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Radial basis function collocation method solution of natural convection in porous media. (English) Zbl 1103.76361
Summary: This paper describes the solution of a steady-state natural convection problem in porous media by the radial basis function collocation method (RBFCM). This mesh-free (polygon-free) numerical method is for a coupled set of mass, momentum, and energy equations in two dimensions structured by the Hardy’s multiquadrics with different shape parameter and different order of polynomial augmentation. The solution is formulated in primitive variables and involves iterative treatment of coupled pressure, velocity, pressure correction, velocity correction, and energy equations. Numerical examples include convergence studies with different collocation point density and arrangements for a two-dimensional differentially heated rectangular cavity problem at filtration Rayleigh numbers \(Ra^*=25\), 50 and 100, and aspect ratios \(A=1/2\), 1, and 2. The solution is assessed by comparison with reference results of the fine-mesh finite volume method in terms of mid-plane velocity components, mid-plane and insulated surface temperatures, streamfunction minimum, and Nusselt number.

MSC:
76M30 Variational methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
76E06 Convection in hydrodynamic stability
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[1] DOI: 10.1002/(SICI)1097-0207(19981030)43:4<713::AID-NME445>3.0.CO;2-8 · Zbl 0948.76050
[2] DOI: 10.1016/0898-1221(91)90123-L · Zbl 0725.65009
[3] Chan, B.K.C., Ivey, C.M. and Barry, J.M. (1970), ”Natural convection in enclosed porous media with rectangular boundaries”,Wärme und Stoffübertragung, Vol. 7, pp. 22-30.
[4] Chen, W. and Tanaka, M. (2000), ”New insights into boundary-only and domain-type RBF methods”,Journal of Nonlinear Science and Numerical Simulation, Vol. 1, pp. 145-51. · Zbl 0954.65084
[5] DOI: 10.1016/S0898-1221(01)00293-0 · Zbl 0999.65142
[6] DOI: 10.1002/(SICI)1099-0887(199902)15:2<137::AID-CNM233>3.0.CO;2-9 · Zbl 0927.65140
[7] DOI: 10.1023/A:1018919824891 · Zbl 0940.65122
[8] Franke, J. (1982), ”Scattered data interpolation: tests of some methods”,Mathematics of Computation, Vol. 48, pp. 181-200. · Zbl 0476.65005
[9] DOI: 10.1016/0017-9310(95)00351-7
[10] DOI: 10.1016/0017-9310(95)00351-7
[11] DOI: 10.1016/S0955-7997(96)00033-1
[12] DOI: 10.1016/0898-1221(90)90272-L · Zbl 0692.65003
[13] DOI: 10.1115/1.3244544
[14] DOI: 10.1002/1097-0363(20010115)35:1<39::AID-FLD81>3.0.CO;2-3 · Zbl 1008.76057
[15] DOI: 10.1016/0898-1221(90)90270-T · Zbl 0692.76003
[16] DOI: 10.1016/0898-1221(90)90271-K · Zbl 0850.76048
[17] DOI: 10.1016/S0955-7997(01)00101-1 · Zbl 1003.65132
[18] DOI: 10.1002/fld.165 · Zbl 1047.76101
[19] DOI: 10.1016/S0893-6080(00)00095-2 · Zbl 02022497
[20] DOI: 10.1016/S0955-7997(01)00092-3 · Zbl 0996.65131
[21] DOI: 10.1016/S0898-1221(01)00305-4 · Zbl 0999.65135
[22] DOI: 10.1115/1.3246629
[23] Sadat, H. and Couturier, S. (2000), ”Performance and accuracy of a meshless method for laminar natural convection”,Numerical Heat Transfer, Vol. 37B, pp. 455-67.
[24] Šarler, B., Perko, J. and Chen, C.S. (2002), ”Natural convection in porous media – radial basis function collocation method solution of the Darcy model”, in Mang, H.A., Rammerstorfer, F.G. and Eberhardsteiner, J. (Eds),Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V).
[25] Šarler, B., Perko, J., Chen, C.S. and Kuhn, G. (2001), ”A meshless approach to natural convection”, in Atluri, S.N. (Ed.),International Conference on Computational Engineering and Sciences – 2001, CD-rom Proceedings, pp. 3-9.
[26] DOI: 10.1002/(SICI)1097-0363(20000530)33:2<279::AID-FLD18>3.0.CO;2-N · Zbl 0972.76071
[27] DOI: 10.1016/S0045-7825(01)00419-4 · Zbl 1065.74074
[28] DOI: 10.1002/(SICI)1097-0207(20000510)48:1<19::AID-NME862>3.0.CO;2-3 · Zbl 0968.65053
[29] DOI: 10.1002/(SICI)1097-0207(19980815)42:7<1263::AID-NME431>3.0.CO;2-I · Zbl 0907.65095
[30] DOI: 10.1007/s004660000181 · Zbl 0986.74079
[31] DOI: 10.1016/S0898-1221(01)00292-9 · Zbl 0999.65143
[32] DOI: 10.1016/S0955-7997(00)00034-5 · Zbl 0994.76058
[33] DOI: 10.1016/S0955-7997(02)00044-9 · Zbl 1032.76628
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