Bragg scattering of waves propagating over a series of poro-elastic submerged breakwaters

This paper presents a mathematical study for the Bragg scattering of water waves propagating over a series of rectangular poro-elastic submerged breakwaters. Based on the linear wave theory and the improved Biot's theory of poro-elastic media with suitable matching conditions, an analytical solution was obtained as a function of elasticity, permeability and geometry of the submerged breakwaters. Experiments were also conducted in a wave flume to verify the present analytical study. In addition, the Bragg resonances for different influence parameters were studied. Satisfactory agreement between the theoretical predictions and the laboratory observations demonstrates the applicability of the present analytical result.     

潜堤对波浪在直立式防浪墙上爬高的影响

对直立式防浪墙前潜堤的空间布置对波浪在直墙上爬高的影响进行了研究.建立简化模型,即直立式防浪墙前海床设计为平底,潜堤设计为直立薄板. 改变潜堤高度及其与防浪墙之间距,研究不同的潜堤布置对波浪在直墙上爬高的影响. 针对线性波浪场,利用数值波浪水槽模拟了潜堤作用下直墙上波浪的爬高现象. 同时建立了理论模型,系统地分析了潜堤布置形式对直墙上波浪爬高的影响. 研究结果表明潜堤相对于水深的高度越大,对波浪在直墙上爬高的影响越大,而在一定的相对高度条件下,潜堤与直立墙之间距对波浪爬高的影响呈现出一定的规律性.      

A model for the scattering of long waves by slotted breakwaters in the presence of currents

Slotted breakwaters have been used to provide economical protection from waves in harbors where surface waves and currents may co-exist. In this paper, the effects of currents on the wave scattering by slotted breakwaters are investigated by using a simple model. The model is based on a long wave approximation. The effects of wave height, barrier geometry and current strength on the reflection and transmission coefficients are examined by the model. The model results are compared with recent experimental data. It is found that both the wave-following and wave-opposing currents can increase the reflection coefficient and reduce the transmission coefficient. The model can be used to study the interaction between long waves and slotted breakwaters in coastal waters. The project partially supported by the Hong Kong Research Grant Council under Grant No. HKUST-DAG03/04.EG39 and HKUST6227/04E.     

Scattering of water waves by vertical porous barriers: An analytical approach

An analytical approach is proposed here to study scattering of deep water waves by a submerged or a surface piercing vertical porous barrier. It involves a connection between two wave potentials of which one is the solution of a boundary value problem associated with wave scattering by the porous barrier and the other is the solution of a complementary type problem where barrier and gap positions are interchanged and solid barrier takes the position of the porous barrier. The connection also involves an auxiliary or a connection wave potential. The potential for the solid barrier problem involves incident wave forcing while the auxiliary potential describes a solid barrier type problem that involves a non-physical forcing. The solution procedure of Ursell (Ursell, 1947) is chosen to solve these boundary value problems explicitly in the case of normal wave incidence as it also determines necessary exact behavior of the potential at the barrier edge. The reflection coefficients are also connected and the reflection amplitudes of the normally incident wave against the vertical porous barriers are obtained analytically. Numerical results for reflection and transmission coefficients are presented.     

Explicit solutions of the scattering problems involving vertical flexible porous structures

Complete analytical solution for the normally incident water wave scattering by a porous flexible vertical elastic plate or tensioned membrane is found. The physical problem in a half-plane is reduced to a couple of equivalent quarter-plane problems by allowing incident waves from either direction of the structure. In the same way, quarter-plane boundary value problems are posed for solid wave potentials that are solutions of the scattering problem involving a rigid structure of the same geometric configuration. Then, two novel integral relations are introduced to establish a link between the required solution wave potentials and few resolvable solid wave potentials. Explicit expressions for the scattering quantities such as the reflection and the transmission coefficients are obtained. Also, the deflection of the flexible vertical structure and the solution potentials are determined analytically. Numerical results for the explicitly derived scattering quantities and structure deflections are presented.     

Explicit formulas for the reflection and transmission coefficients of one-component waves through a stack of an arbitrary number of layers

The transmission and reflection of one-component elastic, acoustic, optical waves on a stack of arbitrary number of different homogeneous layers have been intensively studied in the literature. However, all obtained formulas for the reflection and transmission coefficients are in implicit form. In this paper, we provide the explicit formulas for them. From these formulas we immediately arrive at the explicit formulas for the reflection and transmission coefficients of one-component waves through an FGM layer. Based on the obtained exact formulas, approximate formulas for the reflection and transmission coefficients are established for a stack of thin layers and for a thin FGM layer. It is numerically shown that they are good approximations. Since the obtained formulas are totally explicit, they are useful in evaluating, not only numerically but also analytically, the transmission and reflection coefficients of one-component waves.     

