Large Steepness Research Articles

Fully nonlinear viscous wave generation in numerical wave tanks

A numerical method for simulating the complete physics of the fully nonlinear viscous wave generation phenomenon is presented. To accomplish this objective, the motion of a solid body representing the wave generating mechanism is modeled. In this paper, both the piston-type and flap-type wavemakers are simulated and the results of the model are compared with those of the experiments and analytics. The unsteady, two dimensional Navier–Stokes equations are solved in conjunction with the volume-of-fluid method for treating the free surface. A wide range of waves from linear to nonlinear generated by piston and flap-type wavemakers in intermediate and deep water cases are studied in this paper. The accuracy of the numerical results is verified by a comparison with the results of the wavemaker theory, the available experimental data in the literature, and the experiments preformed in this study. For the cases with small wave steepness, the numerical results agree well with the theoretical and experimental results for both the piston and flap-type wavemakers. However, for cases with large wave steepness, the numerical and experimental wave heights are slightly lower than the analytics. In both the piston and flap-type wavemakers, the numerical results are in good agreement with the measurements.

Whitecaps in deep water

Whitecaps in deep water are located near maximum slopes of the interference pattern of dominant wind waves and the most probable breaking waves are those waves that travel with the speed of the component of the interference pattern in the wind direction. These breaking waves are in the wavelength range 1 to 12 m, much shorter than typical dominant waves. The patterns of large steepness caused by interfering dominant waves move at speeds less than the group speed of the dominant wave unless the wave spectrum is narrow in both wavenumber and azimuth. For spectra with wavenumber and azimuth angle spreads representative of those observed on the ocean, the velocity of propagation of the pattern of steep waves in the wind direction is the same as the speed at the maximum of Phillips Λ(c) function, the speed of the peak of HH Doppler spectra, and the speed measured by acoustic event tracking.

Direct numerical simulation of a turbulent wind over a wavy water surface

Parameterization of the wind‐water‐waves interaction is a key problem of the air‐sea system modeling. Of most importance are water waves with sufficiently large steepness, when nonlinear effects related to the boundary layer separation and vortex generation in the wind flow are well pronounced. Known experimental techniques (contact methods and particle image velocimetry) are not yet able to provide a full, detailed understanding of the wind flow in the viscous sublayer and the buffer region. As an alternative, we consider direct numerical simulations (DNS). In the present paper we discuss numerical algorithm and results of DNS of a turbulent wind flow over a wavy water surface. Waves with maximum steepness of ka = 0.2, wave age 0 < c/u* < 10, and Reynolds number Re = 15,000 are considered. Full, 3‐D Navier‐Stokes equations are solved in curvilinear coordinates in a reference frame moving with the wave phase speed c. DNS results show that an instantaneous velocity field is characterized by the presence of well‐pronounced separation zones in the vicinity of the wave crests whereas the average velocity field is nonseparating. We also perform a comparison of the DNS results with the predictions of a theoretical quasi‐linear model of the wind‐wave interaction.

Direct-simulation-based study of turbulent flow over various waving boundaries

We use direct numerical simulation of stress-driven turbulent Couette flows over waving surfaces to study turbulence in the vicinity of water waves. Mechanistic study is performed through systematic investigation of different wavy surface conditions including plane progressive Airy and Stokes waves with and without wind-induced surface drift, as well as stationary wavy walls and vertically waving walls for comparison. Two different wave steepness valuesak= 0.1 and 0.25 are considered, whereais the wave amplitude andkis the wavenumber. For effects of wave age, defined as the ratio between the wave phase speedcand the turbulence friction velocityu*, we consider three values, namelyc/u*= 2, 14 and 25, corresponding to slow, intermediate and fast waves, respectively. Detailed analysis of turbulence structure and statistics shows their dependence on the above-mentioned parameters. Our result agrees with previous measurement and simulation results and reveals many new features unreported in the literature. Over progressive waves, although no apparent flow separation is found in mean flow, considerable intermittent separations in instantaneous flow are detected in slow waves with large steepness. The near-surface coherent vortical structures are examined. We propose two conceptual vortex structure models: quasi-streamwise and reversed horseshoe vortices for slow waves and bent quasi-streamwise vortices for intermediate and fast waves. Detailed examination of Reynolds stress with quadrant analysis, turbulent kinetic energy (TKE) and TKE budget with a focus on production shows large variation with wave phase; analysis shows that the variation is highly dependent on wave age and wave nonlinearity. Comparison between Airy waves and Stokes waves indicates that although the nonlinearity of surface water waves is a high-order effect compared with the wave age and wave steepness, it still makes an appreciable difference to the turbulence structure. The effect of wave nonlinearity on surface pressure distribution causes substantial difference in the wave growth rate. Wind-induced surface drift can cause a phase shift in the downstream direction and a reduction in turbulence intensity; this effect is appreciable for slow waves but negligible for intermediate and fast waves. In addition to providing detailed information on the turbulence field in the vicinity of wave surfaces, the results obtained in this study suggest the importance of including wave dynamics in the study of wind–wave interaction.

