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Recent Advances in Free Surface Hydrodynamics
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Recent Advances in Free Surface Hydrodynamics

A special issue of Fluids (ISSN 2311-5521).

Deadline for manuscript submissions: closed (31 January 2021) | Viewed by 28458

Special Issue Editor

Prof. Dr. Alexander I. Dyachenko

E-Mail Website
Guest Editor
Landau Institute for Theoretical Physics RAS, 142432 Chernogolovka, Russia
Interests: water waves; freak-waves; hydrodynamics; simulation

Special Issue Information

Dear Colleagues, 

Allow me to present this a Special Issue on "Recent Advances in Free Surface Hydrodynamics." As a part of fluid mechanics, free surface hydrodynamics is one of the foundations of all physics and engineering as a source of physical intuition and mathematical methods since the 18th century. Mathematicians and physicists have long been interested in the subject of surface water waves. The problems formulated in this subject can be considered fundamental, but many questions remain unanswered.

This Special Issue will bring together researchers to present recent advances in the area and may include articles covering nonlinear water wave propagation, freak waves, wave breaking, wave turbulence, solitons and breathers, integrability of free surface hydrodynamics, etc. Laboratory experiments, along with numerical simulation, are also an essential part of these studies. A deep understanding of phenomena occurring on a free surface is important for practical applications such as safe navigation and offshore oil platform design.

Prof. Dr. Alexander I. Dyachenko
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fluids is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • surface wave turbulence
  • freak waves
  • wave breaking
  • kinetic equation for waves
  • extreme events
  • numerical simulation
  • wind-driven water waves

Published Papers (14 papers)

