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.
In this paper, numerical solutions are obtained for steady free convective flow in a rectangular region with discrete wall heat and concentration sources by using the finite volume method. The governing equations consist of the continuity, momentum, energy and mass transfer. These equations conjointly with suitable boundary conditions are solved numerically by using this method. The novel concept in this work is to generalize the SIMPLE algorithm suitably and thereby compute the numerical solutions of the flow variables such as the temperature (θ) and the concentration (C) in addition to the components of velocity and the pressure. All non-dimensional parameters are chosen suitably in accordance with the physical significance of the problem under investigation. With the help of these numerical solutions, we have depicted the profiles of the velocity, pressure, temperature and concentration along the horizontal and vertical directions of the geometric centre of the region. The validity of the numerical solutions are ensured by comparing the present solutions with the benchmark solutions. Code validation has been given for the present problem.
Numerical simulations for 2-D unsteady, incompressible flow with heat transfer in a four-sided lid-driven rectangular domain are reported in the present study. For the four-sided lid-driven rectangular domain, the lower wall is moved to the left, the upper wall is moved to the right, while the right wall is moved upwards and the left wall is moved downwards. All four walls move with equal speed. Different constant temperatures are applied to the left and right moving walls, and thermal insulation is applied to the upper and bottom moving walls. The governing equations are discretized using the QUICK scheme of finite volume methods. The SIMPLE algorithm is adopted to compute the numerical solutions of the flow variables, u-velocity, v-velocity, P, and θ as well as local and average Nusselt numbers for 50 ≤ Re ≤ 1500 and Pr = 6.63. Due to the force generated by moving fluid, the direction of moving walls and the Reynolds number affect fluid flow in the rectangular domain in addition, at different Reynolds numbers along the cold wall of the domain, the variation in average and local Nusselt numbers reveals that overall heat transfer increases isotherms showed that as Reynolds numbers increase, the horizontal temperature gradient near the vertical walls decreases, because of which heat transfer decreases.
In this paper, we have used a streamfunction-vorticity (ψ − ξ) formulation to investigate the problem of 2-D unsteady viscous incompressible flow 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 flow variables: streamfunction ψ, vorticityfunction ξ for low Reynolds numbers Re ≤ 50. 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. Following this, the stability conditions obtained for the time and space steps have been used in numerical computations to arrive at the numerical solutions with desired accuracy. Also investigated was the variation of u and v components of velocity at points closed to left side, top and bottom walls of the square cavity at different time levels for Reynolds number 50. Based on the numerical computations of u-velocity, we found: (i) the u-velocity is almost same near the top and bottom wall of the square cavity above and below the geometric center for Re=15 and 30; (ii) the absolute value of the u-velocity first decreases, then increases, and finally decreases, near top and bottom wall of the square cavity. Based on the numerical computations of the v-velocity, we found: (i) the absolute value of the v-velocity increases uniformly as time level increases' (ii) the v-velocity at the right wall of the square cavity attains a maximum value, and then, decreases towards the geometric center. The numerical solutions of the vorticity vector ξ at Re=50 for different times along the horizontal line through geometric center of the square cavity indicate the following: (i) the vorticity contours decreased in between the left wall boundary to the midpoint of the square domain; (ii) it, then, increased in between the midpoint to the right wall boundary. Based on the numerical solutions of streamfunction ψ, we found: (i) the size of streamline contours decreased when the time increased; (ii) two streamline contours are generated, one above the geometric center, and the other, below the geometric center in clockwise direction.
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