Computational Fluid Mechanics (eBook)

Selected Papers

Alexandre Joel Chorin (Autor)

eBook Download: PDF | EPUB

2014 | 1. Auflage
240 Seiten
Elsevier Science (Verlag)
978-1-4832-7155-2 (ISBN)

Infos

Computational Fluid Mechanics: Selected Papers compiles papers on computational fluid dynamics written between 1967 and 1982. This book emphasizes the numerical solution of the equations of fluid mechanics in circumstances where the viscosity is small. The vortex and projection methods, numerical solution of problems in kinetic theory, combustion theory, and gas dynamics are also discussed. This publication elaborates that turbulence in fluids is dominated by the mechanics of vorticity, and many of the methods are based on vortex representations of the flow. The convergence of vortex calculations in three space dimensions and motion of vortex filaments are likewise deliberated. This compilation is a good source for physicists and students researching on computational fluid mechanics.

Inhalt

Front Cover 1
Computational Fluid Mechanics: Selected Papers 4
Copyright Page 5
Table of Contents 6
Introduction 8
Chapter 1. A Numerical Method for Solving Incompressible Viscous Flow 
18 
ABSTRACT 18
INTRODUCTION 18
THE METHOD OF ARTIFICIAL COMPRESSIBILITY 20
A SIMPLE TEST PROBLEM 24
THERMAL CONVECTION IN A FLUID LAYER HEATED FROM BELOW. 
26 
THERMAL CONVECTION IN THREE SPACE DIMENSIONS 30
REFERENCES 32
Chapter 2. Numerical Solution 
34 
Abstract 34
Introduction 34
Principle of the Method 35
The Dufort-Frankel Scheme and Successive Point Over-Relaxation 39
Solution of a Simple Test Problem 44
Conclusion and Applications. 50
Chapter 3. On the Convergence of Discrete Approximations 
52 
Abstract 52
Introduction 52
Chapter 4. Numerical Solution of Boltzmann's Equation 66
1. Introduction 66
2. Method of Solution 67
3· Flow in One Space Dimension 71
4. Application to a Shock Problem 74
5. Numerical Results 76
6. Conclusions and Generalizations 80
Bibliography 80
Chapter 5. Numerical study of slightly viscous flow 82
1. Introduction 82
2. Principle of the method 83
3. Implementation of the method 85
4. The evaluation of the potential component of the flow 86
5. Heuristic considerations 87
6. Application: flow past a circular cylinder 88
7. Conclusion and further work 92
REFERENCES 92
Chapter 6. Discretization of a Vortex Sheet, 
94 
INTRODUCTION 94
APPROXIMATION OF A VORTEX SHEET BY A FINITE ARRAY OF POINT VORTICES 95
VORTEX SHEET INDUCED BY AN ELLIPTICALLY LOADED WING 96
CONCLUSION 99
REFERENCES 99
Chapter 7. Random Choice Solution of Hyperbolic Systems 102
OUTLINE OF GOAL AND METHOD 102
IMPLEMENTATION OF GLIMM'S METHOD 103
SOLUTION OF A RIEMANN PROBLEM 107
MULTIDIMENSIONAL PROBLEMS 110
EXAMPLES 113
CONCLUSIONS 117
REFERENCES 117
Chapter 8. Random Choice Methods with Applications to 
120 
INTRODUCTION 120
SIMPLE EXAMPLES AND PARTIAL ERROR ESTIMATES 121
BOUNDARY CONDITIONS 125
DETONATIONS AND DEFLAGRATIONS IN A ONE-DIMENSIONAL IDEAL GAS 126
SOLUTION OF A RIEMANN PROBLEM WITH CHEMISTRY 132
NUMERICAL RESULTS 136
CONCLUSIONS 138
REFERENCES 139
Chapter 9. Vortex Sheet Approximation of Boundary Layers 140
INTRODUCTION 140
PRINCIPLE OF THE METHOD 141
VORTICITY CREATION 143
FLOW PAST A SEMI-INFINITE FLAT PLATE 145
A HYBRID ALGORITHM INVOLVING THE RANDOM CHOICE METHOD 148
APPLICATION TO TWO DIMENSIONAL FLOW BEHIND A PISTON 150
CONCLUSION 153
REFERENCES 154
Chapter 10. Flame Advection and Propagation Algorithms 156
OUTLINE OF GOAL AND METHOD 156
A SIMPLE LINE ADVECTION ALGORITHM 157
IMPLEMENTATION OF THE HUYGHENS PRINCIPLE 161
THE EFFECT OF INTERMITTENCY ON THE VELOCITY OF WRINKLED FLAMES 163
ACKNOWLEDGMENT 165
REFERENCES 165
Chapter 11. VORTEX MODELS AND BOUNDARY LAYER INSTABILITY 168
Abstract 168
Introduction 168
The physical problem in two dimensions 169
The numerical methods in two dimensions 170
Application of the numerical method in two dimensions. 174
Numerical results in two dimensions. 176
The physical problem in three space dimensions 179
The numerical methods in three dimensions 180
Application of the numerical methods in three dimensions 183
Numerical results in three dimensions. 183
Conclusions 187
REFERENCES 187
Chapter 12. NUMERICAL METHODS FOR USE IN COMBUSTION MODELING 190
A Reaction-Diffusion Equation 190
The Boundary Layer 
192 
Vortex Method for the Navier-Stokes Equations 194
Conclusions 196
BIBLIOGRAPHY 197
Chapter 13. NUMERICAL MODELLING OF 
198 
INTRODUCTION 198
1. PROBLEM 202
2. PROCEDURE 203
3. VORTEX DYNAMICS 205
4. FLAME PROPAGATION 212
APPENDIX 1. VORTEX MOTION IN THE TRANSFORMED PLANE 218
APPENDIX 2. DERIVATION OF EQUATION 219
APPENDIX 3. CONSERVATION OF CIRCULATION IN A VARIABLE DENSITY FIELD 220
REFERENCES 221
Chapter 14. The Evolution of a Turbulent Vortex 222
1. Introduction 222
2. The Equations of Motion and their Approximate Solution 223
3. Accuracy and Numerical Parameters 226
4. Main Features of the Flow 228
5. Hausdorff Dimension of the Support of Vorticity 230
6. Lognormality of the Vorticity Distribution 235
7. Temporal Intermittency and Higher Statistics 237
8. Conclusion 240
References 240

