Computational methods for inviscid and viscous flows

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Bibliographic Information

Computational methods for inviscid and viscous flows

Charles Hirsch

（Wiley series in numerical methods in engineering, . Numerical computation of internal and external flows ; v. 2）

Wiley, c1990

  • : pbk

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"A Wiley-Interscience publication."

Includes bibliographical references and index

Description and Table of Contents

Volume

ISBN 9780471923510

Description

This is the second of two volumes which together provide a comprehensive account of the numerical computation of internal and external flows. Volume 1 is concerned with descriptions of the methods involved. This second volume complements it by dealing with the application of computational methods to the problems of fluid dynamics. The coverage is comprehensive, including discussion of the equations used and the circumstances for which they are suitable, the numerical techniques used and their solution, and methods for producing computer programs. Volume 1 "Fundamentals of Numerical Discretization" was published in July 1988 and a student edition will be available in July 1989. The text will benefit graduate students and practising engineerings in the field of computational fluid dynamics. Considerable progress has been achieved in the field of Computational Fluid Dynamics (CFD), and it is reaching a stage where the basic methodology is becoming well established. The tremendous development of computer performance has resulted in the present capacity for solving two-dimensional Euler equations in seconds of computer time. Hence more attention can be given to the validation, accuracy and reliability of numerical flow simulations and to their extensions to complex industrial design and analysis applications. This text is directed at students at the graduate level as well as at scientists and engineers already engaged, or starting to be engaged, in Computational Fluid Dynamics.

Table of Contents

  • Part 1 The numerical computation of potential flows: the mathematical formulations of the potential flow model
  • the discretization of the subsonic potential equation
  • the computation of stationary transonic potential flows. Part 2 The numerical solution of the system of Euler equations: the mathematical formulation of the system of Euler equations
  • the Lax-Wendroff family of space centered schemes
  • the central schemes with independent time integration
  • the treatment of boundary conditions
  • upwind schemes for the Euler equations
  • discretization methods for the Navier-Stokes equations
  • second order upwind and high resolution schemes. Part 3 The numerical solution of the Navier-Stokes equations: properties of the system of Navier-Stokes equations.
Volume

: pbk ISBN 9780471924524

Description

Numerical Computation of Internal and External Flows Volume 2: Computational Methods for Inviscid and Viscous Flows C. Hirsch, Vrije Universiteit Brussel, Brussels, Belgium This second volume deals with the applications of computational methods to the problems of fluid dynamics. It complements the first volume to provide an excellent reference source in this vital and fast growing area. The author includes material on the numerical computation of potential flows and on the most up-to-date methods for Euler and Navier-Stokes equations. The coverage is comprehensive and includes detailed discussion of numerical techniques and algorithms, including implementation topics such as boundary conditions. Problems are given at the end of each chapter and there are comprehensive reference lists. Of increasing interest, the subject has powerful implications in such crucial fields as aeronautics and industrial fluid dynamics. Striking a balance between theory and application, the combined volumes will be useful for an increasing number of courses, as well as to practitioners and researchers in computational fluid dynamics. Contents Preface Nomenclature Part V: The Numerical Computation of Potential Flows Chapter 13 The Mathematical Formulations of the Potential Flow Model Chapter 14 The Discretization of the Subsonic Potential Equation Chapter 15 The Computation of Stationary Transonic Potential Flows Part VI: The Numerical Solution of the System of Euler Equations Chapter 16 The Mathematical Formulation of the System of Euler Equations Chapter 17 The Lax - Wendroff Family of Space-centred Schemes Chapter 18 The Central Schemes with Independent Time Integration Chapter 19 The Treatment of Boundary Conditions Chapter 20 Upwind Schemes for the Euler Equations Chapter 21 Second-order Upwind and High-resolution Schemes Part VII: The Numerical Solution of the Navier-Stokes Equations Chapter 22 The Properties of the System of Navier-Stokes Equations Chapter 23 Discretization Methods for the Navier-Stokes Equations Index

