C.A.J Fletcher Computational Techques for Fluid Dvnamics Fundamental and General Techniques Second Edition With 138 Figures Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Springer Series in Computational Physics Editors: R Glowinski M Holt P.Hut H B Keller J Killeen S A Orszag V V Rusanov A Computational Method in Plasma Physics E Bauer, Betancourt, P Garabedian Implementation of Finite Element Methods for Navier-Stokes Equations F Thomasset Finite-DifferenceTechniques for Vectorized Fluid Dynamics Calculations Edited by D Book Unsteady Viscous Flows D I? Telionis Computational Methods for Fluid Flow R Peyret, T D Taylor Computational Methods in Bifurcation Theory and Dissipative Structures M Kubicek, M Marek Optimal Shape Design for Elliptic Systems Pironneau The Method of Differential Approximation Yu I Shokin Computational Galerkin Methods C A J Fletcher Numerical Methods for Nonlinear Variational Problems R Glowinski Numerical Methods in Fluid Dynamics Second Edition M Holt Computer Studies of Phase Transitions and Critical Phenomena 0.G Mouritsen F i i t e Element Methods in Linear Ideal Magnetohydrodynamics R Gruber, J Rappaz Numerical Simulation of Plasmas Y N Dnestrovskii, D P.Kostomarov Computational Methods for Kinetic Models of Magnetically Confied Plasmas J Killeen, G D Kerbel, M C McCoy, A A Mirin Spectral Methods in Fluid Dynamics Second Edition C Canuto, M Y Hussaini, A Quarteroni, T A Zang ComputationalTechniques for Fluid Dynamics Second Edition Fundamental and General Techniques C A J Fletcher ComputationalTechniques for Fluid Dynamics Second Edition Specific Techniques for Different Flow Categories C A J Fletcher Methods for the Localization of Singularities in Numerical Solutions of Gas Dynamics Problems E V Vorozhtsov, N N Yanenko Classical Orthogonal Polynomials of a Discrete Variable A E Nikiforov, S K Suslov, 'I!B Uvarov Flux Coordinates and Magnetic Field Structure: A Guide to a Fundamental Tool of Plasma Theory W D D'haeseleer, W N G Hitchon, J.D Callen, J.L Shohet Dr Clive A J Fletcher Department of Mechanical Engineering, The University of Sydney New South Wales 2006 Australia Editors R Glowinski Institut de Recherche d'Informatique et d'Automatique (INRIA) Domaine de Voluceau Rocquencourt, B P 105 F-78150 Le Chesnay, France H B Keller Applied Mathematics 101-50 Firestone Laboratory California Institute of Technology Pasadena, CA 91125, USA J Killeen Lawrence Livermore Laboratory P Box 808 Livermore, CA 94551, USA M Holt S A Orszag College of Engineering and Mechanical Engineering University of California Berkeley, CA 94720, USA Program in Applied and Computational Mathematics Princeton University, 218 Fine Hall Princeton, NJ 08544-1000, USA V V Rusanov P Hut The Institute for Advanced Study School of Natural Sciences Princeton, NJ 08540, USA Keldysh Institute of Applied Mathematics Miusskaya pl SU-125047 Moscow, USSR ISBN 3-540-53058-4 Auflage Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-53058-4 2nd edition Springer-Verlag NewYork Berlin Heidelberg ISBN 3-540-18151-2 Auflage Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-18151-2 1st edition Springer-Verlag NewYork Berlin Heidelberg Library of Congress Cataloging-in-Publication Data Fletcher, C A J Computational techniques for fluid dynamics I C: A J Fletcher.- 2nd ed p cm.-(Springer series in computational physics) Includes bibliographical references and index Contents: Fundamental and general techniques ISBN 3-540-53058-4 (Springer-Verlag Berlin, Heidelberg, New York).-ISBN 0-387-53058-4 (Springer-Verlag New York, Berlin, Heidelberg) Fluid dynamics-Mathematics Fluid dynamics-Data processing Numerical analysis I Title 11 Series Q C 151.F58 1991 532'.05'0151-dc20 90-22257 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights o f translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its current version, and a copyright fee must always be paid Violations fall under the prosecution act of the German Copyright Law Springer-Verlag Berlin Heidelberg 1988,1991 Printed in Germany The use o f registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: Macmillan India Ltd., India 5513140-543210 -Printed on acid-free paper Springer Series in Computational Physics Editors: R Glowinski M Holt P Hut H B Keller J Killeen S A Orszag V V Rusanov Preface to the Second Edition The purpose and organisation of this book are described in the preface to the first edition (1988).In preparing this edition minor changes have been made, particularly to Chap I to keep it reasonably current However, the rest of the book has required only minor modification to clarify the presentation and to modify or replace individual problems to make them more effective The answers to the problems are available in Solutions Manual for Computational Techniquesfor Fluid Dynamics by C A J Fletcher and K Srinivas, published by Springer-Verlag, Heidelberg, 1991.The computer programs have also been reviewed and tidied up These are available on an IBM-compatible floppy disc direct from the author