SPLINE METHOD OF FRACTIONAL STEPS IN NUMERICAL MODEL OF UNSTEADY NATURAL CONVECTION FLOW AT HIGH RAYLEIGH NUMBER

Abstract

The unsteady natural convection flow in a square cavity at high Rayleigh numbers Ra = 10⁷ and 2 × 10⁷, has been treated by using a spline method of fractional steps in each coordinate direction. The essential features of this method are that in each computation step the problem is treated as a one-dimensional case and in the implicit form, and only one tridiagonal matrix system is evaluated. In general, it is not necessary to evaluate the first and second derivatives, and additional boundary conditions are not required. Hence the computer time can be shortened and the storage requirement reduced. The technique represents an improvement over the spline-alternating direction implicit method. Numerical results have been obtained at high Rayleigh numbers. The main features of the transient flow from zero conditions are in good agreement with the theoretical predictions [1, 2]. The steady-state results have been compared with accurate solutions presented recently for Ra = 10⁷.