scholarly journals The Formation of Parasitic Capillary Ripples on Gravity–Capillary Waves and the Underlying Vortical Structures

2009 ◽  
Vol 39 (2) ◽  
pp. 263-289 ◽  
Author(s):  
Li-Ping Hung ◽  
Wu-Ting Tsai
Keyword(s):  
Capillary Wave ◽  
Wave Train ◽  
Strong Dependence ◽  
Capillary Waves ◽  
Wave Crest ◽  
Carrier Wave ◽  
Vortical Structures ◽  
Steady Stage ◽  
Dimensional Gravity ◽  
Vortical Layer

Abstract The evolution of moderately short, steep two-dimensional gravity–capillary waves, from the onset of the parasitic capillary ripples to a fully developed quasi-steady stage, is studied numerically using a spectrally accurate model. The study focuses on understanding the precise mechanism of capillary generation, and on characterizing surface roughness and the underlying vortical structure associated with parasitic capillary waves. It is found that initiation of the first capillary ripple is triggered by the fore–aft asymmetry of the otherwise symmetric carrier wave, which then forms a localized pressure disturbance on the forward face near the crest, and subsequently develops an oscillatory train of capillary waves. Systematic numerical experiments reveal that there exists a minimum crest curvature of the carrier gravity–capillary wave for the formation of parasitic capillary ripples, and such a threshold curvature (≈0.25 cm−1) is almost independent of the carrier wavelength. The characteristics of the parasitic capillary wave train and the induced underlying vortical structures exhibit a strong dependence on the carrier wavelength. For a steep gravity–capillary wave with a shorter wavelength (e.g., 5 cm), the parasitic capillary wave train is distributed over the entire carrier wave surface at the stage when capillary ripples are fully developed. Immediately underneath the capillary wave train, weak vortices are observed to confine within a thin layer beneath the ripple crests whereas strong vortical layers with opposite orientation of vorticity are shed from the ripple troughs. These strong vortical layers are then convected upstream and accumulate within the carrier wave crest, forming a strong “capillary roller” as postulated by Longuet-Higgins. In contrast, as the wavelength of the gravity–capillary wave increases (e.g., 10 cm), parasitic capillary ripples appear as being trapped in the forward slope of the carrier wave. The strength of the vortical layer shed underneath the parasitic capillaries weakens, and its thickness and extent reduces. The vortices accumulating within the crest of the carrier wave, therefore, are not as pronounced as those observed in the shorter gravity–capillary waves.

scholarly journals Enhanced Energy Dissipation by Parasitic Capillaries on Short Gravity–Capillary Waves

2010 ◽  
Vol 40 (11) ◽  
pp. 2435-2450 ◽  
Author(s):  
Wu-ting Tsai ◽  
Li-ping Hung
Keyword(s):  
Energy Dissipation ◽  
Dissipation Rate ◽  
Capillary Wave ◽  
Wave Train ◽  
Three Dimensional ◽  
Local Maximum ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Carrier Wave ◽  
Order Of Magnitude

Abstract The increased energy dissipation caused by the formation of parasitic capillary wavelets on moderately short, steep gravity–capillary waves is studied numerically. This study focuses on understanding the mechanism leading to dissipation enhancement and on exploring the possible correlation between the enhanced dissipation rate and the characteristic parameters of the parasitic capillaries. The interaction between the parasitic capillary wave train and the underlying dominant flow of the carrier wave induces strong vortex shedding and imposes large straining immediately underneath the troughs of the capillary ripples. These localized strains are very effective in dissipating energy of the carrier gravity–capillary wave. The attenuation rate of the carrier wave can increase by more than one order of magnitude in the presence of capillary wavelets. Systematic simulations for various carrier wavelengths and steepnesses reveal that the enhanced dissipation rate can be quantified well by a simple parameter: the average of all the difference between the local maximum and minimum slopes along the entire carrier wave surface, which is equivalent to the mean slope of the parasitic capillary wave train. The enhanced dissipation rate increases approximately linearly with the carrier gravity–capillary wavenumber for a given mean slope of the capillary wave train. The increased energy dissipation caused by the formation of parasitic capillaries is also found to significantly impact on the characteristics of three-dimensional instabilities of finite-amplitude, uniform gravity–capillary waves.

