveness of the recommendation system, numerical experiments are performed using a PC cluster. Since the PCs do not need to be linked through high-speed line, and in consideration of future trends, a cluster of notebook PCs linked via wireless LAN was constructed. The main parameter mining components in the parameter recommendation system are a parameter collection component for collecting groups of parameters for quasi-optimum solutions using GA, and a schema analysis component. Network communications are used to pass parameters and for collecting evaluation values, but the level of traffic is small. Since the time required for each computation depends on the parameters, the effective performance is determined by the load balancing. For experimental system 1, a computational program was incorporated into the GA program (MPI for

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parallel processing). The GA component generates several initial individuals and sends these to the nodes. Each node decodes the received individual and performs computations using the decoded data as parameters. The results are then returned sequentially to the GA component. Until a single generation has finished, the GA component continues to send new individuals to nodes as they complete processing. This job scheduling is applied to load balancing although it is a passive method. The overall computation proceeds until a certain generation has completed, and then evaluation values and schemas of evaluation values are collected. Rules are extracted from the collected schemas by data mining. Clementine [7] was used as the mining tool. Clementine is a general-purpose data mining tool that serves as a neural network analysis tool, a decision tree analysis tool, and a statistical analysis tool. 5.2

Numerical Experiments Using System 1 and Discussion

Computational efficiency is first checked with respect to parameter mining from the one-dimensional advection equation. The parameters are time integral {explicit method (first order, second order)}, discrete advection term {upwind difference (first order, third order), central difference (second order, fourth order)}, timestep, and grid spacing. The number of starting individuals was varied. The computation time necessary for one job depends on the parameters. The longest time taken to compute the example was about 20 times more than the shortest time. The five PCs (#0–#4) have a performance differences up to two times. Even in this case, however, the processing speed is expected to rise almost linearly with the number of nodes as the number of starting configurations increases. Parameter mining was then performed using the differences of the exact solution from a preset point as an evaluation function. A schema analysis was performed using the remaining parameter groups left after 100 generations as quasi-optimum solutions. As a result, the solution closest to the exact solution was found when the CFL number was 1. With knowledge about numerical computations, this rule is easy to derive. However, since this rule may not be stored in existing knowledge, it will support parameter selection by a CFD beginner. After supervised learning of parameters giving oscillatory and divergent solutions as negative examples and parameters giving definite or highly fitness as positive examples, the results of learning were stored. If the user gives appropriate parameters after some degree of learning, evaluation values can be forecast with high precision. If the problems to be solved exhibit different characteristics, it becomes difficult to use supervised learning. If similarities in the data generation process are considered with nonlinearity in mind, however, data of similar tendencies can be obtained. In this case, it is important to systematically accumulate data that would ordinarily be discarded.

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Lastly, parameter mining along the two-dimensional advection equation produced the existence of rules for the discrete advection term, grid spacing and timestep that cannot be interpreted from numerical stability theories like CFL condition. Since these results were obtained from simple examples, we cannot conclude whether unknown rules can be derived immediately by the proposed method. However, even for simple problems, the behavior of the numerical solution cannot be determined analytically, and so conducting actual computations contributes to accuracy and efficiency enhancements. Although analytical approaches are important, we should not overlook the possibility of extracting rules using the empirical approaches proposed in this paper. To realize this possibility, an enormous number of case studies are necessary. The individual case studies do not require very much computation time or network traffic. With this in mind, experimental system 2 was constructed. 5.3

Experimental System 2

The problems with system 1 are that a dedicated system based on PC clusters is used and that the GA and computational components are controlled by the same program. System 2 simulates a Grid computational environment. This system consists of the computers in System 1, plus an extra computer #5 serving as the database server. Globus Toolkit [8] was installed on all nodes and node #0 was chosen as the GRAM server. OpenPBS [9] was used as the local resource management tool. GA and computational components were separated and the GA program and computing program were made to communicate with each other by file I/O. This makes it easy to incorporate existing computational programs into the recommendation system. When a job is finished in each computer, the client is notified of the job end and is given a URL specifying the location of the parameters, evaluation values and computational data. Up to the end of a single generation, the system continuously sends decoded individuals (parameters) to each computer that has finished a job, and prompts for execution of a new job in the same way as in system 1. After performing genetic breeding operations at the end of each generation, computations are repeated using a new group of starting individuals until a certain generation. The knowledge extraction component is the same as that of system 1. System 2, however, accumulates parameters, evaluation values and computational data have highly fitness as a database for improving data reusability. In terms of data generation process, the computation content is described using computational parameters stored in XML to allow integrated data management. Existing data is searched using parameters as search keys and a URL is returned for each search result. System 2 is thought to be a highly adaptable to wide-area distributed computation and also to be effective as a Grid application.

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6

Conclusion

To construct a recommendation system for computational parameter, the authors examined parameter studies, produced a set of system requirements and designed a system. The system was found to require an efficient method for acquiring and managing evaluation values, a process for continuously determining the next parameter study from analysis of evaluation values, and a method for extracting knowledge from the process data. The framework of the proposed parameter recommendation system can be summarized as follows: • Parameterizing the data process and assigning a hierarchy to the parameters to acquire and manage evaluated values and extract knowledge efficiently, • Linking an optimum parameter search to an optimization problem for user-defined evaluation values, • Acquiring a group of quasi-optimum solutions in the process of solving the optimization problem by genetic algorithm. • Extracting rules by schema analysis, • Forming a feedback process to use the rules for the next parameter search, • Creating a mechanism for executing many parameter searches efficiently. To verify the framework, two experimental systems were constructed and numerical experiments were performed. The basic idea of the parameter recommendation system was thus proved to be feasible. The computational component of the system is highly parallelizable. The small volume of communication traffic enables effective parameter recommendations, even in a loosely coupled parallel environment. In addition, the system was proven to be suitable for wide-area distributed computations like the Grid computing, including data accumulation and management.

References [1] S. Shirayama and K. Kuwahara,“Navier-Stokes Solution of the Flow Field Around a Complete Automobile Configuration”, Proceedings of the 2nd International Conference on Supercomputing in the Automotive Industry, 1988, pp. 293–304. [2] S. Shirayama,“Local Network Method for Incompressible Navier-Stokes Equations”, AIAA-91-1563-CP, June, 1991. [3] T. Ohta and S. Shirayama,“Building an Integrated Environment for CFD with an Object-Oriented Framework”, Transaction of JSCES, Paper No. 19990001, May, 1999, pp. 27–33. [4] ICEM CFD Engineering: http://www.icemcfd.com/ [5] SMARTFIRE: http://fseg.gre.ac.uk/smartfire/home.html [6] Z., Michalewics, Genetic Algorithm + Data Structures = Evolution Programs, Springer-Verlag, 1992.

The Framework of a System for Recommending Computational [7] Clementine: http://www.spss.com/spssbi/clementine/ [8] Globus Toolkit: http://www.globus.org/ [9] OpenPBS: http://www.openpbs.org/

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Gas flow in Close Binary Star Systems T. Matsuda1 , K. Oka1 , I.. Hachisu2 , and H.M.J. Boffin3 1

2

Department of Earth and Planetary Sciences, Kobe University, Kobe 657-8501, Japan [email protected] Department of Earth Science and Astronomy, College of Arts and Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan 3 European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748 Garching-bei-Muenchen, Germany

Summary We first present a summary of our numerical work on accretion discs in close binary systems. Our recent studies on numerical simulations of the surface flow on the mass-losing star in a close binary star is then reviewed.

