Bed-load Formula Research Articles

Dynamics of intertidal gravel dunes

This paper examines the dynamics of intertidal gravel dunes subjected to high‐energy unsteady and reversing tidal currents in the Severn Estuary, UK. The dunes were composed of shale particles with median grain size, D50, around 4 mm and had mean heights and wavelengths of 60 cm and 7 m, respectively. Acoustic instruments were deployed above a dune to measure the streamwise profile and flow turbulence. Radar was used to map a section of the dune field on two occasions. Measurements during the peak spring ebb tides showed that when the measured bed shear stress exceeded the threshold for entrainment of the coarse surface armor, sediments were eroded from the upper surface of the dune. The dune prograded in the direction of the ebb flow a distance O(1 m), and some eroded sediments were deposited on the stoss slope downstream. The dune was then reworked during the following flood tide to reestablish approximately the same pre‐ebb profile shape and location. Radar data showed also that the exposed dunes at low water occupied approximately the same location over a period of 4 days during peak spring tides. The relationship between the depth‐mean current and the net volume of dune sediments moved was found to conform to a power law. The predicted critical bed shear stress, τcrit, for D50 = 4 mm was approximately 1 N/m2. However, owing to a coarse armor layer on the surface of the dune, a measured τcrit value O(4 N/m2) was required to initiate dune erosion. Although the Van Rijn formula was found to overestimate both the height and wavelength of dunes at this site, the dunes appear to be equilibrium bed forms controlled by local hydraulics and sediment supply. Sediment transport predicted by bed load formula were within a factor of two of the observed mass transport of dune sediments for ebb tides.

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 d 50, 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.

Simulating offshore sand waves

Sand waves form a prominent regular pattern in the offshore seabeds of sandy shallow seas and pose a threat to a range of offshore activities. A two-dimensional vertical (2DV) flow and morphological simulation model describing the behaviour of these sand waves has been developed. The simulation model contains the 2DV shallow water equations, with a free water surface and a general bed load formula. The water movement is coupled to the sediment transport equation with a seabed evolution equation. The domain is non-periodic in both directions. The spatial discretisation is performed by a spectral method based on Chebyshev polynomials. A fully implicit method is chosen for the discretisation in time. Firstly, we validate the simulation model mathematically by reproducing the results obtained using a linear stability analysis for infinitely small sand waves. Hereby, we investigate a steady current situation induced by a wind stress applied at the sea surface. The bed forms we find have wavelengths in the order of hundreds of metres when the resistance at the seabed is relatively large. The results show that it is possible to model the initial evolution of sand waves with a numerical simulation model. This paper forms the necessary first step to investigate the intermediate term behaviour of sand waves.

Flow and grain size control of depth‐independent simple subaqueous dunes

In the Grådyb tidal inlet (Danish Wadden Sea) simple medium to large dunes, which are superimposed on large to very large dunes, increase in size with decreasing grain size under quasi‐constant ebb dominated mean flow conditions. The mean height of the superimposed dunes increases from 0.1 m to 0.5 m and the mean dune length from 7 m to 12 m. This happens over a distance of 1.5 km from the inner to the outer part where mean depth (12 m) and the dominant ebb current (1.0 m s−1) remain constant. Over this reach, the grain size changes from 0.6 mm in the inner part to 0.3 mm in the outer part. As the water depth is constant and more than 1 order of magnitude larger than the height of the dunes, these are regarded as being depth‐independent. Relationships between dune dimensions and grain size established in previous studies are transformed into more general statements, bed form dimensions being directly related to the form‐corrected Shields parameter (θ′). In this way, dune dimensions and flow strength/grain size have, for the first time, been directly related to both bed load and free stream velocity in deeper flow systems. The former relationship is based on a recalibrated (form migration) version of the Meyer‐Peter and Müller bed load formula. Four diagrams describing flow conditions and dune migration rates as a function of grain size and dune length are presented.

