Practical sand transport formula for non-breaking waves and currents

Many existing practical sand transport formulae for the coastal marine environment are restricted to a limited range of hydrodynamic and sand conditions. This paper presents a new practical formula for net sand transport induced by non-breaking waves and currents. The formula is especially developed for cross-shore sand transport under wave-dominated conditions and is based on the semi-unsteady, half wave-cycle concept, with bed shear stress as the main forcing parameter. Unsteady phase-lag effects between velocities and concentrations, which are especially important for rippled bed and fine sand sheet-flow conditions, are accounted for through parameterisations. Recently-recognised effects on the net transport rate related to flow acceleration skewness and progressive surface waves are also included. To account for the latter, the formula includes the effects of boundary layer streaming and advection effects which occur under real waves, but not in oscillatory tunnel flows. The formula is developed using a database of 226 net transport rate measurements from large-scale oscillatory flow tunnels and a large wave flume, covering a wide range of full-scale flow conditions and uniform and graded sands with median diameter ranging from 0.13 mm to 0.54 mm. Good overall agreement is obtained between observed and predicted net transport rates with 78% of the predictions falling within a factor 2 of the measurements. For several distinctly different conditions, the behaviour of the net transport with increasing flow strength agrees well with observations, indicating that the most important transport processes in both the rippled bed and sheet flow regime are well captured by the formula. However, for some flow conditions good quantitative agreement could only be obtained by introducing separate calibration parameters. The new formula has been validated against independent net transport rate data for oscillatory flow conditions and steady flow conditions.     

A sheetflow sediment transport model for skewed-asymmetric waves combined with strong opposite currents

Prototype scale physical model tests were conducted to investigate the sheetflow sediment transport of uniform sand under different skewed-asymmetric oscillatory flows with and without the presence of relatively strong currents in the opposite direction against wave propagation. Experiments show that in most cases with fine sands, the “cancelling effect” which balances the on-/off-shore net transport under pure asymmetric/skewed oscillatory flows and results a moderate net transport was developed for combined skewed-asymmetric shaped oscillations. However, under certain conditions (T > 5 s) with coarse sands, the onshore sediment transport was enhanced for combined skewed-asymmetric flows. Additionally, the new experimental data under collinear oscillatory flows and strong currents show that offshore net transport rates increase with decreasing velocity skewness and acceleration skewness. Sediment movement behaviors were investigated through analysis of experimental data obtained from the image analysis technique and attempts were made to estimate and formulate the sheetflow layer thickness. Accordingly, sediment transport under oscillatory sheetflow conditions was studied and successfully explained by comparing the bed shear stress and the phase lag parameter at each half cycle. Consequently, these parameters were incorporated in an improved Dibajinia and Watanabe's type sediment transport model. The formula is calibrated against a comprehensive experimental data (331 in total). Good agreement obtained between predictions and measurements shows that the new formula is fulfilled for practical purposes.     

Database of full-scale laboratory experiments on wave-driven sand transport processes

A new database of laboratory experiments involving sand transport processes over horizontal, mobile sand beds under full-scale non-breaking wave and non-breaking wave-plus-current conditions is described. The database contains details of the flow and bed conditions, information on which quantities were measured and the value of the measured net sand transport rate for 298 experiments conducted in 7 large-scale laboratory facilities. Analysis of the coverage of the experiments and the measured net sand transport rates identified the following gaps in the range of test conditions and/or the type of measurements: (i) graded sand experiments, (ii) wave-plus-current experiments and (iii) intra-wave velocity and concentration measurements in the ripple regime. Furthermore, it highlights two areas requiring further research: (i) the differences in sand transport processes and sand transport rates between real waves and tunnel flows with nominally similar near-bed oscillatory flow conditions and (ii) the effects of acceleration skewness on transport rates. The database is a useful resource for the development and validation of sand transport models for coastal applications.     

Sand transport under combined current and wave conditions: A semi-unsteady,practical model

