Long wave reflection from submerged trapezoidal breakwaters

This study addresses the reflection and transmission of long waves from a trapezoidal breakwater and a series of trapezoidal breakwaters, using the matching method. A systematic shape transfer is derived to determine wave reflection and transmission. The peak Bragg reflection of long waves from a series of trapezoidal breakwaters is shifted toward low frequency. In spite of the spacing between any pair of breakwaters, the top plane width and the arrangement of the series of breakwaters are found to be the two major parameters in designing multiply composite Bragg breakwaters.     

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.     

Experimental study of strong reflection of regular water waves over submerged breakwaters in tandem

A series of laboratory experiments was carried out to investigate the strong reflection of regular water waves over a train of submerged breakwaters. Rectangular and trapezoidal shapes of submerged breakwaters are employed and compared for reflecting capability of incident waves. Measured reflection coefficients of regular waves over impermeable submerged breakwaters are verified by comparing with those of the eigenfunction expansion method. A very good agreement is observed. Reflection coefficients of permeable submerged breakwaters are less than those of impermeable breakwaters. The trapezoidal shape is recommended for a submerged breakwater in terms of reflecting capability and practical application.     

An application of Boussinesq equations to Bragg reflection of irregular waves

A numerical model based on the second-order fully nonlinear Boussinesq equations of Wei et al. [1995. Journal of Waterway, Port, Coastal and Ocean Engineering 121 (5), 251-263] is developed to simulate the Bragg reflection of both regular and irregular surface waves scattered by submerged bars. Particularly for incident regular waves, the computed results are observed to agree very well with the existing experimental data as presented by Davies and Heathershaw [1984. Journal of Fluid Mechanics 144, 419-446] and Kirby and Anton [1990. Proceedings of the 22nd International Conference on Coastal Engineering, ASCE, New York, pp. 757–768). In the case of incident irregular waves, the simulated results reveal that the distribution of Bragg reflection from irregular waves becomes more flat than that of regular waves. Due to lack of experimental data, the numerical results for incident irregular waves are compared with those of the evolution equation of the mild-slope equation [Hsu et al., 2002 Proceedings of the 24th Ocean Engineering Conference in Taiwan, pp. 70–77 (in Chinese)]. In addition, several parameters such as the number of bars, the relative height of bars and the spacing of bars affecting Bragg reflection are also discussed.     

淹没矩形防波堤透反射系数特性研究

采用解析方法研究了斜向入射波作用下淹没矩形防波堤的透反射系数特性.首先利用特征函数展开法导出了绕射势函数的分析解和透反射系数的计算公式,然后利用边界元方法验证了解析解,在此基础上利用解析解分析了若干工况下的防波堤透反射特性.计算结果表明,淹没矩形防波堤截面的宽度、高度和相对位置以及入射角的改变都不同程度影响反射系数和透射系数.在中等深度条件下,对于一定频率的波浪,位置和尺寸适当的淹没矩形堤可以反射大部分斜向入射波.研究结果对设计淹没的矩形防波堤具有重要的参考价值.     

Bragg resonant reflection of carrier waves composing wave groups

Numerical analyses for the Bragg resonant reflection of carrier waves associated long waves due to sinusoidally varying seabeds are performed by using a set of coupled ordinary differential equations derived from the Boussinesq equations. The Boussinesq equations are firstly approximated with the Fourier decomposition. The coupled governing equations are then derived and used to simulate evolution of both short and long wave components. It is also found that wave groups are generated by two carrier waves with slightly different frequencies. The wave energy of the initial wave components is transferred to other harmonic components during propagation over a long distance. Evolution and reflection of both short and long waves were largely affected by nonlinearity.     

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.     