1

Accretion Discs

An accretion disc around a compact star in a close binary star system is an ubiquitous and essential object. Accretion discs play, for example, important roles in cataclysmic variables, nova and X-ray sources. The standard theory to explain the physics in accretion discs is the α-disc model proposed by [1]. In this theory, the accretion disc is in some kind of turbulent state, in which turbulent viscosity is parameterized by a phenomenological parameter α. However, the α-disc model is rather crude approximation to an accretion disc in a close binary system: tidal effects due to the companion star are, for example, not taken into account. To better take these effects into account, one has to rely on numerical simulations. 1.1

Numerical Simulations of Gas Flow in a Close Binary System

A pioneering numerical study of accretion discs in close binary systems was started by [2]. At that time both computers and computational fluid dynamics were not well developed, so his work was only a preliminary one. It should be noted that Prendergast also started a pioneering work on barred galaxies at that time. Prendergast and Taam [3] made a first reliable calculation of gas flow in a close binary system using the beam scheme developed by Prendergast. The beam scheme can be considered as a forerunner of the lattice Boltzmann scheme. In order to solve for the fluid flow, the scheme uses the Boltzmann

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equation rather than the Euler equation as a basic equation. The velocity distribution function has values only at fixed points in velocity space. The original scheme had the drawback of too much artificial viscosity. At that time, [4,5] were working on a numerical study of gas flow in a close binary system, and they were surprised to find the paper by Prendergast and Taam. However, since the size of their mass-accreting star was very large, their model corresponded to the maybe less interesting Algol-type binaries rather than to cataclysmic variables or X-ray stars. Sorensen et al. [4, 5] adopted a much smaller size of the mass-accreting star to simulate a compact star, although the size was still much larger than that of a realistic compact star, i.e. a white dwarf, a neutron star or a black hole. If the numerical size of the compact star is smaller than the so-called circularization radius, an accretion disc may be formed. Sorensen et al. used the Fluid in Cell Method (FLIC) with first order accuracy, and computed the flow only in the orbital plane, using a Cartesian grid. Figure 1 shows the density distribution and velocity vectors of a Rochelobe over-flow in a semi-detached binary system with a mass ratio of one. Gas flows out from a mass-losing star (left) through the L1 point towards a mass-accreting compact star (right). The L1 stream, similar to an elephant trunk is visible but the accretion disc is not well resolved.

Figure 1 Two-dimensional hydrodynamic simulation of accretion disc using FLIC method: Density distribution and velocity vectors on the rotational plane are shown. The left oval shape is a mass-losing companion star, while the dot at the right shows the position of a mass-accreting compact object. Gas from the companion stsar flows through the L1 point towards the compact star due to the gravitational attraction to form a so-called elephant trunk (after [5])

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Lin & Pringle [6] investigated a similar problem using the sticky particle method, which utilizes both particles and cells. Particles entering a cell are assumed to collide, and velocities of particles after the collision are calculated assuming the conservation of momentum and angular momentum. This method may be thought as a forerunner of SPH scheme, which is a particle scheme frequently used in the astrophysics community, but it would be more appropriate to consider it as a forerunner of the Direct Simulation Monte Carlo method (DSMC) developed later by Bird (see [7]). In DSMC, the number of particles in a cell is generally much larger, typically 10–100, and collision pairs are selected randomly based on collision probability. The present authors investigated applications of DSMC to astrophysics. 1.2

Modern Calculation of Accretion Flow

Sawada et al. [8, 9] investigated again two-dimensional calculations of accretion discs using the Osher upwind scheme and Fujitsu VP200/400 vector supercomputers. Figure 2 shows the density distribution in the orbital plane in a semi-detached binary system with unit mass ratio. They first made their calculations using a first-order scheme. When they switched to a second-order scheme, they discovered a pair of spiral shaped shock waves, as seen in the figure. It is very suggestive that using higher order scheme reveals a new feature which could not be seen in a scheme with lower accuracy.

Figure 2 Calculation based on the second-order Osher scheme: Density distribution on the rotational plane is shown. A circle at the center represents a mass-accreting compact object. Gas from the mass-losing companion (at left) flows through the L1 point and forms an accretion disc. A pair of spiral shock in the accretion disc can be seen (after [8, 9])

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Spiral shocks in an accretion disc may represent an interesting possibility to solve a long-standing mystery in the theory of accretion discs, i.e. the problem of angular momentum transfer. In order for accretion to occur, the gas in the accretion disc has to lose its angular momentum. In conventional standard disc model, the disc is supposed to be in a turbulent state and the transfer of angular momentum is supposed to occur through the turbulent viscosity. However, in spite of many efforts to show the disc to be unstable, there has been no success. In the spiral shock model, gas loses angular momentum at the shocks. Nevertheless, the spiral shock model had not attracted much attention from researchers, and there was even an opinion that spiral shocks did not exist in three-dimensional calculations. Sawada & Matsuda [10] performed the first three-dimensional hydrodynamic calculation and obtained spiral shocks. Figure 3 shows our recent calculation by [11]. The figure shows an iso-density surface of an accretion disc around a compact object. Flow-lines on the isodensity surface and on the orbital plane are visualized by the LIC method.

Figure 3 Recent three-dimensional calculation: Iso-density surface and flow-lines on the surface/rotational-plane are shown. Three-dimensional structure of spiral shocks is evident. It is remarkable that the flow from the L1 point penetrates into the disc (after [12])

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Discovery of Spiral Shocks by Observation

As was pointed out earlier, the spiral shock model may solve the long-standing angular momentum problem. Even if not so, if spiral shocks are present, they must have some observational implications. In 1997, they were apparently detected by [13] in the cataclysmic variable, IP Pegasi, using the Doppler tomography technique. Tomography is a technique used, for example, to visualize a cross section of the human body by measuring the absorption of irradiated X-rays. In Doppler tomography, emission lines of hydrogen or helium emitted from hot gas circulating around a compact star are observed and analyzed to give a Doppler map. In X-ray tomography, it is the illuminator that rotates about a human, but in the case of Doppler tomography, use is made of the rotation of the binary system. From temporal variation of the spectrum, one can construct a Doppler map, which is a distribution of emission in the velocity space. From a Doppler map only, it is however not possible to construct uniquely the density distribution in the configuration space. Nevertheless, we may draw useful information from Doppler maps. For example, if spiral structure of hot region emitting spectrum lines exists, it is reflected as a spiral structure in the Doppler map. Steeghs et al. found such a structure. The ring-like structure observed in a Doppler map represents an accretion disc. If the disc is axi-symmetric around a compact star, as is assumed in the standard disc model, the emission structure should be also axi-symmetric. However, the emission structure shows a spiral feature. Interestingly, the surface of the mass-losing companion star is also bright. This is because the surface of the companion is irradiated by a radiation from the hot central part of the accretion disc. Moreover this bright region on the companion start is shifted slightly from a symmetry axis. It may be due to a current on the surface of the companion star.

2 2.1

Flow on a Companion Surface

Flow Pattern

So far we discussed the flow in an accretion disc around a compact star, because there have been lots of works both theoretical and observational. On the other hand the companion star donating gas to the accretion disc has not attracted much attention, because it is difficult to observe the flow on its surface. The only exception was the semi-analytic work by [14], who predicted the existence of an astrostophic wind on the companion surface. But quite recently, surface mapping of the companion in cataclysmic variables became possible (see e.g. [15] for the review). Oka et al. [16] performed a three-dimensional simulation of the surface flow on the companion and discovered three kinds of eddies associated with a high/low pressure on the companion surface: the H-, L1, and L2-eddies.

