Full-Potential Integral Solutions for Steady and Unsteady Transonic Airfoils With and Without Embedded Euler Domains

The integral equation solution of the full-potential equation is presented for steady and unsteady transonic airfoil flow problems. The method is also coupled with an embedded Euler domain solution to treat flows with strong shocks for steady flows. For steady transonic flows, three integral equation schemes are well developed. The first two schemes are based on the integral equation solution of the full-potential equation in terms of the velocity field. The Integral Equation with Shock-Capturing (IE-SC) and the Integral Equation with Shock-Capturing Shock-Fitting (IE-SCSF) schemes have been developed. The IE-SCSF scheme is an extension of the IE-SC scheme, which consists of a shock-capturing (SC) part and a shock-fitting (SF) part in which shock panels are introduced at the shock location. The shock panels are fitted and crossed by using the Rankine-Hugoniot relations in the IE-SCSF scheme. The third scheme is based on coupling the IE-SC integral equation solution of the full-potential equation with the psuedo time integration of the Euler equations in a small embedded domain around the shock within the IE computational domain. The integral solution provides the initial and boundary conditions for the Euler domain. This scheme is named as the Integral Equation-Embedded Euler (IE-EE) scheme. These three schemes are applied to different airfoils over a wide range of Mach numbers, and the results are in good agreement with the experimental data and other computational results. For unsteady transonic flows, the full-potential equation formulation in the moving frame of reference has been used. The steady IE-SC scheme has been extended to treat airfoils undergoing time-dependent motions, and the unsteady IE-SC scheme has thus been developed. The resulting unsteady scheme is applied to a NACA 0012 airfoil undergoing a pitching oscillation around the quarter chord length. The numerical results are compared with the results of an implicit approximately-factored Euler scheme