Wave reflection from partially perforated-wall caisson breakwater

In 1995, Suh and Park developed a numerical model that computes the reflection of regular waves from a fully perforated-wall caisson breakwater. This paper describes how to apply this model to a partially perforated-wall caisson and irregular waves. To examine the performance of the model, existing experimental data are used for regular waves, while a laboratory experiment is conducted in this study for irregular waves. The numerical model based on a linear wave theory tends to over-predict the reflection coefficient of regular waves as the wave nonlinearity increases, but such an over-prediction is not observed in the case of irregular waves. For both regular and irregular waves, the numerical model slightly over- and under-predicts the reflection coefficients at larger and smaller values, respectively, because the model neglects the evanescent waves near the breakwater.     

Reflection of irregular waves from perforated-wall caisson breakwaters

An analytical model has been developed that predicts the reflection of irregular waves normally incident upon a perforated-wall caisson breakwater. To examine the predictability of the developed model, laboratory experiments have been conducted for the reflection of irregular waves of various significant wave heights and periods impinging upon breakwaters having various wave chamber widths. For frequency-averaged reflection coefficients, though the overall agreement is fairly good between measurement and calculation, the model somewhat over-predicts the reflection coefficients at larger values, and under-predicts at smaller values. The model also underestimates the energy loss coefficients as wave reflection becomes larger. These differences occur because the model neglects the evanescent waves near the breakwater, which increase the energy loss at the perforated wall. The frequency-averaged reflection coefficient shows a minimum when the wave chamber width is approximately 0.2 times the significant wavelength, and it decreases with increasing wave steepness. Finally, it is shown that the reflection of irregular waves from a perforated-wall caisson breakwater depends on the wave frequency, so that the reflected wave spectrum shows a frequency dependent oscillatory behavior.     

Simplified analytical solutions for wave interaction with absorbing-type caisson breakwaters

Simplified analytical solutions are presented to model the interaction of linear waves with absorbing-type caisson breakwaters, which possess one, or two, perforated or slotted front faces which result in one, or two, interior fluid regions (chambers). The perforated/slotted surfaces are idealized as thin porous plates. Energy dissipation in the interior fluid region(s) inside the breakwater is modelled through a damping function. Under the assumption of potential flow and linear wave theory a boundary-value problem may then be formulated to describe wave interaction with the idealized structure. A solution to this simplified problem may be obtained by an eigenfunction expansion technique and an explicit analytical expression may be obtained for the reflected wave height. Using the experimental work of previous authors, damping coefficients are determined for both single and double chamber absorbing-type caisson breakwaters. Based on the damping for a single perforated-wall breakwater, a methodology is proposed to enable the estimation of the damping coefficients for a breakwater with two chambers. The theoretical predictions of the reflection coefficients for the two-chamber structures using the present model are compared with those obtained from laboratory experiments by other authors. It is found that the inclusion of the damping in the interior fluid region gives rise to improved agreement between theory and experiment.     

Wave interaction with a perforated wall breakwater with a submerged horizontal porous plate

This study examines the hydrodynamic performance of a new perforated-wall breakwater. The breakwater consists of a perforated front wall, a solid back wall and a submerged horizontal porous plate installed between them. The horizontal porous plate enhances the stability and wave-absorbing capacity of the structure. An analytical solution based on linear potential theory is developed for the interaction of water waves with the new proposed breakwater. According to the division of the structure, the whole fluid domain is divided into three sub-domains, and the velocity potential in each domain is obtained using the matched eigenfunction method. Then the reflection coefficient and the wave forces and moments on the perforated front wall and the submerged horizontal porous plate are calculated. The numerical results obtained for limiting cases are exactly the same as previous predictions for a perforated-wall breakwater with a submerged horizontal solid plate [Yip, T.L., Chwang, A.T., 2000. Perforated wall breakwater with internal horiontal plate. Journal of Engineering Mechanics ASCE 126 (5), 533–538] and a vertical wall with a submerged horizontal porous plate [Wu, J.H., Wan, Z.P., Fang, Y., 1998. Wave reflection by a vertical wall with a horizontal submerged porous plate. Ocean Engineering 25 (9), 767–779]. Numerical results show that with suitable geometric porosity of the front wall and horizontal plate, the reflection coefficient will be always rather small if the relative wave absorbing chamber width (distance between the front and back walls versus incident wavelength) exceeds a certain small value. In addition, the wave force and moment on the horizontal plate decrease significantly with the increase of the plate porosity.     

