Permeable Seabed Research Articles

Surface wave damping by a Robin boundary condition at a permeable seabed

We investigate the damping of inviscid surface gravity waves when they propagate over a permeable seabed. Traditionally, this problem is solved by considering the wave motion in the upper water layer interacting with the Darcy flow in the porous bottom layer through matching of the solutions at the permeable interface. The novel approach in this study is that we describe the interaction between the upper fluid layer and the permeable bottom layer by the application of a Robin boundary condition. By varying the magnitude of the Robin parameter R, we can model the bottom structure of the fluid layer from rigid to completely permeable. In the case when the bottom is a porous medium where Darcy’s law applies, or composed by densely packed vertical Hele-Shaw cells, R is small, and can by determined by comparison with analytical results for a two-layer structure. In this case the wave damping is small. For larger values of R, the damping increases, and for very large R, the wave is almost critically damped. For the nonlinear transport in spatially damped shallow-water waves, increasing bottom permeability (larger R), reduces the magnitude of the horizontal Stokes drift velocity, while the drift profile tends to become parabolic with height. The vertical Stokes drift in the limit of large permeability is linear with height, with a magnitude that is larger than the horizontal drift. It is suggested that the implementation of a permeable bottom bed in some cases could prevent shorelines from damaging erosion by incoming surface waves.

Influence of cnoidal wave-induced seepage force on shields number in shallow water

Sediment incipient motion in shallow water is a major concern for coastal engineers. Wave-induced seepage force has reportedly had an important impact on the sediment incipience. The effect of wave asymmetry and skewness is important for defining nearshore wave motions and the resulting sediment movement, but it has been discounted in most previous studies considering seepage. Using analytical and numerical models, this study examines the influence of two-dimensional nonlinear wave-induced seepage forces on sediment incipient motions in shallow water. A new modified Shields number was derived, which better addresses the problem of seepage. The results indicate that seepage force could stabilize the sand particles by lowering the maximum Shields number by up to 38.0%; in the linear wave cases, the opposite is true. This discrepancy is subject to the nonlinearity and asymmetry of shallow wave motions, as well as the decreasing “suction” force and increasing “injection” force. Besides, disregarding horizontal seepage would greatly underestimate the contribution of seepage to sediment incipient motions, especially in situations with steeper waves or an unsaturated and less permeable seabed. A lower Young's modulus, larger saturation, permeability, and water depth would all considerably decrease the maximum Shields number.

Homotopy analysis of wave transformation over permeable seabeds and porous structures

Based on the Homotopy Analysis Method (HAM), a semi-analytical solution method is proposed to solve the modified mild slope equation (MMSE) for waves propagation over permeable seabeds and porous structures. The present method overcomes the mathematical difficulties in treating the nonlinear dispersion equations introduced by the permeability effect, which limits the previous semi-analytical solutions to impermeable seabed and structures only. The proposed method is applied to both 1-D and 2-D axisymmetric wave transformation problems. The method is validated by systematically comparing the semi-analytical solutions of wave amplitudes and reflection coefficients against available experimental data and numerical results, demonstrating the method's capability for predicting wave transformation over both impermeable and permeable seabeds. It is ready to show that, for impermeable conditions, the present semi-analytical solutions degenerate into the earlier solutions without considering the porous effect. The semi-analytical solution method is further extended to examine the porous effects on wave transformation and reflection in various topographical conditions.

Wave energy reflection by the Bragg breakwater composed of rectangular bars on the horizontal permeable seabed

Wave energy is a kind of renewable and clean energy. When waves meet the breakwaters, wave energy reflection will happen. In this paper, the modified mild slope equation for waves propagating over the permeable seabed is introduced, and the finite difference model to solve the equation is set up. The accuracy and applicability of the model is verified with Zeng et al’s analytical solution for wave reflection by the rectangular Bragg breakwaters on the impermeable seabed. Furthermore, in case of a permeable seabed, the effects of the seabed permeability, the bar width, the bar number and the submergence of the bars on wave reflection coefficient, as well as the difference with the case of an impermeable seabed, is studies in details. The results show the reflection coefficient of the Bragg resonant reflection increases with the increase in the bar number, but decreases with the increase in the seabed permeability and the submergence of the bars. When the bar number increases from 1 to 8, the Bragg resonance reflection coefficient increases from 0.146 to 0.772. When the permeability parameter of the seabed increases from 0.005 s to 0.03 s, the coefficient decreases from 0.403 to 0.347. When the bar submergence increases from 0.5 to 0.875, the coefficient decreases from 0.842 to 0.195. Moreover, there exists a particular value of the bar width, that is half of the distance between two adjacent rectangular bars that maximizes the wave energy reflection.

