Efficient Simulation of Short and Long-Wave Interactions With Applications to Capillary Waves

1994 ◽  
Vol 116 (1) ◽  
pp. 77-82 ◽  
Author(s):  
D. G. Dommermuth
Keyword(s):  
Gravity Waves ◽  
Capillary Wave ◽  
Perturbation Expansion ◽  
Capillary Waves ◽  
Front Face ◽  
Wave Interactions ◽  
Efficient Simulation ◽  
Ripple Formation ◽  
Long Wave ◽  
Unsteady Interaction

A perturbation expansion and a multigrid technique are developed for simulating the fully-nonlinear unsteady-interaction of short waves riding on long gravity waves. Both numerical techniques are capable of simulating wave slopes near breaking and wavelength ratios greater than thirty, but the multigrid technique converges more rapidly and it is more efficient. The results of numerical simulations agree qualitatively with experimental measurements of ripple formation on the front face of a gravity-capillary wave.

Numerical calculations of steady gravity-capillary waves using an integro-differential formulation

1979 ◽  
Vol 94 (4) ◽  
pp. 777-793 ◽  
Author(s):  
James W. Rottman ◽  
D. B. Olfe
Keyword(s):  
Gravity Waves ◽  
Small Amplitude ◽  
Numerical Solutions ◽  
Capillary Wave ◽  
Capillary Waves ◽  
The Other ◽  
Wave Number ◽  
Infinite Depth ◽  
Free Surface Waves ◽  
Differential Formulation

A new integro-differential equation is derived for steady free-surface waves. Numerical solutions of this equation for periodic gravity-capillary waves on a fluid of infinite depth are presented. For the two limiting cases of gravity waves and capillary waves, our results are in excellent agreement with previous calculations. For gravity-capillary waves, detailed calculations are performed near the wave-number at which the classical second-order perturbation solution breaks down. Our calculations yield two solutions in this region, which in the limit of small amplitudes agree with the results obtained by Wilton in 1915; one solution has the small amplitude behaviour of a gravity wave and the other that of a capillary wave, but the numerical results show that at large amplitudes both waves have the characteristics of capillary waves. The calculations also show that the wavenumber range in which two solutions exist increases with increasing wave height.

scholarly journals Parasitic Capillary Waves on Small-Amplitude Gravity Waves with a Linear Shear Current

2021 ◽  
Vol 9 (11) ◽  
pp. 1217
Author(s):  
Sunao Murashige ◽  
Wooyoung Choi
Keyword(s):  
Gravity Waves ◽  
Small Amplitude ◽  
Approximate Model ◽  
Capillary Waves ◽  
Front Face ◽  
Wide Range ◽  
Shear Current ◽  
Linear Shear ◽  
Flow Domain ◽  
Nonlinear Computation

This paper describes a numerical investigation of ripples generated on the front face of deep-water gravity waves progressing on a vertically sheared current with the linearly changing horizontal velocity distribution, namely parasitic capillary waves with a linear shear current. A method of fully nonlinear computation using conformal mapping of the flow domain onto the lower half of a complex plane enables us to obtain highly accurate solutions for this phenomenon with the wide range of parameters. Numerical examples demonstrated that, in the presence of a linear shear current, the curvature of surface of underlying gravity waves depends on the shear strength, the wave energy can be transferred from gravity waves to capillary waves and parasitic capillary waves can be generated even if the wave amplitude is very small. In addition, it is shown that an approximate model valid for small-amplitude gravity waves in a linear shear current can reasonably well reproduce the generation of parasitic capillary waves.

scholarly journals A nonlinear Schrödinger equation for gravity–capillary water waves on arbitrary depth with constant vorticity. Part 1

2018 ◽  
Vol 854 ◽  
pp. 146-163 ◽  
Author(s):  
H. C. Hsu ◽  
C. Kharif ◽  
M. Abid ◽  
Y. Y. Chen
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Gravity Waves ◽  
Capillary Wave ◽  
Modulational Instability ◽  
Capillary Waves ◽  
Rate Of Growth ◽  
Positive Vorticity ◽  
Constant Vorticity ◽  
Wave Trains

A nonlinear Schrödinger equation for the envelope of two-dimensional gravity–capillary waves propagating at the free surface of a vertically sheared current of constant vorticity is derived. In this paper we extend to gravity–capillary wave trains the results of Thomas et al. (Phys. Fluids, 2012, 127102) and complete the stability analysis and stability diagram of Djordjevic & Redekopp (J. Fluid Mech., vol. 79, 1977, pp. 703–714) in the presence of vorticity. The vorticity effect on the modulational instability of weakly nonlinear gravity–capillary wave packets is investigated. It is shown that the vorticity modifies significantly the modulational instability of gravity–capillary wave trains, namely the growth rate and instability bandwidth. It is found that the rate of growth of modulational instability of short gravity waves influenced by surface tension behaves like pure gravity waves: (i) in infinite depth, the growth rate is reduced in the presence of positive vorticity and amplified in the presence of negative vorticity; (ii) in finite depth, it is reduced when the vorticity is positive and amplified and finally reduced when the vorticity is negative. The combined effect of vorticity and surface tension is to increase the rate of growth of modulational instability of short gravity waves influenced by surface tension, namely when the vorticity is negative. The rate of growth of modulational instability of capillary waves is amplified by negative vorticity and attenuated by positive vorticity. Stability diagrams are plotted and it is shown that they are significantly modified by the introduction of the vorticity.

