Wave interaction with cylindrical porous piles

The interaction of waves with arrays of porous circular cylinders is studied theoretically and, under the assumption of potential flow and linear wave theory, an analytical solution is derived. The solution is valid for either submerged or emerged structures. The extension to the cases of unidirectional and multidirectional waves is obtained by means of a transfer function. For specific conditions the model gives the same solution as those previously presented by other authors. Numerical results are presented which exemplify diverse wave and mechanical parameters on the wave transformation due to the presence of a system of circular cylinders.     

Numerical study for waves propagating over a porous seabed around a submerged permeable breakwater: PORO-WSSI II model

The phenomenon of the wave, seabed and structure interactions has attracted great attentions from coastal geotechnical engineers in recent years. Most previous investigations have based on individual approaches, which focused on either flow region or seabed domain. In this study, an integrated model (PORO-WSSI II), based on the Volume-Averaged/Reynolds-Averaged Navier-Stokes (VARANS) equations and Biot's poro-elastic theory, is developed to investigate the mechanism of the wave-permeable structure-porous seabed interactions. The new model is verified with the previous experimental data. Based on the present model, parametric studies have been carried out to investigate the influences of wave, soil and structure parameters on the wave-induced pore pressure. Numerical results indicated: (i) longer wave period and larger wave height will obviously induce a higher magnitude of pore pressure at the leading edge of a breakwater; (ii) after a full wave-structure interaction, the magnitude of pore pressure below the lee side of a breakwater decreases with an increasing structure porosity while it varies dramatically with a change of structure height; and (iii) the seabed thickness, soil permeability and the degree of saturation can also significantly affect the dynamic soil behaviour.     

Water wave interaction with a floating porous cylinder

The interaction of water waves with a freely floating circular cylinder possessing a side-wall that is porous over a portion of its draft is investigated theoretically. The porous side-wall region is bounded top and bottom by impermeable end caps thereby resulting in an enclosed fluid region within the structure. The problem is formulated based on potential flow and linear wave theory and assuming small-amplitude structural oscillations. An eigenfunction expansion approach is then used to obtain semi-analytical expressions for the hydrodynamic excitation and reaction loads on the structure. Numerical results are presented which illustrate the effects of the various wave and structural parameters on these quantities. It is found that the permeability, size and location of the porous region may have a significant influence on the horizontal components of the hydrodynamic excitation and reaction loads, while its influence on the vertical components in most cases is relatively minor.     

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.     

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.     

Wave transformation over submerged permeable breakwater on porous bottom

A numerical model is presented in this study to investigate the wave transformation over a submerged permeable breakwater on a porous slope seabed. For this purpose, the time-dependent mild-slope equation is newly derived for waves propagating over two layers of porous medium. This new mild-slope equation involves the parameters of the porous medium, and it is a type of hyperbolic differential equation, therefore numerically efficient. The validity of the present model is verified based on the comparisons with the previous experiments. The effects of the permeable properties of both the porous seabed and the submerged permeable breakwater are discussed in detail. The geometry of the submerged permeable breakwater to the wave transformation is also investigated based on the numerical solutions.     

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.     

Incompressible SPH flow model for wave interactions with porous media

The paper presents an Incompressible Smoothed Particle Hydrodynamics (ISPH) method to simulate wave interactions with a porous medium. The SPH method is a mesh free particle modeling approach that is capable of tracking the large deformation of free surfaces in an easy and accurate manner. The ISPH method employs a strict incompressible hydrodynamic formulation to solve the fluid pressure and the numerical solution is obtained by using a two-step semi-implicit scheme. The ISPH flow model solves the unsteady 2D Navier–Stokes (NS) equations for the flows outside the porous media and the NS type model equations for the flows inside the porous media. The presence of porous media is considered by including additional friction forces into the equations. The developed ISPH model is first validated by the solitary and regular waves damping over a porous bed and the solitary wave interacting with a submerged porous breakwater. The convergence of the method and the sensitivity of relevant model parameters are discussed. Then the model is applied to the breaking wave interacting with a breakwater covered with a layer of porous materials. The computational results demonstrate that the ISPH flow model could provide a promising simulation tool in coastal hydrodynamic applications.     