For the general purposes of morphodynamic computations in coastal zones, simple formula-based models are usually employed to evaluate sediment transport. Sediment transport rates are computed as a function of the bottom shear stress or the near bed flow velocity and it is generally assumed that the sediment particles react immediately to changes in flow conditions. It has been recognized, through recent laboratory experiments in both rippled and plane bed sheet flow conditions that sediment reacts to the flow in a complex manner, involving non-steady processes resulting from memory and settling/entrainment delay effects. These processes may be important in the cross-shore direction, where sediment transport is mainly caused by the oscillatory motions induced by surface short gravity waves.The aim of the present work is to develop a semi-unsteady, practical model, to predict the total (bed load and suspended load) sediment transport rates in wave or combined wave-current flow conditions that are characteristic of the coastal zone. The unsteady effects are reproduced indirectly by taking into account the delayed settling of sediment particles. The net sediment transport rates are computed from the total bottom shear stress and the model takes into account the velocity and acceleration asymmetries of the waves as they propagate towards the shore.A comparison has been carried out between the computed net sediment transport rates with a large data set of experimental results for different flow conditions (wave-current flows, purely oscillatory flow, skewed waves and steady currents) in different regimes (plane bed and rippled bed) with fine, medium and coarse uniform sand. The numerical results obtained are reasonably accurate within a factor of 2. Based on this analysis, the limits and validity of the present formulation are discussed.     

Modelling and measurement of sand transport processes over full-scale ripples in oscillatory flow

A new series of laboratory experiments was performed in the Aberdeen Oscillatory Flow Tunnel (AOFT) and the Large Oscillating Water Tunnel (LOWT) to investigate time-averaged suspended sand concentrations and transport rates over rippled beds in regular and irregular oscillatory flow. The wave-induced oscillatory near-bed flows were simulated at full-scale. Five series of experiments were carried out. During the two AOFT experimental series, ripple dimensions, ripple migration rates and net sand transport rates were measured under regular and irregular asymmetric flow for two different sand types. The three LOWT experimental series focussed on measurements of the ripple dimensions, ripple migration rates, time-averaged suspended sand concentrations and net sand transport rates under regular asymmetric and irregular weakly asymmetric flow for two different sand types. From analysis of new and other full-scale data, it is concluded that the lower part of the time- and bed-averaged concentration profile (up to two times the ripple height above the ripple crest level) has an exponential profile. A new reference concentration formula is proposed based on the formula of Bosman and Steetzel [Bosman, J.J., Steetzel, H.J., 1986. Time- and bed-averaged concentration under waves. Proc. 20th ICCE Taipei, ASCE, pp. 986–1000], which includes the grain-size influence. Furthermore, it is shown that the concentration decay length is strongly related to the ripple height and that the simple formula R_c = 1.27η gives good agreement with the data. A new transport model is proposed for the wave-related net transport over full-scale ripples based on a modified half wave cycle concept of Dibajnia and Watanabe [Dibajnia, M., Watanabe, A., 1992. Sheet flow under nonlinear waves and currents. Proc. 23rd ICCE Venice, ASCE, pp. 2015–2028; Dibajnia, M., Watanabe, A., 1996. A transport rate formula for mixed sands. Proc. 25th ICCE Orlando, ASCE, pp. 3791–3804]. The magnitudes of the half wave cycle transport contributions are related to the grain-related Shields parameter, the degree of wave asymmetry and a newly defined vortex suspension parameter P, which is the ratio between the ripple height and the median grain-size. The new model has been calibrated using transport data from the new regular flow experiments and has subsequently been validated using other data, including measurements from irregular flow experiments. The new model is seen to perform better overall than existing practical models for ripple regime net sand transport.     

非对称歪斜波引起的片流输沙数值研究

本文用了一个可考虑相位差作用和波浪边界层非对称性的瞬态理论模型和一个两相紊流模型共同研究非对称歪斜波引起的片流输沙现象。为了解速度偏度和加速度偏度对输沙通量和输沙率的贡献,两相流模型为理论模型提供了必要的相位超前、瞬时侵蚀深度和边界层的发展过程。理论模型研究显示了由速度偏度和加速度偏度引起的向岸阶段和离岸阶段的泥沙运动非对称性,解释了净输沙的产生原因。在以往的非对称歪斜波片流输沙研究中,净输沙的产生主要被归结于相位差作用。本文的研究则表明了非对称的边界层发展所产生的净流量和动床面效应在净输沙产生过程中的比相位差作用更为重要。     

Predicting Net Cross-Shore Total Load Transport: A Phase-Averaging, Quasi-Steady Approach Incorporating Undertow Contribution

Wave shapes that induce velocity skewness and acceleration asymmetry are usually responsible for onshore sediment transport, whereas undertow and bottom slope effect normally contribute to offshore sediment transport. By incorporating these counteracting driving forces in a phase-averaged manner, the theoretically-based quasi-steady formula of Wang (2007) is modified to predict the magnitude and direction of net cross-shore total load transport under the coaction of wave and current. The predictions show an excellent agreement with the measurement data on medium and fine sand collected by Dohmen-Janssen and Hanes (2002) and Schretlen (2012) in a full-scale wave flume at the Coastal Research Centre in Hannover, Germany. The modified formula can predict the net onshore transport of fine sand in sheet flows. In particular, it can predict the net offshore transport of medium sand in rippled beds through enlarged bed roughness, as well as the net offshore transport of fine-to-coarse sand in sheet flows with the aid of a new criterion to judge the occurrence of net offshore transport.     

