Coherent structures, self-similarity, and universal roll wave coarsening dynamics

Beyond their inception region from wide-banded inlet noise, roll waves on an inclined plane increase their amplitude and separation downstream in a scale-invariant linear manner by as much as one order of magnitude. Analysis of a set of dissipative hydraulic equations shows that this self-similar coarsening dynamics is driven by irreversible coalescence events between localized roll waves. By exploiting the localized structure of the roll waves and certain symmetries of the hydraulic equations to simple affine coordinate stretching by the substrate thickness, the time-averaged wave properties at every station and wave texture is shown to be self-similar. Moreover, due to weak variations in the substrate thickness and the sensitivity of the coarsening dynamics to it, the downstream evolution along this self-similar family can be captured by a generic cascade coalescence model for all wave separations. The linear coarsening rate of the average wave period is shown to be roughly (root 2-1) times the fraction of "excited" waves, which is only a weak function of a modified Froude number G based on the channel inclination angle. (C) 2000 American Institute of Physics. [S1070-6631(00)00209-9].