Evolution of Small Periodic Disturbances into Roll Waves in Channel Flow with Internal Dissipation*

Abstract

This paper concerns the asymptotic behavior of small disturbances subject to periodic boundary conditions as they evolve in time to produce the quasi‐steady pattern of roll waves. The mathematical problem is rather interesting as solutions of the linearized equations are unstable. We use a combination of perturbation and Fourier series methods to derive evolution equations for asymptotic solution, which corresponds to a single mode initially. Due to the nonlinear wave and wave interactions higher modes are generated at higher order. We show that by including the interactions between the first and second modes, these evolution equations turn out to be a system of one first‐order real equation and two second‐order complex equations. We present asymptotic and numerical results to show that our theory predicts the solution accurately for both transient and quasi‐steady traveling wave phases.