Solitary waves on water of finite depth with a surface or bottom shear layer

1995 ◽  
Vol 7 (5) ◽  
pp. 1048-1055 ◽  
Author(s):  
Huyun Sha ◽  
J.‐M. Vanden‐Broeck
Keyword(s):  
Shear Layer ◽  
Solitary Waves ◽  
Finite Depth

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

The Role of Nonnormality in Overreflection Theory

2010 ◽  
Vol 67 (8) ◽  
pp. 2547-2558 ◽  
Author(s):  
Nikolaos A. Bakas ◽  
Brian F. Farrell
Keyword(s):  
Shear Layer ◽  
Gravity Waves ◽  
Finite Depth ◽  
Stimulated Emission ◽  
Incoming Wave ◽  
Propagating Waves ◽  
Three Stages ◽  
Orr Mechanism ◽  
Weak Stratification

Abstract The role of nonnormality in the overreflection of gravity waves is investigated. In the limit of weak stratification, wave packets having a critical level inside a shear layer of finite depth are reflected with amplified energy. This process, which exhibits the characteristics of stimulated emission, occurs in three stages: first, the incoming wave enters the shear layer and excites nonpropagating perturbations leaning with and against the shear. Subsequently, the energy of perturbations leaning against the shear grows in a manner similar to energy growth of perturbations in constant shear flows, indicating that the Orr mechanism that is slightly modified by stratification underlies the observed growth. Finally, the amplified perturbations excite propagating waves originating from the vicinity of the shear layer boundary. The role of nonnormality in this process is also investigated from the perspective of the associated nonorthogonality of the modes of the dynamical system. It is found that the incident wave packet projects on nonorthogonal analytic modes having the structure of a downward propagating wave in the far field below the shear layer and overreflection expressed by the interaction among these nonorthogonal modes.

Effect of vorticity on steady water waves

2008 ◽  
Vol 608 ◽  
pp. 197-215 ◽  
Author(s):  
JOY KO ◽  
WALTER STRAUSS
Keyword(s):  
Shear Layer ◽  
Stagnation Point ◽  
Mass Flux ◽  
Water Waves ◽  
Large Amplitude ◽  
Maximum Amplitude ◽  
Finite Depth ◽  
Hydraulic Head ◽  
Two Dimensional

Two-dimensional, finite-depth periodic steady water waves with variable vorticity ω=γ(ψ) and large amplitude a are computed for a large number of cases. In particular, the effect of a shear layer at the top, the middle or the bottom is considered. The maximum amplitude amax varies monotonically with the vorticity function γ(ċ). It is increasing if the stagnation point is at the crest, and is decreasing if the stagnation point is in the interior of the fluid or on the bottom. Relationships between the amplitude, hydraulic head, depth and mass flux are investigated.

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.

scholarly journals Effect on Doppler Resonance From a Near-Surface Shear Layer

2017 ◽  
Author(s):  
Benjamin K. Smeltzer ◽  
Yan Li ◽  
Simen Å. Ellingsen
Keyword(s):  
Shear Layer ◽  
Finite Depth ◽  
Gravitational Acceleration ◽  
Surface Shear ◽  
Constant Frequency ◽  
Wave Source ◽  
Near Surface ◽  
Strong Shear ◽  
Uniform Current ◽  
Source Velocity

For waves generated by a wave source which is simultaneously moving and oscillating at a constant frequency ω, a resonance is well known to occur at a particular value τres of the nondimensional frequency τ = ωV/g (V: source velocity relative to the surface, g: gravitational acceleration). For quiescent, deep water, it is well known that τres = 1/4. We study the effect on τres from the presence of a shear flow in a layer near the surface, such as may be generated by wind or tidal currents. Assuming the vorticity is constant within the shear layer, we find that the effects on the resonant frequency can be significant even for sources corresponding to moderate shear and relatively long waves, while for stronger shear and shorter waves the effect is stronger. Even for a situation where the resonant waves have wavelengths about 20 times the width of the shear layer, the resonance frequency can change by ∼ 25% for even a moderately strong shear VS/g = 0.3 (S: vorticity in surface shear layer). Intuition for the problem is built by first considering two simpler geometries: uniform current with finite depth, and Couette flow of finite depth.

scholarly journals Do true elevation gravity–capillary solitary waves exist? A numerical investigation

2002 ◽  
Vol 454 ◽  
pp. 403-417 ◽  
Author(s):  
A. R. CHAMPNEYS ◽  
J.-M. VANDEN-BROECK ◽  
G. J. LORD
Keyword(s):  
Solitary Waves ◽  
Finite Depth ◽  
Bond Number ◽  
Boundary Integral ◽  
Numerical Results ◽  
Boundary Integral Method ◽  
Small Function ◽  
Asymptotic Results ◽  
Water Wave Problem ◽  
Strong Conclusion

This paper extends the numerical results of Hunter & Vanden-Broeck (1983) and Vanden-Broeck (1991) which were concerned with studies of solitary waves on the surface of fluids of finite depth under the action of gravity and surface tension. The aim of this paper is to answer the question of whether small-amplitude elevation solitary waves exist. Several analytical results have proved that bifurcating from Froude number F = 1, for Bond number τ between 0 and 1/3, there are families of ‘generalized’ solitary waves with periodic tails whose minimum amplitude is an exponentially small function of F−1. An open problem (which, for τ sufficiently close to 1/3, was recently proved by Sun 1999 to be false) is whether this amplitude can ever be zero, which would give a truly localized solitary wave.The problem is first addressed in terms of model equations taking the form of generalized fifth-order KdV equations, where it is demonstrated that if such a zero-tail-amplitude solution occurs, it does so along codimension-one lines in the parameter plane. Moreover, along solution paths of generalized solitary waves a topological distinction is found between cases where the tail does vanish and those where it does not. This motivates a new set of numerical results for the full problem, formulated using a boundary integral method, namely to probe the size of the tail amplitude as τ varies for fixed F > 1. The strong conclusion from the numerical results is that true solitary waves of elevation do not exist for the steady gravity–capillary water wave problem, at least for 9/50 < τ < 1=3. This finding confirms and explains previous asymptotic results by Yang & Akylas.

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