A UNIFIED FORMULATION OF THE SEGREGATED CLASS OF ALGORITHMS FOR FLUID FLOW AT ALL SPEEDS

In this article, the segregated SIMPLE algorithm and its variants are reformulated, using a collocated variable approach, to predict fluid flow at all speeds. In the formulation, a unified, compact, and easy-to-understand notation is employed. The SIMPLE, SIMPLER, SIMPLEST, SIMPLEM, SIMPLEC, SIMPLEX, PRIME, and PISO algorithms that are scattered in the literature and appear to a non versed computational fluid dynamics (CFD) user as being unrelated, are shown to share the same essence in their derivations and to be equally applicable for the simulation of incompressible and compressible flows. Moreover, the philosophies behind these algorithms in addition to their similarities and differences are explained.

[1]  Arthur Rizzi,et al.  Computation of inviscid incompressible flow with rotation , 1985, Journal of Fluid Mechanics.

[2]  M. Darwish,et al.  NORMALIZED VARIABLE AND SPACE FORMULATION METHODOLOGY FOR HIGH-RESOLUTION SCHEMES , 1994 .

[3]  C. Rhie,et al.  A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation , 1982 .

[4]  M. Darwish,et al.  A NEW HIGH-RESOLUTION SCHEME BASED ON THE NORMALIZED VARIABLE FORMULATION , 1993 .

[5]  M. Darwish,et al.  New family of adaptive very high resolution schemes , 1998 .

[6]  Åke Björck,et al.  Numerical Methods , 2021, Markov Renewal and Piecewise Deterministic Processes.

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

[8]  H. Aksoy,et al.  NUMERICAL SOLUTION OF NAVIER-STOKES EQUATIONS WITH NONSTAGGERED GRIDS USING FINITE ANALYTIC METHOD , 1992 .

[9]  G T Polley,et al.  Heat transfer and fluid flow , 1976 .

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

[11]  D. Brian Spalding,et al.  Numerical computation of multi-phase fluid flow and heat transfer , 1980 .

[12]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[13]  M. Darwish, F. Moukalled.  B-EXPRESS: A NEW BOUNDED EXTREMUM-PRESERVING STRATEGY FOR CONVECTIVE SCHEMES , 2000 .

[14]  A. Gosman,et al.  Solution of the implicitly discretised reacting flow equations by operator-splitting , 1986 .

[15]  S. Acharya,et al.  Comparison of the Piso, Simpler, and Simplec Algorithms for the Treatment of the Pressure-Velocity Coupling in Steady Flow Problems , 1986 .

[16]  Clovis R. Maliska,et al.  A NONORTHOGONAL FINITE-VOLUME METHOD FOR THE SOLUTION OF ALL SPEED FLOWS USING CO-LOCATED VARIABLES , 1994 .

[17]  Sumanta Acharya,et al.  Improvements to Incompressible Flow Calculation on a Nonstaggered Curvilinear Grid , 1989 .

[18]  D. Kershaw.  The incomplete Cholesky—conjugate gradient method for the iterative solution of systems of linear equations , 1978 .

[19]  D. Spalding,et al.  Two numerical methods for three-dimensional boundary layers , 1972 .

[20]  D. Brian Spalding.  The numerical computation of multi-phase flows , 1985 .

[21]  S. M. H. Karimian,et al.  Pressure-based control-volume finite element method for flow at all speeds , 1995 .

[22]  H. L. Stone.  ITERATIVE SOLUTION OF IMPLICIT APPROXIMATIONS OF MULTIDIMENSIONAL PARTIAL DIFFERENTIAL EQUATIONS , 1968 .

[23]  Milt Chapman,et al.  FRAM—Nonlinear damping algorithms for the continuity equation , 1981 .

[24]  S. P. Vanka,et al.  Fully Coupled Calculation of Fluid Flows with Limited Use of Computer Storage , 1983 .

[25]  D. Spalding,et al.  A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows , 1972 .

[26]  J. Zhu,et al.  A local oscillation-damping algorithm for higher-order convection schemes , 1988 .

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

[28]  A. Brandt.  Guide to multigrid development , 1982 .

[29]  G. Raithby,et al.  A multigrid method based on the additive correction strategy , 1986 .

