Three-dimensional Surface Waves Research Articles

Atmospheric pressure-induced three-dimensional surface wave propagation in the compressible ocean: Effect of static compression

Under the assumptions of linearized water wave theory, we build a three-dimensional mathematical model that couples atmospheric pressure waves and surface ocean waves, including water compressibility and its static part, to simulate Meteotsunami propagation in the ocean. The solution uses the Laplace–Fourier double transformation technique, emphasizing axisymmetry of the mathematical problem and rigorous treatment of a fairly complicated dispersion relation while using inverse transformations. A novel derivation of the axisymmetric atmospheric pressure front is shown. The impact of water compressibility is shown through a comparative graphical representation against the incompressible case. Faster travel of free-surface waves is observed in the incompressible ocean, followed by the cases with and without static compression of the compressible ocean, respectively. The static compression shifts the phase of the acoustic-gravity modes. The locked wave is hardly influenced by the water compressibility and is entangled with the moving pressure front. The model is validated with the observational pressure data and agrees well with our computed pressure profile. Then, the locked wave profile generated from our model agrees well with the deep-ocean assessment and reporting of tsunami data.

A toy model for damped water waves

We consider a toy model for a damped water waves system in a domain [Formula: see text]. The toy model is based on the paradifferential formulation of the water waves system derived in the work of Alazard–Burq–Zuily [On the water-waves equations with surface tension, Duke Math. J. 158 (2011) 413–499]. The form of damping we consider is a modified sponge layer proposed for the [Formula: see text] water waves system by Clamond et al. in [An efficient model for three-dimensional surface wave simulations. Part II: Generation and absorption, J. Comput. Phys. 205 (2005) 686–705]. We show that, in the case of small Cauchy data, solutions to the toy model exhibit a quadratic lifespan. This is done via proving energy estimates with Klainerman vector fields.

Effect of static compression on near-field tsunami waves: Three-dimensional solution

An analytical solution of three-dimensional surface wave profiles due to arbitrary spatio-temporal disturbance of a circular ocean bottom in a compressible ocean is obtained by incorporating the influence of the static ocean background compression under the assumptions of linearized water wave theory. Time-domain simulations of the surface profile in three dimensions and the pressure distribution within the water column for a circular uniform rise and tilt are shown. The corresponding animation movies depict the temporal evolution of the surface profile and pressure field inside the water column eloquently, which was not shown in earlier literature. The impact of static compression is also discussed through the simulations. A novel analytical expression of the potential function for a generic tilted motion (rmcos(mθ),m∈Z) of the circular ocean floor is derived. An efficient numerical code is developed to find surface elevation and pressure distribution, implementing the inverse Fourier integral as matrix multiplication. Validation is performed for the specific case of a rising flat ocean floor, showing the oscillations due to acoustic-gravity modes. Initially, a simplified problem of a flat rising ocean bottom is solved using the eigenfunction matching method, which involves finding a particular solution for the nonhomogeneous ocean bottom condition and the solution for its homogeneous counterpart. Solutions are obtained using a newly developed inner product between the depth-dependent functions. Later, a Green function technique is used to incorporate the impact of the arbitrary spatio-temporal motion of the circular portion of the ocean bed. The solution obtained from the eigenfunction matching method is utilized to obtain the analytical form of Green’s function and, eventually, an expression of surface elevation and pressure distribution inside the ocean water column.

Low-order modelling of three-dimensional surface waves in liquid film flow on a rotating disk

Using low-dimensional numerical simulations, we investigate the characteristics of complex and three-dimensional surface waves in a liquid film flowing over a rotating disk, focusing on large flow rates from a nozzle. Existing integral boundary layer (IBL) models, which are based on spatially averaged variables along the direction normal to the disk surface, have primarily focused on the formation of axisymmetric waves under relatively small flow rates. In this study, an extended IBL model that accounts for both laminar and turbulent regimes is developed by considering the non-uniformity of the local flow rate in the spreading film flow and incorporating closure models dependent on the local Reynolds number. Our numerical results successfully capture the generation of concentric waves by an impinging circular liquid jet and their transition into three-dimensional solitary waves. These findings are in good agreement with visualization images and time-series data of free-surface fluctuations from a displacement sensor. The backscattering of small-scale three-dimensional turbulence into large-scale horizontal turbulence inside the film plays a critical role in determining the transition of wave modes and the nonlinear dynamics of the waves in the turbulent regime. Furthermore, the behaviour of three-dimensional waves in the downstream region, including frequent wave coalescence in the transition region and the breakup of small-scale solitons, is distinct from that of gravity-driven falling film flows. The amplitude of the three-dimensional waves is inversely related to the generalized Reynolds number defined for rotating films.

