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Local radial basis function based gridfree scheme for unsteady incompressible viscous flows. (English) Zbl 1146.76045
Summary: A ‘local’ radial basic function (RBF) based gridfree scheme has been developed to solve unsteady incompressible Navier-Stokes equations in primitive variables. The velocity-pressure decoupling is obtained by making use of a fractional step algorithm. The scheme is validated over a variety of benchmark problems, and a very good agreement is found with the existing results. Comparisons with the benchmark solutions show that the developed local RBF gridfree scheme is stable and produces accurate results on domains discretized even with non-uniform distribution of nodal points.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Kansa, E.J., Multiquadrics – a scattered data approximation scheme with applications to computational fluid dynamics - II solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. math. appl., 19, 147-161, (1990) · Zbl 0850.76048
[2] Fasshauer, G.E., Solving partial differential equations by collocation with radial basis functions, (), 131-138 · Zbl 0938.65140
[3] Wu, Z., Hermite – birkhoff interpolation of scattered data by radial basis functions, Approx. theory appl., 8, 2, 1-10, (1992) · Zbl 0757.41009
[4] Hon, Y.C.; Schaback, R., On unsymmetric collocation by radial basis functions, Appl. math. comput., 119, 2-3, 177-186, (2001) · Zbl 1026.65107
[5] Šarler, B., A radial basis function collocation approach in computational fluid dynamics, CMES comput. model. eng. sci., 7, 2, 185-193, (2005) · Zbl 1189.76380
[6] Chantasiriwan, S., Performance of multiquadric collocation method in solving lid-driven cavity flow problem with low Reynolds number, CMES comput. model. eng. sci., 15, 137-146, (2006)
[7] Mai-Duy, N.; Tran-Cong, T., An efficient indirect RBFN-based method for numerical solution of pdes, Numer. methods partial differ. eq., 21, 4, 770-790, (2005) · Zbl 1077.65125
[8] Mai-Duy, N.; Mai-Cao, L.; Tran-Cong, T., Computation of transient viscous flows using indirect radial basis function networks, CMES comput. model. eng. sci., 18, 1, 59-77, (2007)
[9] Chen, W.; Hon, Y.C., Numerical convergence of boundary knot method in the analysis of Helmholtz, modified Helmholtz and convection – diffusion problems, Comput. methods appl. mech. eng., 192, 1859-1875, (2003) · Zbl 1050.76040
[10] Chen, W.; Tanaka, M., A meshless, integration-free, and boundary-only RBF technique, Comput. math. appl., 43, 379-391, (2002) · Zbl 0999.65142
[11] Chen, C.S.; Brebbia, C.A.; Power, H., Dual reciprocity method for Helmholtz-type operators, Bound. elem., 20, 495-504, (1998) · Zbl 0929.65109
[12] Wright, G.B.; Fornberg, B., Scattered node compact finite difference-type formulas generated from radial basis functions, J. comput. phys., 212, 99-123, (2006) · Zbl 1089.65020
[13] Lee, C.K.; Liu, X.; Fan, S.C., Local multiquadric approximation for solving boundary value problems, Comput. mech., 30, 396-409, (2003) · Zbl 1035.65136
[14] Tolstykh, A.I.; Shirobokov, D.A., On using radial basis functions in a “finite difference mode” with applications to elasticity problems, Comput. mech., 33, 68-79, (2003) · Zbl 1063.74104
[15] Šarler, B.; Vertnik, R., Meshfree explicit local radial basis function collocation method for diffusion problems, Comput. math. appl., 51, 1269-1282, (2006) · Zbl 1168.41003
[16] Vertnik, R.; Šarler, B., Meshless local radial basis function collocation method for convective – diffusive solid – liquid phase change problems, Int. J. numer. methods heat fluid flow, 16, 5, 617-640, (2006) · Zbl 1121.80014
[17] Harlow, F.; Welch, J., Numerical calculation of time dependent viscous flow of fluid with free surface, Phys. fluids, 8, 212-218, (1965)
[18] Chorin, A.J., A numerical method for solving incompressible viscous flow problems, J. comput. phys., 2, 12-26, (1967) · Zbl 0149.44802
[19] Kim, J.; Moin, P., Application of a fractional-step method to incompressible navier – stokes equations, J. comput. phys., 59, 308-323, (1985) · Zbl 0582.76038
[20] Hardy, R.L., Multiquadric equations of topography and other irregular surfaces, J. geophys. res., 76, 1905-1915, (1971)
[21] Franke, R., Scattered data interpolation: test of some methods, Math. comput., 48, 181-200, (1982) · Zbl 0476.65005
[22] Ghia, U.; Ghia, K.N.; Shin, C.T., High-re solutions for incompressible flow using the navier – stokes equations and a multigrid method, J. comput. phys., 48, 387-411, (1982) · Zbl 0511.76031
[23] Kaiktsis, L.; Karniadakis, G.E.; Orszag, S.A., Onset of three-dimensionality, equilibria, and early transition in flow over a backward-facing step, J. fluid mech., 231, 501-528, (1991) · Zbl 0728.76057
[24] A.W. Runchal, Benchmark problems for heat transfer codes, in: B. Blackwell, D.W. Pepper (Eds.), Proceedings of Winter Annual Meeting ASME, Anaheim, CA, 1992, pp. 13-20.
