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.     

Reflected and transmitted waves in a channel with side porous mattresses

The problem of wave propagation and wave damping in a channel with side porous mattresses of arbitrary shape protruding from the walls is studied. The solution was achieved by applying 3-D boundary element method and was employed to study wave field in the channel and to analyze the effect of the geometry of the mattresses and physical and hydraulic properties of porous material on wave damping. The results show that wave damping in the channel strongly depends on wave parameters, especially, on the wave number. Wave reflection and transmission decrease with increasing the wave number. The results also show that the wave field in the channel strongly depends on the geometry of the mattresses as well as on physical and hydraulic properties of porous material used to build these wave dampers. The geometry of the mattresses and physical and hydraulic properties of porous material have a moderate effect on wave reflection and a significant effect on wave transmission. The results show that wave transmission down the channel decreases with increasing the length and thickness of the mattresses. Moreover, wave transmission decreases with increasing the porosity and damping properties of porous media used to build the mattresses. The analysis shows that porous mattresses protruding from the channel walls are very efficient in damping water waves propagating down the channel and may be built in channels to reduce high waves and achieve desired wave conditions. Theoretical results are in reasonable agreement with experimental data.     

A highly wave dissipation offshore breakwater

The aim of this paper is to develop an offshore breakwater, for which coefficients of both the wave reflection and transmission have low values. The breakwater is suggested to compose of n layers of porous materials with different porosities. A complex eigen function method is used in the theoretical analysis. Continuities of both mass flux and fluid pressure are assumed at interfaces between every two adjoining porous materials and at the interface between end materials and water region. Following a series of mathematical processes, the coefficients of the wave transmission and reflection along with the wave energy loss are calculated. The porosity of materials is varied in computations; and results are compared among structures composing of different layers of porous materials. A single layer offshore breakwater is shown to reduce simultaneously the coefficients of transmission and reflection only when the structure is very wide in the direction of wave propagation, and the structure material has a high porosity. A multilayer breakwater, however, can function well in reducing both coefficients at a much narrower width; structure having more layers can be more effective at narrower width. Finally, several experiments are conducted; theoretical computations and experimental results agree well.     

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 interaction with twin plate wave barrier

The wave transmission and reflection characteristics and wave induced pressures on single surface plate and twin plate barriers were investigated experimentally for a wide range of wave heights and periods in regular and random waves. Seven different spacing between the plates were tested. It is found in general, hydrodynamically the twin plate is better than the single surface plate to reduce the wave transmission and increase the wave reflection. It is found that the transmission coefficient of twin plate reduced from 0.8 to 0.3 when the relative plate width is increased from 0.18 to 0.84. Transmission coefficient of twin plate barrier shows oscillating behavior, when relative plate width is increased due to blocking and pumping effect. The reflection coefficient increased from 0.25 to 0.65, when the relative width of the plate is increased from 0.18 to 0.84. The increase in spacing between the plates was also found to increase the reflection coefficient. The transmission coefficient, K_t for 98% probability of non-exceedence was found to be minimum and is about 0.60 when the relative spacing between the plate is about 0.12, compared to K_t=0.76 for single surface plate. The reflection coefficient for 98% probability of non-exceedence was found to exceed 0.66 for single surface plate, whereas it is 0.73 for twin plate with relative spacing of about 0.40. From the investigation with wide range of input parameters, it is found that the twin plate barrier needs to be designed for highest 98% pressure ratio of 2.0, which is equal to the static pressure induced by the design incident wave height.     

Interaction of non-breaking regular waves with a periodic array of artificial porous bars

A computational model is developed to investigate the wave damping characteristics of a periodic array of porous bars. The transmission and reflection coefficients as well as the wave energy dissipation are evaluated relating to the physical properties and geometric factors of bars. It is shown that the porosity, number, width and height of bars all play important roles in the wave damping characteristics, compared to other factors such as the intrinsic permeability. It is observed that like impermeable bars, permeable bars display Bragg phenomenon. However, Bragg reflection produced by permeable bars is smaller than that by impermeable bars. Permeable bars reflect smaller waves, transmit smaller waves and dissipate more wave energy. It is indicated that if the porosity increases, both the reflection and transmission coefficients decrease and more wave energy is dissipated. Further, it is found that the porosity controls the magnitude, but not the oscillation frequency of the reflection coefficient, which depends only on the number of bars.     

Wave reflection from vertical breakwater with porous structure

This paper presents numerical solutions for the wave reflection from submerged porous structures in front of the impermeable vertical breakwater. A new time-dependent mild-slope equation involves the parameters of the porous medium including the porosity, the friction factor and the inertia coefficient, etc. is derived for solving the boundary value problem. A comprehensive comparison between the present model and the existing analytical solution for the case of simple rectangular geometries of the submerged structure is performed first. Then, more complicated cases such as the inclined and trapezoidal submerged porous structures in front of the vertical breakwater with sloping bottom are considered. This study also examines the effects of the permeable properties and the geometric configurations of the porous structure to the wave reflection. It is found that the submerged porous structure with trapezoidal shape has more efficiency to reduce the wave reflection than that of triangular shape. The numerical results show that the minimum wave reflection is occurred when the breakwater is located at the intermediate water depth.     

