Nonlinear distortion of traveling waves in non-uniform gasdynamic flows

A theory of nonlinear wave propagation in quasi-one-dimensional non-uniform gasdynamic flows is developed in order to explain the effect of flow non-uniformity on nonlinear processes such as shock formation and wave quenching. The methodology used is the tracking of the propagating wave front in a hyperbolic system whose evolution is governed by a first order nonlinear ordinary differential equation (Riccati's equation). The present work is an extension of Whitham's wave front expansion method for uniform flow. The equation can be exactly solved for the first derivative of a flow variable at the wave front as a function of the location of wave front in the base flow. This solution is used to develop the criteria for shock formation of a compressive disturbance and quenching of an expansion wave. The solution shows that shock formation depends upon the history of wave propagation. However, quenching of a wave front is independent of the initial slope. The analysis is applied to two important gasdynamic flows: Rayleigh line flow and Fanno line flow. The analysis is a generalization of the authors' previous work in which shock formation in convergent-divergent nozzles was studied (Tyagi and Sujith 2005).