Computing Flow on General Two-Dimensional Nonsmooth Staggered Grids

The classical staggered scheme for the incompressible Navier-Stokes equations is generalized from Cartesian grids to general boundary-fitted structured grids. The resulting discretization is coordinate-invariant. The unknowns are the pressure and the contravariant volume flux components. The grid can be strongly nonuniform and nonorthogonal. The smoothness properties of the coordinate mapping are carefully taken into account. As a result, the accuracy on rough grids is found to be at least as good as for typical finite element and nonstaggered finite volume schemes. Extension to compressible flows results in a scheme with Mach-uniform accuracy and efficiency for Mach numbers ranging from M = 0 to M > 1. Accurate discretization of two-equation turbulence models is also possible.

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

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

[3]  Pieter Wesseling.  Uniform convergence of discretization error for a singular perturbation problem , 1996 .

[4]  Cornelis W. Oosterlee,et al.  Steady incompressible flow around objects in general coordinates with a multigrid solution method , 1994 .

[5]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

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

[7]  D. Assanis,et al.  Comparison of Pressure-Based and Artificial Compressibility Methods for Solving 3D Steady Incompressible Viscous Flows , 1996 .

[8]  Jose C. F. Pereira,et al.  Numerical comparison of momentum interpolation methods and pressure—velocity algorithms using non-staggered grids , 1991 .

[9]  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 .

[10]  Alexandre J. Chorin,et al.  On the Convergence of Discrete Approximations to the Navier-Stokes Equations , 1969 .

[11]  P. Wesseling.  von Neumann stability conditions for the convection-diffusion eqation , 1996 .

[12]  H. Kreiss,et al.  Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations , 1991 .

[13]  John W. Goodrich,et al.  Unsteady solution of incompressible Navier-Stokes equations , 1988 .

[14]  T. Hou,et al.  Second-order convergence of a projection scheme for the incompressible Navier-Stokes equations with boundaries , 1993 .

[15]  Philip L. Roe,et al.  Characteristic time-stepping or local preconditioning of the Euler equations , 1991 .

[16]  Stuart E. Rogers,et al.  Steady and unsteady solutions of the incompressible Navier-Stokes equations , 1991 .

[17]  Cornelis Vuik,et al.  Computing Incompressible Flows in General Domains , 1994 .

[18]  Dochan Kwak,et al.  A three-dimensional incompressible Navier-Stokes flow solver using primitive variables , 1986 .

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

[20]  D. Kwak,et al.  On the method of pseudo compressibility for numerically solving incompressible flows , 1984 .

[21]  J. Ferziger,et al.  An adaptive multigrid technique for the incompressible Navier-Stokes equations , 1989 .

[22]  Lutz Tobiska,et al.  Numerical Methods for Singularly Perturbed Differential Equations , 1996 .

[23]  On accurate discretization of turbulence transport equations in general coordinates , 1995 .

[24]  R. Sani,et al.  On pressure boundary conditions for the incompressible Navier‐Stokes equations , 1987 .

[25]  Wei Shyy,et al.  Pressure-based multigrid algorithm for flow at all speeds , 1992 .

[26]  Cornelis W. Oosterlee,et al.  Multigrid Schemes for Time-Dependent Incompressible Navier-Stokes Equations , 1993, IMPACT Comput. Sci. Eng..

[27]  Cornelis Vuik,et al.  Numerical solution of the incompressible Navier-Stokes equations by Krylov subspace and multigrid methods , 1995, Adv. Comput. Math..

[28]  J. Koseff,et al.  A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates , 1994 .

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

[30]  C. Vuik.  Solution of the discretized incompressible Navier‐Stokes equations with the GMRES method , 1993 .

[31]  Thomas A. Manteuffel,et al.  The numerical solution of second-order boundary value problems on nonuniform meshes , 1986 .

[32]  John J. H. Miller,et al.  A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation , 1996 .

[33]  Laszlo Fuchs,et al.  Solution of three‐dimensional viscous incompressible flows by a multi‐grid method , 1984 .

[34]  P. Gresho.  Some current CFD issues relevant to the incompressible Navier-Stokes equations , 1991 .

[35]  A. Majda.  Compressible fluid flow and systems of conservation laws in several space variables , 1984 .

[36]  P. M. Gresho,et al.  Some Interesting Issues in Incompressible Fluid Dynamics, Both in the Continuum and in Numerical Simulation , 1991 .

