×

An implicit pseudo-spectral multi-domain method for the simulation of incompressible flows. (English) Zbl 1041.76053

Summary: We present a pseudo-spectral method for the solution of incompressible flow problems based on an iterative solver involving an implicit treatment of linearized convective terms. The method allows the treatment of moderately complex geometries by means of a multi-domain approach, and it is able to cope with non-constant fluid properties and non-orthogonal problem domains. In addition, the fully implicit scheme yields improved stability properties as opposed to semi-implicit schemes commonly employed. Key components of the method are a Chebyshev collocation discretization, a special pressure-correction scheme, and a restarted GMRES method with a preconditioner derived from a fast direct solver. The performance of the proposed method is investigated by considering several numerical examples of different complexity, and also includes comparisons to alternative solution approaches based on finite volume discretizations.

MSC:

76M22 Spectral methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Haldenwang, Chebyshev 3d spectral and 2d pseudo-spectral solvers for the Helmholtz equation, Journal of Computational Physics 55 pp 115– (1984) · Zbl 0544.65071 · doi:10.1016/0021-9991(84)90018-4
[2] Serre, Annular and spiral patterns in flows between rotating and stationary discs, Journal of Fluid Mechanics 434 pp 65– (2001) · Zbl 1012.76099 · doi:10.1017/S0022112001003494
[3] Guillard, Iterative methods with spectral preconditioning for elliplic equations, Computational Methods in Applied Mechanics and Engineering 80 pp 305– (1990) · Zbl 0742.65076 · doi:10.1016/0045-7825(90)90034-J
[4] Dimitropoulos, Efficient pseudo-spectral flow simulations in moderately complex geometries, Journal of Computational Physics 144 pp 517– (1998) · Zbl 0929.76094 · doi:10.1006/jcph.1998.6009
[5] Dimitropoulos, An efficient and robust spectral solver for nonseparable elliptic equations, Journal of Computational Physics 133 pp 186– (1997) · Zbl 0877.65074 · doi:10.1006/jcph.1997.5661
[6] Macaraeg, Improvements in spectral collocation through a multiple domain technique, Applied Numerical Mathematics 2 pp 95– (1986) · Zbl 0633.76094 · doi:10.1016/0168-9274(86)90019-X
[7] Lacroix, Proceedings of the 7th GAMM-Conference on Numerical Methods and Fluid Mechanics pp 164– (1988)
[8] Raspo, A spectral multidomain technique for the computation of the Czochralski melt configuration, International Journal of Numerical Methods for Heat and Fluid Flow 6 pp 31– (1996) · Zbl 0863.76058 · doi:10.1108/EUM0000000004097
[9] Raspo, A direct spectral domain decomposition method for the computation of rotating flows in a T-shape geometry, Computers and Fluids 32 pp 431– (2002) · Zbl 1009.76522 · doi:10.1016/S0045-7930(01)00091-3
[10] Bramble, The construction of preconditioners for elliptic problems by substructuring I, Mathematics of Computation 47 pp 103– (1986) · Zbl 0615.65112
[11] Bramble, The construction of preconditioners for elliptic problems by substructuring II, Mathematics of Computation 49 pp 1– (1987) · Zbl 0623.65118
[12] Bramble, The construction of preconditioners for elliptic problems by substructuring III, Mathematics of Computation 51 pp 415– (1988) · Zbl 0701.65070
[13] Ferziger, Computational Methods for Fluid Dynamics (1996)
[14] Canuto, Spectral Methods in Fluid Dynamics (1988)
[15] Vanel, Numerical Methods in Fluid Mechanics pp 463– (1986)
[16] Droll, Computational Fluid Dynamics’98 pp 1240– (1998)
[17] Hugues, An improved projection scheme applied to pseudo-spectral methods for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids 28 pp 501– (1998) · Zbl 0932.76065 · doi:10.1002/(SICI)1097-0363(19980915)28:3<501::AID-FLD730>3.0.CO;2-S
[18] Hackbusch, Theorie und Numerik elliptischer Differentialgleichungen (1996)
[19] Saad, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing 7 pp 856– (1986) · Zbl 0599.65018
[20] da Cunha, A parallel implementation of the restarted GMRES iterative method for nonsymmetric systems of linear equations, Advances in Computer Mathematics 2 (3) pp 261– (1994) · Zbl 0829.65035
[21] De Valerio, Parallel Computing ’91 pp 167– (1992)
[22] Christon, Proceedings of the First MIT Conference on Computational Fluid and Solid Mechanics (2001)
[23] Auteri, Proceedings of the First MIT Conference on Computational Fluid and Solid Mechanics pp 1451– (2001)
[24] Christon, Proceedings of the First MIT Conference on Computational Fluid and Solid Mechanics pp 1460– (2001)
[25] Gresho, Proceedings of the First MIT Conference on Computational Fluid and Solid Mechanics pp 1482– (2001)
[26] Johnston, Proceedings of the First MIT Conference on Computational Fluid and Solid Mechanics pp 1486– (2001)
[27] Xin, Proceedings of the First MIT Conference on Computational Fluid and Solid Mechanics pp 1509– (2001)
[28] The Collected Work of G.I. Taylor 2 (1960)
[29] 1999
[30] Durst, A parallel blockstructured multigrid method for the prediction of incompressible flows, International Journal for Numerical Methods in Fluids 22 pp 549– (1996) · Zbl 0865.76059 · doi:10.1002/(SICI)1097-0363(19960330)22:6<549::AID-FLD366>3.0.CO;2-7
[31] Schäfer, Flow Simulation with High-Performance Computers II pp 547– (1996)
[32] Tric, Computational Fluid Dynamics ’98 pp 979– (1998)
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.