Water Of Infinite Depth Research Articles

Effects of nonlinearity on the crest shape of extreme irregular sea waves: Nonlinear harmonic separation and analysis

This work highlights the effects of nonlinearity on the crest shape of extreme directional water wave events in infinite and finite water depths. HOS-ocean, a High-Order Spectral code developed for solving the deterministic propagation of nonlinear wavefields in open ocean [Ducrozet et al., “HOS-ocean: Open-source solver for nonlinear waves in open ocean based on High-Order Spectral method,” Comput. Phys. Commun. 203, 245–254 (2016)], is used for comparisons with linear simulations. Nonlinear wave harmonics were extracted combining and extending previous methods [Fitzgerald et al., “Phase manipulation and the harmonic components of ringing forces on a surface-piercing column,” Proc. R. Soc. A 470, 20130847 (2014) and Hann et al., “A new set of focused wave linear combinations to extract non-linear wave harmonics,” in Proceedings of 29th International Workshop on Water Waves and Floating Bodies, 2014], showing the contribution of various orders of nonlinearity. The work shows that “walls of water,” [Adcock et al., “Nonlinear dynamics of wave-groups in random seas: Unexpected walls of water in the open ocean,” Proc. R. Soc. A 471, 20150660 (2015) and Adcock et al., “On the shape of large wave-groups on deep water—The influence of bandwidth and spreading,” Phys. Fluids 28, 106601 (2016)], a significant increase in the crest width, are also formulated in finite water depths. Mainly responsible for this is the first order harmonic, with significant energy transfers away from components propagating at an angle, which are more pronounced in finite water depths. As nonlinearity increases, higher order harmonics (>second) become, as expected, more significant. In deep water, rapid energy transfers lead to a broadening of the spectrum which is reflected in the increase in the first order amplitude sum and hence increased crest elevations at the point of the large event. However, in finite water depths, energy transfers are gradually leading to a narrowing of the spectrum around the spectral peak and to a general broadening of the spectrum at the higher harmonics; reflected in the significant decrease in the first order amplitude sum and hence a decrease in crest elevations at the point of the large event, also aided by the larger increase in the out-of-phase difference terms (zeroth harmonic).

Statistical distribution of surface elevation by using quadratic forms of normal random variables

This paper is concerned with the statistical properties of surface elevation, which is crucial for the design and operation of marine and coastal structures and is helpful to the prediction of rogue waves. A semi-analytic quadratic model is proposed to describe the statistical distribution of surface elevation by using the theory of quadratic forms of normal random variables, in which the characteristic function and cumulants are given in analytical form, and the probability density is obtained by the numerical inversion of the characteristic function using the fast Fourier transform algorithm. Compared with Monte Carlo simulations, the quadratic model presents excellent performance in describing the probability density function of surface elevation, especially at the tails, for varying degrees of steepness, bandwidth, and directional spreading in both finite and infinite water depth. The effect of these parameters that characterize sea states on the statistical properties of surface elevation is also investigated. Through a comparative study, the quadratic model is superior to previously existing theoretical models.

Expansion formulae for flexural-gravity wave propagation during wave blocking

Flexural-gravity wave blocking occurs in the presence of lateral compressive force. The existing expansion formulae for the velocity potential were limited in the case of distinct real roots of the dispersion relation with the multiplicity one. This limitation is overcome, and we obtain the complete solution for the flexural-gravity wave-maker problem at blocking and saddle points by deriving the expansion formula for velocity potential using the Fourier transform corresponding to the roots of the dispersion relation with multiplicity two and three, respectively. Further, we check the convergence of the newly derived expansion formulae by using the spectral representation of the associated eigenfunction. It has been proved that the eigenfunctions related to the expansion formulae are linearly dependent and satisfy the orthogonal mode-coupling relation. Next, we solve a model problem of flexural-gravity wave scattering by a crack using the derived expansion formulae and establish the energy balance relation by applying Green’s integral theorem at blocking points for infinite water depth. In addition, we have presented the validation of energy balance relation involving the generalized scattering coefficients.

Stability of Hydroelastic Waves in Deep Water

Two-dimensional periodic travelling hydroelastic waves on water of infinite depth are investigated. A bifurcation branch is tracked that delineates a family of such solutions connecting small amplitude periodic waves to the large amplitude static state for which the wave is at rest and there is no fluid motion. The stability of these periodic waves is then examined using a surface-variable formulation in which a linearised eigenproblem is stated on the basis of Floquet theory and solved numerically. The eigenspectrum is discussed encompassing both superharmonic and subharmonic perturbations. In the former case, the onset of instability via a Tanaka-type collision of eigenvalues at zero is identified. The structure of the eigenvalue spectrum is elucidated as the travelling-wave branch is followed revealing a highly intricate structure.