Sand ripples generated by regular oscillatory flow

The prediction of ripple geometry is a necessary precursor to the prediction of sand transport under waves for ripple regime conditions. The paper begins with a comparison of four existing methods for predicting the geometry of sand ripples generated by oscillatory flow. The comparison points to substantial differences between ripple dimensions predicted by the methods, especially for field-scale conditions. Ripple geometry experiments carried out in a large oscillatory flow tunnel are then described. The experiments involved a range of sand sizes and sinusoidal and asymmetric flows with periods and velocities typical of field conditions. Comparison of measured and predicted ripple geometries leads to the recommendation that the method of Mogridge, Davies and Willis be used to predict ripple geometry for field-scale oscillatory flows. The Nielsen method yields good predictions of ripple length, but the rapid fall-off in ripple steepness predicted by the Nielsen method at high mobility number is not supported by the measurements. The lengths and heights of symmetric ripples produced by sinusoidal flows are found to be similar to the lengths and heights of asymmetric ripples produced by “equivalent” asymmetric flows. Three-dimensional ripples occur with fine sand in long-period flows typical of field conditions. The dimensions of these ripples cannot be predicted using methods developed for two-dimensional ripples. Previously suggested criteria for predicting the occurrence of three-dimensional ripples fail when tested against a wide range of flow and sand conditions. The occurrence of three-dimensional ripples and the effects of ripple and flow history on ripple geometry require further research.     

Transport processes of uniform and mixed sands in oscillatory sheet flow

New experiments were carried out in the Large Oscillating Water Tunnel of WL|Delft Hydraulics (scale 1:1) using asymmetric 2nd-order Stokes waves. The main aim was to gain a better understanding of size-selective sediment transport processes under oscillatory plane-bed/sheet-flow conditions. The new data show that for uniform sand sizes between 0.2 < D < 1.0 mm, measured net transport rates are hardly affected by the grain size and are proportional to the third-order velocity moment. However for finer grains (D = 0.13 mm) net sand transport rates change from the ‘onshore’ direction into the ‘offshore’ direction in the high velocity range. A new measuring technique for sediment concentrations, based on the measurement of electro-resistance (see [McLean, S.R., Ribberink, J.S., Dohmen-Janssen, C.M. and Hassan, W.N.M., 2001. Sediment transport measurements within the sheet flow layer under waves and currents. J. Waterw., Port, Coast., Ocean Eng., ISSN 0733-950X]), was developed further for the improved measurement of sediment dynamics inside the sheet-flow layer. This technique enabled the measurements of particle velocities during the complete wave cycle. It is observed that for long period waves (T = 12.0 s), time-dependent concentrations inside the sheet-flow layer are nearly in phase with the time-dependent flow velocities. As the wave period decreases, the sediment entrainment from the bed as well as the deposition process back to the bed lags behind the wave motion more and more. The new data show that size-gradation has almost no effect on the net total transport rates, provided the grain sizes of the sand mixture are in the range of 0.2 < D < 1.0 mm. However, if very fine grains (D = 0.13 mm) are present in the mixture, net total transport rates of graded sand are generally reduced in comparison with uniform sand with the same D₅₀. The transport rates of individual size fractions of a mixture are strongly influenced by the presence of other fractions in a mixture. Fine particles in sand mixtures are relatively less transported than in that uniform sand case, while the opposite occurs for coarse fractions in a mixture. The relative contribution of the coarse grains to the net total transport is therefore larger than would be expected based on their volume proportion in the original sand mixture. This partial transport behaviour is opposite to what is generally observed in uni-directional (e.g. river) flows. This is caused by vertical sorting of grain sizes in the upper bed layer and in the sheet flow and suspension layers. Kinematic sorting is believed to be responsible for the development of a coarse surface layer on top of a relatively fine sub-layer, providing in this way a relatively large flow exposure for the coarser sizes. Furthermore fine grains are suspended more easily than coarse grains to higher elevations in the flow where they are subject to increasing phase-lag effects (settling lags). The latter also leads to reduced net transport rates of these finer sizes.     

