Mud-Wave Interaction： A Viscoelastic Model

1 .IntroductionThe interaction between waves propagating onthe surface of a body of water andthe bed materialis a long standing coastal problem.The problemis practicallyimportant because the waves can be at-tenuated at a muchfaster rate whenthe bed materi…     

Modelling of wave damping at Guyana mud coast

The Guyana coastal system is characterized by very thick deposits of Amazon mud and high mud concentrations in its coastal waters. The mud deposits can be quite soft and may liquefy under incoming waves. Subsequently, the liquefied mud damps the incoming waves effectively. This paper presents a simple model to predict wave attenuation over soft (fluid) mud beds. This model is based on the two-layer approach by Gade [Gade, H.G., 1958, Effects of a non-rigid, impermeable bottom on plane surface waves in shallow water, Journal of Marine Research, 16 (2) 61–82.] which is implemented in the standard version of the state-of-the-art wave-prediction model SWAN. Input to the mud wave damping module consists of the extension, thickness, density and viscosity of the liquefied (fluid) mud layer.     

Decomposing damped incident and reflected waves using correlation and quasi-linearization methods

Water waves propagating over a layer of soft mud or submerged aquatic vegetation can drastically attenuate over distances comparable to several wave lengths. The attenuation in the case of mud has been found previously to be reasonably described by an exponential decay. Waves reflect from beaches and any structures that they impact. The reflected waves affect wave heights measured in the field or laboratory wave basins.Decomposition of small amplitude waves into incident and reflected waves is a linear problem. However, the presence of the exponential damping introduces nonlinearity to the decomposition problem and requires an iterative process for solving the problem. Despite considerable experimental research on attenuation of waves over mud, none of the existing methods for decomposition of incident and reflected waves have accounted for this case.Here, the Newton Algorithm was used to account for the effect of wave decay over mud by quasi-linearizing the nonlinear equations. Also, a second method using a new error function and a commercial nonlinear solver was proposed in both time and frequency domain. The performance of both methods has been verified using artificial as well as laboratory data.     

A model for the propagation of nonlinear surface waves over viscous muds

The effect of a thin viscous fluid–mud layer on nearshore nonlinear wave–wave interactions is studied using a parabolic frequency-domain nonlinear wave model, modified to incorporate a bottom dissipation mechanism based on a viscous boundary layer approach. The boundary-layer formulation allows for explicit calculation of the mud-induced wave damping rate. The model performed well in tests based on laboratory data. Numerical tests show that damping of high frequency waves occurs, mediated by “difference” nonlinear interactions. Simulations of 2-dimensional wave propagation over a mud “patch” of finite extent show that the wave dissipation causes significant downwave diffraction effects.     

Physical and numerical modeling of reflections from interfaces and thin layers

Basic models for understanding the reflection of seismic waves by the seafloor or the sublayering consist of isolated interfaces and isolated thin layers. An isolated interface model is used to demonstrate a reasonable agreement between physical modeling and two numerical solutions of the elastic wave equation. An isolated thin-layer model is used to show that reflections from it can be regarded as a scattered wavefield caused by three secondary sources proportional toDeltaM,-Deltamu, and,Deltarhoat each element of the layer, whereDelta, indicates the difference between the layer and the surroundings,Mis the compressional wave modulus,muis the shear wave modulus, andrhois the density. This viewpoint leads to a simple explanation of the observed variations in amplitude and phase of the reflected waves as the offset distance from source to receiver increases.     

Mud mass transport due to waves based on an empirical rheology model featured by hysteresis loop

A vertical 2-D water–mud numerical model is developed for estimating the rate of mud mass transport under wave action. A nonlinear semi-empirical rheology model featured by remarkable hysteresis loops in the relationships of the shear stress versus both the shear strain and the rate of shear strain of mud is applied to this water–mud model. A logarithmic grid in the vertical direction is employed for numerical treatment, which increases the resolution of the flow in the neighborhood of both sides of the interface. Model verifications are given through comparisons between the calculated and the measured mud mass transport velocities as well as wave height changes.     

