Transonic flow of dense gases around an airfoil with a parabolic nose

Transonic potential flow of dense gases of retrograde type around the leading edge of a thin airfoil with a parabolic nose is studied. The analysis follows the approach of Rusak (1993) for a perfect gas. Asymptotic expansions of the velocity potential function are constructed in terms of the airfoil thickness ratio in an outer region around the airfoil and in an inner region near the nose. The outer expansion consists of the transonic small-disturbance theory for dense gases, where a leading-edge singularity appears. Analytical expressions are given for this singularity by constructing similarity solutions of the governing nonlinear equation. The inner expansion accounts for the flow around the nose, where a stagnation point exists. A boundary value problem is formulated in the inner region for the solution of an oncoming uniform sonic flow with zero values of the fundamental derivative of gasdynamics (Gamma = 0) and the second nonlinearity parameter (Lambda = 0) around a parabola at zero angle of attack. The numerical solution of the inner problem results in a symmetric flow around the nose. The outer and inner expansions are matched asymptotically resulting in a uniformly valid solution on the entire airfoil surface. In the leading terms, the flow around the nose is symmetric and the stagnation point is located at the leading edge for every transonic Mach number, and small values of Gamma and Lambda of the oncoming flow and any shape and small angle of attack of the airfoil. Furthermore, analysis of the inner region in the immediate neighbourhood of the stagnation point reveals that the flow is purely subsonic, approaching critical conditions in the limit of large (scaled) distances, which excludes the formation of shock discontinuities in the nose region.