A time marching method for two- and three-dimensional blade to blade flows

Abstract

Time marching methods are in principle the most flexible means of calculating blade-to-blade flows in turbomachinery since the same method can be used for subsonic, transonic and supersonic flows with automatic inclusion of time dependence and shock waves. In practice time marching has not lived up to its promise because large amounts of computer time have been necessary to obtain limited accuracy; times of 5-10 minutes being typical for a 2D problem. The basic principle of time marching is to start with a guessed flow distribution and integrate the timedependent equations of motion and energy forward with time until a steady-state solution is obtained. The current 'state of the art' has been reviewed by Gopalakrishnan who concentrates on solving the equations in differential form. An alternative method, described by McDonald, applies the equations for conservation of mass, momentum and energy to elemental control volumes. This integral method is claimed to be more stable than the differential methods since all fluxes are conserved once the steady state is reached. However, as used by McDonald the method requires a complex grid and computer times of the order of 5 minutes. Stability has always been a problem with time marching, limiting the size of time step which can be taken and hence the number of steps required to obtain a steady state solution. Many means of ensuring stability have been devised for the differential form of the equations. Some of these have been compared by Emery, all achieve stability only at the expense of extra computation. The only method published which does not rely on some form of artificial viscosity or smoothing is that of Marsh & Merryweather. Their method, however, required a very large number of time steps and so, although simple, it was still expensive. The extension from two to three dimensions is relatively simple in time marching requiring only the solution of an additional momentum equation. However, Gopalakrishnan recently estimated that a 20 hour computer run would be needed to obtain a 3D solution of comparable accuracy to the 2D solutions. Despite this discouragement, time matching seemed to offer the only possibility for solving for transonic flow through highly flared blade rows where the currently available 2D methods are thought to be inadequate. Hence an attempt was made to develop a much faster form of the method.