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.

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.

ANALYTIC SOLUTION FOR THE VELOCITY FIELD AROUND SUBMERGED PERMEABLE BREAKWATERS

In this paper, an analytic solution to the scattering of water wave by submerged permeable breakwaters in infinite-depth water is presented. The analytic solution of the velocity field around the permeable breakwater under the linear wave train in deep water condition is sought by formulating a nonhomogeneous Riemann-Hilbert problem. To deal with the nonlinearity due to the permeability of the plates, a perturbation method is applied in terms of a small parameter representing permeability. We present the leading order solution (impermeable breakwaters) and the first order solution (permeable breakwaters) of the spatial velocity field. The validity of the leading order solution was verified through comparison with the solutions suggested by Evans (1970). Then, the impermeable and permeable breakwater’s reflection and transmission coefficients are compared in order to investigate the effects of permeability. The analytical solution presented in this research can be used for designing breakwaters, such as calculating the force applied to the breakwaters, and for validation of numerical simulations or experimental results.

Airy-type bichromatic wave pulses on the sea surface and generation of rogue waves

We present a theoretical study of bichromatic weakly non-collinear Airy-type waves propagating with various amplitudes in water of infinite depth. The coupled nonlinear Schrödinger equations are invoked to study the behavior of self- and cross-interacting Airy wave packets. We demonstrate that bichromatic pulses possess the main properties of single Airy wave packets—shape invariance and self-acceleration or self-deceleration—when propagating in a dispersive medium. Accounting for nonlinearity leads to a strong dependence of the structure, stability, and propagation velocity of wave pulses on their amplitude. We show that interacting pulses of Airy-type waves can form giant accelerating and decelerating waves.

Hydrodynamic Tests of Innovative Tourist Submarine

This paper deals with the resistance, towing, seakeeping, and open water propeller tests of an innovative tourist submarine model. Tests were performed in a 276 m long towing tank. As the submarine model is a complex structure composed of various parts attached to the pressure hull, the largest possible model, in the scale of 1:5.0, was produced, considering the towing tank depth and the capabilities of the measurement equipment. Resistance tests were performed in deep water and on the surface in calm water. The tested speed range in both cases was up to 5.5 knots. To ensure the avoidance of free surface effects, resistance tests in deep water were performed for different draughts and then extrapolated to infinite water depth. Smaller effective powers were found for the surface condition. The results are compared to an independently performed computational fluid dynamics (CFD) analysis using OpenFOAM. A fair agreement between the experimentally and numerically predicted effective power is found, while the reasons for the differences found are explained. The free submarine model was towed with a rope performed for the speed range 1.7 kn–3.5 kn, and the towing force in the rope was measured. Seakeeping tests in irregular beam waves at zero speed were performed to check the flooding risk on open hatches. Open water tests of the main thrusters for propelling the submarine were conducted, indicating that both power demand and propeller thrust are slightly larger compared to the initial estimates.

On Survey of the Some Wave Solutions of the Non-Linear Schrödinger Equation (NLSE) in Infinite Water Depth

In this work, we use two different analytic schemes which are the Sine-Gordon expansion technique and the modified exp -expansion function technique to construct novel exact solutions of the non-linear Schrödinger equation, describing gravity waves in infinite deep water, in the sense of conformable derivative. After getting various travelling wave solutions, we plot 3D, 2D and contour surfaces to present behaviours obtained exact solutions.

Hidrodinamička sila otpora trupa turističke podmornice s različitim krajevima uz uporabu OpenFOAM-a

Power reduction is the central goal to maximize cruising duration of tourist underwater vehicles (UV) that can be achieved by shaping the hull. So, in this paper, hydrodynamic force of resistance of the tourist UV’s bare hull is analysed. A numerical model based on Computational Fluid Dynamics in OpenFOAM is developed to simulate the longitudinal movement of an UV in a viscous and incompressible fluid for the infinite water depth. Three head geometries, including both spherical heads (S-S), spherical bow and elliptical stern head (S-E), and UV with both elliptical heads (E-E) are compared. At the first step, the effects of the length-to-diameter ratio and forward speed is studied for the S-S UV. The mesh size is calibrated using Grid Convergence Index, provided by ASME, while the model validation is based on the results for cube and sphere as well as by comparison with resistance coefficient of a SUBOFF bare hull. S-E and E-E UVs are then analysed for typical length-to-diameter ratio, comparing their force of resistance to the S-S type. The elongated elliptical heads are in many cases found favourable compared to the spherical heads. The results of this study may be useful for the conceptual design of tourist UV and for verification of the complex numerical models that are necessary to account for the influence of appendages on the force of resistance of such innovative UV.

