Water Wave Scattering by a Nearly Circular Cylinder Submerged Beneath an Ice-cover

Assuming linear theory, the two-dimensional problem of water wave scattering by a horizontal nearly circular cylinder submerged in infinitely deep water with an ice cover modeled as a thin-elastic plat...     

经过海峡的双层流体由底部波动产生的斜向波散射问题研究(英文)

The problem of oblique wave(internal wave) propagation over a small deformation in a channel flow consisting of two layers was considered.The upper fluid was assumed to be bounded above by a rigid lid,which is an approximation for the free surface,and the lower one was bounded below by an impermeable bottom surface having a small deformation;the channel was unbounded in the horizontal directions.Assuming irrotational motion,the perturbation technique was employed to calculate the first-order corrections of the velocity potential in the two fluids by using Green’s integral theorem suitably with the introduction of appropriate Green’s functions.Those functions help in calculating the reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom deformation.Three-dimensional linear water wave theory was utilized for formulating the relevant boundary value problem.Two special examples of bottom deformation were considered to validate the results.Consideration of a patch of sinusoidal ripples(having the same wave number) shows that the reflection coefficient is an oscillatory function of the ratio of twice the x-component of the wave number to the ripple wave number.When this ratio approaches one,the theory predicts a resonant interaction between the bed and the interface,and the reflection coefficient becomes a multiple of the number of ripples.High reflection of incident wave energy occurs if this number is large.Similar results were observed for a patch of sinusoidal ripples having different wave numbers.It was also observed that for small angles of incidence,the reflected energy is greater compared to other angles of incidence up to.These theoretical observations are supported by graphical results.     

水底地形变化对摇板造波的影响

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.     

Diffraction of oblique water waves by small uneven channel-bed in a two-layer fluid

Obliquely incident water wave scattering by an uneven channel-bed in the form of a small bottom undulation in a two-layer fluid is investigated within the frame work of three-dimensional linear water wave theory. The upper fluid is assumed to be bounded above by a rigid lid, while the lower one is bounded below by a bottom surface having a small deformation and the channel is unbounded in the horizontal directions. Assuming irrotational motion, perturbation technique is employed to calculate the first-order corrections to the velocity potentials in the two fluids by using Fourier transform approximately, and also to calculate the reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom deformation. Consideration of a patch of sinusoidal ripples shows that the reflection coefficient is an oscillatory function of the ratio of twice the component of the wave number along x-axis and the ripple wave number. When this ratio approaches one, the theory predicts a resonant interaction between the bed and interface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of incident wave energy occurs if this number is large.     

Wave Scattering by Porous Bottom Undulation in a Two Layered Channel

The scattering of plane surface waves by bottom undulations in channel flow consisting of two layers is investigated by assuming that the bed of the channel is composed of porous material. The upper surface of the fluid is bounded by a rigid lid and the channel is unbounded in the horizontal directions. There exists only one wave mode corresponding to an internal wave. For small undulations, a simplified perturbation analysis is used to obtain first order reflection and transmission coefficients in terms of integrals involving the shape function describing the bottom. For sinusoidal bottom undulations and exponentially decaying bottom topography, the first order coefficients are computed. In the case of sinusoidal bottom the first order transmission coefficient is found to vanish identically. The numerical results are depicted graphically in a number of figures.     

冰盖下两层流体水波与可渗透起伏海床的作用(英文)

The scattering problem involving water waves by small undulation on the porous ocean-bed in a two-layer fluid,is investigated within the framework of the two-dimensional linear water wave theory where the upper layer is covered by a thin uniform sheet of ice modeled as a thin elastic plate.In such a two-layer fluid there exist waves with two different modes,one with a lower wave number propagate along the ice-cover whilst those with a higher wave number propagate along the interface.An incident wave of a particular wave number gets reflected and transmitted over the bottom undulation into waves of both modes.Perturbation analysis in conjunction with the Fourier transform technique is used to derive the first-order corrections of reflection and transmission coefficients for both the modes due to incident waves of two different modes.One special type of bottom topography is considered as an example to evaluate the related coefficients in detail.These coefficients are depicted in graphical forms to demonstrate the transformation of wave energy between the two modes and also to illustrate the effects of the ice sheet and the porosity of the undulating bed.     

