Fourth-Order Compact Difference For Solving Navier-Stokes Equations And Numerical Simulation Of Natural Convection Heat Transfer

A fourth-order compact finite difference alternating-direction-implicit (ADI) algorithm is proposed for solving the unsteady-state Navier-Stokes (N-S) equations of fluid flow and heat transfer using the streamfunction-vorticity formulation. Point-successive overtaxation iteration (SOR) is used to achieve the steady solutions. The equations of vortex and energy are discretized with alternating-direction-implicit (ADI) scheme in the temporal direction, the convection terms and the viscous terms with fourth-order-accurate symmetrical compact difference approximation in the spatial direction.To check the validity of the present numerical method, the calculated thermally driven flow solutions in a square cavity are compared with the benchmark solutions. The results show that the present algorithms is characterized by its simplicity,efficiency,high order accuracy and stability in computation . This algorithm is applied to modelling the natural convection heat transfer in an enclosure with a heated backward step. Numerical study is concerned with mode-transition and hysteresis phenomena which relate to the geometrical size of enclosure at various Rayleigh numbers.