A spatial dynamics approach to three-dimensional gravity-capillary steady water waves

2001 ◽  
Vol 131 (1) ◽  
pp. 83-136 ◽  
Author(s):  
M. D. Groves ◽  
A. Mielke
Keyword(s):  
Hamiltonian System ◽  
Water Waves ◽  
Spatial Dynamics ◽  
Three Dimensional ◽  
Finite Depth ◽  
Bond Number ◽  
Existence Theory ◽  
Water Wave Problem ◽  
Spatial Direction ◽  
Stokes Waves

This paper contains a rigorous existence theory for three-dimensional steady gravity-capillary finite-depth water waves which are uniformly translating in one horizontal spatial direction x and periodic in the transverse direction z. Physically motivated arguments are used to find a formulation of the problem as an infinite-dimensional Hamiltonian system in which x is the time-like variable, and a centre-manifold reduction technique is applied to demonstrate that the problem is locally equivalent to a finite-dimensional Hamiltonian system. General statements concerning the existence of waves which are periodic or quasiperiodic in x (and periodic in z) are made by applying standard tools in Hamiltonian-systems theory to the reduced equations.A critical curve in Bond number–Froude number parameter space is identified which is associated with bifurcations of generalized solitary waves. These waves are three dimensional but decay to two-dimensional periodic waves (small-amplitude Stokes waves) far upstream and downstream. Their existence as solutions of the water-wave problem confirms previous predictions made on the basis of model equations.

scholarly journals An Existence Theory for Gravity–Capillary Solitary Water Waves

Water Waves ◽  
2021 ◽  
Author(s):  
M. D. Groves
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Applied Mathematics ◽  
Spatial Dynamics ◽  
Existence Theory ◽  
Water Wave Problem ◽  
Reduction Methods ◽  
Centre Manifold Reduction ◽  
Weak Surface ◽  
The One

AbstractIn the applied mathematics literature solitary gravity–capillary water waves are modelled by approximating the standard governing equations for water waves by a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). These formal arguments have been justified by sophisticated techniques such as spatial dynamics and centre-manifold reduction methods on the one hand and variational methods on the other. This article presents a complete, self-contained account of an alternative, simpler approach in which one works directly with the Zakharov–Craig–Sulem formulation of the water-wave problem and uses only rudimentary fixed-point arguments and Fourier analysis.

scholarly journals Three-dimensional surface gravity waves of a broad bandwidth on deep water

2021 ◽  
Vol 926 ◽  
Author(s):  
Yan Li
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Order Approximation ◽  
Three Dimensional ◽  
Finite Depth ◽  
Second Order ◽  
Leading Order ◽  
Broad Bandwidth ◽  
Stokes Waves ◽  
Deep Water Waves

A new nonlinear Schrödinger equation (NLSE) is presented for ocean surface waves. Earlier derivations of NLSEs that describe the evolution of deep-water waves have been limited to a narrow bandwidth, for which the bound waves at second order in wave steepness are described in leading-order approximations. This work generalizes these earlier works to allow for deep-water waves of a broad bandwidth with large directional spreading. The new NLSE permits simple numerical implementations and can be extended in a straightforward manner in order to account for waves on water of finite depth. For the description of second-order waves, this paper proposes a semianalytical approach that can provide accurate and computationally efficient predictions. With a leading-order approximation to the new NLSE, the instability region and energy growth rate of Stokes waves are investigated. Compared with the exact results based on McLean (J. Fluid Mech., vol. 511, 1982, p. 135), predictions by the new NLSE show better agreement than by Trulsen et al. (Phys. Fluids, vol. 12, 2000, pp. 2432–2437). With numerical implementations of the new NLSE, the effects of wave directionality are investigated by examining the evolution of a directionally spread focused wave group. A downward shift of the spectral peak is observed, owing to the asymmetry in the change rate of energy in a more complex manner than that for uniform Stokes waves. Rapid oblique energy transfers near the group at linear focus are observed, likely arising from the instability of uniform Stokes waves appearing in a narrow spectrum subject to oblique sideband disturbances.

