Deep Water Case Research Articles

Apparent singularities of the finite-depth Zakharov equation

The description of weakly nonlinear water-wave evolution over a horizontal bottom by the integro-differential Zakharov equation, because of utilising the underlying Hamiltonian structure, has many advantages over direct use of the Euler equations. However, its application to finite-depth situations is not straightforward since, in contrast to the deep-water case, the kernels governing the four-wave interactions are singular, as well as the kernels in the canonical transformation that removes non-resonant interactions from the original equations of motion. At the singularities, these kernels are finite but not unique. The issue of how to use the Zakharov equation for finite depth and whether it is possible at all was debated intensely in the literature for decades but remains outstanding. Here we show that the absence of a limit of the kernels at the singularities is inconsequential, since in the equations of motion it is only the integral that matters. By applying the definition of the Dirac- $\delta$ , we show that all the integrals involving a trivial manifold singularity are evaluated uniquely. Therefore, the Zakharov evolution equation and the nonlinear canonical transformation are only apparently singular. The findings are validated by application to examples where predictions based on the Zakharov equation are compared with known solutions obtained from the Euler equations.

Experimental Study on the Airborne Noise of Rain

Rainfall noise is investigated experimentally in this paper. The modern high-speed camera and microphone are employed to study both drop dynamics and sound signals. Sound signals produced by single-drop impacts on deep pools, thin liquid films, steel plates, and sandy surfaces were analyzed through a combination of videos and audio. Classical Fourier transforms are performed for converting the original signal to a power spectrum. Results show that in the deep-water cases, the noise is proved to be the entrapment of a tiny breathing bubble that sheds from the dimple. The dominant frequencies of droplets impacting water film, rigid walls, and sandy surfaces are much compared with deep pool cases.

Investigating roughness effects on ship resistance in shallow waters

Shallow waters influence the behaviour and performance of a vessel by modifying the local pressure distribution, wave making, and boundary layer thickness. The boundary layer thickness is also influenced by surface roughness. No previous studies have investigated the combined effects of shallow water contributions and roughness on ship resistance. This study aims to fill this knowledge gap in the literature by using Unsteady Reynolds Averaged Navier-Stokes modelling. Results show that the total resistance coefficient increases between approximately 22% and 36% in the presence of roughness depending on the speed and depth-to-draft ratio. The numerical model used shows that pressure resistance grows at a faster pace than frictional resistance and increases its relative contribution to the total when roughness is applied, contrary to deep water cases.

Pleistocene Shallow-Water Whale-Fall Community from the Omma Formation in Central Japan

We report a single whale bone associated with many molluscan fossils from the Omma Formation, Lower Pleistocene shallow marine deposits, along the Sai-gawa River, Kanazawa City, in central Japan. Most molluscan species which are commonly found in the Omma Formation show disarticulated and/or damaged shells, indicating semi-autochthonous or allochthonous modes of occurrence. However, the assemblage contained chemosynthetic bivalves, such as lucinid, solemyid and thyasirid bivalves, which are rare in the Omma Formation. The lucinids and solemyids show a high articulation ratio, along with some predatory and scavenging gastropods, such as naticids, nassariids and borsoniids whose well-preserved shells indicate an autochthonous mode of occurrence. In addition, most of the lucinid bivalves show an umbo-upward position similar to the life position of Recent species. Recent lucinid, solemyid and most thyasirid bivalves harbor chemosymbiotic bacteria in their gills and are well known members of the chemosynthetic community. These lines of evidence indicate that the community, mainly comprising lucinid bivalves and other autochthonous molluscan species associated with the whale bone, is an ancient whale-fall community. This shallowest fossil whale-fall community differs from deep-water cases in the dominance of infaunal bivalves, such as lucinids, and in the lack of epifaunal and semi-infaunal chemosynthetic bivalves, such as bathymodiolins and vesicomyids. This community supports a previous suggestion that the difference in characteristic species of the whale-fall communities depends on the water depth.

