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Computational Gasdynamics

Book Description

Numerical methods are indispensable tools in the analysis of complex fluid flows. This book focuses on computational techniques for high-speed gas flows, especially gas flows containing shocks and other steep gradients. The book decomposes complicated numerical methods into simple modular parts, showing how each part fits and how each method relates to or differs from others. The text begins with a review of gasdynamics and computational techniques. Next come basic principles of computational gasdynamics. The last two parts cover basic techniques and advanced techniques. Senior and graduate level students, especially in aerospace engineering, as well as researchers and practising engineers, will find a wealth of invaluable information on high-speed gas flows in this text.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. Chapter 1. Introduction
  8. Part I: Gasdynamics Review
    1. Chapter 2. Governing Equations of Gasdynamics
      1. 2.0 Introduction
      2. 2.1 The Integral Form of the Euler Equations
        1. 2.1.1 Conservation of Mass
        2. 2.1.2 Conservation of Momentum
        3. 2.1.3 Conservation of Energy
        4. 2.1.4 Equations of State for a Perfect Gas
        5. 2.1.5 Entropy and the Second Law of Thermodynamics
        6. 2.1.6 Vector Notation
      3. 2.2 The Conservation Form of the Euler Equations
        1. 2.2.1 Vector and Vector–Matrix Notation
        2. 2.2.2 Rankine–Hugoniot Relations
      4. 2.3 The Primitive Variable Form of the Euler Equations
        1. 2.3.1 Vector–Matrix Notation
      5. 2.4 Other Forms of the Euler Equations
    2. Chapter 3. Waves
      1. 3.0 Introduction
      2. 3.1 Waves for a Scalar Model Problem
      3. 3.2 Waves for a Vector Model Problem
      4. 3.3 The Characteristic Form of the Euler Equations
        1. 3.3.1 Examples
        2. 3.3.2 Physical Interpretation
      5. 3.4 Simple Waves
      6. 3.5 Expansion Waves
      7. 3.6 Compression Waves and Shock Waves
      8. 3.7 Contact Discontinuities
    3. Chapter 4. Scalar Conservation Laws
      1. 4.0 Introduction
      2. 4.1 Integral Form
      3. 4.2 Conservation Form
      4. 4.3 Characteristic Form
      5. 4.4 Expansion Waves
      6. 4.5 Compression Waves and Shock Waves
      7. 4.6 Contact Discontinuities
      8. 4.7 Linear Advection Equation
      9. 4.8 Burgers’ Equation
      10. 4.9 Nonconvex Scalar Conservation Laws
      11. 4.10 Entropy Conditions
      12. 4.11 Waveform Preservation, Destruction, and Creation
    4. Chapter 5. The Riemann Problem
      1. 5.0 Introduction
      2. 5.1 The Riemann Problem for the Euler Equations
      3. 5.2 The Riemann Problem for Linear Systems of Equations
      4. 5.3 Three-Wave Linear Approximations – Roe’s Approximate Riemann Solver for the Euler Equations
        1. 5.3.1 Secant Line and Secant Plane Approximations
        2. 5.3.2 Roe Averages
        3. 5.3.3 Algorithm
        4. 5.3.4 Performance
      5. 5.4 One-Wave Linear Approximations
      6. 5.5 Other Approximate Riemann Solvers
      7. 5.6 The Riemann Problem for Scalar Conservation Laws
  9. Part II: Computational Review
    1. Chapter 6. Numerical Error
      1. 6.0 Introduction
      2. 6.1 Norms and Inner Products
      3. 6.2 Round-Off Error
      4. 6.3 Discretization Error
    2. Chapter 7. Orthogonal Functions
      1. 7.0 Introduction
      2. 7.1 Functions as Vectors
      3. 7.2 Legendre Polynomial Series
      4. 7.3 Chebyshev Polynomial Series
      5. 7.4 Fourier Series
    3. Chapter 8. Interpolation
      1. 8.0 Introduction
      2. 8.1 Polynomial Interpolation
        1. 8.1.1 Lagrange Form
        2. 8.1.2 Newton Form
        3. 8.1.3 Taylor Series Form
        4. 8.1.4 Accuracy of Polynomial Interpolation
        5. 8.1.5 Summary of Polynomial Interpolation
      3. 8.2 Trigonometric Interpolation and the Nyquist Sampling Theorem
    4. Chapter 9. Piecewise-Polynomial Reconstruction
      1. 9.0 Introduction
      2. 9.1 Piecewise Interpolation-Polynomial Reconstructions
      3. 9.2 Averaged Interpolation-Polynomial Reconstructions
      4. 9.3 Reconstruction via the Primitive Function
      5. 9.4 Reconstructions with Subcell Resolution
    5. Chapter 10. Numerical Calculus
      1. 10.0 Introduction
      2. 10.1 Numerical Differentiation
        1. 10.1.1 Linear Approximations
        2. 10.1.2 Quadratic Approximations
      3. 10.2 Numerical Integration
        1. 10.2.1 Constant Approximations
        2. 10.2.2 Linear Approximations
      4. 10.3 Runge–Kutta Methods for Solving Ordinary Differential Equations
  10. Part III: Basic Principles of Computational Gasdynamics
    1. Chapter 11. Conservation and Other Basic Principles
      1. 11.0 Introduction
      2. 11.1 Conservative Finite-Volume Methods
        1. 11.1.1 Forward-Time Methods
        2. 11.1.2 Backward-Time Methods
        3. 11.1.3 Central-Time Methods
      3. 11.2 Conservative Finite-Difference Methods
        1. 11.2.1 The Method of Lines
        2. 11.2.2 Formal, Local, and Global Order of Accuracy
      4. 11.3 Transformation to Conservation Form
    2. Chapter 12. The CFL Condition
      1. 12.0 Introduction
      2. 12.1 Scalar Conservation Laws
      3. 12.2 The Euler Equations
    3. Chapter 13. Upwind and Adaptive Stencils
      1. 13.0 Introduction
      2. 13.1 Scalar Conservation Laws
      3. 13.2 The Euler Equations
      4. 13.3 Introduction to Flux Averaging
      5. 13.4 Introduction to Flux Splitting
        1. 13.4.1 Flux Split Form
        2. 13.4.2 Introduction to Flux Reconstruction
      6. 13.5 Introduction to Wave Speed Splitting
        1. 13.5.1 Wave Speed Split Form
      7. 13.6 Introduction to Reconstruction–Evolution Methods
    4. Chapter 14. Artificial Viscosity
      1. 14.0 Introduction
      2. 14.1 Physical Viscosity
      3. 14.2 Artificial Viscosity Form
    5. Chapter 15. Linear Stability
      1. 15.0 Introduction
      2. 15.1 von Neumann Analysis
      3. 15.2 Alternatives to von Neumann Analysis
      4. 15.3 Modified Equations
      5. 15.4 Convergence and Linear Stability
    6. Chapter 16. Nonlinear Stability
      1. 16.0 Introduction
      2. 16.1 Monotonicity Preservation
      3. 16.2 Total Variation Diminishing (TVD)
      4. 16.3 Range Diminishing
      5. 16.4 Positivity
      6. 16.5 Upwind Range Condition
      7. 16.6 Total Variation Bounded (TVB)
      8. 16.7 Essentially Nonoscillatory (ENO)
      9. 16.8 Contraction
      10. 16.9 Monotone Methods
      11. 16.10 A Summary of Nonlinear Stability Conditions
      12. 16.11 Stability and Convergence
      13. 16.12 The Euler Equations
      14. 16.13 Proofs
  11. Part IV: Basic Methods of Computational Gasdynamics
    1. Chapter 17. Basic Numerical Methods for Scalar Conservation Laws
      1. 17.0 Introduction
      2. 17.1 Lax–Friedrichs Method
      3. 17.2 Lax–Wendroff Method
      4. 17.3 First-Order Upwind Methods
        1. 17.3.1 Godunov’s First-Order Upwind Method
        2. 17.3.2 Roe’s First-Order Upwind Method
        3. 17.3.3 Harten’s First-Order Upwind Method
      5. 17.4 Beam–Warming Second-Order Upwind Method
      6. 17.5 Fromm’s Method
    2. Chapter 18. Basic Numerical Methods for the Euler Equations
      1. 18.0 Introduction
      2. 18.1 Flux Approach
        1. 18.1.1 Lax–Friedrichs Method
        2. 18.1.2 Lax–Wendroff Methods
      3. 18.2 Wave Approach I: Flux Vector Splitting
        1. 18.2.1 Steger–Warming Flux Vector Splitting
        2. 18.2.2 Van Leer Flux Vector Splitting
        3. 18.2.3 Liou–Steffen Flux Vector Splitting
        4. 18.2.4 Zha–Bilgen Flux Vector Splitting
