Statistics of Coherent Structures Collisions and their Dynamics on the Surface of Deep Water

2020 ◽  
Author(s):  
Sergey Dremov ◽  
Dmitriy Kachulin ◽  
Alexander Dyachenko
Keyword(s):  
Free Surface ◽  
Incompressible Fluid ◽  
Deep Water ◽  
Coherent Structures ◽  
Water Waves ◽  
Energy Exchange ◽  
Relative Phase ◽  
Collision Probability ◽  
Statistical Characteristics ◽  
Potential Flows

<p>        The present work is devoted to the study of coherent structures collisions dynamics in the models of deep water waves equations: the model of a supercompact equation for deep water unidirectional waves (SCEq) and the model of Dyachenko equations for potential flows of incompressible fluid with free surface. In these models there are special solutions in the form of coherent wave structures called breathers. They can be found numerically by using the Petviashvili method. One can consider the combination of such breathers as a model of rarefied soliton gas, and their paired collisions in this case are a key feature in forming of dynamics and statistics in the model. To describe statistical characteristics of breathers collision Probability Density Function (PDF) is used. PDF of breathers wave amplitudes during their collision was calculated and compared with the known results in the model of Nonlinear Schrodinger equation (NLS). In contrast to the NLS model there is a number of interesting features in the model of SCEq. For instance, the amplitude maximum of wave arising during the collision can exceed the sum of interacting breathers amplitudes, what cannot happen in NLS model. Moreover, it depends on the initial breathers steepness. In addition, it is shown that the breathers acquire phase and space shifts after each collision, and thus their velocity also changes. Depending on the relative phase breathers can give their energy or take it, and as a result their amplitude can be decreased or increased respectively. The same situation can be seen in the model of equations for potential flows of incompressible fluid with free surface. In addition to the dependence on relative phase the duration of the collision also affects the energy exchange. Breathers collisions are accompanied by appearance of little radiation, and its value is relatively less than the value of energy exchange. The results of statistics calculating and dynamics studying in the rarefied gas of coherent structures will be shown in the present work.</p><p>           The work was supported by Russian Science Foundation grant № 18-71-00079.</p>

scholarly journals Bound Coherent Structures Propagating on the Free Surface of Deep Water

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 115
Author(s):  
Dmitry Kachulin ◽  
Sergey Dremov ◽  
Alexander Dyachenko
Keyword(s):  
Free Surface ◽  
Deep Water ◽  
Coherent Structures ◽  
Water Waves ◽  
Nonlinear Models ◽  
Ideal Incompressible Fluid ◽  
Equation Model ◽  
System Of Nonlinear Equations ◽  
Potential Flows ◽  
Deep Water Waves

This article presents a study of bound periodically oscillating coherent structures arising on the free surface of deep water. Such structures resemble the well known bi-soliton solution of the nonlinear Schrödinger equation. The research was carried out in the super-compact Dyachenko-Zakharov equation model for unidirectional deep water waves and the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. The special numerical algorithm that includes a damping procedure of radiation and velocity adjusting was used for obtaining such bound structures. The results showed that in both nonlinear models for deep water waves after the damping is turned off, a periodically oscillating bound structure remains on the fluid surface and propagates stably over hundreds of thousands of characteristic wave periods without losing energy.

Bound Coherent Structures Propagating on the Free Surface of Deep Water

2021 ◽  
Author(s):  
Sergey Dremov ◽  
Dmitry Kachulin ◽  
Alexander Dyachenko
Keyword(s):  
Free Surface ◽  
Incompressible Fluid ◽  
Deep Water ◽  
Coherent Structures ◽  
Water Waves ◽  
Initial Conditions ◽  
Soliton Solutions ◽  
Ideal Incompressible Fluid ◽  
System Of Nonlinear Equations ◽  
Computational Domain

<p><span>               The work presents the results of studying the bound coherent structures propagating on the free surface of ideal incompressible fluid of infinite depth. Examples of such structures are bi-solitons which are exact solutions of the known approximate model for deep water waves — the nonlinear Schrödinger equation (NLSE). Recently, when studying multiple breathers collisions, the occurrence of such objects was found in a more accurate model of the supercompact equation for unidirectional water waves [1]. The aim of this work is obtaining and further studying such structures with different parameters in the supercompact equation and in the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. </span><span>The algorithm used for finding the bound coherent objects was similar to the one described in [2]. As the initial conditions for obtaining such structures in the framework of the above models, the NLSE bi-soliton solutions were used, as well as two single breathers numerically found by the Petviashvili method and placed in a same point of the computational domain. During the evolution calculation the initial structures emitted incoherent waves which were filtered at the boundaries of the domain using the damping procedure. It is shown that after switching off the filtering of radiation, periodically oscillating coherent objects remain on the surface of the liquid, propagate stably during one hundred thousand characteristic wave periods and do not lose energy. The profiles of such structures at different parameters are compared.</span></p><p><span>This work was supported by RSF grant </span><span>19-72-30028</span><span> and </span><span>RFBR grant </span><span>20-31-90093</span><span>.</span></p><p><span>[1] Kachulin D., Dyachenko A., Dremov S. Multiple Soliton Interactions on the Surface of Deep Water //Fluids. – 2020. – Т. 5. – №. 2. – С. 65.</span></p><p><span>[2] Dyachenko A. I., Zakharov V. E. On the formation of freak waves on the surface of deep water //JETP letters. – 2008. – Т. 88. – №. 5. – С. 307.</span></p>