Bragg resonant reflection of water waves by a Bragg breakwater with porous trapezoidal bars on a sloping permeable seabed

For water waves propagating over a permeable sea bottom, a one-dimensional modified mild-slope equation (MMSE) is briefly derived considering the effects of porous media and capable of describing a rapidly varying topography. The thickness of the porous layer is assumed to be infinite to reduce the computational cost. The present numerical model has been validated against Rojanakamthorn et al.’s (1989) experiment for waves transformation over a submerged permeable breakwater on an impermeable seabed, Mase and Takeba's (1994) numerical solution for waves over a porous rippled bed and Liu et al.’s (2020) analytical solution for wave propagation over a series of submerged trapezoidal bars, respectively. A good agreement is obtained. More computational results show that the wave energy will be dissipated in the permeable bottom, and it will achieve the maximal values near the positions of Bragg resonant reflection. The peak Bragg resonance increases with an increasing number of bars and bar slope. However, it decreases with an increase in the permeability, bar submergence, and slope of the seabed. Moreover, there exists a particular value of the bar width that maximizes the Bragg resonant reflection.

Bragg resonant reflection of water waves by a Bragg breakwater with porous rectangular bars on a sloping permeable seabed

A modified mild-slope equation (MMSE) for waves propagation over a porous seabed is derived by using Green's second identity. The MMSE contains both the bottom curvature term and the slope-squared term. The model is computational inexpensive and capable of describing a rapidly varying porous bed. As an example, Bragg resonant reflection of water waves by a Bragg breakwater with an array of porous rectangular bars on a sloping permeable seabed is modeled numerically based on the present MMSE. At first, our numerical solution is validated against the analytical solution for shallow-water waves propagation over an impermeable Bragg breakwater and excellent agreement is obtained. Next, more computational results show that the primary peak Bragg resonance decreases with the increase of the permeability. It is also shown by the present results that the peak value of the primary Bragg resonance increases with increase of the bar number and the bar height. In addition, there exists a particular value of the bar width that maximizes the Bragg resonant reflection.

Characteristics of Breaking Wave Forces on Piles over a Permeable Seabed

Most offshore wind turbines are installed in shallow water and exposed to breaking waves. Previous numerical studies focusing on breaking wave forces generally ignored the seabed permeability. In this paper, a numerical model based on Volume-Averaged Reynolds Averaged Navier–Stokes equations (VARANS) is employed to reveal the process of a solitary wave interacting with a rigid pile over a permeable slope. Through applying the Forchheimer saturated drag equation, effects of seabed permeability on fluid motions are simulated. The reliability of the present model is verified by comparisons between experimentally obtained data and the numerical results. Further, 190 cases are simulated and the effects of different parameters on breaking wave forces on the pile are studied systematically. Results indicate that over a permeable seabed, the maximum breaking wave forces can occur not only when waves break just before the pile, but also when a “secondary wave wall” slams against the pile, after wave breaking. With the initial wave height increasing, breaking wave forces will increase, but the growth can decrease as the slope angle and permeability increase. For inclined piles around the wave breaking point, the maximum breaking wave force usually occurs with an inclination angle of α = −22.5° or 0°.

SOLITARY-WAVE-INDUCED BOUNDARY LAYER FLOWS OVER PERMEABLE BEDS

In the nearshore environment, viscous effects of wave- induced boundary layer flows above sea beds are important in evaluating sediment fluxes and wave damping. These effects need to be included in phase- resolved nearshore models for accurate determination of quantities such as the bed shear stress and the decrease in wave height due to frictional dissipation. In this work, we analyze the boundary layers induced by a solitary wave on permeable sea beds (extending previous work of wave-induced boundary layers on an impermeable bed). We find that the velocity profiles and bed shear stress are sensitive to the hydraulic conductivity of the bed, but not extremely sensitive to the precise boundary condition at the bed-fluid interface.

