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 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.

Étude de la réouverture de la lagune du Havre aux Basques. I. Modélisation numérique des conditions d'écoulement

1995 ◽  
Vol 22 (1) ◽  
pp. 55-71
Author(s):  
Y. Ouellet ◽  
A. Khelifa ◽  
J.-F. Bellemare
Keyword(s):  
Finite Element ◽  
Numerical Study ◽  
Element Model ◽  
Two Dimensional ◽  
Flow Conditions ◽  
Modélisation Numérique ◽  
Tidal Flows ◽  
Dimensional Finite Element ◽  
The Stability ◽  
La Madeleine

A numerical study based on a two-dimensional finite element model has been conducted to analyze flow conditions associated with different possible designs for the reopening of Havre aux Basques lagoon, located in Îles de la Madeleine, in the middle of the Gulf of St. Lawrence. More specifically, the study has been done to better define the depth and geometry of the future channel as well as its orientation with regard to tidal flows within the inlet and the lagoon. Results obtained from the model have been compared and analyzed to put forward some recommendations about choice of a design insuring the stability of the inlet with tidal flows. Key words: numerical model, finite element, lagoon, reopening, Havre aux Basques, Îles de la Madeleine.

Non-normal stability analysis of a shear current under surface gravity waves

2008 ◽  
Vol 609 ◽  
pp. 49-58
Author(s):  
D. AMBROSI ◽  
M. ONORATO
Keyword(s):  
Growth Rate ◽  
Modal Analysis ◽  
Gravity Waves ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Surface Gravity ◽  
Horizontal Shear ◽  
Surface Gravity Waves ◽  
Shear Current ◽  
The Stability

The stability of a horizontal shear current under surface gravity waves is investigated on the basis of the Rayleigh equation. As the differential operator is non-normal, a standard modal analysis is not effective in capturing the transient growth of a perturbation. The representation of the stream function by a suitable basis of bi-orthogonal eigenfunctions allows one to determine the maximum growth rate of a perturbation. It turns out that, in the considered range of parameters, such a growth rate can be two orders of magnitude larger than the maximum eigenvalue obtained by standard modal analysis.

Instability of Shielded Surface Temperature Vortices

2011 ◽  
Vol 68 (5) ◽  
pp. 964-971 ◽  
Author(s):  
Benjamin J. Harvey ◽  
Maarten H. P. Ambaum ◽  
Xavier J. Carton
Keyword(s):  
Surface Temperature ◽  
Smooth Surface ◽  
Numerical Study ◽  
Normal Modes ◽  
Euler System ◽  
The Other ◽  
Two Dimensional ◽  
Dynamics Model ◽  
Contour Dynamics ◽  
The Stability

Abstract The stability characteristics of the surface quasigeostrophic shielded Rankine vortex are found using a linearized contour dynamics model. Both the normal modes and nonmodal evolution of the system are analyzed and the results are compared with two previous studies. One is a numerical study of the instability of smooth surface quasigeostrophic vortices with which qualitative similarities are found and the other is a corresponding study for the two-dimensional Euler system with which several notable differences are highlighted.

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.

Stationary gravity waves on non-uniform free streams: jet-like streams

1975 ◽  
Vol 77 (2) ◽  
pp. 415-438 ◽  
Author(s):  
D. H. Peregrine ◽  
Ronald Smith
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Wave Number ◽  
Explicit Solutions ◽  
Stationary Waves ◽  
Two Dimensional ◽  
Large Wave ◽  
Surface Gravity Waves ◽  
Large Wave Number

AbstractThe basic state considered in this paper is a parallel flow of a jet-like character with the centre of the jet being at or near a free surface which is horizontal. Stationary surface gravity waves may exist on such a flow, and a number of examples are looked at for small amplitude waves. Explicit solutions are given for ‘top-hat’ profile jets and for two-dimensional flows. Asymptotic solutions are developed for stationary waves of large wave-number.

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.

A numerical study of two-dimensional coupled systems and higher order partial differential equations

2019 ◽  
Vol 12 (05) ◽  
pp. 1950071
Author(s):  
R. Rohila ◽  
R. C. Mittal
Keyword(s):  
Differential Equations ◽  
Numerical Study ◽  
Differential Quadrature Method ◽  
Bernstein Polynomials ◽  
Differential Quadrature ◽  
Coupled Systems ◽  
Two Dimensional ◽  
Quadrature Method ◽  
Grid Points ◽  
The Stability

In this paper, a new approach and methodology is developed by incorporating differential quadrature technique with Bernstein polynomials. In differential quadrature method, approximations are done in a way that the derivatives of the function are replaced by a linear sum of functional values at the grid points of the given domain. In Bernstein differential quadrature method (BDQM), Bernstein polynomials are employed for spatial discretization so that a system of ordinary differential equations (ODE’s) is obtained which is solved by SSPRK-43 method. The stability of the method is also studied. The accuracy of the present method is checked by performing numerical experiments on two-dimensional coupled Burgers’ and Brusselator systems and fourth-order extended Fisher Kolmogorov (EFK) equation. Implementation of the method is very easy, efficient and capable of reducing the size of computational efforts.

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