Local RBF Collocation Method for Darcy Flow

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 tempera- ture, 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 derivativesof theRBF's. Theenergy and momen- tum equations are solved through explicit time stepping. The pressure-velocity coupling is cal- culated iteratively, with pressure correction, pre- dicted from the local continuity equation viola- tion. This formulation does not require solu- tion 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 me- dia. The numerical examples include studies with different uniform discretization for differentially heated boundaries at filtration Rayleigh numbers Ra F =25, 50, 10 2 ,1 0 3 ,1 0 4 , and aspect ratios A = 1/2, 1, 2. The solution is assessed by com- parison with reference results of the fine mesh fi- nite volumemethod (FVM) in terms of mid-plane velocities, mid-plane and insulated surface tem- peratures, mid-point streamfunction and Nusselt number. The advantages of the method are sim- plicity, accuracy and straightforwardapplicability in non-uniform node arrangements. Keyword: Darcy flow, primitive variables, natural convection, pressure-velocity coupling, meshless methods, local radial basis function col-