ESSENTIAL COMPUTATIONAL FLUID DYNAMICS

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1 ESSENTIAL COMPUTATIONAL FLUID DYNAMICS Oleg Zikanov WILEY JOHN WILEY & SONS, INC.

2 CONTENTS PREFACE xv 1 What Is CFD? Introduction / Brief History of CFD / Outline of the Book / 6 References and Suggested Reading / 7 I Fundamentals 9 2 Governing Equations of Fluid Dynamics and Heat Transfer Preliminary Concepts / Mass Conservation / Conservation of Chemical Species / Conservation of Momentum / Conservation of Energy / Equation of State / Equations in Integral Form / Equations in Conservation Form / Equations in Vector Form / Boundary Conditions / Rigid Wall Boundary Conditions / Inlet and Exit Boundary Conditions / Other Boundary Conditions / 29 VII

3 VIII CONTENTS References and Suggested Reading / 30 Problems / 30 3 Partial Differential Equations Model Equations; Formulation of a PDE Problem / Model Equations / Domain, Boundary, and Initial Conditions / Equilibrium and Marching Problems / Examples / Mathematical Classification of PDE of Second Order / Classification / Hyperbolic Equations / Parabolic Equations / Elliptic Equations / Numerical Discretization: Different Kinds of CFD / Spectral Methods / Finite Element Methods / Finite Difference and Finite Volume Methods / 49 References and Suggested Reading / 52 Problems / 52 4 Basics of Finite Difference Approximation Computational Grid / Time Discretization / Space Discretization / Finite Differences and Interpolation / Approximation of Э и /Эх / Truncation Error, Consistency, Order of Approximation / Other Formulas for ди/дх: Evaluation of the Order of Approximation / Schemes of Higher Order for First Derivative / Higher-Order Derivatives / Mixed Derivatives / Truncation Error of Linear Interpolation / Approximation of Partial Differential Equations / Approach and Examples / 67

4 CONTENTS ix Interpretation of Truncation Error: Numerical Dissipation and Dispersion / Boundary and Initial Conditions / Consistency of Numerical Approximation / System of Difference Equations / Implicit and Explicit Methods / Development of Finite Difference Schemes / Taylor Series Expansions / Polynomial Fitting / 82 References and Suggested Reading / 83 Problems / 83 5 Finite Volume Method Introduction and Integral Formulation / Finite Volume Grid / Global Conservation Property / Approximation of Integrals / Volume Integrals / Surface Integrals / Methods of Interpolation / Upwind Interpolation / Linear Interpolation / Upwind Interpolation of Higher Order / Interpolation on Nonorthogonal Grids / Boundary Conditions / 101 References and Suggested Reading / 102 Problems / Stability of Transient Solutions Introduction and Definition of Stability / Discretization and Round-off Error / Definition / Stability Analysis / Neumann Method / Matrix Method / Implicit versus Explicit Schemes Stability and Efficiency Considerations / 118 References and Suggested Reading / 120

5 X CONTENTS Problems / Application to Model Equations Linear Convection Equation / Simple Explicit Schemes / Other Schemes / One-Dimensional Heat Equation / Simple Explicit Scheme / Simple Implicit Scheme / Crank-Nicolson Scheme / Burgers and Generic Transport Equations / Method of Lines Approach / Adams Methods / Runge-Kutta Methods / Implicit Schemes: Solution of Tridiagonal Systems by Thomas Algorithm / 136 References and Suggested Reading / 140 Problems / 140 II Methods Steady-State Problems Problems Reducible to Matrix Equations / Elliptic PDE / Implicit Integration of Nonsteady Equations / Direct Methods / Band-Diagonal and Block-Diagonal Matrices / LU Decomposition / Iterative Methods / General Methodology / Jacobi Iterations / Gauss-Seidel Algorithm / Successive Over- and Underrelaxation / Convergence of Iterative Procedures / Multigrid Methods / Pseudo-transient Approach / Systems of Nonlinear Equations / 164

6 CONTENTS Xi Newton's Algorithm / Iteration Methods Using Linearization / Sequential Solution / 168 References and Suggested Reading / 168 Problems / Unsteady Problems of Fluid Flows and Heat Transfer Introduction / Compressible Flows / Overview and General Comments / Explicit MacCormack Method / Beam-Warming Method / Upwinding / Methods for Purely Hyperbolic Systems / Unsteady Conduction Heat Transfer / Simple Methods for Multidimensional Heat Conduction / Approximate Factorization / ADI Method / 191 References and Suggested Reading / 192 Problems / Incompressible Flows 196 Ю.1. General Considerations / Introduction / Role of Pressure / Discretization Approach / Colocated and Staggered Grids / Projection Method for Unsteady Flows / Explicit Schemes / Implicit Schemes / Projection Methods for Steady-State Flows / SIMPLE / SIMPLEC, SIMPLER, and PISO / Other Methods / Vorticity-Streamfunction Formulation for Two-Dimensional Flows /218

7 XU CONTENTS Artificial Compressibility / 222 References and Suggested Reading / 222 Problems / 223 III ArtofCFD Turbulence Introduction / A Few Words About Turbulence / Why Is the Computation of Turbulent Flows Difficult? / Overview of Numerical Approaches / Direct Numerical Simulation (DNS) / Homogeneous Turbulence / Inhomogeneous Turbulence / Reynolds-Averaged Navier-Stokes (RANS) Models / Reynolds-Averaged Equations / Eddy Viscosity Hypothesis / Algebraic Models / Two-Equation Models / Numerical Implementation of RANS Models / Large-Eddy Simulation (LES) / Filtered Equations / Closure Models / Implementation of LES in CFD Analysis: Numerical Resolution and Near-Wall Treatment / 255 References and Suggested Reading / 258 Problems / Computational Grids Introduction: Need for Irregular and Unstructured Grids / Irregular Structured Grids / Generation by Coordinate Transformation / Examples / Grid Quality / 268

8 CONTENTS Xiü Unstructured Grids / Grid Generation / Finite Volume Discretization on Unstructured Grids / Cell Topology / Grid Quality / 275 References and Suggested Reading / 278 Problems / Conducting CFD Analysis Overview: Setting and Solving a CFD Problem / Errors and Uncertainty / Errors in CFD Analysis / Verification and Validation / Adaptive Grids / 293 References and Suggested Reading / 295 INDEX 297

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