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

Keeler, Jack ORCID: https://orcid.org/0000-0002-8653-7970 and Blyth, Mark (2024) On the stability of fully nonlinear hydraulic-fall solutions to the forced water-wave problem. Journal of Fluid Mechanics. ISSN 0022-1120 (In Press)

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Two-dimensional free-surface flow over localised topography is examined with the emphasis on the stability of hydraulic-fall solutions. A Gaussian topography profile is assumed with a positive or negative amplitude modelling a bump or a dip, respectively. Steady hydraulic-fall solutions to the full incompressible, irrotational Euler equations are computed, and their linear and nonlinear stability is analysed by computing eigenspectra of the pertinent linearised operator and by solving an initial value problem. The computations are carried out numerically using a specially developed computational framework based on the finite element method. The Hamiltonian structure of the problem is demonstrated and stability is determined by computing eigenspectra of the pertinent linearised operator. It is found that a hydraulic-fall flow over a bump is spectrally stable. The corresponding flow over a dip is found to be linearly unstable. In the latter case, time-dependent simulations show that the flow ultimately settles into a time-periodic motion that corresponds to an invariant solution in an appropriately defined phase space. Physically, the solution consists of a localised large amplitude wave that pulsates above the dip while simultaneously emitting nonlinear cnoidal waves in the upstream direction and multi-harmonic linear waves in the downstream direction.

Item Type: Article
Faculty \ School: Faculty of Science > School of Mathematics
UEA Research Groups: Faculty of Science > Research Groups > Fluid and Solid Mechanics
Depositing User: LivePure Connector
Date Deposited: 21 Jun 2024 16:30
Last Modified: 21 Jun 2024 16:30
URI: https://ueaeprints.uea.ac.uk/id/eprint/95659

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