Non-linear water waves on shearing flows

Yaosong, Chen; Ling, Guocan; Jiang, Tao

doi:10.1007/bf02486579

Cited by 4 publications

(2 citation statements)

References 2 publications

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“…In the limit as kh 0 → 0, (7.2)-(7.3) becomes (4.2). Moreover, for small amplitude and long waves, after normalization of parameters, (7.2)-( 7.3) has relevance to a variant of the Boussinesq equations (see [YGT94], for instance) (7.4)…”

Section: The Full-dispersion Shallow Water Equationsmentioning

confidence: 99%

Shallow water models with constant vorticity

Hur

2019

European Journal of Mechanics - B/Fluids

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Abstract. We modify the nonlinear shallow water equations, the Kortewegde Vries equation, and the Whitham equation, to permit constant vorticity, and examine wave breaking, or the lack thereof. By wave breaking, we mean that the solution remains bounded but its slope becomes unbounded in finite time. We show that a solution of the vorticity-modified shallow water equations breaks if it carries an increase of elevation; the breaking time decreases to zero as the size of vorticity increases. We propose a full-dispersion shallow water model, which combines the dispersion relation of water waves and the nonlinear shallow water equations in the constant vorticity setting, and which extends the Whitham equation to permit bidirectional propagation. We show that its small amplitude and periodic traveling wave is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin-Feir instability in the irrotational setting; the critical wave number grows unboundedly large with the size of vorticity. The result agrees with that from a multiple scale expansion of the physical problem. We show that vorticity considerably alters the modulational stability and instability in the presence of the effects of surface tension.

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Section: The Full-dispersion Shallow Water Equationsmentioning

confidence: 99%

Shallow water models with constant vorticity

Hur

2019

European Journal of Mechanics - B/Fluids

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“…Model equations of KdV and Boussinesq type for surface waves in the presence of background shear can be derived in a similar fashion as in the irrotational case. The reader may consult for example [15,40,46,54]. Except for a minor modification, the equation to be used in the present study was derived in [40].…”

Section: The Kdv Equation In the Presence Of Shear Flowsmentioning

confidence: 99%

Wave Breaking in Undular Bores with Shear Flows

et al. 2021

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It is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent.

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Particle trajectories in a weakly nonlinear long-wave model on a shear flow

Ige

Kalisch

2024

Applied Numerical Mathematics

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Non-linear water waves on shearing flows

Cited by 4 publications

References 2 publications

Shallow water models with constant vorticity

Shallow water models with constant vorticity

Wave Breaking in Undular Bores with Shear Flows

Particle trajectories in a weakly nonlinear long-wave model on a shear flow

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