A general formula for non-cohesive bed load sediment transport

A bed load transport formula for non-cohesive sediment, based on the bed-shear concept of Meyer-Peter and Müller, was developed and validated for steady flows, oscillatory flows, and combined steady and oscillatory flows. The bed load formula introduced in this study was examined using data from experimental and field measurements for a wide range of flows and sediment conditions, as occurring in river, coastal, and marine environments. More than 1000 steady and 500 oscillatory flow cases were used in the study. The relationship between the bed load transport and the total Shields parameter to the power 1.5 was first confirmed for the steady flows. An exponential factor to take into account the effect of the critical Shields parameter was introduced. The proposed formula for the steady current was expanded and generalized to take into account the effects of oscillatory flows as well as oscillatory flows with a superimposed current at an arbitrary angle. The time-dependent bed load transport was treated in a “quasi-steady” manner using the quadratic value of the instantaneous Shields parameter for the two half-periods of the wave (when the total instantaneous velocity u is in the direction of the wave, u > 0, or in the opposite direction, u < 0). A good correlation was found between the bed load formula and the measurements for colinear oscillatory and steady flows when no phase-lag occurred in the experiments. However, a marked scatter was observed since the Shields parameter had to be estimated and not derived directly from measured data. Finally, the validity and limitations of the obtained bed load transport formula are discussed.

NONLINEAR STABILITY ANALYSIS OF DUNES BY THE USE OF A SEDIMENT TRANSPORT FORMULA INCLUDING THE EFFECT OF THE PRESSURE GRADIENT

In order to enable accurate predictions of the bedform change in rivers, the accurate estimation of the sediment transport rate is important. In this study, we propose a bedload formula taking into account the effect of the pressure gradient in flow caused by the formation of dunes on the basis of Kovacs & Parker's formula including the effect of the local bed inclination. We performed a nonlinear stability analysis by the use of the sediment transport formula derived in this study. It is found that the transition between dune-covered and flat bed transition is charactrized by subcritical bifurcation even if the effect of the local bed inclination is large.

Prediction of Concerted Sediment Flushing

A proprietary one-dimensional numerical model was developed for predicting the amounts of sediment flushed and deposited in the reservoirs in series, the bed evolutions, and variations of the suspended solids concentrations along a river during the concerted sediment flushing events. The model consists of a flow movement module and sediment transport module in which the bed material load is taken as sediment mixture. The nonuniform property of the bed material load is modeled by the introduction of a mixing layer, transition layer, and deposition strata. The model was calibrated on the basis of the field data at Dashidaira and Unazuki reservoirs on the Kurobe River in Japan. The calculated results are in good agreement with the measurements. For the reservoirs out of Japan, the Ashida and Michiue bed load formula used in the model should be verified or replaced by other formulas.

Exponential Formula for Bedload Transport

An exponential formula that does not involve the concept of the critical shear stress is derived in this study for computing bedload transport rates. The formula represents well various experimental data sets ranging from the weak transport to high shear conditions. Comparisons of the present study are also made with many previous bedload formulas commonly cited in the literature.

DEVELOPEMENT OF THE BED-LOAD SEDIMENT OBSERVATION USING THE SOUND PRESSURE

With problems caused by sediment manifesting themselves in recent years, the need for comprehensive sediment management has been rising increasingly. To solve these problems, the appropriate monitoring of sediment transport is imperative. Observation of bed load shifting near river beds, however, is difficult because it is prone to the influence of river flows and river-bed contours. For this reason, experiments on sound pressure generated by bed load were conducted using an experimental channel in this study. As a result, the underwater sound pressure was confirmed to be in proportion to the amount of drifting sand. It was possible to derive an experimental formula for bed load with single particle size of 3mm or larger.

INFLUENCE OF GRAIN SIZE ON SEDIMENT DISCHARGE RATE

Bed-load formulas that have been proposed are roughly classified into two types. One predicts the bed load rate proportional to τ*3/2 and the other calculates it proportional to τ*5/2, in which τ is the non-dimensional bed shear stress. The present study describes the influence of grain size on bed loadrate, based on Egashira et al.'s constitutive relations of debris flows as well as on flume data. The results show the thickness and averagevelocityof bed-load layer are proportional to τ*and τ*3/2, respectively, and suggest that the formula which is a τ*5/2-type is much reasonable from a view point of sediment dynamics.

