Mathematical and Computational Methods for Compressible Flow

, by Feistauer, Miloslav; Felcman, Jiri; Straskraba, Ivan

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9780198505884 | 0198505884
Cover: Hardcover
Copyright: 9/11/2003

Rent

(Recommended)

$172.22

Term

Due

Price

Semester

Dec 12

$181.28

Quarter

Aug 13

$172.22

Short Term

Jul 30

$163.15

*This item is part of an exclusive publisher rental program and requires an additional convenience fee. This fee will be reflected in the shopping bag.
Buy New

Usually Ships in 3-5 Business Days

$232.09

This book is concerned with mathematical and numerical methods for compressible flow. It aims to provide the reader with a sufficiently detailed and extensive, mathematically precise, but comprehensible guide, through a wide spectrum of mathematical and computational methods used in Computational Fluid Dynamics (CFD) for the numerical simulation of compressible flow. Up-to-date techniques applied in the numerical solution of inviscid as well as viscous compressible flow on unstructured meshes are explained, thus allowing the simulation of complex three-dimensional technically relevant problems. Among some of the methods addressed are finite volume methods using approximate Riemann solvers, finite element techniques, such as the streamline diffusion and the discontinuous Galerkin methods, and combined finite volume - finite element schemes. The book gives a complex insight into the numerics of compressible flow, covering the development of numerical schemes and their theoretical mathematical analysis, their verification on test problems and use in solving practical engineering problems. The book will be helpful to specialists coming into contact with CFD - pure and applied mathematicians, aerodynamists, engineers, physicists and natural scientists. It will also be suitable for advanced undergraduate, graduate and postgraduate students of mathematics and technical sciences.

Introduction

(4)

1 Fundamental Concepts and Equations

(52)

1.1 Some mathematical concepts and notation

(8)

1.1.1 Basic notation

(1)

1.1.2 Differential operators

(2)

1.1.3 Transformations of Cartesian coordinates

(1)

1.1.4 Hölder-continuous and Lipschitz-continuous functions

(1)

1.1.5 Symbols 'o' and 'O'

(1)

1.1.6 Measure and integral

(1)

1.1.7 Description of the boundary

(1)

1.1.8 Measure on the boundary of a domain

(1)

1.1.9 Green's theorem.

(1)

1.1.10 Lebesgue spaces

(2)

1.1.11 One auxiliary result

(1)

1.2 Governing equations and relations of gas dynamics

(26)

1.2.1 Description of the flow

(2)

1.2.2 The transport theorem

(3)

1.2.3 The continuity equation

(1)

1.2.4 The equations of motion

(2)

1.2.5 The equations of motion of general fluids

(1)

1.2.6 The law of conservation of the moment of momentum; symmetry of the stress tensor

(1)

1.2.7 The Navier-Stokes equations

(1)

1.2.8 Properties of the viscosity coefficients

(1)

1.2.9 The Reynolds number

(1)

1.2.10 Various forms of the Navier-Stokes equations

(1)

1.2.11 The energy equation

(2)

1.2.12 Thermodynamical relations

(1)

1.2.13 Entropy

(1)

1.2.14 The second law of thermodynamics

(1)

1.2.15 Dissipation form of the energy equation

(2)

1.2.16 Entropy form of the energy equation

(1)

1.2.17 Adiabatic flow

(1)

1.2.18 Barotropic flow

(1)

1.2.19 Complete system describing the flow of a heat conductive gas

(1)

1.2.20 Speed of sound; Mach number

(1)

1.2.21 Simplified models

(1)

1.2.22 Initial and boundary conditions

(2)

1.2.23 Dimensionless form of gas dynamics equations

(2)

1.3 Some advanced mathematical concepts and results

(8)

1.3.1 Spaces of continuous, Hölder-continuous and continuously differentiable functions

(1)

1.3.2 Distributions

(2)

1.3.3 Sobolev spaces

(3)

1.3.4 Functions with values in Banach spaces

(2)

1.4 Survey of concepts and results from functional analysis

(9)

1.4.1 Linear vector spaces

(1)

1.4.2 Normed linear space

(2)

1.4.3 Duals to Banach spaces, weak and weak-* topologies

(2)

1.4.4 Riesz representation theorem

(1)

1.4.5 Operators

(2)

1.4.6 Lax-Milgram lemma

(1)

1.4.7 Imbeddings

(1)

1.4.8 Solution of nonlinear operator equations

(1)

2 Basic facts from the theory of the Euler and Navier Stokes equations

(43)

2.1 Hyperbolic systems and the Euler equations

(1)

2.2 Existence of smooth solutions

(10)

2.2.1 Hyperbolic systems and characteristics

(2)

2.2.2 Formulation of the hyperbolic problem

(1)

2.2.3 Linear scalar equation

(1)

2.2.4 Solution of a linear system

(1)

2.2.5 Nonlinear scalar equation

(2)

2.2.6 Symmetric hyperbolic systems

(1)

2.2.7 Quasilinear system

(1)

2.2.8 Local existence for a quasilinear system

(1)

