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

Murashige, Sunao; Choi, Woo-Young

doi:10.1017/jfm.2019.1021

Cited by 15 publications

(23 citation statements)

References 50 publications

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“…In particular, in light of the canonical Hamiltonian formulation of water waves with constant vorticity due to Wahlen (2007), Tanaka’s instability (Tanaka 1983) should occur in water waves with constant vorticity, as discussed in § 5. This has been confirmed numerically by Murashige & Choi (2019) for wave profiles endowed with reflectional symmetry.…”

Section: Discussionsupporting

confidence: 58%

Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems

Sato

Yamada

2019

J. Fluid Mech.

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The problem of linear instability of a nonlinear traveling wave in a canonical Hamiltonian system with translational symmetry subject to superharmonic perturbations is discussed. It is shown that exchange of stability occurs when energy is stationary as a function of wave speed. This generalizes a result proved by Saffman [3] for traveling wave solutions exhibiting a wave profile with reflectional symmetry. The present argument remains true for any noncanonical Hamiltonian system that can be cast in Darboux form, i.e. a canonical Hamiltonian form on a submanifold defined by constraints, such as a two-dimensional surface wave on a shearing flow, revealing a general feature of Hamiltonian dynamics.

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Section: Discussionsupporting

confidence: 58%

Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems

Sato

Yamada

2019

J. Fluid Mech.

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“…For the present problem, the waves are superharmonically unstable at any amplitude, and they are subharmonically unstable for any amplitude over the entire range of values. We note that if gravity is included along with the background shear, then subharmonic instability occurs only for certain values of as is the case for pure gravity waves (Murashige & Choi 2020).…”

Section: Discussionmentioning

confidence: 76%

“…The base waves of wavelength are assumed to be travelling with constant speed in the positive direction. Following Murashige & Choi (2020) it is convenient to formulate the mathematical problem in a frame of reference that is travelling at the basic wave speed . In this case the condition as holds, where and are the velocity components in the and directions, respectively.…”

Section: Problem Statementmentioning

confidence: 99%

“…By extending the approach of Dyachenko, Zakharov & Kuznetsov (1996), Choi (2009) reformulated the fully nonlinear background shear problem with gravity included in terms of surface variables only and demonstrated the presence of Benjamin–Feir instability via the solution of an initial value problem. Later Murashige & Choi (2020) presented an extensive study of the same gravity–shear problem using an alternative surface variables formulation. They found that, as for pure gravity waves, the system is subharmonically unstable and superharmonic instability may be triggered at sufficiently high amplitude depending on the sign and strength of the background shear.…”

Section: Introductionmentioning

confidence: 99%

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Stability of waves on fluid of infinite depth with constant vorticity

Blyth

Părău

2022

J. Fluid Mech.

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The stability of periodic travelling waves on fluid of infinite depth is examined in the presence of a constant background shear field. The effects of gravity and surface tension are ignored. The base waves are described by an exact solution that was discovered recently by Hur and Wheeler (J. Fluid Mech., vol. 896, 2020). Linear growth rates are calculated using both an asymptotic approach valid for small-amplitude waves and a numerical approach based on a collocation method. Both superharmonic and subharmonic perturbations are considered. Instability is shown to occur for any non-zero amplitude wave.

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“…As an alternative, the method of conformal mapping has been successfully applied to the two‐dimensional linear stability analysis of large‐amplitude waves, for example, by Longuet‐Higgins 8,9 and Tanaka 10 for pure gravity waves, Hogan 11 and Tiron and Choi 12 for pure capillary waves, and Murashige and Choi 13 for gravity waves on a linear shear current. In particular, Tiron and Choi 12 successfully investigated, for all wave amplitudes, the two‐dimensional stability of the capillary wave solution of Crapper, 7 who analytically obtained a closed‐form exact solution in a conformally mapped complex plane.…”

Section: Introductionmentioning

confidence: 99%

Linear stability of transversely modulated finite‐amplitude capillary waves on deep water

Murashige

Choi

2020

Stud Appl Math

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We investigate the three‐dimensional linear stability of the periodic motion of pure capillary waves progressing in permanent form on water of infinite depth for the whole range of wave amplitudes. After introducing a coordinate transformation based on a conformal map for two‐dimensional steady capillary waves, we perform linear stability analysis of finite‐amplitude capillary waves in the transformed space. To solve the linearized equations for small amplitude disturbances, it is assumed that the wavelengths of the disturbances in the transverse direction are much longer than those in the propagation direction and, therefore, the disturbances are weakly three‐dimensional. This assumption along with the periodicity of solutions allows us to write the linearized equations as an eigenvalue problem in matrix form. Following a perturbation theory for matrices, we expand the solutions of this eigenvalue problem in terms of a small parameter measuring the weak three‐dimensionality, and numerically obtain approximate eigenvalues. For weakly three‐dimensional superharmonic disturbances, the numerical results demonstrate that the pure capillary waves are two‐dimensionally stable, but three‐dimensionally unstable for almost all wave amplitudes. On the other hand, for subharmonic disturbances that are known to be two‐dimensionally unstable, it is found that the long‐wavelength disturbances in the transverse direction reduce the two‐dimensional growth rate near the critical amplitude beyond which the pure capillary waves are unstable.

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

Cited by 15 publications

References 50 publications

Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems

Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems

Stability of waves on fluid of infinite depth with constant vorticity

Linear stability of transversely modulated finite‐amplitude capillary waves on deep water

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