Statistically Derived Bedload Formula for Any Fraction of Nonuniform Sediment

Based on a method of combining stochastic processes with mechanics, a new bedload formula for the arbitrary kth size fraction of nonuniform sediment is theoretically developed by using a stochastic model of sediment exchange and the probabilistic distribution of fractional bedload transport rates. The relations, proposed recently by Sun, for the probability of fractional incipient motion and for the average velocity of particle motion are introduced to bedload formula. Plenty of experimental data for the bedload transport rate of uniform sediment are used to determine two constants. The theoretical bedload formula for any fraction of nonuniform sediment possesses several advantages, including a clear physical concept, a strict mathematical derivation, and a self-adaptability to uniform sediment. The formula is verified with natural data expressing the transport of nonuniform sediment under full motion in laboratory flume. The result shows that the experimental observations agree well with the predicted fractional bedload transport rates. Comparison of the theory with field data finds that the proposed formula still applies to partial transport of bedload in gravel-bed streams as long as the immobile percentage of bed material is taken into account.

Equilibrium Near-Bed Concentration of Suspended Sediment

A new approach is presented for calculating the equilibrium near-bed concentration of suspended sediment in an alluvial channel flow. It is formulated from the balance between bed sediment entrainment and suspended sediment deposition across the near-bed boundary. The entrainment flux is determined making use of a turbulent bursting outer-scale-based function and the flux of deposition by the product of near-bed concentration and hindered settling velocity of sediment. A number of flume data records in the literature are analyzed to calibrate and verify the present approach. The observed near-bed concentrations for the data records are obtained by first isolating the suspended load transport rate from the observed total load transport rate using Engelund and Fredsoe’s bed-load formula, and then equating the suspended load transport rate to the shape integration of Dyer and Soulsby. The present approach is shown to perform satisfactorily well compared to the results of data analysis. It is found that the near-bed concentration is evidently dependent on sediment particle size in addition to the Shields parameter due to skin friction. The finding seems to challenge previous relationships that simply represent the near-bed concentration as empirical functions of the purely skin-friction-related Shields parameter.

Flow Resistace and Bed-Load Rate on River Beds Vegetated by Woody Plants

The purpose of this study is to clarify the influence of woody riverine-plants on stream flow and bed-load rate. Characteristics of flow and bed-load rate were investigated based on results of flume experiments conducted by the authors. In the experiments, two types of model plant were used. One was a shrub-type, imitating bush trees such as pussy willows, and the other, a tree-type consisting of stem and crown parts. Verification of model parameters included in a one-dimensional k-ε turbulent flow model was perfomed using velocity measurements taken in the experiments. This model was applied to the estimation of effective tractive-stress of bed-load rate. The relationship between measured bed-load rates and estimated effective tractive-forces was represented well using Ashida-Michiue's bed-load formula.

Bed-load transport for steady flows and unsteady oscillatory flows

The validity of a bed-load transport formula, based on the bed-shear concept of Meyer-Peter and Mueller, was investigated for steady unidirectional flows, oscillatory flows and oscillatory flows with superimposed net currents. The aim of the study was to develop a general bed-load transport concept for a wide range of flow and sediment conditions, as occurring in the marine coastal environment. The results of more than 150 laboratory experiments, including more than 75 recent bed-load transport measurements in oscillating water tunnels, were used for the study. For oscillatory flows, the time-dependent bed-load transport was treated in a `quasi-steady' way and an equivalent Shields parameter θ eq′ was defined in order to enable a comparison between `steady fow' and `unsteady oscillatory flow' measurements. A good correlation was found between the non-dimensional transport parameter Φ b and the non-dimensional excess bed-shear θ eq′− θ c for different sets of oscillatory bed-load transport measurements. However, especially in the lower Shields regime, the oscillatory flow data show a clear deviation from an extensive set of steady flow bed-load data. A good agreement between the (oscillatory flow and the steady flow) data sets could then be obtained by reducing the lower limit of the roughness height k s for oscillatory flows from the (steady flow) value 3 D 90 to D 50. Based on the obtained concept a generalized bed-load formula was derived for both flow types, including oscillatory flows and superimposed currents under an arbitrary angle. A verification, carried out with a set of (co-linear) oscillatory+superimposed current measurements, showed a good agreement between calculated and measured bed-load transport rates. Finally, the validity and limitations of the obtained bed-load model are discussed.

