Flow Similarity in Compressible Convex-Corner Flows

Abstract

T HE Prandtl-Meyer expansion is well-known in supersonic convex-corner flows. It is also possible to formulate an interaction law in an explicit form that would relate the displacement effect of the boundary layer to the pressure induced in the inviscid part of subsonicflows. In the literature, studies that attributed theflow properties near the corner to the viscous-invisvid interaction at the transonic flow regime are scarce [1]. For a laminar boundary layer, an analysis by Ruban and Turkyilmaz [2] indicated that the displacement effect near the corner point is primarily due to the inviscid part. The flow is governed by the main part of the boundary layer and the potential flow region outside the boundary layer (inviscid-inviscid interaction). However, a subsequent study by Ruban et al. [3] highlighted the contribution of viscous-inviscid interaction, in which the displacement thickness near the corner is affected by the overlapping region that lies between the viscous sublayer and main part of the boundary layer. For a turbulent boundary layer, Chung [4] indicated that there would be a mild initial expansion, followed by a strong expansion near the corner apex and then downstream recompression for a compressible convex-corner flow. AtM 0:64 and 0.83, the flow is expanded from subsonic to transonic speed when the similarity parameter M 6:14. Note that the convex-corner angle ranges from 5 to 17 deg. A small separation bubble might also be born at the formation of ah normal shock wave [5]. With an increasing convexcorner angle atM 0:83, there is a slight upstreammovement of the separation position and a downstreammovement of the reattachment position. The shock-induced separated region at M 8:95 is expanded and also induces intense surface-pressure fluctuations, which are associated with the intermittent nature of the surface pressure signals or shock-excursion phenomena. The amplitude of peak surface-pressure fluctuations could also be scaledwithM [6]. In Chung’s study [4], it is known that M cannot be used as a similarity parameter for the test cases at M 0:34. However, it is also known that there exists a relationship (i.e., the Prandtl-Glauert rule) [7] between the minimum pressure coefficients for the incompressible airfoil Q2 Cp;ic and compressible airfoil Cp;c, where Cp;c Cp;ic= 1 M p . According to the hypothesis that small streamline deflections produce proportionally small changes in Mach number and pressure, a hodograph solution for compressible flow past a corner was given by Verhoff et al. [8]. The viscous sublayer is relatively unimportant when the Reynolds number is sufficiently large. In terms of the stream function , the hodograph equation for compressible corner flows is given as follows: