Minimum Energy Hypersonic Nose and Leading Edge Shapes

Through the gradient method, the Pontryagin maximum principle, is applied to a system of first-order differential equations governing the heat transfer (convection and shock layer radiation) and pressure drag of an axisymmetric or two-dimension al body in hypersonic flow. Numerically determined optimum shapes with given ratios of fineness or thickness are found for minimum drag, minimum heat transfer (convection), minimum heat transfer (radiation), and minimum energy. Minimum energy shapes are found which minimize the sum of convection plus drag work, convection plus radiation plus drag work, and convection plus radiation. The axisymmetric cases produced a near f power-law profile for the minimum drag shape and a flat-faced nose for the minimum heat transfer considering convection alone. The optimum-nose shape for a minimum of radiation heating was found to be conical with a cusped tip. The minimum energy shapes were found to result in various combinations of these forms. The over-all results clearly illustrate the geometric means of alleviating the various energy forms. Drag reduction is produced by a slight degree of blunting near the tip. The effect of radiation is concentrated near the tip where the tendency is toward a cone, and minimum convection tends toward the blunted face. Cp = Cv = D = H = h = L = Moo = Nx = Nr = Pr = p = R = R = f, r = rb = S = s, s = T = L/oo = M, v = w =