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Research

11 pages, 1201 KiB  
Article
Phase Evolution of the Time- and Space-Like Peregrine Breather in a Laboratory
by Yuchen He, Pierre Suret and Amin Chabchoub
Fluids 2021, 6(9), 308; https://doi.org/10.3390/fluids6090308 - 31 Aug 2021
Cited by 4 | Viewed by 1595
Abstract
Coherent wave groups are not only characterized by the intrinsic shape of the wave packet, but also by the underlying phase evolution during the propagation. Exact deterministic formulations of hydrodynamic or electromagnetic coherent wave groups can be obtained by solving the nonlinear Schrödinger [...] Read more.
Coherent wave groups are not only characterized by the intrinsic shape of the wave packet, but also by the underlying phase evolution during the propagation. Exact deterministic formulations of hydrodynamic or electromagnetic coherent wave groups can be obtained by solving the nonlinear Schrödinger equation (NLSE). When considering the NLSE, there are two asymptotically equivalent formulations, which can be used to describe the wave dynamics: the time- or space-like NLSE. These differences have been theoretically elaborated upon in the 2016 work of Chabchoub and Grimshaw. In this paper, we address fundamental characteristic differences beyond the shape of wave envelope, which arise in the phase evolution. We use the Peregrine breather as a referenced wave envelope model, whose dynamics is created and tracked in a wave flume using two boundary conditions, namely as defined by the time- and space-like NLSE. It is shown that whichever of the two boundary conditions is used, the corresponding local shape of wave localization is very close and almost identical during the evolution; however, the respective local phase evolution is different. The phase dynamics follows the prediction from the respective NLSE framework adopted in each case. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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35 pages, 7061 KiB  
Article
Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments
by Dan Lucas, Marc Perlin, Dian-Yong Liu, Shane Walsh, Rossen Ivanov and Miguel D. Bustamante
Fluids 2021, 6(6), 205; https://doi.org/10.3390/fluids6060205 - 1 Jun 2021
Cited by 4 | Viewed by 2495
Abstract
In this work we consider the problem of finding the simplest arrangement of resonant deep-water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is [...] Read more.
In this work we consider the problem of finding the simplest arrangement of resonant deep-water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wavevectors K1+K2=K3 (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wavepackets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction in a symmetric configuration. Numerical simulations of the governing equations in natural variables using pseudospectral methods require the inclusion of up to 6-wave interactions, which imposes a strong dealiasing cut-off in order to properly resolve the evolving waves. We study the resonance numerically by looking at a target mode in the base triad and showing that the energy transfer to this mode is more efficient when the system is close to satisfying the resonant conditions. We first look at encountering plane waves with base frequencies in the range 1.32–2.35 Hz and steepnesses below 0.1, and show that the time evolution of the target mode’s energy is dramatically changed at the resonance. We then look at a scenario that is closer to experiments: Encountering wavepackets in a 400-m long numerical tank, where the interaction time is reduced with respect to the plane-wave case but the resonance is still observed; by mimicking a probe measurement of surface elevation we obtain efficiencies of up to 10% in frequency space after including near-resonant contributions. Finally, we perform preliminary experiments of encountering wavepackets in a 35-m long tank, which seem to show that the resonance exists physically. The measured efficiencies via probe measurements of surface elevation are relatively small, indicating that a finer search is needed along with longer wave flumes with much larger amplitudes and lower frequency waves. A further analysis of phases generated from probe data via the analytic signal approach (using the Hilbert transform) shows a strong triad phase synchronisation at the resonance, thus providing independent experimental evidence of the resonance. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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34 pages, 6157 KiB  
Article
Traveling-Standing Water Waves
by Jon Wilkening
Fluids 2021, 6(5), 187; https://doi.org/10.3390/fluids6050187 - 14 May 2021
Cited by 5 | Viewed by 2548
Abstract
We propose a new two-parameter family of hybrid traveling-standing (TS) water waves in infinite depth that evolve to a spatial translation of their initial condition at a later time. We use the square root of the energy as an amplitude parameter and introduce [...] Read more.
