Fourth-order stability analysis for capillary-gravity waves on finite-depth currents with constant vorticity

Dhar, A. K.; Kirby, James T.

doi:10.1063/5.0136002

Cited by 9 publications

(5 citation statements)

References 57 publications

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“…Considering the significant influence of term on stability of wave modulation (Thomas et al. 2012; Francius & Kharif 2017; Dhar & Kirby 2023), wave kinematic and dynamic properties (Pizzo et al. 2023), and the geometry of the fluid particle trajectories (Wang, Guan & Vanden-Broeck 2020), we included this term in the present study.…”

Section: Mathematical Derivationmentioning

confidence: 99%

“…The first-order problem is composed of waves propagating both forwards and backwards, as Mei (1985) demonstrated, and an additional non-propagative mode, B, which represents the wave-induced mean flow and is generated in the process of nonlinear wave modulations in slow spatial and time scales (Thomas et al 2012). Considering the significant influence of term B on stability of wave modulation (Thomas et al 2012;Francius & Kharif 2017;Dhar & Kirby 2023), wave kinematic and dynamic properties (Pizzo et al 2023), and the geometry of the fluid particle trajectories (Wang, Guan & Vanden-Broeck 2020), we included this term in the present study.…”

Section: First-order Problemmentioning

confidence: 99%

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Bragg scattering of nonlinear surface waves by sinusoidal sandbars

Fang,

Tang,

Lin

2024

J. Fluid Mech.

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Based on the multiple-scale expansion technique, a new set of extended nonlinear Schrödinger (ENLS) equations up to the third order is derived to account for the additional high-order bottom and dispersion effects as well as the nonlinear wave interaction on wave transformation over periodic sandbars of sinusoidal geometry. By employing the small-amplitude wave assumption, a closed-form analytical solution for Bragg scattering is obtained from the linearised ENLS equations, which demonstrates that a downshift of wave frequency of the maximum reflection is mainly due to the inclusion of the high-order bottom effect. The factors that affect the downshift of the resonant frequency are identified and a theoretical expression in parabolic form is derived to quantify the downshift magnitude. The fully ENLS equations are further analysed to reveal the additional wave nonlinear effects on Bragg scattering characteristics. Under the condition of infinitesimal sandbar amplitude, the ENLS equations render a theoretical expression of the critical value of $kh$ when the nonlinear wave self-modulation effect and the nonlinear wave cross-modulation effect are equal, whereas the former effect is responsible for wavenumber upshifting and the latter downshifting. When $kh$ is larger than the critical value, the increase of wave nonlinearity will enhance the downshift magnitude of the Bragg resonance, and vice versa. For finite amplitude of the bottom sandbar, the ENLS equations are solved numerically to examine the influence of both wave nonlinearity and sandbar amplitude on the characteristics of Bragg resonance. The results reveal that as the increase of sandbar amplitude, the critical $kh$ increases monotonically.

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Section: Mathematical Derivationmentioning

confidence: 99%

Section: First-order Problemmentioning

confidence: 99%

Bragg scattering of nonlinear surface waves by sinusoidal sandbars

Fang,

Tang,

Lin

2024

J. Fluid Mech.

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“…The new fourth-order outcome shows a remarkable modification in the instability behaviour from the third-order one in deep water. This paper is an extension of the paper by Dhar and Kirby [9] to include the effect of depth uniform current on modulational instability properties.…”

Section: Introductionmentioning

confidence: 98%

“…Hsu et al [16] then elaborated that paper to include capillarity, and studied both the effects of vorticity and capillarity on modulational instability. Later, Dhar and Kirby [9] derived a fourth-order nonlinear evolution equation (NLEE) for GCWs on finite depth with constant vorticity. From the studies on vorticity modified NLSEs of preceding authors, it is revealed that they considered only the effect of vorticity.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Self-Modulation of Gravity-Capillary Waves on Shear Currents in Finite Depth

PAL,

DHAR

2023

ANZIAM J.

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A nonlinear evolution equation correct to fourth order is developed for gravity-capillary waves on linear shear currents in finite water depth. Therefore, this equation covers both effects of depth uniform currents and uniform vorticity. Starting from this equation, an instability analysis is then made for narrow banded uniform Stokes waves. The notable feature is that our investigation due to fourth order shows a remarkable improvement compared with the third-order one, and produces an excellent result compatible with the exact result of Longuet-Higgins. We observe that linear shear currents considerably change the modulational instability properties of capillary-gravity waves, such as the growth rate and bandwidth of instability.

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“…Furthermore, the existence of stagnation points can generate flows where the pressure at the bottom boundary is out of phase with the free surface [11,12]. Many other authors have investigated waves with constant vorticity [11,13,14,15,16,17,18,19,20,21,22]. The readers are referred to the work by Nachbin and Ribeiro-Jr [23] for a review on numerical strategies adopted for capturing the flow beneath waves with constant vorticity.…”

Section: Introductionmentioning

confidence: 99%

An investigation of the flow structure beneath solitary waves with constant vorticity on a conducting fluid under normal electric fields

2023

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The motion of an interface separating two fluids under the effect of electric fields is a subject that has picked the attention of researchers from different areas. While there is an abundance of studies investigating the free surface wave properties, very few works have examined the associated velocity field within the bulk of the fluid. Therefore, in this paper, we investigate numerically the flow structure beneath solitary waves with constant vorticity on an inviscid conducting fluid bounded above by a dielectric gas under normal electric fields in the framework of a weakly nonlinear theory. Elevation and depression solitary waves with constant vorticity are computed by a pseudo-spectral method and a parameter sweep on the intensity of the electric field is carried out in order to study its role in the appearance of stagnation points. We find that for elevation solitary waves the location of stagnation points does not change significantly with variations of the electric field. For depression solitary waves, on the other hand, the electric field acts as a catalyser that makes possible the appearance of stagnation points - in the sense that in its absence there is no stagnation point.

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Fourth-order stability analysis for capillary-gravity waves on finite-depth currents with constant vorticity

Cited by 9 publications

References 57 publications

Bragg scattering of nonlinear surface waves by sinusoidal sandbars

Bragg scattering of nonlinear surface waves by sinusoidal sandbars

Nonlinear Self-Modulation of Gravity-Capillary Waves on Shear Currents in Finite Depth

An investigation of the flow structure beneath solitary waves with constant vorticity on a conducting fluid under normal electric fields

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