Brief Outline of the Equations of Fluid Flow

Abstract

Those who are familiar with compressible fluid flow are aware that the equations of motion are nonlinear and, as such, it is very difficult to obtain analytical solutions. As a consequence, numerical methods are generally employed and the differential equations are approximated by finite difference equations and these in turn are solved in a stepwise manner. In many examples involving compressible fluid flow shocks appear and their presence is a complicating factor since they are characterized by very steep gradients in the variables describing the flow, such as, in the velocity, density, pressure and temperature. In fact, the gradients become infinitely steep when the effects of viscosity and thermal conduction are neglected: this introduces discontinuities in the solutions and, as a result, it requires the application of boundary conditions connecting the values across the shock front but the implementation of this technique can be quite complex. However, the need for any boundary conditions can be avoided by using a method proposed in 1950 by Von Neumann and Richtmyer [1] where an artificially large viscosity is introduced into the numerical calculations: the present text utilizes this technique. Instead of obtaining a discontinuous solution at the shock front, the shock acquires a thickness comparable to the spacing of the grid points used in the numerical procedure so that the shock appears as a near-discontinuity and across which velocity, pressure etc. vary rapidly but continuously.