Correspondent profiles and correspondence law

Abstract

It is well known that the parameter of transonic similitude χ and the corresponding similarity rule for the pressure coefficient C _p are well defined when the Mach number of the free stream, M _∞, is so near to one, that (1 — M _∞)² can be neglected with respect to (1 — M _∞). However to extend the similarity rule to a wider range of M _∞ many different methods have been suggested [1, 2, 3] $\begin{gathered} \chi _S = \frac{{1 - M_\infty ^2 }} {{M_\infty ^{4/3} \left( {\gamma + 1} \right)^{2/3} \theta _0^{2/3} }}; \hfill \\ \chi _G = \frac{{1 - M_\infty ^2 }} {{M_\infty ^{4/3} \left[ {2 + \left( {\gamma - 1} \right)M_\infty ^2 } \right]^{2/3} \theta _0^{2/3} }}; \hfill \\ \chi _O = \frac{{1 - M_\infty ^2 }} {{\left( {\frac{{1 - M_\infty ^2 }} {{1 - q*}}q*} \right)^{2/3} \theta _0^{2/3} }} \hfill \\ \end{gathered}$ being q* the ratio between the velocity q and the critical speed a*, while θ ₀ is the thickness parameter, or a characteristic slope of the profile.