Propagation and scattering of waves by dense arrays of impenetrable cylinders in a waveguide

A coupled numerical scheme, based on modal expansions and boundary integral representations, is developed for treating propagation and scattering by dense arrays of impenetrable cylinders inside a waveguide. Numerical results are presented and discussed concerning reflection and transmission, as well as the wave details both inside and outside the array. The method is applied to water waves propagating over an array of vertical cylinders in constant depth extended all over the water column, operating as a porous breakwater unit in a periodic arrangement (segmented breakwater). Focusing on the reflection and transmission properties, a simplified model is also derived, based on Foldy–Lax theory. The latter provides an equivalent index of refraction of the medium representing the porous structure, modeled as an inclusion in the waveguide. Results obtained by the present fully coupled and approximate models are compared against experimental measurements, collected in wave tank, showing good agreement. The present analysis permits an efficient calculation of the properties of the examined structure, reducing the computational cost and supporting design and optimization studies.     

Complementary mild-slope equations in a two-layer fluid

Mild-slope (MS) type equations are depth-integrated models, which predict under appropriate conditions refraction and diffraction of linear time-harmonic water waves. By using a streamfunction formulation instead of a velocity potential one, the complementary mild-slope equation (CMSE) was shown to give better agreement with exact linear theory compared to other MS-type equations. The main goal of this work is to extend the CMSE model for solving two-layer flow with a free-surface. In order to allow for an exact reference, an analytical solution for a two-layer fluid over a sloping plane beach is derived. This analytical solution is used for validating the results of the approximated MS-type models. It is found that the two-layer CMSE model performs better than the potential based one. In addition, the new model is used for investigating the scattering of linear surface water waves and interfacial ones over variable bathymetry.     

Boundary element method for wave trapping by a multi-layered trapezoidal breakwater near a sloping rigid wall

This study examines the multiple layers in a rubble mound breakwater and their effect on reflection and dissipation of incoming ocean waves. The numerical model is developed using multi-domain boundary element method for oblique water wave trapping near a sloping wall by a multi-layered trapezoidal porous structure, which is utilized to model armour, filter and core layers while examining the hydrodynamics in different configurations. Both, the constant element and linear element approaches to boundary element method are discussed. The cases of bottom-standing porous structures as being submerged and fully extended are considered. The wave hydrodynamics over the structure is described by the reflection and dissipation coefficients along with the forces acting on the sloping wall, and is influenced by wave and structural parametrics of the system. The influence of armour layer in different configurations is highlighted for various structural and wave parameters.

     

Analysis of water wave scattering by a submerged perforated reef ball using multipole method

Meccanica - Based on linear potential theory, this study develops an analytical solution to water wave scattering by a submerged perforated reef ball. The series solutions for velocity potentials...     

A method to study interactions between narrow-banded random waves and multi-chamber perforated structures

A time-domain method, based on linear velocity potential theory, is presented to study the interaction between narrow-banded random waves and perforated structures. A simple relation is derived to estimate the jet length of flows through the perforated wall. The reflection coefficient of narrow banded random waves from perforated structures is calculated by assuming a Rayleigh distribution of the heights of incident random waves. For reflection of narrow-banded waves from a single-chamber perforated breakwater, a comparison of the predicted and measured reflection coefficients shows that the method presented in this paper can provide a prediction better than that of regular waves. Numerical results are also reported on the reflection of narrow-banded waves from multi-chamber perforated breakwaters.The project partially supported by the Hong Kong Research Grant Council (DAG03/04.EG39, DAG04/05.EG32).     

The interaction of surface waves with fixed semi immersed cylinders having symmetric cross sections

The reflection of water waves by a semi immersed cylinder having a symmetric cross section is studied for both Dirichlet and Neumann boundary conditions on the cylinder. The method of conformal transformations as utilized by Ursell and by Tasai for the radiation problem is adapted to the present diffraction problem. The problem is solved by expansions of the reflected wave potential using nonorthogonal functions (wave free potentials). These functions are not complete, and an additional source and a dipole are required. Infinite systems of linear equations are obtained for the unknown expansion coefficients and the unknown strengths of the source and the dipole terms. Numerical results are obtained for the reflection coefficient, transmission coefficient, horizontal force on cylinder, vertical force on cylinder. In the long wave region analytical approximations are obtained for these functions when the cross section is circular. The reflection and transmission coefficients are very different for the two boundary conditions in the long wave region, the Dirichlet reflection coefficient being much larger than the corresponding Neumann coefficient. This behavior is similar to acoustic and electromagnetic diffraction problems in two dimensions. On leave of absence from Itek Corporation, Lexington (Mass.), U.S.A.     

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.     