Analytical solution to wave diffraction by a hemisphere in infinite water depth using the multipole expansion method

Linear water wave diffraction by a hemisphere in infinite water depth is investigated using the multipole expansion method in this paper. In the spherical coordinate system, the diffraction potential is expressed in the form of multipole wave potentials and the wave-free potential, as derived by Hulme (1982) for radiation potentials. The issues associated with the high-order multipole expansion are solved in this paper. The new method can calculate the multipole wave potentials at arbitrary orders and thus successfully calculate the diffraction potentials on the body surface and free surface around a hemisphere. The wave exciting forces acting on hemispheres are calculated using the near-field and far-field methods, which utilizes the radiation potential computed by Hulme's (1982) method to verify the accuracy of the analytical solution in this paper. Compared with the far-field method, the analytical solution in this paper converges more quickly as the truncation term N increases, especially when the wavenumber Ka is large. In this paper, the theory of applying the multipole expansion method to solve hydrodynamic problems with a hemisphere on the free surface is completed, which makes the multipole expansion method no longer limited to low-order expansion calculations and can be applied to more fields.

Scattering of linear gravity-capillary waves by a completely submerged vertical porous elastic plate

The problem of water wave scattering by a vertical porous elastic plate completely submerged in water of infinite depth in the presence of surface tension at the free surface is investigated. Using the concepts of linear water wave theory and Green's Integral theorem, a second kind Fredholm integral equation is obtained which is hypersingular in nature. The obtained hypersingular integral equation is solved numerically. A new mathematical form of the energy identity has been derived which involves an energy loss term that occurs due to the combined effects of surface tension and porous nature of the plate. Numerical estimates for physical quantities like reflection coefficient, energy loss coefficient, hydrodynamic force, and plate deflection are evaluated and then depicted graphically for a certain set of parametric values. The present methodology is validated using the energy identity and the results present in the existing literature. The effect of surface tension on the propagation of gravity waves is analyzed with the help of the various hydrodynamic quantities, which portrays that it is not always insignificant.

On the steady-state exactly resonant, nearly resonant, and non-resonant waves and their relationships

The steady-state exactly resonant, nearly resonant, and non-resonant waves in infinite water depth are investigated, and their relationships are revealed. In the framework of homotopy analysis method (HAM), the two primary wave components' amplitudes in the initial guess of the velocity potential are fixed and the actual frequencies of the primary waves are unknown. For different wavenumber ratio (k2/k1) values, three groups of steady-state wave systems are obtained with the proper auxiliary linear operator and the initial guess. It is found that when the third-order resonance occurs accurately, the energy of each wave group is mainly concentrated in the primary and third-order resonant wave components. When the value of the wavenumber ratio (k2/k1) moves away from the exact resonance, the energy of the whole wave system is either gradually transferred to the two primary or one resonant wave components that finally evolves into the trivial non-resonant wave system, or the energy is more evenly distributed among more wave components that evolves into multiple nearly resonant wave systems. In addition, the results obtained based on HAM are verified and confirmed by means of the Zakharov equation. This work illustrate that the steady-state wave systems are continuous in wavevector space, the normal non-resonant solution on either side of the resonance point comes from the different third-order resonant solutions, and the occurrence of multiple near resonances can significantly increase the nonlinearity of the wave system.

3D hydroelastic modelling of fluid–structure interactions of porous flexible structures

A 3D hydroelastic model subjected to linear wave interaction with horizontal flexible floating and submerged porous structures is developed. The dispersion relation of the aforesaid problem is derived based on Green’s functions approach in water of finite and infinite depths. Three-dimensional Fourier-type expansion formulae for the velocity potentials associated with orthogonal mode-coupling relations are derived in a channel of finite and infinite water depths of finite width. A real physical problem of surface gravity wave interaction with a moored floating and submerged porous plate in finite water depth is analysed by the expansion formula to validate the usefulness of the current method. The numerical accuracy of the present numerical results is checked via convergence test. Further, the effect of different design parameters on the hydroelastic response of floating and a submerged porous plate is studied by analysing the deflection amplitude and structural deformation. It is observed that the present analysis could be useful for a greater insight to design a multi-use flexible floating system.

