The blockage of gravity and capillary waves by longer waves and currents

1990 ◽  
Vol 217 ◽  
pp. 115-141 ◽  
Author(s):  
Jinn-Hwa Shyu ◽  
O. M. Phillips
Keyword(s):  
Multiple Scale ◽  
Finite Depth ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Reflected Waves ◽  
Dominant Wave ◽  
Constant Vorticity ◽  
Waves And Currents ◽  
Uniform Solution ◽  
Group Velocities

Surface waves superimposed upon a larger-scale flow are blocked at the points where the group velocities balance the convection by the larger-scale flow. Two types of blockage, capillary and gravity, are investigated by using a new multiple-scale technique, in which the short waves are treated linearly and the underlying larger-scale flows are assumed steady but can have a considerably curved surface and uniform vorticity. The technique first provides a uniformly valid second-order ordinary differential equation, from which a consistent uniform asymptotic solution can readily be obtained by using a treatment suggested by the result of Smith (1975) who described the phenomenon of gravity blockage in an unsteady current with finite depth.The corresponding WKBJ solution is also derived as a consistent asymptotic expansion of the uniform solution, which is valid at points away from the blockage point. This solution is obviously represented by a linear combination of the incident and reflected waves, and their amplitudes take explicit forms so that it can be shown that even with a significantly varied effective gravity g’ and constant vorticity, wave action will remain conserved for each wave. Furthermore, from the relative amplitudes of the incident and reflected waves, we clearly demonstrate that the action fluxes carried by the two waves towards and away from the blockage point are equal within the present approximation.The blockage of gravity–capillary waves can occur at the forward slopes of a finite-amplitude dominant wave as suggested by Phillips (1981). The results show that the blocked waves will be reflected as extremely short capillaries and then dissipated rapidly by viscosity. Therefore, for a fixed dominant wave, all wavelets shorter than a limiting wavelength will be suppressed by this process. The minimum wavelengths coexisting with the long waves of various wavelengths and slopes are estimated.

scholarly journals Gravity–capillary waves in finite depth on flows of constant vorticity

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160363 ◽  
Author(s):  
Hung-Chu Hsu ◽  
Marc Francius ◽  
Pablo Montalvo ◽  
Christian Kharif
Keyword(s):  
Surface Profile ◽  
Finite Depth ◽  
Bond Number ◽  
Perturbation Series ◽  
Capillary Waves ◽  
Constant Vorticity ◽  
Expansion Procedure ◽  
Shear Current ◽  
Linear Shear

This paper considers two-dimensional periodic gravity–capillary waves propagating steadily in finite depth on a linear shear current (constant vorticity). A perturbation series solution for steady periodic waves, accurate up to the third order, is derived using a classical Stokes expansion procedure, which allows us to include surface tension effects in the analysis of wave–current interactions in the presence of constant vorticity. The analytical results are then compared with numerical computations with the full equations. The main results are (i) the phase velocity is strongly dependent on the value of the vorticity; (ii) the singularities (Wilton singularities) in the Stokes expansion in powers of wave amplitude that correspond to a Bond number of 1/2 and 1/3, which are the consequences of the non-uniformity in the ordering of the Fourier coefficients, are found to be influenced by vorticity; (iii) different surface profiles of capillary–gravity waves are computed and the effect of vorticity on those profiles is shown to be important, in particular that the solutions exhibit type-2-like wave features, characterized by a secondary maximum on the surface profile with a trough between the two maxima.

scholarly journals Two-dimensional stability of finite-amplitude gravity waves on water of finite depth with constant vorticity

2017 ◽  
Vol 830 ◽  
pp. 631-659 ◽  
Author(s):  
M. Francius ◽  
C. Kharif
Keyword(s):  
Growth Rate ◽  
Modulational Instability ◽  
Finite Depth ◽  
Finite Amplitude ◽  
Numerical Results ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Constant Vorticity ◽  
Shear Current ◽  
Modulational Instabilities