Effective Power and Speed Loss of Underwater Vehicles in Close Proximity to Regular Waves

2017 ◽  
Author(s):  
Stefan Daum ◽  
Martin Greve ◽  
Renato Skejic
Keyword(s):  
Free Surface ◽  
Deep Water ◽  
Water Waves ◽  
Underwater Vehicles ◽  
Total Resistance ◽  
Wave Load ◽  
Regular Waves ◽  
Effective Power ◽  
Close Proximity ◽  
Deep Water Waves

The present study is focused on performance issues of underwater vehicles near the free surface and gives insight into the analysis of a speed loss in regular deep water waves. Predictions of the speed loss are based on the evaluation of the total resistance and effective power in calm water and preselected regular wave fields w.r.t. the non-dimensional wave to body length ratio. It has been assumed that the water is sufficiently deep and that the vehicle is operating in a range of small to moderate Froude numbers by moving forward on a straight-line course with a defined encounter angle of incident regular waves. A modified version of the Doctors & Days [1] method as presented in Skejic and Jullumstrø [2] is used for the determination of the total resistance and consequently the effective power. In particular, the wave-making resistance is estimated by using different approaches covering simplified methods, i.e. Michell’s thin ship theory with the inclusion of viscosity effects Tuck [3] and Lazauskas [4] as well as boundary element methods, i.e. 3D Rankine source calculations according to Hess and Smith [5]. These methods are based on the linear potential fluid flow and are compared to fully viscous finite volume methods for selected geometries. The wave resistance models are verified and validated by published data of a prolate spheroid and one appropriate axisymmetric submarine model. Added resistance in regular deep water waves is obtained through evaluation of the surge mean second-order wave load. For this purpose, two different theoretical models based on potential flow theory are used: Loukakis and Sclavounos [6] and Salvesen et. al. [7]. The considered theories cover the whole range of important wavelengths for an underwater vehicle advancing in close proximity to the free surface. Comparisons between the outlined wave load theories and available theoretical and experimental data were carried out for a submerged submarine and a horizontal cylinder. Finally, the effective power and speed loss are discussed from a submarine operational point of view where the mentioned parameters directly influence mission requirements in a seaway. All presented results are carried out from the perspective of accuracy and efficiency within common engineering practice. By concluding current investigations in regular waves an outlook will be drawn to the application of advancing underwater vehicles in more realistic sea conditions.

Nonlinear Fourier Analysis Algorithm and Models for Water Waves in Terms of Surface Elevation, Amplitude Modulations

2019 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Synthetic Aperture Radar ◽  
Surface Waves ◽  
Shallow Water ◽  
Wave Field ◽  
Deep Water ◽  
Coherent Structures ◽  
Water Waves ◽  
Surface Elevation ◽  
Synthetic Aperture ◽  
Wave Measurements

Abstract This paper addresses two issues with regard to nonlinear ocean waves. (1) The first issue relates to the often-confused differences between the coordinates used for the measurement and characterization of ocean surface waves: The surface elevation and the complex modulation of a wave field. (2) The second issue relates to the very different kinds of physical wave behavior that occur in shallow and deep water. Both issues come from the known, very different behaviors of deep and shallow water waves. In shallow water one often uses the Korteweg-deVries that describes the wave surface elevation in terms of cnoidal waves and solitons. In deep water one uses the nonlinear Schrödinger equation whose solutions correspond to the complex envelope of a wave field that has Stokes wave and breather solutions. Here I make clear the relationships between the two ways of characterizing surface waves. Furthermore, and more importantly, I address the issues of matching the two types of wave behavior as the wave motion passes from deep to shallow water, or vice versa. For wave measurements we normally obtain the surface elevation with a wave staff, resistance gauge or pressure recorder for getting time series. Remote sensing applications relate to the use of lidar, radar or synthetic aperture radar for obtaining space series. The two types of wave behavior can therefore crucially depend on where the instrument is placed on the “ground track” or “field” over which the lidar or radar measurements are made. Thus the matching problem from deep to shallow water is not only important for wave measurements, but also for wave modeling. Modern wave models [Osborne, 2010, 2018, 2019a, 2019b] that maintain the coherent structures of wave dynamics (solitons, Stokes waves, breathers, superbreathers, vortices, etc.) must naturally pass from deep to shallow water where the nature of the nonlinear physics, and the form of the coherent structures, change. I address these issues and more herein. This paper is directed towards the development of methods for the real time measurement of waves by shipboard radar and for wave measurements by airplane and helicopter using lidar and synthetic aperture radar. Wave modeling efforts are also underway.