We propose a new two-parameter family of hybrid traveling-standing (TS) water waves in infinite depth that evolve to a spatial translation of their initial condition at a later time. We use the square root of the energy as an amplitude parameter and introduce a traveling parameter that naturally interpolates between pure traveling waves moving in either direction and pure standing waves in one of four natural phase configurations. The problem is formulated as a two-point boundary value problem and a quasi-periodic torus representation is presented that exhibits TS-waves as nonlinear superpositions of counter-propagating traveling waves. We use an overdetermined shooting method to compute nearly 50,000 TS-wave solutions and explore their properties. Examples of waves that periodically form sharp crests with high curvature or dimpled crests with negative curvature are presented. We find that pure traveling waves maximize the magnitude of the horizontal momentum among TS-waves of a given energy. Numerical evidence suggests that the two-parameter family of TS-waves contains many gaps and disconnections where solutions with the given parameters do not exist. Some of these gaps are shown to persist to zero-amplitude in a fourth-order perturbation expansion of the solutions in powers of the amplitude parameter. Analytic formulas for the coefficients of this perturbation expansion are identified using Chebyshev interpolation of solutions computed in quadruple-precision. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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9 pages, 810 KiB  
Article
Turbulence of Capillary Waves on Shallow Water
by Natalia Vladimirova, Ivan Vointsev, Alena Skoba and Gregory Falkovich
Fluids 2021, 6(5), 185; https://doi.org/10.3390/fluids6050185 - 13 May 2021
Cited by 1 | Viewed by 1691
Abstract
We consider the developed turbulence of capillary waves on shallow water. Analytic theory shows that an isotropic cascade spectrum is unstable with respect to small angular perturbations, in particular, to spontaneous breakdown of the reflection symmetry and generation of nonzero momentum. By computer [...] Read more.
We consider the developed turbulence of capillary waves on shallow water. Analytic theory shows that an isotropic cascade spectrum is unstable with respect to small angular perturbations, in particular, to spontaneous breakdown of the reflection symmetry and generation of nonzero momentum. By computer modeling we show that indeed a random pumping, generating on average zero momentum, produces turbulence with a nonzero total momentum. A strongly anisotropic large-scale pumping produces turbulence whose degree of anisotropy decreases along a cascade. It tends to saturation in the inertial interval and then further decreases in the dissipation interval. Surprisingly, neither the direction of the total momentum nor the direction of the compensated spectrum anisotropy is locked by our square box preferred directions (side or diagonal) but fluctuate. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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24 pages, 2391 KiB  
Article
High-Frequency Instabilities of a Boussinesq–Whitham System: A Perturbative Approach
by Ryan Creedon, Bernard Deconinck and Olga Trichtchenko
Fluids 2021, 6(4), 136; https://doi.org/10.3390/fluids6040136 - 1 Apr 2021
Cited by 5 | Viewed by 2034
Abstract
We analyze the spectral stability of small-amplitude, periodic, traveling-wave solutions of a Boussinesq–Whitham system. These solutions are shown numerically to exhibit high-frequency instabilities when subject to bounded perturbations on the real line. We use a formal perturbation method to estimate the asymptotic behavior [...] Read more.
We analyze the spectral stability of small-amplitude, periodic, traveling-wave solutions of a Boussinesq–Whitham system. These solutions are shown numerically to exhibit high-frequency instabilities when subject to bounded perturbations on the real line. We use a formal perturbation method to estimate the asymptotic behavior of these instabilities in the small-amplitude regime. We compare these asymptotic results with direct numerical computations. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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11 pages, 297 KiB  
Article
Derivation of a Viscous Serre–Green–Naghdi Equation: An Impasse?
by Denys Dutykh and Hervé V.J. Le Meur
Fluids 2021, 6(4), 135; https://doi.org/10.3390/fluids6040135 - 1 Apr 2021
Cited by 3 | Viewed by 1680
Abstract
In this article, we present the current status of the derivation of a viscous Serre–Green–Naghdi system. For this goal, the flow domain is separated into two regions. The upper region is governed by inviscid Euler equations, while the bottom region (the so-called boundary [...] Read more.
In this article, we present the current status of the derivation of a viscous Serre–Green–Naghdi system. For this goal, the flow domain is separated into two regions. The upper region is governed by inviscid Euler equations, while the bottom region (the so-called boundary layer) is described by Navier–Stokes equations. We consider a particular regime binding the Reynolds number and the shallowness parameter. The computations presented in this article are performed in the fully nonlinear regime. The boundary layer flow reduces to a Prandtl-like equation that we claim to be irreducible. Further approximations are necessary to obtain a tractable model. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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10 pages, 4010 KiB  
Article
Chaotic Dynamics of the Interface between Dielectric Liquids at the Regime of Stabilized Kelvin-Helmholtz Instability by a Tangential Electric Field
by Evgeny A. Kochurin and Nikolay M. Zubarev
Fluids 2021, 6(3), 125; https://doi.org/10.3390/fluids6030125 - 19 Mar 2021
Cited by 1 | Viewed by 1713
Abstract
The nonlinear dynamics of the interface between two immiscible dielectric liquids at the regime of suppressed Kelvin-Helmholtz instability by external horizontal electric field is studied theoretically. The initial equations of the fluids motion are reduced to a single weakly nonlinear integro-differential equation that [...] Read more.
