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Generalised solutions to the Benjamin problem

Published online by Cambridge University Press:  15 April 2020

Vladimir V. Ostapenko Open the ORCID record for Vladimir V. Ostapenko [Opens in a new window]
Vladimir V. Ostapenko*
Affiliation:
Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences, Lavrentyev Avenue 15, Novosibirsk, 630090, Russia Novosibirsk State University, Pirogova street 2, Novosibirsk, 630090, Russia
*
†Email address for correspondence: [email protected]
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Abstract

We generalise the classical Benjamin solution (Benjamin, J. Fluid Mech., vol. 31, 1968, pp. 209–248) modelling the flow in a horizontal duct of finite depth in situations where the flow contains a region spanning the depth of the duct, and a region in which the surface detaches from the ceiling of the duct as a free surface. It is shown that the Benjamin solution belongs to a one-parameter family of similar solutions, which are divided into two types: solutions that describe potential flows where the free surface of the fluid is deflected from the duct ceiling at a zero angle; and solutions that admit the formation of a vortex flow region in the vicinity of the point of fluid separation from the duct ceiling. It is shown that this one-parameter family of solutions is the limit of a two-parameter family of solutions in which part of the uniform flow energy is converted into energy of the small-scale fluid motion. Based on the local hydrostatic approximation, the applicability of the constructed solutions is discussed.

JFM classification

Waves/Free-surface Flows: Channel flow Vortex Flows: Vortex dynamics
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 893 , 25 June 2020 , R1
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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