Water Of Uniform Finite Depth Research Articles

Wave interaction with multiple thin flexible porous barriers in water of uniform finite depth

The interaction of water waves with multiple flexible porous barriers partially immersed in a uniform finite-depth fluid is investigated in the present study. Using linear theory, the physical problem is mathematically modeled as a boundary value problem. The eigenfunction expansion approach is used to generate the velocity potential function in terms of orthogonal eigenfunctions. The Havelock’s inversion formula is then used to convert the boundary value problem into a set of Fredholm-type integral equations. By choosing a suitable weight function, the multi-term Galerkin’s approximation technique is used to obtain the mathematical solutions of these integral equations in terms of the Chebyshev polynomial. Numerical solutions regarding the reflection and transmission coefficients, hydrodynamic forces, and dissipated wave energy at the barriers are obtained and graphically represented in several figures for various non-dimensional parameters. It is observed from the graphical representations that the permeability and flexibility of the barriers play a crucial role in the modeling of efficient breakwaters.

Optimal Fourier–Kochin flow representations in ship and offshore hydrodynamics: Theory

This study considers the core issue of evaluating flows created by general distributions of sources and dipoles over panels (typically flat/curved triangles/quadrilaterals) used to approximate the surface of a body (ship or offshore structure) in usual implementations of the Green function and boundary integral method in marine hydrodynamics. This crucial basic issue is considered – within the framework of the Fourier–Kochin (FK) method – for four classes of flows in deep-water ship and offshore hydrodynamics: diffraction-radiation of regular waves by an offshore structure, and flow around a ship that advances at a constant speed V in calm water or in regular waves of (encounter) frequency ω in the regimes τ≡ωV/g<1/4 or 0.3≤τ where g is the acceleration of gravity; diffraction-radiation of regular waves by an offshore structure in water of uniform finite depth is also considered. Two notable features of the Green functions G used in the study of these five classes of flows are that (i) they are based on optimal decompositions into Rankine and Fourier components GR and GF, and that (ii) these Green functions are consistent. In particular, a new representation of the Green function for flows around ships advancing in regular waves at τ<1/4 that is consistent with the Green functions for the special cases V=0 or ω=0 is given. The study also provides a complete and largely self-contained account of the FK method for evaluating the Fourier component ϕF in the Rankine–Fourier decomposition ϕR+ϕF of the velocity potential ϕ (and the related velocity) of the flow created by a general distribution of singularities. An essential element of this FK theory is an optimal waves/local-effects (WL) decomposition ϕF=ϕW+ϕL, given for a general dispersion relation associated with a broad class of dispersive waves, in which the component ϕW represents the (far-field and near-field) waves contained in ϕF and the component ϕL corresponds to a non-oscillatory local disturbance that is mostly significant in a small near-field region. Applications of this general WL decomposition to the five classes of flows in marine hydrodynamics of primary interest in the study yield simple analytical flow representations – for general compact distributions of singularities – that offer two notable advantages over the classical direct Green function method: (i) Integration over hull-surface panels within the FK method only involves smooth ordinary functions, which are incomparably simpler than the highly singular functions GF and ∇GF, and (ii) The computations related to GF and ∇GF in the FK theory – implemented in the manner expounded in the study – are proportional to the number N of panels that approximate the body surface, whereas common panel methods require O(N2) computations. Thus, the Fourier–Kochin method and the optimal Rankine–Fourier decompositions and optimal waves/local-flow decompositions given in the study lay the foundation of a new type of computational methods, which offers two compelling advantages over the classical ‘direct Green function method’ steadfastly applied in the past fifty years.

Scattering of water waves by rectangular thick barriers in presence of surface tension

The influence of surface tension over an oblique incident waves in presence of thick rectangular barriers present in water of uniform finite depth is discussed here. Three different structures of a bottom-standing submerged barrier, submerged rectangular block not extending down to the bottom and fully submerged block extending down to the bottom with a finite gap are considered. An appropriate multi-term Galekin approximation technique involving ultraspherical Gegenbauer polynomial is employed for solving the integral equations arising in the mathematical analysis. The reflection and transmission coefficients of the progressive waves for two-dimensional time har- monic motion are evaluated by utilizing linearized potential theory. The theoretical result is validated numerically and explained graphically in a number of figures. The present result will almost match analytically and graphically with those results already available in the literature without considering the effect of surface tension. From the graphical representation, it is clearly visible that the amplitude of reflection coefficient decreases with increasing values of surface tension. It is also seen that the presence of surface tension, the change of width, and the height of the thick barriers affect the nature of the reflection coefficients significantly

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.