The nonlinear dynamics of the interface between two immiscible dielectric liquids at the regime of suppressed Kelvin-Helmholtz instability by external horizontal electric field is studied theoretically. The initial equations of the fluids motion are reduced to a single weakly nonlinear integro-differential equation that describes the interaction of solitary waves (rational solitons) propagating along the interface. The dynamics of two interacting solitons is regular and integrable; they can combine into a stable wave packet (breather). It is shown that the interaction of three solitons becomes complex and, for a wide rang of initial conditions, chaotic. The numerically obtained Poincaré sections demonstrate the destruction of toroidal trajectories in the phase space during the transition of the system to a chaotic regime of fluid motion. Such a behaviour is consistent with the Kolmogorov-Arnold-Moser theory describing quasi-periodic chaotic motion in Hamiltonian systems. At the developed chaotic state, the system fast loses the information on its initial state; the corresponding estimate for Lyapunov exponent is obtained. From the physical point of view, the chaotic behavior of the system is related with structural instability of the soliton triplet. The triplet can decay into a solitary wave and stable breather consisting of two interacting solitons. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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31 pages, 473 KiB  
Article
Wind-Driven Waves on the Air-Water Interface
by Harvey Segur and Soroush Khadem
Fluids 2021, 6(3), 122; https://doi.org/10.3390/fluids6030122 - 16 Mar 2021
Viewed by 1871
Abstract
An ocean swell refers to a train of periodic or nearly periodic waves. The wave train can propagate on the free surface of a body of water over very long distances. A great deal of the current study in the dynamics of water [...] Read more.
An ocean swell refers to a train of periodic or nearly periodic waves. The wave train can propagate on the free surface of a body of water over very long distances. A great deal of the current study in the dynamics of water waves is focused on ocean swells. These swells are typically created initially in the neighborhood of an ocean storm, and then the swell propagates away from the storm in all directions. We consider a different kind of wave, called seas, which are created by and driven entirely by wind. These waves typically have no periodicity, and can rise and fall with changes in the wind. Specifically, this is a two-fluid problem, with air above a moveable interface, and water below it. We focus on the local dynamics at the air-water interface. Various properties at this locality have implications on the waves as a whole, such as pressure differentials and velocity profiles. The following analysis provides insight into the dynamics of seas, and some of the features of these intriguing waves, including a process known as white-capping. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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10 pages, 692 KiB  
Article
Bound Coherent Structures Propagating on the Free Surface of Deep Water
by Dmitry Kachulin, Sergey Dremov and Alexander Dyachenko
Fluids 2021, 6(3), 115; https://doi.org/10.3390/fluids6030115 - 12 Mar 2021
Cited by 2 | Viewed by 1571
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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20 pages, 547 KiB  
Article
Spatial Form of a Hamiltonian Dysthe Equation for Deep-Water Gravity Waves
by Philippe Guyenne, Adilbek Kairzhan, Catherine Sulem and Boyang Xu
Fluids 2021, 6(3), 103; https://doi.org/10.3390/fluids6030103 - 3 Mar 2021
Cited by 5 | Viewed by 1622
Abstract
An overview of a Hamiltonian framework for the description of nonlinear modulation of surface water waves is presented. The main result is the derivation of a Hamiltonian version of Dysthe’s equation for two-dimensional gravity waves on deep water. The reduced problem is obtained [...] Read more.
An overview of a Hamiltonian framework for the description of nonlinear modulation of surface water waves is presented. The main result is the derivation of a Hamiltonian version of Dysthe’s equation for two-dimensional gravity waves on deep water. The reduced problem is obtained via a Birkhoff normal form transformation which not only helps eliminate all non-resonant cubic terms but also yields a non-perturbative procedure for surface reconstruction. The free surface is reconstructed from the wave envelope by solving an inviscid Burgers’ equation with an initial condition given by the modulational Ansatz. Particular attention is paid to the spatial form of this model, which is simulated numerically and tested against laboratory experiments on periodic groups and short-wave packets. Satisfactory agreement is found in all these cases. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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21 pages, 3087 KiB  
Article
Kuramoto-Like Synchronization Mediated through Faraday Surface Waves
by André Nachbin
Fluids 2020, 5(4), 226; https://doi.org/10.3390/fluids5040226 - 29 Nov 2020
Cited by 2 | Viewed by 2782
Abstract
A new class of problems in free surface hydrodynamics appeared after the groundbreaking discovery by Yves Couder and Emmanuel Fort. A bouncing droplet in association with Faraday surface waves gives rise to new nonlinear dynamics, in analogy with the pilot-wave proposed by de [...] Read more.