I would like to take this opportunity to thank the many readers for their usually generous comments about the first edition and particularly those readers who went to the trouble of d r a w i ~ specific errors to my attention In this revised edition considerable effort has been made to remove a number of minor errors that had found their way into the original I express the hope that no errors remain but welcome communication that will help me improve future editions In preparing this revised edition I have received considerable help from Dr K Srinivas, Nam-Hyo Cho, Zili Zhu and Susan Gonzales at the University of sydney and from Professor W Beiglb6ck and his colleagues at Springer-Verlag I am very grateful to all of them Sydney, November 1990 C.A J Fletcher Preface to the First Edition The purpose of this two-volume textbook is to provide students of engineering, science and applied mathematics with the specific techniques, and the framework to develop skill in using them, that have proven effective in the various branches of computational fluid dynamics (CFD) Volume describes both fundamental and general techniques that are relevant to all branches of fluid flow Volume provides specific techniques, applicable to the different categories of engineering flow behaviour, many of which are also appropriate to convective heat transfer An underlying theme of the text is that the competing formulations which are suitable for computational fluid dynamics, e.g the finite difference, finite element, finite volume and spectral methods, are closely related and can be interpreted as part of a unified structure Classroom experience indicates that this approach assists, considerably, the student in acquiring a deeper understanding of the strengths and weaknesses of the alternative computational methods Through the provision of 24 computer programs and associated examples and problems, the present text is also suitable for established research workers and practitioners who wish to acquire computational skills without the benefit of formal instruction The text includes the most up-to-date techniques and is supported by more than 300 figures and 500 references For the conventional student the contents of Vol are suitable for introductory CFD courses at the final-year undergraduate or beginning graduate level The contents of Vol are applicable to specialised graduate courses in the engineering CFD area For the established research worker and practitioner it is recommended that Vol is read and the problems systematically solved before the individual's CFD project is started, if possible The contents of Vol are of greater value after the individual has gained some CFD experience with his own project It is assumed that the reader is familiar with basic computational processes such as the solution of systems of linear algebraic equations, non-linear equations and ordinary differential equations Such material is provided by Dahlquist, Bjorck and Anderson in Numerical Methods; by Forsythe, Malcolm and Moler in Computer Methods for Mathematical Computation; and by Carnaghan, Luther and Wilkes in Applied Numerical Analysis It is also assumed that the reader has some knowledge of fluid dynamics Such knowledge can be obtained from Fluid Mechanics by Streeter and Wylie; from An Indroduction of Fluid Dynamics by Batchelor; or from Incompressible Flow by Panton, amongst others Computer programs are provided in the present text for guidance and to make it easier for the reader to write his own programs, either by using equivalent constructions, or by modifying the programs provided In the sense that the CFD VIII Preface to the First Edition practitioner is as likely to inherit an existing code as to write his own from scratch some practice in modifying existing but simple, programs is desirable An IBMcompatible floppy disk containing the computer programs may be obtained from the author The contents of Vol are arranged in the following way Chapter I contains an introduction to computational fluid dynamics designed to give the reader an appreciation of why CFD is so important the sort of problems it is capable of solving and an overview of how CFD is implemented The equations governing fluid flow are usually expressed as partial differential equations Chapter describes the different classes of partial differential equations and appropriate boundary conditions and briefly reviews traditional methods of solution Obtaining computational solutions consists of two stages: the reduction of the partial differential equations to algebraic equations and the solution of the algebraic equations The first stage, called discretisation is examined in Chap with special emphasis on the accuracy Chapter provides sufficient theoretical background to ensure that computational solutions can be related properly to the usually unknown "exact" solution Weighted residual methods are introduced in Chap.5 as a vehicle for investigating and comparing the finite element finite volume and spectral methods as alternative