Stereo Reconstruction Method for 3D Surface Wave Fields around a Floating Body Using a Marker Net in a Wave Tank

Spatial wave fields around floating bodies are important for the understanding of hydrodynamics, and particularly the wave drift forces, of floating bodies in waves; however, experimental measurement of these fields is challenging. This paper presents a stereo reconstruction method for three-dimensional (3D) surface wave fields around floating bodies in a wave tank. Styrofoam markers were attached to a flexible net in a regular grid, called a marker net, and were placed on the water surface to be used as targets for stereo cameras (SCs). A thin plate spline was applied to the markers detected by the SCs to reconstruct the 3D surface wave profile around a floating body model. The proposed method was validated by measuring the wave field around a cylindrical floating body with a footing at its bottom. These experiments were conducted under regular wave incidence conditions. A wave elevation time series measured using a servo-controlled wave gauge was used as the benchmark data. The 3D surface wave field reconstruction method was applied under three different conditions: without the model, with a fixed model, and with a freely oscillating model. The results showed reliable reconstructions of the scattering and radiation waves. The marker net’s effects on the floating body’s motion and the surrounding wave fields were shown to be negligible by comparing the results acquired with and without the marker net.

An experimental study on the transient runback characteristics of wind-driven film/rivulet flows

A comprehensive experimental campaign was conducted to characterize the dynamic runback process of wind-driven water film/rivulet flows for a better understanding of the transient surface water transport process pertinent to aircraft icing phenomena. The experimental study was conducted by using an open-circuit wind tunnel to generate laminar boundary layer airflows with different freestream wind speeds to drive thin water film/rivulet flows over a flat test plate. A digital image projection (DIP) technique was used to achieve non-intrusive, temporally and spatially resolved measurements of the film thickness distributions to characterize the dynamic runback process of the wind-driven film/rivulet flow under different test conditions. Important characteristics of the wind-driven water runback process—such as the generation of well-organized two-dimensional (i.e., 2D) and more complicated three-dimensional (i.e., 3D) surface waves at air/water interfaces, stumbling runback motion of the film/rivulet heads in “acceleration-and-deceleration” cycles, breaking up the front contact lines to form multiple rivulet flows, meandering and merging of the rivulet flows—were revealed clearly and quantitatively based on the DIP measurements. A comprehensive force balance analysis was also performed to examine the variations of the relevant forces (i.e., the excess pressure forces built inside the film/rivulet flows, aerodynamic drag forces acting on the rivulet heads, and the restraining forces due to the surface tension along the liquid contact lines) and evaluate their importance in the breakoff of the stagnated film/rivulet heads to re-start the runback process of the wind-driven film/rivulet flows.

Numerical methods for modeling focused ultrasound in biomedical problems

The focused ultrasound (FUS) based on the cavitation effect is a promising technology that has a number of drawbacks, and the main one is the unpredictability of side effects. Numerical simulation of the influence of FUS on a human body can help to determine main mechanisms and factors that cause an undesirable tissue damage and adjust the medical equipment to avoid it. In this project, two numerical methods were used and compared within the scope of this problem. The grid-characteristic method allows to manage border and contact conditions of any complexity and implement a wide variety of damage models. The method was verified on experimental data from different areas of industry. In this project, it was used for model problems with simple geometry. The discontinuous Galerkin method is a stable and reliable numerical method that is less convenient in terms of border and contact conditions, but imposes fewer requirements on a computational grid. It allowed us to use grids based on real biological data. Both methods allow to obtain the full three-dimensional elastic wave pattern, including shear and surface waves, which are essential for understanding the influence of high-density inclusions on a FUS procedure. The applicability of both methods and their further development is discussed.

Трёхмерная задача о распространении волн в полупространстве с упруго-стеснённой границей.

In this paper we obtain the dispersion equation of three-dimensional wave propagation problem in half-space with an elastic- restrained border. Research of the problem is simplified by the introduction of potential functions. It is shown that by plane strain the elastic half-space constraint of boundary leads to decreasing of the surface wave localization degree. Three-dimensional surface wave exists only for two kinds of boundary conditions, when the surface of half-space is free from the stresses and the free surface is restrained. In the case of constrained free surface the three-dimensional surface wave has a dispersion property.