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(a)

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(b)

Figure 4 (a) Iso-density surface of the companion star and the streamlines, viewed from the north (top) and those viewed from the negative x direction (bottom). The mass ratio is 2 and the specific heat ratio is γ = 5/3. (b) Iso-density surface of the accretion disc. The specific heat ratio of γ = 1.01 is adopted

The notations H, L1 and L2 denote the high pressure around the pole, the low pressure around the L1 point and the low pressure at the opposite side to the L1 point, respectively. Figure 4a shows streamlines on the surface of the companion star in a semi-detached binary system with mass ratio 2; the companion is assumed to be two times heavier than the mass-accreting compact star. This mass ratio is taken to model a supersoft X-ray source. Figure 4b shows the accretion disc for the mass ratio of 2. Note that Figs. 4a and 4b are not the result of one calculation. In Fig. 4a the ratio of specific heats, γ, is assumed to be 5/3, i.e. an adiabatic gas, while in Fig. 4b, γ = 1.01 is adopted to obtain an accretion disc. In order for an accretion disc to be formed, some kind of cooling is necessary, and we mimic the cooling by lowering γ. These eddies are nothing but the manifestation of the astrostrophic wind predicted by [14]. In a rotating fluid, the pressure gradient force balances the Coriolis force, and therefore the wind blows along isobaric lines. Since gas is withdrawn from the L1 point, a low pressure is inevitably formed near the L1 point. Gas near the equator feels less Coriolis force and easily flows towards the L1 point, and thus the equatorial region becomes a low pressure region. Because of this, a high pressure is formed near the pole regions. The mechanism of the formation of the L2 eddy is much more complicated. 2.2

Doppler Map

Based on the above result, we can construct a Doppler map of the surface flow. This is not an easy task, because the emission lines are emitted from

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the photosphere of regions of hot temperature. We need to know the temperature distribution on the photosphere and have to calculate ionization states of either hydrogen or helium on it. Temperature of the photosphere of the companion star is very much affected by the irradiation from the central region of the accretion disc and the surface of the compact star. Since, in the present calculation, we do not take irradiation effect into account, we cannot construct real observable Doppler map. We use the following convention. As a candidate of the photosphere, we take an iso-density surface, and plot the horizontal velocity components, Vx and Vy , of the gas on the Vx − Vy plane, i.e. Doppler map. Figure 5 shows the so constructed Doppler map. There are a few characteristics to be mentioned. The dark area in and around the companion star is due to the gas on the surface of the companion. The fact that it is not restricted within the oval shape is reminiscent from the surface flow. If there is no flow, the dark region should be within the oval shape. Note that all these

-1 disc companion -0.5

0

Vy

0.5

1

1.5

2

2.5

3 2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

Vx Figure 5 Constructed Doppler map: Vx and Vy are the horizontal component of the gas velocity. The three dots on Vx = 0 axis are the center of mass of the companion, the total system, and the compact star, from top to bottom, respectively. The oval shape denotes the companion surface, and the curved line represents a possible ballistic orbit of particles ejected from the L1 point

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dark area is not observable, since we do not plot the spectral line intensity in this figure. We may argue that the present model Doppler map may be able to explain some observational feature in some of supersoft X-ray sources. The ring-like structure represents the accretion disc. This shape agrees well with observations. The present model Doppler map is of course a very crude one. We have to perform simulations including radiative transfer to construct more realistic Doppler map. This is our future task.

Acknowledgements T. Matsuda was supported by the grant in aid for scientific research of JSPS (13640241). K. Oka was supported by the Research Fellowships of the JSPS for Young Scientists. This work was supported by “The 21st Century COE Program of Origin and Evolution of Planetary Systems” in MEXT.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

[16]

N.I. Shakura and R.A. Sunyaev: Astron. Astrophys.. 24, 1973, 337. K.H. Prendergast: Astrophys. J. 132, 1960, 162. K.H. Prendergast and R.E. Taam: Astrophys. J. 189, 1974, 125. S.-A. Sorensen, T. Matsuda and T. Sakurai: Prog. Theor. Phys. 52, 1974, 333. S.-A. Sorensen, T. Matsuda and T. Sakurai: Astrophys. Space Sci. 33, 1975, 465. D.N.C. Lin and J.E. Pringle: Structure and Evolution of Close Binary Systems, Proc. IAU Symp. 73, 1976, 237. G.A. Bird: Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1994. K. Sawada, T. Matsuda and I. Hachisu: Mon. Not. R. Astron. Soc. 219, 1986, 75 K. Sawada, T. Matsuda , M. Inoue and I. Hachisu: Mon. Not. R. Astron. Soc. 224, 1987, 307. K. Sawada and T. Matsuda: Mon. Not. R. Astron. Soc. 255, 1992, s17. H. Fujiwara, M. Makita, T. Nagae and T. Matsuda: Prog. Theor. Phys. 106, 2001, 729 T. Matsuda, M. Makita, H. Fujiwara, T. Nagae, K. Haraguchi, E. Hayashi and H.M.J. Boffin: Astrophys. Space Sci. 274, 2000, 259. D. Steeghs, T. Harlaftis and K. Horne: Mon. Not. R. Astron. Soc. 290, 1997, L28. S.H. Lubow and F.H. Shu: Astrophys. J. 198, 1975, 383. V.S. Dhillon and C.S. Watson: Proc. of the Astrotomography Workshop, Brussels, July 2000, ed. H. Boffin, D. Steeghs (Springer-Verlag Lecture Notes in Physics), p 94, 2000 K. Oka, T. Nagae, T. Matsuda, H. Fujiwara and H.M.J.Boffin: Astron. Astrophys. 394, 2002, 115.

Flow Simulation using Combined Compact Difference Scheme with Spectral-like Resolution T. Nihei and K. Ishii ITC, Nagoya University, Nagoya 464-8601, JAPAN, [email protected]

Summary We propose a new combined compact difference(CCD) scheme with high resolution for fluid flow simulations. The first order derivative in the new CCD scheme has eighth-(in periodic boundaries) or sixth-(in non-periodic boundaries) order accuracy and spectral-like resolution. In addition, the accuracy of the Poisson equation solver for the pressure evaluation is also improved. We apply the new scheme to the simulation of vortical flow fields in threedimensional lid-driven cavities and evaluate its effectiveness.

1

Introduction

The numerical analyses of vortical flow fields need high resolution as well as high accuracy, since vortices with different spatial scales are generally found in the flow. The combined compact difference (CCD) scheme proposed by the authors has high accuracy with comparatively low computational cost and it is promising for large scale computation with high accuracy [1]. The class of combined compact difference schemes was proposed by Chu and Fan [2]. In the scheme, the derivatives of a function are evaluated by using its values at the grid points and its derivatives at the two adjacent points. Our CCD scheme has eighth-order accuracy and spectral-like resolution for the first order derivative in the problem with the periodic boundary conditions. The order of computational cost for the CCD scheme is the same as that of the ordinary difference scheme in the case of unsteady shallow water problem. In this paper, we consider the flow simulation of the various vortical flows using our scheme. First, we review the flow simulation with high resolution on the shallow water equation on the sphere [1]. It shows that the spectral-like CCD scheme resolves the small-scale vortices well and the whole accuracy is competitive to the spectral method [3, 4] in some nonlinear unsteady cases. Next, the incompressible vortical flow fields in three-dimensional lid-driven cavity are considered. We discuss the combined compact finite difference scheme with Dirichlet and/or Neumann boundary conditions. The Poisson

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207

equation is usually required to solve in the computation of the incompressible fluid flow. It is also shown that the numerical solutions of the 3D Poisson equation with high accuracy can be obtained by a new iterative method in which we use an ADI method and a CCD formulation. It is noted that the parallelization of the scheme is necessary for the large simulation. Since the parallel algorithm for solving tridiagonal matrix proposed by Mattor et al. [5] is easily extended to the CCD scheme, the parallelized CCD scheme has high efficiency. This algorithm is suitable for the domain decomposition.

