scholarly journals Dynamics of steep two-dimensional gravity–capillary solitary waves

2010 ◽  
Vol 664 ◽  
pp. 466-477 ◽  
Author(s):  
PAUL A. MILEWSKI ◽  
J.-M. VANDEN-BROECK ◽  
ZHAN WANG
Keyword(s):  
Solitary Waves ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Amplitude Wave ◽  
Linear Dispersion ◽  
Infinite Depth ◽  
Linear Dispersion Relation ◽  
Dimensional Gravity ◽  
Nonlinear Free Surface ◽  
The Waves

In this paper, the unsteady evolution of two-dimensional fully nonlinear free-surface gravity–capillary solitary waves is computed numerically in infinite depth. Gravity–capillary wavepacket-type solitary waves were found previously for the full Euler equations, bifurcating from the minimum of the linear dispersion relation. Small and moderate amplitude elevation solitary waves, which were known to be linearly unstable, are shown to evolve into stable depression solitary waves, together with a radiated wave field. Depression waves and certain large amplitude elevation waves were found to be robust to numerical perturbations. Two kinds of collisions are computed: head-on collisions whereby the waves are almost unchanged, and overtaking collisions which are either almost elastic if the wave amplitudes are both large or destroy the smaller wave in the case of a small amplitude wave overtaking a large one.

scholarly journals A note on solitary waves with variable surface tension in water of infinite depth

2006 ◽  
Vol 48 (2) ◽  
pp. 225-235 ◽  
Author(s):  
E. Özuğurlu ◽  
J.-M. Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Solitary Waves ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Infinite Depth ◽  
Free Surface Boundary Condition ◽  
Variable Surface ◽  
Dimensional Gravity ◽  
Surface Boundary Condition

AbstractTwo-dimensional gravity-capillary solitary waves propagating at the surface of a fluid of infinite depth are considered. The effects of gravity and of variable surface tension are included in the free-surface boundary condition. The numerical results extend the constant surface tension results of Vanden-Broeck and Dias to situations where the surface tension varies along the free surface.

Internal waves in a viscous atmosphere

1975 ◽  
Vol 72 (4) ◽  
pp. 773-786 ◽  
Author(s):  
W. L. Chang ◽  
T. N. Stevenson
Keyword(s):  
Energy Source ◽  
Incompressible Fluid ◽  
Internal Waves ◽  
Infinite Number ◽  
Similarity Solution ◽  
Small Amplitude ◽  
Horizontal Plane ◽  
Two Dimensional ◽  
The Waves ◽  
Isothermal Atmosphere

The way in which internal waves change in amplitude as they propagate through an incompressible fluid or an isothermal atmosphere is considered. A similarity solution for the small amplitude isolated viscous internal wave which is generated by a localized two-dimensional disturbance or energy source was given by Thomas & Stevenson (1972). It will be shown how summations or superpositions of this solution may be used to examine the behaviour of groups of internal waves. In particular the paper considers the waves produced by an infinite number of sources distributed in a horizontal plane such that they produce a sinusoidal velocity distribution. The results of this analysis lead to a new small perturbation solution of the linearized equations.

scholarly journals On the Dynamics of Two-Dimensional Capillary-Gravity Solitary Waves with a Linear Shear Current

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Dali Guo ◽  
Bo Tao ◽  
Xiaohui Zeng
Keyword(s):  
Solitary Waves ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Numerical Study ◽  
Two Dimensional ◽  
Shear Current ◽  
Linear Shear ◽  
The Stability ◽  
Depression Wave ◽  
Elevation Wave

The numerical study of the dynamics of two-dimensional capillary-gravity solitary waves on a linear shear current is presented in this paper. The numerical method is based on the time-dependent conformal mapping. The stability of different kinds of solitary waves is considered. Both depression wave and large amplitude elevation wave are found to be stable, while small amplitude elevation wave is unstable to the small perturbation, and it finally evolves to be a depression wave with tails, which is similar to the irrotational capillary-gravity 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.

Longitudinal waves in a perpendicular collisionless plasma shock: Part 3. Te ˜ Ti

1971 ◽  
Vol 6 (3) ◽  
pp. 561-566 ◽  
Author(s):  
S. Peter Gary
Keyword(s):  
Collisionless Plasma ◽  
Fixed Ratio ◽  
Linear Dispersion ◽  
Electrostatic Waves ◽  
Longitudinal Waves ◽  
Linear Dispersion Relation ◽  
Drift Instability ◽  
The Waves ◽  
Plasma Shock ◽  
Vlasov Plasma

This paper considers electrostatic waves in a Vlasov plasma of unmagnetized ions and magnetized electrons undergoing an E x B drift. The linear dispersion relation is solved numerically for Te ΰTi. For a fixed ratio of drift velocity to electron thermal velocity, the growth rates of the E x B electron drift instability are smaller, and the waves are stabilized at much smaller values of k. B than in the Te ≫ Ti case.