Analytical Solution for Waves Propagating over a Local Permeable Seabed of Constant Water Depth

Ni, Y.-L.; Xie, T., and Gong, M., 2020. Analytical solution for waves propagating over local permeable seabed of constant water depth. In: Malvárez, G. and Navas, F. (eds.), Global Coastal Issues of 2020. Journal of Coastal Research, Special Issue No. 95, pp. 294-298. Coconut Creek (Florida), ISSN 0749-0208.The present study is concerned with the analytical solution for waves propagating over a local permeable seabed of constant water depth and wave reflection and transmission by the local permeable seabed. The fluid domain is decomposed in three subdomains of which the middle subdomain is permeable and the left and right subdomains are impermeable. Applying the linear wave theory and Darcy's law, the velocity potential of each subdomain is set up, including the effect of evanescent mode. Then, the perturbation method proposed by Mendez is used to obtain the complex wave number and the roots of evanescent mode, and the unknowns in the velocity potential are solved by the continuous conditions at the interfaces between the neighboring subdomains. Based on this analytical model, the effect of permeability coefficient, water depth and length of permeable seabed on wave propagation and transformation is discussed. The results indicate the wave height attenuates increasingly with the increase of permeability coefficient, the length of permeable seabed, and decrease of water depth. And the wave height curve oscillates around the exponential decay curve for the entire permeable seabed proposed by Dean and Dalrymple. This is the result of wave reflection and transmission induced by the local permeable seabed. The reflection coefficient is close to zero when the length of the permeable seabed is an integer multiply of a half wavelength, and it reaches the maximum value when the length of the permeable seabed is odd times of one quarter of the wavelength. The transmission coefficient reduces exponentially with the length increasing in permeable seabed.

Wave force on protected submarine pipelines over porous and impermeable beds using SPH numerical model

A numerical model based on WCSPH method is utilized to simulate wave propagation over a pipeline. A hybrid approach is introduced to absorb outgoing waves based on gradual decrease of fluid velocity in an absorbing layer and using sloping beds simultaneously. The interaction of both regular and irregular waves with pipelines located on porous and impermeable sea-beds has been studied. The effect of pipeline distance to sea-bed on flow pattern and on the applied force is investigated for different geometries of pipeline protection by porous media and side slopes. Irregular wave spectra are studied and the model has been verified in comparison with other numerical and experimental data. The developed model simulates accurately the interaction of waves and a pipeline placed on a porous or an impermeable sea-bed, showing that forces on pipelines on permeable sea-beds are completely different than on impermeable sea-beds.

The influences of Doppler shift on the wave dissipation and soil responses over the porous medium

ABSTRACTThe influences combined the porous seabed and the uniform current on the energy dissipation and the soil responses in the wave–current interaction over the ideal porous seabed are presented in this paper by using the 1D linear wave theory and the linearized porous flow. The relative wave damping rate can be numerically determined by using the complex dispersion relationship. The influences of Doppler shift and the thickness of porous seabed on the wave damping rate, the pore water pressure and its phase shift are discussed, respectively. In addition, the differences using numerous common porous flow model is also compared. Present results find that the energy dissipation is affected by the Doppler shift because the soil responses and the discharge velocity is changed due to the current induced the adjustment of wavenumber or the wavelength. In the case of following current, the interfacial vertical velocity is decreased with the increase of the current velocity. On the other hand, before reaching up the kinematic limit, the magnitude of interfacial vertical velocity is proportional to the flow velocity in the cases of opposing current. Respectively, clarifying the relative importance of the Doppler shift and the thickness of porous media on the energy loss and the soil responses are found that the influence of the thickness of the porous seabed on the energy dissipation has better efficiency. Relative to the thickness of the permeable seabed, the flow velocities and its direction is the important factor on calculating the soil responses.

Effect of porosity on an internal solitary wave propagating over a porous trapezoidal obstacle

A permeable seabed may influence the evolution of an internal solitary wave propagating on a continental shelf and nearshore. In order to study the porous effect, numerical simulations are performed to investigate the flow field and waveform inversion of a large depression ISW propagating over a porous trapezoidal obstacle. A finite volume based Cartesian grid method is adopted to solve the Reynolds averaged Navier-Stokes equations using a k-ε model for the turbulent closure and porous media model together. Numerical results reveal that waveform inversion weakens significantly, as the porosity of the obstacle increases, except when porosity nf<0.2. At the same time, the magnitude of vorticity and turbulent energy decrease remarkably due to acute reduction of the production term induced by percolation and wave-pore interaction. Moreover, a skewed hump shaped relation appears between the maximum vorticity and the porosity. For transmitted wave energy on a horizontal plateau, total energy may decrease as nf<0.2 but increases moderately while nf>0.2 during a strong wave-obstacle interaction.