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.     

A parabolic equation for wave propagation over porous structures

A parabolic wave model was developed to simulate wave transformation over porous structures in the surf zone including wave-breaking and energy dissipation. This model is verified through experimental data for waves passing over a submerged permeable structure with porosities. The numerical results have shown good agreement with the results of laboratory experiments. The model is also applied to the real case of waves propagating over a porous submerged breakwater on a complicated bathymetry.     

Water wave interaction with an array of bottom-mounted surface-piercing porous cylinders

The interaction of water waves with arrays of bottom-mounted, surface-piercing circular cylinders is investigated theoretically. The sidewall of each cylinder is porous and thin. Under the assumptions of potential flow and linear wave theory, a semi-analytical solution is obtained by an eigenfunction expansion approach first proposed for impermeable cylinders by Spring and Monkmeyer (1974), and later simplified by Linton and Evans (1990). Analytical expressions are developed for the wave motion in the exterior and all interior fluid regions. Numerical results are presented which illustrate the effects of various wave and structural parameters on the hydrodynamic loads and the diffracted wave field. It is found that the porosity of the structures may result in a significant reduction in both the hydrodynamic loads experienced by the cylinders and the associated wave runup.     

Harmonic generation past a submerged porous step

Experiments were conducted in a wave flume to study the differences between harmonic evolution of monochromatic waves as they propagate over a submerged impermeable or porous step under non-breaking conditions. Results are used as a preliminary analysis to establish some engineering design criteria on harmonic generation on submerged porous structures. The root-mean-squared wave height evolution is also studied and compared to linear models as a first approximation. It is shown that porous structure increases the effective relative depth and decreases the relative wave height, resulting in a lower Ursell number and a lower chance to generate harmonics. The effective water depth over a step as defined in the paper, provides information to evaluate the potential harmonic generation.     

A new approximate analytic solution for water wave scattering by a submerged horizontal porous disk

This study gives a new approximate analytic solution for water wave scattering by a submerged horizontal porous disk in the context of the linear potential theory. The solution is based on the domain decomposition method. The velocity potentials are determined by two different approaches. One approach is to adopt decompositions for velocity potentials, and the other is to expand the vertical derivative of the velocity potential on the porous disk along the radial direction. Hence the velocity potentials are determined by the matched eigenfunction expansions. Differing from previous solutions with respect to the porous disk, the present solution needs no complex dispersion relations. Thus the new solution is easier for numerical implementation. According to numerical examples, the convergence of the present solution is satisfactory. In addition, the present predictions of the wave surface elevation and the vertical wave force on the disk agree very well with previous results by different approaches. The present solution can also be extended to other structures involving disks, such as a fish cage, a porous disk with finite thickness, and a submerged elastic disk.     

Interaction of cnoidal waves with an array of vertical concentric porous cylinders

An array of large concentric porous cylinder arrays is mounted in shallow water exposed to cnoidal waves. The interactions between waves and cylinders are studied theoretically using an eigenfunction expansion approach. Semi-analytical solutions of hydrodynamic loads and wave run-up on each cylinder are obtained using first approximation to cnoidal waves. The square array configuration of four-legged identical concentric porous cylinder is investigated in present study. Numerical results reveal the variation of dimensionless wave force and wave run-up on individual cylinder with angle of incidence, porosity parameter, spacing between outer and inner cylinders, spacing between concentric porous cylinders and wave parameter. Different mechanism of wave force is found under different range of scattering parameter.     

