Green water impact pressure on a three-dimensional model structure

Green water impact pressure due to plunging breaking waves impinging on a simplified, three-dimensional model structure was investigated in the laboratory. Two breaking wave conditions were tested: one with waves impinging on the vertical wall of the model at the still water level and the other with waves impinging on the horizontal deck surface. Pressure measurements were taken at locations in two vertical planes on the deck surface with one at centerline of deck and the other between the centerline and an edge. Impact pressure was found to be quite different between the two wave conditions even though the incoming waves are essentially identical. Two types of pressure variations were observed??impulsive type and non-impulsive type. Much higher pressure was observed for the deck impingement wave condition, even though the flow velocities were quite close. Void fraction was also measured at selected points. Impact pressure was correlated with the mean kinetic energy calculated based on the measured mean velocities and void fraction. Impact coefficient, defined as the ratio between the maximum pressure at a given point and the corresponding mean kinetic energy, was obtained. For the wall impingement wave condition, the relationship between impact pressure and mean kinetic energy is linear with the impact coefficient close to 1.3. For the deck impingement wave condition, the above relationship does not show good correlation; the impact coefficient was between 0.6 and 7. The impact coefficient was found to be a function of the rate of pressure rise.     

Runup and green water velocities due to breaking wave impinging and overtopping

The present study investigates, through measurements in a 2D wave tank, the velocity fields of a plunging breaking wave impinging on a structure. As the wave breaks and overtops the structure, so-called green water is generated. The flow becomes multi-phased and chaotic as a large aerated region is formed in the flow in the vicinity of the structure while water runs up onto the structure. In this study, particle image velocimetry (PIV) and its derivative, bubble image velocimetry (BIV), were employed to measure the velocity field in front and on top of the structure. Mean and turbulence properties were obtained through ensemble averaging repeated tests. The dominant and maximum velocity of the breaking wave and associated green water are discussed for the three distinct phases of the impingement–runup–overtopping sequence. Initially the flow is mainly horizontal right before the breaking wave impinges on the structure. The flow then becomes primarily vertical and rushes upward along the front wall of the structure right after the impingement. Subsequently, the flow becomes mainly horizontal on top of the structure as the remaining momentum in the wave crest carries the green water through. The distribution of the green water velocity along the top of the structure has a nonlinear profile and the maximum velocity occurs near the front of the fast moving water. Using the measured data and applying dimensional analysis, a similarity profile for the green water flow on top of the structure was obtained, and a prediction equation was formulated. The prediction equation may be used to predict the green water velocity caused by extreme waves in a hurricane.     

Measurements of wave-variance and velocity spectra in breaking waves

Displacements of mechanical waves superposed onto wind waves were measured with a laser displacement gauge in a wind-wave tank. The effects of wave breaking, especially the spilling breaking type, on the wave-variance spectra are investigated. In the absence of wave breaking, the quasi-equilibrium spectrum consists of an f ^–7/3 subrange in the capillary regime, and its spectral density increases with increasing wind speed. When intense spilling breaking occurs, the water surface is saturated with small-scale features that cause not only an increase in the spectral density but also a reduction in the slope of the spectrum at high frequencies. Velocity components under the water surface were measured with a laser Doppler velocimeter. The energy spectra of the vertical and longitudinal velocity components in breaking waves are practically identical in the frequency range near the dominant wave frequency. At higher frequencies, the spectra generally follow Kolmogorov's –5/3 law. In the intermediate frequency range, we observed a higher spectral density for the vertical velocity component than for its longitudinal counterpart. These results suggest that turbulence energy is transferred from the vertical component to the longitudinal component in breaking waves. The acceleration of the water motion becomes as large as gravitational acceleration when intense wave breaking takes place. The flow field in breaking waves is highly dissipative.     

Numerical study of breaking waves by a two‐phase flow model

A two‐phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre‐breaking, overturning and post‐breaking processes. The model is based on the Reynolds‐averaged Navier–Stokes equations with the k ?ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure‐velocity coupling and the air‐water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep‐water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash‐up during wave breaking have been captured by the two‐phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.     

