Highly accurate solutions of the bifurcation structure of mixed-convection heat transfer using spectral method

Abstract

This paper is concerned with producing highly accurate solution and bifurcation structure using the pseudo-spectral method for the two-dimensional pressure-driven flow through a horizontal duct of a square cross-section that is heated by a uniform flux in the axial direction with a uniform temperature on the periphery. Two approaches are presented. In one approach, the streamwise vorticity, streamwise momentum and energy equations are solved for the stream function, axial velocity, and temperature. In the second approach, the streamwise vorticity and a combination of the energy and momentum equations are solved for stream function and temperature only. While the second approach solves less number of equations than the first approach, a grid sensitivity analysis has shown no distinct advantage of one method over the other. The overall solution structure composed of two symmetric and four asymmetric branches in the range of Grashof number (Gr) of 0�2 � 106 for a Prandtl number (Pr) of 0.73 has been computed using the first approach. The computed structure is comparable to that found by Nandakumar and Weinitschke (1991) using a finite difference scheme for Grashof numbers in the range of 0�1�106. The stability properties of some solution branches; however, are different. In particular, the two-cell structure of the isolated symmetric branch that has been found to be unstable by the study of Nandakumar and Weinitschke is found to be stable by the current study.