Extension of von Kármán’s Transonic Similarity Rule

抄録

An attempt is made to extend von Kármán’s transonic similarity theory so as to cover the whole range of Mach number from subsonic to supersonic. First, the equations for the two-dimensional compressible flow past a slender body are reduced to
\frac∂G∂\barζ=B\left(\frac∂φ₁∂ξ\ ight)² (1) or \frac∂²φ₂∂η²=4B\frac∂φ₂∂ξ\frac∂²φ₂∂ξ² (2)
by assuming that the perturbation velocity is small of order ε and neglecting small quantities of O(ε³). Here we have written Φ=U(x+φ), Ψ=U(y+Ψ), ζ=ξ+iη=x+iμy, Ψ=μχ, φ+iχ=A(φ_⊥+iχ₁)=AG, φ₁=φ₂+ξ⁄4B, B=νA⁄4, μ=\sqrt1−M², ν=4M²[1+(γ+1)M²⁄4μ²] where Φ is the stream function. U and M are respectively the free stream velocity and Mach number, γ is the ratio of specific heats, A is a constant of order O(ε), and G(ζ,\barζ), φ₁(ξ,η), χ₁(ξ,η) are O(1).
Then, it is shown that B=O(t) for M\ eweq1, and that (ν⁄μ)t is an appropriate parameter for M\fallingdotseq1, where t is the thickness ratio of the body. For the ease M\ eweq1, either subsonic or supersonic, the procedure of successive approximation can be applied to Eq. (1). It is to be noted that Eq. (2) is exact to the order O(ε²) in contrast to the fact that the similar equation in the usual transonic approximation theory is correct only to the order O(ε). Hence it is suggested that the parameter (ν⁄μ)t may be used with advantage in place of the usual transonic parameter (γ+1)t⁄μ³ or (γ+1)t⁄(1−M²)^3⁄2.
Finally, Liepmann and Bryson’s experimental data on the transonic flow past wedge sections are analyzed using the parameter (ν⁄μ)t.