BASICS OF FLUID MECHANICS ANDINTRODUCTION TO COMPUTATIONALFLUID DYNAMICS

Numerical Methods and AlgorithmsVolume 3

Series Editor:

Claude BrezinskiUniversité des Sciences et Technologies de Lille, France

BASICS OF FLUID MECHANICS ANDINTRODUCTION TO COMPUTATIONALFLUID DYNAMICS

by

TITUS PETRILABabes-Bolyai University, Cluj-Napoca, Romania

DAMIAN TRIFBabes-Bolyai University, Cluj-Napoca, Romania

Springer

Library of Congress Cataloging-in-Publication Data

eBook ISBN: 0-387-23838-7Print ISBN: 0-387-23837-9

Print ©2005 Springer Science + Business Media, Inc.

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©2005 Springer Science + Business Media, Inc.

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Contents

Preface

1. INTRODUCTION TO MECHANICS OF CONTINUA

xiii

1 Kinematics of Continua1.1 The Concept of a Deformable Continuum

13

2 General Principles.

2.1The Stress Tensor and Cauchy’s Fundamental Results

The Forces Acting on a Continuum

3 Constitutive Laws.Inviscid and real fluids3.1 Introductory Notions of Thermodynamics.

First and Second Law of Thermodynamics

1.2

1.3

1

11

4

17

182.2

2.3

2.420

2.52.6

242.7

26

26

323.2

3.33.4

Motion of a Continuum.Lagrangian and Eulerian CoordinatesEuler–Lagrange Criterion.Euler’s and Reynolds’ (Transport) Theorems

Principle of Mass Conservation.The Continuity EquationPrinciple of the Momentum Torsor Variation.The Balance Equations

17

The Cauchy Stress TensorThe Cauchy Motion EquationsPrinciple of Energy Variation.Conservation of EnergyGeneral Conservation Principle

2123

25

Constitutive (Behaviour, “Stresses-Deformations”Relations) LawsInviscid (Ideal) Fluids 34Real Fluids 38

vi

3.5 Shock Waves 4349

2. DYNAMICS OF INVISCID FLUIDS 51

1

2

3

Vorticity and Circulation for Inviscid Fluids.The Bernoulli Theorems

Some Simple Existence and Uniqueness Results

Irrotational Flows of Incompressible Inviscid Fluids.The Plane Case

Conformal Mapping and its Applications within PlaneHydrodynamics4.1 Helmholtz Instability

Principles of the (Wing) Profiles Theory5.1 Flow Past a (Wing) Profile for an Incidence and

a Circulation “a priori” GivenProfiles with Sharp Trailing Edge.Joukovski HypothesisTheory of Joukovski Type ProfilesExampleAn Iterative Method for Numerical Generation ofConformal Mapping

Panel Methods for Incompressible Flow of Inviscid Fluid

6.1 The Source Panel Method for Non-Lifting FlowsOver Arbitrary Two-Dimensional BodiesThe Vortex Panel Method for Lifting Flows OverArbitrary Two-Dimensional BodiesExample

Almost Potential Fluid Flow

Thin Profile Theory8.18.2

Mathematical Formulation of the ProblemSolution Determination

Unsteady Irrotational Flows Generated by the Motion ofa Body in an Inviscid Incompressible Fluid9.19.2

100100The 2-Dimensional (Plane) Case

The Determination of the Fluid Flow Induced bythe Motion of an Obstacle in the Fluid.The Case of the Circular CylinderThe 3-Dimensional Case

The Unique Form of the Fluid Equations3.6

4

5

5.2

5.35.45.5

6.2

6.3

6

7

8

9

9.3102103

51

55

59

6467

70

70

727477

79

81

81

8487

92

959697

Contents vii

9.4 General Method for Determining of the Fluid FlowInduced by the Displacement of an ArbitrarySystem of Profiles Embedded in the Fluid in thePresence of an “A Priori”Given Basic Flow

10 Notions on the Steady Compressible Barotropic Flows

105

11010.1 Immediate Consequences of the Bernoulli

Theorem 11010.210.3

The Equation of Velocity Potential (Steichen) 113Prandtl–Meyer (Simple Wave) Flow 115

10.4 Quasi-Uniform Steady Plane Flows 11710.5 General Formulation of the Linearized Theory 11810.6 Far Field (Infinity) Conditions 11910.7 The Slip-Condition on the Obstacle 12010.8 The Similitude of the Linearized Flows.

