Integral analysis of transonic shock wave/boundary-layer interactionin internal flow

Abstract

An approximate integral viscous-inviscid interaction method ispresented for calculating the development of a turbulent boundary layer subjected to a normal shock wave in an internal flow. The inflow conditions and the downstream pressure are provided for the computation. In the supersonic region of shock pressure rise, the Prandtl-Meyer function is used to couple the viscous and inviscid flows. An analytical model for the coupling process is postulated and appropriate equations are defined. Downstream of the sonic point, onedimensional inviscid flow is assumed for coupling with the viscous flow. The viscous flow is calculated using Green's integral lag-entrainment boundary-layer method. Comparisons of the solutions with the experimental data are made for interactions which are unseparated, near separation, and separated. For comparison purposes, solutions to the time-dependent, mass-averaged Navier-Stokes equations incorporating a two-equation, Wilcox-Rubesin turbulence model are also presented. The computed results from the integral method show good agreement with experimental data for unseparated interactions and reasonable agreement with the trend of the viscous effects when the interaction becomes increasingly separated. Nomenclature C E =entrainment coefficient Cf = skin-friction coefficient, 2r^/p e u 2 e C T = shear-stress coefficient F _ = function of C E and C f H,H l ,H = velocity profile shape parameters H k = kinematic value of H M = Mach number n = exponent in *> ref model P = pressure R = radius of the duct Re -Reynolds number T = temperature u = streamwise velocity x_ = axial distance X = (x-x u )/d u A = mass-flow thickness 6 = boundary-layer thickness 6* = boundary-layer displacement thickness 7 = specific heat ratio (=1.4) 0 = boundary-layer momentum thickness X = scaling parameter for secondary influences on turbulence structure v = Prandtl-Meyer function p = density r = shear stress </ > =flow angle at the boundary-layer edge Subscripts 0 EQ = stagnation condition = equilibrium conditions EQ 0 = equilibrium conditions in absence of secondary influence on turbulence structure e = boundary-layer edge F ="ref = 0 r = recovery condition ref = reference condition u = start of interaction oo = freestream condition at the start of interaction