Generalized Riemann Problems in Computational Fluid Dynamics

Numerical simulation of compressible, inviscid time-dependent flow is a major branch of computational fluid dynamics. Its primary goal is to obtain accurate representation of the time evolution of complex flow patterns, involving interactions of shocks, interfaces, and rarefaction waves. The General...

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Bibliographic Details
Main Authors: Ben-Artzi, Matania, Falcovitz, Joseph
Format: eBook
Language:English
Published: New York Cambridge University Press 10.04.2003
Edition:1
Series:Cambridge Monographs on Applied and Computational Mathematics
Subjects:
Fluid dynamics
Riemann-Hilbert problems
ISBN:9780521173278, 9780521772969, 0521173272, 0521772966
Online Access:Get full text
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Table of Contents:
  • Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- List of Figures -- Preface -- 1 Introduction -- Part I Basic Theory -- 2 Scalar Conservation Laws -- 2.1 Theoretical Background -- Weak Solutions and Jump Conditions -- Shocks, Rarefaction Waves, and Entropy -- The Riemann Problem -- 2.2 Basic Concepts of Numerical Approximation -- Convergence -- 3 The GRP Method for Scalar Conservation Laws -- 3.1 From Godunov to the GRP Method -- 3.2 1-D Sample Problems -- 3.2.1 The Linear Conservation Law -- First-Order Schemes -- Second-Order Schemes -- 3.2.2 The Burgers Nonlinear Conservation Law -- First-Order Computation -- Second-Order Computation -- 3.3 2-D Sample Problems -- The Operator-Splitting Method -- The Linear Conservation Law -- The Nonlinear Burgers Equation -- Case A -- Case B -- Case C -- Case D -- The Guckenheimer Equation -- 4 Systems of Conservation Laws -- 4.1 Nonlinear Hyperbolic Systems in One Space Dimension -- Characteristic Curves and Centered Rarefaction Waves -- Weak Solutions and Jump Discontinuities -- Entropy Conditions, Shock Waves, and Contact Discontinuities -- The Riemann Problem -- 4.2 Euler Equations of Quasi-1-D, Compressible, Inviscid Flow -- The Flow Equations -- Eigenvalues and Characteristic Equations -- Isentropic Flow -- Weak Solutions and Jump Conditions -- Lagrangian Coordinates -- Shock Waves - Detailed Study of the Jump Condition -- Centered Rarefaction Waves -- The Riemann Problem (RP) for Planar Flows -- Perfect (Gamma-Law) Gas -- 5 The Generalized Riemann Problem (GRP) for Compressible Fluid Dynamics -- 5.1 The GRP for Quasi-1-D, Compressible, Inviscid Flow -- Structure of the Solution to the GRP -- The Linear GRP in Lagrangian Coordinates - Setup and Statement of the Main Theorem -- The Acoustic Case -- Resolution of a CRW in the Lagrangian Framework
  • Explicit Formulas for the (Lagrangian) GRP in the Gamma-Law Case -- Concluding the Treatment of the CRW -- Time Derivatives of p, u on the Interface - Proof of the Main Theorem -- Conclusion of the Linear GRP in the Lagrangian Case -- The Linear GRP in the Eulerian Framework -- 5.2 The GRP Numerical Method for Quasi-1-D, Compressible, Inviscid Flow -- The Godunov Scheme -- The Basic GRP Scheme -- The E and L Schemes, Intermediate Schemes, and MUSCL -- Updating the Slopes -- Concluding the GRP Algorithm -- 6 Analytical and Numerical Treatment of Fluid Dynamical Problems -- 6.1 The Shock Tube Problem -- 6.2 Wave Interactions -- 6.2.1 Shock-Contact Interaction -- 6.2.2 Shock-Shock Interaction -- 6.2.3 Shock-CRW Interaction -- 6.2.4 CRW-Contact Interaction -- Approximate Analysis of the Interaction -- Numerical (GRP) Solution -- 6.3 Spherically Converging Flow of Cold Gas -- 6.4 The Flow Induced by an Expanding Sphere -- 6.5 Converging-Diverging Nozzle Flow -- Nozzle Geometry and Steady Flow -- The Finite-Difference Solution -- Part II Numerical Implementation -- 7 From the GRP Algorithm to Scientific Computing -- 7.1 General Discussion -- 7.2 Strang's Operator-Splitting Method -- 7.3 Two-Dimensional Flow in Cartesian Coordinates -- The Linear GRP for a Planar System with Advection -- The Split Scheme for (7.16) -- The Split Scheme and Conservation Form -- 8 Geometric Extensions -- 8.1 Grids That Move in Time -- 8.2 Singularity Tracking -- 8.3 Moving Boundary Tracking (MBT) -- 8.3.1 Basic Setup -- 8.3.2 Survey of the Full MBT Algorithm -- 8.3.3 An Example: Shock Lifting of an Elliptic Disk -- 9 A Physical Extension: Reacting Flow -- 9.1 The Equations of Compressible Reacting Flow -- The Characteristic Relations -- Discontinuities and Centered Rarefaction Waves -- 9.2 The Chapman-Jouguet (C-J) Model -- 9.3 The Z-N-D (Zeldovich-von Neumann-Döring) Solution
  • 9.4 The Linear GRP for the Reacting-Flow System -- The Associated Riemann Problem -- Structure of the Solution to the Linear GRP-The Main Theorem -- The Acoustic Approximation -- Resolution of the Centered Rarefaction Wave -- Conclusion of the Linear GRP -- The Gamma-Law Case -- 9.5 The GRP Scheme for Reacting Flow -- 10 Wave Interaction in a Duct - A Comparative Study -- Appendix A Entropy Conditions for Scalar Conservation Laws -- Appendix B Convergence of the Godunov Scheme -- Appendix C Riemann Solver for a Gamma-Law Gas -- Appendix D The MUSCL Scheme -- Bibliography -- Glossary -- Flow Variables and Thermodynamic Quantities (Section 4.2) -- Coordinates -- Riemann and Generalized Riemann Problem -- Functional Spaces -- Index