Numerical Solution of Natural Convection Problems by a Meshless Method

Abstract

Natural convection is a phenomenon where fluid motion is generated by density changes due to the temperature or concentration variations in a gravity field. The computational modelling of systems with natural convection (Bejan, 2004) has become a highly popular research subject due to its pronounced influence in better understanding of nature as well as in the development of the advanced technologies. Melting of the polar ice caps, the global oceans dynamics, various weather systems, water transport, soil erosion and denudation, magma transport and manufacturing of nano-materials, improving casting processes, energetic studies, exploitation of natural resources, welding, casting and advanced solidifications are two typical contemporary example groups where natural convection plays an important role. This chapter deals with the numerical approach towards solution of this type of problems by a meshless technique. The main part of the solution procedure is focused on the general transport equation treatment and the pressure velocity coupling strategy. The transport phenomena are solved by a local meshless method and explicit time stepping. The local variant of Radial Basis Function Collocation Method (LRBFCM) has been previously developed for diffusion problems (Sarler and Vertnik, 2006), convection-diffusion solid-liquid phase change problems (Vertnik and Sarler, 2006) and subsequently successfully applied in industrial process of direct chill casting (Vertnik, et al., 2006). The fluid flow, which is generally a global problem, is treated by the proposed local iterative method. Instead of solving the pressure Poisson equation or/and pressure correction Poisson equation (Divo and Kassab, 2007) a more simplified local pressure-velocity coupling (LPVC) (Kosec and Sarler, 2008a) algorithm is proposed where the pressure-correction is predicted from the local mass continuity violation similar to the SOLA algorithm (Hong, 2004). The presented solution procedure represents a variant of already developed global approach (Sarler, et al., 2004a, Sarler, 2005). In this chapter, such a local solution procedure is tested with the standard free fluid flow benchmark test (de Vahl Davis natural convection test (de Vahl Davis, 1983)). The test is especially convenient for benchmarking purposes as there are several numerical solutions published in the literature (Divo and Kassab, 2007, Hortmann, et al., 1990, Manzari, 1999, Prax, et al., 1996, Sadat and Couturier, 2000, Sarler, 2005, Wan, et al., 2001).