Finite-amplitude steady-state resonant waves in a circular basin

Yang, Xiaoyan; Dias, Frédéric; Liu, Zeng; Liao, Shijun

doi:10.1017/jfm.2021.165

Cited by 8 publications

(22 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First, triad 7 in table 1 (dark grey row) was explored by Michel [40] for a circular cylinder of radius 9.45 cm and an approximate fluid depth of 3 cm; it follows that the depth-to-radius ratio in Michel's experiment was approximately 0.317, close to the value of 0.30197 reported in table 1. Furthermore, table 1 (grey rows) incorporates two well-known examples of a 1:2 resonance, for which modes 1 and 2 coincide: (i) the critical depth h c = 0.83138 (triad 19) corresponds to the second-harmonic resonance with the fundamental mode [42,43,8,64]; (ii) the critical depth h c = 0.19814 (triad 8) corresponds to a standing wave composed of two resonant axisymmetric modes [34,64]. Finally, triads 1, 2, 3, 5 and 6 (table 1, light grey rows) form an interesting class of resonant triad, for which an axisymmetric mode (m 3 = 0) interacts with two identical counter-propagating non-axisymmetric modes (m 1 = −m 2 = 0 and n 1 = n 2 ).…”

Section: Circular Cylindermentioning

confidence: 99%

“…Of particular interest is the evolution of weakly nonlinear waves steadily propagating around a circular cylinder of unit radius, focusing on the case where the fluid depth is precisely equal to the critical depth of a 1:2 resonance [64]. For the complex-valued eigenmodes defined in equation (19), the correlation condition, D Ψ 2 1 Ψ * 2 dA = 0, determines that the angular wavenumbers satisfy m 2 = 2m 1 [12,64]. By expressing the complex wave amplitudes in polar form, A j (τ ) = a j (τ )e iθ j (τ ) (for j = 1, 2), equation ( 35) may be recast as [42] da…”

Section: The Case Of a 1:2 Resonancementioning

confidence: 99%

“…Miles [42,43] then characterised the weakly nonlinear evolution of such internal resonances, demonstrating that a 1:2 resonance is impossible in a rectangular cylinder [42]. Although Miles' seminal results provide an informative view of the weakly nonlinear dynamics, the influence of fully nonlinear effects was later assessed by Bryant [8] and Yang et al [64]. For the case of a circular cylinder of finite depth, Bryant [8] and Yang et al [64] both characterised new steadily propagating nonlinear waves arising in the vicinity of a 1:2 resonance, and Yang et al [64] also computed nonlinear near-resonant axisymmetric standing waves.…”

Section: Introductionmentioning

confidence: 99%

“…Although Miles' seminal results provide an informative view of the weakly nonlinear dynamics, the influence of fully nonlinear effects was later assessed by Bryant [8] and Yang et al [64]. For the case of a circular cylinder of finite depth, Bryant [8] and Yang et al [64] both characterised new steadily propagating nonlinear waves arising in the vicinity of a 1:2 resonance, and Yang et al [64] also computed nonlinear near-resonant axisymmetric standing waves. Finally, broader mathematical properties of water waves exhibiting O(2) symmetry (of which a circular cylinder is one example) were analysed by Bridges & Dias [6] and Chossat & Dias [12].…”

Section: Introductionmentioning

confidence: 99%

“…Given the restrictive set of critical depths at which a 1:2 resonance may arise [8,64], it is natural to explore the possibility of nonlinear resonance in cylinders whose depth departs from the depths that trigger a 1:2 resonance. To the best of our knowledge, the first and only such study was the seminal experimental investigation performed by Michel [40], who focused on resonant triads arising for free-surface gravity waves confined to a finite-depth circular cylinder.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Resonant triad interactions of gravity waves in cylindrical basins