Observation of Kerosene Injected from a Cavity into a Supersonic Airstream

Kerosene-PLIF and high-speed schiliren are used to visualize flow-field of kerosene injected from a single injector at low plate and cavity walls. PLIF and high-speed schliren images are obtained. The results show that vortex or three-dimensional surface waves appear along boundary of fuel jet plume. With increasing dynamic ratio qj, start point of surface waves move upwards to the injector. In this paper, flapping flows at rear ramp of a cavity lead to oscillation of shock waves generated by shear layer upwards the cavity and separation bubble near shock reflection point on upper walls. Wavy rear ramp of cavity reduces shock strength and separation bubble disappears. PLIF images provide accurate distribution of fuel mixing and penetration depth in contrast to high-speed photography and schliren.

Three-dimensional flow structures in laminar falling liquid films

AbstractFull numerical simulations of the Navier–Stokes equations for four cases of vertically falling liquid films with three-dimensional surface waves have been performed. Flow conditions are based on several previous experimental studies where the streamwise and spanwise wavelengths were imposed, which we exploit by simulating periodic wave segments. The considered flows are laminar but approach conditions at which intermittent wave-induced turbulence has been observed elsewhere. Working liquids range from water to silicone oil and cover a large interval of the Kapitza number ($\textit {Ka}=18\mbox{--}3923$), which relates capillary to viscous forces. Simulations were performed on a supercomputer, using a finite-volume code and the volume of fluid and continuum surface force methods to account for the multiphase nature of the flow. Our results show that surface waves, consisting of large horseshoe-shaped wave humps concentrating most of the liquid and preceded by capillary ripples on a thin residual film, segregate the flow field into two regions: an inertia-dominated one in the large humps, where the local Reynolds number is up to five times larger than its mean value, and a visco-capillary region, where capillary and/or viscous forces dominate. In the inertial region, an intricate structure of different-scale vortices arises, which is more complicated than film thickness variations there suggest. Conversely, the flow in the visco-capillary region of large-$\textit {Ka} $fluids is entirely governed by the local free-surface curvature through the action of capillary forces, which impose the pressure distribution in the liquid film. This results in flow separation zones underneath the capillary troughs and a spanwise cellular flow pattern in the region of capillary wave interference. In some cases, capillary waves bridge the large horseshoe humps in the spanwise direction, coupling the two aforementioned regions and leading the flow to oscillate between three- and two-dimensional wave patterns. This persists over long times, as we show by simulations with the low-dimensional model of Scheidet al. (J. Fluid Mech., vol. 562, 2006, pp. 183–222) after satisfactory comparison with our direct simulations at short times. The governing mechanism is connected to the bridging capillary waves, which drain liquid from the horseshoe humps, decreasing their amplitude and wave speed and causing them to retract in the streamwise direction. Overall, it is observed that spanwise flow structures (not accounted for in two-dimensional investigations) are particularly complex due to the absence of gravity in this direction.

Трехмерная поверхностная волна в полупространстве и кромочные волны в пластинах в случае смешанных граничных условий на поверхности распространения

In the first part of this paper the propagation of sinusoidal three-dimensional surface waves is investigated for an isotropic elastic half-space with mixed boundary conditions. It is assumed that the boundary is fixed in one of the tangential directions and traction free in the other directions. The exact dispersion relation is derived which shows the existence and uniqueness of the three-dimensional surface wave. The speed of this wave depends on the angle of propagation and lies between the shear wave speed and Rayleigh wave speed. The graphs of this dependence are presented for various values of Poisson ratio. In the second part of this paper the three-dimensional edge waves in plates with mixed boundary conditions on the edge are investigated. The faces of the plate are assumed to be traction free. Both symmetric and antisymmetric solutions of three-dimensional dynamic equations of elasticity are considered. It is assumed that the edge is fixed in one of the tangential directions and traction free in the normal and the other tangential direction. Asymptotic analysis is performed, which shows that there is an infinite spectrum of higher order edge waves in such plates. The results of numerical calculations based on the modal expansion method are presented to confirm asymptotic analysis. In addition, by the numerical investigation the fundamental edge wave was found in the symmetric case (the edge is fixed in the tangential direction transversally to the faces). The phase velocity of this wave tends to some limit value depending on the Poisson ratio as the wave number increases. In the antisymmetric case the first higher order wave has the same limit value. The dispersion curves are presented for various values of Poisson ratio.