Textauszug

Introduction

This volume contains 14 papers on computational fluid dynamics written between 1967 and 1982, in particular papers on vortex methods and the projection method, as well as papers on the numerical solution of problems in kinetic theory, combustion theory, and gas dynamics. A great deal of practical experience and theoretical understanding has accumulated in these fields in recent years, and a systematic exposition of current knowledge is difficult to write and would be difficult to read. I believe that some of the ideas in this field are easier to learn if they are presented in the simpler garb that preceded the development of sophisticated implementations and of a general mathematical theory, and this is the motivation for publishing the present collection. These papers explain, among other topics, how one might set up a discrete approximation of the Navier–Stokes equations for an incompressible fluid, build a vortex method, or solve a Riemann problem. Some of these papers are by now quite old, especially by the standards of a rapidly changing field, and the reader should be aware of the existence of a large literature that is more up-to-date. In the next few paragraphs I would like to present a short summary of what the papers collected here contain and make some suggestions for further reading. These suggestions reflect my own interests and do not provide a complete bibliography of computational fluid mechanics or even of the topics covered in this book.

The general theme of these papers is the numerical solution of the equations of fluid mechanics in circumstances where the viscosity is small; the big problem that is looming beyond the specific applications is the problem of turbulence. It is generally understood that turbulence in fluids is dominated by the mechanics of vorticity, and many of the methods are based on vortex representations of the flow. A number of them employ random numbers in some form; the major motivation for studying random algorithms is the belief that they may lead to effective methods for sampling a turbulent flow field. The sequence of papers on vortex methods ends here with a paper that represents a possible starting point for speculations on the structure of turbulent flow. The belief that turbulence can be best understood in physical space, by considering the interactions between physical structures, rather than in wave-number space, where one considers interactions between Fourier modes, is not universally held. The reader interested in turbulence theory should find additional sources of information.

The first paper in the book presents the artificial compressibility method for finding steady-state solutions of the Navier–Stokes equations for an incompressible fluid, with applications to thermal convection. Related ideas have been introduced by Temam [T1] and Yanenko. A recent application with a bibliography can be found in [R2]. The main idea is to find a compressible system that has the same steady state as the given incompressible system, but has the property that its steady state can be reached with less labor. More generally, since the limit of infinite sound speed that leads from a compressible to an incompressible flow is well behaved [K2], one can try to approximate incompressible flow by an appropriate compressible flow even far from a steady state. These ideas are closely related to the penalty method, which is more natural in the context of finite element methods [B9].

