Abstract
We wish to construct airfoils that have the highest free-stream Mach number M∞ for a given set of geometric constraints for which the flow is nowhere supersonic. Nonlifting airfoils which maximize M∞ for a given thickness ratio δ are known to possess long sonic segments at their critical speed. 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. The performance of these airfoils is compared with standard airfoils having the same lift coefficient and δ, and it is shown that the calculated airfoils have a 6%–10% increase in critical M∞.
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