Radiation and diffraction of linear water waves by an infinitely long submerged rectangular structure parallel to a vertical wall

The radiation and the diffraction of linear water waves by an infinitely long floating rectangular structure submerged in water of finite depth with leeward boundary being a vertical wall are analyzed in this paper by using the method of separation of variables. Analytical expressions for the radiated and diffracted potentials are derived as infinite series with unknown coefficients determined by the eigenfunction expansion matching method. The expressions for wave forces and hydrodynamic coefficients are given. A comparison is made between the results obtained by the present analytical solution and those obtained by the boundary element method. By using the present analytical solution, the hydrodynamic influences of the submergence, the width, the thickness of the structure, and the distance between the structure and the wall on the wave forces and hydrodynamic coefficients are discussed in detail.     

A second-order solution of waves passing porous structures

Only linear theoretical analyses of wave interaction with porous structures exist, mainly due to both the complexities of flows inside the porous medium, and the mathematical inhomogeneous boundary-value problem. Since the hydrodynamic flow mechanism is non-linear a non-linear analysis can better describe the characteristic nature of the problem. In this paper, a generalized potential theory is used to describe both the internal and external water flows. An implicit non-linear model is used to describe flow mechanism inside the porous medium. The perturbation method is used to solve the problem analytically up to the second order. The second-order solution is decomposed into time-dependent and time-independent parts. And, correspondingly, the inhomogeneous boundary-value problems are solved analytically. In the analysis, the second-order characteristics of the problem, including the dispersion equation, wave numbers and friction coefficient, as well as wave reflection and transmission, are investigated in detail. It is shown that the mode swapping of the second-order wave numbers only occurs among the evanescent modes. The second-order friction effects become important in shallow-water cases. The comparison of the results of present theory with experimental results shows that the second-order solution is good correction to the linear theory.     

Verification of a VOF-based two-phase flow model for wave breaking and wave–structure interactions

The objective of the present study is to develop a volume of fluid (VOF)-based two-phase flow model and to discuss the applicability of the model to the simulation of wave–structure interactions. First, an overview of the development of VOF-type models for applications in the field of coastal engineering is presented. The numerical VOF-based two-phase flow model has been developed and applied to the simulations of wave interactions with a submerged breakwater as well as of wave breaking on a slope. Numerical results are then compared with laboratory experimental data in order to verify the applicability of the numerical model to the simulations of complex interactions of waves and permeable coastal structures, including the effects of wave breaking. It is concluded that the two-phase flow model with the aid of the advanced VOF technique can provide with acceptably accurate numerical results on the route to practical purposes.     

Forces and moment on a horizontal plate due to regular and irregular waves in the presence of current

This work presents an experimental study of a submerged plate used as a breakwater for coastal areas protection. Questions addressed concern the influence of current on the reflective power of the plate, and its influence on the hydrodynamic loads exerted on it. Results concern both monochromatic and irregular waves. Generally speaking, an influence of the current is found, changing the reflecting power of the structure up to 50%. A homogenized behavior of the loads and moments is found in the presence of currents, meaning that the load values become less sensitive to the frequency. Furthermore, the influence of waves reflected by the wave absorber, representing partially reflective conditions at the shore, is found to be of same order in the absence of current. In any case, the linear behavior of the breakwater is emphasized through the irregular waves approach.     

An extension of the Airy theory for linear waves into shallow water

As a fully developed (Airy) wave propagates from deep into shallow water, its crest becomes more peaked while the trough flattens out. The median crest diameter MCD, defined as the distance between the wave flanks under the crest at a level halfway between the crest and trough, therefore decreases relative to the similarly defined median trough diameter MTD, which remains constant up to the breaking point. The MCD is directly related to other wave characteristics, which enables water particle velocities to be calculated for any water depth without having to recur to more complex, higher-order Stokes, cnoidal or Fenton theories. Over a nearly horizontal bottom, most fully developed wave characteristics can be expressed as functions of the wave period T_w. It is shown that the horizontal particle velocity at the bottom under the breaker crest is at least 9 times faster than under the breaker trough, which explains why sediment is transported landward under fair weather conditions. The proposed equations also shed new light on the formation of spilling, plunging and surging/collapsing breakers.