Bed Load Transport Equations Research Articles

Numerical simulation of graded sediment transport

River engineers commonly need to predict sediment transport rates in the absence of adequate field data, and so make frequent use of numerical sediment transport models. This paper compares the performance of two such computer models for the prediction of graded sediment transport. The intention is to inform model users of the level of accuracy that can be expected for typical applications. One model uses the Parker bedload transport equation and the other uses the Meyer-Peter and Müller sediment transport formula modified for use with graded sediments. The computer model predictions are compared with detailed experimental data collected for the purpose of computer model validation. The Meyer-Peter and Müller-based model produces better predictions of sediment transport rates, whereas the Parker-based model performs better in terms of the grain sizes of both the bedload and the bed material. These results illustrate the limitations of present models and indicate areas requiring further investigation.

Bed-Load Transport Equation for Sheet Flow

When open-channel flows become sufficiently powerful, the mode of bed-load transport changes from saltation to sheet flow. Where there is no suspended sediment, sheet flow consists of a layer of colliding grains whose basal concentration approaches that of the stationary bed. These collisions give rise to a dispersive stress that acts normal to the bed and supports the bed load. An equation for predicting the rate of bed-load transport in sheet flow is developed from an analysis of 55 flume and closed conduit experiments. The equation is i b5v where i b5immersed bed-load transport rate; and v5flow power. That i b5v implies that e b5tan a5ub /u, where e b 5Bagnold's bed-load transport efficiency; ub5mean grain velocity in the sheet-flow layer; and tan a5dynamic internal friction coeffi- cient. Given that tan a'0.6 for natural sand, ub'0.6u, and e b'0.6. This finding is confirmed by an independent analysis of the experimental data. The value of 0.60 for e b is much larger than the value of 0.12 calculated by Bagnold, indicating that sheet flow is a much more efficient mode of bed-load transport than previously thought.

Simple Model for Downstream Variation of Median Sediment Size in Chilean Rivers

A simple theory is developed to account for the observed downstream variation of the median sediment size in Chilean rivers based on a reach-wise equilibrium sediment transport concept. The theory makes use of a bedload transport equation linked with a resistance equation to estimate the median sediment diameter as a function of channel slope, with the flow Reynolds number and bedload concentration as parameters. Both, Meyer-Peter and Müller’s and Ackers and White’s formulas are used alternatively as bedload equations. A Manning–Strickler type of formulation is used as a resistance relationship. The resulting model is validated against field data corresponding to 150 rivers in Central Chile, covering slopes in the range of 0.04–8.61%, with median sediment size in the range of 0.3–250 mm. Despite the simplicity of the present theory and the somewhat bold assumptions made in its derivation, the estimated variation of the median sediment size with channel slope follows the same trend as the field data. Most of the scatter of these data falls within the theoretical limits given by the estimated range of values of the parameters of the model.

Comparison of bed load transport in torrents and gravel bed streams

Data are compiled on bed load transport and discharge in gravel bed rivers and torrents with bed slopes up to 0.17. The transport characteristics of 19 streams are compared with simple bed load transport equations and with measurements from flume experiments. In simple form the bed load transport is a function of an “effective” discharge times a bed slope factor. The efficiency of the streams in transporting bed load is defined as the deviation from the transport function. It varies over several orders of magnitude, particularly for smaller and steeper streams, whereas flume experiments and larger streams are in better agreement with the simple bed load transport relation. The large variation and the strong decrease in efficiency in smaller streams appears to be associated with substantial form losses for relative flow depths (defined by h/d90) smaller than ∼4–6.

Scales of boundary resistance in coarse-grained channels: turbulent velocity profiles and implications

