Application of viscous-inviscid interaction methods to transonic turbulent flows

Two different viscous-inviscid interaction schemes have been developed for the analysis of steady transonic flows. The viscous and inviscid solutions are coupled through the displacement concept using a transpiration velocity. In the semi-inverse interaction method, the viscous and inviscid equations are solved in an explicitly separate manner and the displacement thickness is iteratively updated by a simple coupling algorithm. In the simultaneous interaction method, local solutions of viscous and inviscid equations are treated simultaneously, and the displacement thickness is treated as an unknown and obtained as a part of the solution through a global iteration procedure;The inviscid flow region is described by a direct finite-difference solution of a velocity potential equation in conservative form. The potential equation is solved on a numerically generated mesh by an approximate factorization (AF2) scheme in the semi-inverse method and by a successive line overrelaxation (SLOR) scheme in the simultaneous method. The boundary-layer equations are used for the viscous flow region. The continuity and momentum equations are solved inversely in a coupled manner using a fully implicit finite-difference scheme. The energy equation is solved uncoupled. The FLARE approximation is used in the reversed flow region and its effectiveness is studied by using a windward differencing scheme;The algebraic and one-half equation turbulence models are utilized to describe the Reynolds shear stress in turbulent flow calculations. Parameters affecting the convergence rate of the interaction procedure are discussed. The calculation schemes are evaluated by studying (1) an incompressible laminar flow over a flat plate with a trough, (2) a turbulent transonic flow over an axisymmetric boattail with a cylindrical plume simulator, (3) a turbulent transonic flow over an axisymmetric bump attached to a circular cylinder. The predictions are compared with experimental data and other available numerical results. The simultaneous interaction method becomes more efficient and reliable than the semi-inverse method as the separation size grows. The prediction obtained by the one-half equation turbulence model is generally in good agreement with the measurements, but disagreement is noticeable after the reattachment point