Wave attenuation by multiple thin vertical porous walls in water of uniform finite depth

The analysis for wave attenuation by multiple thin porous walls submerged in water of uniform finite depth is carried out under the acceptance of linear theory of water waves. Two different types of barrier configurations are considered, namely partially immersed walls and bottom-standing walls. A set of coupled Fredholm-type integral equations are obtained for the boundary value problem by employing Havelock's inversion formulae and continuity of water wave potential across the gap below or above the walls. The methodology adopted to solve these integral equations is multi-term Galerkin's approximation with Chebychev's polynomials as a set of basis functions multiplied by suitable weights. Also, the square-root singularity of fluid velocity at the edge of the porous walls is precisely handled. The numerical results are depicted for reflection and transmission coefficients, dissipation of wave energy and dynamic wave force against wavenumber for different parametric values involving in the barrier configurations. This theory demonstrates that the dynamic wave forces exerted on the porous walls are reduced significantly as the magnitude of the porous effect parameter is increased. The present study is ratified with the existing results in the literature of water waves for a single as well as double rigid barriers and single porous barrier.

Hydrodynamic Forces on a Submerged Horizontal Circular Cylinder in Uniform Finite Depth Ice Covered Water

Hydrodynamic forces on a submerged cylinder in uniform finite depth ice-covered water is formulated by using the method of multipoles, the ice-cover being modelled as an elastic plate of very small thickness. The forces (vertical and horizontal) are obtained analytically as well as numerically and depicted graphically for various values of flexural rigidity of the ice-cover to show the effect of its presence. When the flexural rigidity and surface density of the ice-cover are taken to be zero, then the curves for the forces almost coincide with the curves for the case of uniform finite depth water with free surface.

Oblique wave scattering by two thin non-uniform permeable vertical walls with unequal apertures in water of uniform finite depth

ABSTRACT Scattering of surface waves by two permeable vertical walls having apertures at different depths in uniform finite depth water has been studied assuming linear theory. The problem is reduced to a set of coupled integral equations involving the difference of velocity potentials across each wall as unknown function. A multi-term Galerkin's technique is employed to approximate these unknown functions and to obtain approximate solution of the integral equations. Numerical estimates for the reflection and transmission coefficients are then obtained by solving a linear system. It is observed from graphical representations of the reflection coefficient that in comparison with non-identical walls the minima of the reflection coefficient become very close to zero at discrete frequencies in case of identical walls. Very good agreement between some earlier known results and our present results is established.

Water wave scattering by multiple thin vertical barriers

A study of obliquely incident water wave scattering by two, three and four unequal partially immersed vertical barriers in water of uniform finite depth has been carried out in this paper employing Havelock’s expansion of water wave potential. A formulation involving integral equations in terms of either horizontal component of velocities across the gap below each barrier or difference of potentials across each barrier are obtained using the Havelock’s inversion formulae. A multi-term Galerkin approximation technique with Chebychev’s polynomials (multiplied by appropriate weights) as basis functions is adapted to solve these integral equations and to compute the reflection and transmission coefficients numerically. The numerical results are depicted graphically against the wavenumber in several figures for various arrangements of the vertical barriers. From these figures zeros of reflection coefficient are observed when the vertical barriers are immersed upto equal depths below the mean free surface. However, this observation is not true always for non-identical vertical barriers. For two non-identical partially immersed barriers reflection coefficient never vanishes whereas for three non-identical partially immersed barriers reflection coefficient vanishes at discrete frequencies if the two outer barriers have equal lengths of submergence. For four partially immersed barriers arranged symmetrically about a vertical line, zeros of reflection coefficient are always observed. The known results of a single barrier are recovered as special cases so as to establish the correctness of the present method.

Scattering of oblique water waves by two thin unequal barriers with non-uniform permeability

The scattering of obliquely incident water waves by two unequal permeable barriers with variable permeability is investigated for two types of barriers, namely partially immersed barriers and bottom-standing barriers, under the consideration of the theory of linear water waves. The barriers are present in water of uniform finite depth. The velocity potential is expanded by using Havelock’s expansion of water wave potential and employing Havelock’s inversion formula together with the conditions on the permeable barriers; the boundary value problem is reduced to a coupled Fredholm-type vector integral equations. The integral equations are solved using the multi-term Galerkin approximation where the unknown functions are approximated in terms of Chebyshev polynomials. The numerical results for the reflection coefficient, the transmission coefficient and the energy dissipation are depicted graphically. Known results for two identical as well as two non-identical impermeable barriers and for a single and twin permeable barriers are recovered in the limiting cases.