Practical representation of flows due to general singularity distributions for wave diffraction–radiation by offshore structures in finite water depth

The behavior of an offshore structure in regular waves – and the related (linear and nonlinear) wave loads, added-mass and wave-damping coefficients, and body-motions – are commonly analyzed via the Green-function and boundary-integral method associated with potential-flow theory. This realistic, widely-used method requires accurate and efficient numerical evaluation of flows created by distributions of singularities (source, dipole) over (flat or curved) panels of various shapes (triangle, quadrilateral) that are used to approximate the surface of an offshore structure. This basic core element of the theory of diffraction–radiation of regular waves by an offshore structure is considered for water of uniform finite depth. The special case of deep water is also considered. An analytical representation of the flow created by a general distribution of singularities over a hull-surface panel is given. This flow-representation is based on the Fourier–Kochin (FK) approach, in which space-integration over the panel is performed first and Fourier-integration is performed subsequently, unlike the common approach in which the Green function (defined via a Fourier integration) is evaluated first and subsequently integrated over the panel. The analytical and numerical complexities associated with the numerical evaluation and subsequent panel integration of the singular Green function for wave diffraction–radiation by offshore structures are then avoided in the FK approach. In this approach, panel integration merely amounts to integrating an elementary (exponential–trigonometric) function, a trivial task that can be performed accurately and efficiently. The analytical flow-representation given in the study provides a mathematically-exact smooth decomposition of free-surface effects into a non-oscillatory local flow and waves. The waves in this flow decomposition are defined by a regular single Fourier integral, and the local flow is given by a double Fourier integral with a smooth integrand that only involves ordinary functions and is dominant within a compact region near the origin of the Fourier plane. Illustrative numerical applications for typical distributions of sources and dipoles over a panel show that the flow-representation given in the study is well suited for practical numerical evaluations.

The influence of finite depth on the evolution of extreme wave statistics in numerical wave tanks

Model tests with unidirectional random wave fields are common for coastal engineering purposes. In this paper, we investigate the finite water depth effect on both the probability of extreme events and the averaged shape of them in such experimental conditions. We simulate some typical model test conditions with a numerical tool, which solves the fully non-linear wave equations. We find the analytical solution based on Nonlinear Schrödinger equation (NLS) to be accurate for low steepness and relatively deep water cases. However, the analytical solution underestimates the kurtosis at the steady state for high steepness cases in shallow water, and also gives zero value of kurtosis at critical water depth, whereas a small but non-zero value of kurtosis is observed in the numerical tank. We also investigate the averaged shape of the extreme events, which are modified by nonlinear physics over the distance. The horizontal asymmetry is significant initially but is greatly reduced to a steady state, and the contraction of the wave groups grows monotonically until the steady state is reached along the wave tank. Finite water depth has limited effects on the averaged shape of large events, especially for the extremely steep waves.

Integral equation technique for water wave interaction by an array of vertical flexible porous wave barriers

AbstractScattering of obliquely incident surface gravity waves by an array of partial flexible vertical porous wave barriers of varied configurations is studied in finite water depth and deep water cases. For the mathematical modeling purpose, linearized water wave theory is considered. Firstly, the problem associated with double vertical barriers is handled for a solution. To solve the problem, the associated BVP (boundary value problem) is transformed into a system of Fredholm integral equations of the second kind. By using suitable quadrature formulae, these integral equations are converted to a system of linear algebraic equations. These are solved to compute various important physical quantities associated with wave scattering. The energy identities for both finite water depth and deep water cases are derived. These derived energy identities are used to check the computational accuracy and to quantify the loss of wave energy by the barrier system. Subsequently, the study is extended to deal with wave scattering by multiple non‐identical unequally spaced barriers using wide‐spacing approximation and results for equally spaced identical barriers are obtained as a special case. The methodology is further generalized to deal with Bloch problem to incorporate a periodic array of infinite barriers. Occurrences of Bragg resonance in case of multiple barriers are analyzed. From the results associated with the linearized water wave theory, the second‐order mean wave forces are computed and compared with the first‐order wave forces. The phenomena of cloaking for wave scattering by multiple impervious barriers are discussed for a wide variety of wave and structural parameters. The methodology used in this present study can easily be generalized to solve similar problems that arise in mathematical physics.