Two-dimensional vortex structures in the bottom boundary layer of progressive and solitary waves

2013 ◽  
Vol 728 ◽  
pp. 340-361 ◽  
Author(s):  
Pietro Scandura
Keyword(s):  
Boundary Layer ◽  
Solitary Waves ◽  
Small Amplitude ◽  
Flow Instability ◽  
Vortex Structures ◽  
Bottom Boundary Layer ◽  
Two Dimensional ◽  
Bottom Boundary ◽  
Vortex Layer ◽  
Vortex Tubes

AbstractThe two-dimensional vortices characterizing the bottom boundary layer of both progressive and solitary waves, recently discovered by experimental flow visualizations and referred to as vortex tubes, are studied by numerical solution of the governing equations. In the case of progressive waves, the Reynolds numbers investigated belong to the subcritical range, according to Floquet linear stability theory. In such a range the periodic generation of strictly two-dimensional vortex structures is not a self-sustaining phenomenon, being the presence of appropriate ambient disturbances necessary to excite certain modes through a receptivity mechanism. In a physical experiment such disturbances may arise from several coexisting sources, among which the most likely is roughness. Therefore, in the present numerical simulations, wall imperfections of small amplitude are introduced as a source of disturbances for both types of wave, but from a macroscopic point of view the wall can be regarded as flat. The simulations show that even wall imperfections of small amplitude may cause flow instability and lead to the appearance of vortex tubes. These vortices, in turn, interact with a vortex layer adjacent to the wall and characterized by vorticity opposite to that of the vortex tubes. In a first stage such interaction gives rise to corrugation of the vortex layer and this affects the spatial distribution of the wall shear stress. In a second stage the vortex layer rolls up and pairs of counter-rotating vortices are generated, which leave the bottom because of the self-induced velocity.

scholarly journals Stability and dynamics of two-dimensional fully nonlinear gravity–capillary solitary waves in deep water

2016 ◽  
Vol 809 ◽  
pp. 530-552 ◽  
Author(s):  
Z. Wang
Keyword(s):  
Solitary Waves ◽  
Deep Water ◽  
Wave Equations ◽  
Turning Point ◽  
Phase Speed ◽  
Amplitude Curve ◽  
Two Dimensional ◽  
Fully Nonlinear ◽  
Dimensional Gravity ◽  
The Stability

The stability and dynamics of two-dimensional gravity–capillary solitary waves in deep water within the fully nonlinear water-wave equations are numerically studied. It is well known that there are two families of symmetric gravity–capillary solitary waves – depression waves and elevation waves – bifurcating from infinitesimal periodic waves at the minimum of the phase speed. The stability of both branches was previously examined by Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) by means of a numerical spectral analysis. Their results show that the depression solitary waves with single-valued profiles are stable, while the elevation branch experiences a stability exchange at a turning point on the speed–amplitude curve. In the present paper, we provide numerical evidence that the depression solitary waves with an overhanging structure are also stable. On the other hand, Dias et al. (Eur. J. Mech. B, vol. 15, 1996, pp. 17–36) numerically traced the elevation branch and discovered that its speed–amplitude bifurcation curve features a ‘snake-like’ behaviour with many turning points, whereas Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) only considered the stability exchange near the first turning point. Our results reveal that the stability exchange occurs again near the second turning point. A branch of asymmetric solitary waves is also considered and found to be unstable, even when the wave profile consists of a depression wave and a stable elevation one. The excitation of stable gravity–capillary solitary waves is carried out via direct numerical simulations. In particular, the stable elevation waves, which feature two troughs connected by a small dimple, can be excited by moving two fully localised, well-separated pressures on the free surface with the speed slightly below the phase speed minimum and removing the pressures simultaneously after a period of time.

scholarly journals Two-dimensional numerical simulations of shoaling internal solitary waves at the ASIAEX site in the South China Sea

2015 ◽  
Vol 22 (3) ◽  
pp. 289-312 ◽  
Author(s):  
K. G. Lamb ◽  
A. Warn-Varnas
Keyword(s):  
South China Sea ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Gravity Waves ◽  
South China ◽  
Internal Solitary Waves ◽  
Two Dimensional ◽  
Shoaling Waves ◽  
The Waves ◽  
Inertia Gravity Waves

