Compressible modification of boundary-layer momentum thickness localized method in transition model

In current transition models, Menter's correlation between the maximum vorticity Reynolds number Re_vmax in the boundary layer and momentum thickness Reynolds number Re_θ has been widely used in the localization of the boundary layer length scale. However, the existing correlations have problems in correctly reflecting compressibility effects and only relying on local variables. In this paper, the similarity solutions of laminar compressible boundary layer equations are adopted to study the relation between Re_vmax and Re_θ. Through extracting the local Mach number Ma_local and the ratio of wall temperature to the local temperature T_w/T_local right at the position where Re_vmax happens, the influences of the Mach number effect and wall heat transfer effect are characterized, meanwhile a new correlation only relying on local variables between Re_vmax and Re_θ is proposed. For plat-plate boundary layers, the relative error is tiny within a large Mach number and wall-to-adiabatic temperature ratio range and the error caused by the streamwise pressure gradient is not of great concern for general pressure gradients. Furthermore, numerical results over common geometries, including wedges, sharp cones and blunt cones, demonstrate that the proposed correlation could give much better predictions of the relation between Re_vmax and Re_θ than previous correlations. At last, the present correlation is applied to the γ-Re_θ transition model as an application presentation.