means of discretisation Specific techniques to solve the algebraic equations resulting from discretisation are described in Chap Chapters - provide essential background information The one-dimensional diffusion equation considered in Chap provides the simplest model for highly dissipative fluid flows This equation is used to contrast explicit and implicit methods and to discuss the computational representation of derivative boundary conditions If two or more spatial dimensions are present splitting techniques are usually required to obtain computational solutions efficiently Splitting techniques are described in Chap.8 Convective (or advective) aspects of fluid flow and their effective computational prediction are examined in Chap.9 The convective terms are usually nonlinear The additional difficulties that this introduces are considered in Chap 10 The general techniques developed in Chaps - 10 are utilised in constructing specific techniques for the different categories of flow behaviour as is demonstrated in Chaps 14- 18 of Vol In preparing this textbook I have been assisted by many people In particular I would like to thank Dr K Srinivas Nam-Hyo Cho and Zili Zhu for having read the text and made many helpful suggestions I am grateful to June Jeffery for producing illustrations of a very high standard Special thanks are due to Susan Gonzales Lyn Kennedy Marichu Agudo and Shane Gorton for typing the manuscript and revisions with commendable accuracy speed and equilibrium while coping with both an arbitrary author and recalcitrant word processors It is a pleasure to acknowledge the thoughtful assistance and professional competence provided by Professor W Beiglbock Ms Christine Pendl Mr R Michels and colleagues at Springer-Verlag in the production of this textbook Finally I express deep gratitude to my wife, Mary who has been unfailingly supportive while accepting the role of book-widow with her customary good grace Sydney October 1987 C A J Fletcher Contents Computational Fluid Dynamics: An Introduction 1.1 Advantages of Computational Fluid Dynamics 1.2 m i c a l Practical Problems 1.2.1 complex Geometry Simple Physics 1.2.2 Simpler Geometry More Complex Physics 1.2.3 Simple Geometry Complex Physics 1.3 Equation Structure 1.4 Overview of Computational Fluid Dynamics 1.5 Further Reading Partial Differential Equations 2.1 Background 2.1.1 Nature of a Well-Posed Problem 2.1.2 Boundary and Initial Conditions 2.1.3 Classification by Characteristics 2.1.4 Systems of Equations 2.1.5 Classification by Fourier Analysis 2.2 Hyperbolic Partial Differential Equations 2.2.1 Interpretation by Characteristics 2.2.2 Interpretation on a Physical Basis 2.2.3 Appropriate Boundary (and Initial) Conditions 2.3 Parabolic Partial Differential Equations 2.3.1 Interpretation by Characteristics 2.3.2 Interpretation on a Physical Basis 2.3.3 Appropriate Boundary (and Initial) Conditions 2.4 Elliptic Partial Differential Equations 2.4.1 Interpretation by Characteristics 2.4.2 Interpretation on a Physical Basis 2.4.3 Appropriate Boundary Conditions 2.5 Traditional Solution Methods 2.5.1 The Method of Characteristics 2.5.2 Separation of Variables 2.5.3 Green's Function Method 2.6 Closure 2.7 Problems Contents X Contents Preliminary Computational Techniques 3.1 Discretisation 3.1.1 Converting Derivatives to Discrete Algebraic Expressions 3.1.2 Spatial Derivatives 3.1.3 Time Derivatives 3.2 Approximation to Derivatives 3.2.1 Thylor Series Expansion 3.2.2 General Technique 3.2.3 Three-point Asymmetric Formula for [aT/ax]j' 3.3 Accuracy of the Discretisation Process 3.3.1 Higher-Order vs Low-Order Formulae 3.4 Wave Representation 3.4.1 Significance of Grid Coarseness 3.4.2 Accuracy of Representing Waves 3.4.3 Accuracy of Higher-Order Formulae 3.5 Finite Difference Method 3.5.1 Conceptual Implementation 3.5.2 DIFF: Transient Heat Conduction (Diffusion) Problem 3.6 Closure 3.7 Problems Theoretical Background 4.1 Convergence 4.1.1 Lax Equivalence Theorem 4.1.2 Numerical Convergence 4.2 Consistency 4.2.1 FTCS Scheme 4.2.2 Fully Implicit Scheme 4.3 Stability 4.3.1 Matrix Method: FTCS Scheme 4.3.2 Matrix Method: General Two-Level Scheme 4.3.3 Matrix Method: Derivative Boundary Conditions 4.3.4 Von Neumann Method: FTCS Scheme 4.3.5 Von Neumann Method: General Two-Level Scheme 4.4 Solution Accuracy 4.4.1 Richardson Extrapolation 4.5 Computational Efficiency 4.5.1 Operation Count Estimates 4.6 Closure 4.7 Problems Weighted Residual Methods 5.1 General Formulation 5.1.1 Application to an Ordinary Differential Equation 5.2 Finite Volume Method 5.2.1 Equations with First Derivatives Only 5.2.2 Equations with Second Derivatives 5.2.3 FIVOL: Finite Volume Method Applied to Laplace's Equation 5.3 Finite Element Method and Interpolation 5.3.1 Linear Interpolation 5.3.2 Quadratic Interpolation 5.3.3 no-Dimensional Interpolation 5.4 Finite Element Method and the Sturm-Liouville Equation 5.4.1 Detailed Formulation 5.4.2 STURM: Computation of the Sturm-Liouville Equation 5.5 Further Applications of the Finite Element Method 5.5.1 Diffusion Equation 5.5.2 DUCT Viscous Flow in a Rectangular Duct 5.5.3 Distorted Computational Domains: Isoparametric Formulation 5.6 Spectral Method 5.6.1 Diffusion Equation 5.6.2 Neumann Boundary Conditions 5.6.3 Pspdospectral Method 5.7 Closure 5.8 Problems 98 99 101 105 105 Steady Problems 6.1 Nonlinear Steady Problems 6.1:1 Newton's Method 6.1.2 NEWTON: Flat-Plate Collector Temperature Analysis 6.1.3 NEWTBU: no-Dimensional Steady Burgers' Equations 6.1.4 Quasi-Newton Method 6.2 Direct Methods for Linear Systems 6.2.1 FACT/SOLVE: Solution of Dense Systems 6.2.2 aidiagonal Systems: Thomas Algorithm 6.2.3 BANFAC/BANSOL: Narrowly Banded Gauss Elimination 6.2.4 Generalised Thomas Algorithm 6.2.5 Block aidiagonal Systems 6.2.6 Direct Poisson Solvers 6.3 Iterative Methods 6.3.1 General Structure 6.3.2 Duct Flow by Iterative Methods 6.3.3 Strongly Implicit Procedure 6.3.4 Acceleration Techniques 6.3.5 Multigrid Methods 6.4 Pseudotransient Method 6.4.1 ?Lvo.Dimensional Steady Burgers' Equations 6.5 Strategies for Steady Problems 6.6 Closure 6.7 Problems XI 107 Ill 116 117 119 121 126 126 130 135 135 137 143 145 146 149 151 156 156 XI1 Contents Contents 9.2 Numerical Dissipation and Dispersion 9.2.1 Fourier Analysis 9.2.2 Modified Equation Approach 9.2.3 Further Discussion 9.3 Steady Convection-Diffusion Equation 9.3.1 Cell Reynolds Number Effects 9.3.2 Higher-Order Upwind Scheme 9.4 One-Dimensional Transport Equation 9.4.1 Explicit Schemes 9.4.2 Implicit Schemes 9.4.3 TRAN: Convection of a Temperature Front 9.5 Wo-Dimensional Transport Equation 9.5.1 Split Formulations 9.5.2 THERM: Thermal Entry Problem 9.5.3 Cross-Stream Diffusion 9.6 Closure 9.7 Problems 286 288 290 291 293 294 296 299 299 304 305 316 317 318 326 328 329 10 Nonlinear Convection-Dominated Problems 10.1 One-Dimensional Burgers' Equation 10.1.1 Physical Behaviour 10.1.2 Explicit Schemes 10.1.3 Implicit Schemes 10.1.4 BURG: Numerical Comparison 10.1.5 Nonuniform Grid 10.2 Systems of Equations 10.3 Group Finite Element Method 10.3.1 One-Dimensional Group Formulation 10.3.2 Multidimensional Group Formulation 10.4 Tbo-Dimensional Burgers' Equation 10.4.1 Exact Solution 10.4.2 Split Schemes 10.4.3 TWBURG: Numerical Solution 10.5 Closure 10.6 Problems 331 332 332 334 337 339 348 353 355 356 357 360 361 362 364 372 373 One-Dimensional Diffusion Equation 7.1 Explicit Methods 7.1.1 FTCS Scheme 7.1.2 Richardson and DuFort-Frankel Schemes 7.1.3 Three-Level Scheme , 7.1.4 DIFEX: Numerical Results for Explicit Schemes 7.2 Implicit Methods 7.2.1 Fully Implicit Scheme 7.2.2 Crank-Nicolson Scheme 7.2.3 Generalised Three-Level Scheme 7.2.4 Higher-Order Schemes 7.2.5 DIFIM: Numerical Results for Implicit Schemes 7.3 Boundary and Initial Conditions 7.3.1 Neumann Boundary Conditions 7.3.2 Accuracy of Neumann Boundary Condition 7.4 Method of Lines 7.5 Closure 7.6 Problems Multidimensional Diffusion Equation 8.1 Two-Dimensional Diffusion Equation 8.1.1 Explicit Methods 8.1.2 Implicit Method 8.2 Multidimensional Splitting Methods 8.2.1 AD1 Method 8.2.2 Generalised Two-Level Scheme 8.2.3 Generalised Three-Level Scheme 8.3 Splitting Schemes and the Finite Element Method 8.3.1 Finite Element Splitting Constructions Implementation 7.3.3 Initial Conditions 8.3.2 TWDIF: Generalised Finite Difference/ Finite Element Implementation 8.4 Neumann Boundary Conditions 8.4.1 Finite Difference Implementation 8.4.2 Finite Element Implementation 8.5 Method of Fractional Steps 8.6 Closure 8.7 Problems Linear Convection-Dominated Problems 9.1 One-Dimensional Linear Convection Equation 9.1.1 FTCS Scheme 9.1.2 Upwind Differencing and the CFL Condition 9.1.3 Leapfrog and Lax-Wendroff Schemes 9.1.4 Crank-Nicolson Schemes 9.1.5 Linear Convection of a Truncated Sine Wave XI11 Appendix A.l Empirical Determination of the Execution Time of Basic Operations 375 A.2 Mass and Difference Operators 376 Subject Index References 381 389 Contents of Computational Techniques for Fluid Dynamics Specific Techniques for Different Flow Categories 397 Computational Fluid Dynamics: An Introduction This chapter provides an overview of computational fluid dynamics (CFD) with emphasis on its cost-effectiveness in design Some representative applications are described to indicate what CFD is capable of The typical structure of the equations governing fluid dynamics is highlighted and the way in which these equations are converted into computer-executable algorithms is illustrated Finally attention is drawn to some of the important sources of further information 1.1 Advantages of Computational Fluid Dynamics The establishment of the science of fluid dynamics and the practical application of that science has been under way since the time of Newton The theoretical development of fluid dynamics focuses on the construction and solution of the governing equations for the different categories of fluid dynamics and the study of various approximations to those equations The governing equations for Newtonian fluid dynamics, the unsteady Navier~ t o c e sequations, have been known for 150 years or more However, the devec %pment of reduced forms of these equations (Chap 16) is still an active area of research as is the turbulent closure problem for the Reynolds-averaged NavierStokes equations (Sect 11.5.2) For non-Newtonian fluid dynamics, chemically reacting flows and two-phase flows the theoretical development is at a less advanced stage Experimental fluid dynamics has played an important role in validating and delineating the limits of the various approximations to the governing equations The wind tunnel, as a piece of experimental equipment, provides an effective means of simulating real flows Traditionally this has provided a cost-effective alternative to full-scale measurement In the design of equipment that depends critically on the flow behaviour, e.g aircraft design, full-scale measurement as part of the design process is economically unavailable The steady improvement in the speed of computers and the memory size since the 1950s has ledto the emergence of computational fluid dynamics (CFD) This branch of fluid dvnamics complements experimental and theoretical fluid dynamics m v i d i n g an alternative cost-effective means of simulating real flows As such it offers the means of testing theoretical advances for conditions unavailable exper- -_ 1.1 Advantages of Computational Fluid Dynamics Computational Fluid Dynamics: An Introduction imentally For example wind tunnel experiments are limited to a certain r a n g of Reynolds numbers, typically one or two orders of magnitude less than full scale Computational fluid dynamics also provides the convenience of being able to switch off specific terms in the governing equations This permits the testing of theoretical models and, inverting the connection, suggests new paths for theoretical exploration The development of more efficient computers has generated the interest in CFD and, in turn, this has produced a dramatic improvement in the efficiency of the computational techniques Consequently CFD is now the preferred means of testing alternative designs in many branches of the aircraft, flow machinery and, to a lesser extent, automobile industries Following Chapman et al (1975), Chapman (1979,1981), Green (1982), Rubbert (1986) and Jameson (1989) CFD provides five major advantages compared with experimental fluid dynamics: (i) Lead time in design and development is significantly reduced (ii) CFD can simulate flow conditions not reproducible in experimental model tests (iii) CFD provides more detailed and comprehensive information (iv) CFD is increasingly more cost-effective than wind-tunnel testing (v) CFD produces a lower energy consumption Traditionally, large lead times have been caused by the necessary sequence of design, model construction, wind-tunnel testing and redesign Model construction is often the slowest component Using a well-developed CFD code allows alternative designs (different geometric configurations) to be run over a range of parameter values, e.g Reynolds number, Mach number, flow orientation Each case may require 15 runs on a supercomputer, e.g CRAY Y-MP The design optimisation process is essentially limited by the ability of the designer to absorb and assess the computational results In practice CFD is very effective in the early elimination of competing design configurations Final design choices are still confirmed by wind-tunnel testing Rubbert (1986) draws attention to the speed with which CFD can be used to redesign minor components, if the CFD packages have been thoroughly validated, Rubbert cites the example of the redesign of the external contour of the Boeing 757 cab to accommodate the same cockpit components as the Boeing 767 to minimise pilot conversion time Rubbert indicates that CFD provided the external shape which was incorporated into the production schedule before any wind-tunnel verification was undertaken Wind-tunnel testing is typically limited in the Reynolds number it can achieve, usually short of full scale Very high temperatures associated with coupled heat transfer fluid flow problems are beyond the scope of many experimental facilities This is particularly true of combustion problems where the changing chemical composition adds another level of complexity Some categories of unsteady flow motion cannot be properly modelled experimentally, particularly where geometric unsteadiness occurs as in certain categories of biological fluid dynamics Many geophysical fluid dynamic problems are too big or too remote in space or time to simulate experimentally Thus oil reservoir flows are generally inaccessible to detailed experimental measurement Problems of astrophysical fluid dynamics are too remote spatially and weather patterns must be predicted before they occur All of these categories of fluid motion are amenable to the computational approach Experimental facilities, such as wind tunnels, are very effective for obtaining global information, such as the complete lift and drag on a body and the surface pressure distributions at key locations However, to obtain detailed velocity and pressure distributions throughout the region surrounding a body would be prohibitively expensive and very time consuming CFD provides this detailed information at no additional cost and consequently permits a more precise understanding of the flow processes to be obtained Perhaps the most important reason for the growth of CFD is that for much mainstream flow simulation, CFD is significantly cheaper than wind-tunnel testing and will become even more so in the future Improvements in computer hardware performance have occurred hand in hand with a decreasing hardware cost Consequently for a given numerical algorithm and flow problem the relative cost of a computational simulation has decreased significantly historically (Fig 1.1) Paralleling the improvement in computer hardware has been the improvement in the efficiency of computational algorithms for a given problem Current improvements in hardware cost and computational algorithm efficiency show no obvious sign of reaching a limit Consequently these two factors combine to make CFD increasingly cost-effective In contrast the cost of performing experiments continues to increase The improvement in computer hardware and numerical algorithms has also brought about a reduction in energy consumption to obtain computational flow simulations Conversely, the need to simulate more extreme physical conditions, higher Reynolds number, higher Mach number, higher temperature, has brought about an increase in energy consumption associated with experimental testing The chronological development of computers over the last thirty years has been towards faster machines with larger memories A modern supercomputer such as I l l l l l l l l l ~ l l l ~ l ~ ~ l 1 1 1 1 I l 1955 1960 1965 1970 1975 1980 YEAR NEW COMPUTER AVAILABLE 1985 Fig 1.1 Relative cost of computation for a given ~algorithm 1 and flow (after Chapman, 1979; reprinted with permission of AIAA) 372 10.6 Problems 10 Nonlinear Convection-Dominated Problems Table 10.13 Variation of v with x for y/ymaX= 0.4, N X = 6, NY = x - 1.00 -0.60 -0.20 - 0.20 0.60 1.00 Error Exact 0.3249 0.3249 0.1818 0.0000 0.0000 0.0000 - AF-FDM 0.3249 0.3251 0.1774 0.0003 0.0000 0.0000 0.00 18 AF4PU q*= 1.0, q,=O.O 0.3249 0.3246 0.1781 0.0003 0.0000 0.0015 AF-MO 6, = 6, = 0.26 0.3249 0.3250 0.1790 -0.0009 0.0000 0.00 12 -0.0002 0.0005 373 The nonlinear nature of the governing equation makes the application of the conventional Galerkin finite element method uneconomical, particularly in multidimensions and if higher-order interpolation is used This problem is overcome by casting the equations in conservation form and in introducing approximate solutions for the conserved groups The group formulation is often more accurate particularly where the conservation form of the governing equations is preferred on physical grounds 10.6 Problems 1-D Burgers' Equation (Sect 10.1) 10.5 Closure Computational techniques, introduced in conjunction with the diffusion equation in Chaps and 8, and the linear convection and transport equations in Chap have been extended to a family of nonlinear equations, Burgers' equations, in one and two dimensions Burgers' equations contain the same form of convective nonlinearity as in many of the fluid dynamic governing equations Burgers' equations possess readily computable exact solutions for many combinations of initial and boundary conditions For this reason they are appropriate model equations on which to test various computational techniques This feature has been exploited, for the one-dimensional Burgers' equation, in Sects 10.1.4 and 10.3.1 and, for the two-dimensional Burgers' equations, in Sects 10.3.2 and 10.4.3 To take advantage of the Thomas algorithm it is necessary to linearise the convective nonlinearity This is required for both implicit schemes in one dimension (Sect 10.1.3) and for split schemes in multidimensions (Sect 10.4.2) This linearisation extends to systems of equations (Sect 10.2) without difficulty The Fourier analysis which was directly applicable to llnear equations (Sects 9.2.1 and 9.4.3) is still made use of in the von Neumann stability analysis after freezing the nonlinearity he modified equation approach (Sect 9.2.2) is applicable to nonlinear equations However, products of higher-order derivatives appear in sufficient number and magnitude as to make the identification of dissipative and dispersive properties less precise It is usually helpful to analyse the equivalent linear equation as an intermediate step, since this equation often demonstrates qualitatively equivalent dissipative and dispersive behaviour The construction of more accurate schemes often requires a certain amount of empiricism imposed on the guidelines provided by the corresponding linear equations Most of the computational techniques developed for linear eauations extend to nonlinear equations However convective nonlinearities are ofien handled more ecectively in conserved variables, e.g F in (10.3) Conserved variables will be used, 14-18 where ap~ro-ha~s - 10.1 Determine the truncation error of the scheme, (10.20), with the nonlinear coefficients frozen, for the two cases: (i) q = 0, (ii) = Show that the optimum values of and q are given by (10.26) and (10.27), respectively 10.2 Modify program BURG to implement a Crank-Nicolson finite difference discretisation of (10.1) and compare the solution with that of the CrankNicolson finite difference discretisation of (10.3) for the conditions associated with Table 10.5 How does the comparison change when the integration is made to a larger time? 10.3 Modify program BURG to implement (10.26 and 27) and compare solutions for the intermediate cell Reynolds number conditions appropriate to Table 10.4 10.4 Program BURG can be used to obtain steady-state solutions of (10.37) with F=0.5(ii2-c) Implement these changes for the general three-level scheme, equivalent to (8.26), and obtain solutions with i) y = 0, fl = 1, ii) Y = 0, = 0.5, iii) y = 0.5, fl= 1.0 Compare the number of iterations to reach the steady state, for the conditions corresponding to Table 10.9, but with a uniform grid 10.5 For the convection diffusion equation (10.28) code the nonuniform discretisation (10.30-32) and confirm the results shown in Tables 10.6 and 10.7 10.6 Develop a four-point upwind discretisation on a nonuniform grid with one free parameter (equivalent to q) Assume that r, is constant but not equal to unity Apply this discretisation to the modified Burgers' equation (10.37) with the two-level fully implicit marching scheme (10.38) Compare the accuracy of the solutions with those shown in Tables 10.8 and 10.9 for various values of the free parameter Systems of Equations (Sect 10.2) 10.7 Modify program SHOCK (Vol 2, Fig 14.17) to implement the explicit fourpoint upwind scheme introduced in Sect 9.4.3 Obtain solutions corresponding to those shown in Fig 10.9 with sufficient artificial dissipation to suppress excessive oscillations It may be necessary to use a small value of At 374 10 Nonlinear Convection-Dominated Problems (small Courant number, C ) to obtain stable solutions Compare the sharpness of the shock profile with that shown in Fig 10.9 10.8 Extend the Crank-Nicolson finite difference scheme (10.44) to generate a Crank-Nicolson mass operator scheme with variable for the system of equations (10.40) Modify SHOCK (Fig 14.17) to execute this scheme and compare the quality of the solution with that shown in Fig 10.9, for chosen empirically Group Finite Element Method (Sect 10.3) 10.9 Implement the conventional finite element method with Crank-Nicolson time differencing in program BURG, for the one-dimensional Burgers' equation, and compare the solution accuracy with that produced by the group finite element method and the Crank-Nicolson mass operator scheme for the conditions corresponding to Table 10.5 10.10 Estimate the connectivity and residual operation count for the conventional and group finite element formulations applied to the two-dimensional Burgers' equations for quadratic interpolation 10.11 Modify program TWBURG to implement the conventional finite element method applied to (10.57,58) and compare the accuracy and economy with that of the group finite element method applied to (10.52) for the conditions corresponding to Tables 10.12 and 10.13 How does the comparison change on a finer grid? Appendix A.l Empirical Determination of the Execution Time of Basic Operations A computer program, COUNT, is provided in Fig A.l which allows the execution time of various types of operation to be determined approximately The type of operation corresponding to different values of ICT are given in Table A.1 For each operation a nested pair of D O loops is executed The inner loop repeats the given operation 10000 times The outer loop repeats the inner loop N times For N sufficiently large the time from the input of ICT and N (line 15) to the output of ICT and N (line 48) can be determined manually Alternatively a timing subroutine, if available, can be used to measure the CPU time between lines 15 and 48 Table A.1 Owrations considered in Program COUNT Execution time 2-D Burgers' Equation (Sect 10.4) 10.12 Write a program, based on subroutine EXBUR, to generate exact solutions of the two-dimensional Burgers' equations that produce more severe internal gradients 10.13 Obtain solutions using program TWBURG on x 6, 11 x 11 and 21 x 21 grids for the conditions corresponding to Tables 10.12 and 10.13, with OSS,, 6,60.30, Odq,, qy51.0 Decide on optimal choices for S,, 6, and q,, q, for each grid 10.14 Execute program TWBURG with the following choice of parameters: a,=a,=0.01101, a,=a,=O, a,=1.0,1=5, x,=1.0 and v=0.1 This choice produces a moderate boundary gradient adjacent to x = 1.0 What are optimal choices for S,, 6, and q,, q, on an 11 x 11 grid for this case? ICT 10 11 12 Operation (secs) for N = 1000 Empty D O loops Replacement Addition and replacement Subtraction and replacement Multiplication and replacement Divisien and