Solution of Navier-Stokes Equations for the Non-Linear Hydrodynamic Slider by Matrix Algebra Methods

Abstract

The equations of viscous flow, the Navier-Stokes equations, are difficult to solve because they are non-linear, multi-variable, partial differential equations. For isothermal two-dimensional flow three equations have to be solved simultaneously, the momentum equation for each direction and the continuity equation. The usual numerical method is to replace the continuous system with the discrete representation at a number of points by the finite difference approximations to these equations, assuming that the solution to the discrete system approximates and converges with increasing point representation to the continuous solution. An iteration process, whereby the non-linear terms provide corrections to the solution of the linear portion of the equations, would tax many digital computers from the viewpoints of both storage and time. The use of matrix methods discussed in this paper enables considerable simplification and savings in both computer storage and time. Further non-linear terms present no difficulty whatsoever and are*** readily incorporated in the method of solution. This approach permits ready generalization not only to other problems in hydrodynamics but also to other non-linear differential equations. The only knowledge required is that associated with normal matrix methods and finite difference formulae applicable to numerical differentiation and integration. It is thought that the simplicity of the method will appeal to engineers in general. The first part of the paper examines the basic equations by the usual order of magnitude approach and reduces them to their simplest form with the dominant inertia terms retained. The numerical results are shown to be in good agreement with a single perturbation solution. The second part describes the numerical solution of the equations.