The last decade has seen a dramatic increase of our abilities to solve numerically the governing equations of fluid mechanics. In design aerodynamics the classical potential-flow methods have been complemented by higher modelling-level methods. Euler solvers, and for special purposes, already Navier-Stokes solvers are in use. The authors of this book have been working on the solution of the Euler equations for quite some time. While the first two of us have worked mainly on algorithmic problems, the third has been concerned off and on with modelling and application problems of Euler methods.…mehr
The last decade has seen a dramatic increase of our abilities to solve numerically the governing equations of fluid mechanics. In design aerodynamics the classical potential-flow methods have been complemented by higher modelling-level methods. Euler solvers, and for special purposes, already Navier-Stokes solvers are in use. The authors of this book have been working on the solution of the Euler equations for quite some time. While the first two of us have worked mainly on algorithmic problems, the third has been concerned off and on with modelling and application problems of Euler methods. When we started to write this book we decided to put our own work at the center of it. This was done because we thought, and we leave this to the reader to decide, that our work has attained over the years enough substance in order to justify a book. The problem which we soon faced, was that the field still is moving at a fast pace, for instance because hyper sonic computation problems becamemore and more important.
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
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
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