A finite difference approach to solving the Navier-Stokes equations for the 2-D Lid Driven Cavity problem
Introduction
The Navier-Stokes equations are a set of non linear, partial differential equations that govern the viscous motion of fluids. If used correctly, these equations can be used as a powerful tool to model the flow of fluids in many different scenarios, including weather systems, viscous fluids moving around barriers and even in computer game simulations. Due to the non linear nature of these equations there are very few analytically solvable problems that have been found. Due to this, these equations must be approximately solved by numerical methods and their solutions calculated with the help of a computer. I will be using MATLAB to implement and execute my numerical method for solving these equations for my cavity problem.
The Problem
The Lid Driven Cavity problem has been thoroughly studied and results from different numerical methods approaches agree and are well documented. The problem involves a cavity (usually square or rectangular) where there are non slip (zero velocity) boundary conditions on three edges and a constant velocity imposed on the fourth. Figure 1 shows a diagrammatic view of this. This problem is commonly used to test out the accuracy of newly written fluid mechanics software due to its simple boundary and initial conditions. I will be modelling the time evolution of the velocity field for 10 seconds in intervals of 0.01 seconds and superimposing it onto a 15x15 cavity by use of a quiver plot. I will also plot the vorticity of the flow using a contour plot and will produce a separate surface plot of the pressure field for the cavity.