Three-dimensional analysis of submerged, moored, horizontal, rigid cylinders used as breakwaters

Wave attenuation by moored cylinders is considered. The cylinders are submerged with their axes horizontal. Linear potential theory is applied. Three-dimensional motions of the cylinders subjected to normal and oblique monochromatic waves are determined using potential theory and a boundary integral method. Each cylinder has length 9.1 m (30 ft) and radius 1.5 m (5 ft), with its top 1.5 m (5 ft) below the still water line and its bottom 3.0 m (10 ft) above the seabed. Free-surface elevations are obtained for a single cylinder and for two cylinders in series. These configurations are effective wave barriers for a range of wave frequencies and incident angles.     

Wave reflection, transmission and harmonics due to different submerged structures

I~IOXdifferent tabes of submerged structures restricting wave development are extensively appliedin coastal and barber engineering. Generally speaking, the submerged structures are classified intothree types (see Fig. 1 ): a submerged horizontal plate (fixed submerged structure), a submergedbreakwater (rectangular or trapezoidal) and a step-take structure (toPOgraphy). In order to get aclear understanding of the characteristics of wave motion around submemed structures, manyFig. 1. Fol'InS…     

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.     

Bragg reflection of water waves by multiple floating horizontal flexible membranes with submerged rectangular bars on the seabed

Bragg reflection of water waves by multiple floating horizontal flexible membranes is investigated based on the linear wave theory and the assumption of small membrane response. Under the floating horizontal membranes, periodical submerged rectangular bars are arranged on the flat seabed. The total reflection and transmission coefficients are obtained by using the eigenfunction expansion method and the wide spacing approximation. The calculated coefficients are validated with the results available in the literature, which shows that the present method is applicable. The characteristics of Bragg reflection are systematically investigated by changing various parameters including the height of the rectangular bars, the number, the tension, the spacing, and the length of the flexible membranes. The results can help designing multiple floating horizontal flexible membranes as effective floating breakwaters by taking advantage of Bragg reflection.     

堆石防波堤不规则波浪反射系数试验研究

结合物理模型试验,分析斜坡坡度、波陡、相对水深、护面类型和破波参数等因素对堆石防波堤不规则波浪反射系数的影响规律。将常用的Van der Meer公式,Seelig公式,Postma公式和Davison公式计算值和实测值进行比较,并结合试验数据,基于有效波高和平均周期定义的Iribarren数,得出堆石防波堤不规则波浪反射系数经验公式。结果表明,该公式能较好地计算不规则波作用下块石和扭王块体护面堆石防波堤波浪反射系数。     

Performance of solid and perforated U-type breakwaters under regular and irregular waves

An experimental investigation of U-type breakwaters was carried out in a laboratory channel. Both regular and irregular waves were used during testing. Two types of breakwaters such as solid and perforated were studied to analyse the porosity effect of structures. In order to investigate performance of these breakwaters for different immersion depths, four depths of immersions of the solid and perforated breakwaters were selected. Different wave groups were generated over these breakwaters, and the transmission, reflection and energy dissipation characteristics of each breakwater were determined. Three coefficients such as transmission, reflection and energy dissipation coefficients, which were named as C_t, C_r, and C_l, respectively, were used during the evaluation of the test results. The most important parameters governing performance of these breakwaters were determined by using earlier investigations and experimental results. These parameters were expressed as a dimensionless group by using π theory. Based on the test results, empirical expressions were formulated to describe the C_t, C_r, and C_l for different immersion depths of solid and perforated breakwaters under regular and irregular waves.     

Investigation of Π-type breakwaters performance under regular and irregular waves

In this paper, performance of solid and perforated Π-type breakwaters was investigated experimentally. Both regular and irregular waves were used during testing. Four depths of immersions were selected for each breakwater and wave type. Different wave groups were generated over these breakwaters, and the transmission, reflection and energy-dissipation characteristics were determined. The results of the experimental study were evaluated and some empirical expressions based on the results were suggested to define the transmission, reflection and energy-dissipation coefficients for different immersion depths of solid and perforated breakwaters under regular and irregular waves. Moreover, performance of solid and perforated Π-type breakwaters were compared with that of solid and perforated U-type breakwaters investigated by Günaydın and Kabdaşlı [2006. Performance of solid and perforated U-type breakwaters under regular and irregular waves. Ocean Engineering 31, 1377–1405]. These comparisons showed that the most reasonable model and wave type are selected to determine requiring performance parameters.     