[30]  D. R. Liles,et al.  TRAC-PF1: an advanced best-estimate computer program for pressurized water reactor analysis , 1984 .

[31]  A. A. Amsden,et al.  Numerical calculation of multiphase fluid flow , 1975 .

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

[33]  P. J. Zwart,et al.  AN INTEGRATED SPACE-TIME FINITE-VOLUME METHOD FOR MOVING-BOUNDARY PROBLEMS , 1998 .

[34]  C. Rhie.  Pressure-based Navier-Stokes solver using the multigrid method , 1986 .

[35]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

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

[37]  M. Darwish,et al.  NEW BOUNDED SKEW CENTRAL DIFFERENCE SCHEME, PART II: APPLICATION TO NATURAL CONVECTION IN AN ECCENTRIC ANNULUS , 1997 .

[38]  Fue-Sang Lien,et al.  A general non-orthogonal collocated finite volume algorithm for turbulent flow at all speeds incorporating second-moment turbulence-transport closure, Part 1: Computational implementation , 1994 .

[39]  M. Darwish,et al.  A NEW FAMILY OF STREAMLINE-BASED VERY-HIGH-RESOLUTION SCHEMES , 1997 .

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

[41]  K. H. Chen,et al.  Primitive variable, strongly implicit calculation procedure for viscous flows at all speeds , 1991 .

[42]  W. R. Bohl,et al.  SIMMER-II: A computer program for LMFBR disrupted core analysis , 1990 .

[43]  William P. Timlake,et al.  An Accelerated Relaxation Algorithm for Iterative Solution of Elliptic Equations , 1968 .

[44]  M. Darwish,et al.  A high-resolution pressure-based algorithm for fluid flow at all speeds , 2001 .

[45]  M. Darwish,et al.  NEW BOUNDED SKEW CENTRAL DIFFERENCE SCHEME, PART I: FORMULATION AND TESTING , 1997 .

[46]  K. Giannakoglou,et al.  A PRESSURE-BASED ALGORITHM FOR HIGH-SPEED TURBOMACHINERY FLOWS , 1997 .

[47]  G. K. Leaf,et al.  Fully-coupled solution of pressure-linked fluid flow equations , 1983 .

[48]  M. B. Carver.  Numerical computation of phase separation in two fluid flow , 1984 .

[49]  P. Gaskell,et al.  Curvature‐compensated convective transport: SMART, A new boundedness‐ preserving transport algorithm , 1988 .

[50]  M. B. Carver.  A Method of Limiting Intermediate Values of Volume Fraction in Iterative Two-Fluid Computations , 1982 .

[51]  Amin Hassan Ahmed Baghdadi.  Numerical modelling of two-phase flow with inter-phase slip , 1980 .

[52]  S. Patankar.  Numerical Heat Transfer and Fluid Flow , 2018, Lecture Notes in Mechanical Engineering.

[53]  W. T. Hancox,et al.  Numerical Standards for Flow-Boiling Analysis , 1977 .

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

[55]  H. Bruce Stewart.  Fractional step methods for thermohydraulic calculation , 1981 .

[56]  T. Gjesdal,et al.  Comparison of pressure correction smoothers for multigrid solution of incompressible flow , 1997 .

[57]  J. P. V. Doormaal,et al.  ENHANCEMENTS OF THE SIMPLE METHOD FOR PREDICTING INCOMPRESSIBLE FLUID FLOWS , 1984 .

[58]  Ruey-Hor Yen,et al.  ENHANCEMENT OF THE SIMPLE ALGORITHM BY AN ADDITIONAL EXPLICIT CORRECTOR STEP , 1993 .

[59]  J-M Buchlin,et al.  Numerical computation of multi-phase flows , 1981 .

[60]  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 .

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

[62]  H. Bruce Stewart,et al.  Two-phase flow: Models and methods , 1984 .

[63]  D. B. Spalding,et al.  A general purpose computer program for multi-dimensional one- and two-phase flow , 1981 .

[64]  C. Rhie,et al.  A numerical study of the flow past an isolated airfoil with separation , 1981 .

[65]  H. Stewart,et al.  Stability of two-phase flow calculation using two-fluid models , 1979 .

[66]  M. Peric.  A finite volume method for the prediction of three-dimensional fluid flow in complex ducts , 1985 .