Reflection of surface waves by submerged cylinders

Reflection from submerged cylinders are studied by means of integral equations. By expressing the solution as a distribution of vortices, the integral equations become non-singular for closed contours. It is shown that the method gives a short and easy proof for the classical result that no reflection occurs for the circular cylinder. The reflection power for the elliptic contour and the flat plate are studied when the bodies are situated deeply below the surface.     

Radiation and scattering of surface waves by a group of submerged,horizontal, circular cylinders

One of the major methods available for investigating the interaction of water waves with arbitrarily shaped structures is based on the classical theory of Green's functions. For multiple bodies, however, this technique can become expensive in terms of both computer storage and execution time and it is desirable for special geometrics where possible, to seek simpler methods. In this paper, the radiation and scattering of surface waves by a group of parallel, horizontal, circular cylinders, submerged in deep water is studied using a method involving multipole potentials. The method is developed for any number of submerged, parallel, horizontal cylinders with arbitrary positions and radii. In particular, hydrodynamic coefficients are determined for various configurations of two and three cylinders and a comparison is made with results obtained for a cylinder in isolation.     

A finite element model of interaction between viscous free surface waves and submerged cylinders

This paper describes the simulation of the flow of a viscous incompressible Newtonian liquid with a free surface. The Navier–Stokes equations are formulated using a streamline upwind Petrov–Galerkin scheme, and solved on a Q-tree-based finite element mesh that adapts to the moving free surface of the liquid. Special attention is given to fitting the mesh correctly to the free surface and solid wall boundaries. Fully non-linear free surface boundary conditions are implemented. Test cases include sloshing free surface motions in a rectangular tank and progressive waves over submerged cylinders.     

The interaction of surface water waves with submerged breakwaters

This paper concerns the behaviour of nonlinear regular waves interacting with rectangular submerged breakwaters. A new series of experimental results is presented and compared with numerical calculations based upon a Boundary Element Method (BEM) that utilises multiple fluxes to deal with the discontinuities encountered at the corners of the domain. Specifically, comparisons concern both the spatial water surface profiles at various times and the spatial evolution of the harmonics generated by the breakwaters, the latter being an important focus for the paper. The BEM is shown to accurately model both the water surface profile and the harmonic generation, provided the breakwater width is sufficient to ensure that flow separation is not a controlling influence. Furthermore, evidence is provided to confirm that reflection from rectangular submerged breakwaters is fundamentally a linear phenomenon.     

Porosity effects on non-breaking surface waves over permeable submerged breakwaters

This study investigated how the porosity of submerged breakwaters affects non-breaking wave transformations. Eight model geometries each with six different porosities, from 0.421 to 0.912, were also considered. Experimental results reveal that the model width has little effect on wave reflection and transmission when the model heights are fixed. The transmission coefficient is maximum at a kh in the range from 1.3 to 2.0 and minimum at a kh around 0.7. The wave reflection maximum is at kh of near 0.5. The energy loss of the primary waves is maximum near kh=0.81 and minimum when the porosity of the model is large. Porosity does affect wave transformation and its influence becomes significant as the heights of the models increase. For the range of porosities tested, wave energy loss from the primary harmonic was found to be almost constant at around 0.4 when kh >1.3, decreasing slowly when kh <1.3; wave energy loss decreases for porosities above 0.75.     

Scattering of water wave by a submerged horizontal plate and a submerged permeable breakwater

Based on a two-dimensional linear water wave theory, this study develops the boundary element method (BEM) to examine normally incident wave scattering by a fixed, submerged, horizontal, impermeable plate and a submerged permeable breakwater in water of finite depth. Numerical results for the transmission coefficients are also presented. In addition, the numerical technique's accuracy is demonstrated by comparing the numerical results with previously published numerical and experimental ones. According to that comparison, the transmission coefficient relies not only on the submergence of the horizontal impermeable plate and the height of the permeable breakwater, but also on the distance between horizontal plate and permeable breakwater. Results presented herein confirm that the transmission coefficient is minimum for the distance approximately equal to four times the water depth.     

Scattering of water wave by a submerged horizontal plate and a submerged permeable breakwater

Based on a two-dimensional linear water wave theory, this study develops the boundary element method (BEM) to examine normally incident wave scattering by a fixed, submerged, horizontal, impermeable plate and a submerged permeable breakwater in water of finite depth. Numerical results for the transmission coefficients are also presented. In addition, the numerical technique's accuracy is demonstrated by comparing the numerical results with previously published numerical and experimental ones. According to that comparison, the transmission coefficient relies not only on the submergence of the horizontal impermeable plate and the height of the permeable breakwater, but also on the distance between horizontal plate and permeable breakwater. Results presented herein confirm that the transmission coefficient is minimum for the distance approximately equal to four times the water depth.     