2

Combined Compact Difference Scheme

Consider calculating the derivatives of a function f for one direction, in which N grid points are located with uniform spacing h. Let fi , fi , fi and fi be the values of the function and its first-order, second-order and third-order derivatives at i-th grid point xi , respectively. The CCD scheme evaluates these derivatives implicitly from the following approximated relationships:             + b1 h fi+1 + c1 h2 fi+1 + fi−1 − fi−1 + fi−1 fi + a1 fi+1 (1) d1 + (fi+1 − fi−1 ) = 0 , h        a 2      fi+1 + b2 fi+1 + c2 h fi+1 − fi−1 + fi−1 − fi−1 fi + h (2) d2 + 2 (fi+1 − 2fi + fi−1 ) = 0 , h  b3       a3       fi+1 − fi−1 + c3 fi+1 + fi−1 fi + 2 fi+1 + fi−1 + h h (3) d3 + 3 (fi+1 − fi−1 ) = 0 . h The parameters aj , bj , cj and dj , that determine the formal accuracy of the scheme, are determined by minimizing errors in the wavenumber space for most waves excepting very short waves [1, 6]. For example, the CCD scheme whose parameters are given by

a1 = −

8d3 − 195 16d3 − 255 4d3 − 45 8d3 − 315 , b1 = , c1 = − , d1 = , 240 1200 1800 240

a2 = −

11d2 + 15 , 16

a3 = −d3 ,

b2 =

3d2 + 7 , 16

b3 =

8d3 − 15 4d3 − 15 , c3 = − , 20 20

c2 = −

d2 + 3 , 48

(4) d2 = 0.86774 ,

d3 = 2.1721 ,

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Figure 1 Height contours on the Day 15 of the Williamson’s Test Case 5. The grid resolution is 512 × 256. The contour interval is 100 m. Left: Spectral-like 8th-order CCD scheme. Right: Spectral transform method based on the spherical harmonics

has 8th-order accuracy and spectral-like resolution for the 1st-order, 6thorder accuracy for the 2nd-order derivative and has 4th-order accuracy for the 3rd-order derivative, respectively. Figure 1 shows the computational results of the Test Case 5 of the Williamson’s standard test set for the shallow water equations on a sphere [7], that is a test case for a nonlinear unsteady flow, where a cone-shaped mountain is put abruptly into a zonal geostrophic flow on the Day 0. Periodic boundary conditions are used for both longitudinal and latitudinal directions. Due to the spectral-like resolution of the CCD scheme, the result on the Day 15 by using the CCD scheme is indistinguishable from that by using the spectral transform method based on the spherical harmonics with the same grid fineness. For the non-periodic boundaries, we use the relationships f1 + a1,2 f2 + b1,1 hf1 + b1,2 hf2 + c1,1 h2 f1 + c1,2 h2 f2 d1,1 d1,2 d1,3 f1 + f2 + f3 = s1,1 , + h h h a2,1  a2,2  f + f + b2,2 f2 + c2,1 hf1 + c2,2 hf2 f1 + h 1 h 2 d2,1 d2,2 d2,3 + 2 f1 + 2 f2 + 2 f3 = s2,1 , h h h a3,1 a3,2 b3,1  b3,2  f + f + c3,2 f2 f1 + 2 f1 + 2 f2 + h h h 1 h 2 d3,1 d3,2 d3,3 + 3 f1 + 3 f2 + 3 f3 = s3,1 , h h h

(5)

(6)

(7)

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209

where the parameters aj,k , bj,k , cj,k , dj,k and sj,k are determined by using the Hermite expansion near the boundary.

3

Simulation of Incompressible Flows

Consider the incompressible Navier-Stokes equations → → →  → → ∂ u 1 →2 → =− u ·∇ u −∇p+ ∇ u, ∂t Re →

→

∇ · u = 0, →

(8) (9) →

where ∇ is the gradient operator, Re the Reynolds number of the flow, and u and p are the non-dimensional velocity and pressure of the fluid, respectively. The lid-driven cavity flow problem has boundary conditions of

→ →

u

= U w, (10) wall →

where U w is constant vector tangential to the wall on the moving wall, null vector on the other walls. From (9) and (10), we obtain the boundary condition of p as



→

1 →2 →

. (11) = ∇ p

∇ u

Re wall wall The Poisson equation is usually needed to solve for incompressible fluid flow calculations, and its finite difference solver generally takes large portion of the entire computation time to solve the equation. Since CCD schemes implicitly evaluate the derivatives, a method such as SOR, which is often used, is hard to apply to the Poisson equation solver that uses a CCD scheme. Chu and Fan [2] propose an algorithm for a CCD scheme which solves the following 2D elliptic partial differential equation  2  ∂ ∂2 ∂f (x, y) = g (x, y) . (12) + 2 f (x, y) + α ∂x2 ∂y ∂x Their algorithm can be used to solve the Poisson equation by setting α = 0, but its convergence is not fast. We propose new algorithms for the Poisson equation solver with using a CCD scheme. 3.1

The Poisson Equation Solver for Two-Dimensional Problems

The solution of the Poisson equation is obtained as the steady-state solution of ∂2f ∂2f ∂f = + −g . (13) ∂τ ∂x2 ∂y 2

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where τ is a virtual time. By discretizing (13) with the ADI method, we obtain     ∆τ 2 ∆τ ∆τ 2 k+ 12 δx f δy f k − g, (14) = 1+ 1− 2 2 2     1 ∆τ 2 ∆τ 2 ∆τ k+1 δ f δ f k+ 2 − g, (15) 1− = 1+ 2 y 2 x 2 where k is the number of iterative steps, and δx2 and δy2 are the finite difference operator for the 2nd-order derivative that is defined implicitly by the CCD scheme for x and y direction, respectively. ∆τ can be regarded as an acceleration parameter. Equations (14) and (15) can be solved by combining with CCD relationships of (1)–(3) for x (y) direction. This can be solved in parallel by using domain decomposition [8], and therefore, the entire process of the solver can be easily parallelized. When f converges we obtain the solution of (13). The stability of the iterative step is assessed roughly by estimating the amplification factor G of the Fourier mode exp {i (kx x + ky y)} for the iteration, where kx and ky are the wavenumber of the mode for x and y direction, respectively. For simplicity, a rectangle domain with periodic boundaries is assumed and finite difference operators are replaced by exact differential operators. In this case, we obtain    2 − ∆τ kx2 2 − ∆τ ky2  .  (16) G= (2 + ∆τ kx2 ) 2 + ∆τ ky2 Since |G| < 1 for all ∆τ > 0, the iterative step is unconditionally stable. By minimizing max |G| for all possible Fourier modes, the optimum value of ∆τ is estimated as ∆τopt =

max (Nx hx , Ny hy ) min (hx , hy ) , π2

(17)

where Nx , hx , Ny and hy are the number of grid points and the grid spacing for x and y direction, respectively. Figure 2 shows the convergence property of the algorithm for the problem  2  ∂ ∂2f ∂2 ∂2f 2 + 2 = + 2 {(x(1 − x)y(1 − y)} on Ω , (18) ∂x2 ∂y ∂x2 ∂y ∂f = 0 on ∂Ω , ∂n f = 0 on ∂Ω ,

(19) (20)

where Ω = [0, 1] × [0, 1], and n is the local direction normal to ∂Ω. The exact 2 solution f  of (18)–(20) is f = {(x(1− x)y(1 − y)} . The computation grid k   of the size 65 × 65 is used. f − f 2 / f 2 is plotted for each iterative