Laboratory Study of Unidirectional Focusing Waves in Intermediate Depth Water

The results of laboratory measurements of large focusing wave groups, which were generated using the New Wave theory, are presented. The influences of both the steepness and frequency bandwidth on focused wave characteristics were examined. The influence of frequency bandwidth on focused wave groups with small and moderate steepness was very small. However, for cases with the large steepness, the nonlinearity increased with increasing bandwidth frequency and widened free-wave regimes are identified for those cases with large steepness at the focal location. The underlying nonlinear phase coupling of focused waves was examined using wavelet-based bicoherence and biphase, which can detect nonlinear phase coupling in a short time series. For wave groups with large initial steepness, as wave groups approached the focal location, the values of bicoherence between primary waves and its higher harmonics progressively increased to 1 and the corresponding biphase was gradually close to zero, suggesting that an extreme wave event can be produced by considering Stokes-like nonlinearity to very high-order. Furthermore, the fast change of bicoherence of focused wave groups indicates that the nonlinear energy transfer within focusing waves is faster than that of nonfocusing wave trains.

Estimation of the influence of surface waves on the sea surface radiative balance

The sea roughness effect on insolation and sun rays reflection for different sun heights above the horizon and unclouded sky is considered. Numerical assessments for albedo, insolation, and absorbed radiation are received for Stokes waves. Normal ray falling at the wave crest is suggested. The ray trace approach is used that allowed to take into account such effects as the dependence of albedo on the wave tilt, double light reflection from neighboring wave crests and the trough shading by the crest. It is concluded that the main roughness effect on the radiative balance is the insolation increasing and albedo diminishing at large wave steepness when the sun height above the horizon is less than 40°.

Numerical simulation of the Benjamin-Feir instability and its consequences

Full nonlinear equations for one-dimensional potential surface waves were used for investigation of the evolution of an initially homogeneous train of exact Stokes waves with steepness AK=0.01–0.42. The numerical algorithm for the integration of nonstationary equations and the calculation of exact Stokes waves is described. Since the instability of the exact Stokes waves develops slowly, a random small-amplitude noise was introduced in initial conditions. The development of instability occurs in two stages: in the first stage the growth rate of disturbances was close to that established for small steepness by Benjamin and Feir [J. Fluid. Mech. 27, 417 (1967)] and for medium steepness [McLean, J. Fluid Mech. 114, 315 (1982)]. For any steepness, the Stokes waves disintegrate and create random superposition of waves. For AK<0.13, waves do not show a tendency to breaking, which is recognized by approaching a surface to non-single-value shape. Sooner or later, if AK>0.13, one of the waves increases its height, and finally it comes to the breaking point. For large steepness of AK>0.35 the rate of growth is slower than for medium steepness. The data for spectral composition of disturbances and their frequencies are given.