Hydrodynamics of two-dimensional compressible fluid with broken parity: Variational principle and free surface dynamics in the absence of dissipation

We consider an isotropic compressible non-dissipative fluid with broken parity subject to free surface boundary conditions in two spatial dimensions. The hydrodynamic equations describing the bulk dynamics of the fluid as well as the free surface boundary conditions depend explicitly on the parity breaking non-dissipative odd viscosity term. We construct a variational principle in the form of an effective action which gives both bulk hydrodynamic equations and free surface boundary conditions. The free surface boundary conditions require an additional boundary term in the action which resembles a $1+1D$ chiral boson field coupled to the background geometry. We solve the linearized hydrodynamic equations for the deep water case and derive the dispersion of chiral surface waves. We show that in the long wavelength limit the flow profile exhibits an oscillating vortical boundary layer near the free surface. The thickness of the layer is controlled by the length scale given by the ratio of odd viscosity to the sound velocity $\delta \sim \nu_o/c_s$. In the incompressible limit, $c_s\to \infty$ the vortical boundary layer becomes singular with the vorticity within the layer diverging as $\omega \sim c_s$. The boundary layer is formed by odd viscosity coupling the divergence of velocity $\boldsymbol\nabla \cdot \boldsymbol{v}$ to vorticity $\boldsymbol\nabla \times \boldsymbol{v}$. It results in non-trivial chiral free surface dynamics even in the absence of external forces. The structure of the odd viscosity induced boundary layer is very different from the conventional free surface boundary layer associated with dissipative shear viscosity.

Satellite-derived bathymetry in optically complex waters using a model inversion approach and Sentinel-2 data

This study presents an assessment of a model inversion approach to derive shallow water bathymetry in optically complex waters, with the aim of both understanding localised capability and contributing to the global evaluation of Sentinel-2 for coastal monitoring. A dataset of 12 Sentinel-2 MSI images, in three different study areas along the Irish coast, has been analysed. Before the application of the bathymetric model two atmospheric correction procedures were tested: Deep Water Correction (DWC) and Case 2 Regional Coastal Color (C2RCC) processor. DWC outperformed C2RCC in the majority of the satellite images showing more consistent results. Using DWC for atmospheric correction before the application of the bathymetric model, the lowest average RMSE was found in Dublin Bay (RMSE = 1.60, bias = −0.51), followed by Mulroy Bay (RMSE = 1.66, bias = 1.30) while Brandon Bay showed the highest average error (RMSE = 2.43, bias = 1.86). However, when the optimal imagery selection was considered, depth estimations with a bias less than 0.1 m and a spread of ±1.40 m were achieved up to 10 m. These results were comparable to those achieved by empirical tuning methods, despite not relying on any in situ depth data. This conclusion is of particular relevance as model inversion approaches might allow future modifications in crucial parts of the processing chain leading to improved results. Atmospheric correction, the selection of optimal images (e.g. low turbidity), the definition of suitably limited ranges for the per-pixel occurrence of optical constituents (phytoplankton, CDOM, backscatter) and seabed reflectances, in combination with the understanding of the specifics characteristics at each particular site, were critical steps in the derivation of satellite bathymetry.

Time domain simulation of focused waves by High-Level Irrotational Green–Naghdi equations and Harmonic Polynomial Cell method

Wave focusing is one of the main mechanisms that can generate freak waves in deep water. The strongly nonlinear High-Level Irrotational Green–Naghdi (HLIGN) equations and the Harmonic Polynomial Cell (HPC) method are used here to simulate the focused waves. Numerical simulations are performed in intermediate water depth as well as in deep water. In intermediate water, the HLIGN free surface and wave kinematics are then compared with the well-accepted laboratory data and the results of the HPC method. The effect of water depth on focused waves is also carried out. It is demonstrated that both the HLIGN equations and the HPC method can accurately simulate the surface elevation and particle velocity of focused waves compared with the experiments. The velocity distribution is distinct for the focused waves with the same crest elevations but in different water depths. The HLIGN equations are considered to be an accurate and efficient wave model to investigate focused waves in the intermediate and deep water cases.

