GDQ METHOD FOR NATURAL CONVECTION IN A SQUARE CAVITY USING VELOCITY–VORTICITY FORMULATION

Abstract

ABSTRACT This article describes a compact numerical algorithm based on the generalized differential quadrature (GDQ) method for the numerical analysis of natural convection in a differentially heated square cavity. The velocity–vorticity form of the Navier–Stokes equations and energy equation are used to represent the mass, momentum, and energy conservations of the fluid medium in the cavity. The GDQ form of the governing equations and the vorticity definition at the boundaries are solved by a coupled solution algorithm using a global matrix scheme for all the field variables. The vorticity values at the boundary are correctly imposed using the GDQ method, which approximates a given space derivative with higher-order accuracy compared to the existing schemes based on Taylor's series expansion. This has assured a divergence-free solution for the flow field by satisfying the continuity constraint, though the pressure term is not used directly in the present formulation. The proposed algorithm is validated for a lid-driven cavity flow for Reynolds number Re = 100, 400, and 1,000, and the predicted velocity profiles are in excellent agreement with the benchmark solutions. The algorithm is then used to compute the average Nusselt number and flow parameters for natural convection in a square cavity for Rayleigh number Ra = 103, 104, 105, and 106. These results are in better agreement with the benchmark solutions than the results obtained by other numerical schemes, which used much finer grids compared to the present scheme.