Research on transonic flow of nozzle contractions for providing an asymptotic solution

Abstract

The effect of a contraction on the flow in a supersonic nozzle is investigated, and improvements are made to provide an accurate asymptotic solution of transonic flow. Owing to the elliptical characteristic of the governing equation, supersonic nozzle contractions are not rigorously designed from aerodynamics theory. The contraction effect is often masked by the design defects of the supersonic nozzle contours and, therefore, is hard to evaluate independently. In this study, the authors use a sufficiently advanced supersonic contour to support numerical and theoretical research on various contractions. The results show that the commonly used contractions lead to abrupt changes in transonic flow field parameters, and the degree does not vary with the nozzle Mach number. Based on the work of Hall, the accurate asymptotic solution of the sonic line in the nozzle is derived. Most practical sonic lines provided by various contractions are inconsistent with the accurate asymptotic solution, which further affects the uniformity of the supersonic flow. By iterating the shift of the Witoszynski contraction, the end point curvature is found to be the core parameter. On this basis, three improved contractions are proposed. These new contractions allow discontinuities in wall curvature and avoid the problem inherent in Sivells' cylindrical-quartic–conical-quartic contraction. Each improved contraction successfully coincides with theoretical transonic flow, helping the supersonic nozzle eliminate waves completely and achieve a high degree of flow uniformity.