2.2.9 Local existence for equations of inviscid barotropic flow

(1)

2.3 Weak solutions

(24)

2.3.1 Blow up of classical solutions

(1)

2.3.2 Generalized formulation for systems of conservation laws

(3)

2.3.3 Examples of piecewise smooth weak solutions

(3)

2.3.4 Entropy condition

(3)

2.3.5 Entropy in fluid mechanics

(1)

2.3.6 Method of artificial viscosity

(1)

2.3.7 Existence and uniqueness of weak entropy solutions for scalar conservation laws

(1)

2.3.8 Riemann problem

(1)

2.3.9 Linear Riemann problem

(1)

2.3.10 Nonlinear Riemann problem

(5)

2.3.11 Existence result for the 2 x 2 Euler system of barotropic flow

(3)

2.3.12 Global existence results for general 1D systems

(1)

2.4 Nonstationary Navier-Stokes equations of compressible flow

(4)

2.4.1 Results for the full system of compressible Navier Stokes equations

(2)

2.4.2 Results for equations of barotropic flow

(2)

2.5 Existence results for stationary compressible Navier Stokes equations

(4)

2.5.1 Existence of a regular solution for small data

(1)

2.5.2 Existence of weak solutions for barotropic flow

(2)

3 Finite difference and finite volume methods for nonlinear hyperbolic systems and the Euler equations

100

(216)

3.1 Further properties of the Euler equations

101

(40)

3.1.1 The Euler equations

101

(3)

3.1.2 Diagonalization of the Jacobi matrix

104

(3)

3.1.3 Rotational invariance of the Euler equations in 3D

107

(1)

3.1.4 Hyperbolicity of the Euler equations in 3D

108

(1)

3.1.5 The 2D case

109

(3)

3.1.6 Solution of the Riemann problem for 1D Euler equations

112

(26)

3.1.7 Solution of the Riemann problem for the split 3D Euler equations

138

(3)

3.2 Numerical methods for hyperbolic systems with one space variable

141

(42)

3.2.1 Example of a nonconservative scheme

143

(1)

3.2.2 Semidiscretization in space

144

(1)

3.2.3 Space-time nonuniform grid

145

(1)

3.2.4 Qualitative properties of numerical schemes for conservation laws

145

(1)

3.2.5 Order of the scheme

146

(1)

3.2.6 Stability of the scheme

147

(2)

3.2.7 Stability in a nonlinear case

149

(1)

3.2.8 Lax-Friedrichs scheme

150

(1)

3.2.9 Stability of the Lax-Friedrichs scheme

150

(4)

3.2.10 Lax-Wendroff scheme

154

(1)

3.2.11 Stability of the Lax-Wendroff scheme

155

(1)

3.2.12 The Godunov method

156

(1)

3.2.13 Riemann solver for a scalar equation

156

(1)

3.2.14 Engquist-Ocher scheme

157

(1)

3.2.15 Riemann numerical flux for a linear system

158

(4)

3.2.16 Riemann solver for a nonlinear hyperbolic system

162

(1)

3.2.17 Flux vector splitting schemes for the Euter equations with one space variable

162

(2)

3.2.18 The Roe scheme

164

(8)

3.2.19 Consistency of the schemes

172

(1)

3.2.20 Linear stability of flux vector splitting schemes

172

(1)

3.2.21 Higher order schemes

172

(6)

3.2.22 ENO and WENO schemes

178

(5)

3.3 The finite volume method for the multidimensional Euler equations

183

(36)

3.3.1 Finite volume mesh

185

(10)

3.3.2 Derivation of a general finite volume scheme

195

(2)

3.3.3 Properties of the numerical flux

197

(1)

3.3.4 Construction of some numerical fluxes

197

(1)

3.3.5 Another construction of the multidimensional numerical flux

198

(1)

3.3.6 Boundary conditions

199

(4)

3.3.7 Stability of the finite volume schemes

203

(1)

3.3.8 FV schemes on 2D uniform rectangular meshes

204

(1)

3.3.9 Von Neumann linear stability

205

(1)

3.3.10 Application to the Lax-Friedrichs scheme

206

(5)

3.3.11 Extension of the stability conditions to the Euler equations

211

(1)

3.3.12 Convergence of the finite volume method

212

(1)

3.3.13 Entropy condition

213

(2)

3.3.14 Implicit FV methods

215

(4)

3.4 Ocher-Solomon scheme

219

(16)

3.4.1 Approximate Riemann solver

220

(1)

3.4.2 The Jacobi matrix of f 1

221

(1)

3.4.3 Riemann invariants

222

(1)

3.4.4 Integration of the eigenvectors rl

223

(1)

3.4.5 Integration path in the admissible state set

224

(1)

3.4.6 Ocher-Solomon approximate Riemann solver

225

(2)

3.4.7 Inlet/outlet boundary conditions

227

(5)

3.4.8 Solid wall boundary conditions

232

(1)

3.4.9 Ocher-Solomon scheme for the 2D Euter equations

233

(2)

3.5 Higher order finite volume schemes

235

(8)

