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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Communicated by M.Y. Hussaini
This research was supported by the Air Force Office of Scientific Research under Grant F49620-93-1-0022DEF and by the National Science Foundation under Grant DMS-9157546. Research support for M.C.A.K. was given by the IBM Corporation under a Graduate Research Fellowship.
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Kropinski, M.C.A., Schwendeman, D.W. & Cole, J.D. Hodograph design of lifting airfoils with high critical mach numbers. Theoret. Comput. Fluid Dynamics 7, 173–188 (1995). https://doi.org/10.1007/BF00312361
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DOI: https://doi.org/10.1007/BF00312361