Using Lagrangian particle saltation observations for bedload sediment transport modelling

A Lagrangian model for the saltation of sand in water is proposed. Simulated saltation trajectories neglecting particle rotation and turbulence effects compare fairly well with experimental observations. The model for particle motion is coupled with a stochastic model for particle collision with the bed, such that a number of realizations of the saltation process can be simulated numerically. Model predictions of mean values and standard deviations of saltation height, length and streamwise particle velocity agree fairly well with experimental observations. Model predictions of the dynamic friction coefficient are also in good agreement with experimental observations, but they underestimate the value of 0·63 proposed by Bagnold for this coefficient. The saltation model is applied to the estimation of bedload transport rates of sand using a Bagnoldean formulation. Modelled values of the bedload transport rates overestimate those predicted by commonly used bedload formulae, which appears to be a consequence of problems in the definition of the dynamic friction coefficient. These results seem to indicate a few problems with the Bagnoldean formulation, particularly regarding the continuum assumption for the bedload layer, which would be valid only for very high particle concentrations and small particle diameters, and also regarding the evaluation of the shear stress exerted on the bed by the saltating particles. © 1998 John Wiley & Sons, Ltd.

Bed-Load Rate on Movable Beds with Vegetation

This paper is concerned with bed-load rate on movable beds ac companied by vegetation. Firstly, some flume experiments are carried out with two kinds of plant-models which are cylindrical and curved to downstream, and branched and inclined also to downstream. Secondly, a one-dimensional κ-e turbulent flow model is composed in order to estimate the efficient tractive force of bed-load. Model qualification is performed based on velocity data collected in the above experiments. Relation between the efficient tractive force evaluated by the present κ-e model and measured bed-load rate is represented satisfactorily well by Ashida-Michiue's bed-load formula.

EXPERIMENTAL INVESTIGATION AND ANALYSIS OF HEC-6 RIVER MORPHOLOGICAL MODEL

Only comparatively few experimental studies have been carried out to investigate the performance of the HEC-6 river morphological model. The model was developed by the Hydrologic Engineering Center of the US Army Corps of Engineers. In this study, experiments were carried out in a 20 m long concrete flume 0.6 m wide with varying rectangular cross-sections. The channel bed is paved with uniform sand of D50 = 0.9 mm and D90 = 1.2 mm within the test reach of 12 m. Two types of experiments were carried out with sediment transport, one under steady uniform flow and another under steady non-uniform flow conditions. Nine steady uniform flow experiments were carried out to compare the measured equilibrium relationship of flow and sediment transport rate with two bedload formulae, namely, Du Boys and Meyer–Peter and Muller, and with three total load formulae, namely, Toffaleti, Laursen and Yang. It was found that even though the sediment transport consists of a certain portion of bedload, the total load formulae give satisfactory results and better agreement than the two bedload formulae. Five steady non-uniform flow experiments were carried out under various conditions of varying bed profile and channel width and also with sediment addition and withdrawal. The measured transient water surface and bed profiles are compared with the computed results from the HEC-6 model. It was found that the Toffaleti and Yang total load formulae used in the HEC-6 model give the most satisfactory prediction of actual bed profiles under various conditions of non-uniform flow and sediment transport. The effects of Manning's n, variations of sediment inflow, various sediment transport formulae, sediment grain size and the model numerical parameters, i.e. distance interval Δx and numerical weighting factor, on the computed water surface and bed profiles were determined. It was found that the selection of the sediment transport formulae has the most significant effect on the computed results. It can be concluded that the HEC-6 model can predict satisfactorily a long-term average pattern of local scour and deposition along a channel with either a small abrupt change in geometry or gradually varying cross-sections. However, the accuracy of the model prediction is reduced in the regions where highly non-uniform flow occurs.

Scouring of Sand Bars Due to Non-equilibrium Sediment Transport

The scouring process of sand bars due to overflow is experimentally, theoretically and numerically studied. The bed load formula of Meyer Peter-Miuller is extended to include the effects of local slope and pressure gradient, by using the experimental results. The equation of motion is derived to describe the sand motion under non-equilibrium condition. Numerical computations based upon these relationships simulate very well the temporal change of the measured profiles of sand bars.

Derivation of Bed-load Formula from a View Point of Continuum Body

Derivation of Bed-load Formula from a View Point of Continuum Body

On transport of bed load by waves

The Bagnold's energy concept for sediment transport is used to estimate the volume of sediment mobilized by strong wave action. The results are compared with wave tunnel data for sheet flow conditions and with bed load formulae by Sleath and by Madsen and Grant. The predictions differ excessively and agreement with wave tunnel data is poor. The results indicate that only a small percentage ofthe energy dissipation at the bed is due to the sediment in motion.