Hydrodynamic efficiency of partially immersed caissons supported on piles

The efficiency of the breakwater, which consists of caissons supported on two or three rows of piles, was studied using physical models. The efficiency of the breakwater is presented as a function of the transmission, reflection and the wave energy dissipation coefficients. Regular waves with wide ranges of wave heights and periods and constant water depth were used. Different characteristics of the caisson structure and the supporting pile system were also tested. It was found that, the transmission coefficient (k_t) decreases with increasing the relative breakwater draft D/L, increasing the relative breakwater width B/h, and decreasing the piles gap-diameter ratio G/d. It is possible to achieve k_t values less than 0.25 when D/L≥0.1. The reflection coefficient takes the opposite trend especially when D/L≤0.15. The proposed breakwater dissipates about 10-25% of the incident wave energy. Also, simple empirical equations are developed for estimating the wave transmission and reflection. In addition, the proposed breakwater model is efficient compared with other floating breakwaters.     

The theoretical study on diagonal wave interaction with perforated-wall breakwater with rock fill

The interaction of diagonal waves with perforated-wall breakwater partially filled with rock fill is studied using the linear potential theory. By means of the matched eigenfunction expansion method, an analytical method is presented to calculate the reflection coefficient and the wave force coefficient of the breakwater. The calculated results of the reflection coefficient for limiting cases are the same to the existing results. The main effect factors of the reflection coefficient and the wave force coefficient are analyzed by numerical examples. With the increasing of thickness of rock fill, the wave force coefficient on the perforated wall generally decreases, while the reflection coefficient increases. With the increasing of the incident angle of the wave, the reflection coefficient of the breakwater first decreases, reaches its minimum, and then increases monotonously.     

Wave reflection from perforated-wall caisson breakwaters

Analytical models for predicting wave reflection from a perforated-wall caisson breakwater have been developed. Most of the existing models deal with the case in which the waves are normally incident to the caisson lying on a flat sea bottom. In the present paper, using the Galerkin-eigenfunction method, an analytical model is developed that can predict the reflection coefficient of a perforatedwall caisson mounted on a rubble mound foundation when waves are obliquely incident to the breakwater at an arbitrary angle. The developed model is compared with other theoretical results and hydraulic experimental data.     

Performance characteristics of a conceptual ring-shaped spar-type VLFS with double-layered perforated-wall breakwater

A ring-shaped spar-type Very Large Floating Structure (VLFS) is proposed as an intermediate base for supporting deepwater resource exploitation far away from the coast line. The proposed VLFS is composed of eight rigidly connected deep-draft spar-type modules and an inside harbor. A double-layered perforated-wall breakwater is vertically attached to the VLFS and pierces through the water surface to attenuate the wave energy in the inside harbor. The hydrodynamic performance characteristics of the ring-shaped VLFS was experimentally evaluated in the present study, focusing on the motion responses, wave elevations, and wave run-ups. The natural periods of the motions in vertical plane were determined to be larger than 40 s, which is much larger than common wave periods. This enhanced the motion performance in vertical plane and afforded favorable habitation and operation condition on the VLFS. A large surge damping was induced by the vertical breakwater, which tended to significantly affect the surge and pitch motions, but had a negligible effect on the heave motion. The component frequencies of the wave elevations in the inside harbor and the wave run-ups were identical with those of the incident waves. The wave attenuation was frequency-dependent and effective for the common wave frequencies, with a smaller loss coefficient observed in higher sea state. The wave attenuation and wave run-ups tended to improve in the absence of the leeward walls.     

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.     