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211

Figure 2 Convergence of the relative error with respect to iterations for the Poisson solver based on the ADI method. (a): The solver proposed by Chu and Fan. (b): The solver proposed by the authors

step k starting from the initial guess of f 1 = 0, where · 2 denotes the 2 norm. Solution by using this algorithm converges faster than that by using the algorithm proposed by Chu and Fan. It is noted that the computation time required for an iteration of this algorithm is almost the same as that proposed by Chu and Fan. 3.2

The Poisson Equation Solver for Three-Dimensional Problems

A straightforward extension of the two-dimensional algorithm for threedimensional problems is the following:     1 ∆τ 2 k− 1 ∆τ 2 ∆τ ∆τ 2 δx f k+ 3 = δy f 3 + 1 + δz f k − g, (21) 1− 3 3 3 3     2 1 ∆τ 2 ∆τ 2 ∆τ 2 k ∆τ δy f k+ 3 = 1 + δx f k+ 3 + δ f − g, (22) 1− 3 3 3 z 3     2 ∆τ 2 ∆τ 2 k+ 1 ∆τ 2 ∆τ k+1 3 δz f δx f δy f k+ 3 − g, (23) 1− = + 1+ 3 3 3 3 2

where f 3 = f 1 . Unlike the two-dimensional algorithm, this algorithm is conditionally stable. The amplification factor is obtained as     1  A ± A2 − 4∆τ 3 kx2 ky2 kz2 3 + ∆τ kx2 3 + ∆τ ky2 3 + ∆τ kz2 2   , (24) G= 2 (3 + ∆τ kx2 ) 3 + ∆τ ky2 (3 + ∆τ kz2 )

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    A = 27 − 18∆τ kx2 + ky2 + kz2 + 3∆τ 2 kx2 ky2 + ky2 kz2 + kz2 kx2 + 2∆τ 3 kx2 ky2 kz2 , (25) where kx , ky and kz are the wavenumber of the Fourier mode for x, y and z direction, respectively. For the requirement of stability, |G| < 1, ∆τ must satisfy   6 (26) 0 < ∆τ < 2 min h2x , h2y , h2z . π This condition makes the algorithm impractical for large calculations with fine grid. In order to get a stable iterative algorithm for solving the Poisson equation, we consider the discretized equation     k+1     ∆τ  2 δx + δy2 + δz2 f − f k = ∆τ δx2 + δy2 + δz2 f k − g . 1− 2 (27) By the ADI algorithm, we get      1 ∆τ 2 δx ∆f k+ 3 = ∆τ δx2 + δy2 + δz2 f k − g , (28) 1− 2   2 1 ∆τ 2 δy ∆f k+ 3 = ∆f k+ 3 , 1− (29) 2   2 ∆τ 2 δ ∆f k+1 = ∆f k+ 3 , (30) 1− 2 z and f k+1 = f k + ∆f k+1 .

(31)

Like the two-dimensional algorithm, (28)–(30) can be solved by combining with CCD relationships, and the algorithm can be parallelized easily. The amplification factor is obtained as   ∆τ kx2 + ky2 + kz2   . G=1−  (32) 2 2 2 1 + ∆τ 1 + ∆τ 1 + ∆τ 2 kx 2 ky 2 kz Since |G| < 1 for all ∆τ > 0, this algorithm is unconditionally stable. 3.3

Three-dimensional incompressible lid-driven cavity flow

The three-dimensional incompressible lid-driven cubic cavity flows at Re = 100, 200 and 400 are calculated by using a CCD scheme. The parameters of the CCD scheme are determined by (4) with exception of d2 = −9.12992, d3 =

165d2 + 450 . 28d2 + 80

(33)

Flow simulation using CCD Scheme

213

Figure 3 The cross-section of streamlines of the lid-driven cavity flow at crossflow center plane of the cavity. The number of grid points is 333 . Re = 100

The CCD scheme in this case has 8th-order accuracy and high resolution for the 1st-order and the 2nd-order derivative and has 4th-order accuracy for the 3rd-order derivative, respectively. The boundary scheme used here has 6thorder accuracy, in which the boundary condition of (10) or (11) is included. The fourth-order Runge-Kutta method is used for time integration of the Navier-Stokes equation. To maintain the incompressibility, the MAC method is used for each sub-step of the Runge-Kutta method. The computation is done by using 33 × 33 × 33 uniform grid. Figure 3 shows the cross-section of streamlines at cross-flow center plane of the cavity at Re = 100. The results in all cases agree well with that of Ishii and Iwatsu [9], where 81 × 81 × 81 non-uniform grid is used.

4

Conclusions

A new combined compact difference(CCD) scheme with high resolution for fluid flow simulations is proposed. Solvers of the Poisson equation are also proposed, and they converge faster than that previously proposed by Chu and Fan. The result of the three-dimensional incompressible lid-driven cavity flow calculation shows good agreement with the conventional result obtained by using about 14 times grid points. Our scheme can be parallelized easily, and is promising for large-scale flow simulations with high accuracy.

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Acknowledgement Computer resources are provided by the Information Technology Center of Nagoya University. This work is supported by the Japan Science and Technology Agency and Grand-in-Aid for Scientific Research from the Japan Ministry of Education, Science and Culture.

References [1] T. Nihei and K. Ishii: “A Fast Solver of the Shallow Water Equations on a Sphere using a Combined Compact Difference Scheme”. J. Comput. Phys. 187, 2003, pp. 639–659. [2] P. C. Chu and C. Fan: “A Three-point Combined Compact Difference Scheme”. J. Comput. Phys. 140, 1998, pp. 370–399. [3] R. Jakob-Chien, J. J. Hack and D. L. Williamson: “Spectral Transform Solutions to the Shallow Water Test Set”. J. Comput. Phys. 119, 1995, pp. 164–187. [4] H. -B. Cheong: “Application of Double Fourier Series to the Shallow-Water Equations on a Sphere”. J. Comput. Phys. 165, 2000, pp. 261–287. [5] N. Mattor, T. J. Williams and D. W. Hewtt: “Algorithm for Solving Tridiagonal Matrix Problems in Parallel”. Parallel Computing 21, 1995, pp. 1769–1782. [6] S. K. Lele: “Compact Finite Difference Schemes with Spectral-like Resolution”. J. Comput. Phys. 103, 1992, pp. 16–42. [7] D. L. Williamson: “A Standard Test Set for Numerical Approximations to the Shallow Water Equations in Spherical Geometry”. J. Comput. Phys. 102, 1992, pp. 221–224. [8] T. Nihei and K. Ishii: “Parallelization of a Highly Accurate Finite Difference Scheme for Fluid Flow Calculations”. Theor. Appl. Mech. Jap. 52, 2003, pp. 71–81. [9] K. Ishii and R. Iwatsu: “Numerical Simulation of the Lagrangian Flow Structure in a Driven-Cavity”. Topological Fluid Mechanics, (ed. H. K. Mofatt and A. Tsinpbar, Cambridge Univ. Press), 1989, pp. 54–63.

A New High Order Finite Volume Method for the Euler Equations on Unstructured Grids Z.J. Wang 2555 Engineering Building, Michigan State University East Lansing, MI 48824, U.S.A.

Summary A new high-order finite volume method – spectral volume (SV) method – has been recently developed for unstructured grids. The SV method has been shown to be more efficient than the k-exact finite volume method, and possess higher resolution for discontinuities than the discontinuous Galerkin method. This paper reviews the method and presents some computational results for a benchmark computational aeroacoustic problem.

