Unified methods for computing incompressible and compressible flow

If there are regions in the flow domain where the Mach number M is not small, the incompressible Navier-Stocks or Euler equations cannot be applied. In principle, the compressible equations of motion are uniformly valid as the Mach number ranges from zero to supersonic (until real gas effects set in). Therefor it suffices to forego the simplifications that incompressibility brings, and all one has to do is to employ the compressible equations of motion. However, as will be discussed below, the standard numerical methods that have been developed for compressible flow (discussed in Chap. 10 and 12) break down or do not function properly when M ≲ 0.2. Methods to remedy this will be discussed in the present chapter. It would be ideal if a unified method could be found that is accurate and efficient for both compressible and incompressible flows. Such unified methods with accuracy and efficiency more or less uniform in the Mach number have indeed been proposed, and will be discussed below.

[1]  Eli Turkel,et al.  Preconditioning and the Limit to the Incompressible Flow Equations , 1993 .

[2]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[3]  M. Liou,et al.  A New Flux Splitting Scheme , 1993 .

[4]  A. A. Amsden,et al.  Numerical calculation of almost incompressible flow , 1968 .

[5]  C. Merkle,et al.  Computational modeling of the dynamics of sheet cavitation , 1998 .

[6]  J. J. Leendertse,et al.  Aspects of a computational model for long-period water-wave propagation , 1967 .

[7]  G. Stelling,et al.  On the construction of computational methods for shallow water flow problems , 1983 .

[8]  A. D. Gosman,et al.  The computation of compressible and incompressible recirculating flows by a non-iterative implicit scheme , 1986 .

[9]  C. Vuik,et al.  A staggered scheme for hyperbolic conservation laws applied to unsteady sheet cavitation , 1999 .

[10]  Cornelis Vuik,et al.  Segregated solution methods for compressible flow , 2000 .

[11]  Masatsugu Maeda,et al.  Unsteady Structure Measurement of Cloud Cavitation on a Foil Section Using Conditional Sampling Technique , 1989 .

[12]  Ben P. Sommeijer,et al.  Time integration of three-dimensional numerical transport models , 1993 .

[13]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[14]  W. Hansen.  Theorie zur Errechnung des Wasserstandes und der Strömungen in Randmeeren nebst Anwendungen , 1956 .

[15]  Hester Bijl,et al.  A Unified Method for Computing Incompressible and Compressible Flows in Boundary-Fitted Coordinates , 1998 .

[16]  S. Patankar,et al.  Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations , 1988 .

[17]  H. Guillard,et al.  On the behaviour of upwind schemes in the low Mach number limit , 1999 .

[18]  C. Merkle,et al.  Application of time-iterative schemes to incompressible flow , 1985 .

[19]  H. Bijl,et al.  Computing Flow on General Two-Dimensional Nonsmooth Staggered Grids , 1998 .

[20]  C. L. Merkle,et al.  The application of preconditioning in viscous flows , 1993 .

[21]  C. Rhie,et al.  Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation , 1983 .

[22]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[23]  B. Leer,et al.  Flux-vector splitting for the Euler equations , 1997 .

[24]  Lewis F. Richardson,et al.  Weather Prediction by Numerical Process , 1922 .

[25]  G. D. Raithby,et al.  The segregated approach to predicting viscous compressible fluid flows , 1986 .

[26]  Burton Wendroff,et al.  The Riemann problem for materials with nonconvex equations of state I: Isentropic flow☆ , 1972 .

[27]  A. A. Amsden,et al.  A numerical fluid dynamics calculation method for all flow speeds , 1971 .

[28]  M. Perić,et al.  A collocated finite volume method for predicting flows at all speeds , 1993 .

[29]  Christopher J. Roy,et al.  Preconditioned multigrid methods for two-dimensional combustion calculations at all speeds , 1998 .

[30]  Pieter Wesseling,et al.  Computing Flows on General Three-Dimensional Nonsmooth Staggered Grids , 1999 .

[31]  G. D. van Albada,et al.  A comparative study of computational methods in cosmic gas dynamics , 1982 .

[32]  Bram van Leer,et al.  On the Relation Between the Upwind-Differencing Schemes of Godunov, Engquist–Osher and Roe , 1984 .

[33]  S. Osher,et al.  Upwind difference schemes for hyperbolic systems of conservation laws , 1982 .

[34]  R. LeVeque.  Approximate Riemann Solvers , 1992 .

[35]  Wei Shyy,et al.  Adaptive grid computation for inviscid compressible flows using a pressure correction method , 1988 .

[36]  Wayne A. Smith,et al.  Preconditioning Applied to Variable and Constant Density Flows , 1995 .

[37]  B. Koren.  Improving Euler Computations at Low Mach Numbers , 1996 .

[38]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[39]  Eli Turkel,et al.  Review of preconditioning methods for fluid dynamics , 1993 .

[40]  Guus S. Stelling,et al.  Practical Aspects of Accurate Tidal Computations , 1986 .

[41]  James J. McGuirk,et al.  Shock capturing using a pressure-correction method , 1989 .

[42]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[43]  David A. Randall,et al.  Geostrophic Adjustment and the Finite-Difference Shallow-Water Equations , 1994 .

[44]  Barry Koren,et al.  Analysis of preconditioning and multigrid for Euler flows with low-subsonic regions , 1995, Adv. Comput. Math..

[45]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[46]  F. W. Schmidt,et al.  USE OF A PRESSURE-WEIGHTED INTERPOLATION METHOD FOR THE SOLUTION OF THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON A NONSTAGGERED GRID SYSTEM , 1988 .

[47]  Eli Turkel,et al.  Assessment of Preconditioning Methods for Multidimensional Aerodynamics , 1997 .

[48]  S. Osher.  Numerical Solution of Singular Perturbation Problems and Hyperbolic Systems of Conservation Laws , 1981 .