Fully Localised Three-Dimensional Gravity-Capillary Solitary Waves on Water of Infinite Depth

Fully localised three-dimensional solitary waves are steady water waves which are evanescent in every horizontal direction. Existence theories for fully localised three-dimensional solitary waves on water of finite depth have recently been published, and in this paper we establish their existence on deep water. The governing equations are reduced to a perturbation of the two-dimensional nonlinear Schrödinger equation, which admits a family of localised solutions. Two of these solutions are symmetric in both horizontal directions and an application of a suitable variant of the implicit-function theorem shows that they persist under perturbations.

Three-dimensional waves under ice computed with novel preconditioning methods

In this work, we present three-dimensional, nonlinear traveling wave solutions for water waves under a sheet of ice, i.e., flexural-gravity waves. The ice is modeled as a thin elastic plate on top of water of infinite depth and the equations are formulated as a boundary integral method. Depending on the velocity of the moving disturbance generating the flow, different deflection patterns of the floating ice sheet are observed. In order to compute solutions as efficiently as possible, we introduce a novel hybrid preconditioning technique used in an iterative Newton-Krylov solver. This technique is able to significantly increase the grid refinement and decrease the computational time of our solutions in comparison to methods that are presently used in the literature. We show how this approach is generalizable to three-dimensional ice wave patterns in different velocity regimes.

Natural frequencies and modal shapes of three-dimensional moonpool with recess in infinite-depth and finite-depth waters

In this study new extensions of the theoretical models developed by Molin (2001) and Molin et al. (2018) are proposed to solve the resonance problem for three-dimensional circular and rectangular moonpools with recesses. Eigenfunction expansions are adopted to describe the velocity potentials in all the subdomains. Then, the velocity potentials and normal velocities are matched at the common boundaries such that the eigenvalue problems are formulated and solved, yielding the natural frequencies and associated modal shapes of the free surface. Both the models for infinite and finite water depths are derived for circular moonpools with recesses. Applications are also made for the three-dimensional rectangular moonpools with recesses. In addition, frozen-mode approximation is derived, which yields simple formulas for prediction of the natural frequencies for piston-mode resonances. The proposed models in this study are validated by comparing the obtained results with the experimental data and the results using other numerical models. Moreover, for the first sloshing mode in rectangular moonpool, simple approximation model is proposed based on the assumption that the half of the wavelength of a standing wave is the same as the moonpool length. The validity of the model is examined by comparing the solutions with the results by the diffraction–radiation code WAMIT.

Convergence of eigenfunction expansions for flexural gravity waves in infinite water depth

In the present paper, point-wise convergence of the eigenfunction expansion to the velocity potential associated with the flexural gravity waves problem in water wave theory is established for infinite water depth case. To take into account the hydroelastic boundary condition at the free surface, a flexible membrane is assumed to float in water waves. In this context, firstly the eigenfunction expansion for the velocity potentials is obtained. Thereafter, an appropriate Green’s function is constructed for the associated boundary value problem. Using suitable properties of the Green’s functions, the vertical components of the eigenfunction expansion is written in terms of the Dirac delta function. Finally, using the property of the Dirac delta function, the convergence of the eigenfunction expansion to the velocity potential is shown.

Surface gravity wave interaction with a horizontal flexible floating plate and submerged flexible porous plate

An analytical study associated with surface gravity wave interaction with a horizontal flexible floating and a submerged porous plate is presented under linearized water wave theory and small amplitude structural response. The expansion formulae of the referred problem based on Green's function technique is derived using the two-dimensional Laplace equation and fundamental solution of source potentials in finite and infinite water depths. The velocity potential for the free oscillations in a closed basin is derived and the hydroelastic wave characteristics in floating and submerged flexural modes are analyzed in specific cases. Further, the usefulness of the expansion formula is demonstrated by analyzing a real physical problem of wave interaction with a moored flexible floating plate and a moored submerged flexible porous plate of finite dimensions in finite water depth. The convergence of the present solution is checked and the effect of a moored finite submerged porous plate on the moored finite flexible floating plate with different design parameters are analyzed. It is observed that the present analysis may be helpful for better understanding in designing a floating and submerged flexible porous plate system for application as a breakwater.