Non‐hydrostatic 3D free surface layer‐structured finite volume model for short wave propagation

In this paper a layer‐structured finite volume model for non‐hydrostatic 3D environmental free surface flow is presented and applied to several test cases, which involve the computation of gravity waves. The 3D unsteady momentum and mass conservation equations are solved in a collocated grid made of polyhedrons, which are built from a 2D horizontal unstructured mesh, by just adding several horizontal layers. The mesh built in such a way is unstructured in the horizontal plane, but structured in the vertical direction. This procedure simplifies the mesh generation and at the same time it produces a well‐oriented mesh for stratified flows, which are common in environmental problems. The model reduces to a 2D depth‐averaged shallow water model when one single layer is defined in the mesh. Pressure–velocity coupling is achieved by the Semi‐Implicit Method for Pressure‐Linked Equations algorithm, using Rhie–Chow interpolation to stabilize the pressure field. An attractive property of the model proposed is the ability to compute the propagation of short waves with a rather coarse vertical discretization. Several test cases are solved in order to show the capabilities and numerical stability of the model, including a rectangular free oscillating basin, a radially symmetric wave, short wave propagation over a 1D bar, solitary wave runup on a vertical wall, and short wave refraction over a 2D shoal. In all the cases the numerical results are compared either with analytical or with experimental data. Copyright © 2008 John Wiley & Sons, Ltd.     

Generation of internal waves by a turbulent jet in a stratified fluid

The process of generation of internal waves by an initially cylindrical, turbulent jet with a Gaussian profile of the average horizontal velocity component in a fluid with stable linear density stratification is investigated by direct numerical simulation. It is shown that on time intervals Nt < 30, where N is the buoyancy frequency, the vertical velocity pulsations collapse, which is accompanied by the generation of internal waves whose spatial period is close to the wavelength of the spiral mode of jet instability in a homogeneous fluid. The wave dynamics and kinematics can be satisfactorily described by the linear theory for a pulsed source and their parameters are in good agreement with the parameters of the “coherent” internal waves generated by a stratified wake in a laboratory experiment. At large times the wave generation ceases and the variations of the fluid density are localized in the neighborhood of the centers of large-scale vortices formed in the horizontal plane in the neighborhood of the jet.     

Three-dimensional hydrodynamics on finite elements. Part II: Non-linear time-stepping model

The development and application of a non-linear 3D hydrodynamic model are described. The model is based on the wave equation rearrangement of the primitive 3D shallow water equations with a general eddy viscosity formulation for the vertical shear. A Galerkin procedure is used to discretize these on simple sixnode elements: linear triangles in the horizontal with linear variations in the vertical. Resolution of surface, bottom and interfacial boundary layers is facilitated and total flexibility is preserved for specifying spatial and temporal variations in the vertical viscosity and density fields. A semi-implicit time-stepping algorithm allows the solutions for elevation and velocity to be uncoupled during each time step. The elevation solution is essentially a 2D wave equation calculation with a stationary sparse matrix representing the gravity waves. With nodal quadrature the subsequent velocity calculation is achieved by factoring only a tridiagonal diffusion matrix representing the vertical viscous terms. As a result the overall calculation scales computationally as only a 2D problem but provides the full 3D solution. Application to field-scale problems is illustrated for the English Channel/Southern Bight system and the Lake Maracaibo system.     

Assessment of assembly shape and geometric effects on amplification of shear waves using discrete element method

The discrete element method (DEM) is a capable tool used to simulate shear wave propagation in granular assemblies for many years. Researchers have studied assembly shapes such as rectangles (in 2D simulations) or cylinders and cubes (in 3D simulations). This paper aimed to qualify the effect of assembly shape on the shear wave propagation and maximum amplification in the vertical plane (horizontal and vertical directions) caused by this propagation. To this end, shear wave propagations in different assembly shapes such as rectangle, trapezium, and triangle with rigid boundary conditions were simulated. A sine wave pulse was applied with a point source by moving a particle as the transmitter particle. To evaluate the shear wave velocity of the assemblies, the transmitter and receiver particles were simulated. All the simulations were performed with 2D DEM which is a useful tool to determine the amount and location of the maximum amplification factor of the assembly in both horizontal and vertical directions. An advantage of this study was assessing the effect of parameters such as input wave frequency, assembly height, shape, and aspect ratios on the amplification of the input waves.     