A numerical investigation of normal-mode perturbations of a two-dimensional periodic finite-amplitude gravity wave propagating on a vertically sheared current of constant vorticity is considered. For this purpose, an extension of the method developed by Rienecker & Fenton (J. Fluid Mech., vol. 104, 1981, pp. 119–137) is used for the numerical computations of the finite-amplitude waves on a linear shear current. This method enables to compute accurately waves with or without critical layers and pressure anomalies. The numerical results of the linear stability analysis extend the weakly nonlinear analytical results of Thomas et al. (Phys. Fluids, vol. 24, 2012, 127102) to fully nonlinear waves. In particular, the restabilization of the Benjamin–Feir modulational instability, whatever the depth, for an opposite shear current is confirmed. For these sideband instabilities, the numerical results show some deviations with the weakly nonlinear theory as the wave steepness of the basic wave and vorticity are increased. Besides the modulational instabilities, new instability bands corresponding to quartet and quintet instabilities, which are not sideband disturbances, are discovered. The present numerical results show that with opposite shear currents, increasing the shear reduces the growth rate of the most unstable sideband instabilities but enhances the growth rate of these quartet instabilities, which eventually dominate the Benjamin–Feir modulational instabilities.

On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation

2021 ◽  
pp. 1-23
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Small Amplitude ◽  
Type Equation ◽  
Discrete Systems ◽  
Finite Depth ◽  
Capillary Waves ◽  
Classical Solutions ◽  
Well Posedness ◽  
Kawahara Equation ◽  
The Cauchy Problem

The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

Disturbances in semi-infinite heterogeneous media generated by torsional sources. I

1972 ◽  
Vol 62 (2) ◽  
pp. 541-550
Author(s):  
R. S. Sidhu
Keyword(s):  
Heterogeneous Media ◽  
Formal Solution ◽  
Homogeneous Medium ◽  
Finite Depth ◽  
Free Boundaries ◽  
Axially Symmetric ◽  
Sh Waves ◽  
Reflected Waves ◽  
Buried Point ◽  
Heterogeneous Layer

abstract This paper studies the generation of axially symmetric transient SH waves in semi-infinite heterogeneous media in which μ and ρ vary with depth. The sources generating these waves are taken in the form of time-dependent torsional-body forces of finite dimensions. The solution is obtained using Hankel and Laplace transforms and Green's function. The disturbance from a buried point source of impulsive type is discussed in two cases, (a) μ = μo(1 + ɛz)2, ρ = ρo (1 + ɛz)2, (b) μ = μoe2az, ρ = ρoe2az. It is shown that, in contrast to the results for a homogeneous medium, in case (i), the wave reflected by the free surface generates secondary disturbances which trail behind the wave front and die out as t increases; the incident wave in this medium generates no such disturbance. In case (ii), however, both the incident as well as the reflected waves generate secondary disturbances. Formal solution for the disturbance in a heterogeneous layer of finite depth with stress-free boundaries is discussed in Appendix II.

A new family of capillary waves

1980 ◽  
Vol 98 (1) ◽  
pp. 161-169 ◽  
Author(s):  
Jean-Marc Vanden-Broeck ◽  
Joseph B. Keller
Keyword(s):  
Integral Equation ◽  
Nonlinear Integral Equation ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Wave Steepness ◽  
Algebraic Equations ◽  
Infinite Depth ◽  
New Family ◽  
Nonlinear Algebraic Equations ◽  
Finite Set

A new family of finite-amplitude periodic progressive capillary waves is presented. They occur on the surface of a fluid of infinite depth in the absence of gravity. Each pair of adjacent waves touch at one point and enclose a bubble at pressure P. P depends upon the wave steepness s, which is the vertical distance from trough to crest divided by the wavelength. Previously Crapper found a family of waves without bubbles for 0 [les ] s [les ] s* = 0·730. Our solutions occur for all s > s** = 0·663, with the trough taken to be the bottom of the bubble. As s → ∞, the bubbles become long and narrow, while the top surface tends to a periodic array of semicircles in contact with one another. The solutions were obtained by formulating the problem as a nonlinear integral equation for the free surface. By introducing a mesh and difference method, we converted this equation into a finite set of nonlinear algebraic equations. These equations were solved by Newton's method. Graphs and tables of the results are included. These waves enlarge the class of phenomena which can occur in an ideal fluid, but they do not seem to have been observed.

