Application of Two-Dimensional Cellular Automaton Lattice-Gas Models to the Simulation of Hydrodynamics Brian J N Wylie Sthg 0f(àflb Doctor of Philosophy 1990 Abstract YDRODYNAMIC equations are notoriously difficult to solve, both analytically II and computationally, therefore less complicated methods, drawing on the power ofcellular automata and lattice-gases are introduced. Their combined ability to capture the fundamental properties of fluid dynamics in an inherently simple man- ner is discussed. The FHP7 cellular automaton lattice-gas model of Frisch, Has- slacher and Pomeau, which will form the basis for the subsequent simulations, is described in detail, with a more general covering of the associated models. HE SCALABLE and flexible computational power of the transputer-based ECS T multicomputer, and how this may be applied to the lattice-gas simulations at hand is addressed. The distributed multiprocessor architecture provides unique challenges, such that the implementation might achieve its potential. It is found that a straightforward one-dimensional geometric decomposition of the lattice, in conjunction with the loosely-synchronous nature of the distributed update, provides a natural load-balancing, and highly scalable efficiency. ISUALISATION of the development of the hydrodynamic features captured by the V simulations, such that their content may be clearly extracted is also addressed. Many interesting transient and dynamic features, often occuring on time-scales which make their analysis by other methods difficult, are easily identified. Those occasionally found to be the result of artifacts, perhaps in the initialisation of the simulation are quickly identified, such that the simulations may be refined. LEMENTARY static systems and flows are designed, such that the ability of E the FHP7 lattice-gas to model incompressible hydrodynamics, and its multi- computer implementation, are verified against the theoretically and experimentally expected behaviour. Subsequently, more complex flow configurations involving ob- structions and jets, generally beyond the limits of current analytic techniques are constructed, and found to qualitatively match experimental visualisations. 0 LATTICE-GASES are currently known to accurately model compressible fluid Ndynamics completely, and the ultimate cause of this limitation still requires clarification. The behaviour of the FHP7 lattice-gas, in regimes where compressibil- ity effects are expected to be important, is investigated with the aim of identifying those aspects of its microdynamics which cause breakdown of its macrodynamics. Acknowledgements I would like to thank my supervisors, Richard Kenway and David McComb, for providing enthusiasm and direction when this was lacking, and assistance when it was needed. And also 'Uncle' David, for a timely rescue when I was lost, and accepting the need for Erdbeerzeit. Bruce Boghosian is also gratefully thanked for his helpful explanations and insight. Thanks are also due to the managers, and the service and operations staff, of the various computing facilities used in the course of the research and preparation of this thesis, for their assistance, generally beyond the call of duty. Also for their tolerance of my excesses, in the (ab)use of same. A final big round of thanks, go to all my friends and colleagues - the Goths, members of the Department of Physics and the computing community in Edinburgh and the dwellers of the 'Net all over the world who freely gave of their time and skills in pandering my flights of fancy and made the time pass enjoyably and, oh so, quickly. Last, but not least, the welcome tranquility of the hills and the sea, the still of the night and the freshness of the rain, and the illumination and magic of the moon and the stars, all contributed significantly to provide inspiration to see this work through to its conclusion. That and the 'Eyes' and the 'Hiker' keeping an eye on me. The Edinburgh Parallel Computing Centre is a multidisciplinary project supported by major grants from the Department of Trade and Industry, the Computer Board and the Science and Engineering Research Council, and also substantial support from the University of Edinburgh and Industrial Affiliates. Part of this work was supported by a research studentship from the Science and Engineering Research Council. The production of this thesis, and the sustainance of its author during its drafting, was also supported by my parents. Use of any trademark is not intended in any way to infringe on the rights of the trademark holder. Can it be there's some sort of error, Hard to stop the surmounting terror, Is it really the end, not some crazy dream ? [1M82] 11 Ut saIQruoJJ sues QT[ ShtOU ptr lift S'MON,, ow vqlv-7 a.tdym,v '3flOJ(fl LOLI Contents 0 Prologue i 1 Lattice-gas hydrodynamics 4 1.1 Origins ..................................5 1.1.1 The hydrodynamic equations .................5 1.1.2 Cellular automata ........................7 1.1.3 Elementary gases ........................9 1.2 Cellular automaton lattice-gas models .................9 1.2.1 HPP ...............................9 1.2.2 FHP6 ..............................12 1.2.3 F}1P7 ..............................14 1.2.4 FCHC ..............................17 1.2.5 Others ..............................19 1.2.6 The models investigated ....................21 1.2.6.1 21) models .......................21 1.2.6.2 Why not 3D ? 