Scattering of surface waves in a basin with a statistically uneven bottom

The propagation of surface waves over a statistically uneven bottom is considered on the basis of equations of the linear potential theory. The expression which emphasizes the dispersion influence is obtained for the spatial decrement of the average field. The north-east Pacific is considered as an example.Translated by Mikhail M. Trufanov.     

Harmonic generation by waves propagating over a submerged step

Harmonic generation by waves propagating over a two-dimensional submerged step is investigated. A nonlinear theory correct to second order is presented for steps of infinite and finite lengths subjected to single harmonic waves.The boundary value problem for the second-order scattered velocity potential is linearly decomposed into two separate boundary value problems, each having only one inhomogeneous boundary condition.Theoretical results indicate that the higher harmonics are generated in the shallow-water region over a step and then are transmitted to the deeper water as free waves.Numerical calculations compare favourably with existing experimental data.     

Wave force coefficients for horizontally submerged rectangular cylinders

The results of wave force measurements carried out on a section of horizontally submerged rectangular cylinders, which are used as pontoons in many offshore structures, are reported in this paper. Two rectangular cylinders with aspect (depth–breadth) ratios equal to $\frac{1}{2}$ and $\frac{3}{4}$ and a square section (aspect ratio=1.0) cylinder are chosen for this study. Experiments are carried out in a wave tank at a water depth of 2.2 m at low Keulegan–Carpenter (KC) numbers to measure the horizontal and vertical wave forces acting on a 100 mm section, located at mid-length of the cylinders. For each cylinder, tests are carried out for two relative depths of submergence of 2.68 and 4.68. Measured wave forces in regular and irregular waves are then used to derive drag (C_D) and inertia coefficients (C_M). The analysis show that at very low KC numbers the inertia coefficients for all cylinders approached the potential flow values for both horizontal and vertical forces. The drag coefficients at low KC numbers exhibited large values and they decreased sharply with increase in KC number. For the square cylinder, where relatively a large KC number is obtained compared to other cylinders, inertia coefficients reached minimum values in the range of KC of about 3–4 and increased thereafter. In this range, C_M values are about 50% or so, smaller than the same at KC close to zero. The results of the experiments reveal that aspect ratio has large influence on hydrodynamic coefficients.     

Horizontal and vertical cylinders in waves

The present investigation examines a vertical cylinder and a horizontal cylinder in progressive waves. The physical differences between the flows are explored and experimental results are compared to previous planar harmonic flow measurements. It is found that modifications to the usual Morison approach are required in some cases to adequately account for the orbital motions of the fluid and to account for the orientation of the orbits with respect to the cylinder axis. The axial variations of the wave force on vertical cylinders are considered in order to evaluate the common practice of assuming constant values of $\text{C}{}_{\mathrm{m}}\text{and}\text{C}{}_{\mathrm{D}}$ over the entire span. Lastly, the methods of computing force transfer coefficients from a force record are examined and several sources of error are identified and briefly discussed.     

Bragg reflection of water waves by multiple submerged semi-circular breakwaters

This study examines the Bragg reflection of water waves by multiple submerged semi-circular breakwaters. The multipole expansions combined with the shift of polar coordinates are used to develop full linear potential solutions of the problem. In the full solutions, the obliquely and normally incident waves are independently considered. Experimental tests are carried out to measure the reflection and transmission coefficients of the breakwaters at different wave periods and body spacings. The analytical results are in reasonable agreement with the experimental data. The peak reflection coefficient of multiple submerged semi-circular breakwaters and the bandwidth of Bragg reflection are carefully examined by numerical examples. Some significant results for practical application are discussed.     

Scattering of gravity waves by multiple surface-piercing floating membrane

Interaction of surface gravity waves with multiple vertically moored surface-piercing membrane breakwaters in finite water depth is analyzed based on the linearized theory of water waves. The study is carried out using least square approximation method to understand the effect of the vertical membrane as effective breakwater. Initially the problem is studied for a single membrane wave barrier but for the case of multiple membrane breakwaters the study is carried out using the method of wide-spacing approximation. In the present study, it is observed that the deflection of the membrane is reduced with the increase in the stiffness parameter of the mooring lines attached to the membrane. In the case of single surface-piercing membrane with moored and fixed edge conditions, the reflection and transmission coefficients are compared and analyzed in detail. The resonating pattern in the reflection coefficients are also observed for multiple floating membrane which can also be referred as Bragg's resonance. In the presence of the porosity constant the wave reflection is also observed to be decreasing and the change in the distance between the vertical floating breakwaters also helps in the attenuation of wave height. It is observed that the presence of multiple floating breakwater helps in the reduction of wave height in the transmitted region.     