Catalogue of rogue waves occurred in the World Ocean from 2011 to 2018 reported by mass media sources

A catalogue of anomalously large waves (rogue or freak waves) occurred in the World Ocean during 2011–2018 reported in mass media sources and scientific literature has been compiled and analyzed. It includes 210 hazardous events caused damages or human losses. The majority of events is based on eyewitness accounts, and as a rule is not confirmed by direct measurements. All collected events divided into deep water cases, shallow water cases and occurrences on the coast (gentle beach or rocks). The following parameters have been determined: date, location, damage, description, reference, and weather conditions. The most dangerous areas in the World Ocean in terms of freak waves are highlighted.

Shallow water effect of tandem flapping foils on renewable energy production

ABSTRACTStudies about the flapping foil hydrokinetics turbines, as a new method to extract energy from incoming flow field, have recently increased significantly. Studies on the effect of shallow water conditions on the performance of the turbine have not been seen so far. This study investigates the effect of working environment on the performance of flapping foil hydrokinetic turbine. The unsteady and incompressible flow around two flapping foils in tandem operate in shallow water is simulated using Computational Fluid Dynamic (CFD) method. The results of shallow water conditions are compared with deep-water case. It is observed that the shallow water condition heavily affects the performance of the system and the total efficiency decreases considerably. Particularly, when the kinematic parameters are optimum for performance, the total power extraction efficiency of the system for h0 = 6c, h0 = 3c and h0 = 1.5c has a reduction of 7.24, 8.5, and 10.14%, respectively, compared with deep-water case. The interaction between the boundary layer of the sea floor and the flapping foils was found to be the main factor of efficiency reduction.

Elastic weak turbulence: From the vibrating plate to the drum.

Weak wave turbulence has been observed on a thin elastic plates since the work by Düring etal. [Phys. Rev. Lett. 97, 025503 (2006)PRLTAO0031-900710.1103/PhysRevLett.97.025503]. Here we report theoretical, experimental, and numerical studies of wave turbulence in a forced thin elastic plate submitted to increasing tension. When increasing the tension (or decreasing the bending stiffness of the plate) the plate evolves progressively from a plate into an elastic membrane as in drums. We first consider a thin plate and increase the tension in experiments and numerical simulations. We observe that the system remains in a state of weak turbulence of weakly dispersive waves. This observation is in contrast with what has been observed in water waves when decreasing the water depth, which also changes the waves from dispersive to weakly dispersive. The weak turbulence observed in the deep water case evolves into a solitonic regime. Here no such transition is observed for the stretched plate. We then apply the weak turbulence theory to the membrane case and show with numerical simulations that indeed the weak turbulence framework remains valid for the membrane and no formation of singular structures (shocks) should be expected in contrast with acoustic wave turbulence.

Renewable energy from the high seas: Geo-spatial modelling of resource potential and legal implications for developing offshore wind projects beyond the national jurisdiction of coastal States

Offshore wind energy projects are currently restricted to the exclusive economic zones of coastal States. Recent advances in technology are raising the prospect of utilising excellent wind resources on the high seas.Using a global geo-spatial model we identify potential resource areas for this. In the shallow water case for bottom fixed foundations the largest locations are found on the Mascarene Plateau in the Indian Ocean and the Grand Banks in the North Atlantic. The deep water case for floating platforms identifies the largest regions on the Grand Banks/Flemish Cap and Rockall Bank/Hatton Ridge, both in the North Atlantic. The overall legal framework for wind energy projects on the high seas is the United Nations Convention in the Law of the Sea. Flag states will play a central regulatory role for high seas wind energy developments. There is the danger that flags of convenience might evolve and unduly undercut environmental and safety standards that are in place for projects on the territorial sea and EEZ. Such abuse of high seas freedom could compromise the UNCLOS principle of ‘due regard’. Marine spatial planning approaches and the establishment of cooperative mechanisms, led by the IMO, could safeguard against such potential misappropriation.

Numerical study on characteristics of dam-break wave

The influence of the downstream water depth on dam-break wave is systematically and comparatively studied by simulating the physical experiments in this study. Commercial CFD software package FLOW-3D is chosen as the simulation tool and is validated firstly. Three experimental cases of different initial downstream water depth, including dry-bed case (r = 0), shallow downstream water case (r = 0.1) and deep downstream water case (r = 0.4) carried out by Ozmen-Cagatay and Kocaman (2010) are then simulated. The generation and propagation process of dam-break wave is divided into two and three stages, respectively, for dry-bed and wet-bed cases. And the characteristics of the dam-break wave including free surface profile, wave front velocity, mark point movement and trajectory, interface between upstream and downstream water, bottom resistance and turbulent kinetic energy are compared and analyzed elaborately in each stage for each case. The differences of the characteristics between different downstream water depth cases are achieved and the reasons of the differences are explored, which present a comprehensive understanding of dam-break wave generation and propagation.

