Hsueh-Chia Chang and Evgeny A. Demekhin

University of Notre Dame

Abstract

A classical experiment of Brock (1969) o roll waves on an inclined plane indicates that the wave period increases linearly downstream as the interfacial waves coalesce irreversible. We examine this unique nonlinear wave coarsening dynamics in the context of coherent structure theory. The dominant roll wave structures are captured by shallow-water equations as localized hydraulic jumps. Moreover, due to the similarity between the mass and moment conservation equations, every hydraulic jump can be scaled into a unique normalized mother structure. This similarity is verified against Brock's data and against numerical simulation over a long plane. Moreover, the wave coarsening dynamics are observed to be driven by a few "excited" jumps which result from earlier coalescence events. These excited jumps contain more liquid and travel faster than the unexcited equilibrium ones. They hence capture the slower equilibrium jumps in a cascade of coalescence events. There is also a faster equilibrium dynamics of remaining jumps after the disappearance of every jump to ensure global mass conservation. Using a new "resonance pole" spectral theory on the mother jump, we determine that the speed of the excited jump increases linearly with respect to the liquid it gains after each coalescence. This linear scaling between wave speed and liquid mass and self-similar scaling of every jump render the coarsening dynamics to be also self-similar at every downstream station and at every level of wave texture (period). The resulting scaling yields the observed linear coarsening rate downstream and quantitatively accurate coarsening rates for all Froude numbers.