规则波下刚性植物根茎对边界层最大剪切力特性影响研究

基于考虑根茎影响的近岸植物消波实验,对边界层的最大剪切力特性进行了研究。利用前人提出的规则波下含植物水流的边界层流速计算公式,探究了单独根和根茎组合两种植物模型的边界层最大剪切力特性,分析了两种模型的剪切力的沿程变化、植物对剪切力衰减特性的影响,拟合了衰减系数的公式。研究表明,波浪在通过植物带时,剪切力会出现一定幅度增大,随后逐渐降低。植物对剪切力的消减效果随入射波高的增大而增大,且茎有助于根群对边界层最大剪切力的消减作用,其消减系数的范围为0.06~0.61,拟合的公式更加适用于水深较浅的情况。     

The study on the bottom friction and the breaking coefficient for typhoon waves in radial sand ridges—the Lanshayang Channel as an example

Owing to the interactions among the complex terrain, bottom materials, and the complicate hydrodynamics, typhoon waves show special characteristics as big waves appeared at the high water level(HWL) and small waves emerged at low and middle water levels(LWL and MWL) in radial sand ridges(RSR). It is assumed that the mud damping, sandy bed friction and wave breaking effects have a great influence on the typhoon wave propagation in this area. Under the low wave energy, a mud layer will form and transport into the shallow area, thus the mud damping effects dominate at the LWL and the MWL. And high Collins coefficient(c around 1) can be applied to computing the damping effects at the LWL and the MWL. But under the high wave energy, the bottom sediment will be stirred and suspended, and then the damping effects disappear at the HWL. Thus the varying Collins coefficient with the water level method(VCWL) is implemented into the SWAN to model the typhoon wave process in the Lanshayang Channel(LSYC) of the RSR, the observed wave data under "Winnie"("9711") typhoon was used as validation. The results show that the typhoon wave in the RSR area is able to be simulated by the VCWL method concisely, and a constant wave breaking coefficient(γ) equaling 0.78 is better for the RSR where wide tidal flats and gentle bed slopes exist.     

非淹没刚性植物对海啸作用下海堤水动力特性影响数值模拟研究

基于非静压单相流模型NHWAVE建立了高精度数值波浪水槽。通过设计不同的计算工况,系统研究了非淹没刚性植物对海啸作用下海堤周围水动力特性的影响。着重分析了不同入射波高、不同植物分布密度以及不同植物分布宽度条件下海啸波沿程波形特征以及海堤堤顶越浪流空间分布特征。结果表明:不同入射波高、植物分布密度以及植物宽度条件下,堤顶流厚度和水体流速具有明显单调的变化趋势,并且随着植物分布密度和宽度增大,波能衰减增大;随着入射波非线性增强,植物分布密度和分布宽度对堤顶前段水流厚度的影响也随之增强,而对堤顶后段水流厚度的影响则减弱,且堤顶后缘水流厚度约为堤顶前缘厚度的二分之一;在波浪非线性较大情况下,植物的存在对堤顶流速度的空间分布趋势几乎没有影响,但相对堤顶流速度增加程度均大于无植物情况,且堤顶后缘水流速度约为堤顶前缘的1.6倍。     

On the wave damping due to a permeable seabed

Laboratory experiments were performed to study the wave damping induced by a porous bed. During the propagation of waves over a porous medium the wave characteristics change: a significant wave height attenuation of about 20–30% is observed and, in almost all cases, an increase in wavelength. The wave decay is found to depend on the wave characteristics like the wave height, the wavelength and the wave shape. We have also studied the influence of the geometric properties of the porous bed (i.e. thickness and length) on the wave dissipation. It is found that the attenuation of the wave height increases with the permeable bed thickness and that there is a maximum wave dissipation for a length of the porous seabed equal to 2.0–2.5 times the wavelength. A comparison is also made of our findings with available literature results. A parametric study of the wave damping has been performed by varying the values of the resistance coefficients derived by both literature and experiments. Literature analytical models have been applied by using the resistance coefficients that better describe our flow conditions. All models in use underpredict the observed wave attenuation for any sensible values of the resistance coefficients.     