Basic models of farfield ship waves in shallow water

Basic models of the farfield waves created by a monohull ship or a catamaran that travels at a constant speed in calm water of uniform finite depth are considered. A classical basic model is the 1-point ship model introduced by Kelvin in 1887 for deep water and extended to uniform finite water depth by Havelock in 1908. Another basic model is the 2-point ship model, which considers the two dominant waves created by the bow and the stern of a monohull ship or by the twin bows of a catamaran, and therefore accounts for main wave interference effects. These 2-point ship models contain information that cannot be found in the classical patterns for the 1-point ship model commonly used since Kelvin. In particular, whereas Havelock’s classical wave patterns for the 1-point model only depend on the Froude number based on the water depth, the farfield waves for the 2-point models of a monohull ship or a catamaran also depend on the Froude number associated with the length of the monohull ship or the spacing between the twin hulls of the catamaran. Moreover, the ‘apparent wake angles’ associated with the highest waves that result from constructive wave interferences can greatly differ from the cusp or asymptote angles of Havelock’s wave patterns.

Interaction of surface water waves with a vertical elastic plate: a hypersingular integral equation approach

In this paper, we present an alternative method to investigate scattering of water waves by a submerged thin vertical elastic plate in the context of linear theory. The plate is submerged either in deep water or in the water of uniform finite depth. Using the condition on the plate, together with the end conditions, the derivative of the velocity potential in the direction of normal to the plate is expressed in terms of a Green’s function. This expression is compared with that obtained by employing Green’s integral theorem to the scattered velocity potential and the Green’s function for the fluid region. This produces a hypersingular integral equation of the first kind in the difference in potential across the plate. The reflection coefficients are computed using the solution of the hypersingular integral equation. We find good agreement when the results for these quantities are compared with those for a vertical elastic plate and submerged and partially immersed rigid plates. New results for the hydrodynamic force on the plate, the shear stress and the shear strain of the vertical elastic plate are also evaluated and represented graphically.

Farfield waves created by a catamaran in shallow water

The farfield waves created by a catamaran with identical twin hulls that advances at constant speed along a straight path in calm water of uniform finite depth are considered. The catamaran is represented as two point sources at the twin bows of the catamaran. This simple approximation of a catamaran as a 2-point wavemaker is realistic for the high-speed regime considered here. Constructive interferences between the divergent waves created by the two point sources result in an apparent wake angle, where the highest waves are found, that can be much narrower than the wake angle associated with the cusps or the asymptotes of the well-known wave patterns radiated from a single point. The farfield wave pattern of a high-speed catamaran can then differ markedly from the wave pattern obtained in the classical analysis of the waves created by a ship modeled as a 1-point wavemaker, in which interference effects are ignored. Moreover, the highest waves created by a catamaran can be much shorter than the waves at the cusps of the wave pattern, and significantly shorter than the ship length. Indeed, the wavelength of the highest waves is equal to the spacing between the twin hulls of the catamaran at high Froude numbers. Wave interferences between the twin bows of a catamaran (dominant at high Froude numbers) are compared to wave interferences between the bow and the stern of a monohull ship or a catamaran (significant at moderate Froude numbers).

Oblique wave-free potentials for water waves in constant finite depth

In this paper, a method to construct oblique wave-free potentials in the linearised theory of water waves for water with uniform finite depth is presented in a systematic manner. The water has either a free surface or an ice-cover modelled as a thin elastic plate. For the case of free surface, the effect of surface tension may be neglected or taken into account. Here, the wave-free potentials are singular solutions of the modified Helmholtz equation, having singularity at a point in the fluid region and they satisfy the conditions at the upper surface and the bottom of water region and decay rapidly away from the point of singularity. These are useful in obtaining solutions to oblique water wave problems involving bodies with circular cross-sections such as long horizontal cylinders submerged or half-immersed in water of uniform finite depth with a free surface or an ice-cover modelled as a floating elastic plate. Finally, the forms of the upper surface related to the wave-free potentials constructed here are depicted graphically in a number of figures to visualize the wave motion. The results for non-oblique wave-free potentials and the upper surface wave-free potentials are obtained. The wave-free potentials constructed here will be useful in the mathematical study of water wave problems involving infinitely long horizontal cylinders, either half-immersed or completely immersed in water.

Farfield waves created by a monohull ship in shallow water

Interference effects between the dominant waves created by the bow and the stern of a monohull ship, of length L, that advances at constant speed V along a straight path in calm water of uniform finite depth D are considered. The largest waves due to constructive interference result in an apparent wake angle ψmax that can differ greatly from the cusp or asymptote angles associated with the wave pattern of a ship when interference effects are ignored, as in Kelvin’s classical analysis. Thus, wave interference has a very large effect on the wave signature of a ship in shallow water and cannot be ignored. Water-depth effects on the wake angle ψmax are found to be insignificant for water depths dL≡D/L greater than ddeepL≈0.89 or for Froude numbers F≡V/gL greater than Fdeep≈0.5/dL. Furthermore, the water depth dL has only a relatively small influence on the wake angle ψmax for 1.5<F, and the deep-water approximation ψmax≈0.14/F2 can then be used.

