On roll waves down an open inclined channel

Roll waves occur in turbulent flow down open inclined channels, e.g. spillways from dams, open aqueducts, and runoff channels. The equations of shallow water theory, augmented by the Chezy formula are used; energy dissipation is expressed through tangential shear, and a further term is used to express the effect of energy dissipation by shearing normal to the flow. It is shown that the inclusion of such a term does not alter the condition for the stability of the uniform flow and that when the uniform flow is unstable, a one parameter family of quasi steady periodic solutions exists, appearing as a Hopf bifurcation out of the uniform flow at a given critical value. Valid expansions of the periodic solutions are obtained using the Krylov-Bogoliubov-Mitropolski averaging method. The results are also extended to larger amplitudes by numerical integration (from paper)