Interfacial hydroelastic hydraulic falls and trapped waves over bottom obstacles

Steady nonlinear flexural-gravity hydraulic falls on the interface of a two-layer density stratified flow past a submerged obstruction on the bottom of a channel are considered. The fluid is assumed to be ideal, and the flow is irrotational. We extend the previous works (Dias and Vanden-Broeck, 2002; Wang et al., 2022) by including the effect of hydroelasticity. The interface is modeled as a thin elastic shell with the Cosserat theory. The boundary integral equation techniques are employed to find steady solutions by numerically solving the full Euler equations. New solutions characterized by subcritical flow upstream with different depth ratios (thick upper layer, thick bottom layer, or critical depth) are found, and the effects of the aspect ratio of obstruction are investigated. By introducing a second obstruction downstream, solutions characterized by a train of trapped waves are sought with wavelength coherent with the prediction of the linear dispersion relation. In addition, solutions with a soliton-like form and oscillatory decaying tails are found when the sheet rigidity is small and the second obstruction is placed upstream.