Numerical simulation of lowest-order short-crested wave instabilities

A numerical study of doubly periodic deep-water short-crested wave instabilities, arising from various quartet resonant interactions, is conducted using a high-order Boussinesq-type model. The model is first verified through a series of simulations involving classical class I plane wave instabilities. These correctly lead to well-known (nearly symmetric) recurrence cycles below a previously established breaking threshold steepness, and to an asymmetric evolution (characterized by a permanent transfer of energy to the lower side-band) above this threshold, with dissipation from a smoothing filter promoting this behaviour in these cases. A series of class Ia short-crested wave instabilities, near the plane wave limit, are then considered, covering a wide range of incident wave steepness. A close match with theoretical growth rates is demonstrated near the inception. It is shown that the unstable evolution of these initially three-dimensional waves leads to an asymmetric evolution, even for weakly nonlinear cases presumably well below breaking. This is characterized by an energy transfer to the lower side-band, which is also accompanied by a similar transfer to more distant upper side-bands. At larger steepness, the evolution leads to a permanent downshift of both the mean and peak frequencies, driven in part by dissipation, effectively breaking the quasi-recurrence cycle. A single case involving a class Ib short-crested wave instability at relatively large steepness is also considered, which demonstrates a reasonably similar evolution. These simulations consider the simplest physical situations involving three-dimensional instabilities of genuinely three-dimensional progressive waves, revealing qualitative differences from classical two-dimensional descriptions. This study is therefore of fundamental importance in understanding the development of three-dimensional wave spectra.

Lagrangian simulation of oil droplets transport due to regular waves

Dispersed oils (i.e., oil droplets) at sea are transported by convection due to waves and buoyancy and by turbulent diffusion. This work follows the common approach in the oil community of using a Lagrangian approach instead of the Eulerian approach. Our focus was on small scale simulation of oil plumes subjected to regular waves. Stokes' theory was used to obtain analytical expressions for wave kinematics. The velocity above the Mean Water Level was obtained using a second order Taylor's expansion of the velocity at the MWL. Five hundred droplets were used to simulate the plume for a duration of 60 wave periods. A Monte Carlo framework (300 simulations) was used to compute theoretical mean and variance of plumes. In addition, we introduced a novel dimensionless formulation, whose main advantage was to allow one to report distances in terms of the wave length and times in terms of the wave period. We found that the Stokes' drift was the major mechanism for horizontal transport. We also found that lighter oils propagate faster but spread less than heavier oils. Increasing turbulent diffusion caused the plume to disperse deeper in the water column and to propagate less forward. The spreading in both vertical and horizontal directions increased with an increase in turbulent diffusion. The increase in wave slope (or wave steepness) caused, in general, an increase in the downward and horizontal transport. In the context of mixing in the water column, the dimensionless formulation showed that small steepness waves with a large turbulent diffusion coefficient could result in essentially the same spreading as large steepness waves with a small turbulent diffusion coefficient.

Dielectric study of the viscous and glassy states of a binary, nifedipine-based pharmaceutical alloy

An equimolar mixture of nifedipine and acetaminophen was studied in its supercooled liquid and in its glassy state using dielectric spectroscopy. It is found that this alloy is characterized by a relatively large steepness index m=83. The low-temperature relaxation of pure nifedipine is studied and compared with that of the mixture.

Three-dimensional instabilities of periodic gravity waves in shallow water

A linear stability analysis of finite-amplitude periodic progressive gravity waves on water of finite depth has extended existing results to steeper waves and shallower water. Some new types of instability are found for shallow water. When the water depth decreases, higher-order resonances lead to the dominant instabilities. In contrast with the deep water case, we have found that in shallow water the dominant instabilities are usually associated with resonant interactions between five, six, seven and eight waves. For small steepness, dominant instabilities are quasi two-dimensional. For moderate and large steepness, the dominant instabilities are three-dimensional and phased-locked with the unperturbed nonlinear wave. At the margin of instability diagrams, these results suggest the existence of new bifurcated three-dimensional steady waves.

Nonlinear evolution of components of an electromagnetic field of helicoidal waves in plasma

We derive model equations of the interaction between circular polarized helicon wavetrains in a two-dimensional plasma. For such quasi-one-dimensional waves a nonlinear resonance results in N-wave interactions. The corresponding system of equations is known to be integrable. A novel feature of the study, in the context of soliton theory, is the account of the wave asynchronism that seems to be essential for wavetrains with wavefronts of large steepness, observed in experiments for dense helicon plasma. To generate solutions we use recent results of the binary Darboux transformation technique, applying it to three-wave interactions. The solutions have been analysed numerically and plotted.