Gravel-bed surfaces are characterized by morphological features occurring at different roughness scales. The total shear stress generated by the flow above such surfaces is balanced by the sum of friction drag (grain stress) and form drag components (created by bed forms). To facilitate a better understanding of total resistance and bed load transport processes, there is a need to mathematically separate shear stress into its component parts. One way to do so is to examine the properties of vertical velocity profiles above such surfaces. These profiles are characterized by an inner layer that reflects grain resistance and an outer layer that reflects total resistance. A flume-based project was conducted to address these concerns through systematically comparing different roughness scales to ascertain how increased roughness affects the properties of vertical velocity profiles. Great care was taken to create natural roughness features and to obtain flow data at a high spatial and temporal resolution using an Acoustic Doppler Velocimeter. Average vertical velocity profiles above each roughness scale were clearly segmented. The vertical extent of the inner flow region was directly related to the scale of roughness present on the bed (and independent of flow depth), increasing with increased roughness. On a rough but rather uniform “plane” bed made of heterogeneous coarse sediments (with no bed forms), the shape of the velocity profile was clearly dominated by the local variations in grain characteristics. When pebble clusters were superimposed, the average shear stress in the outer flow region increased by 100% from the plane bed conditions. The ratio of inner grain shear stress to outer total shear stress for this pebble cluster experiment was 0.18 under shallow flow conditions and 0.3 under deep flow conditions. The grain stress component that should be used in bed load transport equations therefore appears to vary in these experiments between 15% and 30% of the total channel stress, increasing with decreased resistance. Roughness height ( K s/ D 50) values at the grain scale for the plane bed and pebble cluster experiments were 0.73 and 0.63, respectively. These are values that should be used in flow resistance equations to predict grain resistance and grain stress for bed load transport modeling.

Modelling bedload yield in braided gravel bed rivers

This paper outlines an approach for estimating the annual bedload yield of a braided channel. This procedure is based on the extension of theory of flow and sediment transport in braided rivers recently presented by Paola (1996). The revised approach accounts explicitly for the relationship between increasing discharge and varying channel hydraulics, and is suitable for use in obtaining bedload transport rate estimates over a range of discharges. Integration of such estimates using flow-duration data allows annual bedload yield to be determined. Model parameterisation is achieved using topographic survey data for the Waimakariri River, New Zealand. Comparison of modelled bedload yield with values estimated from repeated topographic surveys indicates that the model is able to accurately predict both the medium-term (c. 30 years) mean annual bedload yield of the Waimakariri at Crossbank (the section 17.8 km upstream of the river mouth), and also short-term fluctuations in bedload yield associated with varying annual flow statistics. Streamwise patterns of volumetric erosion and deposition determined for a 45-km length of the Waimakariri using the model are also in broad agreement with trends identified in topographic survey data for the period 1961–1997. However, significant deviations between modelled and surveyed volumes of cut and fill are evident at some locations. Comparison of model performance with conventional applications of bedload transport equations, which tend to underestimate transport rates for braided channels, suggests that the approach presented here may represent a significant improvement. This is the case because the model quantifies the relationship between braid intensity and spatial variability in flow hydraulics at a cross-section. Output from the model suggests that braided rivers may transport a significant proportion of their annual bedload at lower discharges than those indicated by earlier theoretical approaches. Results also provide quantitative support for the argument that increased intensity of braiding may promote higher rates of bedload transport in gravel bed rivers.

SIMULATING SEDIMENT TRANSPORT IN THE CARSON RIVER AND LAHONTAN RESERVOIR, NEVADA, USA1

ABSTRACT: The discovery of the Comstock Lode in western Nevada in 1859 initiated the use of liquid mercury (Hg) or “quicksilver” to remove gold (Au) and silver (Ag) from crushed ore. Today, Hg is present in historic mill tailings piles, in alluvial deposits adjacent to the Carson River, and in Lahontan Reservoir. Mercury concentrations in Carson River water have been reported as high as 61 μg/L by the U.S. Geological Survey. Fish from Lahontan Reservoir have methylmorcury (MeHg) concentrations as much as four times the 1.0 μg/g limit for human consumption. Since more than 95 percent of total Hg in water can be associated with particulates, the transport of sediment must be quantified to understand the fate of Hg in the system. By linking the RIVMOD hydrodynamic model with the WASP5 water quality model, and using suspended sediment rating curves along with bedload transport equations, reliable predictions of sediment transport can be made. Measured suspended sediment data from the Carson River, and an estimate of annual sediment loading to Lahontan Reservoir, were used to create a calibrated sediment transport model. Model simulations predicted the long term transport of sediment into Lahontan Reservoir, the transport of sediment into Lahontan Reservoir during a flood year (1986 water year), and concentrations of total Hg in the Carson River using an estimate of sediment Hg concentrations. This research will eventually be used with an Hg model to predict the fate of Hg in the river and reservoir system.

