Unsteady Hypersonic Flows

Extension to unsteady motions involving small time dependent oscillations of a thin body exposed to a steady uniform stream is straightforward. The full problem has a total derivative including local time derivative term, i. e.

with a reduced time introduced where t = Ut/L, the continuity equation for example becomes

(12.72)

The equation for u remains uncoupled from the other equations. However, u can no longer be found in terms of the other variables since there is no counterpart of the Bernoulli’s equation. If required, U can be obtained form solving the x-momentum equation.

Notice that x and Ї derivatives appear only in the combination

д d д Ї + dx

hence, introducing the coordinates

x = x – Ut, or x = x – Ut

reduces the unsteady small disturbance problem to exactly the form of the steady problem. This means that Hayes’ analogy remains valid for unsteady motion if the variation of the contour with time is taken into account. For more details on this point, see Hamaker and Wong [48], and Lighthill [49].