FIFTH-ORDER EVOLUTION EQUATION OF GRAVITY–CAPILLARY WAVES

2017 ◽  
Vol 59 (1) ◽  
pp. 103-114
Author(s):  
DIPANKAR CHOWDHURY ◽  
SUMA DEBSARMA
Keyword(s):  
Evolution Equation ◽  
Evolution Equations ◽  
Capillary Wave ◽  
Wave Train ◽  
Critical Value ◽  
Capillary Waves ◽  
Wave Number ◽  
Region Of Stability ◽  
Fifth Order ◽  
Nonlinear Gravity

We extend the evolution equation for weak nonlinear gravity–capillary waves by including fifth-order nonlinear terms. Stability properties of a uniform Stokes gravity–capillary wave train is studied using the evolution equation obtained here. The region of stability in the perturbed wave-number plane determined by the fifth-order evolution equation is compared with that determined by third- and fourth-order evolution equations. We find that if the wave number of longitudinal perturbations exceeds a certain critical value, a uniform gravity–capillary wave train becomes unstable. This critical value increases as the wave steepness increases.

Alternative Methodologies for Subsea Flexible Strength Analysis

2018 ◽  
Author(s):  
Anskey A. Miranda ◽  
Fred P. Turner ◽  
Nigel Barltrop
Keyword(s):  
Real World ◽  
Wave Train ◽  
Irregular Waves ◽  
Wave Crest ◽  
Strength Analysis ◽  
New Wave ◽  
Regular Wave ◽  
Wave Analysis ◽  
Sea State ◽  
Irregular Wave

This paper presents a study of the analysis methodologies used to predict the most likely response of flexibles in a subsea environment, with the aim of determining an efficient and reliable prediction methodology. The most accurate method involves simulating multiple wave realisations of a real world sea state, i.e. irregular waves, and post-processing the results to determine the most probable maximum (MPM). Due to the computationally intensive nature of this approach, however, regular wave analysis is typically used to determine flexible response. This approach considers the maximum wave within a design storm at a desired period; the choice of periods may leave room for uncertainty in the conservatism of the approach. With proper screening, regular wave analysis can be a valid yet overly conservative approach resulting in over design and additional cost. However, if screened incorrectly, there is a possibility that the choice of periods could give results that are under conservative. In addition to regular wave analysis, the paper presents two alternative methodologies to determine the most likely response, with the focus on reducing the computational resources required. The first alternative is an ‘Irregular Wave Screen’ approach in which the wave train is screened at areas of interest for waves within a user defined threshold of the maximum wave height, in addition to other user defined parameters. Only waves within these parameters are simulated to determine responses. The second alternative is the ‘New Wave’ approach, which models the most probable wave elevation around the maximum wave crest. The calculated new wave is then placed at the desired location to determine responses. The responses of the Regular, Irregular Wave Screen and New Wave methodologies are compared with the Irregular MPM approach to determine their feasibility to predict the response of flexibles in a real world irregular sea state with lower computational requirements.

scholarly journals Travelling wave solutions on an axisymmetric ferrofluid jet

2019 ◽  
Vol 865 ◽  
pp. 414-439 ◽  
Author(s):  
A. Doak ◽  
J.-M. Vanden-Broeck
Keyword(s):  
Copper Wire ◽  
Classical Problem ◽  
Solution Space ◽  
Travelling Wave ◽  
Capillary Waves ◽  
Azimuthal Magnetic Field ◽  
Wave Solutions ◽  
Dimensional Gravity ◽  
Capillary Jets ◽  
Potential Flow Model