[25] D. Choudhury, Benchmark problems for heat transfer codes, in: B. Blackwell, D.W. Pepper (Eds.), Proceedings of Winter Annual Meeting ASME, Anaheim, CA, 1992, pp. 53-59.
[26] Gartling, D.A., A test problem for outflow boundary conditions – flow over a backward-facing step, Int. J. numer. methods fluids, 11, 953-967, (1990)
[27] Gresho, P.M.; Gartling, D.K.; Torczynski, J.R.; Cliffe, K.A.; Winters, K.H.; Garratt, T.J.; Spence, A.; Goodrich, J.W., Is the steady viscous incompressible two-dimensional flow over a backward-facing step at re=800 stable?, Int. J. numer. methods fluids, 17, 501-541, (1993) · Zbl 0784.76050
[28] Keskar, J.; Lyn, D.A., Computations of a laminar backward-facing step flow at re=800 with a spectral domain decomposition method, Int. J. numer. methods fluids, 29, 411-427, (1999) · Zbl 0948.76061
[29] Comini, G.; Manzan, M.; Nonino, C., Finite element solution of the streamfunction – vorticity equations for incompressible two-dimensional flows, Int. J. numer. methods fluids, 19, 513-525, (1994) · Zbl 0813.76038
[30] Sheu, T.W.H.; Tsai, S.F., Consistent petrov – galerkin finite element simulation of channel flows, Int. J. numer. methods fluids, 31, 1297-1310, (1999) · Zbl 0993.76047
[31] Sani, R.L.; Gresho, P.M., Resume and remarks on the open boundary condition minisymposium, Int. J. numer. methods fluids, 18, 983-1008, (1994) · Zbl 0806.76072
[32] Cruchaga, M.A., A study of the backward-facing step problem using a generalized streamline formulation, Commun. numer. methods eng., 14, 697-708, (1998) · Zbl 0913.76044
[33] Rams˘ak, M.; S˘kerget, L., A subdomain boundary element method for high-Reynolds laminar flow using stream function – vorticity formulation, Int. J. numer. methods fluids, 46, 815-847, (2004) · Zbl 1060.76609
[34] Lee, J.S.; Fung, Y.C., Flow in locally constricted tubes at low Reynolds numbers, J. appl. mech., 37, (1970) · Zbl 0191.56105
[35] Young, D.F.; Tsai, F.Y., Flow characteristics in models of arterial stenosis - I. steady flow, J. biomech., 6, 395-410, (1973)
[36] Deshpande, M.D.; Giddens, D.P.; Mabon, R.F., Steady laminar flow through modelled vascular stenosis, J. biomech., 9, 165-174, (1976)
[37] Lee, T.S., Numerical studies of fluid flow through tubes with double constrictions, Int. J. numer. methods fluids, 11, 1113-1126, (1990) · Zbl 0715.76069
[38] Lee, T.S., A false transient approach to steady state solution of fluid flow through vascular constrictions, Comput. mech., 7, 269-277, (1991) · Zbl 0734.76100
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