Interaction of oblique waves with infinite number of perforated caissons

An analytic solution based on the division of the fluid domain is developed for the interaction of obliquely incident waves with infinite number of perforated caissons. The whole fluid domain is firstly divided into infinite sub-domains according to the division of structures, and subsequently eigenfunction expansion is employed to represent the velocity potential in each domain. A phase relation is utilized for the analysis of wave oscillation in each caisson, and the character of structure geometry is considered in setting up the mathematical model of reflection waves. The reflection waves from the present analysis include many propagation waves traveling in different directions when the incident wave frequency is high. Benchmark examinations show that the continuous condition of water particle velocity is satisfied at the front walls of caissons, and the reflection coefficients keep agreement with the energy conservation relation very well when porous effect parameter is infinite. Numerical results show that the reflection coefficients of obliquely incident waves are smaller when the length of caissons is shorter at low frequency. The wave reflection coefficients and the wave forces normal to caissons decrease and the wave forces along caissons increase with the increase of the wave incident angle.     

Virtual wave gauges based upon stereo imaging for measuring surface wave characteristics

A virtual wave gauge (VWG) technique based on stereo imaging is developed to remotely measure water wave height, period, and direction. VWG minimizes computational costs by directly tracking the elevation of the water surface at selected points of interest using a Eulerian based dynamic searching algorithm. Results show that the VWG technique developed in this paper dramatically improves efficiency by two orders of magnitude compared to the traditional Lagrangian–Eulerian based point cloud method of stereo image processing. VWG is tested against traditional wave wire gauges to within 98% accuracy for significant wave height. Furthermore, the flexibility of the VWG is demonstrated in two field applications. First in an offshore breaking wave case, an array of VWGs is used to efficiently measure wave directionality. Second to investigate the reflection coefficient of a rock-mounted structure interacting with nearshore waves, linear and spatial VWG arrays are designed and implemented based on a priori information of the wave field from a preliminary VWG measurement. Overall, we demonstrate that the flexible and computational efficient VWG technique has the potential to make real-time remote stereo imaging wave measurements a reality.     

Characteristic friction coefficient and scale effects in oscillatory porous flow

This work presents a simple method to evaluate the performance of a porous breakwater when it is impinged with normal incidence by a non-breaking monochromatic wave train. It is based on: 1) a potential flow model for wave interaction with permeable structures and 2) a set of experimental tests on a rectangular porous structure with uniform granular distribution. A characteristic friction diagram is obtained considering wave energy balance in a control volume, minimising the error between the numerical model and the experimental results for the wave transmission coefficient. Results show that, for large breakwater widths, the reflection process reaches a saturation regime before the waves exit the structure at a distance from the seaside between the interval 0.2 < x/L < 0.45. For larger breakwater widths, the reflection coefficient is almost constant (except for “resonant” conditions) and wave transmission decreases exponentially. Under such conditions, the wave propagation through the porous medium depends on the relative diameter D/L and the porosity of the material; the dependence on the relative breakwater width B/L and the ratio diameter wave height D/H is weak. This diagram intends to be useful for preliminary engineering studies of breakwater's efficiency and performance and as an adequate selection criteria of the experimental stone diameter to minimize scale effects in laboratory studies.     

Wave motion over a breakwater system of a horizontal plate and a vertical porous wall

A two-dimensional analytical solution is presented to study the reflection and transmission of linear water waves propagating past a submerged horizontal plate and through a vertical porous wall. The velocity potential in each fluid domain is formulated using three sets of orthogonal eigenfunctions and the unknown coefficients are determined from the matching conditions. Wave elevations and hydrodynamic forces acting on the porous wall are computed. Reflection and transmission coefficients are presented to examine the performance of the breakwater system. The present analytical solutions are found in fairly good agreement with the available laboratory data. The results indicate that the plate length, the porous-effect, the gap between plate and porous wall, and the submerged depth of the plate all show a significant influence on the reflected and transmitted wave fields. It is also interesting to note that the submerged plate plays an important role in reducing the transmitted wave height, especially for long incident waves.     