Prediction of Bed-Load Transport by Desert Flash Floods

A number of predictive bed-load sediment transport equations are rated against a unique set of field data collected by automatic slot samplers during flash floods in a desert wadi. The Meyer-Peter and Müller equation is shown to perform well, providing a median ratio of calculated to observed ( C /O) bed-load flux of 1.18. The Bagnold equation is shown to underpredict considerably, with a median C /O of 0.44. The Parker equation performs better, though it still underpredicts with a median C /O of 0.78. The apparent success of the Meyer-Peter and Müller formula is attributed to the ready supply of sediment to the channel system in desert and semidesert environments. This ensures that the channel bed remains unarmored, in contrast to supply-limited, armored, perennial rivers of humid zones. It also ensures that bed-load flux responds to changing hydraulic conditions in a comparatively simple fashion.

A reappraisal of the Engelund bed load equation

Abstract Many bed load transport equations have been developed from differing standpoints. The Engelund equation has been selected for scrutiny since, although it accords well with experimental data in general, it has limitations under conditions of high bed load transport rates. The application of two phase flow theory has been used to derive an alternative “Engelund” equation which may be applied to both high and low transport rates.

Comparisons between acoustic measurements and predictions of the bedload transport of marine gravels

ABSTRACTContinuous, detailed records of marine gravel transport have been obtained acoustically and compared with bedload transport rates (qb) predicted by five bedload transport equations using measurements of the near‐bed turbulent current flow. When mean flow data are used in these equations, total qb estimates are similar to those measured. However, when instantaneous flow data are used, total qb is over‐estimated by approximately one order‐of‐magnitude. Based on the acoustic measurements, an empirical equation has been obtained that gives accurate estimates of total qb over a tidal cycle and simulates well the intermittent characteristics of marine bedload transport.

Closure to “ Laminar Einstein Bed Load Transport Equation for Overland Sheet Flow ” by Weihua Zhang and Terrance W. Cundy (December, 1987, Vol. 113, No. 12)

Closure to “ <i>Laminar Einstein Bed Load Transport Equation for Overland Sheet Flow</i> ” by Weihua Zhang and Terrance W. Cundy (December, 1987, Vol. 113, No. 12)

Discussion of “ Laminar Einstein Bed Load Transport Equation for Overland Sheet Flow ” by Weihua Zhang and Terrance W. Cundy (December, 1987, Vol. 113, No. 12)

Discussion of “ <i>Laminar Einstein Bed Load Transport Equation for Overland Sheet Flow</i> ” by Weihua Zhang and Terrance W. Cundy (December, 1987, Vol. 113, No. 12)

Theoretical Study on the Coexistence of Ripples and Dunes

Formation of ripples and dunes on alluvial river bed is studied by river bed stability analysis, basing on two-dimensional shear flow equation and a Meyer-Peter and Wüller type bed load transport equation modified by the inclusion of the lag distance of bed load relative to local bed shear. Dominant dimensionless wave-numbers of sand wave are analyzed. The effect of sediment particle diameter and lag distance on the dominant wavenumbers is examined, and the difference between ripples and dunes, with respect to their cause of occurrence, is discussed.

Laminar Einstein Bed Load Transport Equation for Overland Sheet Flow

Einstein’s bed load transport equation was modified for laminar overland flow under rainfall conditions. The modifications consisted of using a laminar velocity profile and a lift force calculated from the fluid drag force. The parameter λ\N was fitted from available experimental data. The results show improvement over the original Einstein equation. This sediment transport model was then coupled with an overland flow model and tested on data from a sand filled flume.