ON BREAKING WAVES AND WAVE-CURRENT INTERACTION IN SHALLOW WATER: A 2DH FINITE ELEMENT MODEL

A two-dimensional (horizontal plane) coastal and estuarine region model, capable of predicting the combined effects of gravity surface shallow- water waves (shoaling, refraction, diffraction, reflection and breaking), and steady currents, is described and numerical results are compared with those obtained experimentally. Two series of observations within a wave flume and a combined wave-current facility were developed. In the first case, the wave was generated via a hinged paddle located within a deepened section at one end of the channel, as, in the second case, the wave propagating with or against the current was generated by a plunger-type wavemaker; the re-circulating current was introduced via one passing tank connected to a centrifugal pump. Several comparisons for a number of 1D situations and one 2D horizontal plane case are presented.     

The structure of the turbulent boundary layer over a mobile and deformable boundary

The structure of the velocity field above a propagating water wave of fixed frequency was investigated in order to evaluate the transport of wind momentum to water waves and the influence of a mobile and deformable boundary on the bursting cycle. The vertical and horizontal velocities were measured in a transformed Eulerian wave-following frame of reference with the aid of a cross hot film, in a wind-wave research facility at Stanford University.The mean velocity profiles have a log-linear form with a wake free-stream characteristic. The wave-coherent motion in the free-stream is irrotational; in the boundary layer, it has a strong shear behavior related to the wave-associated stress. The wave-induced velocity field and the wave-perturbed turbulence depend strongly on the ratio of the wave-speed to the mean free-stream velocity, c/U ₀.The presence of the propagating waves affects the bursting cycle, making the contribution of sweeps and ejections almost equal and dependent on the ratio c/U ₀. The magnitudes of the contribution of the bursting events are generally enhanced by the presence of water waves. The time interval between ejections or sweeps does not scale with either the inner and/or outer flow variables.This paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

A coupled-mode model for water wave scattering by vertically sheared currents in variable bathymetry regions

A coupled-mode model is developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, sheared current with linear vertical velocity profile, over general bottom topography. The wave potential is represented by a series of local vertical modes containing the propagating and evanescent modes, plus additional terms accounting for the satisfaction of the boundary conditions. Using the above representation, in conjunction with a variational principle, a coupled system of differential equations on the horizontal plane is derived, with respect to the unknown modal amplitudes. In the case of small-amplitude waves, a linearized version of the above coupled-mode system is obtained, extending previous analysis by Belibassakis et al. (2011) to the propagation of water waves over variable bathymetry regions in the presence of vertically sheared currents. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system reduces to a one-equation model, that is shown to extend known mild-slope mild vertical shear equation concerning wave–current interaction over slowly varying topography. After additional simplifications, the latter model is shown to be compatible with the extended mild-slope mild-shear equation by Touboul et al. (2016). Results are presented for various representative test cases demonstrating the usefulness of the present coupled mode system and the importance of various terms in the modal expansion, and compared against experimental data collected in wave flume validating the present method. The analytical structure of the present system facilitates extensions to model non-linear effects and applications concerning wave scattering by inhomogeneous currents in coastal regions with general 3D bottom topography.     

A New Lagrangian Asymptotic Solution for Gravity–Capillary Waves in Water of Finite Depth

A third-order Lagrangian asymptotic solution is derived for gravity–capillary waves in water of finite depth. The explicit parametric solution gives the trajectory of a water particle and the wave kinematics for Lagrangian points above the mean water level, and in a water column. The water particle orbits and mass transport velocity as functions of the surface tension are obtained. Some remarkable trajectories may contain one or multiple sub-loops for steep waves and large surface tension. Overall, an increase in surface tension tends to increase the motions of surface particles including the relative horizontal distance travelled by a particle as well as the time-averaged drift velocity     

Internal wave reflections and transmissions arising from a non-uniform mesh. Part I: An analysis for the Crank–Nicolson linear finite element scheme