The Glauert–Prandtl Rule 121

11 Mach Lines. Weak Discontinuity Surfaces 123

12 Direct and Hodograph Methods for the Study of theCompressible Inviscid Fluid Equations 127

12812.112.2

A Direct Method [115]Chaplygin Hodograph Method.Molenbroek–Chaplygin equation 129

3. VISCOUS INCOMPRESSIBLE FLUID DYNAMICS 133

1 The Equation of Vorticity (Rotation) and the CirculationVariation 133

2 Some Existence and Uniqueness Results 136

3 The Stokes System 138

4 Equivalent Formulations for the Navier–StokesEquations in Primitive Variables 1404.1 Pressure Formulation4.2 Pressure-Velocity Formulation

140142

5 Equivalent Formulations for the Navier–StokesEquations in “Non-Primitive” Variables 1435.1 Navier–Stokes Equations in Orthogonal Generalized

Coordinates. Stream Function Formulation 1445.2 A “Coupled” Formulation in Vorticity and Stream

Function 1515.3 The Separated (Uncoupled) Formulation in

Vorticity and Stream Function 152

viii

5.4 An Integro-Differential Formulation 157

6 Similarity of the Viscous Incompressible Fluid Flows 1591626.1 The Steady Flows Case

7 Flows With Low Reynolds Number. Stokes Theory 1647.1 The Oseen Model in the Case of the Flows Past a

Thin Profile 167

8 Flows With High (Large) Reynolds Number8.18.28.38.48.5

Mathematical ModelThe Boundary Layer EquationsProbabilistic Algorithm for the Prandtl EquationsExampleDynamic Boundary Layer with Sliding on a PlanePlaque

172

191

4. INTRODUCTION TO NUMERICAL SOLUTIONS FORORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS 197

1 Introduction 197

2 Discretization of a Simple Equation 2032.1 Using the Finite Difference Method

3 The Cauchy Problem for Ordinary Differential Equations3.1 Examples

4 Partial Differential Equations4.1 Classification of Partial Differential Equations

2.2 Using the Finite Element Method203203

2.3 Using the Finite Volume Method 2052.4 Comparison of the Discretization Techniques 206

207216

226226

4.2 The Behaviour of Different Types of PDE 2284.3 Burgers’ Equation 2314.4 Stokes’ Problem 2364.5 The Navier–Stokes System 239

5. FINITE-DIFFERENCE METHODS 247

1 Boundary Value Problems for OrdinaryDifferential Equations 2471.1 Supersonic Flow Past a Circular Cylindrical

Airfoil 249

2 Discretization of the Partial Differential Equations 253

3 The Linear Advection Equation 2573.1 Discretization of the Linear Advection Equation 257

173174180187

Contents ix

3.2 Numerical Dispersion and Numerical Diffusion 2643.3 Lax, Lax–Wendroff and MacCormack Methods 266

4 Diffusion Equation 2724.1 Forward-Time Scheme 2724.2 Centered-Time Scheme 2734.3 Backward-Time Scheme 2744.4 Increasing the Scheme’s Accuracy 2754.5 Numerical Example 275

5 Burgers Equation Without Shock 2775.1 Lax Scheme 2775.2 Leap-Frog Scheme 2785.3 Lax–Wendroff Scheme 279

6 Hyperbolic Equations 2806.1 Discretization of Hyperbolic Equations 2806.2 Discretization in the Presence of a Shock 2856.3 Method of Characteristics 291

7 Elliptic Equations 2957.1 Iterative Methods 2967.2 Direct Method 3047.3 Transonic Flows 3077.4 Stokes’ Problem 312

8 Compact Finite Differences 3208.1 The Compact Finite Differences Method (CFDM) 3208.2 Approximation of the Derivatives 3218.3 Fourier Analysis of the Errors 3268.4 Combined Compact Differences Schemes 3298.5 Supercompact Difference Schemes 333

9 Coordinate Transformation 3359.1 Coordinate Stretching 3389.2 Boundary-Fitted Coordinate Systems 3399.3 Adaptive Grids 344