Effects of the interface position of water-air flow on turbulent wall structures

Direct numerical simulations (DNS) are performed to investigate the effect of the interface position of water-air flow on turbulence statistics and flow structures in wall-bounded water-air turbulent flow through a straight channel. Water depths of 90 and 180 viscous wall units, referred to as shallow-water and deep-water cases, respectively, are examined. Water-to-air density and viscosity ratios of 831.7 and 55.56 are considered to model a realistic flow condition at temperature of 25 oC and pressure of 1 atm. The Reynolds number and Froude number are set to 180 and 1.22 x 10-4, respectively, for both shallow-water and deep-water case, based on the friction velocity at the bottom wall, the half depth of the channel, and water density and viscosity. The Navier-Stokes equations are solved using a timesplitting projection method on an octree grid structure, while the deformation of the interface between water and air is computed using a volume-of-fluid method. With the presence of the water-air interface, velocity profiles in deep-water and shallow-water cases are found to slightly deviate from the log-law profile for a single phase turbulent flow in a channel. The deviation is magnified when the interface is placed closer to the log-law region. Turbulent velocity fluctuations in the water stream are found to be associated with quasi-streamwise vortices and hairpin vortices. The quasi-streamwise vortices which are attached close to the wall are found in both deep-water and shallow-water cases. However, the hairpin vortices of which leading portions are lifted away from the wall are found to be diminished in the shallow-water case while they are clearly observed in the deep-water case.

Numerical Study into the Effect of Working Environment on Energy Extraction Performance of Tandem Arranged Flapping Foils

More recently, the oscillating foil hydrokinetics turbines, as a new method to extract energy from incoming flow field, have been studied significantly. However, the focus of previous investigations is directed towered the fundamental parameters which affect the performance of the system. In this work, the impact of working environment on the efficiency of flapping foil hydrokinetic turbine is investigated. The system is placed in proximity the floor sea and the mean distance between the foil pitching axis and the seabed, h 0 , are changed as h 0 =6c, 3c and 1.5c. The results show that the total energy extraction efficiency of shallow water condition has decreased considerably to compare with deep water case because of interaction between the boundary layer of the seabed and the flapping foils. In addition, the upstream foil losses more energy than downstream foil.

Turbulent horizontal convection under spatially periodic forcing: a regime governed by interior inertia

Differential heating applied at a single horizontal boundary forces ‘horizontal convection’, even when there is no net heat flux through the boundary. However, almost all studies of horizontal convection have been limited to a special class of problem in which temperature or heat flux differences were applied in only one direction and over the horizontal length of a box (the Rossby problem; Rossby, Deep-Sea Res., vol. 12, 1965, pp. 9–16). These conditions strongly constrain the flow. Here we report laboratory experiments and direct numerical simulations (DNS) extending the results of Griffiths & Gayen (Phys. Rev. Lett., vol. 115, 2015, 204301) for horizontal convection forced by boundary conditions imposed in a two-dimensional periodic array at a horizontal boundary. The experiments use saline and freshwater fluxes at a permeable base with the imposed boundary salinity having a horizontal length scale one quarter of the width of the box. The flow reaches a state in which the net boundary buoyancy flux vanishes and the bulk of the fluid shows an inertial range of turbulence length scales. A regime transition is seen for increasing water depth, from an array of individual coherent plumes on the forcing scale to convection dominated by emergent larger scales of overturning. The DNS explore the analogous thermally forced case with sinusoidal boundary temperature of wavenumber $n=4$, and are used to examine the Rayleigh number ($Ra$) dependence for shallow- and deep-water cases. For shallow water the flow transitions with increasing $Ra$ from laminar to turbulent boundary layer regimes that are familiar from the Rossby problem and which have normalised heat transport scaling as $Nu\sim Ra^{1/5}$ and $Nu\sim (Ra\,Pr)^{1/5}$, with $Nu$ the Nusselt number and $Pr$ the Prandtl number, in this case maintaining a stable array of coherent turbulent plumes. For deep-water and large $Ra$ the laminar scaling transitions to $Nu\sim (Ra\,Pr)^{1/4}$, with the scales of turbulence extending to the dimensions of the box. The $1/4$ power law regime is explained in terms of the momentum of symmetric, inviscid large scales of motion in the interior coupled to diffusive loss of heat through stabilised parts of the boundary layer. The turbulence production is predominantly by shear instability rather than convection, with viscous dissipation distributed throughout the bulk of the fluid. These conditions are not seen in the highly asymmetric flow in the Rossby problem even at Rayleigh numbers up to six orders of magnitude greater than the transition found here. The new inertial interior regime has the rate of supply of available potential energy, and its removal by mixing of density, increasing as $Ra^{5/4}$, which is faster than $Ra^{6/5}$ in the Rossby problem. Irreversible mixing is confined close to the forcing boundary and is very much larger than the viscous dissipation, which is proportional to $Ra$.

