Survey of Methods for Exhaust-Nozzle Flow Analysis

accurate prediction of the range, payload, and operational economy of conventional, V/STOL, and supersonic jet aircraft, accurate knowledge of the velocity and discharge coefficients of the exhaust nozzle and the variation of these coefficients with nozzle pressure ratio is required. The need for high accuracy has become increasingly important because of the higher gross-to-net thrust ratios of today's high-bypass ratio turbofan engines and tomorrow's advanced supersonic jet engines. These high gross-to-net thrust ratios have the effect of greatly magnifying errors made in the prediction of the nozzle velocity coefficient. For example, studies of a Mach 2.2 supersonic transport aircraft conducted at the Douglas Aircraft Company have shown that a 1% variation in the nozzle thrust coefficient results in a 3.1% change in the direct operating cost of the airplane and a 2.3797o change in the specific fuel consumption. Thus, the analytical calculations that are used to provide initial design information, to guide the modification of existing nozzles, and to identify geometries for detailed experimental testing must be as accurate as possible. In addition, the wide variety of nozzle configurations that are now being considered dictates that the analytical methods must be flexible in the geometries that they can handle, and economic constraints dictate that the methods must be fast from the standpoint of computer time. In order to have a chance of developing an analytical method with good computational economy, it is necessary to exclude from consideration those phenomena which exercise a secondary effect on nozzle performance. Therefore, unless specifically mentioned, the methods considered in this paper will be restricted to those capable only of solving the problems of two-dimensional (planar or axisymmetric) isentropic (inviscid and shock-free) flow of a perfect gas. Because of the high pressure ratios at which modern jet engines operate, it is necessary that analytical methods be capable of handling mixed flows, that is, flows in which both subsonic and supersonic flow regions are present. Methods that are capable of treating such problems are commonly called transonic flow methods because there are regions where the flow is sonic and near sonic. Because nozzle analysis methods must have this capability, the method of characteristics will not be discussed (since it is limited to supersonic flows), nor will methods that are appropriate only for subsonic flows (such as the methods of classical hydrodynamics). Despite the mixed (subsonic-supersonic) nature of the flowfield, much progress has been made in recent years in the development of methods of solution appropriate for propulsion nozzle analysis. As an indication of this, the recent bibliography of Newman and Allison can be cited which, although restricted to external transonic flow problems, contains over 650 entries. Indeed one finds, as eloquently expressed by Murphy^ that " methods of obtaining numerical solutions of the equations of fluid flow have proliferated to the point where the number of different methods nearly equals the number of active workers in the field." Clearly against such a background, it is impossible to pretend that the survey presented here is complete. It is not the intention nor even the inclination of the authors to present a complete listing of all transonic nozzle analysis methods. The purpose of this survey, rather, is to classify and present critical com-