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

• Published in: Physical review. E, Statistical, nonlinear, and soft matter physics Vol. 85; no. 3 Pt 2; p. 036703
• Main Authors: Yu, Pei Xiang, Tian, Zhen F
• Format: Journal Article
• Language: English
• Published: United States 01-03-2012
• ISSN: 1539-3755; 1550-2376
• DOI: 10.1103/PhysRevE.85.036703

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 (ψ-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.