A mathematical and numerical study of roll waves

In this paper we analyze the gravity-driven laminar flow of a shallow fluid layer down an uneven incline with the principal objective of investigating the effect of bottom topography on the instability of the flow. The equations of motion are approximations to the Navier-Stokes equations which exploit the assumed relative shallowness of the fluid layer and fast laminar flow. The explicit dependence on the cross-stream coordinate is eliminated by depth-integrating the equations of motion. A linear stability analysis of the steady flow is carried out by taking, advantage of Floquet-Bloch theory. A numerical scheme is devised for solving the nonlinear governing equations and is used to calculate the evolution of the perturbed steady flow. The results are used to confirm the analytical predictions and to investigate the structure of roll waves. The effect of bottom topography for cases where the bottom profile corresponds to periodic undulations is discussed.