replacement Integer IF statement Power and replacement Square root and replacement Sin and replacement Exponential and replacement Array addition and replacement 12.3 15.8 19.1 19.1 19.0 34.8 15.3 81.9 111 112 123 25.1 The execution times shown in Table A.l are appropriate to the Supermicrocomputer (SUN Sparc Stl) in Table 4.4 The execution time for ICT = is subtracted from cases ICT = and For all other cases the execution time for ICT = is subtracted For the fixed point operations (FX) shown in Table 4.4 the integer statement (line 6) is activated and the library function evaluations (lines 38, 40 and 42) are replaced with labelled CONTINUE statements to satisfy the G O T statement (line 21) It is recommended that program COUNT be compiled in an unoptimised mode to obtain consistent results 376 10 11 12 13 14 15 16 Appendix C C C C C A.2 COWIT FACILITATES THE DETERUIUATIOU OF THE EXECUTIOU TINE FOR SPECIFIC OPERATIOUS DEPEUDIUC 01 THE VALUE OF ICT U IS THE UUUBER OF OUTER LOOPS DIUEUSIOU E(lOOOO).F(lOOOO).C(10000) INTEGER A.B.C.E.F,G = 3.0 C = 2.0 IA = DO 21 I = 1.10000 F(1) = 2.0 21G(J)=1.0 YRITE(*.22) 22 FORUAT(SX.'VALUE FOR ICT AID U',/) READ(*,*)ICT,U An approximate solution is introduced for Mass and Difference Operators 377 T as where 4,(x, y) are two-dimensional bilinear Lagrange interpolating functions, equivalent to (5.60) Equation (A.2) is substituted into (A.l), the Galerkin weighted residual integral (5.5 and 10) is evaluated and the result can be written M , Q M ~ T + U M ~ Q L , T + V MQ , L , T - ~ , M ~ Q L ~ ~ T - ~ ~ M , (A.3) QL~~T= C 17 DO 14 I = 1.U 18 DO 13 = 1,10000 19 C 20 C~~(~.2.~.4,5,8,7,8,9,10,11,12),1~~ 21 COUTIUUE 22 COT0 13 23 2A.C 24 GOTO 13 25 A = B + C 28 corn 13 27 A = B - C 28 corn 13 29 A = B*C 30 corn 13 31 A = 8/C 32 corn 13 33 IF(1 CT 0)CDTO 13 corn 13 34 35 A = B**IA 36 COT0 13 37 A = SQRT(B) 38 COT0 13 39 10 A = SIU(B) 40 GOTO 13 41 11 A = EXP(B) 42 COT0 13 43 12 E(J) = F(J) t C(1) 44 13 COUTIUUE 45 14 COFTIUUE 46 C 47 YRITE(*,l5)ICT,U 48 15 FORUAT(5X.' ICT=',17,5X.' U='.I7) 49 STOP 50 END where Q is the tensor product and T = dT/dt The terms M, and My are directional mass operators and are defined by where A = 2/(1+ r,) , B = 2/(1+ r,) The directional difference operators are given by where r, and ry are grid growth ratios rx = Fig A.1 Listing of Program COUNT A.2 Mass and Difference Operators It is indicated in Sects 5.5.1 and 5.5.2 that the finite element method can be interpreted as providing a term-by-term discretisation if directional mass and differenceoperators are identified explicitly The origin of these operators within the Galerkin finite element formulation, Sects 5.1 and 5.3, will be indicated here in relation to the two-dimensional transport equation (9.81) X j + ~ - ~ i ,r y = X j - xj- Yk+l-Yk , Yk-Yk-1 so that on a uniform grid r, = r, = It can be seen that there is a term-by-term correspondence between the original equation (A.l) and the discretised equation (A.3) The source of this correspondence can be found by considering a single term a?;jax in (A.l) Application of the finite element method to (A.l) includes the following contribution from aF/ax: where x denotes the contributions from the elements adjacent to node rn The e Lagrange interpolating functions 4, can be written as the product of onedimensional interpolating functions 4, = 4P'4kY'3 (A.8) A.2 Mass and Difference Operators Appendix 378 where 42), are given by (5.45 and 46) Consequently the contributions to the integral in (A.7) can be split into directional components integral nature of the Galerkin method Therefore, in a three-dimensional problem a?;/axwould be discretised as ~T/~X+M,@M,@L,T As a result it is convenient to introduce the operators (A.10) 379 (A 16) For brick elements with trilinear interpolation this implies that My @ M, @ L, would be at most a 27-point operator On a uniform grid there is a connection between the role of the mass operators and Fad&differencing A fourth-order evaluation of aT/ax using Pad&differencing is obtained by solving the tridiagonal system and A d4y) ~ F I L,, = - J Axj-1/2 e (A.17) ~ ~ X (A 11) , where it is assumed that the mth Galerkin node coincides with the global grid point (j,k), so that (A 18) AX^ ^/^ = x j - x j - ~ and A Y ~ - I , z=yk-yk-l (A.12) Thus the term aT/ax in (A.l) is discretised as My L,T On the global grid, this has the form It may be noted that here the mass operator is applied in the same direction as the derivative is taken In contrast in the Galerkin finite element method, d T / a x + M , @ L,T , (A 13) Where second derivatives occur in the governing equation (A.l), application of integration by parts leads to the following definition of the operator L,,, at an interior point: (A.14) Comparable definitions to those given in (A.lO, 11 and 14) are available for M,, L, and L,,, respectively In practice the integrals in (A.10), etc., are evaluated by introducing element-based coordinates (t, q ) as in (5.58-60) For linear elements, 4): = 0.5(1+ 55,) with t, = +1 and - 15