A three-dimensional surface wave–ocean circulation coupled model and its initial testing

A theoretical framework to include the influences of nonbreaking surface waves in ocean general circulation models is established based on Reynolds stresses and fluxes terms derived from surface wave-induced fluctuation. An expression for the wave-induced viscosity and diffusivity as a function of the wave number spectrum is derived for infinite and finite water depths; this derivation allows the coupling of ocean circulation models with a wave number spectrum numerical model. In the case of monochromatic surface wave, the wave-induced viscosity and diffusivity are functions of the Stokes drift. The influence of the wave-induced mixing scheme on global ocean circulation models was tested with the Princeton Ocean Model, indicating significant improvement in upper ocean thermal structure and mixed layer depth compared with mixing obtained by the Mellor–Yamada scheme without the wave influence. For example, the model–observation correlation coefficient of the upper 100-m temperature along 35° N increases from 0.68 without wave influence to 0.93 with wave influence. The wave-induced Reynolds stress can reach up to about 5% of the wind stress in high latitudes, and drive 2–3 Sv transport in the global ocean in the form of mesoscale eddies with diameter of 500–1,000 km. The surface wave-induced mixing is more pronounced in middle and high latitudes during the summer in the Northern Hemisphere and in middle latitudes in the Southern Hemisphere.

Three-dimensional surface wave tomography for the upper crustal velocity structure of southern Korea using seismic noise correlations

The cross-correlation technique was applied to ambient seismic noises to investigate the upper crustal velocity structure of the southern Korean Peninsula based on surface wave group velocities. We used seismograms recorded continuously during August 2005 at 91 accelerograph stations in the southern Korean Peninsula. From the correlation results of 4095 data pairs, the arrival times of Green’s functions along paths between stations of pairs were measured in discrete bins of frequency bands with central frequencies of 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, and 0.45 Hz. The spatial distribution of arrival times for each frequency band bin was inverted to obtain a two-dimensional tomographic map of Rayleigh wave group velocity. For each tomographic cell of the maps, we derived dispersion curves by extracting group velocities at center frequencies of the 7 frequency band bins. Shear-wave velocity obtained by depth inversion of the dispersion curve at each cell produced a one-dimensional velocity profile at depths. After aligning the profiles corresponding to tomographic cells, we obtained a three-dimensional uppermost crustal velocity model at depths down to approximately 8 km in the southern Korean Peninsula. These results show low-velocity distributions in the Gyeongsang Basin, western Gyeonggi Massif, and the area of Jeju Island and its northeastern coast. The Okcheon Fold Belt and Yeongnam Massif have relatively high velocity distributions. The low-velocity distribution in the Gyeongsang Basin shrinks in northsouth direction at depths around 4 km and widens again with a tendency to migrate to the east as the depth is increased. However, the linear trend of low velocities in NNE-SSW direction at deeper depths (6.25–8.25 km) in the western area distinctively coincides with the western boundary of the Gyeongsang Basin. The background noise analysis in this study provided the constraining information for determination of depth variation in the shear-wave velocity distribution.

Computation formulas by FFT of the nonlinear orbital velocity in three-dimensional surface wave fields

Accurate representations of the surface potential and the orbital velocity of nonlinear water waves are obtained, given the spatial wave-elevation field and its time derivative along two-dimensional sections of the ocean surface. The effect of a horizontal current is accounted for. The method is three-dimensional. The kernel of an integral equation and its right-hand side, both nonlinear functions of the elevation, are obtained in series expansions, and expressed explicitly by Fourier transform (FFT). Calculations for a periodic sine-wave and non-periodic model directional irregular wave field over swaths (wave slope in the range ±0.3 in both cases) illustrate the formulas. The Gibbs phenomenon along the boundaries affects the very high-order contributions in the non-periodic case.

Linear surface capillary-gravity short-crested waves on a current

One of the forward situations in the study of water waves is the basic three-dimensional surface wave motion of short-crested waves. Capillary waves result in rich effects concerned closely with remote sensing in the open ocean. Ocean currents experience a complete process in surface wave motion. Based on the above ideas, a linear dynamical system of surface capillary-gravity short-crested waves is developed by considering the current effects, thus leading to the following analytical expressions of the kinematic and dynamic variables: the wave height, the wave steepness, the phase velocity, the wave-particle velocities, accelerations and trajectories and the wave pressure. A number of the classical, typical and latest special wave cases can arise from these expressions.