Table of Contents

Preface xv Nomenclature xix Part V: The Numerical Computation of Potential Flows 1 Chapter 13 The Mathematical Formulations of the Potential Flow Model 4 13.1 Conservative Form of the Potential Equation 4 13.2 The Non-conservative Form of the Isentropic Potential Flow Model 6 13.2.1 Small-perturbation potential equation 7 13.3 The Mathematical Properties of the Potential Equation 9 13.3.1 Unsteady potential flow 9 13.3.2 Steady potential flow 9 13.4 Boundary Conditions 14 13.4.1 Solid wall boundary condition 14 13.4.2 Far field conditions 15 13.4.3 Cascade and channel flows 17 13.4.4 Circulation and Kutta condition 18 13.5 Integral or Weak Formulation of the Potential Model 18 13.5.1 Bateman variational principle 19 13.5.2 Analysis of some properties of the variational integral 20 Chapter 14 The Discretization of the Subsonic Potential Equation 26 14.1 Finite Difference Formulation 27 14.1.1 Numerical estimation of the density 29 14.1.2 Curvilinear mesh 31 14.1.3 Consistency of the discretization of metric coefficients 34 14.1.4 Boundary conditions-curved solid wall 36 14.2 Finite Volume Formulation 38 14.2.1 Jameson and Caughey's finite volume method 39 14.3 Finite Element Formulation 42 14.3.1 The finite element-Galerkin method 43 14.3.2 Least squares or optimal control approach 47 14.4 Iteration Scheme for the Density 47 Chapter 15 The Computation of Stationary Transonic Potential Flows 57 15.1 The Treatment of the Supersonic Region: Artificial Viscosity-Density and Flux Upwinding 61 15.1.1 Artificial viscosity-non-conservative potential equation 62 15.1.2 Artificial viscosity-conservative potential equation 66 15.1.3 Artificial compressibility 67 15.1.4 Artificial flux or flux upwinding 70 15.2 Iteration Schemes for Potential Flow Computations 77 15.2.1 Line relaxation schemes 77 15.2.2 Guidelines for resolution of the discretized potential equation 81 15.2.3 The alternating direction implicit method-approximate factorization schemes 88 15.2.4 Other techniques-multigrid methods 98 15.3 Non-uniqueness and Non-isentropic Potential Models 104 15.3.1 Isentropic shocks 105 15.3.2 Non-uniqueness and breakdown of the transonic potential flow model 105 15.3.3 Non-isentropic potential models 112 15.4 Conclusions 117 Part VI: The Numerical Solution of the System of Euler Equations 125 Chapter 16 The Mathematical Formulation of the System of Euler Equations 132 16.1 The Conservative Formulation of the Euler Equations 132 16.1.1 Integral conservative formulation of the Euler equations 133 16.1.2 Differential conservative formulation 134 16.1.3 Cartesian system of coordinates 134 16.1.4 Discontinuities and Rankine-Hugoniot relations-entropy condition 135 16.2 The Quasi-linear Formulation of the Euler Equations 138 16.2.l The Jacobian matrices for conservative variables 138 16.2.2 The Jacobian matrices for primitive variables 145 16.2.3 Transformation matrices between conservative and non-conservative variables 147 16.3 The Characteristic Formulation of the Euler Equations-Eigenvalues and Compatibility Relations 150 16.3.1 General properties of characteristics 151 16.3.2 Diagonalization of the Jacobian matrices 153 16.3.3 Compatibility equations 154 16.4 Characteristic Variables and Eigenvalues for One-dimensional Flows 157 16.4.1 Eigenvalues and eigenvectors of Jacobian matrix 158 16.4.2 Characteristic variables 162 16.4.3 Characteristics in the xt-plane-shocks and contact discontinuities 168 16.4.4 Physical boundary conditions 171 16.4.5 Characteristics and simple wave solutions 173 16.5 Eigenvalues and