1

Introduction

The spectral volume method is a new high-order finite volume method developed recently for hyperbolic conservation laws on unstructured grids [1–3]. Its design philosophy is related to the high-order k-exact finite volume (FV) method [4], and the discontinuous Galerkin (DG) method [5]. The SV method is a Godunov-type finite volume method [6, 7], which has become the-stateof-the-art for the numerical solution of hyperbolic conservation laws. What distinguishes the SV method from the k-exact finite volume method is in the data reconstruction. Instead of using a large stencil of neighboring cells to perform a high-order polynomial reconstruction, the unstructured grid cell – called a spectral volume – is partitioned into a “structured” set of sub-cells called control volumes (CVs), such as those shown in Fig. 1. Then the cellaverages on these sub-cells are used to reconstruct a high-order polynomial inside the macrocell – the SV, which is to say all the CVs in the SV share the same polynomial reconstructed from the subcell averages. After that, the SV method proceeds exactly as a FV method for the CVs. If all the triangular spectral volumes are partitioned in a geometrically similar manner, the reconstruction problems for all the SVs are identical, and can be solved analytically. This is why the SV method is more efficient than the k-exact FV method. Because monotonicity limiters can be designed for the subcells, the SV method was shown to have higher resolution for discontinuities than the DG method [3].

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Z.J. Wang

d (a) linear Figure 1

d (b) quadratic

(c) 4th Order – Cubic

Partitions of various orders in a triangular spectral volume

In this paper, we apply a 2D SV Euler solver to study several benchmark problems in computational aeroacoustics. In the next section, we first present the basic SV method for the 2D Euler equations. In Sect. 3, several numerical tests are used to assess the performance of the SV method. A second-order FV scheme is compared to the SV schemes for relative cost to achieve the same solution quality. Finally, conclusions are summarized in Sect. 4.

2

Basic Idea of the Spectral Volume Method

The unsteady 2D Euler equation in conservative form can be written as ∂Q ∂E ∂F + + =S, ∂t ∂x ∂y

(1)

where Q is the vector of conserved variables, E and F are the inviscid flux vectors and S is the source vector. Given a desired order of accuracy k for (1), each spectral volume Si is then partitioned into m = k(k + 1)/2 control volumes (CV s), and the j-th CV of Si is then denoted by Ci,j . Figure 1 displays several possible partitions for linear, quadratic and cubic SVs. Given the cell-averaged conservative variables for all the CV s in Si , a polynomial pi (x, y) ∈ P k−1 (the space of polynomials of degree at most k − 1) can be reconstructed such that it is a k-th order accurate approximation to the function Q(x, y) inside Si : pi (x, y) = Q(x, y) + O(hk ), (x, y) ∈ Si .

(2)

This reconstruction can be solved analytically by satisfying the following conditions:  pi (x, y)dxdy Ci,j ¯ i,j . =Q (3) Vi,j

A New High Order Finite Volume Method

217

The reconstruction can be more conveniently expressed as pi (x, y) =

m


¯ i,j , Lj (x, y)Q

(4)

j=1

where Lj (x, y) ∈ P k−1 are the “shape” functions which satisfy  Lm (x, y)dxdy Ci,j

Vi,j

= δjm .

(5)

The high-order reconstruction is then used to generate high-order updates for the cell-averaged state variable on the CVs. Integrating (1) in Ci,j , we obtain the following integral equation for the CV-averaged mean  K  ¯ i,j dQ 1
1 + (f • n)dA = SdV , (6) dt Vi,j r=1 Vi,j Ar

Ci,j

where f = (E, F ), K is the number of faces in Ci,j , and Ar represents the r-th face of Ci,j . Both the surface and volume integrals can be computed with a k-th order accurate Gauss quadrature formula. With the polynomial distribution on each SV , the state variable is discontinuous across the SV boundaries. Therefore, the flux at the interface depends on two discontinuous state variables just to the left and right of a face. This flux is computed with a Riemann solver or flux splitting procedures, i.e., f (Q) • nr ≈ fRiem (QL , QR , nr ) ,

(7)

where QL and QR are the vector of conservative variables just to the left and right of a face. It is the Riemann solver which introduces the “upwinding” and dissipation into the SV method such that the SV method is not only highorder accurate, but also stable. For time integration, we use the third-order TVD Runge-Kutta scheme from [8].

3 3.1

Numerical Results

Accuracy Study with Vortex Evolution Problem

This is an idealized problem for the Euler equations in 2D used by Shu [9]. The mean flow is {ρ, u, v, p} = {1, 1, 1, 1}. An isotropic vortex is then added to the mean flow, i.e., with perturbations in u, v, and temperature T = p/ρ, and no perturbation in entropy S = p/ργ : ε 0.5(1−r2 ) e (−y, x) , 2π (γ − 1)ε2 1−r2 δT = − e , 8γπ 2

(δu, δv) =

δS = 0 ,

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Z.J. Wang

Figure 2

An Irregular (10×10×2) Grid Used in the Vortex Propagation Problem

where r2 = x2 + y 2 , and the vortex strength ε = 5. If the computational domain is infinitely big, the exact solution of the Euler equations with the above initial conditions is just the passive convection of the isotropic vortex with the mean velocity (1, 1). In the numerical simulation, the computational domain is taken to be [−5, 5] × [−5, 5], with characteristic inflow and outflow boundary conditions imposed on the boundaries. The numerical simulations were carried out until t = 2 on an irregular grid shown in Fig. 2. The finer irregular grids are generated recursively by cutting each coarser grid cell into four finer grid cells. In Table 1, the L1 and L∞ norms in the CV-averaged density are presented for SV schemes of second to fourth orders. The errors presented in the Table are made time step independent by using sufficiently small time steps. Note that all the simulations have reached the desired order of accuracy in the L1 norm. The L∞ norms degrade about half an order on the irregular mesh probably because the mesh is not smooth. In order to compare the SV method with a second-order finite volume (FV) method, simulations were also performed on the sub-cell grids generated by the various SV partitions. These grids are named SV2, SV3 and SV4 grids respectively as shown in Fig. 3. In other words, the numbers of unknowns in the finite volume simulations are the same as those of the corresponding SV simulations. The L1 and L∞ errors computed using a second-order FV method [10] on the sub-cell grids are presented in Table 2. It is obvious that a second-order accuracy has been achieved in both the L1 and L∞ norms by the finite volume solver on all the grids. Note that the solution errors on the SV3 grids are only slightly lower than those on the SV2 grids although the SV3 grids have twice as many control volumes as the SV2 grids. Furthermore, the errors on the SV4 grids are larger than those on the SV2 grids although the former have more than three times the number of control volumes. This can be attributed to the non-smoothness of the control volumes in the SV4

A New High Order Finite Volume Method Table 1

Accuracy Study of SV Schemes with Propagating Vortex Problem

Order

2

3

4

Grid

L∞ error

L∞ order

L1 error

L1 order

10 × 10 × 2

8.48e-2

–

5.49e-3

–

20 × 20 × 2

2.72e-2

1.64

1.38e-3

1.99

40 × 40 × 2

7.37e-3

1.88

3.79e-4

1.87

80 × 80 × 2

2.22e-3

1.73

9.91e-5

1.94

10 × 10 × 2

2.74e-2

–

1.52e-3

–

20 × 20 × 2

6.03e-3

2.18

2.78e-4

2.46

40 × 40 × 2

8.51e-4

2.83

4.36e-5

2.68

80 × 80 × 2

1.61e-4

2.40

6.79e-6

2.68

10 × 10 × 2

7.90e-3

–

3.47e-4

–

20 × 20 × 2

4.53e-4

4.12

3.00e-5

3.53

40 × 40 × 2

4.74e-5

3.26

2.12e-6

3.82

80 × 80 × 2

3.82e-6

3.63

1.45e-7

3.87

(a) SV2 Figure 3

219

(b) SV3

(c) SV4

Control volumes for the second to fourth order spectral volume schemes

grids. It is interesting to see that the second-order finite volume method is more accurate than the second-order SV method with the same number of DOFs. This is because the linear reconstruction is for the macro-cell in the SV method, but for the sub-cell in the FV method. The third and fourth order SV schemes are more accurate than the second-order FV scheme with the same DOFs. For example, the 3rd order SV method is a factor of 3.81 more accurate (in terms of L1 error) than the 2nd order FV scheme on the finest mesh, while the 4th order SV scheme is a factor of 435 more accurate than the 2nd -order FV scheme with the same number of DOFs.