Computing Acceleration Loads on Free-Fall Lifeboat Occupants: Consequences of Including Nonlinearities in Water Waves and Mother Vessel Motions

2010 ◽  
Author(s):  
Neil Luxcey ◽  
Se´bastien Fouques ◽  
Thomas Sauder
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Free Fall ◽  
Momentum Conservation ◽  
Irregular Waves ◽  
Stokes Waves ◽  
Induced Motion ◽  
Wave Models ◽  
Wave Induced ◽  
The Impact

The safety of occupants in free-fall lifeboats (FFL) launched from a skid is addressed, and the focus is on numerical evaluation of acceleration loads during water impact. This paper investigates the required level of detail when modeling the physics of a lifeboat launch in waves. The first part emphasizes the importance of the non-linearity of the wave surface. Severity of impacts in linear (Airy) waves is compared to impacts in regular Stokes waves of the 5th order. Correspondingly, severity of impacts in irregular waves of the 2nd order is statistically compared to impacts in linear irregular waves. Theory of the two wave models are also briefly presented. The second part discusses the importance of a more detailed modeling of the launching system. This concerns especially cases for which damage to the mother vessel induces major lifeboat heel angles. A three-dimensional skid model is presented, along with validation against experimental measurements. In addition, the wave induced motion of the mother vessel is included. Consequences on the severity of the impact of the lifeboat in regular waves are discussed. This study is based on MARINTEK’s impact simulator for free-fall lifeboats, in which slamming loads are evaluated based on momentum conservation, a long wave approximation, and a von Karman type of approach. It is coupled here to the SIMO software, also developed at MARINTEK. Performance of this coupling is discussed.

On the stability of lumps and wave collapse in water waves

2008 ◽  
Vol 366 (1876) ◽  
pp. 2761-2774 ◽  
Author(s):  
T.R Akylas ◽  
Yeunwoo Cho
Keyword(s):  
Ground State ◽  
Water Waves ◽  
Initial Conditions ◽  
Finite Depth ◽  
Equation System ◽  
Capillary Waves ◽  
Mean Flow ◽  
Water Wave Problem ◽  
Petviashvili Equation ◽  
Wave Collapse

In the classical water-wave problem, fully localized nonlinear waves of permanent form, commonly referred to as lumps, are possible only if both gravity and surface tension are present. While much attention has been paid to shallow-water lumps, which are generalizations of Korteweg–de Vries solitary waves, the present study is concerned with a distinct class of gravity–capillary lumps recently found on water of finite or infinite depth. In the near linear limit, these lumps resemble locally confined wave packets with envelope and wave crests moving at the same speed, and they can be approximated in terms of a particular steady solution (ground state) of an elliptic equation system of the Benney–Roskes–Davey–Stewartson (BRDS) type, which governs the coupled evolution of the envelope along with the induced mean flow. According to the BRDS equations, however, initial conditions above a certain threshold develop a singularity in finite time, known as wave collapse, due to nonlinear focusing; the ground state, in fact, being exactly at the threshold for collapse suggests that the newly discovered lumps are unstable. In an effort to understand the role of this singularity in the dynamics of lumps, here we consider the fifth-order Kadomtsev–Petviashvili equation, a model for weakly nonlinear gravity–capillary waves on water of finite depth when the Bond number is close to one-third, which also admits lumps of the wave packet type. It is found that an exchange of stability occurs at a certain finite wave steepness, lumps being unstable below but stable above this critical value. As a result, a small-amplitude lump, which is linearly unstable and according to the BRDS equations would be prone to wave collapse, depending on the perturbation, either decays into dispersive waves or evolves into an oscillatory state near a finite-amplitude stable lump.

