Finite-amplitude steady-state wave groups with multiple near-resonances in finite water depth

Finite-amplitude wave groups with multiple near-resonances are investigated to extend the existing results due to Liu et al. (J. Fluid Mech., vol. 835, 2018, pp. 624–653) from steady-state wave groups in deep water to steady-state wave groups in finite water depth. The slow convergence rate of the series solution in the homotopy analysis method and extra unpredictable high-frequency components in finite water depth make it hard to obtain finite-amplitude wave groups accurately. To overcome these difficulties, a solution procedure that combines the homotopy analysis method-based analytical approach and Galerkin method-based numerical approaches has been used. For weakly nonlinear wave groups, the continuum of steady-state resonance from deep water to finite water depth is established. As nonlinearity increases, the frequency bands broaden and more steady-state wave groups are obtained. Finite-amplitude wave groups with steepness no less than $0.20$ are obtained and the resonant sets configuration of steady-state wave groups are analysed in different water depths. For waves in deep water, the majority of non-trivial components appear around the primary ones due to four-wave, six-wave, eight-wave or even ten-wave resonant interactions. The dominant role of four-wave resonant interactions for steady-state wave groups in deep water is demonstrated. For waves in finite water depth, additional non-trivial high-frequency components appear in the spectra due to three-wave, four-wave, five-wave or even six-wave resonant interactions with the components around the primary ones. The amplitude of these high-frequency components increases further as the water depth decreases. Resonances composed by components only around the primary ones are suppressed while resonances composed by components around the primary ones and from the high-frequency domain are enhanced. The spectrum of steady-state resonant wave groups changes with the water depth and the significant role of three-wave resonant interactions in finite water depth is demonstrated.

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Finite amplitude steady-state wave groups with multiple near resonances in deep water

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.787 ◽

2017 ◽

Vol 835 ◽

pp. 624-653 ◽

Cited By ~ 11

Author(s):

Z. Liu ◽

D. L. Xu ◽

S. J. Liao

Keyword(s):

Steady State ◽

Linear Operator ◽

Deep Water ◽

Wave Equations ◽

Finite Amplitude ◽

Wave Groups ◽

Homogeneous Solutions ◽

State Wave ◽

Resonant Interactions ◽

Auxiliary Linear Operator

In this paper, finite amplitude steady-state wave groups with multiple nearly resonant interactions in deep water are investigated theoretically. The nonlinear water wave equations are solved by the homotopy analysis method (HAM), which imposes no constraint on either the number or the amplitude of the wave components, to resolve the small-divisor problems caused by near resonances. A new kind of auxiliary linear operator in the framework of the HAM is proposed to transform the small divisors associated with the non-trivial nearly resonant components to singularities associated with the exactly resonant ones. Primary components, exactly resonant components together with nearly resonant components are considered as the initial non-trivial components, since all of them are homogeneous solutions to the auxiliary linear operator. For wave groups with weak nonlinearity, the energy transfer between nearby nearly resonant components is remarkable. As the nonlinearity increases, the number of steady-state wave groups increases as more components join the near resonance. This indicates that the probability of existence of steady-state resonant waves increases with the nonlinearity of wave groups. The frequency band broadens and spectral asymmetry becomes more and more pronounced. The amplitude of each component may either increase or decrease with the nonlinearity of wave groups, while the amplitude of the whole wave group increases continuously and finite amplitude wave groups are obtained. This work shows the wide existence of steady-state waves when multiple nearly resonant interactions are considered.

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On the near resonances of collinear steady-state wave groups in finite water depth

Ocean Engineering ◽

10.1016/j.oceaneng.2019.04.042 ◽

2019 ◽

Vol 182 ◽

pp. 584-593 ◽

Cited By ~ 2

Author(s):

Zeng Liu ◽

Jianglong Sun ◽

De Xie ◽

Zhiliang Lin

Keyword(s):

Steady State ◽

Water Depth ◽

Wave Groups ◽

State Wave ◽

Finite Water Depth

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Mass, momentum, and energy flux conservation between linear and nonlinear steady-state wave groups

Physics of Fluids ◽

10.1063/1.4998764 ◽

2017 ◽

Vol 29 (12) ◽

pp. 127104 ◽

Cited By ~ 6

Author(s):

Zeng Liu ◽

Dali Xu ◽

Shijun Liao

Keyword(s):

Steady State ◽

Energy Flux ◽

Wave Groups ◽

State Wave ◽

Linear And Nonlinear ◽

Flux Conservation

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Ship waves on uniform shear current at finite depth: wave resistance and critical velocity

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.20 ◽

2016 ◽

Vol 791 ◽

pp. 539-567 ◽

Cited By ~ 17

Author(s):

Yan Li ◽

Simen Å Ellingsen

Keyword(s):

