Stokes waves with constant vorticity: I. Numerical computation

Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES...

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Bibliographic Details
Published in:Studies in applied mathematics (Cambridge) Vol. 142; no. 2; pp. 162 - 189
Main Authors: Dyachenko, Sergey A., Hur, Vera Mikyoung
Format: Journal Article
Language:English
Published: Cambridge Blackwell Publishing Ltd 01.02.2019
Subjects:
Amplitudes
Computational fluid dynamics
Conformal mapping
Hilbert transformation
mathematical physics
nonlinear waves
Numerical analysis
Rigid structures
Rotating bodies
Rotating disks
Rotating fluids
Traveling waves
Vorticity
water waves and fluid dynamics
ISSN:0022-2526, 1467-9590
Online Access:Get full text
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Summary:Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.
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ISSN:0022-2526
1467-9590
DOI:10.1111/sapm.12250