Water Of Finite Depth Research Articles

Numerical Method for Calculating Fourier Coefficients and Properties of Water Waves with Shear Current and Vorticity in Finite Depth

Many numerical methods have been developed since 1961, but unresolved issues remain. This study developed a numerical method to address these issues and determine the coefficients and properties of rotational waves with a shear current in a finite water depth. The number of unknown constants was reduced significantly by introducing a wavelength-independent coordinate system. The reference depth was calculated independently using the shooting method. Therefore, there was no need for partial derivatives with respect to the wavelength and the reference depth, which simplified the numerical formulation. This method had less than half of the unknown constants of the other method because Newton's method only determines the coefficients. The breaking limit was calculated for verification, and the result agreed with the Miche formula. The water particle velocities were calculated, and the results were consistent with the experimental data. Dispersion relations were calculated, and the results are consistent with other numerical findings. The convergence of this method was examined. Although the required series order was reduced significantly, the total error was smaller, with a faster convergence speed.

Research on the methods for separating wind sea and swell from directional wave spectra in finite-depth waters

Research on the methods for separating wind sea and swell from directional wave spectra in finite-depth waters

Hydroelastic Response of a Moored Floating Flexible Circular Structure Applying BIEM

A hydroelastic model associated with the interaction between a surface wave and a floating circular structure connected with mooring lines in finite water depth is developed using BIEM. The BIEM solution is achieved using free surface Green’s function and Green’s theorem. Furthermore, the algebraic equations for circular structural displacement are derived from the integro-differential equation. The correctness of the BIEM code is verified with the results of shear force and deflection amplitude existing in the literature, and the hydroelastic response of the circular structure is analyzed. The comparison results show a good level of agreement between the present results and those from other calculations. It is observed that the shear force, bending moment, and deflection decrease for higher values of stiffness of the mooring lines. The current study may be supportive of the visualization of the effect of mooring stiffness and to generalize articulated circular structure models for ocean space utilization.

Wave Radiation by a Floating Body in Water of Finite Depth Using an Exact DtN Boundary Condition

Wave Radiation by a Floating Body in Water of Finite Depth Using an Exact DtN Boundary Condition

Crest-height statistics in finite water depth. Part 1: The role of the nonlinear interactions in uni-directional seas

This paper explores the competing nonlinear processes that define the largest crest heights in uni-directional random seas. In deep water, the third-order near-resonant interactions produce a focusing of the free-wave energy and hence larger crest elevations. However, as the effective water depth reduces, theoretical considerations, based upon the assumption that the frequency spectrum is narrow-banded, suggest that this process weakens and below kpd=1.363 (kp being the wavenumber of the spectral peak frequency and d the water depth) energy defocusing occurs. This paper first explores how the near-resonant interactions affect the crest heights arising in broad-banded, non-breaking, uni-directional seas in a wide range of effective water depths. It also quantifies the role of the bound-wave interactions. The numerical calculations conclude that kpd=1.363 indeed defines the boundary between energy focusing and defocusing for realistic jonswap sea-states, irrespective of the spectral bandwidth and steepness. However, for kpd<1.363, the bound-wave contributions increase the largest crest heights, while the near-resonant interactions reduce them. The tail of the crest-height distributions is therefore defined by two competing nonlinear processes. The present results have important implications for both the interpretation of laboratory data describing crest-height distributions and the appropriateness of second-order models for practical engineering calculations.

Generalized hydrodynamic coefficients of twin connected circular cylinders in finite water depth

In this work the generalized hydrodynamic coefficients of connected twin circular cylinders are studied. Firstly, a matched boundary element method with eigen-function expansion is applied to study the generalized hydrodynamic coefficients of floating and submerged tunnels with twin horizontal cylinders in finite water depth. By numerical examination it is found that gap resonance problem can be neglected for the submerged floating tunnels with a certain submergence depth, and the axial wave number play an important role for the added mass. Then, analytic solutions are derived for the high frequency approximations of the generalized added masses of twin connected cylinders by the eigenfunction expansion in the polar coordinate system with aid of their images about the x-axis in the infinite fluid domain. The high frequency approximations of generalized added masses decrease with increasing vibration wave number. When T/a and B/a are larger than 3 (where T is the cylinder submergence, B is the spacing distance between the twin cylinders and a is the radius of the cross-section of the cylinder), the hydrodynamic interference between the two cylinders can be neglected and the generalized added mass coefficients of connected twin cylinders can be derived from these of single isolated cylinders.

