Nose Drag in Free-Molecule Flow and Its Minimization

Abstract

The superaerodynamic nose drag of a body in a free-molecule flow involves two parameters: the speed ratio 5 between ordered and random molecular motions (modified Mach Number), and the temperature ratio Tr/T between the solid surface and undisturbed gas. Simplifications of the drag formula are obtained at hypersonic as well as low-subsonic extremes. To minimize the drag on a nose of specified length and base radius, the ordinary method of calculus of variation was found inadequate. A generalized approach has, accordingly, been developed, and the specification of end conditions is discussed at length. Results of the present investigation indicate that in all cases an optimum nose requires a flat tip. The optimum nose curve for the hypersonic extreme does not depend on the temperature ratio, but that for the low-subsonic extreme varies in the following manner: for a hot body the curve is convex; for a cold body, concave. An optimum solution exists in a restricted range of specification only. With prescribed tip and base radii the admissible nose length is bounded below for the cases of hypersonic and low-subsonic hot body and bounded above for the case of low-subsonic cold body. A vanishing tip radius leads to an infinitely long nose in the former and a vanishing nose in the latter case. Optimum nose curves for several temperature ratios at the lowsubsonic extreme, as well as the one for hypersonic extreme, are presented. I t is observed that at the low-subsonic extreme, with Tr/T —> co, the hot-body solution asymptotically approaches the hypersonic solution—i.e., a slender conventional warhead with a flat t ip; whereas with Tr/T —*0, the cold-body solution asymptotically approaches the minimal-surface solution—i.e., (a) with prescribed tip radius, a catenoid; (b) with unspecified tip radius, a flat disc.