Nonlinear particle kinematics of ocean waves

A fundamental relation is derived governing the Lagrangian kinematics of fluid particles on the surface of nonlinear ocean waves which may be known only stochastically. The horizontal trajectories of fluid particles on the free surface are shown to obey a pair of coupled nonlinear Ricatti-type ordinary differential equations driven by the temporal and spatial gradients of the free-surface elevation defined relative to an Eulerian frame. This equation is explicit in that it does not require the solution of a fully nonlinear potential flow free-surface problem and may be viewed as a deterministic or stochastic equation depending on the interpretation of the definition of the free-surface elevation. It is free of empirical corrections often used to estimate the particle kinematics above the calm water surface, is valid in potential flow and for waves of large steepness in two and three dimensions and in waters of all depths and may be used for the evaluation of the extreme unsteady loads exerted on surface piercing vertical circular cylinders by steep random waves.

The run-up on a cylinder in progressive surface gravity waves: harmonic components

Wave run-up is the vertical uprush of water that occurs when an incident wave impinges on a partially immersed body. In this present work, wave run-up is studied on a fixed vertical cylinder in plane progressive waves. These progressive waves are of a form suitable for description by Stokes' wave theory whereby the typical energy content of a wave train consists of one fundamental harmonic and corresponding phase locked Fourier components. The limitations of canonical wave diffraction theory—whereby the free-surface boundary condition is treated by a Stokes expansion—in predicting the harmonic components of the wave run-up are discussed. An experimental campaign is described and the choice of monochromatic waves is indicative of the diffraction regime for large volume structures where the assumption of potential flow theory is applicable, or more formally A/a<O(1) (A and a being the wave amplitude and cylinder radius, respectively). The wave environment is represented by a parametric variation of the scattering parameter ka and wave steepness kA (where k denotes the wave number). The zeroth-, first-, second- and third-harmonics of the wave run-up are examined to determine the importance of each with regard to local wave diffraction and incident wave nonlinearities. It is shown that the complete wave run-up is not well accounted for by second-order diffraction theory. This is, however, dependent upon the coupling of ka and kA. In particular, whilst the modulus and phase of the second-harmonic are moderately predicted, the mean set-up is not well predicted by a second-order diffraction theory. Experimental evidence suggests this to be caused by higher than second-order nonlinear diffraction. Moreover, these effects most noticeably operate at the first-harmonic in waves of moderate to large steepness when ka≪1.

On the disappearance of the lowest-order instability for steep gravity waves in finite depth

This paper extends to very large steepness the results of the linear stability analysis of two-dimensional periodic gravity waves over a bottom of uniform depth. Both subharmonic and superharmonic instabilities are investigated. For the case considered herein, it is shown how the lowest-order instability (class I) completely disappears at a critical wave steepness. These results also confirm numerically a theorem connected with the appearance of superharmonic instabilities in finite depth.

Theoretical prediction of a surface-induced spin-reorientation phase transition in BaFe12O19 nanocrystals

The equation of the magnetization of a hexagonal crystal is derived for the first time for an arbitrary orientation of the external magnetic field relative to the crystallographic c axis. In order to clarify the magnetization mechanism for a real ensemble of small particles in the framework of the given problem, surface anisotropy (which is significant for nanosize objects) was taken into account along with crystalline magnetic anisotropy and anisotropy in the particle shape. Model computer experiments prove that the magnetization curves for nanocrystals oriented in a polar angle range of 65–90° exhibit an anomaly in the form of a jump, indicating a first-order spin-reorientation phase transition. This explains a larger steepness of the experimental curve reconstructed taking into account the interaction between particles as compared to the theoretical dependence obtained by Stoner and Wohlfarth [IEEE Trans. Magn. MAG 27 (4), 3469 (1991)]. An analysis of variation of the characteristic anisotropy surface and its cross section with increasing ratio |K2|/K1 of the crystalline magnetic anisotropy constants upon a transition from a macroscopic to a nanoscopic crystal shows that surface anisotropy leads to a change in the magnetic structure. As a result, an additional easy magnetization direction emerges in the basal plane apart from the easiest magnetization direction (along the c axis). The direction of hard magnetization emerges from the basal plane, the angle of its orientation relative to the c axis being a function of the ratio | K2|/K1.