Interference effects between flexible cylinders in waves

An approximate method is presented to estimate the hydrodynamic loading and associated structural response of each of a pair of free-standing, bottom-nounted, flexible circular cylinders subjected to a regular train of linear surface waves. The cylinders are aligned parallel to the incident wave direction and the response of each is assumed to be one-dimensional and governed by a beam equation. The solution technique for the fluid velocity potential involves replacing scattered waves by equivalent plane waves together with non-planar, first-correction terms, and can be shown to be a large-spacing approximation. Numerical results are presented which show the influence of the various wave and structural parameters on the hydrodynamic loading and dynamic response of the individual cylinders.     

An iterative procedure for the diffraction of water waves by multiple cylinders

Diffraction of water waves by an array of vertical cylinders of circular cross section is studied. In order to account for first order interaction among the cylinders, the body boundary condition is satisfied for each cylinder considering the scattered wave field from other cylinders in an iterative way. After each iteration, coefficients in the partial wave decomposition of the wave potential are modified. Convergence is fast for the whole range of frequencies and for a large number of bodies, compared with exact algebraic methods of Linton and Evans [J. Fluid Mech. 46 (1990) 549] and Kagemoto and Yue [J. Fluid Mech. 166 (1) (1986) 189].     

Experimental investigation of the harmonic generation by waves over a submerged plate

Experiments in a wave flume have been performed to analyse the nonlinear interaction between regular gravity waves and a submerged horizontal plate used as breakwater. A new method, based on the Doppler shift generated by a moving probes, has been used to discriminate the incident fundamental mode and the reflected fundamental mode. The relationships of the reflection and transmission coefficients to the wave number at different submergence depth ratios are presented. The accurate discrimination, by this method, of the phase-locked and free modes allows the quantification of the higher harmonics generated by the breakwater and the analysis of the nonlinear interaction between the waves and the submerged plate. The transfer of energy from the fundamental mode to higher harmonics is very large in the cases of small submergence depth ratios. The vortices produced at the edges take part in the production of higher harmonics by interaction with the free surface but involve, at the same time, a dissipation process that increases the efficiency of the breakwater.     

Modelling of waves and currents around submerged breakwaters

Recent experimental data collected during the DELOS project are used to validate two approaches for simulating waves and currents in the vicinity of submerged breakwaters.The first approach is a phase-averaged method in which a wave model is used to simulate wave transformation and calculate radiation stresses, while a flow model (2-dimensional depth averaged or quasi-3D) is used to calculate the resulting wave driven currents. The second approach is a phase resolving method in which a high order 2DH-Boussinesq-type model is used to calculate the waves and flow.The models predict wave heights that are comparable to measurements if the wave breaking sub-model is properly tuned for dissipation over the submerged breakwater. It is shown that the simulated flow pattern using both approaches is qualitatively similar to that observed in the experiments. Furthermore, the phase-resolving model shows good agreement between measured and simulated instantaneous surface elevations in wave flume tests.     

Scattering of obliquely incident water waves by partially reflecting non-transmitting breakwaters

In the present paper, analytic solutions are derived for scattering of water waves obliquely incident to a partially reflecting semi-infinite breakwater or breakwater gap. In order to examine the correctness of the derived solutions, they are compared with the solutions derived by McIver (1999) and Bowen and McIver (2002) for a semi-infinite breakwater and a breakwater gap, respectively, in the case of perfect reflection. The derived analytic solutions are used to investigate the effect of reflection coefficient of the breakwater and wave incident angle upon the tranquility at harbor entrance. The tranquility is deteriorated by the reflected waves as the reflection coefficient increases and as waves are incident more obliquely.     

Hydrodynamics of a submerged hydrofoil advancing in waves

The hydrodynamic problem of a hydrofoil travelling at constant speed in water waves has been investigated through velocity potential theory. The boundary conditions on the free surface have been linearized, and the effects are accounted for through the Green function. The overall problem is decomposed into the steady forward speed problem and periodic wave radiation and diffraction problems. Each of these problems is solved using the boundary integral equation over the hydrofoil surface together with a vortex sheet behind the trailing edge. The body surface boundary condition is imposed on its mean position. As a result the steady potential will contribute a well-known m_j term to the body surface boundary condition on the radiation problem. The numerical difficulty in dealing with this term is effectively resolved through a difference method. The effects of the thickness on the wave radiation and diffraction are investigated. The applicability of various reciprocity relationships in this problem is discussed.