可渗透海床上具有小底部变形的斜弯曲重力波的相互作用:格林函数法（英文）

The interaction of oblique incident water waves with a small bottom deformation on a porous ocean-bed is examined analytically here within the framework of linear water wave theory. The upper surface of the ocean is assumed to be covered by an infinitely extended thin uniform elastic plate, while the lower surface is bounded by a porous bottom surface having a small deformation. By employing a simplified perturbation analysis, involving a small parameter δ(1), which measures the smallness of the deformation, the governing Boundary Value Problem(BVP) is reduced to a simpler BVP for the first-order correction of the potential function. This BVP is solved using a method based on Green's integral theorem with the introduction of suitable Green's function to obtain the first-order potential, and this potential function is then utilized to calculate the first-order reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom deformation. Consideration of a patch of sinusoidal ripples shows that when the quotient of twice the component of the incident field wave number propagating just below the elastic plate and the ripple wave number approaches one, the theory predicts a resonant interaction between the bed and the surface below the elastic plate. Again, for small angles of incidence, the reflected wave energy is more as compared to the other angles of incidence. It is also observed that the reflected wave energy is somewhat sensitive to the changes in the flexural rigidity of the elastic plate, the porosity of the bed and the ripple wave numbers. The main advantage of the present study is that the results for the values of reflection and transmission coefficients obtained are found to satisfy the energy-balance relation almost accurately.     

Scattering of surface waves by the edge of a small undulation on a porous bed in an ocean with ice-cover

Scattering of surface waves by the edge of a small undulation on a porous bed in an ocean of finite depth, where the free surface has an ice-cover being modelled as an elastic plate of very small thickness, is investigated within the framework of linearized water wave theory. The effect of surface tension at the surface below the ice-cover is neglected. There exists only one wave number propagating at just below the ice-cover. A perturbation analysis is employed to solve the boundary value problem governed by Laplace＇s equation by a method based on Green＇s integral theorem with the introduction of appropriate Green＇s function and thereby evaluating the reflection and transmission coefficients approximately up to first order. A patch of sinusoidal ripples is considered as an example and the related coefficients are determined.     

Scattering of Oblique Surface Waves by the Edge of Small Deformation on a Porous Ocean Bed

The scattering of oblique incident surface waves by the edge of a small cylindrical deformation on a porous bed in an ocean of finite depth, is investigated here within the framework of linearized water wave theory. Using perturbation analysis, the corresponding problem governed by modified Helmholtz equation is reduced to a boundary value problem for the first-order correction of the potential function. The first-order potential and, hence, the reflection and transmission coefficients are obtained by a method based on Green’s integral theorem with the introduction of appropriate Green’s function. Consideration of a patch of sinusoidal ripples shows that when the quotient of twice the component of the incident field wave number along x-direction and the ripple wave number approaches one, the theory predicts a resonant interaction between the bed and the free-surface, and the reflection coefficient becomes a multiple of the number of ripples. Again, for small angles of incidence, the reflected energy is more as compared to the other angles of incidence. It is also observed that the reflected energy is somewhat sensitive to the changes in the porosity of the ocean bed. From the derived results, the solutions for problems with impermeable ocean bed can be obtained as particular cases.     

