Numerical Methods for Unsteady Compressible Flow Problems

✍ Scribed by Philipp Birken

• Publisher: CRC Press
• Tongue: English
• Leaves: 242
• Series: Numerical Analysis and Scientific Computing
• Category: Library

✦ Synopsis

Numerical Methods for Unsteady Compressible Flow Problems is written to give both mathematicians and engineers an overview of the state of the art in the field, as well as of new developments. The focus is on methods for the compressible Navier-Stokes equations, the solutions of which can exhibit shocks, boundary layers and turbulence. The idea of the text is to explain the important ideas to the reader, while giving enough detail and pointers to literature to facilitate implementation of methods and application of concepts.

The book covers high order methods in space, such as Discontinuous Galerkin methods, and high order methods in time, in particular implicit ones. A large part of the text is reserved to discuss iterative methods for the arising large nonlinear and linear equation systems. Ample space is given to both state-of-the-art multigrid and preconditioned Newton-Krylov schemes.

Features

• Applications to aerospace, high-speed vehicles, heat transfer, and more besides

• Suitable as a textbook for graduate-level courses in CFD, or as a reference for practitioners in the field

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface and acknowledgments
1. Thermal Fluid Structure Interaction
1.1. The method of lines
1.2. Hardware
1.3. Notation
1.4. Outline
2. The governing equations
2.1. The Navier-Stokes equations
2.1.1. Basic form of conservation laws
2.1.2. Conservation of mass
2.1.3. Conservation of momentum
2.1.4. Conservation of energy
2.1.5. Equation of state and Sutherland law
2.2. Nondimensionalization
2.3. Source terms
2.4. Simplifications of the Navier-Stokes equations
2.5. The Euler equations
2.5.1. Properties of the Euler equations
2.6. Solution theory
2.7. Boundary and initial conditions
2.7.1. Initial conditions
2.7.2. Fixed wall conditions
2.7.3. Moving walls
2.7.4. Periodic boundaries
2.7.5. Farfield boundaries
2.8. Boundary layers
2.9. Laminar and turbulent flows
2.9.1. Turbulence models
2.9.1.1. RANS equations
2.9.1.2. Large Eddy Simulation
2.9.1.3. Detached Eddy Simulation
3. The space discretization
3.1. Structured and unstructured grids
3.2. Finite volume methods
3.3. The line integrals and numerical flux functions
3.3.1. Discretization of the inviscid fluxes
3.3.1.1. HLLC
3.3.1.2. AUSMDV
3.3.2. Low Mach numbers
3.3.3. Discretization of the viscous fluxes
3.4. Convergence theory for finite volume methods
3.4.1. Hyperbolic conservation laws
3.4.2. Parabolic conservation laws
3.5. Source terms
3.6. Finite volume methods of higher order
3.6.1. Convergence theory for higher-order finite volume schemes
3.6.2. Linear reconstruction
3.6.3. Limiters
3.7. Discontinuous Galerkin methods
3.7.1. Polymorphic modal-nodal scheme
3.7.2. DG Spectral Element Method
3.7.3. Discretization of the viscous fluxes
3.8. Convergence theory for DG methods
3.9. Boundary conditions
3.9.1. Implementation
3.9.2. Stability and the SBP-SAT technique
3.9.3. Fixed wall
3.9.4. Inflow and outflow boundaries
3.9.5. Periodic boundaries
3.10. Spatial adaptation
4. Time integration schemes
4.1. Order of convergence and order of consistency
4.2. Stability
4.2.1. The linear test equation, A- and L-stability
4.2.2. TVD stability and SSP methods
4.2.3. The CFL condition, von Neumann stability analysis and related topics
4.3. Stiff problems
4.4. Backward differentiation formulas
4.5. Runge-Kutta methods
4.5.1. Explicit Runge-Kutta methods
4.5.2. Fully implicit RK methods
4.5.3. DIRK methods
4.5.4. Additive Runge-Kutta methods
4.6. Rosenbrock-type methods
4.6.1. Exponential integrators
4.7. Adaptive time step size selection
4.8. Operator splittings
4.9. Alternatives to the method of lines
4.9.1. Space-time methods
4.9.2. Local time stepping Predictor-Corrector-DG
4.10. Parallelization in time
5. Solving equation systems
5.1. The nonlinear systems
5.2. The linear systems
5.3. Rate of convergence and error
5.4. Termination criteria
5.5. Fixed point methods
5.5.1. Stationary linear methods
5.5.2. Nonlinear variants of stationary methods
5.6. Multigrid methods
5.6.1. Multigrid for linear problems
5.6.2. Full Approximation Schemes
5.6.3. Smoothers
5.6.3.1. Pseudo time iterations and dual time stepping
5.6.3.2. Alternatives
5.6.4. Residual averaging and smoothed aggregation
5.6.5. Multi-p methods
5.6.6. State of the art
5.6.7. Analysis and construction
5.6.7.1. Smoothing factors
5.6.7.2. Example
5.6.7.3. Optimizing the spectral radius
5.6.7.4. Example
5.6.7.5. Numerical results
5.6.7.6. Local Fourier analysis
5.6.7.7. Generalized locally Toeplitz sequences
5.7. Newton's method
5.7.1. Simplified Newton's method
5.7.2. Methods of Newton type
5.7.3. Inexact Newton methods
5.7.4. Choice of initial guess
5.7.5. Globally convergent Newton methods
5.7.6. Computation and storage of the Jacobian
5.8. Krylov subspace methods
5.8.1. GMRES and related methods
5.8.1.1. GCR
5.8.2. BiCGSTAB
5.9. Jacobian-free Newton-Krylov methods
5.10. Comparison of GMRES and BiCGSTAB
5.11. Comparison of variants of Newton's method
6. Preconditioning linear systems
6.1. Preconditioning for JFNK schemes
6.2. Specific preconditioners
6.2.1. Block preconditioners
6.2.2. Stationary linear methods
6.2.3. ROBO-SGS
6.2.4. ILU preconditioning
6.2.5. Multilevel preconditioners
6.2.6. Nonlinear preconditioners
6.2.7. Other preconditioners
6.2.8. Comparison of preconditioners
6.3. Preconditioning in parallel
6.4. Sequences of linear systems
6.4.1. Freezing and recomputing
6.4.2. Triangular preconditioner updates
6.4.3. Numerical results
6.5. Discretization for the preconditioner
7. The final schemes
7.1. DIRK scheme
7.2. Rosenbrock scheme
7.3. Parallelization
7.4. Efficiency of nite volume schemes
7.5. Efficiency of Discontinuous Galerkin schemes
7.5.1. Polymorphic modal-nodal DG
7.5.2. DG-SEM
8. Thermal Fluid Structure Interaction
8.1. Gas quenching
8.2. The mathematical model
8.3. Space discretization
8.4. Coupled time integration
8.5. Dirichlet-Neumann iteration
8.5.1. Extrapolation from time integration
8.6. Alternative solvers
8.7. Numerical Results
8.7.1. Cooling of a flanged shaft
A. Test problems
A.1. Shu-vortex
A.2. Supersonic flow around a cylinder
A.3. Wind turbine
A.4. Vortex shedding behind a sphere
B. Coefficients of time integration methods
Bibliography
Index