On Bagnold's sediment transport equation in tidal marine environments and the practical definition of bedload

Bagnold's sediment transport equation has proved to be important in studying tidal marine environments. This paper discusses three problems concerning Bagnold's transport equation and its practical application: 1Bagnold's suspended-load transport equation and the total-load transport equation with are incorrect from the viewpoint of energy conservation. In these equations the energy loss due to bedload transport has been counted twice. The correct form should be for suspended-load transport and for total-load transport with 2The commonly used Bagnold's transport coefficient K varies as a non-linear function of the dimensionless excess shear stress, which can be represented best by the power law , where the coefficient A and exponent B depend on sediment grain size D. The empirical values of A and B for fine to medium grained sands are determined using Guy et al.'s (1966) flume-experiment data. 3The sediment transport rates predicted from this equation are compared with bedform migration measurements in the flume and the field. This comparison shows that the sediment transport rates measured from bedform migrations are higher than the predicted bedload transport rates, but comparable to the calculated total-load (bedload plus intermittent suspended-load) transport rates. This indicates that bedform migration involves both bedload and intermittent suspended-load transport. As a logical conclusion, bedform migration data should be compared with Bagnold's total-load transport equation rather than with his bedload transport equation. In this respect the term ‘bed material’ might be more appropriate than the term ‘bedload’ for estimating sediment transport rate from bedform migration data. The sediment transport rates predicted from this modified Bagnold transport equation are in good agreement with field measurements of bedform migration rates in four individual tidal marine environments, which cover a wide range of sediment grain size, flow velocity and bedform conditions (ranging from small ripples, megaripples to sandwaves).

Unit Stream Power Equation for Gravel

The basic assumptions used in the derivation of bedload or gravel transport equations are reviewed and examined to determine their validities. Laboratory data indicate that unit stream power is more desirable than other variables for the determination of gravel transport rate or concentration. A dimensionless unit stream power equation for gravel transportation is developed based on well established theories in turbulence and fluid mechanics. Comparison between this new equation and other well‐known equations indicates that the new dimensionless unit stream power equation for gravel transportation is more accurate than others for the gravel data compared in this paper.

Reply [to “Comment on ‘A reexamination of Bagnold's granular‐fluid model and bed load transport equation’ by J. A. Bailard and D. L. Inman”

Reply [to “Comment on ‘A reexamination of Bagnold's granular‐fluid model and bed load transport equation’ by J. A. Bailard and D. L. Inman”

Comment [on “A reexamination of Bagnold's granular‐fluid model and bed load transport equation” by J. A. Bailard and D. L. Inman

Bailard and Inman (1979) have raised a number of important questions in their study challenging the classic work of Bagnold (1954) on the flow behavior of granular materials. However, we believe that neither the theoretical arguments nor the experimental results of Bailard and Inman offer compelling reasons to reject Bagnold's basic conclusions. In the following remarks, we shall address several spects of Bailard and Inman's work, retaining their notation for ease of comparison.

Bed-load transport under waves and currents

The data from the flume experiments of Kalkanis (1964) and Abou-Seida (1965), giving the rates of accumulation of sand in recessed traps in oscillating sand beds, have been reassessed. In contrast to the work of Madsen and Grant (1976), the accumulation rate is interpreted as a measure of the average concentration of bed-load per unit area during the period of oscillation for which the Shields Number ψ ( t) exceeds the threshold Shields Number ψ TH. For the experimental runs where the Reynolds Number > 35, the volume concentration of bed-load C∗(t) (cm 3 cm −2) was found to be directly proportional to the excess Shields Number ψ EX( t) = ψ ( t) − ψ TH): C ∗(t)=(0.09±0.03)ψ EX t and the volume transport rate q (t) under instantaneous flow velocity u (t) was therefore: q(t)=0.09±0.03)ψ EX (t)u(t) for ψ( t)> ψTH The rate of transport predicted by this equation was tested against the data of Manohar (1955) and good agreement found, showing a closer match between calculated and observed transport, at high rates, than found by the equation of Madsen and Grant (1976). The form of the transport rate equation proposed here: q(t)∞ [u 2(t)−u 2 TH ]u(t) is compared with the form of the bedload transport equation for unidirectional flow and is found to have a firmer physical basis than some other wave transport equations presently in use.

A reexamination of Bagnold's Granular‐Fluid model and bed load transport equation

Bagnold's bed load transport model is reexamined by using only the momentum equation, the Coulomb yield criteria and the definition of the rate of energy dissipation for a shearing fluid. The general form of Bagnold's bed load equation is rederived but with two specific differences: Coulomb's yield criteria tan ϕ = T/P holds only at the interface between regions of shearing and nonshearing grains, and the ratio of the mean to maximum granular velocity Ū/Umax enters as an additional factor.