On the response of a Coulomb‐damped poroelastic bed to water waves

Abstract

The problem of forced vibration of a slightly inelastic porous bed by water waves is treated analytically on the basis of a linearized expression of the nonlinear damping term for the grain‐to‐grain friction in bed soils and the linear theory by Biot (1962a [Jour. Appl. Physics, 33:1482–1498]) on the elastic wave propagation in porous media. A dispersion relation of water waves is obtained as a function of wave frequency, water depth, permeability, Poisson's ratio, rigidity, and specific loss of bed soil. Three types of elastic waves are induced in a bed by water waves: a shear wave and a compressional wave in the skeletal frame of soil, and a compressional wave in the pore fluid. The compressional wave, due to the motion of the pore fluid relative to the skeletal frame of soil, is highly damped by the viscosity of pore fluid and only a short range effect near the boundaries of discontinuity, such as a sea‐seabed interface. The seabed response to water waves is characterized by the two Mach numbers, i.e., the ratio of water‐wave speed to shear‐wave speed in soil and the ratio of water‐wave speed to compressional‐wave speed in soil. Most of the water‐wave propagation problems fall into the subsonic flow condition, where elastic waves in the bed travel faster than water waves.

For sandy beds, generally the speeds of compressional and shear waves are much higher than the phase velocity of the water wave. For this case, the solution of the Coulomb‐damped poroelastic bed response presented in this paper approaches the solution of the massless poroelastic bed response in Yamamoto et al. (1978 [Jour. Fluid Mech., 87(1): 193–206]). The damping of water waves due to internal grain‐to‐grain friction is equally or more significant than the damping due to percolation in sand beds.

For clay beds, the speed of the shear wave in soil becomes low and comparable to the phase speed of the water wave. The bed motion for this case is considerably amplified due to the near‐resonance vibration of shear mode of bed vibration. The water wavelength on a clay bed is significantly shortened compared to the water wavelength over a rigid bed. The water wave damping due to internal grain‐to‐grain friction in soil becomes much larger compared to the water wave damping due to percolation in clay beds. Long water waves over a soft clayey bed attenuate within several wavelengths of travel distance.     

一种分层海底反向散射模型

In order to predict the bottom backscattering strength more accurately, the stratified structure of the seafloor is considered. The seafloor is viewed as an elastic half-space basement covered by a fluid sediment layer with finite thickness. On the basis of calculating acoustic field in the water, the sediment layer, and the basement, four kinds of scattering mechanisms are taken into account, including roughness scattering from the water-sediment interface, volume scattering from the sediment layer, roughness scattering from the sediment-basement interface,and volume scattering from the basement. Then a backscattering model for a stratified seafloor applying to low frequency(0.1–10 kHz) is established. The simulation results show that the roughness scattering from the sediment-basement interface and the volume scattering from the basement are more prominent at relative low frequency(below 1.0 kHz). While with the increase of the frequency, the contribution of them to total bottom scattering gradually becomes weak. And the results ultimately approach to the predictions of the high-frequency(10–100 kHz) bottom scattering model. When the sound speed and attenuation of the shear wave in the basement gradually decrease, the prediction of the model tends to that of the full fluid model, which validates the backscattering model for the stratified seafloor in another aspect.     

Hamiltonian description of shear and gravity shear waves in an ideal incompressible fluid

Methods of studying the dynamics of wave disturbances in st;ratified shear flows of an ideal incompressible fluid are considered. The equations governing the motions of interest represent Hamilton equations and are derived by writing the velocity field in terms of Clebsch potentials. Equations written in terms of semi-Lagrangian variables are integrodifferential equations, which make it possible to consider both continuous and discontinuous solutions, as well as the cases where the parameters of the undisturbed medium are step functions. Two dynamic systems are presented. The first, canonical system of equations is most suitable for describing gravity waves in a shear flow in the case where the undisturbed medium is characterized by sharp gradients of density and flow velocity. The simplest model in which disturbances obey this system of equations is the well-known Kelvin-Helmholtz model. The second dynamic system describes, in particular, gravity-shear waves and, in the case of a homogeneous medium, shear waves in a two-dimensional flow. This system is most suitable for studying the dynamics of disturbances in models with sharp gradients of vorticity. On the basis of the approach developed in this study, the problem of the dynamics of disturbances in a flow with a continuous distribution of vorticity in a finite-thickness layer is solved. If the thickness of this layer is small compared to the characteristic wavelength and the gradient of the undisturbed vorticity in this layer is large, the solution has the form of a mode whose frequency is close to the frequency of the shear wave on a vorticity jump that would be obtained by letting the layer’s thickness approach zero. The results obtained allow, in particular, the estimation of the range of validity of finite-layer approximations for models with smooth profiles of flow and density. In addition, these results can be interpreted as the basis for the development of nonlinear aspects of the theory of hydrodynamic stability.     