scholarly journals Turbulence of capillary waves forced by steep gravity waves

2018 ◽  
Vol 850 ◽  
pp. 803-843 ◽  
Author(s):  
M. Berhanu ◽  
E. Falcon ◽  
L. Deike
Keyword(s):  
Nonlinear Wave ◽  
Power Law ◽  
Gravity Waves ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Turbulence Theory ◽  
Wave Turbulence ◽  
Small Scale ◽  
Turbulent Regime ◽  
Strong Nonlinearity

We study experimentally the dynamics and statistics of capillary waves forced by random steep gravity waves mechanically generated in the laboratory. Capillary waves are produced here by gravity waves from nonlinear wave interactions. Using a spatio-temporal measurement of the free surface, we characterize statistically the random regimes of capillary waves in the spatial and temporal Fourier spaces. For a significant wave steepness (0.2–0.3), power-law spectra are observed both in space and time, defining a turbulent regime of capillary waves transferring energy from the large scale to the small scale. Analysis of temporal fluctuations of the spatial spectrum demonstrates that the capillary power-law spectra result from the temporal averaging over intermittent and strong nonlinear events transferring energy to the small scale in a fast time scale, when capillary wave trains are generated in a way similar to the parasitic capillary wave generation mechanism. The frequency and wavenumber power-law exponents of the wave spectra are found to be in agreement with those of the weakly nonlinear wave turbulence theory. However, the energy flux is not constant through the scales and the wave spectrum scaling with this flux is not in good agreement with wave turbulence theory. These results suggest that theoretical developments beyond the classic wave turbulence theory are necessary to describe the dynamics and statistics of capillary waves in a natural environment. In particular, in the presence of broad-scale viscous dissipation and strong nonlinearity, the role of non-local and non-resonant interactions should be reconsidered.

The Vortical Structure of Parasitic Capillary Waves

1995 ◽  
Vol 117 (3) ◽  
pp. 355-361 ◽  
Author(s):  
R. C. Y. Mui ◽  
D. G. Dommermuth
Keyword(s):  
Free Surface ◽  
Capillary Wave ◽  
Surface Current ◽  
Breaking Waves ◽  
Capillary Waves ◽  
Front Face ◽  
Surface Boundary ◽  
Surface Boundary Layer ◽  
Complex Boundary ◽  
Grid Points

A two-dimensional numerical simulation of the parasitic capillary waves that form on a 5 cm gravity-capillary wave is performed. A robust numerical algorithm is developed to simulate flows with complex boundary conditions and topologies. The free-surface boundary layer is resolved at the full-scale Reynolds, Froude, and Weber numbers. Seventeen million grid points are used to resolve the flow to within 6 × 10–4 cm. The numerical method is used to investigate the formation of parasitic capillary waves on the front face of a gravity-capillary wave. The parasitic capillary waves shed vorticity that induces surface currents that exceed twenty-five percent of the phase velocity of the gravity-capillary wave when the steepness of the parasitic capillary waves is approximately 0.8 and the total wave steepness is 1.1. A mean surface current develops in the direction of the wave’s propagation and is concentrated on the front face of the gravity-capillary wave. This current enhances mixing, and remnants of this surface current are probably present in post-breaking waves. Regions of high vorticity occur on the back sides of the troughs of the parasitic capillary waves. The vorticity separates from the free surface in regions where the wave-induced velocities exceed the vorticity-induced velocities. The rate of energy dissipation of the gravity-capillary wave with parasitic capillaries riding on top is twenty-two times greater than that of the gravity-capillary wave alone.

Measurement of high frequency capillary waves on steep gravity waves

1978 ◽  
Vol 86 (3) ◽  
pp. 401-413 ◽  
Author(s):  
John H. Chang ◽  
Richard N. Wagner ◽  
Henry C. Yuen
Keyword(s):  
Surface Tension ◽  
High Resolution ◽  
Gravity Wave ◽  
Gravity Waves ◽  
Deep Water ◽  
High Frequency ◽  
Qualitative Agreement ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Good Qualitative Agreement

The properties of high frequency capillary waves generated by steep gravity waves on deep water have been measured with a high resolution laser optical slope gauge. The results have been compared with the steady theory of Longuet-Higgins (1963). Good qualitative agreement is obtained. However, the quantitative predictions of the capillary wave slopes cannot be verified by the data because the theory requires knowledge of an idealized quantity - the crest curvature of the gravity wave in the absence of surface tension - which cannot be measured experimentally.

