Hostname: page-component-84b7d79bbc-tsvsl Total loading time: 0 Render date: 2024-07-29T21:12:26.096Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential asymptotics and the generation of free-surface flows by submerged line vortices

Published online by Cambridge University Press:  06 March 2023

Josh Shelton Open the ORCID record for Josh Shelton [Opens in a new window]  and
Philippe H. Trinh Open the ORCID record for Philippe H. Trinh [Opens in a new window]
Josh Shelton*
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
Philippe H. Trinh*
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
*
†Email addresses for correspondence: j.shelton@bath.ac.uk, p.trinh@bath.ac.uk
†Email addresses for correspondence: j.shelton@bath.ac.uk, p.trinh@bath.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

There has been significant recent interest in the study of water waves coupled with non-zero vorticity. We derive analytical approximations for the exponentially small free-surface waves generated in two dimensions by one or several submerged point vortices when driven at low Froude numbers. The vortices are fixed in place, and a boundary-integral formulation in the arclength along the surface allows the study of nonlinear waves and strong point vortices. We demonstrate that, for a single point vortex, techniques in exponential asymptotics prescribe the formation of waves in connection with the presence of Stokes lines originating from the vortex. When multiple point vortices are placed within the fluid, trapped waves may occur, which are confined to lie between the vortices. We also demonstrate that, for the two-vortex problem, the phenomenon of trapped waves occurs for a countably infinite set of values of the Froude number. This work will form a basis for other asymptotic investigations of wave–structure interactions where vorticity plays a key role in the formation of surface waves.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 958 , 10 March 2023 , A29
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Binder, B.J., Vanden-Broeck, J.-M. & Dias, F. 2005 Forced solitary waves and fronts past submerged obstacles. Chaos 15 (3), 037106.CrossRefGoogle ScholarPubMed
Chapman, S.J., Dallaston, M.C., Kalliadasis, S., Trinh, P.H. & Witelski, T.P. 2022 The role of exponential asymptotics and complex singularities in transitions and branch merging of nonlinear dynamics. In review.Google Scholar
Chapman, S.J. & Vanden-Broeck, J.-M. 2002 Exponential asymptotics and capillary waves. SIAM J. Appl. Maths 62 (6), 1872–1898.Google Scholar
Chapman, S.J. & Vanden-Broeck, J.-M. 2006 Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299–326.CrossRefGoogle Scholar
Crowdy, D.G. 2023 Exact solutions for steadily travelling water waves with submerged point vortices. J. Fluid Mech. 954, A47.CrossRefGoogle Scholar
Crowdy, D.G. & Nelson, R. 2010 Steady interaction of a vortex street with a shear flow. Phys. Fluids 22 (9), 096601.CrossRefGoogle Scholar
Crowdy, D.G. & Roenby, J. 2014 Hollow vortices, capillary water waves and double quadrature domains. Fluid Dyn. Res. 46 (3), 031424.CrossRefGoogle Scholar
Dias, F. & Vanden-Broeck, J.-M. 2004 Trapped waves between submerged obstacles. J. Fluid Mech. 509, 93–102.CrossRefGoogle Scholar
Dingle, R.B. 1973 Asymptotic Expansions: Their Derivation and Interpretation. Academic Press.Google Scholar
Elcrat, A.R. & Miller, K.G. 2006 Free surface waves in equilibrium with a vortex. Eur. J. Mech. (B/Fluids) 25 (2), 255–266.CrossRefGoogle Scholar
Forbes, L.K. 1985 On the effects of non-linearity in free-surface flow about a submerged point vortex. J. Engng Maths 19 (2), 139–155.CrossRefGoogle Scholar
Gazdar, A.S. 1973 Generation of waves of small amplitude by an obstacle placed on the bottom of a running stream. J. Phys. Soc. Japan 34 (2), 530–538.CrossRefGoogle Scholar
Haziot, S., Hur, V., Strauss, W., Toland, J., Wahlén, E., Walsh, S. & Wheeler, M. 2022 Traveling water waves–the ebb and flow of two centuries. Q. Appl. Maths 80 (2), 317–401.CrossRefGoogle Scholar
Hocking, G.C., Holmes, R.J. & Forbes, L.K. 2013 A note on waveless subcritical flow past a submerged semi-ellipse. J. Engng Maths 81 (1), 1–8.CrossRefGoogle Scholar
Holmes, R.J., Hocking, G.C., Forbes, L.K. & Baillard, N.Y. 2013 Waveless subcritical flow past symmetric bottom topography. Eur. J. Appl. Maths 24 (2), 213–230.CrossRefGoogle Scholar
Lustri, C.J., McCue, S.W. & Binder, B.J. 2012 Free surface flow past topography: a beyond-all-orders approach. Eur. J. Appl. Maths 23 (4), 441–467.CrossRefGoogle Scholar
Miksis, M., Vanden-Broeck, J.-M. & Keller, J.B. 1981 Axisymmetric bubble or drop in a uniform flow. J. Fluid Mech. 108, 89–100.CrossRefGoogle Scholar
Moston-Duggan, A.J., Porter, M.A. & Lustri, C.J. 2023 Nanoptera in higher-order nonlinear Schrödinger equations: effects of discretization. J. Nonlinear Sci. 33 (1), 1–47.CrossRefGoogle Scholar
Shelton, J., Milewski, P. & Trinh, P.H. 2021 On the structure of steady parasitic gravity-capillary waves in the small surface tension limit. J. Fluid Mech. 922, A16.CrossRefGoogle Scholar
Shelton, J. & Trinh, P.H. 2022 Exponential asymptotics for steady parasitic capillary ripples on steep gravity waves. J. Fluid Mech. 939, A17.CrossRefGoogle Scholar
Tuck, E.O. & Scullen, D.C. 1998 Tandem submerged cylinders each subject to zero drag. J. Fluid Mech. 364, 211–220.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. & Tuck, E.O. 1985 Waveless free-surface pressure distributions. J. Ship Res. 29 (03), 151–158.CrossRefGoogle Scholar
Xie, X. & Tanveer, S. 2002 Analyticity and nonexistence of classical steady Hele-Shaw fingers. Commun. Pure Appl. Maths 56 (3), 353–402.Google Scholar