The stability of hydraulic jumps: analysis, computation, and experiment

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The dynamics of hydraulic jumps is described by hyperbolic partial differential equations with source terms due to frictional losses and channel-bed variations. The analysis of hyperbolic systems has traditionally been focused on systems without non-linear source terms, as the nonlinearities in the hyperbolic operators themselves are rather intricate, and play an important role in the nature of the solutions. Significantly less attention has been paid to the role of nonlinear source terms such as those arising in the shallow-water equations and in the reactive Euler equations. Such source terms are responsible for a rich variety of phenomena, including the complex dynamical features of detonation shock fronts. In this project, we investigate the role of the nonlinear wave interactions arising from source terms in the shallow-water system, which may be responsible for the formation of polygonal hydraulic jumps.This project concerns the dynamics of peculiar flow structures that may emerge from the circular hydraulic jump. When a vertical jet impinges on a flat solid surface, the jet fluid spreads radially in a thinning film until reaching a critical radius at which the film thickness increases dramatically in what is termed a `hydraulic jump'. In certain parameter regimes, despite the axisymmetric source conditions, striking asymmetric flows emerge, including polygonal hydraulic jumps, the explanation for which remains elusive. Our combined theoretical, numerical and experimental project will be focused towards rationalizing these subtle flows by developing the mathematical analogy between hydraulic jumps and detonation shock fronts. A new area of mathematical analysis will be initiated and explored.