Numerical solutions of an unsteady 2-D incompressible flow with heat and mass transfer at low, moderate, and high Reynolds numbers

In this paper, we have proposed a modified Marker-And-Cell (MAC) method to investigate the problem of an unsteady 2-D incompressible flow with heat and mass transfer at low, moderate, and high Reynolds numbers with no-slip and slip boundary conditions. We have used this method to solve the governing equations along with the boundary conditions and thereby to compute the flow variables, viz. $u$-velocity, $v$-velocity, $P$, $T$, and $C$. We have used the staggered grid approach of this method to discretize the governing equations of the problem. A modified MAC algorithm was proposed and used to compute the numerical solutions of the flow variables for Reynolds numbers $Re = 10$, 500, and 50,000 in consonance with low, moderate, and high Reynolds numbers. We have also used appropriate Prandtl $(Pr)$ and Schmidt $(Sc)$ numbers in consistence with relevancy of the physical problem considered. We have executed this modified MAC algorithm with the aid of a computer program developed and run in C compiler. We have also computed numerical solutions of local Nusselt $(Nu)$ and Sherwood $(Sh)$ numbers along the horizontal line through the geometric center at low, moderate, and high Reynolds numbers for fixed $Pr = 6.62$ and $Sc = 340$ for two grid systems at time $t = 0.0001s$. Our numerical solutions for u and v velocities along the vertical and horizontal line through the geometric center of the square cavity for $Re = 100$ has been compared with benchmark solutions available in the literature and it has been found that they are in good agreement. The present numerical results indicate that, as we move along the horizontal line through the geometric center of the domain, we observed that, the heat and mass transfer decreases up to the geometric center. It, then, increases symmetrically.