We consider a potential flow model of axisymmetric waves travelling on a ferrofluid jet. The ferrofluid coats a copper wire, through which an electric current is run. The induced azimuthal magnetic field magnetises the ferrofluid, which in turn stabilises the well known Plateau–Rayleigh instability seen in axisymmetric capillary jets. This model is of interest because the stabilising mechanism allows for axisymmetric magnetohydrodynamical solitary waves. A numerical scheme capable of computing steady periodic, solitary and generalised solitary wave solutions is presented. It is found that the solution space for the model is very similar to that of the classical problem of two-dimensional gravity–capillary waves.

scholarly journals Transverse instability of gravity–capillary solitary waves on deep water in the presence of constant vorticity

2019 ◽  
Vol 871 ◽  
pp. 1028-1043
Author(s):  
M. Abid ◽  
C. Kharif ◽  
H.-C. Hsu ◽  
Y.-Y. Chen
Keyword(s):  
Growth Rate ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Deep Water ◽  
Capillary Waves ◽  
Two Dimensional ◽  
Transverse Instability ◽  
Constant Vorticity ◽  
Dimensional Gravity ◽  
Wave Properties

The bifurcation of two-dimensional gravity–capillary waves into solitary waves when the phase velocity and group velocity are nearly equal is investigated in the presence of constant vorticity. We found that gravity–capillary solitary waves with decaying oscillatory tails exist in deep water in the presence of vorticity. Furthermore we found that the presence of vorticity influences strongly (i) the solitary wave properties and (ii) the growth rate of unstable transverse perturbations. The growth rate and bandwidth instability are given numerically and analytically as a function of the vorticity.

Experimental Variation of Focusing Wave Groups for the Investigation of Their Predictability

2009 ◽  
Author(s):  
Janou Hennig ◽  
Christian E. Schmittner
Keyword(s):  
Deterministic Model ◽  
Focal Point ◽  
Wave Train ◽  
Wave Crest ◽  
Intermediate Water ◽  
Test Results ◽  
Wave Groups ◽  
Experimental Variation ◽  
Wave Phase Velocity ◽  
Target Wave

In deterministic model testing, focusing wave groups are used for the simulation of dedicated wave environments. They are characterized by the transient appearance of one relatively steep wave crest. The phasing of the wave components which leads to an exact focusing in one point in time and space is strongly dependent on the correct modeling of the wave phase velocity while the position of the focusing point depends on the wave group celerity. For wave generation purposes, the calculation of a wave maker control signal based on a target wave train at a desired position in the tank (inverse or backward modeling) is of crucial importance. Numerical wave tanks and empirical approaches are often calibrated based on wave characteristics measured in a particular tank. This paper presents model test results for the variation of frequency range, steepness and focal point of focusing wave groups at intermediate water depth. The measured characteristics are compared to predicted parameters.

Instability of modulation wave train and disturbance of time period in slightly stable media for unstable nonlinear Schrödinger dynamical equation

2020 ◽  
pp. 2150010
Author(s):  
Mujahid Iqbal ◽  
Aly R. Seadawy
Keyword(s):  
Fiber Optics ◽  
Wave Train ◽  
Soliton Solutions ◽  
Three Dimensions ◽  
Nonlinear Pdes ◽  
Carrier Wave ◽  
Physical Sciences ◽  
Nonlinear Schrödinger ◽  
Dark Solitons ◽  
Time Period

In this study, we constructed the exact traveling and soliton solutions for unstable nonlinear Schrödinger equation (UNLSE) by a modified mathematical method. The UNLSE explains the disorder of a time period in slightly unstable and stable medium, and also accomplishes the instability for modulated carrier wave. The demonstrated soliton solutions represent to bright-dark solitons, periodic solitary wave, and traveling waves. This technique is more efficient and powerful to investigate various kinds of soliton solutions for higher order unstable nonlinear PDEs. By applying symbolic computations, the physical interpretations of soliton solutions represent by two- and three-dimensions graphical. Our obtained results may helpful in the study of nonlinear optics, fiber optics, solitonic dynamic, and various kinds of physical sciences.

