Analysis of unsteady transonic channel flow with shock waves

The inviscid, unsteady flow in a two-dimensional channel with a sonic throat is analyzed using an asymptotic expansion for the velocity potential in terms of a small parameter E, which is a measure of the nondimensional perturbation velocity. The analysis includes the case where shock waves exist in a channel with arbitrary wall shape, with arbitrary disturbances imposed at a given downstream location. The problem considered in detail is one where the channel walls are fixed and the flow is steady in first order, but unsteady in second order terms, with the shock motion relative to an initial steady state shock position. Solutions for the first and second order perturbation potential are derived. Numerical results for the flow field in an accelerating nozzle flow with a shock wave, where the nozzle back pressure is oscillating sinusoidally, show the resulting shock wave motion and unsteady flow downstream of the shock wave. The results indicate that in a channel with a variable wall contour over a substantial portion of its length, the imposed initial wave form is modified as it propagates forward.