COMPUTATION OF TRANSIENT NATURAL CONVECTION IN A SQUARE CAVITY BY AN IMPLICIT FINITE-DIFFERENCE SCHEME IN TERMS OF THE STREAM FUNCTION AND TEMPERATURE

Abstract

In this article we discuss the application of an implicit scheme to the solution by finite differences of transient natural convection in terms of the stream function and temperature. The second-order energy differential and fourth-order momentum equations are discretized according to the well-known alternate direction implicit (ADI) method. Then the temperature field solution is built based on the classic tridiagonal matrix algorithm (TDMA), and the stream function solution is built based on the two original hypotheses proposed here together with the penta diagonal matrix algorithm (PDMA). The time step in the algorithm is obtained analytically as a function of the Prandtl number and the size of the grid imposing the diagonal dominance condition in the pentadiagonal matrix generated by the hypothesis. We verify the hypothesis and the algorithm using the transient natural convection solution in the following parameter ranges: 10 3 r Ra r 10 6 , 10 -2 r Pr r 10 2 , 21 2 21 r grid r 61 2 61. The transient solution to the problem is presented using the average Nusselt number and local Nusselt numbers on the hot wall, CPU time, the temperature and velocity profiles in the y = 0.5 section, and the temperature in the central point of the cavity. The results of the permanent solution are presented using the maximum value of the stream function within the domain and the average Nusselt number on the cold wall as a function of the Rayleigh number, the relaxation parameter, Prandtl number, and the size of the grid. Finally, both the transient and permanent solution results are compared with the results of five other published studies.