Dynamics of elastically moored floating objects

The two-dimensional problem of wave transformation by, and motions of, moored floating objects is solved numerically as a boundary value problem by direct use of Green's identity formula for a potential function. The cross-sectional shape of the floating object, the bottom configuration and the mooring arrangements may be all arbitrary. For a given incident wave, the three modes of body motion, the wave system and mooring forces are all solved at the same time. A laboratory experiment is conducted to verify the theory. Generally good agreements between the theory and experiments are obtained as long as the viscous damping due to flow separation is small. A numerical experiment indicates that a conventional sluck mooring is to worsen the wave attenuation by a floating breakwater and that a properly arranged elastic mooring can considerably improve the wave attenuation by a floating breakwater.     

Hydroelastic response of a circular plate in waves on a two-layer fluid of finite depth

The hydroelastic response of a circular, very large floating structure （VLFS）, idealized as a floating circular elastic thin plate, is investigated for the case of time-harmonic incident waves of the surface and interfacial wave modes, of a given wave frequency, on a two-layer fluid of finite and constant depth. In linear potential-flow theory, with the aid of angular eigenfunction expansions, the diffraction potentials can be expressed by the Bessel functions. A system of simultaneous equations is derived by matching the velocity and the pressure between the open-water and the plate-covered regions, while incorporating the edge conditions of the plate. Then the complex nested series are simplified by utilizing the orthogonality of the vertical eigenfunctions in the open-water region. Numerical computations are presentedto investigate the effects of different physical quantities, such as the thickness of the plate, Young＇s modulus, the ratios ofthe densities and of the layer depths, on the dispersion relations of the flexural-gravity waves for the two-layer fluid.Rapid convergence of the method is observed, but is slower at higher wave frequency. At high frequency, it is found that there is some energy transferred from the interfacial mode to the surface mode.     

Sea Floor Sediment and Its Acouso-Physical Properties in the Southeast Open Sea Area of Hainan Island in China

Acouso-physical properties of sea floor sediments in the southeast offshore sea area of Hainan Island on the northern continental shelf of the South China Sea are analyzed. In many cruises, conductivity-temperature-depth measurements of seawater, measurements of shallow stratum and side-scan sonar have been made. Acoustic parameters, basic sedimentary parameters, physical-mechanical parameters and ¹⁴C age, etc., have been measured. The sediment elastic parameters, including Young's modulus, bulk modulus, constrained modulus, rigidity modulus, Poisson's ratio, Lames constant, etc., have been calculated. Results show that the compression wave velocity of the seafloor sediment in the sea area ranges from 1474–1700 m/s, and there are high and low sound velocity sediment types in the different sea areas; the shear wave velocity is 150–600 m/s; at 100 kHz the sediment sound attenuation is 35–260 dB/m, the sediment density is 1.4–2.0 g/cm³; the sediment porosity is 42–88%. Sound field parameters and describing sound reciprocity between sea and seafloor are described.     

Experimental Study on Ocean Internal Wave Force on Vertical Cylinders in Different Depths

A series of experimental studies about the force of internal solitary wave and internal periodic wave on vertical cylinders have been carried out in a two-dimensional layered internal wave flume. The internal solitary waves are produced by means of gravitational collapse at the layer thickness ratio of 0.2, and the internal periodic waves are produced with rocker-flap wave maker at the layer thickness ratio of 0.93. The wave parameters are obtained through dyeing photography. The vertical cylinders of the same size are arranged in different depths. The horizontal force on each cylinder is measured and the vertical distribution rules are researched. The internal wave heights are changed to study the impact of wave heights on the force. The results show that the horizontal force of concave type internal solitary wave on vertical cylinder in the upper-layer fluid has the same direction as the wave propagating, while it has an opposite direction in the lower-layer. The horizontal force is not evenly distributed in the lower fluid. And the force at different depths increases along with wave height. Internal solitary wave can produce an impact load on the entire pile. The horizontal force of internal periodic waves on the vertical cylinders is periodically changed at the frequency of waves. The direction of the force is opposite in the upper and lower layers, and the value is close. In the upper layer except the depth close to the interface, the force is evenly distributed; but it tends to decrease with the deeper depth in the lower layer. A periodic shear load can be produced on the entire pile by internal periodic waves, and it may cause fatigue damage to structures.     