A two-phase numerical model for sediment transport prediction under oscillatory sheet flows

To predict sediment transport under oscillatory sheet flow condition, especially for fine sand, is still a challenging research subject in coastal engineering. This paper describes a newly-developed numerical model based on two-phase theory with the use of a one-equation turbulence closure, and its applications in predicting fine sediment suspension in near-prototype oscillatory sheet flow conditions. Model results were compared with comprehensive laboratory measurements of flow velocity and sediment concentration under both symmetrical and asymmetrical oscillatory sheet flows from a large-scale water tunnel. Good agreements between the model results and measurements were achieved and the results demonstrated that the model is capable of reproducing detailed characteristics of sediment entrainment process in the sheet flow regime. The comparisons also revealed the fact that the concentration peaks at flow reversal is associated with the strong vertical sediment transport flux in the pickup layer, which has been widely observed in many laboratory experiments. The effects of flow reversal events on total sediment transport were also discussed.     

Bed shear stress under skewed and asymmetric oscillatory flows

We present a new formulation to predict the bed shear stress under skewed/asymmetric oscillatory flows (with or without a co-linear mean current), extending the work of Nielsen (1992). The nonlinearity of the oscillatory flow is incorporated through the use of two parameters: the index of skewness or nonlinearity, and the waveform parameter. The new formulation is tested against the bed shear stress estimated from the log-fit and momentum-integral methods, using oscillatory data from oscillating water tunnel experiments. The new formulation and the momentum-integral method agree well, but differ from those with the log-fit method, possibly because both methodologies lead to different results for the phase lead between the bed shear stress and the free-stream velocity. The new bed shear stress formulation is incorporated in a quasi-steady bedload formula, and accurately reproduces net transport rates under non-linear, nonbreaking waves with and without an opposing current.     

Phase-lag effects in sheet flow transport

The inception of the sheet flow regime as well as the effects of the phase lag when the sheet flow regime is established were investigated for oscillatory flows and combined steady and oscillatory flows. A new criterion for the inception of sheet flow is proposed based on around 300 oscillatory flow cases from experiments. This criterion was introduced in the Camenen and Larson [Camenen, B., Larson, M., 2005. A bedload sediment transport formula for the nearshore. Estuarine, Coastal and Shelf Science 63, 249–260.] bed load formula in order to take into account phase-lag effects in the sheet flow regime. The modification of the Camenen and Larson formula significantly improves the overall agreement with data and yields a correct behavior in relation to some of the main governing parameters, which are the median grain size d₅₀, the orbital wave velocity U_w, and the wave period T_w. The calibration of the new formula was based on more than 200 experimental data values on the net sediment transport rate for a full wave cycle. A conceptual model was also proposed to estimate the ratio between sediment transport rate with and without phase lag, (r_pl = q_s,net / q_s,net,ϕ=0). This simple model provides accurate results and may be used together with any quasi-steady model for bed load transport.     

The dimensions of sand ripples in full-scale oscillatory flows

New large-scale experiments have been carried out in two oscillatory flow tunnels to study ripple regime sand suspension and net sand transport processes in full-scale oscillatory flows. The paper focuses on ripple dimensions and the new data are combined with existing data to make a large dataset of ripple heights and lengths for flows with field-scale amplitudes and periods. A feature of the new experiments is a focus on the effect of flow irregularity. The combined dataset is analysed to examine the range of hydraulic conditions under which oscillatory flow ripples occur, to examine the effects of flow irregularity and ripple three-dimensionality on ripple dimensions and to test and improve existing methods for predicting ripple dimensions.The following are the main conclusions. (1) The highest velocities in a flow time-series play an important role in determining the type of bedform occurring in oscillatory flow. Bedform regime is well characterised by mobility number based on maximum velocity in the case of regular flow and based on the mean of the highest one tenth peak velocities in the case of irregular flow. (2) For field-scale flows, sand size is the primary factor determining whether equilibrium ripples will be 2D or 3D. 2D ripples occur when the sand D₅₀ ≥ 0.30 mm and 3D ripples occur when D₅₀ ≤ 0.22 mm (except when the flow orbital diameter is low). (3) Ripple type (2D or 3D) is the same for regular and irregular flows and ripple dimensions produced by equivalent regular and irregular flows follow a similar functional dependence on mobility number, with mobility number based on maximum velocity in the case of regular flow and based on the mean of the highest one tenth velocities in the case of irregular flow. For much of the ripple regime, ripple dimensions have weak dependency on mobility number and ripple dimensions are similar for regular and irregular flows with the same flow orbital amplitude. However, differences in ripples produced by equivalent regular and irregular flows become significant at the high mobility end of the ripple regime. (4) Ripple dimensions predicted using the Wiberg and Harris formulae are in poor agreement with measured ripple dimensions from the large-scale experiments. Predictions based on the Mogridge et al. and the Nielsen formulae show better overall agreement with the data but also show systematic differences in cases of 3D ripples and ripples generated by irregular flows. (5) Based on the combined large-scale data, modifications to the Nielsen ripple dimension equations are proposed for the heights and lengths of 2D ripples. The same equations apply to regular and irregular flows, but with mobility number appropriately defined. 3D ripples are generally smaller than 2D ripples and estimates of 3D ripple height and length may be obtained by applying multipliers of 0.55 and 0.73 respectively to the 2D formulae. The proposed modified Nielsen formulae provide an improved fit to the large-scale data, accounting for flow irregularity and ripple three-dimensionality.     