This is the first of a series of three related papers dealing with some of the consequences of non-uniform meshes in a numerical model. In this paper the accuracy of the Crank–Nicolson linear finite element scheme, which is applied to the linear shallow water equations, is examined in the context of a single abrupt change in nodal spacing. The (in)accuracy is quantified in terms of reflection and transmission coefficients. An incident wave impinging on the interface between two regions with different nodal spacings is shown to give rise to no reflected waves and two transmitted waves. The analysis is verified using three different wavelengths (2Δx, 4Δx 8Δx) in three ‘hot-start’ numerical experiments with a mesh expansion factor of 2 and three experiments with a mesh contraction factor of 1/2. An energy flux analysis based on the concept of group velocity shows that energy is conserved across the interface.     

2D and 3D Turbulent Fluctuations in Open Channel Flow with Re τ = 590 Studied by Large Eddy Simulation

Well-resolved 3D Large Eddy Simulations (LES) are presented for open channel flow at a Reynolds number Re _τ  = 590 based on friction velocity u _τ and water depth h. The results are depth-averaged and thereby information is obtained on the 2D horizontal fluctuations in the channel. The total turbulence is decomposed into 2D and 3D fluctuations and the energy content of these as well as their spectral distribution is studied. It is found that only 15% of the fluctuating energy is contained in the 2D fluctuations and that these are mostly of scales larger than the water depth while the 3D fluctuations are restricted by the limited vertical extent of the water body and have scales smaller than the water depth. Information is obtained on the dispersion terms arising from the depth-averaging procedure, and scalar transport due to a vertical line source of tracer is studied thereby investigating the contribution of the 2D and 3D fluctuations to the transverse mixing of the scalar.     

A laboratory study of irregular shoaling waves

The present research aims to investigate the dynamics of a single laboratory irregular wave, characterized by a narrow-banded spectrum and developing on a sloping sand bottom, in intermediate waters up to the surf zone. Experiments focused on the wave shoaling region, in order to examine how the wave is affected by breaking induced turbulence offshore the surf zone. A 3D acoustic Doppler velocimeter was used to measure the three wave velocity components, which were all processed to evaluate the time-averaged vertical distributions of orbital velocities, wave and turbulent Reynolds shear stresses and turbulent intensities. The vertical distributions of the phase-averaged velocity components, turbulent kinetic energy and transport of turbulence were also analysed. The adopted phase-averaging technique was applied to each investigated measurement point. Therefore, the crucial element of the study is that all the analysed values derive directly from real measurements and are not approximated by any kind of interpolation. The study confirmed some dynamic behaviour in the shoaling zone already known in the literature, such as the typical cell-type flow pattern of the mean flow and the necessity to evaluate the turbulent kinetic energy with all the three velocity components, when available, which would otherwise be underestimated. Referring to the time-averaged wave and Reynolds shear stresses, a contribution was added to the open debate on their order of magnitude. The measured wave Reynolds shear stresses were also compared with the results of the model by Zou et al. (J Geophys Res 111:C09032, 2006), confirming the behaviour typical of dissipative breaking waves. The analysis of turbulence transport in the shoaling zone revealed that it is seaward directed close to the surface and landward directed close to the bottom. The results presented in the paper can be extended only to other analogous flow conditions.     

Numerical simulations of wave interactions with vertical wave barriers using the SPH method

This paper focuses on the fluid boundary separation problem of the conventional dynamic solid boundary treatment (DSBT) and proposes a modified DSBT (MDSBT). Classic 2D free dam break flows and 3D dam break flows against a rectangular box are used to assess the performance of this MDSBT in free surface flow and violent fluid–structure interaction, respectively. Another test, water column oscillations in a U‐tube, is specially designed to reveal the applicability of dealing with two types of particular boundaries: the wet–dry solid boundary and the large‐curvature solid boundary. A comparison between the numerical results and the experimental data shows that the MDSBT is capable of eliminating the fluid boundary separation, improving the accuracy of the solid boundary pressure calculations and preventing the unphysical penetration of fluid particles. Using a 2D SPH numerical wave tank with MDSBT, the interactions between regular waves and a simplified vertical wave barrier are simulated. The numerical results reveal that the maximum horizontal force occurs at the endpoint of the vertical board, and with the enlargement of the relative submerged board length, the maximum moment grows linearly; furthermore, the relative average mass transportation under the breakwater initially increases to 11.14 per wave strike but is later reduced. The numerical simulation of a full‐scale 3D wave barrier with two vertical boards shows that the wave and structure interactions in the practical project are far more complicated than in the simplified 2D models. The SPH model using the MDSBT is capable of providing a reference for engineering designs. Copyright © 2014 John Wiley & Sons, Ltd.     

