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.

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.

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 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 Fourth Order Nonlinear Evolution Equation For Interfacial Gravity Waves In The Presence Of Air Flowing Over Water And A Basic Current Shear

2015 ◽  
Vol 20 (3) ◽  
pp. 517-530
Author(s):  
D.P. Majumder ◽  
A.K. Dhar
Keyword(s):  
Evolution Equation ◽  
Gravity Waves ◽  
Fourth Order ◽  
Water Waves ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Wave Train ◽  
Wave Number ◽  
Marginal Stability ◽  
Starting Point

Abstract A fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves as first pointed out by Dysthe (1979) is derived for gravity waves propagating at the interface of two superposed fluids of infinite depth in the presence of air flowing over water and a basic current shear. A stability analysis is then made for a uniform Stokes gravity wave train. Graphs are plotted for the maximum growth rate of instability and for wave number at marginal stability against wave steepness for different values of air flow velocity and basic current shears. Significant deviations are noticed from the results obtained from the third order evolution equation, which is the nonlinear Schrödinger equation.

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 ◽  
pp. 103
Author(s):  
Dipankar Chowdhury ◽  
Suma Debsarma
Keyword(s):  
Evolution Equation ◽  
Capillary Waves ◽  
Fifth Order

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

PROBABILITY DENSITY EVOLUTION EQUATIONS — A HISTORICAL INVESTIGATION

2009 ◽  
Vol 03 (03) ◽  
pp. 209-226 ◽  
Author(s):  
LI JIE ◽  
CHEN JIANBING
Keyword(s):  
Evolution Equation ◽  
Probability Density ◽  
Evolution Equations ◽  
Planck Equation ◽  
Point Of View ◽  
Lagrangian Description ◽  
Density Evolution ◽  
Physical Point ◽  
Probability Density Evolution ◽  
Generalized Probability

The paper aims at clarifying the essential relationship between traditional probability density evolution equations and the generalized probability density evolution equation which is developed by the authors in recent years. Using the principle of preservation of probability as a uniform fundamental, the probability density evolution equations, including the Liouville equation, Fokker–Planck equation and the Dostupov–Pugachev equation, are derived from the physical point of view. It is pointed out that combining with Eulerian or Lagrangian description of the associated dynamical system will lead to different probability density evolution equations. Particularly, when both the principle and dynamical systems are viewed from Lagrangian description, we are led to the generalized probability density evolution equation.

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