A two-phase flow model for asymmetric sheetflow conditions

A newly developed two-phase flow model was applied to simulate the sediment movement under 2nd-order Stokes wave sheetflow conditions with different sediment sizes and wave periods. As for the distribution of eddy viscosity and sediment diffusion coefficient, the difference between onshore and offshore phases was considered by using an equivalent sinusoidal velocity amplitude for the asymmetric velocity profile. Sophisticated comparisons between laboratory measurements [O'Donoghue, T., Wright, S., 2004b. Flow tunnel measurements of velocities and sand flux in oscillatory sheetflow for well-sorted and graded sands. Coast. Eng., 51 (11–12), 1163–1184.] and the present numerical simulation were performed for sediment concentration, sediment velocity, sand flux and net transport rate. Four existing engineering models, together with the present two-phase flow model, were introduced for net transport rate prediction. Taking both the net sand transport rate magnitude and direction into account, the present process-based two-phase flow model provided the best estimations, which can simulate both the onshore net transport for medium/coarse sand cases and offshore net transport for fine sand cases with the agreement by a factor of 2 for almost all the considered cases.     

Modelling of sand transport under wave-generated sheet flows with a RANS diffusion model

A 1DV-RANS diffusion model is used to study sand transport processes in oscillatory flat-bed/sheet flow conditions. The central aim is the verification of the model with laboratory data and to identify processes controlling the magnitude and direction (‘onshore’/‘offshore’) of the net time-averaged sand transport. The model is verified with a large series of measured net sand transport rates, as collected in different wave tunnels for a range of wave-current conditions and grain sizes. Although not all sheet flow details are represented in the 1DV-model, it is shown that the model is able to give a correct representation of the observed trends in the data with respect to the influence of the velocity, wave period and grain diameter. Also detailed mean sediment flux profiles in the sheet flow layer are well reproduced by the model, including the direction change from ‘onshore’ to ‘offshore’ due to a difference in grain size from 0.34 mm (medium sand) to 0.13 mm (fine sand). A model sensitivity study with a selected series of net transport data shows that the stirring height of the suspended sediment ε_s/w_s strongly controls the magnitude and direction of the net sediment transport. Inclusion of both hindered settling and density stratification appears to be necessary to correctly represent the sand fluxes for waves alone and for waves + a superimposed current. The best agreement with a large dataset of net transport measurements is obtained with the 1DV-RANS model in its original settings using a Prandtl–Schmidt number σ_ρ = 0.5.     

前进波作用下沙纹床面的净输沙方向与净输沙强度

在沙纹床面输沙过程中，假设水流从涡中取出并搬运的悬移质数量与推移质运动的沙星成比例，由12组细沙实验结果得到了沙纹床面净输沙的方向和输沙强度公式，并与他人的实验结果进行对比。     

Sheet flow and suspension of sand in oscillatory boundary layers

after revisionTime-dependent measurements of flow velocities and sediment concentrations were conducted in a large oscillating water tunnel. The measurements were aimed at the flow and sediment dynamics in and above an oscillatory boundary layer in plane bed and sheet-flow conditions. Two asymmetric waves and one sinusoidal wave were imposed using quartz sand with D₅₀ = 0.21 mm. A new electro-resistance probe with a large resolving power was developed for the measurement of the large sediment concentrations in the sheet-flow layer. The measurements revealed a three layer transport system consisting of a pick-up/deposition layer, an upper sheet flow layer and a suspension layer.In the asymmetric wave cases the total net transport was directed “onshore” and was mainly concentrated in the thin sheet flow layer (< 0.5 cm) at the bed. A small net sediment flux was directed “offhore” in the upper suspension layer. The measured flow velocities, sediment concentrations and sedimenl fluxes showed a good qualitative agreement with the results of a (numerical) 1DV boundary-layer flow and transport model. Although the model did not describe all the observed processes in the sheet-flow and suspension layer, the computational results showed a reasonable agreement with measured net transport rates in a wide range of asymmetric wave conditions.