Scattering of surface water waves involving semi-infinite floating elastic plates on water of finite depth

Two problems of scattering of surface water waves involving a semi-infinite elastic plate and a pair of semi-infinite elastic plates,separated by a gap of finite width,floating horizontally on water of finite depth,are investigated in the present work for a two-dimensional time-harmonic case.Within the frame of linear water wave theory,the solutions of the two boundary value problems under consideration have been represented in the forms of eigenfunction expansions.Approximate values of the reflection and transmission coefficients are obtained by solving an over-determined system of linear algebraic equations in each problem.In both the problems,the method of least squares as well as the singular value decomposition have been employed and tables of numerical values of the reflection and transmission coefficients are presented for specific choices of the parameters for modelling the elastic plates.Our main aim is to check the energy balance relation in each problem which plays a very important role in the present approach of solutions of mixed boundary value problems involving Laplace equations.The main advantage of the present approach of solutions is that the results for the values of reflection and transmission coefficients obtained by using both the methods are found to satisfy the energy-balance relations associated with the respective scattering problems under consideration.The absolute values of the reflection and transmission coefficients are presented graphically against different values of the wave numbers.     

Fast Prediction of Acoustic Radiation from a Hemi-capped Cylindrical Shell in Waveguide

In order to predict acoustic radiation from a structure in waveguide, a method based on wave superposition is proposed, in which the free-space Green＇s function is used to match the strength of equivalent sources. In addition, in order to neglect the effect of sound reflection from boundaries, necessary treatment is conducted, which makes the method more efficient. Moreover, this method is combined with the sound propagation algorithms to predict the sound radiated from a cylindrical shell in waveguide. Numerical simulations show the effect of how reflections can be neglected if the distance between the structure and the boundary exceeds the maximum linear dimension of the structure. It also shows that the reflection from the bottom of the waveguide can be approximated by plane wave conditionally. The proposed method is more robust and efficient in computation, which can be used to predict the acoustic radiation in waveguide.     

Damping of Oblique Ocean Waves by a Vertical Porous Structure Placed on a Multi-step Bottom

Oblique ocean wave damping by a vertical porous structure placed on a multi-step bottom topography is studied with the help of linear water wave theory. Some portion of the oblique wave, incident on the porous structure, gets reflected by the multi-step bottom and the porous structure, and the rest propagates into the water medium following the porous structure. Two cases are considered： first a solid vertical wall placed at a finite distance from the porous structure in the water medium following the porous structure and then a special case of an unbounded water medium following the porous structure. In both cases, boundary value problems are set up in three different media, the first medium being water, the second medium being the porous structure consisting ofp vertical regions-one above each step and the third medium being water again. By using the matching conditions along the virtualvertical boundaries, a system of linear equations is deduced. The behavior of the reflection coefficient and the dimensionless amplitude of the transmitted progressive wave due to different relevant parameters are studied. Energy loss due to the propagation of oblique water wave through the porous structure is also carried out. The effects of various parameters, such as number of evanescent modes, porosity, friction factor, structure width, number of steps and angle of incidence, on the reflection coefficient and the dimensionless amplitude of the transmitted wave are studied graphically for both cases. Number of evanescent modes merely affects the scattering phenomenon. But higher values of porosity show relatively lower reflection than that for lower porosity. Oscillation in the reflection coefficient is observed for lower values of friction factor but it disappears with an increase in the value of friction factor. Amplitude of the transmitted progressive wave is independent of the porosity of the structure. But lower value of friction factor causes higher transmission. The investigation is then carried out for the second ca     

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.     

半无限惯性表面产生的波浪散射问题研究的另一种方法(英文)

A new method to solve the boundary value problem arising in the study of scattering of two-dimensional surface water waves by a discontinuity in the surface boundary conditions is presented in this paper. The discontinuity arises due to the floating of two semi-infinite inertial surfaces of different surface densities. Applying Green’s second identity to the potential functions and appropriate Green’s functions, this problem is reduced to solving two coupled Fredholm integral equations with regular kernels. The solutions to these integral equations are used to determine the reflection and the transmission coefficients. The results for the reflection coefficient are presented graphically and are compared to those obtained earlier using other research methods. It is observed from the graphs that the results computed from the present analysis match exactly with the previous results.     