scholarly journals PAIRWISE INTERACTIONS OF COHERENT STRUCTURES ON THE SURFACE OF DEEP WATER

2019 ◽  
Vol 47 (1) ◽  
pp. 66-68
Author(s):  
D.I. Kachulin ◽  
A.A. Gelash ◽  
A.I. Dyachenko ◽  
V.E. Zakharov
Keyword(s):  
Wave Field ◽  
Deep Water ◽  
Coherent Structures ◽  
Relative Phase ◽  
Field Amplitude ◽  
Energy Losses ◽  
Exact Model ◽  
Pairwise Interactions ◽  
The Moment ◽  
Maximum Amplification

The interactions of coherent structures (different types of solitary wave groups) on the surface of deep water is an important nonlinear wave process, which has been studied both theoretically and experimentally (Dyachenko et al., 2013a, b; Slunyaev et al., 2017). At the moment, a complete theoretical description of such interactions is known only for the simplest model – the nonlinear Schrödinger equation (NSE) where exact multi-soliton solutions are found. In the work (Kachulin, Gelash, 2018), the dynamics of pairwise interactions of coherent structures (breathers) on the surface of deep water were numerically investigated in the framework of the Dyachenko-Zakharov model. Significant differences were found in the collision dynamics of breathers of the compact Dyachenko-Zakharov equation when compared to the behavior of the NLSE solitons. It was found that in a more precise model of gravitational surface waves, in contrast to the NLSE, the maximum amplification of the wave field amplitude during the collision process of coherent structures can exceed the sum of the initial amplitudes of the breathers. In addition, the maximum amplification of the wave field amplitude increases with increasing steepness of the interacting breathers and exceeds unity by 20% at the value of the wave steepness m ≈ 0.2. It was revealed that an important parameter determining the dynamics of pairwise collisions of breathers is the relative phase of these objects at the moment of interaction. The interaction of breathers in the non-integrable Dyachenko-Zakharov model leads to the appearance of small radiation, which was discovered earlier in 2013 (Dyachenko et al., 2013a, b). In the work (Kachulin, Gelash, 2018) we demonstrate that the magnitude of the energy losses of the colliding solitons to radiation also depends on their relative phase. Maximum of the energy losses is observed at the same relative phase, at which the amplitude amplification maximum is observed. In addition, depending on the value of the relative phase, solitons can both gain and lose the energy, which results in increase or decrease of their amplitude after a collision. It was found that, in contrast to the NSE model, the spatial shifts of solitons in a more precise model can be both positive and negative. We use the exact breather solutions of the Dyachenko-Zakharov model and the canonical transformation to physical variables (the free surface profile and the potential on the liquid surface) to find approximate solutions in the form of breathers within the framework of exact nonlinear equations for potential incompressible fluid flows. The preliminary results of our numerical experiments in the exact model demonstrate similar dynamics of the interaction of breathers, which indicates that the theoretical picture of the interaction of coherent structures presented here is universal and can be observed in laboratory experiments. The study of the dynamics of breather interactions in the exact model performed by D.I. Kachulin was supported by the Russian Science Foundation (Grant No. 18-71-00079). The work of V.E. Zakharov and A.I. Dyachenko on the dynamics of breather interactions in approximate models was supported by the state assignment “Dynamics of the complex materials”.

scholarly journals Equations for Deep Water Counter Streaming Waves and New Integrals of Motion

Fluids ◽  
2019 ◽  
Vol 4 (1) ◽  
pp. 47 ◽  
Author(s):  
Alexander Dyachenko
Keyword(s):  
Free Surface ◽  
Deep Water ◽  
Water Waves ◽  
Integrals Of Motion ◽  
Wave Interactions ◽  
The Right ◽  
The Waves

The waves on a free surface of 2D deep water can be split into two groups: the waves moving to the right, and the waves moving to the left. A specific feature of the four-wave interactions of water waves allows to describe the evolution of the two groups as a system of two equations. The fundamental consequence of this decomposition is the conservation of the “number of waves” in each particular group. The envelope approximation for the waves in each group of counter streaming waves is obtained.

scholarly journals On the formation of water waves by wind (second paper)

1926 ◽  
Vol 110 (754) ◽  
pp. 241-247 ◽  
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Deep Water ◽  
Water Waves ◽  
Velocity Potential ◽  
Systematic Difference ◽  
Small Extent ◽  
Short Waves ◽  
First Approximation ◽  
The Waves