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.     

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.     

Wave interaction with ‘’-type breakwaters

The wave transmission, reflection and energy dissipation characteristics of ‘’-type breakwaters were studied using physical models. Regular and random waves in a wide range of wave heights and periods and a constant water depth were used. Five different depths of immersion (two emerged, one surface flushing and two submerged conditions) of this breakwater were selected. The coefficient of transmission, K_t, and coefficient of reflection, K_r, were obtained from the measurements, and the coefficient of energy loss, K_l was calculated using the law of balance of energy. It was found that the wave transmission is significantly reduced with increased relative water depth, d/L, whether the vertical barrier of the breakwater is surface piercing or submerged, where ‘d’ is the water depth and ‘L’ is the wave length. The wave reflection decreases and energy loss increases with increased wave steepness, especially when the top tip of the vertical barrier of this breakwater is kept at still water level (SWL). For any incident wave climate (moderate or storm waves), the wave transmission consistently decreases and the reflection increases with increased relative depth of immersion, Δ/d from −0.142 to 0.142. K_t values less than 0.3 can be easily obtained for the case of Δ/d=+0.071 and 0.142, where Δ is the height of exposure (+ve) or depth of immersion (−ve) of the top tip of the vertical barrier. This breakwater is capable of dissipating wave energy to an extent of 50–80%. The overall performance of this breakwater was found to be better in the random wave fields than in the regular waves. A comparison of the hydrodynamic performance of ‘’-type and ‘T’-type shows that ‘T’-type breakwater is better than ‘’-type by about 20–30% under identical conditions.     

Wave interaction with T-type breakwaters

The wave transmission, reflection and energy dissipation characteristics of partially submerged ‘T'-type breakwaters (Fig. 1) were studied using physical models. Regular and random waves, with wide ranges of wave heights and periods and a constant water depth were used. Five different depths of immersions of the ‘T'-type breakwater were selected. The coefficient of transmission, K_t, coefficient reflection, K_r, were obtained from the measurements and the coefficient of energy loss, K_l is calculated using the law of conservation of energy. It is found that the coefficient of transmission generally reduces with increased wave steepness and increased relative water depth, d/L. This breakwater is found to be effective closer to deep-water conditions. K_t values less than 0.35 is obtained for both normal and high input wave energy levels, when the horizontal barrier of the T type breakwater is immersed to about 7% of the water depth. This breakwater is also found to be very efficient in dissipating the incident wave energy to an extent of about 65% (i.e. K_l>0.8), especially for high input wave energy levels. The wave climate in front of the breakwater is also measured and studied.

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Fig. 1. Schematic view of the T-type breakwater.     

Breaking solitary wave evolution over a porous underwater step

Solitary wave evolution over a shelf including porous damping is investigated using Volume-Averaged Reynolds Averaged Navier–Stokes equations. Porous media induced damping is determined based on empirical formulations for relevant parameters, and numerical results are compared with experimental information available in the literature. The aim of this work is to investigate the effect of wave damping on soliton disintegration and evolution along the step for both breaking and non-breaking solitary waves. The influence of several parameters such as geometrical configuration (step height and still water level), porous media properties (porosity and nominal diameter) or solitary wave characteristics (wave height) is analyzed. Numerical simulations show the porous bed induced wave damping is able to modify wave evolution along the step. Step height is observed as a relevant parameter to influence wave evolution. Depth ratio upstream and downstream of the edge appears to be the more relevant parameter in the transmission and reflection coefficients than porosity or the ratio of wave height–water depth. Porous step also modifies the fission and the solitary wave disintegration process although the number of solitons is observed to be the same in both porous and impermeable steps. In the absence of breaking, porous bed triggers a faster fission of the incident wave into a second and a third soliton, and the leading and the second soliton reduces their amplitude while propagating. This decrement is observed to increase with porosity. Moreover, the second soliton is released before on an impermeable step. Breaking process is observed to dominate over the wave dissipation at the porous bottom. Fission is first produced on a porous bed revealing a clear influence of the bottom characteristics on the soliton generation. The amplitude of the second and third solitons is very similar in both impermeable and porous steps but they evolved differently due to the effect of bed damping.     

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.     

斜向波与带填料开孔板式防波堤相互作用的研究

基于线性势流理论,对斜向波与消浪室内带部分填料的开孔板式防波堤结构的相互作用进行了理论研究.利用匹配特征函数展开法给出了开孔防波堤结构反射系数和波浪力系数的理论计算方法,在极限情况下的计算结果与已有结果一致.利用数值算例分析了开孔结构反射系数和波浪力系数的主要影响因素.增加消浪室内填料厚度通常可以降低开孔墙结构的波浪力系数,加大结构的反射系数.随着波浪入射角度的增加,带填料防波堤结构的反射系数值先减小,达到最小值后又单调增加.     