Durey¹,

Milewski²

2023

Preprint

View full text Add to dashboard Cite

We present the results of a theoretical investigation into the existence, evolution and excitation of resonant triads of nonlinear free-surface gravity waves confined to a cylinder of finite depth. It is well known that resonant triads are impossible for gravity waves in laterally unbounded domains; we demonstrate, however, that horizontal confinement of the fluid may induce resonant triads for particular fluid depths. For any three correlated wave modes arising in a cylinder of arbitrary cross-section, we prove necessary and sufficient conditions for the existence of a depth at which nonlinear resonance may arise, and show that the resultant critical depth is unique. We enumerate the low-frequency triads for circular cylinders, including a new class of resonances between standing and counter-propagating waves, and also briefly discuss annular and rectangular cylinders. Upon deriving the triad amplitude equations for a finite-depth cylinder of arbitrary cross-section, we deduce that the triad evolution is always periodic, and determine parameters controlling the efficiency of energy exchange. In order to excite a particular triad, we explore the influence of external forcing; in this case, the triad evolution may be periodic, quasi-periodic, or chaotic. Finally, our results have potential implications on resonant water waves in man-made and natural basins, such as industrial-scale fluid tanks, harbours and bays.

show abstract

Section: Circular Cylindermentioning

confidence: 99%

Section: The Case Of a 1:2 Resonancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Resonant triad interactions of gravity waves in cylindrical basins

Durey¹,

Milewski²

2023

Preprint

View full text Add to dashboard Cite

show abstract

Resonant triad interactions of gravity waves in cylindrical basins

Durey

Milewski

2023

J. Fluid Mech.

View full text Add to dashboard Cite

We present the results of a theoretical investigation into the existence, evolution and excitation of resonant triads of nonlinear free-surface gravity waves confined to a cylinder of finite depth. It is well known that resonant triads are impossible for gravity waves in laterally unbounded domains; we demonstrate, however, that horizontal confinement of the fluid may induce resonant triads for particular fluid depths. For any three correlated wave modes arising in a cylinder of arbitrary cross-section, we prove necessary and sufficient conditions for the existence of a depth at which nonlinear resonance may arise, and show that the resultant critical depth is unique. We enumerate the low-frequency triads for circular cylinders, including a new class of resonances between standing and counter-propagating waves, and also briefly discuss annular and rectangular cylinders. Upon deriving the triad amplitude equations for a finite-depth cylinder of arbitrary cross-section, we deduce that the triad evolution is always periodic, and determine parameters controlling the efficiency of energy exchange. In order to excite a particular triad, we explore the influence of external forcing; in this case, the triad evolution may be periodic, quasi-periodic or chaotic. Finally, our results have potential implications on resonant water waves in man-made and natural basins, such as industrial-scale fluid tanks, harbours and bays.

show abstract

On bi-chromatic steady-state gravity waves with an arbitrary included angle

Yang

2022

Physics of Fluids

View full text Add to dashboard Cite

Being encouraged by the recent study by Yang et al. [“On collinear steady-state gravity waves with an infinite number of exact resonances,” Phys. Fluids 31, 122109 (2019)] on two primary waves traveling in the same/opposite direction, we investigate two primary waves traveling with an arbitrary included angle by solving the water-wave equations as a nonlinear boundary-value problem. When the included angle is small, there exists an infinite number of nearly resonant wave components corresponding to an infinite number of small denominators in the framework of the classical analytic approximation methods, like perturbation methods. At a certain included angle, it will cause the classical third-order resonance, corresponding to a singularity. Fortunately, based on the homotopy analysis method (HAM), this type of problem can be solved uniformly by choosing a proper auxiliary linear operator with which the small denominators and singularity can be avoided once and for all. Approximate homotopy-series solutions can be obtained for the two primary waves traveling with an arbitrary included angle. The solutions bifurcate at the angle of classical third-order resonance. Regardless of the resonance at the third-order, it is found that when the acute angle between the two primary wave components becomes smaller, the wave energy slowly shifts from the primary wave components to the high-order wave components, and the increase in wave amplitude strengthens this energy transfer. Moreover, when the included angle is close to or smaller than the third-order resonant angle, the third-order quasi-resonant interaction has a greater influence on the energy distribution, especially if the overall amplitude is relatively large. All of this illustrates the validity and potential of the HAM to be applied to rather complicated nonlinear problems that may have an infinite number of singularities or small denominators.

show abstract

Finite-amplitude steady-state resonant waves in a circular basin

Abstract: Abstract

Cited by 8 publications

References 42 publications

Resonant triad interactions of gravity waves in cylindrical basins

Resonant triad interactions of gravity waves in cylindrical basins

Resonant triad interactions of gravity waves in cylindrical basins

On bi-chromatic steady-state gravity waves with an arbitrary included angle

Contact Info

Product

Resources

About