On the stability of fully nonlinear hydraulic-fall solutions to the forced water wave problem (Q6614118)

This study concerns the stability of hydraulic-fall solutions, which are steady flows where the water level drops from a higher upstream level to a lower downstream level over localised topographic features. While previous studies have explored such flows predominantly over bumps, the stability of flows over dips has remained underexplored. This study aims to fill this gap by rigorously analysing both linear and nonlinear stability of these flows, using a Gaussian-shaped topography to model the bump or dip, and thereby provide a more comprehensive understanding of the dynamics at play.\N\NTo address this problem, the authors employ a combination of numerical and analytical methods. They utilise a computational framework based on the finite element method to simulate the flow and calculate stability properties. The methodology involved solving the full incompressible, irrotational Euler equations, which describe the fluid dynamics, and conducting both linear stability analysis (via eigenvalue problems) and nonlinear stability analysis (through time-dependent simulations). By employing the Hamiltonian structure of the governing equations, the authors are able to deduce the stability conditions and interpret the eigenvalue spectra. Linear stability is evaluated by computing eigenspectra of a linearised operator around the steady-state solution, while nonlinear stability is assessed by solving initial value problems, allowing the researchers to observe the long-term evolution of perturbations applied to steady hydraulic-fall solutions.\N\NThe main findings of the study reveal a distinct difference in stability between flows over bumps and dips. Hydraulic falls over a bump are found to be spectrally stable, indicating that small perturbations do not grow over time, thus preserving the steady flow state. In contrast, flows over a dip are found to be linearly unstable, as small disturbances could grow, leading to a time-dependent behaviour. Time-dependent simulations further show that the unstable flows over dips eventually settled into a time-periodic state, characterised by a large, localised wave pulsating above the dip. This periodic wave emit nonlinear cnoidal waves upstream and multi-harmonic linear waves downstream, demonstrating a clear transition from the initial steady state to a new dynamic equilibrium.\N\NThe significance of this research lies in its contribution to the fundamental understanding of hydraulic falls, which have been observed experimentally and are relevant in natural and engineered water systems. By elucidating the differences in stability between flows over bumps and dips, the study provides insights into how localised topography can impact water waves, which is essential for predicting the behaviour of surface waves in channels, rivers, and coastal regions. Additionally, the use of a Hamiltonian framework and advanced numerical techniques presents a robust approach for further exploration of similar fluid dynamic problems, potentially extending to more complex three-dimensional settings or flows influenced by additional physical factors such as viscosity and surface tension.