A fourth-order compact difference algorithm for numerical solution of natural convection in an inclined square enclosure

Abstract

In this work, a fourth-order compact finite difference (FD) algorithm on the nine-point 2-D stencil is proposed to simulate numerically natural convection in an inclined square enclosure using the vorticity-stream function form of the Naiver-Stokes equations. The key feature of the proposed algorithm is that for all Rayleigh numbers Ra of physical interest, the point-successive overrelaxation or point-successive underrelaxation (SOR) iteration can be used. The numerical capability of the presented algorithm is demonstrated by the application to natural convection in an inclined square enclosure. The angle of inclination of the cavity is varied from $-{90}^o$ (heated from above) to ${90}^o$ (heated from below) in steps of ${15}^o.$ Computations are performed for Rayleigh numbers equal to 10³, 10⁴, 10⁵, 10⁶ and 10⁷ while the Prandtl number is kept constant (Pr = 0.71). Test results, which are presented in terms of streamlines, isotherms, isovorticities and local and average Nusselt number, indicate that the present algorithm could predict the benchmark results for temperature and flow fields on relatively coarser grids.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China under Grand 11472010, 61762073, 11872151 and 11372075, and by the Natural Science Foundation of Ningxia Province of China under Grand NZ16050.