Unsteady Analogy

Van Dyke considered the shock as an internal boundary (shock fitting) and wrote the equations in non-conservative form for the smooth flow downstream of the shock wave.

He also replaced the energy equation by the condition that the entropy is constant along streamlines (but not across the shock)

(12.69)

It is clear now, that the x-momentum equation for u is uncoupled from the others. The continuity, the y – and z-momentum equations and the entropy equation can be solved for p, v, w and p, independently of її. If required, її can be determined from Bernoulli’s law

(12.70)

Hence, Van Dyke confirmed Hayes’ observation that the reduced problem is com­pletely equivalent to a full problem for unsteady flow in one less space dimension.

The latter is the unsteady motion in the (y, z) plane due to a moving solid boundary described by B = 0, where x is interpreted as the time. For example, the steady hypersonic flow over a slender pointed body of revolution is equivalent to the prob­lem of unsteady planar motion due to a circular cylinder whose radius varies with time, growing from zero at time x = 0 (see also Goldsworthy [46]).

In fact, for two dimensional steady flows, the equations of motion reduce to those of unsteady one dimensional flow if one of the velocity components remains constant everywhere (see Bird [47]).