Experimental investigating on the reflected waves from the caisson-type vertical porous seawalls

The hydrodynamic efficiencies of caisson-type vertical porous seawalls used for protecting coastal areas were calculated in this study. Physical models were developed to compare the wave reflection from vertical plane, semi-porous, and porous seawalls caused by both regular and random waves. Tests were carried out for a wide range of wave heights, wave periods, and different water depths (d=0.165, 0.270 and 0.375 m). The performance regarding the reflected waves from porous and semi-porous seawalls showed improvement when compared with those from the plane seawall. The reflection coefficients of the porous and semi-porous seawalls were calculated as 0.6 and 0.75, respectively, while the coefficient for the fully reflecting plane vertical wall was significantly higher (0.9). It was also observed that the reflection coefficient decreases with increase in wave steepness and relative water depth. In addition, the reduction in the reflection coefficient of porous and semi-porous seawalls, as compared to that of a plane seawall, was observed for both regular and random waves. New equations were also proposed to calculate the reflection coefficient of different types of seawalls with the aid of laboratory experiments. By verifying the developed equations using some other experimental data, it was validated that the equations could be used for practical situations. The results of the present study can be applied to optimize the design of vertical seawalls and for coastal protecting schemes.     

Wave reflection in 3D conditions

Based on recent experiments carried out in wave basin on breakwaters with armour layer of rocks and cubes, this paper examines the dependence of the reflection coefficient on wave directional spreading and obliquity. Results suggest that long-crested and short-crested waves give similar reflection. The reflection coefficient is markedly dependent on the wave angle of incidence. The performance of formulae available in the literature is checked against the new dataset and a significant improvement is proposed by including the wave obliquity factor that appears in the traditional expression for the overtopping discharge.     

A coupled numerical model for simulation of wave breaking and hydraulic performances of a composite seawall

A numerical model is developed by combining a porous flow model and a two-phase flow model to simulate wave transformation in porous structure and hydraulic performances of a composite type low-crest seawall. The structure consists of a wide submerged reef, a porous terrace at the top and an impermeable rear wall. The porous flow model is based on the extended Navier-Stokes equations for wave motion in porous media and k−ε turbulence equations. The two-phase flow model combines the water domain with the air zone of finite thickness above water surface. A unique solution domain is established by satisfying kinematic boundary condition at the interface of air and water. The free surface advection of water wave is modeled by the volume of fluid method with newly developed fluid advection algorithm. Comparison of computed and measured wave properties shows reasonably good agreement. The influence of terrace width and structure porosity is investigated based on numerical results. It is concluded that there exist optimum value of terrace width and porosity that can maximize hydraulic performances. The velocity distributions inside and in front of the structure are also investigated.     

Wave transmission and reflection characteristics of a rigid surface and submerged horizontal plate

The wave transmission and reflection characteristics of a rigidly fixed surface and submerged horizontal plate were investigated experimentally in detail for a wide range of incident wave steepnesses and for different depths of submerge of the plate in deep water conditions in regular water wave fields. The experiments were conducted at the Ocean Engineering Centre, Indian Institute of Technology, Madras, India, in a wave flume 10 m long, 0.3 m wide and in a constant water depth of 0.8 m. The horizontal plate is 0.22 m thick and 1.2 m in length, covering the enrire width of the flume. From the present investigation, it is found that for a rigid surface plate, the coefficient of transmission is a minimum and the coefficient of reflection is a maximum, but the maximum value of the coefficient of energy loss occurs for plates submerged closer to the still-water level (SWL) and not for the surface plate. It is also found that the value of the coefficient of reflection increases with the increase in the value of the Reddy-Neelamani (RN) number, the ratio of horizontal water particle excursion at the bottom of the plate in its absence to the length of the plate. The coefficient of transmission is found to decrease rapidly with increase in the value of RN number up to 0.1. The wave transmission is only 5% for RN from 0.1 to 0.2. It is also found that for RN number greater than 0.04, the minimum energy dissipation is consistently about 60% of the incident wave energy.     

Wave Propagation in A Converging Channel of Arbitrary Configuration

A numerical solution was derived to determine wave field in a converging channel bounded by rubble-mound jetties. The solution was achieved by applying boundary element method. The model was applied to analyze the effect of channel convergence, the cross-section of the jetties and their physical and damping properties on wave field in the channel. The study reveals numerous non-intuitive results specific for jetted and convergent channels. The analysis shows that wave reflection is usually low and is of secondary practical importance. Wave transmission strongly depends on the channel geometry and transmitted waves may be higher than incident waves, despite reflection and damping processes. Moreover, wave transmission depends on physical and damping properties of rubble jetties and the results show that wave transmission may increase with the increasing damping properties of jetties, which is a non-intuitive feature of wave fields in jetted channels. The analysis reveals several novel results of practical importance. It is shown that the rubble-mound jetties should be constructed from the material of high porosity, which ensures low transmission. More attention should be devoted to hydraulic properties of porous materials. It is recommended to use the material of moderate damping properties. The material of high damping properties often increases the wave transmission. It is possible, by a selection of rubble-mound material, to obtain lower transmission level for steep waves than for waves of moderate steepness. A series of laboratory experiments were conducted in the wave flume to verify the theoretical results. The comparisons show that theoretical results are in fairly good agreement with experimental data.     

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.