Asymptotic solution for the one dimensional Euler equations for isentropic flow in a variable area duct

The objective of this paper is to derive an asymptotic solution to the one-dimensional Euler equations for isentropic flow through ducts with slowly varying area. The homogeneous (or constant-area) problem is generally handled using Riemann's method of characteristics. We solve the variable-area problem by an asymptotic expansion about this homogeneous solution. A length scale characterizing the area variation is introduced and an asymptotic power series in increasing powers of the inverse of the length scale s are constructed for the unknowns. The problem reduces to solving the homogeneous Euler equation, and coupled linear PDEs for successive correction terms in the asymptotic series. An integration methodology is also presented for simple wave regions. As an illustration, we obtain closed-form analytical expressions for the first order perturbation terms in the case of an exponential duct for a sample simple wave. Nonlinear distortion of the wavefront is captured accurately in the analytical solution, as verified by comparison with numerical results from CLAWPACK, a finite-volume simulation package for conservation laws. Other issues such as asymptoticness and convergence of the series are discussed.