A Numerical Simulation of Unsteady Lid Driven Cavity Flow With Moving Boundaries Using CMSIP

A two-dimensional, mathematical model is adopted to investigate the development of circulation patterns for compressible, lid driven flow inside a rectangular cavity. The bottom of the cavity is free to move at a specified speed. The vertical sides and the bottom of the cavity are assumed insulated. The cavity is filled with a compressible fluid with Pr =1. The physics based mathematical model for this problem consists of conservation of mass, momentum (two-dimensional Navier-Stokes equations) and energy equations for the enclosed fluid subjected to appropriate boundary conditions. The density of the fluid is related to pressure and temperature of the fluid through a simple ideal gas relation. The solution procedure requires that a time and space dependent transformation is applied to the governing equations to obtain a rigid (non-moving) solution domain.  The transformed equations are discretized using second order accurate differencing for spacial derivatives and first order forward finite differencing for time derivatives where the computational domain is represented by a uniform and orthogonal mesh. The resulting nonlinear equations are linearized using Newton’s linearization method. The set of algebraic equations that result from this process are then put into a matrix form and solved using the Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns (primitive variables) of the problem.  An in-house CFD code is developed based on the numerical solution algorithm (CMSIP) explained above. Various case studies were carried using the in-house CFD code and a commercial CFD code (Fluent) to validate the accuracy of the CFD code. Two of these were benchmark cases that are widely used in steady laminar incompressible flow modeling: driven cavity flow and laminar channel flow. (The benchmark case studies reported in the literature and used here for comparison purposes are due to Ghia for the driven cavity flow and due to Morihara for the laminar channel flow.) The results of the benchmark case studies show that the velocity predictions by the in-hose code (CMSIP) fairs well when compared to the values given by the commercial CFD code and those reported in the literature. Also, a grid independence study and a time increment independence study were carried out to establish the most efficient (but accurate enough) mesh size and time increment to be used in the unsteady cavity flow simulations. Numerical experiments were carried out using the proposed compressible flow model to simulate the development of lid driven circulation patterns for various Reynolds numbers (100 to 3200) and for a fixed aspect ratio (one in this case) to verify the accuracy of the solution algorithm. Finally, a numerical experiment was carried out for the unsteady, moving bottom driven cavity problem (Re=400) where the aspect ratio of the cavity is changed from 1 to 1.5 at a constant speed. The results indicate the formation of a primary circulation cell developing at the center of the cavity with secondary (small) circulation patterns (vortices) developing at the bottom corners of the cavity. During the downward displacement of the bottom (of the cavity), the lower right vortex moves to the center while growing in strength and size resulting in a secondary circulation cell just below the primary one.