Comment on “An optimal system, invariant solutions, conservation laws, and complete classification of Lie group symmetries for a generalized (2+1)-dimensional Davey–Stewartson system of equations for the wave propagation in water of finite depth”

Comment on “An optimal system, invariant solutions, conservation laws, and complete classification of Lie group symmetries for a generalized (2+1)-dimensional Davey–Stewartson system of equations for the wave propagation in water of finite depth”

Investigation of acoustic radiation from a sphere vibrating on the free surface of a finite depth water using a boundary element method

In this study, a boundary element method (BEM) is applied to investigate acoustic radiation from a sphere vibrating in pulsating mode on the free surface of finite or infinite depth water. Effect of the free surface is introduced by employing a half-space Green’s function. A modified version of the Helmholtz integral equation (HIE) is used to calculate acoustic radiation from the sphere vibrating in pulsating mode on the free surface. Free-terms of the HIE are calculated using two different forms of integrals and “dummy” boundary elements. Moreover, to simulate finite depth fluid medium, a chain image-source method is used to derive a waveguide Green’s function. To demonstrate applicability of the method presented, calculated acoustic pressures are compared with those by finite element method (FEM) and analytical calculations. Additionally, the effects of submergence depth and vibration frequency on acoustic radiation are investigated for infinitely deep water together with those of water depth and field point distance on acoustic radiation for finite water depth medium. The calculations show that there is a good agreement between BEM, FEM and analytical solutions. Also, it is observed that field point distance significantly affects the convergence behavior of waveguide Green’s function. Furthermore, it is noted that submergence depth, domain depth and vibration frequency have pronounce influence on radiated pressure amplitude and pressure field pattern.

Motion of a submerged body under a free surface and an ice cover in finite water depth conditions

In this study, we consider the motion of a submerged body near the free surface of a liquid and a liquid covered with an ice cover in finite water depth conditions. The influence of water depth on the nature of motion of a submerged body has been experimentally and theoretically analysed. The experiments were performed in an ice tank. The use of a cable towing system allowed the researchers to measure the magnitude of the relative vertical displacement of the submerged body resulting from the action of the lift force. The influence of ice cover on the relationship between the vertical displacement, trim angle, criterion for ice breaking, and speed of movement of the submerged body model was established. For theoretical research, the ice cover was modelled by a viscoelastic floating plate, and the submarine was modelled by a source–sinks system in a fluid flow. The influence of water depth, the presence of an ice cover, changes in the submergence depth, and the speed of movement of the submerged body on the wave resistance, lift force, and trimming moment were analysed. Theoretical results are supported by experimental data.

Zero-frequency added mass of submerged bodies with near-surface effects

The added mass of ships, submarines, and marine platforms is a primary quantity of interest to understand vessel dynamics and motion. For low-frequency maneuvers at low forward speed, the water surface acts as a wall and the added-mass depends on the proximity to the free-surface. In this paper a method for efficiently determining the full six-degree-of-freedom zero-speed added-mass tensor is presented for bodies that are on or near a wall boundary. The method is based on solving the unsteady Euler equations from rest for one time step to find the generalized added-mass force. The method is based on the work of el Moctar (2022) but in this paper it is simplified to use a static grid, and thereby extended for general six degree-of-freedom motion. The method is validated for a series of canonical problems with analytical solutions that include a circular cylinder that is both deeply submerged and near a wall, and a spheroid that extends from a wall. The method is further validated by computing the added mass for a very-large-crude carrier that is in finite-depth water. Finally the method is used to compute the full six degree-of-freedom added-mass tensor for the Joubert BB2 Submarine for deep submergence and near the water surface.