Directed fluid sheets

1976 ◽  
Vol 347 (1651) ◽  
pp. 447-473 ◽  
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Steady Motion ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Inviscid Fluid ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Inviscid Fluids ◽  
Two Dimensional Flows

This paper is concerned mainly with incompressible inviscid fluid sheets but the incompressible linearly viscous fluid sheet is also considered. Our development is based on a direct formulation using the two dimensional theory of directed media called Cosserat surfaces . The first part of the paper deals with the formulation of appropriate nonlinear equations (which may include the effects of gravity and surface tension) governing the two dimensional motion of incompressible inviscid media for two categories, namely those ( a ) for two dimensional flows confined to a plane perpendicular to a specified direction and ( b ) for propagation of fairly long waves in a stream of variable initial depth. The latter development is a generalization of an earlier direct formulation of a theory of water waves when the fixed bottom of the stream is level (Green, Laws & Naghdi 1974). In the second part of the paper, special attention is given to a demonstration of the relevance and applicability of the present direct formulation to a variety of two dimensional problems of inviscid fluid sheets. These include, among others, the steady motion of a class of two-dimensional flows in a stream of finite depth in which the bed of the stream may change from one constant level to another, the related problem of hydraulic jumps, and a class of exact solutions which characterize the main features of the time-dependent free surface flows in the three dimensional theory of incompressible inviscid fluids.

scholarly journals Non-existence of three-dimensional travelling water waves with constant non-zero vorticity

2014 ◽  
Vol 746 ◽  
Author(s):  
E. Wahlén
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Elliptic Equations ◽  
Three Dimensional ◽  
Unique Continuation ◽  
Finite Depth ◽  
Capillary Waves ◽  
Liouville Type ◽  
Liouville Type Results

AbstractWe prove that there are no three-dimensional bounded travelling gravity waves with constant non-zero vorticity on water of finite depth. The result also holds for gravity–capillary waves under a certain condition on the pressure at the surface, which is satisfied by sufficiently small waves. The proof relies on unique continuation arguments and Liouville-type results for elliptic equations.

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.

Integrals with a large parameter: water waves on finite depth due to an impulse

1995 ◽  
Vol 450 (1938) ◽  
pp. 67-87 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Stationary Phase ◽  
Water Waves ◽  
Three Dimensional ◽  
Finite Depth ◽  
Three Dimensions ◽  
Dimensional Problem ◽  
Airy Functions ◽  
Asymptotic Results ◽  
Conventional Solution

The classical Cauchy–Poisson problem, of water waves generated by an impulsive disturbance on the free surface, is treated in three dimensions for finite constant depth. The conventional solution is in the form of a Fourier-Bessel transform. We wish to find its asymptotic behaviour at large distances r and large times t . Difficulties arise at the wavefront, where r/t is equal to the maximum group velocity. In the analogous two-dimensional problem the waves near the front are associated with two nearly coincident points of stationary phase and described asymptotically by Airy functions. In the three-dimensional problem the solution is expressed as a double integral, where there are four nearly coincident points of stationary phase. A systematic procedure is given for successive terms in an asymptotic expansion, involving the square of an Airy function and its derivatives. In both two and three dimensions it is important to use an appropriate transform of the variable(s) of integration, to achieve uniform validity of the asymptotic expansion. Calculations are performed to illustrate the utility of the asymptotic results.

Steady water waves with vorticity: an analysis of the dispersion equation

2014 ◽  
Vol 751 ◽  
Author(s):  
V. Kozlov ◽  
N. Kuznetsov ◽  
E. Lokharu
Keyword(s):  
Dispersion Equation ◽  
Gravity Waves ◽  
Water Waves ◽  
Liouville Problem ◽  
Finite Depth ◽  
Stokes Waves ◽  
General Dispersion ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Appl Math

AbstractTwo-dimensional steady gravity waves with vorticity are considered on water of finite depth. The dispersion equation is analysed for general vorticity distributions, but under assumptions valid only for unidirectional shear flows. It is shown that for these flows (i) the general dispersion equation is equivalent to the Sturm–Liouville problem considered by Constantin & Strauss (Commun. Pure Appl. Math., vol. 57, 2004, pp. 481–527; Arch. Rat. Mech. Anal., vol. 202, 2011, pp. 133–175), (ii) the condition guaranteeing bifurcation of Stokes waves with constant wavelength is fulfilled. Moreover, a necessary and sufficient condition that the Sturm–Liouville problem mentioned in (i) has an eigenvalue is obtained.

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