Comments on “The Three-Dimensional Current and Surface Wave Equations”

Abstract The lowest order sigma-transformed momentum equation given by Mellor takes into account a phase-averaged wave forcing based on Airy wave theory. This equation is shown to be generally inconsistent because of inadequate approximations of the wave motion. Indeed the evaluation of the vertical flux of momentum requires an estimation of the pressure p and coordinate transformation function s to first order in parameters that define the large-scale evolution of the wave field, such as the bottom slope. Unfortunately, there is no analytical expression for p and s at that order. A numerical correction method is thus proposed and verified. Alternative coordinate transforms that allow a separation of wave and mean flow momenta do not suffer from this inconsistency nor do they require a numerical estimation of the wave forcing. Indeed, the problematic vertical flux is part of the wave momentum flux, thus distinct from the mean flow momentum flux, and not directly relevant to the mean flow evolution.

Numerical simulation of non-linear regular and focused waves in an infinite water-depth

Inviscid three-dimensional free surface wave motions are simulated using a novel quadratic higher order boundary element model (HOBEM) based on potential theory for irrotational, incompressible fluid flow in an infinite water-depth. The free surface boundary conditions are fully non-linear. Based on the use of images, a channel Green function is developed and applied to the present model so that two lateral surfaces of an infinite-depth wave tank can be excluded from the calculation domain. In order to generate incident waves and dissipate outgoing waves, a non-reflective wave generator, composed of a series of vertically aligned point sources in the computational domain, is used in conjunction with upstream and downstream damping layers. Numerical experiments are carried out, with linear and fully non-linear, regular and focused waves. It can be seen from the results that the present approach is effective in generating a specified wave profile in an infinite water-depth without reflection at the open boundaries, and fully non-linear numerical simulations compare well with theoretical solutions. The present numerical technique is aimed at efficient modelling of the non-linear wave interactions with ocean structures in deep water.

Conditional second-order short-crested water waves applied to extreme wave episodes

A derivation of the mean second-order short-crested wave pattern and associated wave kinematics, conditional on a given magnitude of the wave crest, is presented. The analysis is based on the second-order Sharma and Dean finite-water wave theory. A comparison with a measured extreme wave profile, the Draupner New Year Wave, shows a good agreement in the mean, indicating that this second-order wave can be a good identifier of the shape and occurrence of extreme wave events. A discussion on its use as an initial condition for a fully nonlinear three-dimensional surface wave analysis is given.

Some Consequences of the Three-Dimensional Current and Surface Wave Equations

AbstractThree-dimensional, interacting current and surface gravity wave equations have recently been derived and compared with their counterpart vertically integrated equations; they are in the form of sigma-coordinate equations. The purpose of this paper is to examine some of the consequences of these equations including energy transfer between mean energy, wave energy, and turbulence energy, to frame some outstanding research issues, to provide a Cartesian version of the sigma-coordinate equations, and to compare with other formulations of wave–current interaction. In general, the paper is intended to set the stage for the development of numerical coupled surface wave and three-dimensional general circulation models. These models often include a flow-dependent turbulence-based viscosity.

Automated trinocular stereo imaging system for three-dimensional surface wave measurements

A novel automated trinocular stereo imaging system (ATSIS) is developed for non-intrusively measuring the temporal evolution of three-dimensional wave characteristics. The system consists of three progressive digital cameras to provide three independent stereo-pairs, i.e. left–right, left–center, and center–right, for accurately estimating depth of a scene. A third camera assists to resolve correspondence problems due to specular reflection on the water surface and provides additional constraints on image matching, dramatically reducing the chance of a mismatch. An oblique configuration for the trinocular system effectively increases spatial coverage, allowing observations of wave phenomena over a broad range of spatial scales. The height resolution is increased with the optical axes of the cameras pointed at an oblique angle with respect to vertical surface wave displacements. A new exterior calibration procedure is developed in this paper to determine the orientation of cameras in the field. Field experiments demonstrate that ATSIS can robustly measure hundreds of matched image points in seconds, allowing fast extraction of the temporal evolution of a three-dimensional surface wave field.