Similarity laws for supersonic flows

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Abstract
The non-linear differential equation for the velocity potential of three-dimensional steady irrotational supersonic flow past wings of finite span has been investigated. It is found that the whole Mach number range from 1 to ∞ \infty may be divided into two regions (not strictly divided), in each of which similarity laws are obtained, with two parameters K 1 = ( M 2 − 1 ) 1 / 2 / τ n {K_1} = {\left ( {{M^2} - 1} \right )^{1/2}}/{\tau ^n} and K 2 = A ( M 2 − 1 ) 1 / 2 {K_2} = A{\left ( {{M^2} - 1} \right )^{1/2}} ; τ \tau is the non-dimensional thickness ratio, A A the aspect ratio of the wing, M M the Mach number of the uniform stream in which the wing is placed. The factor n n is given explicitly as a function of M M and τ \tau ; in the lower region of Mach numbers it tends to 1 / 3 1/3 as M → 1 M \to 1 , for all τ \tau , giving the ordinary transonic rule, and in the upper region it tends to − 1 - 1 as M → ∞ M \to \infty , for all τ \tau , as in the ordinary hypersonic rule.

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