Compatibility Relations in Multidimensional Flows 176 16.5.1 Jacobian eigenvalues and eigenvectors in primitive variables 177 16.5.2 Diagonalization of the conservative Jacobians 180 16.5.3 Mach cone and compatibility relations 184 16.5.4 Boundary conditions 191 16.6 Some Simple Exact Reference Solutions for One-dimensional Inviscid Flows 196 16.6.1 The linear wave equation 196 16.6.2 The inviscid Burgers equation 196 16.6.3 The shock tube problem or Riemann problem 204 16.6.4 The quasi-one-dimensional nozzle flow 211 Chapter 17 The Lax-Wendroff Family of Space-centred Schemes 224 17.1 The Space-centred Explicit Schemes of First Order 226 17.1.1 The one-dimensional Lax-Friedrichs scheme 226 17.1.2 The two-dimensional Lax-Friedrichs scheme 229 17.1.3 Corrected viscosity scheme 233 17.2 The Space-centred Explicit Schemes of Second Order 234 17.2.1 The basic one-dimensional Lax-Wendroff scheme 234 17.2.2 The two-step Lax-Wendroff schemes in one dimension 238 17.2.3 Lerat and Peyret's family of non-linear two-step Lax-Wendroff schemes 246 17.2.4 One-step Lax-Wendroff schemes in two dimensions 251 17.2.5 Two-step Lax-Wendroff schemes in two dimensions 258 17.3 The Concept of Artificial Dissipation or Artificial Viscosity 272 17.3.1 General form of artificial dissipation terms 273 17.3.2 Von Neumann-Richtmyer artificial viscosity 274 17.3.3 Higher-order artificial viscosities 279 17.4 Lerat's Implicit Schemes of Lax-Wendroff Type 283 17.4.1 Analysis for linear systems in one dimension 285 17.4.2 Construction of the family of schemes 288 17.4.3 Extension to non-linear systems in conservation form 292 17.4.4 Extension to multi-dimensional flows 296 17.5 Summary 296 Chapter 18 The Central Schemes with Independent Time Integration 307 18.1 The Central Second-order Implicit Schemes of Beam and Warming in One Dimension 309 18.1.1 The basic Beam and Warming schemes 310 18.1.2 Addition of artificial viscosity 315 18.2 The Multidimensional Implicit Beam and Warming Schemes 326 18.2.1 The diagonal variant of Pulliam and Chaussee 328 18.3 Jameson's Multistage Method 334 18.3.1 Time integration 334 18.3.2 Convergence acceleration to steady state 335 Chapter 19 The Treatment of Boundary Conditions 344 19.1 One-dimensional Boundary Treatment for Euler Equations 345 19.1.1 Characteristic boundary conditions 346 19.1.2 Compatibility relations 347 19.1.3 Characteristic boundary conditions as a function of conservative and primitive variables 349 19.1.4 Extrapolation methods 353 19.1.5 Practical implementation methods for numerical boundary conditions 357 19.1.6 Nonreflecting boundary conditions 369 19.2 Multidimensional Boundary Treatment 372 19.2.1 Physical and numerical boundary conditions 372 19.2.2 Multidimensional compatibility relations 376 19.2.3 Farfield treatment for steadystate flows 377 19.2.4 Solid wall boundary 379 19.2.5 Nonreflective boundary conditions 384 19.3 The Far-field Boundary Corrections 385 19.4 The Kutta Condition 395 19.5 Summary 401 Chapter 20 Upwind Schemes for the Euler Equations 408 20.1 The Basic Principles of Upwind Schemes 409 20.2 One-dimensional Flux Vector Splitting 415 20.2.1 Steger and Warming flux vector splitting 415 20.2.2 Properties of split flux vectors 417 20.2.3 Van Leer's flux splitting 420 20.2.4 Non-reflective boundary conditions and split fluxes 425 20.3 One-dimensional Upwind Discretizations Based on Flux Vector Splitting 426 20.3.1 First-order explicit upwind schemes 426 20.3.2 Stability conditions for first-order flux vector splitting schemes 428 20.3.3 Non-conservative firstorder upwind