Wave-Current Loading on a Vertical Slender Cylinder by Two Different Numerical Models

2006 ◽  
Author(s):  
M. H. Zaman ◽  
R. E. Baddour
Keyword(s):  
Numerical Models ◽  
Numerical Technique ◽  
Three Dimensional ◽  
Vertical Cylinder ◽  
Finite Depth ◽  
Offshore Structure ◽  
Current Field ◽  
Oblique Wave ◽  
Waves And Currents ◽  
Current Loading

The study of the effects resulting from the interaction of a combined wave-current field with any ocean structure is important for the design and performance evaluation of that structure. The prudent computation of forces exerted by waves and currents is an essential task in the study of the stability of an offshore structure. A study on the loading of an oblique wave and a current field on a fixed vertical slender cylinder in a 3D flow frame is illustrated in Zaman and Baddour (2004). The three dimensional expressions describing the characteristics of the combined wave-current field in terms of mass, momentum and energy flux conservation equations are formulated. The parameters before the interaction of the oblique wave-free uniform current and current-free wave are used to formulate the kinematics of the flow field. These expressions are also employed to formulate and calculate the loads imparted by the wave-current fluid flow on a bottom mounted slender vertical cylinder. In this work a 2D version of the above 3D model called here Model-I has been used for the numerical computations presented in this paper. The second model denoted model-II in the present paper is based on Euler equations. This model is formulated through the vertical integration of the continuity equation and the equations of motions, Zaman et al (1997). A semi-implicit numerical technique is employed for the numerical solution. In the present paper comparisons are made between the results obtained from the 2D version of the above models in finite depth. Both models are then compared with some relevant experimental data. Morison et al equation (1950) is deployed for the load computations in all cases.

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.

scholarly journals Ship waves in the presence of uniform vorticity

2014 ◽  
Vol 742 ◽  
Author(s):  
Simen Å. Ellingsen
Keyword(s):  
Shear Flow ◽  
Finite Depth ◽  
Critical Value ◽  
Wave Problem ◽  
Transverse Waves ◽  
Ship Waves ◽  
Striking Result ◽  
Constant Vorticity ◽  
Shear Current ◽  
Half Angle

AbstractLord Kelvin’s result that waves behind a ship lie within a half-angle $\phi _{\mathit{K}}\approx 19^{\circ }28'$ is perhaps the most famous and striking result in the field of surface waves. We solve the linear ship wave problem in the presence of a shear current of constant vorticity $S$, and show that the Kelvin angles (one each side of wake) as well as other aspects of the wake depend closely on the ‘shear Froude number’ $\mathit{Fr}_{\mathit{s}}=VS/g$ (based on length $g/S^2$ and the ship’s speed $V$), and on the angle between current and the ship’s line of motion. In all directions except exactly along the shear flow there exists a critical value of $\mathit{Fr}_{\mathit{s}}$ beyond which no transverse waves are produced, and where the full wake angle reaches $180^\circ $. Such critical behaviour is previously known from waves at finite depth. For side-on shear, one Kelvin angle can exceed $90^\circ $. On the other hand, the angle of maximum wave amplitude scales as $\mathit{Fr}^{-1}$ ($\mathit{Fr}$ based on size of ship) when $\mathit{Fr}\gg 1$, a scaling virtually unaffected by the shear flow.

scholarly journals Topographically forced long waves on a sheared coastal current. Part 2. Finite amplitude waves

1997 ◽  
Vol 343 ◽  
pp. 153-168 ◽  
Author(s):  
S. R. CLARKE ◽  
E. R. JOHNSON
Keyword(s):  
Wave Equation ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Coastal Current ◽  
Constant Vorticity ◽  
Long Wave ◽  
Long Time ◽  
Wave Limit ◽  
Parameter Values ◽  
Upstream Flow

This paper analyses the finite-amplitude flow of a constant-vorticity current past coastal topography in the long-wave limit. A forced finite-amplitude long-wave equation is derived to describe the evolution of the vorticity interface. An analysis of this equation shows that three distinct near-critical regimes occur. In the first the upstream flow is restricted, with overturning of the vorticity interface for sufficiently large topography. In the second quasi-steady nonlinear waves form downstream of the topography with weak upstream influence. In the third regime the upstream rotational fluid is partially blocked. Blocking and overturning are enhanced at headlands with steep rear faces and decreased at headlands with steep forward faces. For certain parameter values both overturning and partially blocked solutions are possible and the long-time evolution is critically dependent on the initial conditions. The reduction of the problem to a one-dimensional nonlinear wave equation allows solutions to be followed to much longer times and parameter space to be explored more finely than in the related pioneering contour-dynamical integrations of Stern (1991).

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