Effects of Cnoidal Wave-Induced Seepage on Vertical Breakwater Resting on Permeable Elastic Seabed

Theoretical investigations on cnoidal waves interacting with breakwater resting on permeable elastic seabed are presented in this paper. Based on the shallow water reflected wave theory and Biot consolidation theory on wave-induced seepage pressure, the analytical solutions to first order cnoidal wave reflection and wave-induced seepage pressure are obtained by the eigenfunction expansion approach. Numerical results are presented to show the effects of depth of water, breakwater geometry on cnoidal wave-induced seepage uplift force and overturning moment. Compared with Airy wave theory, in certain shallow water conditions, the shallow water wave theory can more effectively illustrate wave nonlinearity effect in wave load prediction.

海底地盤の透過性の効果を考慮した人工リーフ周辺の流動場に関する研究

海底地盤の透過性の効果を考慮した人工リーフ周辺の流動場に関する研究

Tilting While Lifting a Large Object from a Rigid Porous Seabed

AbstractThis study investigates the flow field and the hydrodynamic forces induced by the tilting lift of a large object from a rigid porous seabed using an analytical approach. The flow in the corners between the object and the seabed is a corner flow with a low Reynolds number, and the porous media flow in the seabed obeys the Brinkman equations. The complete boundary conditions for viscous flow, including the continuity of velocities and stresses, are utilized at the seabed-water interface. The Helmholtz decomposition theorem, which decomposes the flow field into irrotational and rotational parts, and a perturbation expansion, are employed to solve the boundary-value problem. Perturbation solutions with first-order corrections are presented, indicating that the flow inside the wedged corner is a Stokes flow driven by the liftup of the object, and the flow within the permeable seabed is a boundary-layer flow. Furthermore, in the wedged corner, the flow induced by the slip velocity at the porous interfac...

Modelling flow in the porous bottom of the Barents Sea shelf

In their recent paper, Węsławski et al. (2012) showed that the Svalbardbanken area of the Barents Sea is characterized by a high organic carbon settlement to the permeable sea bed, which consists of gravel and shell fragments of glacial origin. In the present paper, which can be considered as a supplement to the Węsławski et al. paper, two potential hydrodynamic mechanisms of downward pore water transport into porous media are discussed in detail. In particular, estimated statistical characteristics of the pore water flow, induced by storm surface waves, indicate that the discharge of water flow can be substantial, even at large water depths. During stormy weather (wind velocity V=15ms−1 and wind fetch X=200km) as much as 117.2 and 26.1m3hour−1 of water filter through the upper 5m of the shell pit at water depths of 30 and 50m respectively. For a porous layer of greater thickness, the mean flow discharge is even bigger.The second possible mechanism of flow penetration in the porous layer is based on the concept of geostrophic flow and spiral formation within the Ekman layer. Assuming that the current velocity in the near-bottom water layer is u¯=1m, the resulting mean discharge through this layer becomes as large as 0.99 and 0.09m3s−1 for downstream and transverse flows respectively.

Water-wave induced groundwater pressure and flow in variable bathymetry regions and sandy beaches by an enhanced coupled-mode model

An enhanced coupled-mode system is developed for modelling wave-induced pressure and groundwater flow in variable bathymetry coastal regions, in the layer under the permeable seabed. The sea bottom is assumed of general shape, characterised by sloping parts and undulations, without restrictions concerning mildness of bottom slope and curvature. The present model is based on a simplified version of the storage and momentum equations, corresponding to large values of stiffness ratio of the porous medium, permitting prediction of wave-induced groundwater pressure and flow in the case when the elastic motion of the seabed is very small and could be approximately neglected. The representation of the wave pressure field is based on a local-mode series expansion in the water column and the porous medium, consisted of propagating and evanescent modes, and defined by formulating and solving vertical eigenvalue problems at each local horizontal position. The latter series is enhanced by appropriate terms (the sloping-interface mode) enabling consistent satisfaction of the interface conditions at the seabed and providing fast convergence. Numerical results are presented and compared against predictions by other methods and measured data, illustrating the applicability of the present model to variable bathymetry coastal regions and sandy beaches.

