DRBEM Solution for Unsteady Natural Convection Flow in Primitive Variables with Fractional Step Time Advancement

In this study, two-dimensional, transient flow of an incompressible, laminar, viscous fluid in a cavity is considered with the occurance of heat flux (temperature is not constant). The governing equations which are continuity, momentum and energy equations, define natural convection in differentially heated cavities in terms of primitive variables (velocities, temperature, and pressure). The no-slip condition for the velocity is imposed on the cavity walls. Left and right walls are heated and cooled, respectively, and top and bottom walls are adiabatic. The dual reciprocity boundary element method (DRBEM) is employed for solving natural convection flow equations utilizing fractional step for the time derivatives. This uncouples velocities from the pressure. Then, the predicted velocity and the pseudo-pressure equations together with the energy equation are all solved by using DRBEM with constant elements. DRBEM transforms directly the differential equations to the boundary integral equations, and thus, only the boundary of the problem has to be discretized. This saves considerable computational work. Velocities and the pressure are obtained iteratively in the time direction with a predictor-corrector scheme. Temperature is also obtained in the iteration procedure by using relaxation parameter between consecutive time levels. In the iterative procedure the nonlinear convective terms are approximated explicitly from the two previous steps. The present numerical procedure gives quite accurate results for Rayleigh number values up to 10(4). It has the advantage of treating directly the primitive unknowns and obtaining the pressure field also. Since the time derivatives are discretized at the beginning of the procedure, the solution is obtained iteratively at all transient levels, and also at steady-state with considerably large time increment compared to other explicit time integration procedures. The proposed numerical scheme is also computationally cheap since DRBEM discretizes only the boundary of the region, resulting with small sized systems, compared to other domain-type numerical methods