A new class of problems in free surface hydrodynamics appeared after the groundbreaking discovery by Yves Couder and Emmanuel Fort. A bouncing droplet in association with Faraday surface waves gives rise to new nonlinear dynamics, in analogy with the pilot-wave proposed by de Broglie. The droplet and the underlying vibrating bath are of silicon oil. A weakly viscous potential theory model should be used. Numerical simulations are presented with one and two bouncing droplets oscillating while confined to their cavities. These oscillators are implicitly coupled by the underlying surface wave field. In certain regimes, the oscillators can spontaneously synchronize, even when placed at a distance. Cavity parameters are varied in order to highlight the sensitive wave-mediated coupling. The present nonlinear wave-mediated oscillator synchronization is more general than that displayed by the celebrated Kuramoto model and therefore of general interest. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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18 pages, 1793 KiB  
Article
Wave Patterns of Gravity–Capillary Waves from Moving Localized Sources
by Vladimir Gnevyshev and Sergei Badulin
Fluids 2020, 5(4), 219; https://doi.org/10.3390/fluids5040219 - 24 Nov 2020
Cited by 14 | Viewed by 2179
Abstract
We study wave patterns of gravity–capillary waves from moving localized sources within the classic setup of the problem of ship wakes. The focus is on the co-existence of two wave systems with opposite signatures of group velocity relative to the localized source. It [...] Read more.
We study wave patterns of gravity–capillary waves from moving localized sources within the classic setup of the problem of ship wakes. The focus is on the co-existence of two wave systems with opposite signatures of group velocity relative to the localized source. It leads to the problem of choice of signs for phase functions of the gravity (“slow”) and capillary (“fast”) branches of the dispersion relation: the question generally ignored when constructing phase patterns of the solutions. We detail characteristic angles of the wake patterns: (i) angle of demarcation of gravity and capillary waves—“the phase Mach” cone, (ii) angle of the minimal group velocity of gravity–capillary waves—“the group Mach” cone, (iii, iv) angles of cusps of isophases that appear after a threshold current speed. The outer cusp cone is naturally associated with the classic cone of Kelvin for pure gravity waves. The inner one results from the effect of capillarity and tends to the “group Mach” pattern at high speeds of current. Amplitudes of the wave patterns are estimated within the recently proposed approach of reference functions for the problem of propagation of packets of linear dispersive waves. The effect of shape is discussed for elliptic reference sources. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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9 pages, 2002 KiB  
Article
Interaction Features of Internal Wave Breathers in a Stratified Ocean
by Ekaterina Didenkulova and Efim Pelinovsky
Fluids 2020, 5(4), 205; https://doi.org/10.3390/fluids5040205 - 10 Nov 2020
Cited by 3 | Viewed by 1705
Abstract
Oscillating wave packets (breathers) are a significant part of the dynamics of internal gravity waves in a stratified ocean. The formation of these waves can be provoked, in particular, by the decay of long internal tidal waves. Breather interactions can significantly change the [...] Read more.
Oscillating wave packets (breathers) are a significant part of the dynamics of internal gravity waves in a stratified ocean. The formation of these waves can be provoked, in particular, by the decay of long internal tidal waves. Breather interactions can significantly change the dynamics of the wave fields. In the present study, a series of numerical experiments on the interaction of breathers in the frameworks of the etalon equation of internal waves—the modified Korteweg–de Vries equation (mKdV)—were conducted. Wave field extrema, spectra, and statistical moments up to the fourth order were calculated. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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12 pages, 6907 KiB  
Article
Faraday Waves in a Square Cell Network: The Effects of Varying the Cell Size
by Franklin Peña-Polo, Ignacio Carvajal-Mariscal, Carlos A. Vargas and Leonardo Di G. Sigalotti
Fluids 2020, 5(4), 192; https://doi.org/10.3390/fluids5040192 - 29 Oct 2020
Cited by 2 | Viewed by 1874
Abstract
We have conducted experiments of the Faraday instability in a network of square cells filled with water for driving frequencies and amplitudes in the intervals 10≤F≤22 Hz and 0.1≤A≤3 mm, respectively. The experiments were aimed [...] Read more.
We have conducted experiments of the Faraday instability in a network of square cells filled with water for driving frequencies and amplitudes in the intervals 10≤F≤22 Hz and 0.1≤A≤3 mm, respectively. The experiments were aimed at studying the effects of varying the size of the cells on the surface wave patterns. Images of the surface wave patterns were recorded with a high-speed camera. The time series of photographs composing each video was Fourier analyzed, and information about the waveforms was obtained by using a Pearson correlation analysis. For small square cells of side length l=2.5 cm, adjacent cells collaborate synchronously to form regular patterns of liquid bumps over the entire grid, while ordered matrices of oscillons are formed at higher frequencies. As the size of the cells is increased to l=5 cm, collective cell behaviour at lower frequencies is no longer observed. As the frequency is increased, a transition from three triangularly arranged oscillons within each cell to three, or even four, irregularly arranged oscillons is observed. The wave patterns, the waveforms and the energy content necessary to excite Faraday waves are seen to depend on the cell size. Full article
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
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