Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems

Sato, Naoki; Yamada, Michio

doi:10.1017/jfm.2019.565

Cited by 3 publications

(4 citation statements)

References 27 publications

(59 reference statements)

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“…For superharmonic disturbances (p = 0), it was shown that, even in the presence of a linear shear current (Ω = 0), steady waves lose stability at the critical amplitude, where the wave energy is an extremum, similarly to the Tanaka instability for irrotational waves (Ω = 0), as shown in figures 5 and 6. This agrees with the theoretical result by Sato & Yamada (2019). In addition, it was found that the critical wave steepness a c or the corresponding parameter γ c changes with Ω and, in particular, γ c of downstream propagating waves (Ω < 0) significantly decreases with increase of the magnitude |Ω| of the shear strength, as shown in figure 7.…”

Section: Discussionsupporting

confidence: 90%

“…From these, we found that, with change of Ω, the critical point moves, but the above three properties (P1), (P2) and P3 a c 0.2229). These agree with the theoretical results by Sato & Yamada (2019). It should be also emphasized that, for downstream propagating waves (Ω < 0), the value of γ c significantly decreases with increase of |Ω|, and this may be related to the onset of wave breaking due to a sharp change of the slope Θ of the water surface shown in figure 2(b).…”

Section: Superharmonic Instabilitysupporting

confidence: 91%

“…Wahlén (2007) derived a canonical Hamiltonian system for water waves on a linear shear current by generalizing Zakharov's formulation with a suitable choice of canonical variables. Sato & Yamada (2019) pointed out that translational symmetry in a canonical Hamiltonian system is essential in the Tanaka instability, and analytically predicted from the result by Wahlén (2007) that the same type of instability can occur even in the presence of a linear shear current.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability analysis of deep-water waves on a linear shear current using unsteady conformal mapping

Murashige

Choi

2020

J. Fluid Mech.

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Section: Discussionsupporting

confidence: 90%

Section: Superharmonic Instabilitysupporting

confidence: 91%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stability analysis of deep-water waves on a linear shear current using unsteady conformal mapping

Murashige

Choi

2020

J. Fluid Mech.

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“…Breaking waves are typically observed for H/λ < 1, as reviewed by Perlin, Choi & Tian (2013). Similar superharmonic instabilities also apply to the GCWs of interest here (MacKay & Saffman 1986;Sato & Yamada 2019). In the absence of surface tension, the classical KH instability associated with steady constant background flows can lead to spontaneous formation of curvature singularities along an inviscid two-fluid interface, consequently causing surface rolling and breaking (Moore 1979;Siegel 1995;Cowley, Baker & Tanveer 1999;DeVoria & Mohseni 2018).…”

Section: Nonlinear Dynamics and Simulation: Unsteady Vortex Methodssupporting

confidence: 65%

On the role of unsteadiness in impulsive-flow-driven shear instabilities: precursors of fragmentation

Shen,

Bourouiba

2023

J. Fluid Mech.

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Shear instabilities at the interface of two fluids, such as classical Kelvin–Helmholtz instability (KHI), is the precursor of interface destabilization, leading to fluid fragmentation critical in a wide range of applications. While many insights into such instabilities are derived for steady background forcing flow, unsteady impulse flows are ubiquitous in environmental and physiological processes. Yet, little is understood on how unsteadiness shapes the initial interface amplification necessary for the onset of its topological change enabling subsequent fragmentation. In this combined theoretical, numerical and experimental study, we focus on an air-on-liquid interface exposed to canonical unsteady shear flow profiles. Evolution of the perturbed interface is formulated theoretically as an impulse-driven initial value problem using both linearized potential flow and nonlinear boundary integral methods. We show that the unsteady airflow forcing can amplify the interface's inherent gravity–capillary wave, up to wave-breaking transition, even if the configuration is classically KH stable. For impulses much shorter than the gravity–capillary wave period, it is the cumulative action, akin to total energy, that determines amplification, independent of the details of the impulse profile. However, for longer impulses, the details of the impulse profile become important. In this limit, akin to a resonance, it is the entangled history of the interaction of the forcing, i.e. the impulse, that changes rapidly in amplitude, and the response of the oscillating interface that matters. The insights gained are discussed and experimentally illustrated in the context of interface distortion and destabilization relevant for upper respiratory mucosalivary fluid fragmentation in violent exhalations.

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The superharmonic instability and wave breaking in Whitham equations

Carter,

Francius,

Kharif

et al. 2023

Physics of Fluids

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The Whitham equation is a model for the evolution of surface waves on shallow water that combines the unidirectional linear dispersion relation of the Euler equations with a weakly nonlinear approximation based on the Korteweg–De Vries equation. We show that large-amplitude, periodic, traveling-wave solutions to the Whitham equation and its higher-order generalization, the cubic Whitham equation, are unstable with respect to the superharmonic instability (i.e., a perturbation with the same period as the solution). The threshold between superharmonic stability and instability occurs at the maxima of the Hamiltonian and L2-norm. We examine the onset of wave breaking in traveling-wave solutions subject to the modulational and superharmonic instabilities. We present new instability results for the Euler equations in finite depth and compare them with the Whitham results. We show that the Whitham equation more accurately approximates the wave steepness threshold for the superharmonic instability of the Euler equations than does the cubic Whitham equation. However, the cubic Whitham equation more accurately approximates the wave steepness threshold for the modulational instability of the Euler equations than does the Whitham equation.

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Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems

Cited by 3 publications

References 27 publications

Stability analysis of deep-water waves on a linear shear current using unsteady conformal mapping

Stability analysis of deep-water waves on a linear shear current using unsteady conformal mapping

On the role of unsteadiness in impulsive-flow-driven shear instabilities: precursors of fragmentation

The superharmonic instability and wave breaking in Whitham equations

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