[37]  A. Chorin.  A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

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

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

[40]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[41]  P. Wesseling.  An Introduction to Multigrid Methods , 1992 .

[42]  Phillip Colella,et al.  Application of the Godunov method and its second-order extension to cascade flow modeling , 1984 .

[43]  P. Hartwich,et al.  High-resolution upwind schemes for the three-dimensional incompressible Navier-Stokes equations , 1987 .

[44]  A. E. Mynett,et al.  The ISNaS incompressible Navier-Stokes solver : invariant discretization , 1994 .

[45]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[46]  Cornelis W. Oosterlee,et al.  Invariant discretization of the incompressible Navier‐Stokes equations in boundary fitted co‐ordinates , 1992 .

[47]  H. Daiguji,et al.  Application of an implicit time-marching scheme to a three-dimensional incompressible flow problem in curvilinear coordinate systems , 1992 .

[48]  S. Armfield.  Ellipticity, accuracy, and convergence of the discrete Navier-Stokes equations , 1994 .

[49]  P. Wesseling,et al.  Benchmark solutions for the incompressible Navier–Stokes equations in general co‐ordinates on staggered grids , 1993 .

[50]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[51]  A. Weiser,et al.  On convergence of block-centered finite differences for elliptic-problems , 1988 .

[52]  Wei Shyy,et al.  Numerical Recirculating Flow Calculation Using a Body-Fitted Coordinate System , 1985 .

[53]  W. Rodi,et al.  Finite volume methods for two-dimensional incompressible flows with complex boundaries , 1989 .

[54]  W. Shyy,et al.  Development of a pressure-correction/ staggered-grid based multigrid solver for incompressible recirculating flows , 1993 .

[55]  B. Lakshminarayana,et al.  Computation of Unsteady Viscous Flow Using a Pressure-Based Algorithm , 1993 .

[56]  Bertil Gustafsson,et al.  A numerical method for incompressible and compressible flow problems with smooth solutions , 1986 .

[57]  Marcel Zijlema,et al.  Finite volume computation of incompressible turbulent flows in general co‐ordinates on staggered grids , 1995 .

[58]  P. Wesseling,et al.  Finite volume discretization of the incompressible Navier-Stokes equations in general coordinates on staggered grids , 1991 .

[59]  Cornelis W. Oosterlee,et al.  A Robust Multigrid Method for a Discretization of the Incompressible Navier-Stokes Equations in General Coordinates , 1993, IMPACT Comput. Sci. Eng..

[60]  Marcel Zijlema,et al.  Invariant discretization of the - model in general co-ordinates for prediction of turbulent flow in complicated geometries , 1995 .

[61]  S. Koshizuka,et al.  A staggered differencing technique on boundary-ditted curvilinear grids for incompressible flows along curvilinear or slant walls , 1990 .

[62]  Cornelis Vuik,et al.  FAST ITERATIVE SOLVERS FOR THE DISCRETIZED INCOMPRESSIBLE NAVIER–STOKES EQUATIONS , 1996 .

[63]  Toshiaki Ikohagi,et al.  Finite-difference schemes for steady incompressible Navier-Stokes equations in general curvilinear coordinates , 1991 .

[64]  J. B. Perot,et al.  An analysis of the fractional step method , 1993 .

[66]  D. Kwak,et al.  A fractional step solution method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems , 1991 .

[67]  J. Dukowicz,et al.  Approximate factorization as a high order splitting for the implicit incompressible flow equations , 1992 .

[68]  Wei Shyy,et al.  On the adoption of velocity variable and grid system for fluid flow computation in curvilinear coordinates , 1990 .

[69]  L. Davidson,et al.  Mathematical derivation of a finite volume formulation for laminar flow in complex geometries , 1989 .

[70]  P. Colella,et al.  A second-order projection method for the incompressible navier-stokes equations , 1989 .

[71]  P. Vanbeek,et al.  Aspects of Nonsmoothness in Flow Computations , 1994 .

[72]  P. Wesseling,et al.  A numerical method for the computation of compressible flows with low Mach number regions , 1996 .

[73]  R. Kessler,et al.  Comparison of finite-volume numerical methods with staggered and colocated grids , 1988 .

[74]  T. A. Porsching,et al.  An unconditionally stable convergent finite difference method for Navier-Stokes problems on curved domains , 1987 .

[75]  T. Taylor,et al.  Computational methods for fluid flow , 1982 .

[76]  Peter A. Forsyth,et al.  Quadratic convergence for cell-centered grids , 1988 .