The effects of reflection coefficient of the harbour sidewall on the performance of floating breakwaters

This paper studies the interaction of linear water waves with a moored floating breakwater with a leeward boundary composed from a vertical wall. This describes a real modelling for the case of ports in contrary to the problems of unbounded domains. It involves the application of a partial reflection boundary condition for this sidewall. In fact, the partial reflection problem in practical application is of great importance in the design of a harbour or breakwater and mainly for short waves. The reflection coefficient of the harbour boundary (sidewall) plays an important role in modifying the performance of the floating breakwater. Moreover, it reduces the resonant peaks appearing inside the ports due to the energy accumulation in an enclosed domain. The effects of the variation of structural parameters of the breakwater on the transmitted wave height are discussed in details for various values of partial reflection.     

Wave Power Focusing due to the Bragg Resonance

Wave energy has drawn much attention as an achievable way to exploit the renewable energy. At present, in order to enhance the wave energy extraction, most efforts have been concentrated on optimizing the wave energy convertor and the power take-off system mechanically and electrically. However, focusing the wave power in specific wave field could also be an alternative to improve the wave energy extraction. In this experimental study, the Bragg resonance effect is applied to focus the wave energy. Because the Bragg resonance effect of the rippled bottom largely amplifies the wave reflection, leading to a significant increase of wave focusing. Achieved with an energy conversion system consisting of a point absorber and a permanent magnet single phase linear motor, the wave energy extracted in the wave flume with and without Bragg resonance effect was measured and compared quantitatively in experiment. It shows that energy extraction by a point absorber from a standing wave field resulted from Bragg resonance effect can be remarkably increased compared with that from a propagating wave field (without Bragg resonance effect).     

恒定流对潜堤波浪反射系数影响研究

为研究恒定均匀流对直立潜堤波浪反射系数的影响,采用了高阶边界元方法和混合欧拉-拉格朗日时间步进方法更新瞬时自由水面,运用基于二次形状函数的网格重分方案结合四阶龙格库塔方法更新时间积分,并运用两点法得到淹没潜体前的反射系数。将数值模拟得到的波面数据与物理模型试验结果进行对比,两者吻合良好。通过数值模拟研究给出了入射波周期、水流流速、潜堤相对宽度和相对淹没深度对反射系数的影响。结果表明,水流流速和潜堤相对淹没深度对反射系数影响较大,潜堤相对宽度的影响相对较小;水流的存在会使入射波周期等参数对反射系数的影响程度发生改变,在实际工程设计时应考虑水流对潜堤防浪效果的影响。     

透空式防波堤周围的非线性波浪传播的数值模拟

对非线形波浪在透空式防波堤周围的波浪变形进行了数值模拟,在Boussinesq波浪方程中加入与透空建筑物有关的新的耗散项,从而界定了透空建筑物引起的部分反射和透射,波浪折射衍射的传播过程通过控制方程求解。波浪控制方程通过有限差分方法求解。模型应用于模拟波浪经过具有部分反射的群桩式透空结构,结果表明透空式防波堤可以有效地衰减波浪,是重力式结构的一种替代形式。     

Wave height transformation and set-up between a submerged permeable breakwater and a seawall

In this study,we investigated wave transformation and wave set-up between a submerged permeable breakwater and a seawall.Modified time-dependent mild-slope equations,which involve parameters of the porous medium,were used to calculate the wave height transformation and the mean water level change around a submerged breakwater.The numerical solution is verified with experimental data.The simulated results show that modulations of the wave profile and wave set-up are clearly observed between the submerged breakwater and the seawall.In contrast to cases without a seawall,the node or pseudo-node of wave height evolution can be found between the submerged breakwater and the seawall.Higher wave set-up occurs if the nodal or pseudo-nodal point appears near the submerged breakwater.We also examined the influence of the porosity and friction factor of the submerged permeable breakwater on wave transformation and set-up.