Three-dimensional wave-free potentials in the theory of water waves

The method of constructing wave-free potentials in three dimensions is presented here in a systematic manner for a number of situations such as deep water and water of uniform finite depth with a free surface, neglecting or taking into account the effect of surface tension, or with an ice-cover modelled as a thin elastic plate floating on water. doi:10.1017/S1446181113000357

Water wave scattering by an elastic thin vertical plate submerged in finite depth water

The problem of water wave scattering by a thin vertical elastic plate submerged in uniform finite depth water is investigated here. The boundary condition on the elastic plate is derived from the Bernoulli-Euler equation of motion satisfied by the plate. Using the Green's function technique, from this boundary condition, the normal velocity of the plate is expressed in terms of the difference between the velocity potentials (unknown) across the plate. The two ends of the plate are either clamped or free. The reflection and transmission coefficients are obtained in terms of the integrals' involving combinations of the unknown velocity potential on the two sides of the plate, which satisfy three simultaneous integral equations and are solved numerically. These coefficients are computed numerically for various values of different parameters and depicted graphically against the wave number in a number of figures.

THREE-DIMENSIONAL WAVE-FREE POTENTIALS IN THE THEORY OF WATER WAVES

AbstractProblems of wave interaction with a body with arbitrary shape floating or submerged in water are of immense importance in the literature on the linearized theory of water waves. Wave-free potentials are used to construct solutions to these problems involving bodies with circular geometry, such as a submerged or half-immersed long horizontal circular cylinder (in two dimensions) or sphere (in three dimensions). These are singular solutions of Laplace’s equation satisfying the free surface condition and decaying rapidly away from the point of singularity. Wave-free potentials in two and three dimensions for infinitely deep water as well as water of uniform finite depth with a free surface are known in the literature. The method of constructing wave-free potentials in three dimensions is presented here in a systematic manner, neglecting or taking into account the effect of surface tension at the free surface or for water with an ice cover modelled as a thin elastic plate floating on the water. The forms of the wave motion at the upper surface (free surface or ice-covered surface) related to these wave-free potentials are depicted graphically in a number of figures for all the cases considered.

Water wave scattering by a sphere submerged in uniform finite depth water with an ice-cover

Using the multipole method, we formulate the problem of water wave scattering by a submerged sphere in uniform finite water depth with an ice-cover, the ice-cover being modelled as an elastic plate of very small thickness. This leads to an infinite system of linear equations which are solved numerically by standard techniques. The vertical and horizontal forces on the sphere are obtained and depicted graphically against the wave number for various values of the depth of water and flexural rigidity of the ice-cover to show the effect of the presence of ice-cover and also the effect of varying depth of water on these quantities. When the flexural rigidity is taken to be zero, the numerical results exactly coincide with the curves of the vertical and horizontal forces on the sphere for the cases of uniform finite depth water with a free surface.

Effect of bottom undulation on the waves generated due to rolling of a plate

In the present paper, the effect of a small bottom undulation of the sea bed in the form of periodic bed form on the surface waves generated due to a rolling oscillation of a vertical barrier either partially immersed or completely submerged in water of non uniform finite depth is investigated. A simplified perturbation technique involving a non dimensional parameter characterizing the smallness of the bottom deformation is applied to reduce the given boundary value problem to two independent boundary value problems upto first order. The first boundary value problem corresponds to the problem of water wave generation due to rolling oscillation of a vertical barrier either partially immersed or completely submerged in water of uniform finite depth. This is a well known problem whose solution is available in the literature. From the second boundary value problem, the first order correction to the wave amplitude at infinity is evaluated in terms of the shape function characterizing the bottom undulation, by employing Green’s integral theorem. For a patch of sinusoidal ripples at the sea bottom, the first order correction to the wave amplitude at infinity for both the configuration of the barrier is then evaluated numerically and illustrated graphically for various values of the wave number. It is observed that resonant interaction of the wave generated, with the sinusoidal bottom undulation occurs when the ratio of twice the wavelength of the sinusoidal ripple to the wave length of waves generated, approaches unity. Also it is found that the resonance increases as the length of the barrier increases.

Water wave radiation by a sphere submerged in water with an ice-cover

Using the multipoles method, we formulate the problems of radiation (both heave and sway) of water waves by a submerged sphere in deep as well as in uniform finite depth water with an ice-cover, with the ice-cover being modelled as an elastic plate of very small thickness. In each case this leads to an infinite system of linear equations which are solved numerically by standard techniques. The added-mass and damping coefficients for a heaving and swaying sphere are obtained and depicted graphically against the wave number for various values of the radius of the submerged sphere and flexural rigidity of the ice-cover to show the effect of the presence of ice-cover on these quantities. When the flexural rigidity is taken to be zero, the numerical results for the added-mass and damping coefficient for water with a free surface are recovered.