COMPARISON OF A DISCRETE-SHOCK, FINITE-DfFFEf^tNCi^ 1ECHNIQUE AND THE METHOD OF CHARACTERISTICS FOR CALCULATING INTERNAL SUPERSONIC FLOWS

Abstract

A numerical method for calculating the inviscid flow within a supersonic inlet is presented. The most unique feature is that existing shock waves or shock waves produced by discrete surface slope changes are propagated as discontinuities through the fixed grid which spans the distance between the walls of the inlet. MacCormack's second-order, predictor-corrector, finite difference scheme Is used to advance the field points on either side of the shock wave. Results are presented for simple two-dimensional configurations and show excellent agreement with the method of characteristics. Introduction Supersonic internal flows vary greatly in complexity, ranging from the simple flow in nozzles to the very complex flows in jetengine internal-compression inlets. Traditionally, solution to such internal flows has relied on the method of characteristics as a computational technique!^^However, as the complexity of the flow increases, the internal logic of a method of characteristics program that can effect a solution becomes increasingly detailed. For extensive design calculations, a technique that incorporates simpler logic yet provides a realistic prediction of complex internal flows would be useful. Shock-capturing, finite-difference techniques represent an attractive alternative to the method of characteristics, particularly if simple shock waves emanating from surface discontinuities can be maintained as discrete-shock waves throughout the internal flow passage while complex imbedded shock waves are captured. This paper describes the development of a discrete-shock, finite-difference technique and its application to the calculation of internal flows. The analysis considers only two-dimensional flow with one shock wave, produced by a surface discontinuity, allowed to propagate into the channel. In general, the technique follows closely the analysis of Kutler,'^^wherein the outer bow shock on a space-shuttle-type vehicle was maintained as a discrete shock and all other shocks in the flow were captured by the finite-difference technique. However, In Kutler's approach, the bow shock represented the outer boundary of the solution, whereas in this analysis, the discrete shock is allowed to propagate through the computational mesh, bounded by both an upper and a lower surface. Presently, shock waves formed by either a curved wall, or a secondary discrete slope change, are retained as imbedded or captured shock waves in the solution, and the change in flow properties are spread over several mesh points. Solutions, obtained by the discrete-shock, finite-difference technique, are compared to those given by the method of characteristics to demonstrate the ability of the finite-difference technique to provide physically realistic flow-field solutions. Mathematical Modeling Review of Basic Eouations In a Cartesian coordinate system, the steady inviscid, twodimensional, fluid dynamic equations (continuity, x and y momentum) can be written In conservative form as: U x ^ ^ = (1) The three components of the vectors 0 and P, which reffresent the conservative variables, are defined as: U = pu kp + pu^ puv ; F = pv puv kp + pv (2) where p is the pressure; p, the mass density; u, the x velocity component; v, the y velocity component; k = 7 I/27; and 7 is the ratio of specific heats of the gas (taken to be 7/5 herein). The units of these equations were normalized by dividing both pressure and density by their respective stagnation conditions, while the velocities were divided by the maximum adiabatic velocity. With this normalization and the further restriction that the flow is adiabatic, the energy equation can be written as: