Characteristics of the boundary layer at the bottom of a solitary wave

The results of direct numerical simulations of the boundary layer generated at the bottom of a solitary wave are described. The numerical results, which agree with the laboratory measurements of Sumer et al. (2010) show that the flow regime in the boundary layer can be laminar, laminar with coherent vortices and turbulent. The average velocity and bottom shear stress are computed and the results obtained show that the logarithmic law can approximate the velocity profile only in a restricted range of the parameters and at particular phases of the wave cycle. Moreover, the maximum value of the bottom shear stress is found to depend on the dimensionless wave height only, while the minimum (negative) value depends also on the dimensionless boundary layer thickness. Diagrams and simple formulae are proposed to evaluate the minimum and maximum bottom shear stresses and their phase shift with respect to the wave crest.     

The intertidal erosion rate of cohesive sediment: a case study from Long Island Sound

Over the past decades, many attempts have been made to generate useful bottom erosion models for the study of cohesive sediment movement. This study addresses some of the key questions involved in determining the functional relationship between erosion rate and bottom shear stress. Current, wave, and turbidity data were collected from a bottom mounted instrument array in a moderately energetic estuarine environment. The bottom shear stress was calculated from a wave–current interaction model. The erosion rate was derived from the observed sediment concentration using a vertical mixing model. Examination of the relationship between erosion rate and bottom stress showed that the erosion rate varied at intertidal frequency. When averaged over the tidal fluctuation, the erosion rate remained approximately constant at low stress, but increased sharply when the shear stress rose above a critical value. This suggests two-stage erosion. The bed has a layered structure, in which a thin layer of loose, high water content material overlies a more consolidated bed. The top layer of high water content material (fluff) was easily disturbed and re-suspended by tidal currents, but the consolidated bottom layer was eroded only under conditions of high shear stress.     

Seabed shear stresses under irregular waves plus current from Monte Carlo simulations of parameterized models

Shear stresses on a rough seabed under irregular waves plus current are calculated. Parameterized models valid for regular waves plus current have been used in Monte Carlo simulations, assuming the wave amplitudes to be Rayleigh-distributed. Numerical estimates of the probability distribution functions are presented. For waves only, the shear stress maxima follow a Weibull distribution, while for waves plus current, both the maximum and time-averaged shear stresses are well represented by a three-parameter Weibull distribution. The behaviour of the maximum shear stresses under a wide range of wave-current conditions has been investigated, and it appears that under certain conditions, the current has a significant influence on the maximum shear stresses. Results of comparison between predictions and measurements of the maximum bottom shear stresses from laboratory and field experiments are presented.     

Laboratory observation of boundary layer flow under spilling breakers in surf zone using particle image velocimetry

A boundary layer flow under spilling breakers in a laboratory surf zone with a smooth bottom is investigated using a high resolution particle image velocimetry (PIV) technique. By cross-correlating the images, oscillatory velocity profiles within a viscous boundary layer of O(1) mm in thickness are resolved over ten points. Using PIV measurements taken for an earlier study and the present study, flow properties in the wave bottom boundary layer (WBBL) over the laboratory surf zone are obtained, including the mean velocities, turbulence intensity, Reynolds stresses, and intermittency of coherent events. The data are then used to estimate the boundary layer thickness, phase variation, and bottom shear stress. It is found that while the time averaged mass transport inside the WBBL is onshore in the outer surf zone, it changes to offshore in the inner surf zone. The zero Eulerian mass transport occurs at h/h_b ≈ 0.92 in the outer surf zone. The maximum overshoot of the streamwise velocity and boundary layer thickness are not constant across the surf zone. The bottom shear stress is mainly contributed by the viscous stress through mean velocity gradient while the Reynolds stress is small and negligible. The turbulence level is higher in the inner surf zone than that in the outer surf zone, although only a slight increase of turbulent intensity is observed inside the WBBL from the outer surf zone to the inner surf zone. The variation of phase inside and outside the WBBL was examined through the spatial velocity distribution. It is found the phase lead is not constant and its value is significantly smaller than previous thought. By analyzing instantaneous velocity and vorticity fields, a remarkable number of intermittent turbulent eddies are observed to penetrate into the WBBL in the inner surf zone. The size of the observed large eddies is about 0.11 to 0.16 times the local water depth. Its energy spectra follow the − 5/3 slope in the inertial subrange and decay exponentially in the dissipation subrange.     

Simulation of Random Waves and Associated Laminar Bottom Shear Stresses

This work presents a new approach for simulating the random waves in viscous fluids and the associated bottom shear stresses. By generating the incident random waves in a numerical wave flume and solving the unsteady two-dimensional Navier-Stokes equations and the fully nonlinear free surface boundaiy conditions for the fluid flows in the flume, the viscous flows and laminar bottom shear stresses induced by random waves axe determined. The deterministic spectral amplitude method implemented by use of the fast Fourier transform algorithm was adopted to generate the incident random waves. The accuracy of the numerical scheme is confirmed by comparing the predicted wave spectrum with the target spectrum and by comparing the nanlerical transfer function between the shear stress and the surface elevation with the theoretical transfer function. The maximum bottom shear stress caused by random waves, computed by this wave model, is compared with that obtained by Myrhaug＇ s model （1995）. The transfer function method is also employed to determine the maximum shear stress, and is proved accurate.     

Intercomparison of wave-driven current models

Seven numerical models which simulate waves and currents in the surf-zone are tested for the case of a reduced-scale detached breakwater subjected to the action of regular waves with normal incidence. The computed wave heights, water levels and velocities are compared with measurements collected in an experimental wave basin. The wave height decay in the surf-zone is predicted reasonably well. Set-up and currents appear to be less well predicted. This intercomparison exercise shows that radiation stresses are systematically overestimated by formulations used in the models, mean bottom shear stresses are not always co-linear with the mean bottom velocity vector in shallow water, and turbulence modelling in the surf-zone requires a sophisticated     

Bottom friction and its effects on periodic long wave propagation

A new set of Boussinesq-type equations describing the free surface evolution and the corresponding depth-integrated horizontal velocity is derived with the bottom boundary layer effects included. Inside the boundary layer the eddy viscosity gradient model is employed to characterize Reynolds stresses and the eddy viscosity is further approximated as a linear function of the distance measured from the seafloor. Boundary-layer velocities are coupled with the irrotational velocity in the core region through boundary conditions. The leading order boundary layer effects on wave propagation appear in the depth-integrated continuity equation to account for the velocity deficit inside the boundary layer. This formulation is different from the conventional approach in which a bottom stress term is inserted in the momentum equation. An iterative scheme is developed to solve the new model equations for the free surface elevation, depth-integrated velocity, the bottom stress, the boundary layer thickness and the magnitude of the turbulent eddy viscosity. A numerical example for the evolution of periodic waves propagating in one-dimensional channel is discussed to illustrate the numerical procedure and physics involved. The differences between the conventional approach and the present formulation are discussed in terms of the bottom frictional stress and the free surface profiles.     

Modeling sediment transport in the swash zone: A review

A critical review of conceptual and mathematical models developed in recent decades on sediment transport in the swash zone is presented. Numerous studies of the hydrodynamics and sediment transport in the swash zone in recent years have pointed out the importance of swash processes in terms of science advancement and practical applications. Evidently, the hydrodynamics of the swash zone are complex and not fully understood. Key hydrodynamic processes include both high-frequency bores and low-frequency infragravity motions, and are affected by wave breaking and turbulence, shear stresses and bottom friction. The prediction of sediment transport that results from these complex and interacting processes is a challenging task. Besides, sediment transport in this oscillatory environment is affected by high-order processes such as the beach groundwater flow. Most relationships between sediment transport and flow characteristics are empirical, based on laboratory experiments and/or field measurements. Analytical solutions incorporating key factors such as sediment characteristics and concentration, waves and coastal aquifer interactions are unavailable. Therefore, numerical models for wave and sediment transport are widely used by coastal engineers. This review covers mechanisms of sediment transport, important forcing factors, governing equations of wave-induced flow, groundwater interactions, empirical and numerical relations of cross-shore and longshore sediment transport in the swash zone. Major advantages and shortcomings of various numerical models and approaches are highlighted and reviewed. These will provide coastal modelers an impetus for further detailed investigations of fluid and sediment transport in the swash zone.     

Dynamic instability of laminated sandwich plates subjected to in-plane partial edge loading

The dynamic instability of laminated sandwich plates subjected to in-plane partial edge loading is studied for the first time using an efficient finite element plate model. The plate model is based on a refined higher order shear deformation plate theory, where the transverse shear stresses are continuous at the layer interfaces with stress free conditions at plate top and bottom surfaces. Interestingly the plate model having all these refined features requires unknowns at the reference plane only. However, this theory requires C¹ continuity of transverse displacement, which is difficult to satisfy arbitrarily in any existing finite element. To deal with this, a new triangular element developed by the authors is used in the present paper.     

An analytic solution to the wave bottom boundary layer governing equation under arbitrary wave forcing

Existing models of the wave bottom boundary layer have focused on the vertical and temporal dynamics associated with monochromatic forcing. While these models have made significant advances, they do not address the more complicated dynamics of random wave forcing, commonly found in natural environments such as the surf zone. In the closed form solution presented here, the eddy viscosity is assumed to vary temporally with the bed shear velocity and linearly with depth, however, the solution technique is valid for any eddy viscosity which is separable in time and space. A transformation of the cross-shore velocity to a distorted spatial domain leads to time-independent boundary conditions, allowing for the derivation of an analytic expression for the temporal and vertical structure of the cross-shore velocity under an arbitrary wave field. The model is compared with two independent laboratory observations. Model calculations of the bed shear velocity are in good agreement with laboratory measurements made by Jonsson and Carlsen (1976, J. Hydraul. Res., 14, 45–60). A variety of monochromatic, skewed, and asymmetric wave forcing conditions, characteristic of those found in the surf zone, are used to evaluate the relative effects on the bed shear. Because the temporal variation of the eddy viscosity is assumed proportional to the bottom shear, a weakly nonlinear interaction is created, and a fraction of the input monochromatic wave energy is transferred to the odd harmonics. For a monochromatic input wave, the ratio of the third harmonic of velocity at the bed to the first is <10%. However, for a skewed and asymmetric input wave, this ratio can be as large as 30% and is shown to increase with increasing root-mean-square input wave acceleration. The work done by the fluid on the bed is shown to be a maximum under purely skewed waves and is directed onshore. Under purely asymmetric waves, the work done is significantly smaller and directed offshore.     

基于潮间带现场数据的底部切应力算法对比

底部切应力作为水动力和泥沙输移模型中的关键参数,对底床泥沙起动、侵蚀淤积速率的研究十分重要.目前基于现场实测流速数据计算底部切应力的理论方法有6种:LP-mean法、LP-max法、TKE法、TKE W法、RS法和ID法,这些方法都有其特定的适用条件.河口海岸浅水区域水流和波浪作用复杂,遴选合适的方法计算底部切应力非常...     

Numerical study on vertical structures of undertow inside and outside the surf zone

Nearshore shoaling and breaking waves can drive a complex circulation system of wave-induced currents. In the cross-shore direction, the local vertical imbalance between the gradient of radiation stress and that of pressure due to the setup drives an offshore flow near the bottom, called ‘undertow’, which plays a significant role in the beach profile evolution and the structure stability in coastal regions. A 1DV undertow model was developed based on the relationship between the turbulent shear stress and t...     

Long-term distribution of seabed shear stresses beneath random waves

The long-term distribution of seabed shear stresses under random waves is presented. The approach combines short-term distribution of maximum bottom shear stresses with a joint frequency table of significant wave height and peak period. An example of application is given where the long-term probability of exceeding a given level of the maximum bottom shear stress in the central North Sea is presented. The example includes estimation of the return period of the critical shear stress for sheet flow conditions, as well as the bottom shear stresses associated with the 1, 10 and 100 years return periods.     

A model of profile evolution on wave-dominated mud coast

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海岸破波带内水底悬沙浓度形成机理实验研究

通过大尺度水槽波浪引起泥沙悬移的动床模型实验,研究了沙坝海岸破波带内水底悬沙浓度形成机理,通过比较时间平均水底悬沙浓度与时间平均水底波浪水质点动能或时间平均水底湍动能之间的相关性,论证了利用时间平均湍动能比利用时间平均波浪水质点动能计算时间平均水底悬沙浓度更为适用,并提出了以上时间平均水底悬沙浓度与水底湍动能之间的关系也可以用来近似表达时间变化的水底悬沙浓度与时间变化的水底湍动能之间的关系。研究针对规则波、波群和不规则波3种波浪形态进行,并分别对破波带内的爬坡区、内破波区和沙坝区3个区域实验结果进行讨论。     

Equilibrium beach profiles under breaking and non-breaking waves

Simple theoretical models to determine the equilibrium profile shape under breaking and non-breaking waves are presented. For the case of breaking waves, it is assumed that the seaward transport in the undertow is locally balanced by a net vertical sedimentation, so that no bottom changes occur at equilibrium. The parameterization of the water and sediment flux in the surf zone yields a power curve for the equilibrium profile with a power of 2/3, which is in agreement with previous field investigations on surf zone profile shapes. Three different models were developed to derive the profile shape under non-breaking waves, namely (1) a variational formulation where the wave energy dissipation in the bottom boundary layer is minimized over the part of the profile affected by non-breaking waves, (2) an integration of a small-scale sediment transport formula over a wave period where the slope conditions that yield zero net transport determine equilibrium, and (3) a conceptual formulation of mechanisms for onshore and offshore sediment transport where a balance between the mechanisms defines equilibrium conditions. All three models produced equilibrium profile shapes of power-type with the power typically in the range 0.15–0.30. Comparison with field data supported the results obtained indicating different powers for the equilibrium profile shape under breaking and non-breaking waves.     

Modeling wave�Ccurrent bottom boundary layers beneath shoaling and breaking waves

The boundary layer characteristics beneath waves transforming on a natural beach are affected by both waves and wave-induced currents, and their predictability is more difficult and challenging than for those observed over a seabed of uniform depth. In this research, a first-order boundary layer model is developed to investigate the characteristics of bottom boundary layers in a wave–current coexisting environment beneath shoaling and breaking waves. The main difference between the present modeling approach and previous methods is in the mathematical formulation for the mean horizontal pressure gradient term in the governing equations for the cross-shore wave-induced currents. This term is obtained from the wave-averaged momentum equation, and its magnitude depends on the balance between the wave excess momentum flux gradient and the hydrostatic pressure gradient due to spatial variations in the wave field of propagating waves and mean water level fluctuations. A turbulence closure scheme is used with a modified low Reynolds number k-ε model. The model was validated with two published experimental datasets for normally incident shoaling and breaking waves over a sloping seabed. For shoaling waves, model results agree well with data for the instantaneous velocity profiles, oscillatory wave amplitudes, and mean velocity profiles. For breaking waves, a good agreement is obtained between model and data for the vertical distribution of mean shear stress. In particular, the model reproduced the local onshore mean flow near the bottom beneath shoaling waves, and the vertically decreasing pattern of mean shear stress beneath breaking waves. These successful demonstrations for wave–current bottom boundary layers are attributed to a novel formulation of the mean pressure gradient incorporated in the present model. The proposed new formulation plays an important role in modeling the boundary layer characteristics beneath shoaling and breaking waves, and ensuring that the present model is applicable to nearshore sediment transport and morphology evolution.     

On a theoretical model of rough turbulent wave boundary layers

An analytical theory which describes the motion in a turbulent wave boundary layer near a rough sea bottom by using a two-layer time invariant eddy viscosity model is presented. The eddy viscosity in the inner layer increases quadratically with the height above the sea bottom. In the outer layer the eddy viscosity is taken as a constant. The mean velocity and shear stress profiles, the bottom shear stress and the bottom friction coefficient are presented, and comparisons are made with experimental results.     

Analysis of some key parametrizations in a beach profile morphodynamical model

A numerical process-based model to forecast beach profile morphodynamics has been developed. In the present paper, an analysis of various modelling approaches and key parametrizations involved in the estimation of the wave-driven current and the suspended sediment concentration is carried out.Several resolution techniques for the 1DV horizontal (i.e., in the x-direction perpendicular to coastline) momentum equation governing the Mean Horizontal Velocity (MHV) are analysed. In the first kind of techniques, the mean horizontal velocity is computed from the momentum equation, whereas the Mean Water Level (MWL) is computed using a parametrization of the depth-averaged momentum equation. Two boundary or integral conditions are thus needed. In the second kind, both mean horizontal velocity and mean water level gradient in the x-direction are the unknowns of the momentum equation, thus, three boundary or integral conditions are needed. Various additional conditions are discussed. We show that using a technique of the first kind is equivalent to imposing the difference between the surface and the bottom shear stresses in the 1D vertical equation. Both techniques lead to results that are in good agreement with the Delta Flume experimental data, provided the Stokes drift flow discharge is imposed as an additional condition. The influence of the breaking roller model and of the turbulent viscosity parametrization are also analysed.Suspended sediment transport by the mean current and wave-induced bedload transport are taken into account in the sediment flux. Three turbulent diffusivity parametrizations are compared for suspended sediment concentration estimations. A linear profile for the turbulent diffusivity taking into account the wave bottom shear stress and the surface wave breaking turbulence production is shown to give the best results. Using experimental data, we put forward the poor estimation of the bottom sediment concentration given by the three implemented parametrizations. We thus propose a new parametrization relying on a Shields parameter based on the breaking roller induced surface shear stress. Using this new parametrization, the bottom profile used in the tests keeps its two bars which disappear otherwise. However, the morphodynamical model still overestimates the bars offshore motion, a bias already observed in other models.