A case study on tandem configured oscillating foils in shallow water

Previous research on the oscillating-foil turbine system has demonstrated its great potential for energy extraction. However, not much is known about the interaction of this device with its working environment. To determine the performance and environmental impact of an oscillating-foil turbine in shallow water, a case study have been conducted which was made of the dual oscillating energy extraction foils system with a tandem configuration which operates at two different water depths: i.e., D = 5c and D = 10c. The performance and the environmental effects of the device were compared between shallow-water and deep-water cases. The results show a 10% efficiency loss in the D = 5c case compared with that of the deep water case, because of the interaction between the oscillating-foils and the seabed. It is also observed that the foil vortices dissipation rate of the D = 5c case is 13% less than that of the deep-water case due to the free-surface effect. The water level also rises 23% around the oscillating-foils location of the D = 5c case because of the blockage effect of the device.

Analysis of the linear version of a highly dispersive potential water wave model using a spectral approach in the vertical

The properties and accuracy of the linearized version of the fully dispersive and nonlinear wave model developed in Yates and Benoit (2015) and Raoult et al. (2016) are analyzed for both flat and variable bottom bathymetries. This model considers only a single layer of fluid and uses a basis of orthogonal Chebyshev polynomials to project the vertical structure of the potential. This approach results in an exponential convergence rate with the maximum degree of the Chebyshev polynomial, denoted NT, while only first- and second-order derivatives in space need to be evaluated.For the constant water depth case, the linear dispersion relation of the model is derived analytically, and expressions are established for NT ranging from 2 to 15. The analysis shows a rapid increase in accuracy in the deep water range with increasing NT. For instance, the relative error in the calculated wave celerity (in comparison with Stokes’ analytical solution) remains smaller than 2.5% for deep water cases with kh up to 100 using NT≥9 (k and h are the representative wavenumber and water depth, respectively). The wave kinematics, vertical profiles of the horizontal and vertical orbital velocities, converge to the Stokes profiles for kh up to 60 when using a sufficiently high value of NT. The vertically-averaged relative errors of the horizontal and vertical velocities remain below 6% and 3%, respectively, for kh up to 60 when using NT≥11. The presented model shows better dispersive properties in deep water than several high-order Boussinesq-type models.For variable bottom bathymetries, the shoaling properties of the model are studied numerically, exhibiting good agreement with results from Stokes linear theory in the case of mild bottom slopes, using a sufficiently high value of NT with respect to the offshore relative water depth. For an offshore water depth of kh=10 (i.e. more than 3 times the deep water limit), accurate wave heights in shallow water (kh=0.25) are obtained with NT=6 (or higher). Finally, the linear version of the model is validated with comparisons to analytical solutions of the reflection and transmission coefficients of regular waves over Roseau-type bathymetric profiles. Two bottom profiles are considered, including one with a steep slope, whose maximum value reaches about 1:0.7 (i.e. an angle of about 54.9 deg.). Using NT=7, small differences (<0.4%) with the analytical solution are observed for the four considered cases, confirming the ability of the linear model to represent accurately the effects of steep bottom gradients on wave propagation dynamics.