Paper 2 presents the projection method for incompressible flow. The idea is that the equation of continuity can be viewed as a constraint, and one can solve the equations step-by-step by first ignoring the constraint and then projecting the result on the space of incompressible flows. Some subtlety is required to formulate the boundary conditions for the intermediate step. This construction was partially inspired by the existence theory for the Navier–Stokes equations presented in [F1] and has by now become standard (and indeed, in [C5] I showed that any consistent and stable approximation to the Navier–Stokes equations in pressure–-velocity variables is essentially equivalent to the projection method). The correct formulation of the projection solves the problem of finding numerical boundary conditions for the pressure. Standard references in which this method is discussed include [C13], [G9], [P2], and [T1]. In the paper the projection was implemented by relaxation with a checkerboard pattern, which was a reasonable methodology for the time; however, faster Laplace solvers can be adapted for this purpose, see, e.g., [A4]. As a sidelight, I would like to mention that this paper contains a discussion of the relations between the DuFort–Frankel scheme, checkerboard relaxation and matrix condition (A). An interesting second-order variant of the projection method is presented in [B7]. Further discussion and finite element versions can be found, e.g., in [G9] and [T1].

Paper 3 contains a convergence proof for the projection method. Further work can be found, e.g., in [T1]. The analysis in the paper has two elements that are still of current interest: It shows that convergence and stability depend on the approximations for the gradient and for minus the divergence being at least approximately adjoint, and it contains a model of error growth which says basically that the error accumulates slowly as long as it is small, but if it ceases to be small all hell can break loose. A fixed-point theorem is needed to get convergence in a strong sense. This analysis shows that it is important to obtain error bounds rather than merely weak convergence results. Furthermore, there is a large recent literature in which it is claimed that very poor approximations can provide qualitative models of turbulence; the analysis in this paper suggests that such claims should be viewed with great caution. There is an interesting proof of the validity of a time-discretized projection in [E1].

Paper 4 is a numerical study of Boltzmann’s equation by a method that can be viewed as a numerical generalization of Grad’s thirteen moment approximation. I am not sure that this is a very good method of solution (but I do not know what is). The method is also an early example of a combined expansion–collocation method (as in the pseudo-spectral method), it provides some insight into the usefulness of some of the standard analytical approximation for Boltzmann’s equation, and it provides an interesting contrast to the lack of convergence of the seemingly related Wiener–Hermite expansions of turbulence theory (see, e.g., [C6]).

Paper 5 is the original paper on the random vortex method. An incompressible flow can be approximated by a collection of vortex elements. The main ideas in this paper are: the use of vortices with finite core to improve convergence, the use of a random walk to approximate diffusion, and vorticity creation at boundaries to represent the no-slip boundary condition. For reviews of the applications of this method, see, e.g., [A6], [G10], [M3], and [M4]. The method has been greatly improved in the years since this paper appeared and has been the object of a great amount of outstanding theoretical analysis. The generalizations to three space dimensions and more accurate boundary conditions will be discussed below. I would like to provide here references to some of the major developments:

(i) A beautiful convergence theory has been developed; for inviscid flow without boundaries, see [A5], [B2], [B3], [B4], [B5], [B6], [C18], [G6], [H1], [M4], and [R1]; ([A5] contains a review). For viscous flow, see [G5], [L1], and [L2]. The theory has suggested better choices of smoothing than the one I used in this paper (see in particular [B6] and [P1]). Convergence results for related problems can be found in [R3].

(ii) The algorithm in the paper is O(N2), i.e., it takes O(N2) operations to sum the interactions of TV vortex elements. It turns out that one can sum these interactions, to within machine accuracy, by using only O(N) or O(NlogN) operations, thus greatly increasing the power of the method (see e.g. [A2], [G8], and [K1]).

(iii) Elaborate numerical checks on the accuracy of the method can be found, e.g., in [C1], [C2], [C3], [P1], and [S4], and applications include [B11] and [C2], [C3] and [G2] as well as the papers mentioned below. Interesting mixed methods are presented in [S5] and [V1].

(iv) General discussions of the role of vortex dynamics in the study of turbulence and more generally in fluid mechanics can be found in [C6], [M1], and [M2].

Paper 6 (with P. Bernard) describes a calculation of the motion of a vortex sheet represented by a collection of vortex blobs (i.e. vortex elements smoothed by a finite cut-off). Later papers that expanded and developed this approach include [A1], [C17], and [K4]. These latter exceptionally fine calculations have also led to important theoretical developments [D1].

Papers 7 and 8 describe applications of the random choice method to problems in compressible gas dynamics and...

Produkt-Details

• Erscheint lt. Verlag: 28.6.2014
• Sprache: englisch
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 1-4832-7155-2 / 1483271552
• ISBN-13: 978-1-4832-7155-2 / 9781483271552