Local RBF Collocation Method for Darcy Flow
G. Kosec; and B. Sarler

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 25, No. 3, pp. 197-208, 2008
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Keywords Darcy flow, primitive variables, natural convection, pressure-velocity coupling, meshless methods, local radial basis function collocation method, multiquadrics.
Abstract This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) in solution of coupled heat transfer and fluid flow problems in Darcy porous media. The involved temperature, velocity and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF's. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation. This formulation does not require solution of pressure Poisson or pressure correction Poisson equations and thus much simplifies the Kassab and Divo formulation [Divo and Kassab (2007)]. The solution procedure is represented for a steady natural convection problem in a rectangular cavity, filled with Darcy porous media. The numerical examples include studies with different uniform discretization for differentially heated boundaries at filtration Rayleigh numbers$\unhbox \voidb@x \hbox {Ra}_{\unhbox \voidb@x \hbox {F}} =$25, 50,$10^2$,$10^3$,$10^4,$ and aspect ratios$A=1/2$, 1, 2. The solution is assessed by comparison with reference results of the fine mesh finite volume method (FVM) in terms of mid-plane velocities, mid-plane and insulated surface temperatures, mid-point streamfunction and Nusselt number. The advantages of the method are simplicity, accuracy and straightforward applicability in non-uniform node arrangements.
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