21 1.2.7 Model extensions ........................21 1.2.7.1 Multi-species .....................22 1.2.7.2 Miscellaneous .....................22 1.3 Model fundamentals ..........................23 1.3.1 Minimal basis ..........................23 1.3.1.1 Basic properties ....................23 1.3.1.2 Consequences .....................23 1.3.1.3 Conservables ......................24 1.3.2 Achieving macrodynamics ...................26 1.3.2.1 Cell averaging .....................26 1.3.2.2 Properties .......................27 1.4 Summary ................................30 2 Multicomputer implementation & visualisation 33 2.1 Introduction ...............................34 2.2 Distributed parallelism ........................... 34 2.2.1 Concepts of concurrency .....................34 2.2.1.1 Computer architectures ................34 2.2.1.2 MIMD architectures .................36 2.2.2 Multicomputer architecture ..................37 2.2.2.1 Why a multiprocessor computer ? 37 2.2.2.2 The transputer ....................38 2.2.2.3 The occam model ...................38 2.2.3 Problem decomposition .....................39 lv 2.2.3.1 Algebraic decomposition ...............39 2.2.3.2 Data decomposition .................39 2.2.3.3 Geometric decomposition ..............40 2.2.4 Problems with parallelism ...................40 2.2.4.1 Communication ....................40 2.2.4.2 Load imbalance ....................41 2.3 Implementation of model ........................42 2.3.1 Data structures .........................42 2.3.1.1 Sites ..........................42 2.3.1.2 Lattice ..........................42 2.3.2 Site update ...........................43 2.3.2.1 Generation ......................43 2.3.2.2 Propagation ......................44 2.3.2.3 Collision ........................44 2.3.2.4 Update cost ......................45 2.3.3 Distribution of work ......................47 2.3.3.1 Partition .......................47 2.3.3.2 Borders ........................48 2.3.3.3 Border swapping ...................49 2.3.4 Collection and analysis of data ................50 2.3.4.1 Cell averaging .....................50 2.3.4.2 Graphics update ...................50 2.3.4.3 Data analysis .....................51 2.3.5 Flow configuration .......................51 2.3.5.1 Barriers ........................51 2.3.5.2 Sources ...........................51 2.3.5.3 Configuration .....................52 2.3.6 Lattice design .........................53 2.3.7 Comments .............................54 2.3.7.1 Update speed .....................54 2.3.7.2 Load balance ......................55 2.4 Flow visualisation .............................56 2.4.1 Introduction ...........................56 2.4.1.1 Conventional techniques ...............56 2.4.1.2 The computational role ................57 2.4.1.3 The visualisation process ...............58 2.4.2 Visualisation techniques ....................58 2.4.2.1 Direction as icons ....................59 2.4.2.2 Attribute mapping of magnitude ..........59 2.4.2.3 Composite mapping of the flow ...........60 2.4.3 Image optimisation .......................60 2.4.3.1 Scaling ........................61 2.4.3.2 The palette ......................61 2.4.4 Image construction ... 63 2.4.4.1 Display composition .................63 2.4.4.2 Data collection ....................63 V 2.4.4.3 Rendering 63 2.4.5 Image analysis ..........................64 2.4.5.1 Image cycling .....................64 2.4.5.2 Colour cycling ....................64 2.4.5.3 Recall .........................64 2.4.6 Visualisation summary .....................65 2.5 Summary ................................65 3 Fundamental & phenomenological flow simulations 67 3.1 Introduction ................................68 3.2 Fluids at rest ..............................68 3.2.1 A contained isotropic fluid ..................68 3.2.1.1 Determination of the optimal averaging cell size . 68 3.2.2 A pressure wave in a contained fluid .............70 3.2.2.1 Measurement of the isotropy of propagation . 72 3.3 Channel flows ..............................73 3.3.1 Introduction ...........................73 3.3.2 Laminar flow in