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Table 2 Accuracy Study of a Second-Order FV Scheme on the Control Volumes Generated Using Various SV Partitions with Propagating Vortex Problem Grid

SV2

SV3

SV4

DOFs

L∞ error

L∞ order

L1 error

L1 order

10 × 10 × 2 × 3

7.16e-2

–

3.79e-3

–

20 × 20 × 2 × 3

1.80e-2

1.99

8.84e-4

2.10

40 × 40 × 2 × 3

4.57e-3

1.98

1.94e-4

2.19

80 × 80 × 2 × 3

1.15e-3

1.99

4.41e-5

2.14

10 × 10 × 2 × 6

3.51e-2

–

2.05e-3

–

20 × 20 × 2 × 6

1.20e-2

1.55

4.66e-4

2.14

40 × 40 × 2 × 6

2.76e-3

2.12

1.03e-4

2.18

80 × 80 × 2 × 6

1.06e-3

1.38

2.59e-5

1.99

10 × 10 × 2 × 10

1.06e-1

–

3.95e-3

–

20 × 20 × 2 × 10

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Next we compare the relative cost of the SV and FV schemes. In this paper, the relative cost is defined roughly as the relative CPU time needed to achieve the same solution error. The following CPU times were measured on a 2.0 GHz Intel Pentium 4 computer. Forty residual computations on the finest mesh with a 2nd -order SV scheme took 4.25 seconds. The 2nd -order FV scheme took 5.13 seconds, the 3rd -order SV scheme took 14.8 seconds and the 4th -order SV scheme took 35.2 seconds. For a 2nd -order FV scheme in 2D, the error should decrease linearly with the increase in the number of cells. The relative cost is estimated assuming that the SV2 grid is chosen for the 2nd -order FV scheme. Therefore, the 2nd -order SV scheme has a cost of 1.86 (4.25/5.13*9.91e-5/4.41e-5) relative to the 2nd -order FV scheme (i.e., takes 1.86 times the CPU time to achieve the same error). The 3rd -order SV scheme has a relative cost of 0.44 (14.8/5.13*6.79e-6/4.41e-5), and the 4th -order SV has a relative cost of 0.023 (35.2/5.13*1.45e-7/4.41e-5). Clearly the high-order SV schemes are much more efficient than the 2nd -order FV scheme in achieving the same solution quality. 3.2

Scattering Of Periodic Acoustic Source from Two Cylinders

This case is selected from [11]. The scattering from two circular cylinders of a periodic acoustic source is considered. The acoustic source can be expressed

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in the following form (x−xc )2 +(y−yc )2

S = e− ln 2  b2 sin(ωt)f (t)  3  t . f (t) = min 1, t0 The following parameters are chosen in the present study: xc = 0, yc = 0,  = 6π, t0 = 4 and b = 0.2. The two cylinders both have a diameter of 1 and are located at (4, 0) and (−4, 0). Due to the problem symmetry, only a quadrant of the physical domain is considered in the computation. A coarse mesh with an approximate size of 0.14 is presented in Fig. 4. Since each triangle is further partitioned into 10 sub-cells (control volumes) in the spectral volume method, the equivalent number of point-per-wave (PPW) is about 7.5 (sqrt(10)(1/3)/0.14). The entire computational domain extends to r = 13, with grid expansion for r/D > 9. The coarse mesh has 13 points on the half cylinder, the medium mesh has 17 points and the fine mesh has 24 points. The initial conditions are ρ = 1, p = 1/1.4, u = v = 0. The computation was carried out until t = 30, when a periodic solution seemed to have been obtained as shown in the pressure history plot in Fig. 5. The computed acoustic pressure on the medium mesh is displayed in Fig. 6. The computed mean-squared fluctuating pressure (MSFP) is compared with the analytical values [11] in Figs. 7 and 8. The MSFP has been normalized using the analytical magnitude of the incident fluctuating pressure field at a distance from the source corresponding to the center of the cylinder, which is 0.000186533

Figure 4

Coarse grid for the two cylinder case, mesh size = 0.14

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Figure 6 Computed acoustic pressure on the medium mesh (mesh size = 0.1) with a source of frequency 6π

in the present case. Note that the agreement between the computational and analytical solutions is very good at a grid resolution of about 15 PPW. The solution quality seems to be comparable to that in [11] with the same number of unknowns. However, the problem setup time is significantly lower in the present study due to the use of unstructured grids.

4

Conclusions

An accuracy study was performed for the SV method, and it was shown that the high-order SV schemes are much more efficient than the second-order FV scheme in achieving the same solution quality. The 4th order SV scheme is then used to simulate a CAA benchmark problem, and excellent agreement was obtained with the analytical solution.

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Exact 4th Order SV, 148,750 DOFs

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Computed mean-squared fluctuation pressures

Acknowledgement The author thanks Drs. Miguel Visbal and Scott Sherer of Air Force Research Laboratory for many helpful discussions, and for providing the benchmark solutions. Part of the work was performed while the author visited the AFRL under a summer faculty program.

References [1] Z.J. Wang, Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J. Comput. Phys. 178, 210 (2002). [2] Z.J. Wang and Yen Liu, Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation, J. Comput. Phys. 179, 665-697 (2002). [3] Z.J. Wang and Yen Liu, Spectral (finite) volume method for conservation laws on unstructured grids III: one-dimensional systems and partition optimization, J. of Scientific Computing, to appear. [4] T.J. Barth and P.O. Frederickson, High-order solution of the Euler equations on unstructured grids using quadratic reconstruction,” AIAA Paper No. 900013, 1990. [5] B. Cockburn and C.-W. Shu, The Runge-Kutta discontinuous Garlerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 199–224, (1998). [6] S.K. Godunov, A finite-difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47, 271 (1959).

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[7] B. van Leer, Towards the ultimate conservative difference scheme, Journal of Computational Physics, 1979, vol. 32: 101. [8] C.-W. Shu, Total-Variation-Diminishing time discretizations, SIAM Journal on Scientific and Statistical Computing 9, 1073–1084 (1988). [9] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni), Lecture Notes in Mathematics, volume 1697, Springer, 1998, pp. 325–432. [10] Z.J. Wang, A fast nested multi-grid viscous flow solver for adaptive Cartesian/quad grids, International Journal for Numerical Methods in Fluids, vol. 33, No. 5, pp. 657-680, 2000. [11] S. Sherer and M. Visbal, Computational study of acoustic scattering from multiple bodies using a high-order overset grid approach, AIAA Paper No. 2003-3203.