Global Bifurcation and Highest Waves on Water of Finite Depth

We consider the two-dimensional problem for steady water waves with vorticity on water of finite depth. While neglecting the effects of surface tension we construct connected families of large amplitude periodic waves approaching a limiting wave, which is either a solitary wave, the highest solitary wave, the highest Stokes wave or a Stokes wave with a breaking profile. In particular, when the vorticity is nonnegative we prove the existence of highest Stokes waves with an included angle of 120^circ . In contrast to previous studies, we fix the Bernoulli constant and consider the wavelength as a bifurcation parameter, which guarantees that the limiting wave has a finite depth. In fact, this is the first rigorous proof of the existence of extreme Stokes waves with vorticity on water of finite depth. Aside from the existence of highest waves, we provide a new result about the regularity of Stokes waves of arbitrary amplitude (including extreme waves). Furthermore, we prove several new facts about steady waves, such as a lower bound for the wavelength of Stokes waves, while also eliminating a possibility of the wave breaking for waves with non-negative vorticity.

A new probability density function for the surface elevation in irregular seas

To date, the predominant means for computing the probability density function (p.d.f.) for the free surface elevation of a nonlinear, irregular water wave field, free of assumptions involving narrow-bandedness and small directionality, is the approximate Gram–Charlier series solution of Longuet-Higgins (J. Fluid Mech., vol. 17, 1963, pp. 459–480, hereafter LH63). In this paper we re-visit the derivation of this p.d.f. to second order in the wave steepness, utilizing both moment and cumulant generating functions. We show that LH63's approximate solution based on the cumulant generating function, in fact, matches that derived from the moment generating function. Moreover, through a change of variables coupled with complex analysis, it is shown that the approximation employed by LH63 is unnecessary, and the second-order p.d.f. stemming from the cumulant generating function can be represented exactly in terms of the Airy function. The new second-order p.d.f. predicts increased probability of extreme positive surface elevations typical of e.g. rogue waves, relative to both second- and third-order solutions of LH63. This heavy positive tail is inherent, and is explained through comparison of the asymptotic limits of the p.d.f.s for large surface elevations. A semi-theoretical method is also proposed for remedying non-physical spurious oscillations that arise in the negative tail, based on the envelope of the Airy function with negative arguments. This modified negative tail is valid for irregular wave fields having skewness less than or equal to 0.2. The new p.d.f.s are compared against those based on data sets generated from second-order irregular wave theory as well as a fully nonlinear, spectrally accurate numerical wave model. Good accuracy is collectively demonstrated for directionally spread irregular seas in both finite and infinite water depths for a range of directional spreading.

Modulation effect of linear shear flow, wind, and dissipation on freak wave generation in finite water depth

This paper studies the modulation effect of linear shear flow (LSF), comprising a uniform flow and a shear flow with constant vorticity, combined with wind and dissipation on freak wave generation in water of finite depth. A nonlinear Schrödinger equation (NLSE) modified by LSF, strong wind, and dissipation is derived. This can be reduced to consider the effects of LSF, light wind, and dissipation, and further reduced to include only LSF. The relation between modulational instability (MI) of the NLSE and freak waves represented as a modified Peregrine Breather solution is analyzed. When considering only LSF, the convergence (divergence) effect of uniform up-flow (down-flow) and positive (negative) vorticity increases (decreases) the MI growth rate and promotes (inhibits) freak wave generation. The combined effect of LSF and light wind shows that a light adverse (tail) wind can restrain (amplify) MI and bury (trigger) freak waves. Under the effect of a light tailwind, LSF has the same effect on the MI growth rate and freak wave generation as the case without any wind. The combination of LSF and strong wind enables both adverse and tail winds to amplify MI and trigger freak waves. In the presence of strong wind, LSF has the opposite effect to the case of a light tailwind.

