COMPUTATIONAL FLUID DYNAMICS The Basics with Applications

Transcription

1 COMPUTATIONAL FLUID DYNAMICS The Basics with Applications John D. Anderson, Jr. Department of Aerospace Engineering University of Maryland McGraw-Hill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto

2 CONTENTS Part I Preface Basic Thoughts and Equations 1 Philosophy of Computational Fluid Dynamics Computational Fluid Dynamics : Why? Computational Fluid Dynamics as a Research Tool Computational Fluid Dynamics as a Design Tool The Impact of Computational Fluid Dynamics-Some Other 9 Examples Automobile and Engine Applications Industrial Manufacturing Applications Civil Engineering Applications Environmental Engineering Applications Naval Architecture Applications (Submarine Example) Computational Fluid Dynamics: What Is It? The Purpose of This Book 32 : 2 The Governing Equations of Fluid Dynamics : Their Derivation, a Discussion of Their Physical Meaning, and a Presentation of Forms Particularly Suitable to CFD Introduction Models of the Flow Finite Control Volume Infinitesimal Fluid Element Some Comments The Substantial Derivative (Time Rate of Change Following a Moving Fluid Element The Divergence of the Velocity Its Physical Meaning A Comment 48

3 XU CONTENTS 2.5 The Continuity Equation Model of the Finite Control Volume Fixed in Space Model of the Finite Control Volume Moving with the Fluid Model of an Infinitesimally Small Element Fixed in Space Model of an Infinitesimally Small Fluid Element Moving with the Flow All the Equations Are One : Some Manipulations Integral versus Differential Form of the Equations : An Important Comment The Momentum Equation The Energy Equation Summary of the Governing Equations for Fluid Dynamics: With Comments Equations for Viscous Flow (the Navier-Stokes Equations) Equations for Inviscid Flow (the Euler Equations) Comments on the Governing Equations Physical Boundary Conditions Forms of the Governing Equations Particularly Suited for CFD: Comments on the Conservation Form, Shock Fitting, and Shock Capturing Summary 92 Problems 93 3 Mathematical Behavior of Partial Differential Equations : The Impact on CFD Introduction Classification of Quasi-Linear Partial Differential Equations A General Method of Determining the Classification of Partial Differential Equations : The Eigenvalue Method General Behavior of the Different Classes of Partial Differential Equations: Impact on Physical and Computational Fluid Dynamics Hyperbolic Equations Parabolic Equations Elliptic Equations Some Comments : The Supersonic Blunt Body Problem Revisited Well-Posed Problems Summary 121 Problems 121 Part II Basics of the Numerics 4 Basic Aspects of Discretization Introduction Introduction to Finite Differences 128

4 CONTENTS %IIl 4.3 Difference Equations Explicit and Implicit Approaches : Definitions and Contrasts Errors and an Analysis of Stability Stability Analysis : A Broader Perspective Summary 165 GUIDEPOST 166 Problems Grids with Appropriate Transformations Introduction General Transformation of the Equations Metrics and Jacobians Form of the Governing Equations Particularly Suited 5.5 for CFD Revisited : The Transformed Version 183 A Comment Stretched (Compressed) Grids Boundary-Fitted Coordinate Systems ; Elliptic Grid Generation 192 GUIDEPOST Adaptive Grids Some Modern Developments in Grid Generation Some Modern Developments in Finite-Volume Mesh Generation: Unstructured Meshes and a Return to Cartesian Meshes Summary 212 Problems Some Simple CFD Techniques : A Beginning Introduction The Lax-Wendroff Technique MacCormack's Technique 222 GUIDEPOST Some Comments : Viscous Flows, Conservation Form, and Space Marching Viscous Flows Conservation Form Space Marching The Relaxation Technique and Its Use with Low-Speed Inviscid Flow Aspects of Numerical Dissipation and Dispersion; Artificial Viscosity The Alternating-Direction-Implicit (ADI) Technique 6.8 The Pressure Correction Technique : Application 243 to Incompressible Viscous Flow Some Comments on the Incompressible 247 Navier-Stokes Equations 248