In a previous paper I investigated the problem of the formation of waves on deep water by wind, and found that the available data were consistent with the hypothesis that the growth of the waves is due principally to a systematic difference between the pressures of the air on the front and rear slopes. Lamb had already discussed the maintenance of waves against viscosity by an approximate method, but without obtaining numerical results. Being under the incorrect impression that Lamb’s approximation would not hold for the short waves I was chiefly considering, I proceeded on more elaborate lines. It now appears, however, that Lamb’s method is not only applicable to the problem of waves on deep water, but is readily extended to cover the case when the water is comparatively shallow, and to allow for surface tension. The fundamental approximations are first, the usual one that squares of the displacements from the steady state can be neglected, and second, that viscosity modifies the motion of the water to only a small extent. The motion of the water can then, to a first approximation, be considered as irrotational. With the previous notation, let ζ be the elevation of the free surface x, y, z the position co-ordinates, t the time, U the undisturbed velocity of the water, h the depth, and φ the velocity potential. Also let σ, p, q , and ϑ denote respectively ∂/∂ t , ∂/∂ x , ∂/∂ y , and ∂/∂ z , and write p 2 + q 2 = - r 2 .

The trajectories of particles in steep, symmetric gravity waves

1979 ◽  
Vol 94 (3) ◽  
pp. 497-517 ◽  
Author(s):  
M. S. Longuet-Higgins
Keyword(s):  
Free Surface ◽  
Solitary Waves ◽  
Deep Water ◽  
Water Waves ◽  
Orbital Motion ◽  
Vertical Gradient ◽  
Wave Crest ◽  
Maximum Height ◽  
Corner Flow ◽  
Deep Water Waves

To gain insight into the orbital motion in waves on the point of breaking, we first study the trajectories of particles in some ideal irrotational flows, including Stokes’ 120° corner-flow, the motion in an almost-highest wave, in periodic deep-water waves of maximum height, and in steep, solitary waves.In Stokes’ corner-flow the particles move as though under the action of a constant force directed away from the crest. The orbits are expressible in terms of an elliptic integral. The trajectory has a loop or not according as q [sqcup ] c where q is the particle speed at the summit of each trajectory, in a reference frame moving with speed c. When q = c, the trajectory has a cusp. For particles near the free surface there is a sharp vertical gradient of the horizontal displacement.The trajectories of particles in almost-highest waves are generally similar to those in the Stokes corner-flow, except that the sharp drift gradient at the free surface is now absent.In deep-water irrotational waves of maximum steepness, it is shown that the surface particles advance at a mean speed U equal to 0·274c, where c is the phase-speed. In solitary waves of maximum amplitude, a particle at the surface advances a total distance 4·23 times the depth h during the passage of each wave. The initial angle α which the trajectory makes with the horizontal is close to 60°.The orbits of subsurface particles are calculated using the ‘hexagon’ approximation for deep-water waves. Near the free surface the drift has the appearance of a thin forwards jet, arising mainly from the flow near the wave crest. The vertical gradient is so sharp, however, that at a mean depth of only 0.01L below the surface (where L is the wavelength) the forwards drift is reduced to less than half its surface value. Under the action of viscosity and turbulence, this sharp gradient will be modified. Nevertheless the orbital motion may contribute appreciably to the observed ‘winddrift current’.Implications for the drift motions of buoys and other floating bodies are also discussed.

Surfing surface gravity waves

2017 ◽  
Vol 823 ◽  
pp. 316-328 ◽  
Author(s):  
Nick E. Pizzo
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Water Wave ◽  
Phase Speed ◽  
Surface Gravity ◽  
Simple Criterion ◽  
Surface Gravity Wave ◽  
Surface Gravity Waves

A simple criterion for water particles to surf an underlying surface gravity wave is presented. It is found that particles travelling near the phase speed of the wave, in a geometrically confined region on the forward face of the crest, increase in speed. The criterion is derived using the equation of John (Commun. Pure Appl. Maths, vol. 6, 1953, pp. 497–503) for the motion of a zero-stress free surface under the action of gravity. As an example, a breaking water wave is theoretically and numerically examined. Implications for upper-ocean processes, for both shallow- and deep-water waves, are discussed.

scholarly journals An alternative approach to study irrotational periodic gravity water waves

2021 ◽  
Vol 72 (4) ◽  
Author(s):  
Biswajit Basu ◽  
Calin I. Martin
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Water Wave ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Stagnation Points ◽  
Alternative Approach ◽  
New Formulation ◽  
Bounded Below ◽  
Surface Dynamic

AbstractWe are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph assumption of the free surface nor does require the absence of stagnation points. By way of this alternative approach we prove the existence of a local curve of solutions to the water wave problem with fixed flow force and more relaxed assumptions.

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