Froude Number ◽

Critical Velocity ◽

Water Depth ◽

Deep Water ◽

Wave Resistance ◽

Ship Motion ◽

Ship Waves ◽

Lateral Wave ◽

Shear Current ◽

Finite Water Depth

We present a comprehensive theory for linear gravity-driven ship waves in the presence of a shear current with uniform vorticity, including the effects of finite water depth. The wave resistance in the presence of shear current is calculated for the first time, containing in general a non-zero lateral component. While formally apparently a straightforward extension of existing deep water theory, the introduction of finite water depth is physically non-trivial, since the surface waves are now affected by a subtle interplay of the effects of the current and the sea bed. This becomes particularly pronounced when considering the phenomenon of critical velocity, the velocity at which transversely propagating waves become unable to keep up with the moving source. The phenomenon is well known for shallow water, and was recently shown to exist also in deep water in the presence of a shear current (Ellingsen, J. Fluid Mech., vol. 742, 2014, R2). We derive the exact criterion for criticality as a function of an intrinsic shear Froude number $S\sqrt{b/g}$ ($S$ is uniform vorticity, $b$ size of source), the water depth and the angle between the shear current and the ship’s motion. Formulae for both the normal and lateral wave resistance forces are derived, and we analyse their dependence on the source velocity (or Froude number $Fr$) for different amounts of shear and different directions of motion. The effect of the shear current is to increase wave resistance for upstream ship motion and decrease it for downstream motion. Also the value of $Fr$ at which $R$ is maximal is lowered for upstream and increased for downstream directions of ship motion. For oblique angles between ship motion and current there is a lateral wave resistance component which can amount to 10–20 % of the normal wave resistance for side-on shear and $S\sqrt{b/g}$ of order unity. The theory is fully laid out and far-field contributions are carefully separated off by means of Cauchy’s integral theorem, exposing potential pitfalls associated with a slightly different method (Sokhotsky–Plemelj) used in several previous works.

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Derivation of Water Particle Kinematics in the Near Surface Zone by the Effective Water Depth Procedure

29th International Conference on Ocean, Offshore and Arctic Engineering: Volume 4 ◽

10.1115/omae2010-20507 ◽

2010 ◽

Author(s):

N. I. Mohd Zaki ◽

G. Najafian

Keyword(s):

Water Depth ◽

High Frequency ◽

Random Wave ◽

Surface Zone ◽

Offshore Structure ◽

Near Surface ◽

Water Particle ◽

Frequency Components ◽

Water Particle Kinematics ◽

Effective Water

Linear Random Wave Theory (LRWT) is frequently used to simulate water particle kinematics at different nodes of an offshore structure from a reference surface elevation record. However, it is well known that LRWT leads to water particle kinematics with exaggerated high-frequency components in the vicinity of mean water level (MWL). To avoid this problem, empirical techniques (such as Wheeler & vertical stretching methods) are frequently used to provide a more realistic representation of the wave kinematics in the near surface zone. In this paper, a modified version of LRWT, based on the derivation of an effective water depth, is introduced. The proposed technique leads to predicted kinematics (in the near surface zone) which lie between corresponding values from the Wheeler and the vertical stretching methods. Furthermore, it does not suffer from exaggerated high-frequency components in the near surface zone.

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HIGH ORDER NONLINEAR WAVE INTERACTIONS FROM DEEP TO FINITE WATER DEPTH WITH BOTTOM TOPOGRAPHY CHANGE

Coastal Engineering Proceedings ◽

10.9753/icce.v36v.waves.24 ◽

2020 ◽

pp. 24

Author(s):

Zuorui Lyu ◽

Hiroaki Kashima ◽

Nobuhito Mori

Keyword(s):

Nonlinear Wave ◽

Water Depth ◽

Deep Water ◽

Modulational Instability ◽

High Order ◽

Long Distance ◽

Freak Wave ◽

Wave Evolution ◽

Finite Water Depth ◽

Skewness And Kurtosis

In recent years, freak wave/rouge wave has become an important problem in science and engineering. Modulational instability is considered to be an important factor leading to freak wave in the wave evolution of deep water, and Janssen (2003) defined Benjamin-Feir index (BFI) to reflect it. Mori and Janssen (2006) gave the occurrence probability of freak waves based on a weakly non-Gaussian theory, and distribution of wave height is determined by skewness and kurtosis of surface elevation to a considerable extent in deep water. According to observational record, freak wave has not only been found in deep water in the ocean, but also been observed in shallow water and coastal areas. In the process of water wave entering continental shelf, water depth is changing with mild slope after a long distance propagation. This study focus on investigating how water depth affect skewness and kurtosis in the high order nonlinear wave evolution from deep water to finite water depth in two-dimension.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/a8LiJvXWRrw

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Ship Springing Response in Finite Water Depth

Journal of Ship Research ◽

10.5957/jsr.2012.56.2.80 ◽

2012 ◽

Vol 56 (02) ◽

pp. 80-90

Author(s):

Jelena Vidic-Perunovic

Keyword(s):

Relative Motion ◽

Water Depth ◽

Deep Water ◽

Bending Moment ◽

Intermediate Water ◽

Short Term ◽

Strip Theory ◽

Influence Of Water ◽

Finite Water Depth ◽

Wave Induced

The influence of water depth on the vertical wave induced bending moment acting on a hull has been studied. The deep-water second-order nonlinear hydroelastic strip theory, which is based on the relative motion concept, has been generalized to account for a finite water depth. Results for an analytical beam and for a tanker ship are presented and discussed. Short-term load predictions that account for a range of different sea states are given for a tanker ship. As seen from the present study, the effect of intermediate water depth may be a significant factor in calculations of ship springing response.

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