Hodograph design of lifting airfoils with high critical Mach numbers (Q1892195)

We wish to construct airfoils that have the highest free-stream Mach number \(M_ \infty\) for a given set of geometric constraints for which the flow is nowhere supersonic. To construct lifting airfoils, we proceed under the conjecture that the optimal airfoil satisfying a given set of constraints is the one possessing the longest possible arc length of sonic velocity. A boundary value problem is formulated in the hodograph plane using transonic small-disturbance theory whose solution determines an airfoil with long sonic arcs. For small lift coefficients, the hodograph domain covers two Riemann sheets, and a finite difference method is used to solve the boundary value problem on this domain. A numerical integration of the solution around the boundary yields an airfoil shape, and three examples are discussed.