Exact solutions for submerged von Kármán point vortex streets cotravelling with a wave on a linear shear current

Keeler, Jack S. ORCID: https://orcid.org/0000-0002-8653-7970 and Crowdy, Darren G. (2023) Exact solutions for submerged von Kármán point vortex streets cotravelling with a wave on a linear shear current. Journal of Fluid Mechanics, 969. ISSN 0022-1120

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New exact solutions are presented to the problem of steadily travelling water waves with vorticity wherein a submerged von Kármán point vortex street cotravels with a wave on a linear shear current. Surface tension and gravity are ignored. The work generalizes an earlier study by Crowdy & Nelson (Phys. Fluids, vol. 22, 2010, 096601) who found analytical solutions for a single point vortex row cotravelling with a water wave in a linear shear current. The main theoretical tool is the Schwarz function of the wave, and the work builds on a novel framework set out recently by Crowdy (J. Fluid Mech., vol. 954, 2022, A47). Conformal mapping theory is used to construct Schwarz functions with the requisite properties and to parametrize the waveform. A two-parameter family of solutions is found by solving a pair of nonlinear algebraic equations. This system of equations has intriguing properties: indeed, it is degenerate, which radically reduces the number of possible solutions, although the space of physically admissible equilibria is still found to be rich and diverse. For inline vortex streets, where the two vortex rows are aligned vertically, there is generally a single physically admissible solution. However, for staggered streets, where the two vortex rows are offset horizontally, certain parameter regimes produce multiple solutions. An important outcome of the work is that while only degenerate von Kármán point vortex streets can exist in an unbounded simple shear current, a broad array of such equilibria is possible in a shear current beneath a cotravelling wave on a free surface.

Item Type: Article
Additional Information: Funding information: J.S.K. acknowledges funding from the Leverhulme Trust, ECF-2021-017. Data availability statement: The code that supports the findings of this study is openly available at https://doi.org/10.6084/m9.figshare.22128548.v3.
Uncontrolled Keywords: general fluid mechanics,vortex interactions,condensed matter physics,mechanics of materials,mechanical engineering,applied mathematics ,/dk/atira/pure/subjectarea/asjc/3100/3104
Faculty \ School: Faculty of Science > School of Mathematics
UEA Research Groups: Faculty of Science > Research Groups > Fluid and Solid Mechanics
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Depositing User: LivePure Connector
Date Deposited: 11 Aug 2023 08:30
Last Modified: 07 Sep 2023 11:30
URI: https://ueaeprints.uea.ac.uk/id/eprint/92814
DOI: 10.1017/jfm.2023.551


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