‘H-states’: exact solutions for a rotating hollow vortex

Crowdy, Darren; Nelson, Rhodri; Krishnamurthy, Vikas S.

doi:10.1017/jfm.2021.55

Cited by 6 publications

(6 citation statements)

References 35 publications

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“…For example, some of the radial geometry analogues are as follows: Crowdy (2002 a ) found exact solutions for a relative equilibrium comprising a central finite-area vortex patch in solid body rotation with corotating point vortices (case 1); Crowdy & Roenby (2014) found exact solutions for an equilibrium comprising a central hollow vortex with satellite point vortices (case 2); Crowdy et al. (2021) found analytical solutions for a rotating hollow vortex equilibrium, which were given the name ‘H-states’, with no satellite point vortices (case 3). These H-states are now understood to be the radial analogues of the constant-vorticity water-wave solutions found by Hur & Wheeler (2020).…”

Section: Discussionmentioning

confidence: 99%

“…Table 1 summarizes how published results already in the literature can now be understood within the case 1–3 taxonomy just described; both the periodic ‘water wave’ geometry and the radial ‘vortex’ geometry are surveyed in this table. Of particular note is that the ‘radial analogues’ of the case 3 water-wave solutions found by Hur & Wheeler (2020) have recently been identified by Crowdy, Nelson & Krishnamurthy (2021) and given the designation ‘H-states’ in analogy with the name ‘V-states’ given to rotating vortex patches. The table also indicates the many known solutions in the radial vortex geometry within the case 1 category and, as already mentioned, it is expected that analogous solutions exist in the water-wave geometry.…”

Section: Water Waves and The Schwarz Functionmentioning

confidence: 96%

See 1 more Smart Citation

Exact solutions for steadily travelling water waves with submerged point vortices

Crowdy

2023

J. Fluid Mech.

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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 also provides a route to finding new exact solutions with this paper focussing on new solutions falling within the case 2 category. Among several presented here are a submerged point vortex pair cotravelling with a solitary deep-water wave, von Kármán point vortex streets cotravelling with a periodic deep-water wave and a point vortex row cotravelling with a wave in water of finite depth. Some other more exotic waveforms are also constructed. All these new solutions generalize those of Crowdy & Roenby (Fluid Dyn. Res., vol. 46, 2014) who found steady waves in deep water cotravelling with a submerged point vortex row for which the free surface shapes turn out to coincide with those of pure capillary waves on deep water found by Crapper (J. Fluid Mech., vol. 2, 1957). The new exact solutions are likely to provide a useful basis for asymptotic or numerical studies when additional effects such as gravity and capillarity are incorporated.

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Section: Discussionmentioning

confidence: 99%

Section: Water Waves and The Schwarz Functionmentioning

confidence: 96%

Exact solutions for steadily travelling water waves with submerged point vortices

Crowdy

2023

J. Fluid Mech.

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“…Previously, similar solutions (uniformly rotating, nonradial) had been found for the two-dimensional Euler equations in the context of vortex patches or smooth, compactly supported functions [7][8][9][10][11][12]. Recently, exact solutions for a rotating hollow vortex (called 'H-states') were found in closed form by Crowdy et al [13], which are rotating vortex sheets with an extra vortex patch in the interior of the sheet. Unlike the rotating sheet, the H-states have zero velocity in the interior of the sheet in the rotating frame.…”

Section: Introductionmentioning

confidence: 69%

“…Recently, exact solutions for a rotating hollow vortex (called ‘H-states’) were found in closed form by Crowdy et al . [13], which are rotating vortex sheets with an extra vortex patch in the interior of the sheet. Unlike the rotating sheet, the H-states have zero velocity in the interior of the sheet in the rotating frame.…”

Section: Introductionmentioning

confidence: 99%

Remarks on stationary and uniformly rotating vortex sheets: flexibility results

Gómez-Serrano

Park

Shi

et al. 2022

Phil. Trans. R. Soc. A.

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“…This complex representation of potential flows continues to be an integral part of the recent research in fluid mechanics, both for canonical [2][3][4][5][6] and applied [7][8][9][10][11][12] flows. In particular, the complex potential is used in both classical and modern monographs about vortex dynamics [13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Stability of two-dimensional potential flows using bicomplex numbers

2022

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The use of the complex velocity potential and the complex velocity is widely disseminated in the study of two-dimensional incompressible potential flows. The advantages of working with complex analytical functions made this representation of the flow ubiquitous in the field of theoretical aerodynamics. However, this representation is not usually employed in linear stability studies, where the representation of the velocity as real vectors is preferred by most authors, in order to allow the representation of the perturbation as the complex exponential function. Some of the classical attempts to use the complex velocity potential in stability studies suffer from formal errors. In this work, we present a framework that reconciles these two complex representations using bicomplex numbers. This framework is applied to the stability of the von Kármán vortex street and a generalized formula is found. It is shown that the classical results of the symmetric and staggered von Kármán vortex streets are just particular cases of the generalized dynamical system in bicomplex formulation.

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‘H-states’: exact solutions for a rotating hollow vortex

Abstract: Abstract

Cited by 6 publications

References 35 publications

Exact solutions for steadily travelling water waves with submerged point vortices

Exact solutions for steadily travelling water waves with submerged point vortices

Remarks on stationary and uniformly rotating vortex sheets: flexibility results

Stability of two-dimensional potential flows using bicomplex numbers

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