Capillary‐gravity solitary waves on water of finite depth interacting with a linear shear current

2021 ◽  
Author(s):  
T. Gao ◽  
P. A. Milewski ◽  
Z. Wang
Keyword(s):  
Solitary Waves ◽  
Finite Depth ◽  
Shear Current ◽  
Linear Shear

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 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.

Steep solitary waves in water of finite depth with constant vorticity

1994 ◽  
Vol 274 ◽  
pp. 339-348 ◽  
Author(s):  
J.-M. Vanden-Broeck
Keyword(s):  
Integral Equation ◽  
Solitary Waves ◽  
Numerical Scheme ◽  
Integral Equation Method ◽  
Finite Depth ◽  
Boundary Integral ◽  
Constant Vorticity ◽  
Uniform Shear ◽  
Uniform Shear Flow ◽  
Shear Current

Solitary waves with constant vorticity in water of finite depth are calculated numerically by a boundary integral equation method. Previous calculations are confirmed and extended. It is shown that there are branches of solutions which bifurcate from a uniform shear current. Some of these branches are characterized by a limiting configuration with a 120° angle at the crest of the wave. Other branches extend for arbitrary large values of the amplitude of the wave. The corresponding solutions ultimately approach closed regions of constant vorticity in contact with the bottom of the channel. A numerical scheme is presented to calculate directly these closed regions of constant vorticity. In addition, it is shown that there are branches of solutions which do not bifurcate from a uniform shear flow.

Comparison principles for free-surface flows with gravity

1991 ◽  
Vol 230 ◽  
pp. 231-243 ◽  
Author(s):  
Walter Craig ◽  
Peter Sternberg
Keyword(s):  
Solitary Waves ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Immiscible Fluids ◽  
Steady Flows ◽  
Two Dimensional ◽  
Surface Flows ◽  
Ship Hulls ◽  
Geometrical Information ◽  
Supercritical Flows

This article considers certain two-dimensional, irrotational, steady flows in fluid regions of finite depth and infinite horizontal extent. Geometrical information about these flows and their singularities is obtained, using a variant of a classical comparison principle. The results are applied to three types of problems: (i) supercritical solitary waves carrying planing surfaces or surfboards, (ii) supercritical flows past ship hulls and (iii) supercritical interfacial solitary waves in systems consisting of two immiscible fluids.

scholarly journals Finite-depth effects on solitary waves in a floating ice sheet

2014 ◽  
Vol 49 ◽  
pp. 242-262 ◽  
Author(s):  
Philippe Guyenne ◽  
Emilian I. Părău
Keyword(s):  
Solitary Waves ◽  
Ice Sheet ◽  
Finite Depth ◽  
Depth Effects

scholarly journals Internal gravity-capillary solitary waves in finite depth

2016 ◽  
Vol 40 (4) ◽  
pp. 1053-1080 ◽  
Author(s):  
Dag Viktor Nilsson
Keyword(s):  
Solitary Waves ◽  
Finite Depth

The evolution equation for the second-order internal solitary waves in stratified fluids of finite depth

2006 ◽  
Vol 18 (S1) ◽  
pp. 300-305
Author(s):  
Youliang Cheng ◽  
Zhongyao Fan
Keyword(s):  
Evolution Equation ◽  
Solitary Waves ◽  
Finite Depth ◽  
Second Order ◽  
Internal Solitary Waves ◽  
Stratified Fluids

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.

Oblique runup of non-breaking solitary waves on an inclined plane

2011 ◽  
Vol 668 ◽  
pp. 582-606 ◽  
Author(s):  
GEIR K. PEDERSEN
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Evolution Equations ◽  
Boussinesq Equations ◽  
Normal Incidence ◽  
Finite Depth ◽  
Breaking Waves ◽  
Wave Pattern ◽  
Angle Of Incidence ◽  
Inclined Plane

When a wave of permanent form is obliquely incident on an inclined plane, the wave pattern becomes stationary in a frame of reference which moves along the shore. This enables a simplified mathematical description of the problem which is used herein as a basis for efficient and accurate numerical simulations. First, a nonlinear and weakly dispersive set of Boussinesq equations for the downstream evolution of such stationary patterns is derived. In the hydrostatic approximation, streamline-based Lagrangian versions of the evolution equations are developed for automatic tracing of the shoreline. Both equation sets are, in their present form, developed for non-breaking waves only. Finite difference models for both equation sets are designed. These methods are then coupled dynamically to obtain a single nonlinear model with dispersive wave propagation in finite depth and an accurate runup representation. The models are tested by runup of waves at normal incidence and comparison with a more general model for the refraction of a solitary wave on a slope. Finally, a set of runup computations for oblique solitary waves is performed and compared with estimates of oblique runup heights obtained from a combination of an analytic solution for normal incidence and optics. We find that the runup heights decrease in proportion to the square of the angle of incidence for angles up to 45°, for which the height is reduced by around 12% relative to that of normal incidence. In Appendix A, the validity of the downstream formulation is discussed in the light of solitary wave optics and wave jumps.

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