Analytical and numerical solution for a elastic pipe bend at in-plane bending with consideration for the end effect

The authors proposed an analytical method for the analysis of the end effect in a pipe bend loaded by a bending moment with consideration for the action of internal pressure. The method is based on the use of simplifying hypotheses and is reduced to the solution of a system of fourth-order differential equations along the axial coordinate with respect to unknown coefficients in the expansion for tangential displacements. An approximate analytical solution, which has a trapezoidal structure and is written in terms of Krylov’s functions, has been obtained. Boundary conditions are formulated in terms of the tangential and longitudinal displacements and axial and shearing stress resultant. For the flexibility factor, analytical solutions are presented in the case where a bend is approximated by a rigid restraint on both ends. To verify the analytical solution and its applicability limits, two numerical procedures were developed, which are based on the finite difference method and the reduction to the Kochi problem by the expansion of the unknowns in the Fourier series over the circumferential coordinate. The authors compare the results obtained with data from the literature, discuss the advantages and disadvantages of the methods, and present recommendations for their practical application.     

Hydroelastic analysis of gravity wave interaction with submerged horizontal flexible porous plate

The present study deals with the surface gravity wave interaction with submerged horizontal flexible porous plate under the assumption of small amplitude water wave theory and structural response. The flexible porous plate is modeled using the thin plate theory and wave past porous structure is based on the generalized porous wavemaker theory. The wave characteristics due to the interaction of gravity waves with submerged flexible porous structure are studied by analyzing the complex dispersion relation using contour plots. Three different problems such as (i) wave scattering by a submerged flexible porous plate, (ii) wave trapping by submerged flexible porous plate placed at a finite distance from a rigid wall and (iii) wave reflection by a rigid wall in the presence of a submerged flexible porous plate are analyzed. The role of flexible porous plate in attenuating wave height and creating a tranquility zone is studied by analyzing the reflection, transmission and dissipation coefficients for various wave and structural parameters such as angle of incidence, depth of submergence, plate length, compression force and structural flexibility. In the case of wave trapping, the optimum distance between the porous plate and rigid wall for wave reflection is analyzed in different cases. In addition, effects of various physical parameters on free surface elevation, plate deflection, wave load on the plate and rigid wall are studied. The present approach can be extended to deal with acoustic wave interaction with flexible porous plates.     

Plane problem of hydrodynamic rocking of a body submerged in a two-layer fluid without forward speed

The results of numerically solving the linear problem of the rocking of a submerged body in an inviscid incompressible fluid bounded either by a rigid top or by a free surface are presented. In each case the equivalence relations connecting the solutions of the radiation and diffraction problems are derived. The apparent mass and damping coefficients are calculated and a comparison with an approximate solution is given.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 144–155, May–June, 1994.     

Modeling the interaction of solitary waves and semi-circular breakwaters by using unsteady reynolds equations

A vertical 2 -D numerical wave model was developed based on unsteady Reynolds equations. In this model, the k-epsilon models were used to close the Reynolds equations, and volume of fluid( VOF) method was used to reconstruct the free surface. The model was verified by experimental data. Then the model was used to simulate solitary wave interaction with submerged, alternative submerged and emerged semi-circular breakwaters. The process of velocity field, pressure field and the wave surface near the breakwaters was obtained. It is found that when the semi-circular breakwater is submerged, a large vortex will be generated at the bottom of the lee side wall of the breakwater ; when the still water depth is equal to the radius of the semi-circular breakwater, a pair of large vortices will be generated near the shoreward wall of the semi-circular breakwater due to wave impacting, but the velocity near the bottom of the lee side wall of the breakwater is always relatively small. When the semi-circular breakwater is emerged, and solitary wave cannot overtop it, the solitary wave surface will run up and down secondarily during reflecting from the breakwater. It can be further used to estate the diffusing and transportation of the contamination and transportation of suspended sediment.     

Approximate analytical solutions and approximate value of skin friction coefficient for boundary layer of power law fluids

This paper presents a theoretical analysis for laminar boundary layer flow in a power law non-Newtonian fluids.The Adomian analytical decomposition technique is presented and an approximate analytical solution is obtained.The approximate analytical solution can be expressed in terms of a rapid convergent power series with easily computable terms.Reliability and efficiency of the approximate solution are verified by comparing with numerical solutions in the literature.Moreover,the approximate solution can be successfully applied to provide values for the skin friction coefficient of the laminar boundary layer flow in power law non-Newtonian fluids.     

The numerical study of roll‐waves in inclined open channels and solitary wave run‐up

In this paper, we introduce a finite‐volume kinetic BGK scheme and its applications to the study of roll and solitary waves. The current scheme is based on the numerical solution of the gas‐kinetic Bhatnagar–Gross–Krook model in the flux evaluation across each cell interface. An intrinsic connection between the BGK model and time‐dependent, non‐linear, non‐homogeneous shallow‐water equations enables us to solve shallow‐water equations automatically with our kinetic scheme. The analytical solution, experimental measurements, and numerical calculations for problems associated with roll‐waves down an inclined open channel and solitary waves incident on a sloped beach are also presented. Copyright © 2005 John Wiley & Sons, Ltd.