Flexural-gravity wave scattering due to a crack in a floating ice sheet in the perspective of wave blocking in a three-layer fluid

Flexural-gravity wave blocking is demonstrated in a three-layer fluid having a floating ice-covered surface in the water of finite depth. The study reveals that wave blocking occurs for higher values of compressive force in either of the modes, which depends on the density ratio and the wave frequency. Two (three) roots of the dispersion relation coalesce at the blocking (saddle) frequencies. Further, within the blocking frequencies, the dispersion relation possesses five propagating wave modes, of which three are affiliated with the layer where blocking occurs, whilst the other two modes are connected with the wave motion propagating in the respective layers. Dead water analogue is established due to the propagation of internal waves along the interface when the thickness of an individual layer is either thin or the density of two consecutive layers is close to each other. Subsequently, the role of blocking dynamics on the flexural-gravity wave scattering by a linear crack in an ice sheet floating in a three-layer fluid is studied. The canonical energy balance relation is generalized to account for multiple propagating wave modes. The velocity potentials are expanded in terms of a scattering matrix to account for the multiple incidents and scattered wave modes within the blocking frequencies. The study depicts the occurrence of removable/jump discontinuities in the scattering coefficients, both at the points where blocking/mode conversion takes place. Irregularities in the ice sheet's deflection and interface elevations are due to the superposition of multiple propagating wave modes within the blocking frequencies.

Nonlinear analytical solution for radiation stress of higher-order Stokes waves on a flat bottom

The radiation stress of water surface waves plays an important role in many aspects of marine hydrodynamics. The nonlinear effects of highly nonlinear waves on the radiation stress were investigated on a flat bottom under finite water depth and deep water conditions. Analytical solutions for radiation stress were derived based on different order nonlinear Stokes wave theory, respectively. The results indicate that the nonlinear radiation stresses are significantly greater than those based on the linear wave, and they increase as the nonlinearity strengthens. Through an inter-comparison of the expressions of nonlinear radiation stress of various orders of Stokes waves, it turned out that the results based on the second-order Stokes wave overestimated the radiation stress, while the expressions based on Fenton's fifth-order Stokes wave were more reasonable in reflecting the nonlinear effect of waves. In the case of extreme wave steepness, the fifth-order Stokes wave radiation stresses Sxx5 was approximately 17% larger than that of the linear wave. Finally, using the expressions of the fifth-order Stokes wave radiation stress, bound infragravity waves excited by the bichromatic wave groups formed by the superposition of two fifth-order Stokes waves were obtained.

Dynamic responses of a transversely isotropic poroelastic seabed subjected to nonlinear random waves with emphasis on liquefaction depth

Seabed instability is one of the important causes that lead to the failures of marine structures. Most previous works on seabed instability are limited to regular surface waves and linear random waves while nonlinear random waves are seldom touched. Hence, in this study, the hydrodynamic pressure of nonlinear random waves up to the second order is derived first. Then the analytical solutions of the dynamic responses of a transversely isotropic poroelastic seabed under the nonlinear hydrodynamic pressure are developed. The solutions are validated against analytical and experimental works in literature. The frequency response functions of pore water pressure, shear stress, vertical effective stress and horizontal effective stress are presented and discussed in the frequency domain. The influences of significant wave height, peak wave period and water depth on the dynamic responses and liquefaction depths of soil are analyzed in the time domain. The nonlinear random waves, linear random waves, linear regular waves and the second-order Stokes waves are compared in both time and frequency domains in terms of their influences on the dynamic responses and liquefaction depths of soil. The results reveal that the dynamic responses induced by second-order random waves are always greater than that induced by linear random waves, and much greater than that induced by equivalent linear waves and the second-order Stokes waves. The seabed may not liquefy under regular waves but does liquefy under random waves. Furthermore, the instantaneous liquefaction depths of soil caused by the second-order random waves are up to 23.5% larger than that induced by linear random waves. Therefore, the nonlinear random waves are recommended for the finite water depth use in engineering practice. In addition, the frequency-difference term plays an important role in soil liquefaction.

