Numerical Solutions of 2-D Unsteady Incompressible Flow in a Driven Square Cavity Using Streamfunction-Vorticity Formulation

Abstract

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.

A Hybrid Analytical Approximate Technique for Solving Two-dimensional Incompressible Flow in Lid-driven Square Cavity Problem

“…Hubert [17], explored the states of incompressible flow in a four-sided lid-driven square cavity by using the hyperbolic method. Ambethkara and Kumarb [18], used the flow-vortex (𝜓 − 𝜉) method to solve the problem of steady two-dimensional incompressible viscous flow in a driven square cavity with moving top and bottom walls. Gürbüz and Tezer [19], applied the Stokes approximation method to the Stokes equations for twodimensional magneto hydrodynamics to study the effect of the magnetic field on flow in a cap-driven cavity.Ali et al [20][21], the Forchheimer model's porous medium with changeable viscosity was used to theoretically examine the continuous dependency of double-diffusive convection .…”

Numerical Solutions of Two-Dimensional Vorticity Transport Equation Using Crank-Nicolson Method

“…In fact, this problem cannot be solved analytically due to the nonlinear terms that appear in equation (1). So that since the last decades, problem (1)-(2), with different initial-boundary conditions, has been solved numerically by some authors using several methods, such as the Petrov-Galerkin finite element method 10 , finite difference schemes, see for instance [11][12][13][14][15][16] , and the boundarydomain integral method 17 . Because of the poor stability properties of explicit finite difference methods, the implicit methods are more recommended to compute the numerical solutions of initial-boundary value problems in two or more dimensions-space.…”

A numerical method for viscous flow in a driven cavity with heat and concentration sources placed on its side wall

“…Natural convection in a rectangular enclosure by a discrete heat source was presented computationally by Qarnia et al [23]. Ambethkar and Manoj Kumar [24] have presented numerical solutions of a 2-D incompressible flow in a driven square cavity. Alleborn et al [25] have investigated a lid-driven cavity with heat and mass transport.…”

“…Alleborn et al [25] have investigated a lid-driven cavity with heat and mass transport. Ambethkar et al [26] have investigated numerical solutions of a 2-D incompressible flow in a driven square cavity using streamfunction-vorticity formulation. Nithyadevi et al [27] have investigated the effect of a Prandtl number on natural convection in a rectangular enclosure with discrete heaters.…”