6. FINITE ELEMENT AND BOUNDARY ELEMENTMETHODS 3451 Finite Element Method (FEM) 345

1.1 Flow in the Presence of a Permeable Wall 3491.2 PDE-Toolbox of MATLAB 354

2 Least-Squares Finite Element Method (LSFEM) 3562.1 First Order Model Problem 356

x

2.2 The Mathematical Foundation of the Least-SquaresFinite Element Method 363

2.3 Div-Curl (Rot) Systems 3702.4 Div-Curl (Rot)-Grad System 3752.5 Stokes’ Problem 377

3 Boundary Element Method (BEM) 3803.1 Abstract Formulation of the Boundary Element

Method 3813.2 Variant of the Complex Variables Boundary

Element Method [112] 3853.3 The Motion of a Dirigible Balloon 3893.4 Coupling of the Boundary Element Method and

the Finite Element Method 391

7. THE FINITE VOLUME METHOD ANDTHE GENERALIZED DIFFERENCE METHOD 3971 ENO Finite Volume Schemes 398

1.1 ENO Finite Volume Scheme in One Dimension 3991.2 ENO Finite Volume Scheme in Multi-Dimensions 406

2 Generalized Difference Method 4112.1 Two-Point Boundary Value Problems 4112.2 Second Order Elliptic Problems 4242.3 Parabolic Equations 4292.4 Application 433

8. SPECTRAL METHODS 439

1 Fourier Series 4421.1 The Discretization 4421.2 Approximation of the Derivatives 445

2 Orthogonal Polynomials 4472.1 Discrete Polynomial Transforms 4472.2 Legendre Polynomials 4502.3 Chebyshev Polynomials 452

3 Spectral Methods for PDE 4553.1 Fourier–Galerkin Method 4553.2 Fourier-Collocation 4563.3 Chebyshev-Tau Method 4573.4 Chebyshev-Collocation Method 4583.5 The Calculation of the Convolution Sums 4593.6 Complete Discretization 460

Contents xi

4 Liapunov–Schmidt (LS) Methods 462

5 Examples 4725.1 Stokes’ Problem 4725.2 Correction in the Dominant Space 479

Appendix AVectorial-Tensorial Formulas 483

References 487

Index 497

Preface

The present book – through the topics and the problems approach– aims at filling a gap, a real need in our literature concerning CFD(Computational Fluid Dynamics). Our presentation results from a largedocumentation and focuses on reviewing the present day most importantnumerical and computational methods in CFD.

Many theoreticians and experts in the field have expressed their in-terest in and need for such an enterprise. This was the motivation forcarrying out our study and writing this book. It contains an importantsystematic collection of numerical working instruments in Fluid Dynam-ics.

Our current approach to CFD started ten years ago when the Univer-sity of Paris XI suggested a collaboration in the field of spectral methodsfor fluid dynamics. Soon after – preeminently studying the numericalapproaches to Navier–Stokes nonlinearities – we completed a numberof research projects which we presented at the most important interna-tional conferences in the field, to gratifying appreciation.

An important qualitative step in our work was provided by the devel-opment of a computational basis and by access to a number of expertsoftwares. This fact allowed us to generate effective working programsfor most of the problems and examples presented in the book, an as-pect which was not taken into account in most similar studies that havealready appeared all over the world.

What makes this book special, in comparison with other similar en-terprises?

This book reviews the main theoretical aspects of the area, emphasizesvarious formulations of the involved equations and models (focussing onoptimal methods in CFD) in order to point out systematically the mostutilized numerical methods for fluid dynamics. This kind of analysis –leaving out the demonstration details – takes notice of the convergence

xiv

and error aspects which are less prominent in other studies. Logically,our study goes on with some basic examples of effective applications ofthe methods we have presented and implemented (MATLAB).

The book contains examples and practical applications from fluid dy-namics and hydraulics that were treated numerically and computation-ally – most of them having attached working programs. The inviscidand viscous, incompresible fluids are considered; practical applicationshave important theoretical outcomes.

Our study is not extended to real compresible fluid dynamics, or toturbulence phenomena. The attached MATLAB 6 programs are con-ceived to facilitate understanding of the algorithms, without optimizingintentions.

Through the above mentioned aspects, our study is intended to be aninvitation to a more complete search: it starts with the formulation andstudy of mathematical models of fluid dynamics, continues with analysisof numerical solving methods and ends with computer simulation of thementioned phenomena.

As for the future, we hope to extend our study and to present a newmore complete edition, taking into account constructive suggestions andobservations from interested readers.

We cannot end this short presentation without expressing our grat-itude to our families who have supported us in creating this work insuch a short time, by offering us peace and by acquitting us from oureveryday duties.

The authors