Flexural gravity wave interaction with an articulated heterogeneous plate within the paradigm of blocking dynamics

Surface gravity waves interact with the flexural waves to generate the flexural gravity waves whose characteristics are triggered for higher values of lateral compressive stress to generate multiple propagating wave modes. This investigation examines the scattering of obliquely incident flexural gravity waves due to articulation in two semi-infinite heterogeneous floating elastic plates in finite water depth within a blocking dynamics regime. The dispersion curve demonstrates the existence of three propagating wave modes within the blocking limits. The canonical eigenfunction expansion method used for a single propagating mode is generalized to account for multiple propagating wave modes within the limits of blocking periods. The energy relation is established using the conservation of wave energy flux and Snell's law of refraction, which depends upon the angles and amplitude of the scattered waves along with the wave energy transfer rates. The amplitude of scattering coefficients (energy transfer rate) goes beyond the unit, where the corresponding energy transfer rate (scattering coefficients) diminishes for specific wave periods. Subsequently, complete wave reflection occurs for oblique waves beyond a critical angle of incidence for a fixed period and prior to a critical angle of incidence at a higher angle of incidence. Removable discontinuities occur at the blocking and saddle points, while a jump discontinuity appears due to a change in the incident wave mode in the scattering coefficients. Surface plots reveal the irregular pattern of plate deflection for the period within the blocking limits. Linear time-dependent plate displacement is simulated in two and three dimensions.

Flexural-gravity wave interaction with multiple vertical cylinders of arbitrary cross section frozen in an ice sheet

The interaction problem of flexural-gravity wave with multiple vertical cylinders frozen in an ice sheet on the surface of water with finite water depth is considered. The linearized velocity potential theory is adopted for fluid flow, and the thin elastic plate model is applied for ice sheet deflection. Each cylinder is bottom-mounted, and the shape of its cross section can be arbitrary while remaining constant in the vertical direction. The velocity potential is expanded into an eigenfunction series in the vertical direction, which satisfies the boundary condition on the ice sheet automatically. The horizontal modes, which satisfy the Helmholtz equations, are then transformed into a series of boundary integral equations along the ice sheet edges or the intersection of the ice sheet with the cylinders. The problem is then solved numerically by imposing the ice sheet edge condition together with the impermeable condition on the cylinders. The solution is exact in the sense that the error is only due to numerical discretization and truncation. Computations are first carried out for single and multiple vertical circular cylinders, and good agreements are obtained with the semi-analytical solution. To resolve the difficulty of excessive computation at a large number of cylinders, the effect of the evanescent wave of a cylinder on those at large distance is ignored. This allows for the case of a large number of cylinders in different arrangements to be simulated. Extensive results are provided. Their physics and practical relevance are discussed.

Mean flow modeling in high-order nonlinear Schrödinger equations

The evaluation and consideration of the mean flow in wave evolution equations are necessary for the accurate prediction of fluid particle trajectories under wave groups, with relevant implications in several domains, from the transport of pollutants in the ocean to the estimation of energy and momentum exchanges between the waves at small scales and the ocean circulation at large scale. We derive an expression of the mean flow at a finite water depth, which, in contrast to other approximations in the literature, accurately accords with the deep-water limit at third order in steepness and is equivalent to second-order formulations in intermediate water. We also provide envelope evolution equations at fourth order in steepness for the propagation of unidirectional wave groups either in time or space that include the respective mean flow term. The latter, in particular, is required for accurately modeling experiments in water wave flumes in arbitrary depths.

Wind-generated waves on a water layer of finite depth

In this paper, we study the linear stability of a two-dimensional shear flow of an air layer overriding a water layer of finite depth. The air layer is considered to be of an infinite extent with an exponential velocity profile. Three different background conditions are considered in the finite-depth water layer: a quiescent background, a linear velocity profile and a quadratic velocity profile. It is known that the cases of the quiescent water layer and the linear velocity profile allow for analytical treatment. We further provide an analytical solution for the case of the quadratic velocity field: we specifically consider a flow-reversal profile, although the result could be generalized to other quadratic profiles as well. The role of water layer depth on the growth rate of the Miles and rippling instabilities is studied in each of the three cases. Using asymptotic analysis, with the air–water density ratio being a small parameter, we obtain an analytical expression for the growth rate of the Miles mode and discuss the condition for the existence of a long-wave cutoff for these profiles. We provide analytical expressions for the stability boundary in the parameter space of inverse squared Froude number and wavenumber. In scenarios where a long-wave cutoff does not exist, we have carried out a long-wave asymptotic study to obtain the growth rate behaviour in that regime.