Flow structure beneath rotational water waves with stagnation points

Ribeiro, Roberto; Milewski, Paul A.; Nachbin, André

doi:10.1017/jfm.2016.820

Cited by 65 publications

(109 citation statements)

References 36 publications

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“…This critical layer has a Kelvin cat eye structure together with the respective formation of critical points, as numerically illustrated by the authors in Ref. .…”

Section: Introductionmentioning

confidence: 62%

“…The paper by Teles and Peregrine conjectured flow features which were later investigated by both numerical and theoretical researchers. These include numerical studies with traveling waves, such as that by Vasan and Oliveras as well as by Ribeiro et al Applied analysis researchers recently made efforts in understanding the impact of constant vorticity on nonlinear traveling wave solutions. A good source of recent articles is found in the theme volume edited by Constantin .…”

Section: Introductionmentioning

confidence: 99%

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Rotational waves generated by current‐topography interaction

2019

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We study nonlinear free‐surface rotational waves generated through the interaction of a vertically sheared current with a topography. Equivalently, the waves may be generated by a pressure distribution along the free surface. A forced Korteweg–de Vries equation (fKdV) is deduced incorporating these features. The weakly nonlinear, weakly dispersive reduced model is valid for small amplitude topographies. To study the effect of gradually increasing the topography amplitude, the free surface Euler equations are formulated in the presence of a variable depth and a sheared current of constant vorticity. Under constant vorticity, the harmonic velocity component is formulated in a simplified canonical domain, through the use of a conformal mapping which flattens both the free surface as well as the bottom topography. Critical, supercritical, and subcritical Froude number regimes are considered, while the bottom amplitude is gradually increased in both the irrotational and rotational wave regimes. Solutions to the fKdV model are compared to those from the Euler equations. We show that for rotational waves the critical Froude number is shifted away from 1. New stationary solutions are found and their stability tested numerically.

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“…This critical layer has a Kelvin cat eye structure together with the respective formation of critical points, as numerically illustrated by the authors in Ref. .…”

Section: Introductionmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

Rotational waves generated by current‐topography interaction

2019

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“…However, even for constant vorticity, Refs. and others numerically observed overhanging profiles and interior stagnation points.…”

Section: Introductionmentioning

confidence: 75%

Stokes waves with constant vorticity: I. Numerical computation

Dyachenko

Hur

2019

Stud Appl Math

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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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“…This contrasts with many other numerical methods, such as those based on surface integro-differential equations (VandenBroeck 1996; Ashton & Fokas 2011) or on the conformal mapping technique (Choi 2009;Ribeiro-Jr et al 2017), which require the results to be mapped back to the physical domain.…”

Section: Computation Of Steady Wavesmentioning

confidence: 99%

“…For a review on the recent rigorous results, the reader can refer to Constantin & Varvaruca (2011) and Kozlov & Kuznetsov (2014). Among the authors using asymptotic methods or purely numerical methods, on can cite Tsao (1959), Dalrymple (1974), Brevik (1979), Simmen & Saffman (1985), Teles da Silva & Peregrine (1988) , Kishida & Sobey (1988), Vanden-Broeck (1996), Swan & James (2001), Ko & Strauss (2008), Pak & Chow (2009), Cheng, Cang & Liao (2009 , Moreira & Chacaltana (2015), Hsu, Francius, Montalvo & Kharif (2016), Ribeiro-Jr, Milewski & Nachbin (2017). Although the recent important theoretical developments have confirmed that periodic waves can exist over flows with arbitrary vorticity, it appears that their stability to infinitesimal disturbances and their subsequent nonlinear evolution have not been studied extensively so far.…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional stability of finite-amplitude gravity waves on water of finite depth with constant vorticity

Francius

Kharif

2017

J. Fluid Mech.

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A numerical investigation of normal-mode perturbations of a two-dimensional periodic finite-amplitude gravity wave propagating on a vertically sheared current of constant vorticity is considered. For this purpose, an extension of the method developed by Rienecker & Fenton (1981) is used for the numerical computations of the finite amplitude waves on a linear shear current. This method enables to compute accurately waves with or without critical layers and pressure anomalies. The numerical results of the linear stability analysis extend the weakly nonlinear, analytical results of Thomas et al. (2012) to fully nonlinear waves. In particular, the restabilization of the Benjamin-Feir modulational instability, whatever the depth, for an opposite shear current is confirmed. For these side-band instabilities, the numerical results show some deviations with the weakly nonlinear theory as the wave steepness of the basic wave and vorticity are increased. Besides the modulational instabilities new instability bands corresponding to quartet and quintet instabilities, which are not side-band disturbances, are discovered. The present numerical results show that with opposite shear currents increasing the shear reduces the growth rate of the most unstable side-band instabilities, but enhances the growth rate of these quartet instabilities, which eventually dominate the Benjamin-Feir modulational instabilities.

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Flow structure beneath rotational water waves with stagnation points

Cited by 65 publications

References 36 publications

Rotational waves generated by current‐topography interaction

Rotational waves generated by current‐topography interaction

Stokes waves with constant vorticity: I. Numerical computation

Two-dimensional stability of finite-amplitude gravity waves on water of finite depth with constant vorticity

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