TRANSONIC FLOW AROUND OPTIMUM CRITICAL AIRFOILS

Optimum critical nonlifting airfoils that have the highest free stream Mach number M(infinity) 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(infinity) --> 1 and delta --> 0 consists of a sonic are 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.