有限水深中垂直下潜弹性薄板的水波散射(英文)

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.     

Exciting forces for a wave energy device consisting of a pair of coaxial cylinders in water of finite depth

Two coaxial vertical cylinders-one is a riding hollow cylinder and the other a solid cylinder of greater radius at some distance above an impermeable horizontal bottom,were considered.This problem of diffraction by these two cylinders,which were considered as idealization of a buoy and a circular plate,can be considered as a wave energy device.The wave energy that is created and transferred by this device can be appropriately used in many applications in lieu of conventional energy.Method of separation of variables was used to obtain the analytical expressions for the diffracted potentials in four clearly identified regions.By applying the appropriate matching conditions along the three virtual boundaries between the regions,a system of linear equations was obtained,which was solved for the unknown coefficients.The potentials allowed us to obtain the exciting forces acting on both cylinders.Sets of exciting forces were obtained for different radii of the cylinders and for different gaps between the cylinders.It was observed that changes in radius and the gap had significant effect on the forces.It was found that mostly the exciting forces were significant only at lower frequencies.The exciting forces almost vanished at higher frequencies.The problem was also investigated for the base case of no plate arrangement,i.e.,the case having only the floating cylinder tethered to the sea-bed.Comparison of forces for both arrangements was carried out.In order to take care of the radiation of the cylinders due to surge motion,the corresponding added mass and the damping coefficients for both cylinders were also computed.All the results were depicted graphically and compared with available results.     

Optimization of bow shape for a non ballast water ship

In this research,a commercial CFD code ＂Fluent＂ was applied to optimization of bulbous bow shape for a non ballast water ships（NBS）.The ship was developed at the Laboratory of the authors in Osaka Prefecture University,Japan.At first,accuracy of the CFD code was validated by comparing the CFD results with experimental results at towing tank of Osaka Prefecture University.In the optimizing process,the resistances acting on ships in calm water and in regular head waves were defined as the object function.Following features of bulbous bow shapes were considered as design parameters： volume of bulbous bow,height of its volume center,angle of bow bottom,and length of bulbous bow.When referring to the computed results given by the CFD like resistance,pressure and wave pattern made by ships in calm water and in waves,an optimal bow shape for ships was discovered by comparing the results in the series of bow shapes.In the computation on waves,the ship is in fully captured condition because shorter waves,λ/Lpp 0.6,are assumed.     

Coupled motions of two ships in irregular waves in time domain

A three-dimensional time domain approach is used to study the coupled motions of two ships with forward speed in waves. In this approach, the boundary condition is satisfied on the mean wetted hull surface of the moving bodies and the free surface condition is linearized. The problem is solved by using a transient free-surface Green function source distribution on the submerged hulls. After solving the response amplitude operator, the method of spectral analysis is employed to clearly express the motion energy spectrum and significant amplitude of two ships. For verifying the code, two same circular cylinders at beam wave are selected to calculate coupled motions by comparison with the results obtained by 3D frequency method which has been proved to be efficient for solving such problems.Two Wigley ships of different sizes with the same forward speed are chosen for numerical calculation of the interaction effect, and some useful suggestions are obtained for underway replenishment at sea.     

Wave analysis of porous geometry with linear resistance law

The wave diffraction-radiation problem of a porous geometry of arbitrary shape located in the free surface of a fluid is formulated by a set of integral equations, assuming a linear resistance law at the geometry. The linear forces, the energy relation and the mean horizontal drift force are evaluated for non-porous and porous geometries. A geometry of large porosity has an almost vanishing added mass. The exciting forces are a factor of 5–20 smaller compared to a solid geometry. In the long wave regime, the porous geometry significantly enhances both the damping and the mean drift force, where the latter grows linearly with the wavenumber. The calculated mean drift force on a porous hemisphere and a vertical truncated cylinder, relevant to the construction of fish cages, is compared to available published results.