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Sheared free-surface flow over three-dimensional obstructions of finite amplitude

Published online by Cambridge University Press:  18 September 2019

Andreas H. Akselsen Open the ORCID record for Andreas H. Akselsen [Opens in a new window]  and
Simen Å. Ellingsen Open the ORCID record for Simen Å. Ellingsen [Opens in a new window]
Andreas H. Akselsen*
Affiliation:
Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Simen Å. Ellingsen
Affiliation:
Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
*
†Email address for correspondence: andreas.h.akselsen@ntnu.no
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Abstract

When shallow water flows over uneven bathymetry, the water surface is modulated. This type of problem has been revisited numerous times since it was first studied by Lord Kelvin in 1886. Our study analytically examines currents whose unperturbed velocity profile $U(z)$ follows a power law $z^{q}$, flowing over a three-dimensional uneven bed. This particular form of $U$, which can model a miscellany of realistic flows, allows explicit analytical solutions. Arbitrary bed shapes can readily be imposed via Fourier’s theorem provided their steepness is moderate. Three-dimensional vorticity–bathymetry interaction effects are evident when the flow makes an oblique angle with a sinusoidally corrugated bed. Streamlines are found to twist and the fluid particle drift is redirected away from the direction of the unperturbed current. Furthermore, a perturbation technique is developed which satisfies the bottom boundary condition to arbitrary order also for large-amplitude obstructions which penetrate well into the current profile. This introduces higher-order harmonics of the bathymetry amplitude. States of resonance for first- and higher-order harmonics are readily calculated. Although the method is theoretically restricted to bathymetries of moderate inclination, a wide variety of steeper obstructions are satisfactorily represented by the method, even provoking occurrences of recirculation. All expressions are analytically explicit and sequential fast Fourier transformations ensure quick and easy computation for arbitrary three-dimensional bathymetries. A method for separating near and far fields ensures computational convergence under the appropriate radiation condition.

JFM classification

Geophysical and Geological Flows: River dynamics Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Shear waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 878 , 10 November 2019 , pp. 740 - 767
Copyright
© 2019 Cambridge University Press 

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