Inviscid transonic flow field analysis

Abstract

This paper describes a method for analysing inviscid transonic flow. This method is based on the fact that the angle made by the streamline of the transonic flow and of the corresponding incompressible flow is usually small. By using curvilinear coordinates, the differential equation of the stream function of an inviscid compressible flow is simplified and a general solution of the equation obtained.As examples of the method, transonic solutions are given for flow through twodimensional and axisymmetric Laval nozzles of different throat wall radii together with sonic lines and iso-Mach lines. To determine the discharge coefficients of Laval nozzles, an integral relation is developed. The general behaviour of the transonic flow in the throat region is presented, and the effect of the mass discharge on the Mach number distribution in the nozzle analysed. The effects of the ratio of the specific heats on the characteristics of the flow in the throat region are discussed. For transonic flow around a circular cylinder and a sphere, sonic lines and iso-Mach lines are presented for free-stream Mach number varying from the subcritical to the supercritical, including a free-stream Mach number of one.Part of the results obtained are compared with those available in current literature. For the two-dimensional hyperbolic Laval nozzles, the iso-Mach lines are compared with those given by Cherry (1959) and Serra (1972). For, axisymmetric Laval nozzles, the discharge coefficient and the Mach number at the throatsection for various throat wall radii are compared with those given by Sauer (1944), Hall (1962), Kliegel & Levine (1969), and Klopfer & Holt (1975). The theoretical discharge coefficients are compared with the experimental results by Backet al (1975), Durham (1955), Norton & Shelton (1969) etc. For the transonic flow around a circular cylinder, the iso-Mach lines are compared with Cherry’s exact solution for the quasi-circular cylinder for.M∞ equal to 0.51. The. Mach number distributions on the surface of the circular cylinder are compared with those given by Imai (1941) forM∞ equal to 0.4, by Cherry (1947) forM∞ equal to 0.51, by Dorodnicyn (1956) forM∞ equal to 1, and by Hafez, South & Murman (1979) forM∞ equal to 0.51.The present method has a much wider scope of application, requires simpler computation and gives results with good accuracy. It is being used to analyse supercritical wings and cascades, and we expect to extend its application to the field of transonic unsteady flow.