In this paper, we have used a streamfunction-vorticity (ψ-ξ) formulation to investigate the problem of 2-D unsteady viscous incompressible flow with heat transfer in a driven square cavity with moving top and bottom walls. We used this formulation to solve the governing equations along with no-slip and slip wall boundary conditions. A general algorithm was used for this formulation in order to compute the numerical solutions for the low Reynolds numbers Re ≤ 50. The numerical solutions of temperature are calculated for different Prandtl numbers 0.7 (for air) and 6.75 (for water). We have executed this with the aid of a computer programme developed and run in C++ compiler. We have proved the stability and convergence of the numerical scheme using matrix method. Heat transfer is studied by using the local Nusselt number. The uvelocity, v-velocity, pressure, temperature profiles along the horizontal and vertical line through geometric center of the square cavity, isotherms and isobars at different Reynolds numbers Re = 15 and 50 have been depicted.
This study presents the mixed convection inside a four-sided lid-driven square porous cavity whose right wall is maintained at a sinusoidal temperature condition, the left wall of the cavity is maintained at a cold temperature, while the top and the bottom walls are adiabatic. We have discussed two different cases depending upon the direction of the moving walls. Brinkmann-extended Darcy model is represented in terms of and using the stream function-vorticity formulation to simulate the momentum transfer in the porous medium. This formulation is used to solve the governing equations as a coupled system of equations which consists of the field variables, vorticity () , stream function () , and temperature (T). The velocity components (u, v) are derived from the stream function () whereas the average Nusselt number is derived from temperature. The stability and consistency of the applied numerical scheme to the considered problem has been proven by matrix method. The numerical results are investigated by ranging the various dimensionless numbers such as Grashof number (10 3 ≤ Gr ≤ 10 5) , Darcy number (10 −1 ≤ Da ≤ 10 −5) , Reynolds number (10 ≤ Re ≤ 1000) and keeping the Prandtl number (Pr = 0.7) fixed.
In this paper, an unsteady 2-D incompressible fluid flow with heat and mass transfer in a four-sided lid driven square cavity is investigated numerically. The top, bottom, left, and right walls of the square cavity move to the right, left, downward and upward respectively. All four sides of the cavity move with a uniform velocity. The flow variables are simulated below the critical Reynolds numbers with isothermal and mass-transfer conditions in the square cavity. We have used a streamfunction-vorticity (ψ − ξ) formulation to investigate the fluid flow in terms of flow variables ψ, ξ , T and C at low Reynolds numbers (Re). The Prandtl number (Pr) and Schmidt number (Sc) have been chosen as 6.62 and 10, 50, 100, 150 respectively, in order to calculate the numerical solutions of T and C. The matrix method has been used to evaluate the stability and convergence of the numerical scheme. The conditions obtained from the matrix method have been used to arrive at the numerical solutions with desired accuracy.
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