        5. 18.2.5 First-Order Upwind Methods
        6. 18.2.6 Beam–Warming Second-Order Upwind Method
      4. 18.3 Wave Approach II: Reconstruction–Evolution
        1. 18.3.1 Godunov’s First-Order Upwind Method
        2. 18.3.2 Roe’s First-Order Upwind Method
        3. 18.3.3 Harten’s First-Order Upwind Method
        4. 18.3.4 First-Order Upwind Method Based on One-Wave Solver
        5. 18.3.5 Second- and Higher-Order Accurate Methods
    3. Chapter 19. Boundary Treatments
      1. 19.0 Introduction
      2. 19.1 Stability
      3. 19.2 Solid Boundaries
      4. 19.3 Far-Field Boundaries
  12. Part V: Advanced Methods of Computational Gasdynamics
    1. Chapter 20. Flux Averaging I: Flux-Limited Methods
      1. 20.0 Introduction
      2. 20.1 Van Leer’s Flux-Limited Method
      3. 20.2 Sweby’s Flux-Limited Method (TVD)
        1. 20.2.1 The Linear Advection Equation with a > 0
        2. 20.2.2 The Linear Advection Equation with a < 0
        3. 20.2.3 Nonlinear Scalar Conservation Laws with a(u) > 0
        4. 20.2.4 Nonlinear Scalar Conservation Laws with a(u) < 0
        5. 20.2.5 Nonlinear Scalar Conservation Laws at Sonic Points
        6. 20.2.6 The Euler Equations
      4. 20.3 Chakravarthy–Osher Flux-Limited Methods (TVD)
        1. 20.3.1 A Second-Order Accurate Method: A Semidiscrete Version of Sweby’s Flux-Limited Method
        2. 20.3.2 Another Second-Order Accurate Method
        3. 20.3.3 Second- and Third-Order Accurate Methods
        4. 20.3.4 Higher-Order Accurate Methods
      5. 20.4 Davis–Roe Flux-Limited Method (TVD)
        1. 20.4.1 Scalar Conservation Laws
        2. 20.4.2 The Euler Equations
      6. 20.5 Yee–Roe Flux-Limited Method (TVD)
        1. 20.5.1 Scalar Conservation Laws
        2. 20.5.2 The Euler Equations
    2. Chapter 21. Flux Averaging II: Flux-Corrected Methods
      1. 21.0 Introduction
      2. 21.1 Boris–Book Flux-Corrected Method (FCT)
      3. 21.2 Zalesak’s Flux-Corrected Methods (FCT)
      4. 21.3 Harten’s Flux-Corrected Method (TVD)
        1. 21.3.1 Scalar Conservation Laws
        2. 21.3.2 The Euler Equations
      5. 21.4 Shu–Osher Methods (ENO)
    3. Chapter 22. Flux Averaging III: Self-Adjusting Hybrid Methods
      1. 22.0 Introduction
      2. 22.1 Harten–Zwas Self-Adjusting Hybrid Method
      3. 22.2 Harten’s Self-Adjusting Hybrid Method
      4. 22.3 Jameson’s Self-Adjusting Hybrid Method
        1. 22.3.1 Scalar Conservation Laws
        2. 22.3.2 The Euler Equations
    4. Chapter 23. Solution Averaging: Reconstruction–Evolution Methods
      1. 23.0 Introduction
      2. 23.1 Van Leer’s Reconstruction–Evolution Method (MUSCL)
        1. 23.1.1 Linear Advection Equation
        2. 23.1.2 The Lagrange Equations
      3. 23.2 Colella–Woodward Reconstruction–Evolution Method (PPM)
      4. 23.3 Anderson–Thomas–Van Leer Reconstruction–Evolution Methods (TVD/MUSCL): Finite-Volume Versions of the Chakravarthy–Osher Flux-Corrected Methods
        1. 23.3.1 A Second-Order Accurate Method
        2. 23.3.2 Second- and Third-Order Accurate Methods
      5. 23.4 Harten–Osher Reconstruction–Evolution Method (UNO)
        1. 23.4.1 The Linear Advection Equation
        2. 23.4.2 Nonlinear Scalar Conservation Laws
        3. 23.4.3 The Euler Equations
      6. 23.5 Harten–Engquist–Osher–Chakravarthy Reconstruction–Evolution Methods (ENO)
        1. 23.5.1 Second-Order Accurate Temporal Evolution for Scalar Conservation Laws
        2. 23.5.2 Third-Order Accurate Temporal Evolution for Scalar Conservation Laws
        3. 23.5.3 Second-Order Accurate Temporal Evolution for the Euler Equations
    5. Chapter 24. A Brief Introduction to Multidimensions
      1. 24.0 Introduction
      2. 24.1 Governing Equations
      3. 24.2 Waves
      4. 24.3 Conservation and Other Numerical Principles
  13. Index