Participants Photograph

Participants List

International 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Jay Boris, Naval Research Laboratory, USA C. K. CHU, Columbia University, USA Peter R. EISEMAN, Program Development Company, USA Kirti N. GHIA, University of Cincinnati, USA Urmila GHIA, University of Cincinnati, USA James C. Hill, Iowa State University, USA Jae Min HYUN, Korea Advanced Institute of Science and Technology, SOUTH KOREA Ho Sang KWAK, Kumoh National Institute of Technology, Korea Matthias MEINKE, Aerodynamische Institute of Rheinisch– Westflischen Hochschule, Germany Elaine ORAN, Naval Research Laboratory, USA Z. J. WANG, Michigan State University, USA Jaw-Yen YANG, National Taiwan University, TAIWAN Norman J. ZABUSKY, Rutgers University, USA

Domestic 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Angel M. Bethancourt, Institute of Computational Fluid Dynamics Satoshi CHIBA, Yokkaichi University Noriko DOI, Institute of Computational Fluid Dynamics Kozo FUJII, The Institute of Space and Astronautical Science Yasuhiko FUJIKAWA, VINAS Co. Seiji FUJINO, Kyushu University Yu FUKUNISHI, Tohoku University Toru FUSEGI, Beit Lab Inc. Toshiyuki GOTOH, Nagoya Institute of Technology Itaru HATAUE, Kumamoto University Ryutaro HIMENO, Institute of Physical and Chemical Research Kiyosi HORIUTI, Tokyo Institute of Technology Osamu INOUE, Tohoku University Katsuya ISHII, Nagoya University Takashi ISHIKAMI, VINAS Co., Ltd. Reima IWATSU, Tokyo Denki University Nobuyasu ITO, The University of Tokyo

228

31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54.

Participants List

Tetsuya KAWAMURA, Ochanomizu University Siro KITAMURA, Institute of Computational Fluid Dynamics Masaru KIYA, Kushiro National College of Technology Satoko KOMURASAKI, Nihon University Kunio KUWAHARA, The Institute of Space and Astronautical Science Takuya MATSUDA, Kobe University Hideo MIYACHI, KGT Inc. Mikio NAGASAWA, KGT Inc. Ken NAITOH, Yamagata University Kazuhiro NAKAHASHI, Tohoku University Shigeru OBAYASHI, Tohoku University Satoru OGAWA, National Aerospace Laboratory of Japan Junichi OOIDA, Institute of Computational Fluid Dynamics Takuya SAKURAGI, YKK Co. Yuko SATO, Ochanomizu University Keisuke SAWADA, Tohoku University Susumu SHIRAYAMA, The University of Tokyo Hiroshi SUITO, Okayama University Kunihiko TAKEDA, Nagoya University Tetsuro TAMURA, Tokyo Institute of Technology Yoshiaki TAMURA, Toyo University Mitsuhiro WASHIDA, VINAS Co., Ltd. Tadashi WATANABE, NEC Solutions Co. Yoshio WATANABE, Ricoh Co.

Sponsors We wish to thank the followings for their contribution to the success of this conference: 1. Asian Office of Aerospace Research and Development, US Air Force Office of Scientific Research 2. Fluids Engineering Division, The Japan Society of Mechanical Engineers 3. RIKEN (The Institute of Physical and Chemical Research) 4. Fujitsu Limited 5. NEC Corporation 6. VINAS Co., Ltd.

The Sixth International Nobeyama Workshop on the New Century of Computational Fluid Dynamics

April 21-24, 2003 at Nobeyama, Japan Workshop Program

April 21 (Monday) Session 1 (14:30-15:30) Z. J. Wang (Michigan State Univ.) 21-1-1 Peter R. Eiseman (Program Development Company) Grid Topology Automation 21-1-2 Kazuhiro Nakahashi (Tohoku University) Unstructured Mesh Generation for High-Fidelity Geometry CFD Session 2 (15:50-16:50) K. N. Ghia (Univ. of Cincinnati) 21-2-1 Jay P. Boris (US Naval Research Laboratory) Computational Fluid Dynamics for Nanoscale Atmospheric Modeling 21-2-2 Ken Naitoh (Yamagata University) An Approach on DNA Computing Using Actual Living Organism April 22 (Tuesday) Session 1 (09:00-10:00) S. Obayashi (Tohoku Univ.) 22-1-1 Kunio Kuwahara (The Institute of Space and Astronautical Science) Development of Implicit Large Eddy Simulation Session 2 (10:20-11:50) E. S. Oran (US Naval Research Lab.) 22-2-1 Jaw-Yen Yang (National Taiwan University) An Efficient Spectral Collocation Method with Domain Decomposition for Optical Waveguides Analysis 22-2-2 Katsuya Ishii (Nagoya University) Flow Simulation Using Combined Compact Difference Scheme with Spectral-Like Resolution

230

The Sixth International Nobeyama Workshop

22-2-3 Z. J. Wang (Michigan State University) A New High-Order Finite Volume Method for the Euler Equations on Triangular Grids Session 3 (14:00-15:30) U. Ghia (Univ. of Cincinnati) 22-3-1 Kirti N. Ghia (University of Cincinnati) Large-Eddy Simulation of Separated Flows inside Turbomachinery Cascades 22-3-2 Ho Sang Kwak (Kumoh National Inst. of Tech.) and Jae Min Hyun (Korea Advanced Inst. of Science and Tech.) Numerical Simulations of Time-Dependent Buoyancy-Driven Convection in an Enclosure 22-3-3 Kozo Fujii (The Institute of Space and Astronautical Science) Some Remarks on the Reliability of CFD Simulations in Aerospace (How can we trust CFD simulation result?) Session 4 (15:50-16:50) J. M. Hyun (KAIST) 22-4-1 Shigeru Obayashi (Tohoku University) Evolutionary Multi-Objective Optimization and Visualization 22-4-2 Urmila Ghia (University of Cincinnati) On Fluid-Structure Interactions April 23 (Wednesday) Session 1 (09:00-10:00) J. Y. Yang (National Taiwan Univ.) 23-1-1 Takuya Matsuda (Kobe University) Source-Sink Flow on a Rotating Stellar Surface 23-1-2 Elaine S. Oran (US Naval Research Laboratory) Numerical Simulation of the Deflagration State of Type Ia Supernovae Session 2 (10:20-11:50) N. J. Zabusky (Rutgers Univ.) 23-2-1 Ryutaro Himeno (Institute of Physical and Chemical Research) Computation of Flows around a Baseball

The Sixth International Nobeyama Workshop

23-2-2 Matthias Meinke (RWTH Aachen) Some Aachen Contributions to Computations of Vortical Flows 23-2-3 Reima Iwatsu (Tokyo Denki University) Vortex Breakdown Flows in Cylindrical Geometry Session 3 (14:00-15:30) J. C. Hill (Iowa State Univ.) 23-3-1 Osamu Inoue (Tohoku University) Direct Numerical Simulation of Acoustic Field 23-3-2 Kiyoshi Horiuchi (Tokyo Institute of Technology) A Process for Formation of Turbulent Structures along the Vortex Sheet 23-3-3 Tetsuya Kawamura (Ochanomizu University) Numerical Simulation of Formation and Movement of Various Sand Dunes Session 4 (15:50-17:20) K. Fujii (ISAS) 23-4P Panel discussion: ”CFD in Future: Structured overset, Unstructured, or Cartesian?” Kunio Kuwahara (ISAS), Kazuhiro Nakahashi (Tohoku Univ.), Ryutaro Himeno (RIKEN), and Z. J. Wang (Michigan State Univ.) April 24 (Thursday) Session 1 (09:00-10:00) P. R. Eiseman (PDC) 24-1-1 Norman J. Zabusky (Rutgers University) Science and Art of Fluid Motion 24-1-2 Hideo Miyachi (KGT Inc.) Parallel Visualization Systems Session 2 (10:20-11:50) J. P. Boris (US Naval Research Lab.) 24-2-1 James C. Hill (Iowa State University) Computational Challenges in Chemical Reaction Engineering: Turbulent Mixing with Chemical Reaction

231

232

The Sixth International Nobeyama Workshop

24-2-2 Nobuyasu Ito (The University of Tokyo) Statistical Physics Approach for Multiphase Flow Simulation 24-2-3 Keisuke Sawada (Tohoku University) Fully Conservative Algorithm for Capturing Contact Surface

Notes on Numerical and Fluid Mechanics and Multidisciplinary Design Available Volumes

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