Solution of Phase Change Problems by Collocation with Local Pressure Correction

This paper explores an application of a novel mesh-free Local Radial Basis Function Collocation Method (LRBFCM) [Sarler and Vertnik (2006)] in solution of coupled heat transfer and fluid flow problems with solid-liquid phase change. The melting/freezing of a pure substance is solved in primitive variables on a fixed grid with convection suppression, proportional to the amount of the solid fraction. 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 [Kosec and Sarler (2008a)]. The solution procedure is assessed on the classical rectangular 2D cavity melting benchmark test [Gobin and Le Quéré (2000)] which encompasses a low Prandtl 0.02 and Stefan number 0.01 situation (metal) with Rayleigh numbers 2.51e4 and 2.5e5, and a high Prandtl 50 and Stefan number 0.1 situation (paraffin wax) with Rayleigh numbers 10e7 and 10e8. The results of the mesh free simulation of the related four cases have been compared with the results of a spectra of different numerical methods [Bertrand, Binet, Combeau, Couturier, Delannoy, Gobin, Lacroix, Quéré Médale, Mencinger, Sadat and Vieira (1998)] in terms of liquid-solid interphase position at a fixed time, and time evolution of the average hot side Nusselt number and average cavity liquid fraction. The results show good agreement with other approaches in terms of the dynamics of the interphase boundary and complicated flow structure, despite the simplest LRBFCM implementation. The advantages of the method are simplicity, accuracy, and straightforward applicability in non-uniform node arrangements.