Notes on Numerical Fluid Mechanics

I Historical Origins of the Inviscid Model.- 1.1 From Antiquity to the Renaissance.- 1.2 The Enlightenment: the Age of Reason.- 1.2.1 Leonhard Euler.- 1.3 The 19th Century: Mathematical Fluid Mechanics.- 1.3.1 Vortex Discontinuities and Resistance.- 1.3.2 Shock Waves.- 1.4 The 20th Century: The Computational Era.- 1.4.1 Early Methods.- 1.4.2 Methods to Solve the Euler Equations: 1950-1970.- 1.4.3 Methods to Solve the Euler Equations: 1970-1990.- 1.5 Brief Overview of Field.- 1.5.1 Secondary Reference Sources.- 1.5.2 Three Categories of Methods.- 1.6 Outline of the Remaining Chapters.- 1.7 References.- II The Euler Equations.- 2.1 The Classical Euler Equations in Gas Dynamics.- 2.2 Basic Results of the Non-Conservative Equations.- 2.2.1 Isentropic Flow.- 2.2.2 Homentropic Flow.- 2.3 Basic Results of the Conservative Equations.- 2.3.1 Isenthalpic Flow.- 2.3.2 Shock Flow.- 2.3.3 Speed of Sound.- 2.3.4 Eigenvalues of Pressure Waves.- 2.3.5 Homogeneous Property of the Euler Equations.- 2.4 Coordinate Transformations.- 2.5 Stokes' Integral.- 2.6 Physical Boundary Conditions.- 2.7 Other Forms of the Euler Equations.- 2.8 References.- III Fundamentals of Discrete Solution Methods.- 3.1 Hyperbolic Equations and Waves.- 3.2 Characteristics.- 3.3 Wavefronts Bounding a Constant State.- 3.4 Riemann Invariants.- 3.5 Well-Posed and Unique Solutions.- 3.6 Initial Boundary-Value Problems.- 3.7 Weak Solutions and Shocks.- 3.8 Discrete Solution Methods.- 3.9 Classical Finite-Difference Approximations to Derivatives.- 3.10 Computational Grid and Accuracy.- 3.11 Local Truncation Error.- 3.12 Consistency.- 3.13 Convergence and Stability.- 3.14 Notion of Convergence.- 3.15 Notion of Stability.- 3.15.1 A Bound for the Spectral Radius.- 3.16 Von Neumann Method.- 3.17 Matrix Method.- 3.18 The Energy Method.- 3.19 Schemes for Non-Linear Equations.- 3.20 References.- IV The Finite Volume Concept.- 4.1 Coordinate Transformations.- 4.1.1 The Differential Approach.- 4.2 The Finite-Volume Approach.- 4.2.1 Continuum Equations.- 4.2.2 Coordinate Geometry.- 4.2.3 Spatial Finite-Volume Discretization.- 4.2.4 Flux Evaluation.- 4.2.5 Stability and Accuracy at Mesh Singularities.- 4.3 Relationship to Finite Differences.- 4.4 Numerical Conservation.- 4.4.1 Uniform Free Stream.- 4.5 Cell Vertex Methods.- 4.6 Boundary Conditions for the Continuous Problem.- 4.6.1 Coordinate Cuts.- 4.6.2 Solid Walls.- 4.6.3 Zero-Flux Transport.- 4.6.4 Inflow/Outflow Boundary.- 4.7 Discretization of the Flow Domain.- 4.7.1 Resolution of Scales.- 4.7.2 Topology of Grid-Point Patterns.- 4.8 Finite-Volume Truncation Error.- 4.9 Multi-Block Meshes.- 4.10 Boundary Conditions for the Discrete Problem.- 4.10.1 Accuracy and Stability.- 4.10.2 Empirical Rule for Boundary-Condition Accuracy.- 4.10.3 Farfield Boundary Conditions.- 4.11 References.- V Centered Differencing.- 5.1 Flux-Averaged Methods.- 5.2 Local Fourier Stability.- 5.3 Local Time-Step Scaling.- 5.4 Artificial-Viscosity Model.- 5.4.1 Non-Linear Artificial Viscosity.- 5.4.2 Linear Artificial Viscosity.- 5.4.3 Boundary Conditions.- 5.5 Time Integration and Convergence to Steady State.- 5.5.1 Steady State Operator.- 5.5.2 Eigenspectrum of Centered Schemes.- 5.6 References.- VI Principles of Upwinding.- 6.1 Initial Considerations.- 6.2 Foundation of Upwinding.- 6.3 A Local Solution to the Model Equation.- 6.4 Conservative Upwinding.- 6.5 Accuracy of Three-Point Schemes.- 6.6 Stability Considerations for Three-Point Schemes.- 6.7 The Finite-Volume Cell-Face Concept.- 6.8 The Riemann Probem at a Finite-Volume Cell Face.- 6.9 The Characteristic Derivative.- 6.10 The Scalar Invariant.- 6.11 Characteristic Condition.- 6.12 Eigenvalues and Invariants.- 6.13 A Simple Linear Riemann Solver.- 6.14 A Near Exact Riemann Solver.- 6.15 The Isentropic Riemann Solver.- 6.16 An Osher-Type Riemann Solver.- 6.17 A Linear Riemann Solver Using Primitive Variables.- 6.18 The Exact Non-Conservative Riemann Solver.- 6.19 An Alternative Osher-Type Approximate Riemann Solver.- 6.20 Asymmetric Osher-Type Approximate Riemann Solvers.- 6.21 A Linear Newton-Type Riemann Solver.- 6.22 A Quadratic Newton-Type Riemann Solver.- 6.23 A Linear Conservative Riemann Solver.