×

A higher-order compact finite difference algorithm for solving the incompressible Navier-Stokes equations. (English) Zbl 1242.76216

Summary: On the basis of the projection method, a higher order compact finite difference algorithm, which possesses a good spatial behavior, is developed for solving the 2D unsteady incompressible Navier-Stokes equations in primitive variable. The present method is established on a staggered grid system and is at least third-order accurate in space. A third-order accurate upwind compact difference approximation is used to discretize the non-linear convective terms, a fourth-order symmetrical compact difference approximation is used to discretize the viscous terms, and a fourth-order compact difference approximation on a cell-centered mesh is used to discretize the first derivatives in the continuity equation. The pressure Poisson equation is approximated using a fourth-order compact difference scheme constructed currently on the nine-point 2D stencil. New fourth-order compact difference schemes for explicit computing of the pressure gradient are also developed on the nine-point 2D stencil. For the assessment of the effectiveness and accuracy of the method, particularly its spatial behavior, a problem with analytical solution and another one with a steep gradient are numerically solved. Finally, steady and unsteady solutions for the lid-driven cavity flow are also used to assess the efficiency of this algorithm.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Chorin, Numeriacl solution of the Navier-Stokes equations, Mathematics of Computation 22 pp 745– (1968) · doi:10.1090/S0025-5718-1968-0242392-2
[2] Chorin, On the convergence of discrete approximation to the Navier-Stokes equations, Mathematics of Computation 23 pp 341– (1969) · Zbl 0184.20103 · doi:10.1090/S0025-5718-1969-0242393-5
[3] Bell, A second-order projection method for the incompressible Navier-Stokes equations, Journal of Computational Physics 85 pp 257– (1989) · Zbl 0681.76030 · doi:10.1016/0021-9991(89)90151-4
[4] E, Projection method I: convergence and numerical boundary layers, SIAM Journal on Numerical Analysis 33 pp 1017– (1995) · Zbl 0842.76052 · doi:10.1137/0732047
[5] E, Projection method II: Godunov-Ryabenki analysis, SIAM Journal on Numerical Analysis 33 pp 1597– (1996) · Zbl 0920.76056 · doi:10.1137/S003614299426450X
[6] Botella, On the solution of the Navier-Stokes equations using Chebyshev projection schemes with third-order accuracy in time, Computer and Fluids 26 pp 107– (1997) · Zbl 0898.76077 · doi:10.1016/S0045-7930(96)00032-1
[7] Brown, Accurate projection methods for the incompressible Navier-Stokes equations, Journal of Computational Physics 168 pp 464– (2001) · Zbl 1153.76339 · doi:10.1006/jcph.2001.6715
[8] Lopez, An efficient spectral-projection method for the Navier-Stokes equations in cylindrical geometries II. Three-dimensional cases, Journal of Computational Physics 176 pp 384– (2002) · Zbl 1130.76392 · doi:10.1006/jcph.2002.6993
[9] Auteri, A mixed-basis spectral projection methods, Journal of Computational Physics 175 pp 1– (2002) · Zbl 1039.76049 · doi:10.1006/jcph.2001.6855
[10] Ferziger, Computational Methods for Fluid Dynamics (1996) · doi:10.1007/978-3-642-97651-3
[11] Li, A compact fourth-order finite difference scheme for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids 20 pp 1137– (1995) · Zbl 0836.76060 · doi:10.1002/fld.1650201003
[12] Li, A compact fourth-order finite difference scheme for unsteady viscous incompressible flows, Journal of Scientific Computing 16 pp 29– (2001) · Zbl 1172.76344 · doi:10.1023/A:1011146429794
[13] Ma, Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme, International Journal for Numerical Methods in Fluids 30 pp 509– (1999) · Zbl 0946.76062 · doi:10.1002/(SICI)1097-0363(19990715)30:5<509::AID-FLD851>3.0.CO;2-E
