A Meshless Semi-Analytical Method for Solving Convection Problems in Complex-Shaped Closed Cavities

摘要

Recently, a number of meshless techniques were proposed for solving convection-diffusion problems in arbitrarily shaped cavities. Unfortunately, most of them, including the finite difference or finite element methods, do not allow one to get a solution exactly satisfying boundary conditions for domains of irregular geometry. In this report a novel approach is proposed, which gives the possibility to avoid the necessity of constructing a complicated mesh in the neighborhood of the domain boundary. The technique is based on the R-function method combining the means of analytical geometry and projection techniques of mathematical physics, in particular the Galerkin method. Analytical expressions in the form of expansions in certain bases are obtained for the temperature, vorticity, and stream functions. Unlike other mesh and meshless techniques, these semi-analytical expressions satisfy boundary conditions exactly and approximate the temperature and velocity fields inside the cavity with accuracy depending on the number of terms of the expansion.