A depth‐integrated non‐hydrostatic finite element model for wave propagation

This paper presents a two‐dimensional finite element model for simulating dynamic propagation of weakly dispersive waves. Shallow water equations including extra non‐hydrostatic pressure terms and a depth‐integrated vertical momentum equation are solved with linear distributions assumed in the vertical direction for the non‐hydrostatic pressure and the vertical velocity. The model is developed based on the platform of a finite element model, CCHE2D. A physically bounded upwind scheme for the advection term discretization is developed, and the quasi second‐order differential operators of this scheme result in no oscillation and little numerical diffusion. The depth‐integrated non‐hydrostatic wave model is solved semi‐implicitly: the provisional flow velocity is first implicitly solved using the shallow water equations; the non‐hydrostatic pressure, which is implicitly obtained by ensuring a divergence‐free velocity field, is used to correct the provisional velocity, and finally the depth‐integrated continuity equation is explicitly solved to satisfy global mass conservation. The developed wave model is verified by an analytical solution and validated by laboratory experiments, and the computed results show that the wave model can properly handle linear and nonlinear dispersive waves, wave shoaling, diffraction, refraction and focusing. Copyright © 2013 John Wiley & Sons, Ltd.     

Boundary element modeling for defect characterization potential in a wave guide

Wave scattering analysis implemented by boundary element methods (BEM) and the normal mode expansion technique is used to study the sizing potential of two-dimensional shaped defects in a wave guide. Surface breaking half-elliptical shaped defects of three opening lengths (0.3, 6.35 and 12.7 mm) and through-wall depths of 10–90% on a 10 mm thick steel plate were considered. The reflection and transmission coefficients of both Lamb and shear horizontal (SH) waves over a frequency range 0.05–2 MHz were studied. A powerfully practical result was obtained whereby the numerical results for the S₀ mode Lamb wave and n₀ mode SH wave at low frequencies showed a monotonic increase in signal amplitude with an increase in the defect through-wall depth. At high frequency (usually above the cut-off frequency of the A₁ mode for Lamb waves and the n₁ mode for SH waves, respectively), the monotonic trend does not hold in general due to the energy redistribution to the higher order wave modes. Guided waves impinging onto an internal stringer-like an inclusion were also studied. Both the Lamb and SH waves were shown to be insensitive to the stringer internal inclusions at low frequency. Experiments with piezoelectric Lamb wave transducers and non-contact SH wave electro-magnetic acoustic transducers (EMAT) verified some of the theoretical results.     

Green water void fraction due to breaking wave impinging and overtopping

The present study uses laboratory measurements to investigate the void fraction of an overtopping flow on a structure. The overtopping flow, also called green water, was generated by the impingement of a plunging breaking wave on the structure following the Froude similarity of an extreme hurricane wave and a simplified offshore structure. The flow is multi-phased and turbulent with significant aeration. A fiber optic reflectometer (FOR) and bubble image velocimetry (BIV) were employed to measure the void fraction and velocity in the flow, respectively, and to determine the water level on the deck. Mean properties of void fraction and velocity were obtained by ensemble-averaging and time-averaging the repeated instantaneous measurements. The temporal and spatial distributions of void fraction reveal that the flow is very highly aerated near the front of green water and has relatively low aeration near the deck surface. The mean void fraction and velocity distributions were also depth-averaged for simplicity and potential use in engineering applications. Using the measured data, similarity profiles for depth-averaged void fraction, depth-averaged velocity, and water level were found. The study suggests that using only the velocity data is insufficient if the flow momentum or the flow rate is to be determined. The accuracy of the void fraction measurements was validated by comparing the directly measured water volume of the overtopping flow with the calculated water volume based on the measured velocity and void fraction.