The inviscid hypersonic similitude was given first by Tsien  as discussed earlier. Based on the equivalence principle by Hayes , the similitude is applicable to general equations of state and three dimensional bodies. Hayes and Probstein , extended the similitude to take into account the interaction effect of the displacement thickness of the boundary layer and in this case, the total drag, including frictional drag, obeys the similarity law for the pressure drag. In viscous hypersonic similitude, the fluid is assumed however to be a perfect gas with additional assumptions on the dependence of the viscosity on temperature.
More recently, Viviand  studied the general similitude for the full Navier – Stokes equations for hypersonic flows involving high temperature real gas effects, whether in thermodynamic equilibrium or not, and showed that exact similitude is not possible in general. Viviand however introduced the concept of approximate similitude and applied it for simple models.
In the following the main important results of these studies are summarized.
A starting point is the Mach number Independence Principle by Oswatitsch . In the case of a perfect gas, a limiting solution of the governing equations and boundary conditions is obtained as M0 approaches infinity. For a general fluid, the free stream density p0 and velocity U must be fixed and the quantities a0, p0, T0 and h0 approach zero. If M0 is sufficiently large, the solution behind the bow shock wave becomes independent of M0 and hence of p0 and T0. Hayes and Probstein  considered this principle is more than a similitude: “Two flows behind the bow shock waves, with different sufficiently high values of M0, are not merely similar but are essentially identical”.
The Mach Number Independence Principle, with p0 and U fixed applies to real fluids, including viscosity, heat conduction, relaxation, diffusion and rarefied gas effects. (For a perfect gas with constant y, the requirement that p0 and U be fixed can be relaxed provided the gas is inviscid). Hayes and Probstein  introduced the “Hypersonic Boundary Layer Independence” where they showed that the principal part of a hypersonic boundary layer, with given pressure and wall temperature distributions and free stream total enthalpy is independent of the external Mach number distribution outside the boundary layer. (The dependence upon Ms and ps appears only within a thin transitional layer at the outer edge of the boundary layer).
Again, the above principle is considered more than a similitude. The viscous hypersonic similitude was then developed taking into consideration the interaction between the displacement thickness distribution of the boundary layer and the external flow field. it is required that both the body and the displaced body are affinely related, hence
– = f (12.240)
t c U
Hayes and Probstein  concluded that x, the interaction parameter, must be invariant, where
X = Mlji: (12,241)
with c = jiwTo/(^oTw) and Re0 = poUl/^o.
The result for the surface pressure distribution is given, for two-dimensional flows, by
C = F(j’K’^Tk’-<’Pr) (12242)
The skin friction and heat transfer coefficient become
7 = G (7 – K-X H Y, Pr) <12,243)
C = H(j-K-x-hTTH–<-Pr) (12244)
Similarly, the total force coefficients CL/т2, CD/т3 as well as the pitching moment coefficient obey similar laws. (Notice, it is necessary that the base pressure follow the similarity law for the base pressure drag to follow the similarity law of the drag).
Formulas for both weak and strong viscous interactions, in terms of the parameter X, have been discussed earlier.