[14] Spotz, High-order compact scheme for the steady stream-function vorticity equations, International Journal for Numerical Methods in Engineering 38 pp 3497– (1995) · Zbl 0836.76065 · doi:10.1002/nme.1620382008
[15] Fu, Computational Fluid Dynamics Review pp 234– (1995)
[16] Sanyasiraju, Higher order semi compact scheme to solve transient incompressible Navier-Stokes equations, Computational Mechanics 35 pp 441– (2005) · Zbl 1096.76034 · doi:10.1007/s00466-004-0633-6
[17] Tian, A Fourth-order compact finite difference scheme for the steady streamfunction-vorticity formulation of the Navier-Stokes/Boussinesq equations, International Journal for Numerical Methods in Fluids 41 pp 495– (2003) · Zbl 1038.76029 · doi:10.1002/fld.444
[18] Ferreira de Sousa, Fourth- and tenth-order compact finite difference solutions of perturbed circular vortex flows, International Journal for Numerical Methods in Fluids 49 pp 603– (2005) · Zbl 1236.76038 · doi:10.1002/fld.1017
[19] Lele, Compact finite difference schemes with spectral-like resolution, Journal of Computational Physics 103 pp 16– (1992) · Zbl 0759.65006 · doi:10.1016/0021-9991(92)90324-R
[20] Hou, Simulation of cavity flow by the Lattice Boltzmann method, Journal of Computational Physics 118 pp 329– (1995) · Zbl 0821.76060 · doi:10.1006/jcph.1995.1103
[21] Bruneau, An efficient scheme for solving steady incompressible Navier-Stokes equations, Journal of Computational Physics 89 pp 389– (1990) · Zbl 0699.76034 · doi:10.1016/0021-9991(90)90149-U
[22] Ben-Artzi, A pure-compact scheme for the streamfunction formulation of Navier-Stokes equations, Journal of Computational Physics 205 pp 640– (2005) · Zbl 1087.76025 · doi:10.1016/j.jcp.2004.11.024
[23] Gupta, A new paradigm for solving Navier-Stokes equations: streamfunction-velocity formulation, Journal of Computational Physicis 207 pp 52– (2005) · Zbl 1177.76257 · doi:10.1016/j.jcp.2005.01.002
[24] Stéphane, A 2D compact fourth-order projection decomposition method, Journal of Computational Physics 206 pp 252– (2005) · Zbl 1087.76083 · doi:10.1016/j.jcp.2004.12.005
[25] Ghia, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, Journal of Computational Physics 48 pp 387– (1982) · Zbl 0511.76031 · doi:10.1016/0021-9991(82)90058-4
[26] Deng, Incompressible-flow calculations with a consistent physical interpolations finite-volume approach, Computers and Fluids 23 (8) pp 1029– (1994) · Zbl 0816.76066 · doi:10.1016/0045-7930(94)90003-5
[27] Sahin, A novel fully impicit finit volume method applied to the lid-driven cavity problem-Part I: High Reynolds number flow calculations, International Journal for Numerical Methods in Fluids 42 pp 57– (2003) · Zbl 1078.76046 · doi:10.1002/fld.442
[28] Schreiber, Driven cavity flows by efficient numerical techniques, Journal of Computational Physics 49 pp 310– (1983) · Zbl 0503.76040 · doi:10.1016/0021-9991(83)90129-8
[29] Kim, Application of a fractional-step method to incompressible Navier-Stokes equations, Journal of Computational Physics 59 pp 308– (1985) · Zbl 0582.76038 · doi:10.1016/0021-9991(85)90148-2
[30] Auteri, Numerical investigation on the stability of singular driven cavity flow, Journal of Computational Physics 183 pp 1– (2002) · Zbl 1021.76040 · doi:10.1006/jcph.2002.7145
[31] Gamet, Compact finite difference schemes on non-uniform meshes. Application to direct numerical simulations of compressible flows, International Journal for Numerical Methods in Fluids 29 (2) pp 159– (1999) · Zbl 0939.76060 · doi:10.1002/(SICI)1097-0363(19990130)29:2<159::AID-FLD781>3.0.CO;2-9
[32] Knikker, Study of a staggered fourth-order compact scheme for unsteady incompressible viscous flows, International Journal for Numerical Methods in Fluids 59 pp 1063– (2009) · Zbl 1158.76380 · doi:10.1002/fld.1854
[33] Ben-Artzi, A compact difference scheme for the biharmonic equation in planar irregular domains, SIAM Journal of Numerical Analysis 47 pp 3087– (2009) · Zbl 1202.65138 · doi:10.1137/080718784
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.