Slip Factor and Slip Velocity on a Permeable Bed

Abstract CHEN, Y.-Y.; CHEN, G.-Y., and LIN, C.-H. 2012. The slip factor and slip velocity on a permeable bed. In the present study, a new, simple method for determining the slip velocity (also called the bed velocity) at the solid-fluid interface in the wave boundary layer is proposed. Based on experimental and numerical results, when waves travel over a rigid permeable seabed, a nonzero slip velocity exists at the interface. The defect of a small slip velocity has been found to occur in previous studies and is usually encountered in fluid–porous layer problems. In the wave–rigid permeable seabed problem, the slip effect depends on the properties of the seabed. The slip velocity boundary condition (SVBC) is one specification of the slip conditions and is usually applied to explain the slip phenomenon in a fluid–porous layer problem. However, the traditional SVBC or the slip velocity is only considered in a single flow, and application of SVBC in harmonic motion is still an open problem that necessitates a...

Wave Damping over a Multilayered, Permeable Seabed

Abstract In this study, a wave-damping rate was derived for waves propagating over a multilayered, permeable seabed of finite depth. For a single-layer bed, a monogram is presented for the relationship between the wave-damping rate and the relative water depth for various depths of the permeable bed. The derived damping rate was used in the extended mild-slope equation to calculate wave dissipation over the permeable bed. For a single-layer bed, the result from the extended mild-slope equation model was compared with the result from the integral equation method and showed good agreement. The extended mild-slope equation model was also used to examine the wave transmission over a permeable shoal with a various number of layers, each with a different permeability. In general, short waves are more influenced by the permeability of the shoal than long waves unless the water depth is so great that the influence of the permeable bed on the surface water waves disappears. For a multilayered bed, the influences o...

투과성구조물에 작용하는 지진해일파압에 관한 연구

본 연구에서는 투과성구조물에서 우수한 파랑제어기능을 발휘하는 것으로 잘 알려져 있는 슬리트케이슨제 중에 기본형인 연직벽형 횡슬리트케이슨제가 불투과성지반 및 투과성지반상에 설치된 조건하에 구조물에 작용하는 단주기파랑 및 지진해일파에 의한 파압의 특성을 논의한다. 투과성지반의 해석에서는 공극율을 편의상 0.4로 가정하여 2차원 및 3차원수치해석을 수행하고, 불투과성의 경우와 그의 차이를 비교 검토한다. 이 때, 입사파랑의 조건으로는 설계파랑으로 일반적으로 고려되는 단주기파랑과 그의 진폭과 동일한 크기로 내습하는 지진해일파(고립파 혹은 단파)를 고려하며, 슬리트케이슨제의 전면유공부 및 유수실 내부의 불투과벽체에 작용하는 파압을 대상으로 한다. 해석에서는 기체와 액체의 혼상동적현상을 하나의 지배방정식으로 해석하는 이상류(二相流) 모델에 기초한 2차원 및 3차원수치파동수로를 각각 적용한다. 얻어진 수치해석결과에 의하면, 슬리트케이슨제의 전면유공부에서는 단주기 파랑에 비해 지진해일파의 작용파압이 약 3~5배 높은 값을 나타내고, 유수실 내부벽에서는 약 2~4배 높은 값을 각각 나타낸다. In this study, wave pressure of short-period gravity waves and tsunami acting on the upright section of the horizontal-slit type caisson placed on the impermeable or permeable seabed, which is a well-known permeable breakwater with a good wave controlling ability, are investigated via numerical simulations. Further, the permeable seabed was modeled as the porous media with porosity of 0.4. Using the numerical results, the effects of the seabed conditions on the wave pressure on the front wall and inside wall of the chamber have been studied. In the numerical simulations, short-period gravity waves and tsunami(solitary wave or bore) with the same amplitude to the gravity wave are considered. A numerical wave tank is used, which is able to consider a gas-liquid two-phase flow in the same calculation zone. Numerical results show that the wave pressure of the tsunami was 3~5 times higher than the short-period gravity waves acting on the front wall and it was 2~4 times higher than the short-period gravity waves acting on the inner wall.