An analytical solution for wave-induced seabed response in a multi-layered poro-elastic seabed

In this study, a new analytical solution for the wave-induced seabed response in a multi-layered poro-elastic seabed is developed. The seabed is treated as a multi-layered porous medium and characterized by Biot’s theory. The displacements of the solid skeleton and the pore pressure are expressed in terms of two scalar potentials and one vector. Then, the Biot’s dynamic equation can be solved using Fourier transformation and reducing to Helmholtz equations. To obtain the general solutions for the multi-layered poro-elastic seabed in the frequency-wave-number domain, the transmission and reflection matrices (TRM) method is used to form the equivalent stiffness. Using the boundary conditions and continuous conditions, the frequency-wave-number domain solutions are obtained. Finally, the time-space domain solutions for the multi-layered poro-elastic seabed are obtained by means of the inverse Fourier transformation with respect to the horizontal coordinate. Based on the new solution, a parametric study is carried out to examine the effects of soil characteristics (number of layers, permeability and shear modulus) and wave characteristics (water depth and wave steepness) on seabed responses. The results indicate that the seabed response is affected significantly by permeability, shear modulus and relative water depth.     

Modeling wave-mud interaction on the central chenier-plain coast,western Louisiana Shelf,USA

The strong coupling between hydrodynamics and seafloors on shallow muddy shelves, and resulting bed reworking, have been extensively documented. On these shelves, spectral wave transformation is driven by a complex combination of forcing mechanisms that include nonlinear wave interactions and wave energy dissipation induced by fluid-mud at a range of frequencies. Wave-mud interaction is investigated herein by using a previously validated nonlinear spectral wave model and observations of waves and near-bed conditions on a mildly-sloping seafloor off the muddy central chenier-plain coast, western Louisiana Shelf, United States. Measurements were made along a cross-shelf transect spanning 1 km between 4 and 3 m water depths. The high-resolution observations of waves and near-bed conditions suggest presence of a fluid mud layer with thickness sometimes exceeding 10 cm under strong long wave action (1 meter wave height with 7 s peak period at 4 meter depth). Spectral wave transformation is modeled using the stochastic formulation of the nonlinear Mild Slope Equation, modified to account for wave-breaking and mud-induced dissipation. The model is used in an inverse manner in order to estimate the viscosity of the fluid mud layer, which is a key parameter controlling mud-induced wave dissipation but complicated to measure in the field during major wave events. Estimated kinematic viscosities vary between 10⁻⁴-10⁻³ m²/s. Combining these results of the wave model simulations with in-depth analysis of near-bed conditions and boundary layer modeling allows for a detailed investigation of the interaction of nonlinear wave propagation and mud characteristics. The results indicate that mud-induced dissipation is most efficient when the wave-induced resuspensions of concentrations  > 10 g/L settle due to relatively small bottom stresses to form a fluid mud layer that is not as thin and viscous as a consolidated seafloor in absence of wave action but also not as thick and soft as a near-bed high concentration layer that forms during strong wave action.     

风暴过程中潮滩悬沙浓度和悬沙输运的变化及其动力机制——以长江三角洲南汇潮滩为例

悬沙浓度是淤泥质海岸重要的环境指标。为探讨潮滩悬沙浓度和悬沙输运对风暴事件的响应过程及其动力机制,于2014年9月"凤凰"台风过境前、中、后在长江三角洲南汇潮滩进行了现场观测,获得同步高分辨率的水深、波高、近底流速和浊度剖面时间序列(9个潮周期)。结果表明,风暴中平均和最大波高、波-流联合底床剪切应力、悬沙浓度和悬沙输运率可比平静天气高数倍;风暴期间高潮位低流速阶段悬沙沉降导致近底发育数十厘米厚的浮泥层(悬沙浓度大于10 g/L)。研究认为风暴事件中淤泥质海岸悬沙浓度和悬沙输运的剧烈变化其根本动力机制是风暴把巨大能量传递给近岸水体,进而显著增大波-流联合底床剪切应力,导致细颗粒泥沙再悬浮。