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A three-dimensional Dirichlet-to-Neumann operator for water waves over topography

Published online by Cambridge University Press:  25 April 2018

D. Andrade Open the ORCID record for D. Andrade [Opens in a new window]  and
A. Nachbin
D. Andrade*
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, Estrada D. Castorina 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil
A. Nachbin*
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, Estrada D. Castorina 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil
*
†Email addresses for correspondence: nachbin@impa.br, dandrade@impa.br
†Email addresses for correspondence: nachbin@impa.br, dandrade@impa.br
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Abstract

Surface water waves are considered propagating over highly variable non-smooth topographies. For this three-dimensional problem a Dirichlet-to-Neumann (DtN) operator is constructed reducing the numerical modelling and evolution to the two-dimensional free surface. The corresponding discrete Fourier integral operator is defined through a matrix decomposition. The topographic component of the decomposition requires special care, and a Galerkin method is provided accordingly. One-dimensional numerical simulations, along the free surface, validate the DtN formulation in the presence of a large-amplitude rapidly varying topography. An alternative conformal-mapping-based method is used for benchmarking. A two-dimensional simulation in the presence of a Luneburg lens (a particular submerged mound) illustrates the accurate performance of the three-dimensional DtN operator.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Topographic effects
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 845 , 25 June 2018 , pp. 321 - 345
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Copyright
© 2018 Cambridge University Press 

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Andrade and Nachbin supplementary movie

Focusing by the Luneburg lens: evolution and refraction of the velocity potential by the submerged circular mound.

Download Andrade and Nachbin supplementary movie(Video)
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