Experiments on nonlinear gravity–capillary waves

1999 ◽  
Vol 380 ◽  
pp. 205-232 ◽  
Author(s):  
LEV SHEMER ◽  
MELAD CHAMESSE
Keyword(s):  
Narrow Band ◽  
Numerical Study ◽  
Three Dimensional ◽  
Capillary Waves ◽  
Carrier Wave ◽  
Nls Equation ◽  
Zakharov Equation ◽  
The Stability ◽  
Nonlinear Gravity

Benjamin–Feir instability of nonlinear gravity–capillary waves is studied experimentally. The experimental results are compared with computations performed for values of wavelength and steepness identical to those employed in the experiments. The theoretical approach is based on the Zakharov nonlinear equation which is modified here to incorporate weak viscous dissipation. Experiments are performed in a wave ume which has an accurately controlled wavemaker for generation of the carrier wave, as well as an additional independent conical wavemaker for generation of controlled three-dimensional disturbances. The approach adopted in the present experimental investigation allows therefore the determination of the actual boundaries of the instability domain, and not just the most unstable disturbances. Instantaneous surface elevation measurements are performed with capacitance-type wave gauges. Multipoint measurements make it possible to determine the angular dependence of the amplitude of the forced and unforced disturbances, as well as their variation along the tank. The limits of the instability domains obtained experimentally for each set of carrier wave parameters agree favourably with those computed numerically using the model equation. The numerical study shows that application of the Zakharov equation, which is free of the narrow-band approximation adopted in the derivation of the nonlinear Schrödinger (NLS) equation, may lead to qualitatively different results regarding the stability of nonlinear gravity–capillary waves. The present experiments support the results of the numerical investigation.

scholarly journals Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water

2002 ◽  
Vol 43 (4) ◽  
pp. 513-524 ◽  
Author(s):  
Suma Debsarma ◽  
K.P. Das
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Nonlinear Evolution Equations ◽  
Marginal Stability ◽  
Stability And Instability ◽  
The Stability

AbstractFor a three-dimensional gravity capillary wave packet in the presence of a thin thermocline in deep water two coupled nonlinear evolution equations correct to fourth order in wave steepness are obtained. Reducing these two equations to a single equation for oblique plane wave perturbation, the stability of a uniform gravity-capillary wave train is investigated. The stability and instability regions are identified. Expressions for the maximum growth rate of instability and the wavenumber at marginal stability are obtained. The results are shown graphically.

Gentle spilling breakers: crest profile evolution

1999 ◽  
Vol 379 ◽  
pp. 191-222 ◽  
Author(s):  
JAMES H. DUNCAN ◽  
HAIBING QIAO ◽  
VASANTH PHILOMIN ◽  
ALEXANDRA WENZ
Keyword(s):  
Surface Profile ◽  
Leading Edge ◽  
Capillary Waves ◽  
Wave Crest ◽  
Fixed Position ◽  
Left Behind ◽  
The Face ◽  
Mean Water Level ◽  
The Mean ◽  
Spilling Breakers

The surface profile histories of gentle spilling breakers generated mechanically with a dispersive focusing technique are studied experimentally. Froude-scaled generation conditions are used to produce waves with three average frequencies: f0=1.42, 1.26, and 1.15 Hz. At each frequency, the strength of the breaker is varied by varying the overall amplitude of the wavemaker motion. It is found that in all cases the beginning of the breaking process is marked by the formation of a bulge in the profile at the crest on the forward face of the wave. The leading edge of this bulge is called the toe. As the breaking process continues, the bulge becomes more pronounced while the toe remains in nearly a fixed position relative to the crest. Capillary waves form ahead of the toe. At a time of about 0.1/f0 after the bulge first becomes visible, the toe begins to move down the face of the wave and very quickly accelerates to a constant velocity which scales with the wave crest speed. During this phase of the breaker evolution, the surface profile between the toe and the crest develops ripples which eventually are left behind the wave crest. It is found that the height of the toe above the mean water level scales with the nominal wavelength λ0=g/(2πf20) of the breaker, while the size and shape of the bulge and the length of the capillary waves ahead of the toe are independent of f0.

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