schemes 438 20.4 Multi-dimensional Flux Vector Splitting 438 20.4.1 Steger and Warming flux splitting 440 20.4.2 Van Leer flux splitting 440 20.4.3 Arbitrary meshes 441 20.5 The Godunov-type Schemes 443 20.5.1 The basic Godunov scheme 444 20.5.2 Osher's approximate Riemann solver 453 20.5.3 Roe's approximate Riemann solver 460 20.5.4 Other Godunov-type methods 469 20.5.5 Summary 472 20.6 First-order Implicit Upwind Schemes 473 20.7 Multi-dimensional First-order Upwind Schemes 475 Chapter 21 Second-order Upwind and High-resolution Schemes 493 21.1 General Formulation of Higher-order Upwind Schemes 494 21.1.1 Higher-order projection stages-variable extrapolation or MUSCL approach 495 21.1.2 Numerical flux for higher-order upwind schemes 498 21.1.3 Second-order space- and time-accurate upwind schemes based on variable extrapolation 499 21.1.4 Linearized analysis of second-order upwind schemes 502 21.1.5 Numerical flux for higher-order upwind schemes-flux extrapolation 504 21.1.6 Implicit second-order upwind schemes 512 21.1.7 Implicit second-order upwind schemes in two dimensions 514 21.1.8 Summary 516 21.2 The Definition of High-resolution Schemes 517 21.2.1 The generalized entropy condition for inviscid equations 519 21.2.2 Monotonicity condition 525 21.2.3 Total variation diminishing (TVD)schemes 528 21.3 Second-order TVD Semi-discretized Schemes with Limiters 536 21.3.1 Definition of limiters for the linear convection equation 537 21.3.2 General definition of flux limiters 550 21.3.3 Limiters for variable extrapolation-MUSCL-method 552 21.4 Timeintegration Methods for TVD Schemes 556 21.4.1 Explicit TVD schemes of first-order accuracy in time 557 21.4.2 Implicit TVD schemes 558 21.4.3 Explicit second-order TVD schemes 560 21.4.4 TVD schemes and artificial dissipation 564 21.4.5 TVD limiters and the entropy condition 568 21.5 Extension to Non-linear Systems and to Multi-dimensions 570 21.6 Conclusions to Part VI 583 Part VII: The Numerical Solution of the Navier-Stokes Equations 595 Chapter 22 The Properties of the System of Navier-Stokes Equations 597 22.1 Mathematical Formulation of the Navier-Stokes Equations 597 22.1.1 Conservative form of the Navier-Stokes equations 597 22.1.2 Integral form of the Navier-Stokes equations 599 22.1.3 Shock waves and contact layers 600 22.1.4 Mathematical properties and boundary conditions 601 22.2 Reynolds-averaged Navier-Stokes Equations 603 22.2.1 Turbulent-averaged energy equation 604 22.3 Turbulence Models 606 22.3.1 Algebraic models 608 22.3.2 One- and two-equation models-k- models 613 22.3.3 Algebraic Reynolds stress models 615 22.4 Some Exact One-dimensional Solutions 618 22.4.1 Solutions to the linear convection-diffusion equation 618 22.4.2 Solutions to Burgers equation 620 22.4.3 Other simple test cases 621 Chapter 23 Discretization Methods for the Navier-Stokes Equations 624 23.1 Discretization of Viscous and Heat Conduction Terms 625 23.2 Time-dependent Methods for Compressible Navier-Stokes Equations 627 23.2.1 First-order explicit central schemes 628 23.2.2 One-step Lax-Wendroff schemes 629 23.2.3 Two-step Lax-Wendroff schemes 630 23.2.4 Central schemes with separate space and time discretization 636 23.2.5 Upwind schemes 648 23.3 Discretization of the Incompressible Navier-Stokes Equations 654 23.3.1 Incompressible Navier-Stokes equations 654 23.3.2 Pseudo-compressibility method 656 23.3.3 Pressure correction methods 661 23.3.4 Selection of the space discretization 666 23.4 Conclusions to Part VII 674 Index 685

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