2023
DOI: 10.1017/jfm.2022.1058
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Exact solutions for steadily travelling water waves with submerged point vortices

Abstract: This paper presents a novel theoretical framework, based on the concept of the Schwarz function of a wave, for understanding water waves with vorticity in the absence of gravity and capillarity. The framework leads naturally to a taxonomy of three subcases, herein referred to as cases 1, 2 and 3, into which fall three existing studies of water waves incorporating uniform vorticity and submerged point vortices. This provides a theoretical unification of several seemingly unrelated results in the literature. It … Show more

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Cited by 7 publications
(17 citation statements)
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“…Finally, we note that, in this paper, the forcing mechanism producing the waves was via the complex-plane singularities associated with the point vortices -then, we found that the waves were singularly perturbed due to the inertial term in Bernoulli's equation, thus producing exponentially small waves, scaling as exp(−const./F 2 ). Recently, analytical solutions have been developed for pure-vorticity-driven water waves, notably in the works by Crowdy & Nelson (2010), Crowdy & Roenby (2014) and Crowdy (2023). In essence, we believe these solutions can serve as leading-order approximations in the regime of small surface tension; it might be expected that exponentially small parasitic ripples then exist on the surface of such vorticity-driven profiles.…”
Section: Discussionmentioning
confidence: 99%
See 1 more Smart Citation
“…Finally, we note that, in this paper, the forcing mechanism producing the waves was via the complex-plane singularities associated with the point vortices -then, we found that the waves were singularly perturbed due to the inertial term in Bernoulli's equation, thus producing exponentially small waves, scaling as exp(−const./F 2 ). Recently, analytical solutions have been developed for pure-vorticity-driven water waves, notably in the works by Crowdy & Nelson (2010), Crowdy & Roenby (2014) and Crowdy (2023). In essence, we believe these solutions can serve as leading-order approximations in the regime of small surface tension; it might be expected that exponentially small parasitic ripples then exist on the surface of such vorticity-driven profiles.…”
Section: Discussionmentioning
confidence: 99%
“…The exponential asymptotics techniques developed in this work can also be extended to situations where capillary ripples are forced on the surface of steep vortex-driven waves. The leading-order solution in these asymptotic regimes would then be known analytically from the works of Crowdy & Nelson (2010), Crowdy & Roenby (2014) and Crowdy (2023), for example. We shall discuss these and other exciting future directions in § 6.…”
Section: Introductionmentioning
confidence: 99%
“…Once the expression (1.4) for the complex velocity field has been derived in terms of the Schwarz function S(z) of the wave profile, the case 1 category of solutions follows simply by taking q = 0, which means that the form of the complex velocity field reduces to (1.8), as explained above. This is the generalized viewpoint espoused by Crowdy (2022). However, because the present paper focuses on only case 1 solutions, it is possible to defer to the earlier work of Crowdy & Nelson (2010) and offer a more direct formulation in this case.…”
Section: Case 1 Category Of Solutionsmentioning
confidence: 99%
“…For present purposes, it is enough to point out that after describing the general framework, Crowdy (2022) focused on producing a range of new solutions falling within the case 2 category. Among these are solutions describing two submerged vortex rows, also known as von Kármán vortex streets, cotravelling with a free surface wave but where the flow was otherwise irrotational; the earlier work of Crowdy & Roenby (2014) had found steady waves cotravelling with a single submerged point vortex row.…”
Section: Introductionmentioning
confidence: 99%
“…The transformation of the boundary condition can then be interpreted (at least for analytic boundaries) as an embedding in the Schwarz function of the boundary curve (Shapiro 1992; Ablowitz & Fokas 2003). Such encoding of information using Schwarz functions has been recently studied in the context of travelling water waves (Crowdy 2023).…”
Section: Effective Boundary Technique For Finite Anchoring Strengthmentioning
confidence: 99%