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Boundary integrals for oscillating bodies in stratified fluids

Published online by Cambridge University Press:  21 September 2021

Bruno Voisin Open the ORCID record for Bruno Voisin [Opens in a new window]
Bruno Voisin*
Affiliation:
Laboratoire des Écoulements Géophysiques et Industriels, Université Grenoble Alpes, CNRS, Grenoble INP, 38000Grenoble, France
*
†Email address for correspondence: bruno.voisin@univ-grenoble-alpes.fr
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Abstract

The theoretical foundations of the boundary integral method are considered for inviscid monochromatic internal waves, and an analytical approach is presented for the solution of the boundary integral equation for oscillating bodies of simple shape: an elliptic cylinder in two dimensions, and a spheroid in three dimensions. The method combines the coordinate stretching introduced by Bryan and Hurley in the frequency range of evanescent waves, with analytic continuation to the range of propagating waves by Lighthill's radiation condition. Not only are the waves obtained for arbitrary oscillations of the body, with application to radial pulsations and rigid vibrations, but also the distribution of singularities equivalent to the body, allowing later inclusion of viscosity in the theory. Both a direct representation of the body as a Kirchhoff–Helmholtz integral involving single and double layers together, and an indirect representation involving a single layer alone, are considered. The indirect representation is seen to require a certain degree of symmetry of the body with respect to the horizontal and the vertical. As the surface of the body is approached the single- and double-layer potentials exhibit the same discontinuities as in classical potential theory.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
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Journal of Fluid Mechanics , Volume 927 , 25 November 2021 , A3
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© The Author(s), 2021. Published by Cambridge University Press

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