Abstract. The interaction of barotropic tides with Luzon Strait topography generates some of the world's largest internal solitary waves which eventually shoal and dissipate on the western side of the northern South China Sea. Two-dimensional numerical simulations of the shoaling of a single internal solitary wave at the site of the Asian Seas International Acoustic Experiment (ASIAEX) have been undertaken in order to investigate the sensitivity of the shoaling process to the stratification and the underlying bathymetry and to explore the influence of rotation. The bulk of the simulations are inviscid; however, exploratory simulations using a vertical eddy-viscosity confined to a near bottom layer, along with a no-slip boundary condition, suggest that viscous effects may become important in water shallower than about 200 m. A shoaling solitary wave fissions into several waves. At depths of 200–300 m the front of the leading waves become nearly parallel to the bottom and develop a very steep back as has been observed. The leading waves are followed by waves of elevation (pedestals) that are conjugate to the waves of depression ahead and behind them. Horizontal resolutions of at least 50 m are required to simulate these well. Wave breaking was found to occur behind the second or third of the leading solitary waves, never at the back of the leading wave. Comparisons of the shoaling of waves started at depths of 1000 and 3000 m show significant differences and the shoaling waves can be significantly non-adiabatic even at depths greater than 2000 m. When waves reach a depth of 200 m, their amplitudes can be more than 50% larger than the largest possible solitary wave at that depth. The shoaling behaviour is sensitive to the presence of small-scale features in the bathymetry: a 200 m high bump at 700 m depth can result in the generation of many mode-two waves and of higher mode waves. Sensitivity to the stratification is considered by using three stratifications based on summer observations. They primarily differ in the depth of the thermocline. The generation of mode-two waves and the behaviour of the waves in shallow water is sensitive to this depth. Rotation affects the shoaling waves by reducing the amplitude of the leading waves via the radiation of long trailing inertia-gravity waves. The nonlinear-dispersive evolution of these inertia-gravity waves results in the formation of secondary mode-one wave packets.

Nonlinear resonances in a laminar wall jet: ejection of dipolar vortices

2007 ◽  
Vol 588 ◽  
pp. 279-308 ◽  
Author(s):  
STEFAN WERNZ ◽  
HERMANN F. FASEL
Keyword(s):  
Wave Packet ◽  
Large Amplitude ◽  
Small Amplitude ◽  
Three Dimensional ◽  
Resonant Interaction ◽  
Navier Stokes ◽  
Wall Jet ◽  
Two Dimensional ◽  
Amplitude Wave ◽  
Amplitude Level

Nonlinear mechanisms leading to the ejection of dipolar vortices from a laminar wall jet are being investigated using highly accurate Navier–Stokes simulations. With a set of well-defined numerical experiments for a forced Glauert wall jet, the nonlinear resonant interaction between the large-amplitude harmonic disturbance and a small-amplitude wave packet is systematically explored using two-dimensional simulations. Generated by a small-amplitude pulse, the wave packet experiences rapid resonant growth in the subharmonic part of its spectrum resulting in vortex mergings and, ultimately, the ejection of a pair of counter-rotating vortices from the wall jet. This two-dimensional subharmonic instability, if not mitigated by competing three-dimensional instabilities, can lead to the detachment of the entire wall jet from the surface. As shown using three-dimensional direct numerical simulations, vortex ejection still occurs in a forced transitional wall jet if the two-dimensional wave packet can reach a large amplitude level upstream of the region of three-dimensional turbulent breakdown. Movies are available with the online version of the paper.

Solitary waves on a vorticity layer

1994 ◽  
Vol 264 ◽  
pp. 303-319
Author(s):  
F. J. Higuera ◽  
J. Jiménez
Keyword(s):  
Solitary Waves ◽  
Large Amplitude ◽  
Small Amplitude ◽  
Symmetry Axis ◽  
Vries Equation ◽  
Continuous Family ◽  
Planar Case ◽  
Logarithmic Corrections ◽  
Spherical Vortex ◽  
The Waves

Contour dynamics methods are used to determine the shapes and speeds of planar, steadily propagating, solitary waves on a two-dimensional layer of uniform vorticity adjacent to a free-slip plane wall in an, otherwise irrotational, unbounded incompressible fluid, as well as of axisymmetric solitary waves propagating on a tube of azimuthal vorticity proportional to the distance to the symmetry axis. A continuous family of solutions of the Euler equations is found in each case. In the planar case they range from small-amplitude solitons of the Benjamin–Ono equation to large-amplitude waves that tend to one member of the touching pair of counter-rotating vortices of Pierrehumbert (1980), but this convergence is slow in two small regions near the tips of the waves, for which an asymptotic analysis is presented. In the axisymmetric case, the small-amplitude waves obey a Korteweg–de Vries equation with small logarithmic corrections, and the large-amplitude waves tend to Hill's spherical vortex.

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