Unified Supersonic-Hypersonic Small Disturbance Theory

It is clear, in the case of the nonlinear transonic small disturbance theory, that subsonic and supersonic theories are special cases.

For hypersonic flow, a connection with the adjoining supersonic range will be advantageous. The difficulty arises in the continuity equation. In the hypersonic theory, it becomes

(12.75)

while in the linearized supersonic theory it reduces instead to

dp du d v dw

UdX + P0 dX + dy + Tz = 0

The term p0 dU must be retained in linearized supersonic theory, and it must be neglected in the hypersonic theory in order to achieve similitude.

Van Dyke proved, however, that the small disturbance hypersonic theory covers the linearized supersonic theory if it is interpreted in accordance with the similarity rule of the latter. The solutions of the hypersonic small disturbance theory remain valid at small values of the parameter M0t provided that the latter is replaced by pr,

where в = m2 – 1 and the pressure and density are scaled as follows

p = p()(yM02r2p – , and p = p0 ^в0p – _L^ (12.77)

The pressure coefficient becomes

The error in the unified theory is O (r2) or O (r/в) whichever is the greater. See Van Dyke [45].