Regular wave pressures and forces on submerged pipelines near a sloping boundary

The hydrodynamic pressures induced by regular waves around the circumference of a pipeline normal to the wave direction and near a rigid bed of slope 1:10 have been investigated in a wave flume. The pressures were integrated to obtain the force time history, from which the peak horizontal and vertical forces are evaluated. The maximum and root mean square horizontal and transverse force coefficients are correlated with the Keulegan–Carpenter (KC) number. The effect of the distance between the sloping bed and the pipeline on the force coefficients is discussed. The force coefficients are found to decrease with an increase in KC number and with the decrease in the relative clearance of the pipeline from the boundary. In addition, the reflection characteristics of the sloping bed in the presence of the pipeline as a function of surf similarity parameter and their comparison with the results from existing literature are also reported. The details of the model setup, experimental procedure, results and discussion are presented in this paper.     

On the radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas

The radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas of finite depth is investigated. The analytical expressions for the radiated and diffracted potentials are derived as infinite series by use of the method of separation of variables. The unknown coefficients in the series are determined by the eigenfunction expansion matching method. The expressions for wave forces, hydrodynamic coefficients and reflection and transmission coefficients are given and verified by the boundary element method. Using the present analytical solution, the hydrodynamic influences of the angle of incidence, the submergence, the width and the thickness of the structure on the wave forces, hydrodynamic coefficients, and reflection and transmission coefficients are discussed in detail.     

Wave reflection from coastal structures in design conditions

Based on an extensive database of more than 4000 data, this paper analyses wave reflection for various types of coastal structures in design conditions, such as smooth, rock and armour unit slopes. A new simple formula has been developed that relates the reflection coefficient to the breaker parameter and seems to fit all kinds of revetment materials by changing two coefficients. These coefficients depend only on the correct roughness factor, provided by recent research on overtopping discharges. The effects on reflection due to composite slopes and to low crests are also examined and proper extensions of the formula are provided.     

Full wave solution for hydrodynamic behaviors of pile breakwater

Rayleigh expansion is used to study the water-wave interaction with a row of pile breakwater in finite water depth. Evanescent waves, the wave energy dissipated on the fluid resistance and the thickness of the breakwater are totally included in the model. The formulae of wave reflection and transmission coefficients are obtained. The accuracy of the present model is verified by a comparison with existing results. It is found that the predicted wave reflection and transmission coefficients for the zero order are all highly consistent with the experimental data (Hagiwara, 1984; Isaacson et al., 1998) and plane wave solutions (Zhu, 2011). The losses of the wave energy for the fluid passing through slits play an important role, which removes the phenomena of enhanced wave transmission.     

Wave interaction with a wave absorbing double curtain-wall breakwater

This study examines the hydrodynamic performance of a wave absorbing double curtain-wall breakwater. The breakwater consists of a seaward perforated wall and a shoreward impermeable wall. Both walls extend from above the seawater to some distance above the seabed. Then the below gap allows the seawater exchange, the sediment transport and the fish passage. By means of the eigenfunction expansion method and a least square approach, a linear analytical solution is developed for the interaction of water waves with the breakwater. Then the reflection coefficient, the transmission coefficient and the wave forces acting on the walls are calculated. The numerical results obtained for limiting cases agree very well with previous predictions for a single partially immersed impermeable wall, the double partially immersed impermeable walls and the bottom-standing Jarlan-type breakwater. The predicted reflection coefficients for the present breakwater also agree reasonable with previous experimental results. Numerical results show that with appropriate structure parameters, the reflection and transmission coefficients of the breakwater may be both below 0.5 at a wide range of the relative water depth. At the same time, the magnitude of wave force acting on each wall is small. This is significant for practical engineering.     

Interaction of flexural gravity waves with shear current in shallow water

In the present study, the effect of shear current on the propagation of flexural gravity waves is analyzed under the assumptions of linearized shallow-water theory. Explicit expressions for the reflection and transmission coefficients associated with flexural gravity wave scattering by a step discontinuity in both water depth and current speed are derived. Further, trapping and scattering of flexural gravity waves by a jet-like shear current with a top-hat profile are examined and certain limiting conditions for the waves to exist are derived. The effects of change in water depth, current speed, incident wavelength and the angle of incidence on the group and phase velocities as well as on the reflection and transmission characteristics are analyzed through different numerical results.