5 XIv CONTENTS Some Comments on Central Differencing of the Incompressible Navier-Stokes Equations ; The Need for a Staggered Grid The Philosophy of the Pressure Correction Method The Pressure Correction Formula The Numerical Procedure : The SEMPLE Algorithm Boundary Conditions for the Pressure Correction Method 262 GUIDEPOST Some Computer Graphic Techniques Used in CFD xy Plots Contour Plots Vector and Streamline Plots Scatter Plots Mesh Plots Composite Plots Summary on Computer Graphics Summary 277 Problems 278 Part III Some Applications 7 Numerical Solutions of Quasi-One-Dimensional Nozzle Flows Introduction: The Format for Chapters in Part III Introduction to the Physical Problem: Subsonic-Supersonic Insentropic Flow CFD Solution of Subsonic-Supersonic Isentropic Nozzle Flow: MacCormack's Technique The Setup Intermediate Results : The First Few Steps Final Numerical Results: The Steady-State Solution CFD Solution of Purely Subsonic Isentropic Nozzle Flow The Setup: Boundary and Initial Conditions Final Numerical Results : MacCormack's Technique The Anatomy of a Failed Solution The Subsonic-Supersonic Isentropic Nozzle Solution Revisited : The Use of the Governing Equations in Conservation Form The Basic Equations in Conservation Form The Setup Intermediate Calculations : The First Time Step Final Numerical Results : The Steady State Solution 351

6 CONTENTS xv 7.6 A Case with Shock Capturing The Setup The Intermediate Time-Marching Procedure: The Need for Artificial Viscosity Numerical Results Summary Numerical Solution of a Two-Dimensional Supersonic Flow: Prandtl-Meyer Expansion Wave Introduction Introduction to the Physical Problem: Prandtl-Meyer Expansion Wave-Exact Analytical Solution The Numerical Solution of a Prandtl-Meyer Expansion Wave Flow Field The Governing Equations The Setup Intermediate Results Final Results Summary Incompressible Couette Flow : Numerical Solutions by Means of an Implicit Method and the Pressure Correction Method Introduction The Physical Problem and Its Exact Analytical Solution The Numerical Approach : Implicit Crank-Nicholson Technique The Numerical Formulation The Setup Intermediate Results Final Results Another Numerical Approach: The Pressure Correction Method The Setup Results Summary 445 Problem Supersonic Flow over a Flat Plate : Numerical Solution by Solving the Complete Navier-Stokes Equations Introduction The Physical Problem The Numerical Approach : Explicit Finite-Difference Solution of the Two-Dimensional Complete Navier-Stokes Equations The Governing Flow Equations The Setup 452

7 %Vi CONTENTS The Finite-Difference Equations Calculation of Step Sizes in Space and Time Initial and Boundary Conditions Organization of Your Navier-Stokes Code Overview The Main Program The MacCormack Subroutine Final Remarks Final Numerical Results : The Steady State-Solution Summary 474 Part IV Other Topics 11 Some Advanced Topics in Modern CFD : A Discussion Introduction The Conservation Form of the Governing Flow Equations Revisited: The Jacobians of the System Specialization to One-Dimensional Flow Interim Summary Additional Considerations for Implicit Methods Linearization of the Equations: The Beam and Warming Method The Multidimensional Problem: Approximate Factorization Block Tridiagonal Matrices Interim Summary Upwind Schemes Flux-Vector Splitting The Godunov Approach General Comment Second-Order Upwind Schemes High-Resolution Schemes : TVD and Flux Limiters 11.7 Some 509 Results Multigrid Method Summary Problems The Future of CFD 12.1 The 515 Importance of CFD Revisited Computer Graphics in CFD 12.3 The 516 Future of CID: Enhancing the Design Process 12.4 The 517 Future of CFD : Enhancing Understanding Conclusion 533

8 CONTENTS XVII Appendix A Thomas' Algorithm for the Solution of a Tridiagonal System of Equations 534 References 539