- 6.24 The Exact Conservative Riemann Solver.- 6.25 Roe's Average.- 6.26 Riemann Solvers Based on Fluxes: The Steger-Warming Fluxes.- 6.27 Generalized Steger-Warming Fluxes.- 6.28 Einfeldt-Type Fluxes.- 6.29 Van Leer-Type Fluxes.- 6.30 1D-Mass Flux of van Leer Type.- 6.31 1D-Momentum Flux of van Leer Type.- 6.32 1D-Energy Flux of van Leer Type.- 6.33 3D-Mass Flux.- 6.34 3D-Momentum Fluxes.- 6.35 3D-Energy Flux.- 6.36 The Use of the Conservative Riemann Solver for Splitting Flux Differences.- 6.37 The Use of the Conservative Riemann Solver for Splitting Flux Differences by Projection.- 6.38 Evaluation of Eigenvalues by Projection.- 6.39 Non-Oscillating Interpolation: Introduction.- 6.40 Five-Point Schemes.- 6.41 First-Order Upwind Scheme.- 6.42 Second-Order Upwind Scheme.- 6.43 Third-Order Biased Upwind Scheme.- 6.44 Fourth-Order Centered Scheme.- 6.45 The von Neumann Stability Test for Upwind Schemes.- 6.46 Criticism on the von Neumann Stability Test for Upwind Schemes.- 6.47 Extremum Principles for Upwind Schemes.- 6.48 Foundation of Flux Limiting.- 6.49 Flux Limiting by Sensing Functions.- 6.50 Flux Limiting by Biased Differences.- 6.51 Flux Limiting with Minimum Dispersion.- 6.52 Limiters.- 6.53 Seven-Point Schemes.- 6.54 Truth Functions.- 6.55 Non-oscillating Interpolation and Riemann Solvers.- 6.56 Riemann Solvers and Strong Shocks.- 6.57 Riemann Solvers and Boundary Conditions.- 6.58 Other Updates.- 6.59 Lax-Wendroff (L-W) Type Updates.- 6.60 Implicit Updates.- 6.61 Implicit Formulation.- 6.62 The Split Matrix.- 6.63 The Homogeneous Implicit Solution.- 6.64 Matrix Conditioning.- 6.65 References.- VII Convergence to Steady State.- 7.1 Introduction.- 7.2 Mathematical Understanding of Convergence.- 7.2.1 The Continuous Problem.- 7.2.2 Linear Semi-Discrete Problem.- 7.2.3 Linearized Euler Equations.- 7.2.4 Effect of Discrete Space Operator.- 7.3 Multi-Grid Scheme.- 7.4 Enthalpy Damping.- 7.5 Residual Averaging.- 7.6 Mesh Sequencing.- 7.7 References.- VIII A Note on the Use of Supercomputers.- 8.1 Supercomputers as Driver of Computational Fluid Dynamics.- 8.2 Future Developments in Supercomputing: Parallel Processing.- 8.3 References.- IX Coupling of Euler Solutions to Viscous Models.- 9.1 Diffusive Transport Effects in Fluid Flows.- 9.2 Treatment of Weak Interaction Flow Problems.- 9.3 Treatment of Strong Interaction Flow Problems.- 9.4 References.- X Modelling of Vortex Flows: Vorticity in Euler Solutions.- 10.1 Boundary Layers, Wakes and Vortices in their Inviscid Limit.- 10.2 The Lifting Wing as Inviscid Computation Problem.- 10.3 The Structure of the Wake of a Lifting Wing.- 10.4 Vorticity Creation and Entropy Rise in Euler Solutions for Lifting Wings.- 10.5 The State of the Art: a Critical Evaluation.- 10.6 A Note on the Solution of the Navier-Stokes Equations.- 10.7 References.- XI Methods in Practical Applications.- 11.1 Near-Incompressible Flow.- 11.1.1 Transverse Circular Cylinder.- 11.1.2 Some Numerical Experiments on Transverse Cylinders.- 11.1.3 Airfoil with Lift.- 11.1.4 Vortex Flow Over Sharp-Edged Delta Wing.- 11.1.5 Flow Through a Francis Water Turbine.- 11.1.6 Flow Past an Automobile.- 11.2 Subsonic/Transonic Flow.- 11.2.1 Comparison of Different Riemann Solvers.- 11.2.2 Flow Around Airfoils.- 11.2.3 Vortex Flow Over Sharp-Edged Delta Wings.- 11.2.4 Vortex Flow Over Round-Edged Delta Wings.- 11.2.5 Analysis of Flow Around a Project Wing.- 11.2.6 Flow Through Ducts.- 11.3 Supersonic/Hypersonic Flow.- 11.3.1 Leeside Flow Using Centered Scheme.- 11.3.2 Computation of Leeside Flow Using an Upwind Scheme.- 11.3.3 Supersonic Flow Around Delta Wings.- 11.4 Flow Past Complex Configuratons.- 11.4.1 Generic Fighter Configuration at Transonic and Supersonic Speed.- 11.4.2 Flow past Hypersonic Generic Aircraft.- 11.4.3 Equilibrium Real-Gas Solution for a Reentry Configuration.- 11.5 Coupling with Viscous Models.- 11.6 A Note on Unsteady Applications.- 11.7 References.- XII Future Prospects.- 12.1 General Considerations.- 12.2 Beyond Dimensional Splitting.- 12.3 Finite Element Formulations.- 12.4 Geometric Complexity.- 12.5 Interdisciplinary Problems.- 12.6 References.- XIII List of Symbols.- XIV Index of Authors.- XV Subject Index.