Viscous-inviscid analysis of transonic and low Reynolds number airfoils

Abstract

A method of accurately calculating transonic and low Reynolds number airfoil flows, implemented in the viscous-inviscid design/analysis code ISES, is presented. The Euler equations are discretized on a conservative streamline grid and are strongly coupled to a two-equation integral boundary-layer formulation, using the displacement thickness concept. A transition prediction formulation of the e9 type is derived and incorporated into the viscous formulation. The entire discrete equation set, including the viscous and transition formulations, is solved as a fully coupled nonlinear system by a global Newton method. This is a rapid and reliable method for dealing with strong viscous-inviscid interactions, which invariably occur in transonic and low Reynolds number airfoil flows. The results presented demonstrate the ability of the ISES code to predict transitioning separation bubbles and their associated losses. The rapid airfoil performance degradation with decreasing Reynolds number is thus accurately predicted. Also presented is a transonic airfoil calculation involving shock-induced separation, showing the robustness of the global Newton solution procedure. Good agreement with experiment is obtained, further demonstrating the performance of the present integral boundary-layer formulation. Nomenclature CD = dissipation coefficient, (l/peul)^T(du/dri)dTrj Cf = skin-friction coefficient, 2rwall/pew2 CT = shear stress coefficient, rmax/peul hQ = stagnation enthalpy H = shape parameter, d*/6 H * — kinetic energy shape parameter, 0*/6 II* * = density shape parameter, d**/6 Hk = kinematic shape parameter, f [ 1- (u/ue) ] dr/ + l ( u / u e) [ l- ( u / u e) ] d i i Me = boundary-layer edge Mach number n = transition disturbance amplification variable Ree = momentum thickness Reynolds number, peued/^e p = pressure q =speed ue = boundary-layer edge velocity ur = wall shear velocity, Vrwall/p 5 * = displacement thickness, { [ 1- (pu/peue) ] dy