Transonic Flow Around Optimum Critical Airfoils

Abstract

Optimum critical nonlifting airfoils that have the highest free stream Mach number $M_\infty $ for a given thickness ratio $\delta $ have recently been presented by Schwendeman, Kropinski, and Cole [Z. Angew. Math. Phys., 44 (1993), pp. 556–571]. The asymptotic shape of these airfoils as $M_\infty \to 1$ and $\delta \to 0$ consists of a sonic arc with a $( x^{2/ 5} )$ nose. In the present paper the transonic potential flow around these asymptotic shapes is analyzed. Asymptotic expansions of the velocity potential function are constructed in terms of $\delta $ 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, where a leading-edge singularity appears. The inner expansion accounts for the flow around the nose, where a stagnation point exists. The matching between the expansions results in a boundary value problem in the inner region for the solution of a uniform sonic flow about a two-dimensional $( x^{2/ 5} )$ surface. The numerical solution of the inner flow results in the pressure distribution on the nose. Finally, a uniformly valid pressure distribution over the entire airfoil surface is derived.