3.5.1 General form of a 'second order' MUSCL-type FV scheme

235

(1)

3.5.2 Computation of the approximate gradient

236

(2)

3.5.3 Linear extrapolation

238

(1)

3.5.4 Limitation procedure

238

(1)

3.5.5 The choice of variables ul

239

(1)

3.5.6 Test of the accuracy of MUSCL-type schemes

240

(3)

3.6 Adaptive methods

243

(30)

3.6.1 Geometrical data structure

243

(1)

3.6.2 Adaptation algorithm

244

(1)

3.6.3 Mesh refinement

245

(4)

3.6.4 Adaptation techniques based on constant recovery

249

(8)

3.6.5 Adaptation techniques based on linear recovery

257

(2)

3.6.6 Data reinitialization for the mesh refinement

259

(1)

3.6.7 Anisotropic mesh adaptation

259

(13)

3.6.8 Data reinitialization for anisotropic mesh refinement

272

(1)

3.7 Examples of finite volume simulations

273

(43)

3.7.1 Shock-tube problem

275

(2)

3.7.2 GAMM channel

277

(7)

3.7.3 The 3D channel -10 % cylindrical bump

284

(6)

3.7.4 The 3D channel with 25 % spherical bump

290

(6)

3.7.5 Flow past NACA 0012 airfoil

296

(7)

3.7.6 Flow past a cascade of profiles

303

(8)

3.7.7 Scramjet

311

(5)

4 Finite element solution of compressible flow

316

(191)

4.1 Finite element method - elementary treatment

317

(50)

4.1.1 Elliptic problems

318

(2)

4.1.2 Finite element discretization of the elliptic problem

320

(5)

4.1.3 Convergence of the FEM

325

(5)

4.1.4 Several additional remarks

330

(1)

4.1.5 Parabolic problems

331

(2)

4.1.6 Finite element discretization of the parabolic problem

333

(2)

4.1.7 Stability and convergence

335

(6)

4.1.8 Mass lumping

341

(1)

4.1.9 Singularly perturbed and hyperbolic problems

341

(5)

4.1.10 Streamline diffusion method

346

(10)

4.1.11 Discontinuous Galerkin FEM for a linear hyperbolic problem

356

(5)

4.1.12 Adaptive mesh refinement and a posteriori error estimates

361

(6)

4.2 Finite element solution of viscous barotropic flow

367

(8)

4.2.1 Continuous problem

367

(2)

4.2.2 Discrete problem

369

(1)

4.2.3 Existence and uniqueness of the approximate solution

370

(2)

4.2.4 Discrete problem with nonhomogeneous boundary conditions

372

(3)

4.3 Finite element solution of a heat-conductive gas flow

375

(32)

4.3.1 Continuous problem

376

(7)

4.3.2 Symmetrization of the Euler and Navier-Stokes equations

383

(2)

4.3.3 Galerkin finite element space semidiscretization and its stabilization

385

(3)

4.3.4 Analysis of a linear model system with one space variable

388

(5)

4.3.5 Multidimensional problems

393

(10)

4.3.6 Time discretization

403

(4)

4.4 Combined finite volume-finite element method for viscous compressible flow

407

(27)

4.4.1 Computational grids

408

(1)

4.4.2 FV and FE spaces

409

(3)

4.4.3 Space semidiscretization of the problem

412

(3)

4.4.4 Time discretization

415

(1)

4.4.5 Realization of boundary conditions in the convective form bh

416

(1)

4.4.6 Operator splitting

416

(3)

4.4.7 Applications of the combined FV-FE methods

419

(9)

4.4.8 Computation of the drag and lift

428

(6)

4.5 Theory of the combined FV-FE method

434

(31)

4.5.1 Continuous problem

435

(4)

4.5.2 Semi-implicit method combining dual finite volumes with conforming finite elements

439

(4)

4.5.3 Convergence of the semi-implicit scheme

443

(9)

4.5.4 Explicit method combining conforming finite elements with dual finite volumes

452

(6)

4.5.5 Combination of barycentric finite volumes with nonconforming finite elements

458

(7)

4.6 Discontinuous Galerkin finite element method

465

(27)

4.6.1 DGFEM for a scalar conservation law with one space variable

466

(3)

4.6.2 Realization of the discrete problem

469

(1)

4.6.3 Investigation of the order of the DGFEM

470

(5)

4.6.4 DGFEM for multidimensional problems

475

(4)

4.6.5 An example of implementation

479

(2)

4.6.6 Problem with periodic boundary conditions

481

(1)

4.6.7 Limiting of the order of accuracy

482

(7)

4.6.8 Approximation of the boundary

489

(3)

4.6.9 DGFEM for convection-diffusion problems and viscous flow

492

(15)

4.6.10 Numerical examples

501

(6)

References

507

(22)

Index

529

Please wait while the item is added to your bag...

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Mathematical and Computational Methods for Compressible Flow

Summary

Author Biography

Table of Contents

Supplemental Materials

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Mathematical and Computational Methods for Compressible Flow

Summary

Author Biography

Table of Contents

Supplemental Materials

Digital License