Nonlinear Four-Wave Interactions and Freak Waves

Four-wave interactions are shown to play an important role in the evolution of the spectrum of surface gravity waves. This fact follows from direct simulations of an ensemble of ocean waves using the Zakharov equation. The theory of homogeneous four-wave interactions, extended to include effects of nonresonant transfer, compares favorably with the ensemble-averaged results of the Monte Carlo simulations. In particular, there is good agreement regarding spectral shape. Also, the kurtosis of the surface elevation probability distribution is determined well by theory even for waves with a narrow spectrum and large steepness. These extreme conditions are favorable for the occurrence of freak waves.

Wave-induced drift of an elliptical surface film

This study investigates experimentally the drift velocity of an elliptical surface film advected by deep-water waves. Thin polyethylene sheets were used to simulate the inextensible surface film. The drift velocities were obtained by recording and analyzing a sequence of images captured using a video camera. The results show that the drift velocity increases with the longitudinal length of the polyethylene sheet until approximately 0.8 times the wavelength. Beyond that, further increment would not result in substantial increase in the drift velocity. The effect due to the normalized transverse width is found to be significant within a particular range. At large wave steepness, the drift velocity appears to be limited by the Stokes drift. A set of best-fit empirical equations based on the sigmoidal function is introduced for oil spill trajectory prediction.

Modélisation numérique du déferlement plongeant

ABSTRACT Wave breaking is simulated using the Navier-Stokes equations and the volume tracking algorithm PLIC-VOF. The method is first validated on the case of non linear wave propagation on flat bottom. The plunging motion of a sinusoidal wave of large initial steepness is next studied. The characteristics of the wave up to the impact are found to be well described by the method. First results obtained after the impact encourage us to go on in this direction.

On the Badulin, Kharif and Shrira model of resonant water waves

This paper is a reappraisal of the Hamiltonian model derived by Shrira, Badulin and Kharif (BKS) for three-dimensional nonlinear water waves. The model was introduced in [J. Fluid Mech. 318 (1996) 375] in an effort to describe the formation of traveling waves with crescent-shaped features that arise from the instability of the Stokes wave train at moderately large steepness. There have been observations of such traveling waves in wave tank experiments by Su et al. [J. Fluid Mech. 124 (1982) 45–72] and Su [J. Fluid Mech. 124 (1982) 73–108]. Some of the regimes described in these papers are of lightly breaking waves, which are asymmetric, with all crescents facing forward. Other regimes that they observe apparently give rise to traveling waves which have asymmetric crescent-shaped features facing both forwards and backwards. We show that the BKS model describes the Stokes wave train and its loss of stability at moderate amplitudes as a Hamiltonian saddle-node bifurcation, which corresponds to the formation of a stable three-dimensional wave pattern which exhibits asymmetric crescent-shaped elements. The model also produces a family of solutions homoclinic to the unstable Stokes wave train, which surrounds the orbit of crescent-shaped wave patterns and which provides a mechanism for transition. Other traveling wave solutions of the BKS model having nonzero transverse momentum are good candidates for the skew wave patterns possessing characteristic hexagonal shaped structures separated by quiescent stripes which are produced to the sides of the experiments in wave tanks. The BKS model has solutions which satisfy two of the three characteristics specified in [J. Fluid Mech. 318 (1996) 375] for nonlinear crescent-shaped waves, avoiding the introduction of a dissipative mechanism to describe features of these familiar wave patterns. The one weakness of the BKS model is that the crescent-shaped wave patterns are transformed to themselves under time reversal composed with a phase shift. Therefore all of the wave patterns described by the BKS model possess forward and backward facing crescent-shaped elements simultaneously, associated with alternating crests. These solutions reproduce the features of some but not all of the wave patterns in the observations of Su et al. [J. Fluid Mech. 124 (1982) 45–72] and Su [J. Fluid Mech. 124 (1982) 73–108]. In the deep water case, we introduce and analyze a new and more realistic four degrees of freedom Hamiltonian model of water waves which has two principal five wave interactions. While being more complicated and not completely integrable, nonetheless this model has traveling wave solutions with similar crescent-shaped elements, and others with the hexagonal features of the BKS model.