Weakly nonlinear shock propagation in slowly varying one-dimensional flows

Abstract

A theoretical model for the linear and nonlinear evolution of acoustic perturbations in nonuniform one-dimensional isentropic flows is developed. On the assumption that the length scale of the mean flow greatly exceeds that of the imposed disturbances, it is shown that the propagation of linear waves can be described in a closed form that is consistent with the conservation of acoustic power. This permits the application of a “nonlinearization” procedure (Landau [J. Phys. USSR 9, 496 (1945)],Whitham [Proc. R. Soc. London, Ser. A 201, 89 (1950)]), wherein the linear functional forms are assumed to hold for weakly nonlinear simple-wave disturbances, and the effects of amplitude on the wave speed are included to leading order. When nonlinear steepening results in the formation of shocks, they are fitted into the solution using the Rankine-Hugoniot relations. The method is applicable to arbitrary waveforms and, in the present study, the evolution of hump-like and periodic disturbances through nonuniform flows is examined. Focusing on the latter class of disturbances, the parametric range of applicability of the asymptotic theory is assessed by comparing the model predictions with numerical solutions of the Euler equations for different excitation frequencies and amplitudes. The usefulness of the present approach is illustrated by its application to periodic shocks propagating through accelerating and diffusive mean flows.