scholarly journals Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments

Fluids ◽  
2021 ◽  
Vol 6 (6) ◽  
pp. 205
Author(s):  
Dan Lucas ◽  
Marc Perlin ◽  
Dian-Yong Liu ◽  
Shane Walsh ◽  
Rossen Ivanov ◽  
...  
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Gravity Waves ◽  
Deep Water ◽  
Plane Waves ◽  
Surface Elevation ◽  
Wave Interactions ◽  
Frequency Space ◽  
Target Mode ◽  
The Hilbert Transform

In this work we consider the problem of finding the simplest arrangement of resonant deep-water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wavevectors K1+K2=K3 (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wavepackets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction in a symmetric configuration. Numerical simulations of the governing equations in natural variables using pseudospectral methods require the inclusion of up to 6-wave interactions, which imposes a strong dealiasing cut-off in order to properly resolve the evolving waves. We study the resonance numerically by looking at a target mode in the base triad and showing that the energy transfer to this mode is more efficient when the system is close to satisfying the resonant conditions. We first look at encountering plane waves with base frequencies in the range 1.32–2.35 Hz and steepnesses below 0.1, and show that the time evolution of the target mode’s energy is dramatically changed at the resonance. We then look at a scenario that is closer to experiments: Encountering wavepackets in a 400-m long numerical tank, where the interaction time is reduced with respect to the plane-wave case but the resonance is still observed; by mimicking a probe measurement of surface elevation we obtain efficiencies of up to 10% in frequency space after including near-resonant contributions. Finally, we perform preliminary experiments of encountering wavepackets in a 35-m long tank, which seem to show that the resonance exists physically. The measured efficiencies via probe measurements of surface elevation are relatively small, indicating that a finer search is needed along with longer wave flumes with much larger amplitudes and lower frequency waves. A further analysis of phases generated from probe data via the analytic signal approach (using the Hilbert transform) shows a strong triad phase synchronisation at the resonance, thus providing independent experimental evidence of the resonance.

Resonant wave interactions in a stratified shear flow

1988 ◽  
Vol 190 ◽  
pp. 357-374 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Shear Flow ◽  
Gravity Waves ◽  
Critical Level ◽  
Internal Gravity Waves ◽  
Wave Interactions ◽  
Resonant Wave ◽  
Resonant Interactions ◽  
Stratified Shear Flow ◽  
Weak Resonance ◽  
Background Shear

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

Radiation of short waves from the resonantly excited capillary–gravity waves

2016 ◽  
Vol 810 ◽  
pp. 5-24 ◽  
Author(s):  
M. Hirata ◽  
S. Okino ◽  
H. Hanazaki
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Gravity Waves ◽  
Wave Pattern ◽  
Short Wave ◽  
Bond Number ◽  
Linear Wave ◽  
Cnoidal Wave ◽  
Short Waves ◽  
Long Wave

Capillary–gravity waves resonantly excited by an obstacle (Froude number: $Fr=1$) are investigated by the numerical solution of the Euler equations. The radiation of short waves from the long nonlinear waves is observed when the capillary effects are weak (Bond number: $Bo<1/3$). The upstream-advancing solitary wave radiates a short linear wave whose phase velocity is equal to the solitary waves and group velocity is faster than the solitary wave (soliton radiation). Therefore, the short wave is observed upstream of the foremost solitary wave. The downstream cnoidal wave also radiates a short wave which propagates upstream in the depression region between the obstacle and the cnoidal wave. The short wave interacts with the long wave above the obstacle, and generates a second short wave which propagates downstream. These generation processes will be repeated, and the number of wavenumber components in the depression region increases with time to generate a complicated wave pattern. The upstream soliton radiation can be predicted qualitatively by the fifth-order forced Korteweg–de Vries equation, but the equation overestimates the wavelength since it is based on a long-wave approximation. At a large Bond number of $Bo=2/3$, the wave pattern has the rotation symmetry against the pattern at $Bo=0$, and the depression solitary waves propagate downstream.

Crest instabilities of gravity waves. Part 1. The almost-highest wave

1994 ◽  
Vol 258 ◽  
pp. 115-129 ◽  
Author(s):  
Michael S. Longuet-Higgins ◽  
R. P. Cleaver
Keyword(s):  
Gravity Wave ◽  
Gravity Waves ◽  
Capillary Waves ◽  
Radius Of Curvature ◽  
Horizontal Distance ◽  
Exponential Rate ◽  
Basic Mode ◽  
Initial Stage ◽  
Small Scales ◽  
Spilling Breaker

It is shown theoretically that the crest of a steep, irrotational gravity wave, considered in isolation, is unstable. There exists just one basic mode of instability, whose exponential rate of growth β equals 0.123(g / R)½, where g denotes gravity and R is the radius of curvature at the undisturbed crest. A volume of water near the crest is shifted towards the forward face of the wave; the ‘toe’ of the instability is at a horizontal distance 0.45R ahead of the crest. The instability may represent the initial stage of a spilling breaker. On small scales, the ‘toe’ may be a source of parasitic capillary waves.

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