Compact computations based on a stream-function-velocity formulation of two-dimensional steady laminar natural convection in a square cavity

A class of compact second-order finite difference algorithms is proposed for solving steady-state laminar natural convection in a square cavity using the stream-function-velocity (psi-u) form of Navier-Stokes equations. The stream-function-velocity equation and the energy equation are all solved as a coupled system of equations for the four field variables consisting of stream function, two velocities, and temperature. Two strategies are considered for the discretizaton of the temperature equation, which are a second-order five-point compact scheme and a fourth-order nine-point compact scheme, respectively. The numerical capability of the presented algorithm is demonstrated by the application to natural convection in a square enclosure for a wide range of Rayleigh numbers (from 10(3) to 10(8)) and compared with some of the accurate results available in the literature. The presented schemes not only show second-order accurate, but also prove effective. For larger Rayleigh numbers, the algorithm combining the second-order compact scheme for the stream-function-velocity equation with the fourth-order compact scheme for the temperature equation performs more stably and effectively.