Bed Load Transport Formula Research Articles

Slope Correction for Calculation of Bedload Sediment Transport Rates in Steep Channels

AbstractClassical theories developed for investigating bedload sediment transport are applicable largely for channel beds of a gentle slope. Examples are the well-known Shields diagram for the evaluation of the critical condition for initial sediment motion and the classical bedload transport formulas derived with deterministic or probabilistic approaches. How to apply the existing bedload formulas for steep channels remains challenging. In this study, a simple procedure is proposed for correcting slope effects so that the existing bedload formulas can be extended for steep channels. To verify the proposed correction factor, two representative bedload formulas are modified and checked against published data of sediment transport rates that were significantly affected by steep slopes.

NUMERICAL SIMULATION OF FLOW AND LOCAL SCOUR AT TWO SUBMERGED-EMERGENT TANDEM CYLINDRICAL PIERS

In this paper, the flow and local scour variation around two submerged and un-submerged tandem piers are studied using 3D flow model where the upstream pier is submerged while the downstream pier is emergent. The model uses a finite-volume method to solve the non-transi ent Navier-Stokes equations for three dimensions on a general non-orthogonal grid. The   k turbulence model is used to solve the Reynolds-stress term. The numerical model solves the sediment continuity equation in conjunction with van-Rijn’s bed-load sediment transport formula to simulate the bed evolution. The 3D flow model is verified through experimental study in a non cohesive bed material in an experimental flume. The different causes of local scour around two submerged and unsubmerged piers are simulated well, such as bow flow, down flow, horseshoe vortex, pressure variation and lee-wake vortex. It is found from this study that the maximum local scour depth by interaction between two tandem submerged unsubmerged piers depends on submersion ratio of upstream pier, the densimetric Froude number, the longitudinal distance between piers and the ratio of pier diameter to channel bed width. The maximum scour depth decreases by increasing the submerged pier height then begin to increase by increasing the submerged pier to a height larger than half the water depth and in general the maximum scour depth is less than that of two unsubmerged piers. The results show good agreement between simulation and experimental results. Also, empirical equations are developed for computing the maximum scour depth due to the interaction between two submerged unsubmerged piers with circular shapes as a function of submergence ratio, piers spacing, densimetric Froude number and channel width to pier diameter ratio.

Numerical modelling of non-cohesive embankment breach with the dual-mesh approach

The numerical model BASEMENT, based on the two-dimensional shallow-water equations, is applied to breaching processes of non-cohesive earth embankments due to overtopping. The governing equations are solved using an explicit finite-volume method combined with a Godunov-type approach. For spatial discretization, a mass-conserving dual-mesh approach with separate, unstructured meshes for hydrodynamic and sediment computations is developed allowing for accurate terrain representation. The surface erosion is modelled with the Exner and sorting equations for multiple grain classes in combination with empirical bed-load transport formulas and advection–diffusion equations for suspended-load transport. Additionally, the lateral breach widening caused by gravitationally-induced slope failures is considered using a novel algorithm adapted for unstructured grids. The model is successfully applied to two recent laboratory experiments including plane and spatial dike breaches, and a field-scale embankment breach. Detailed comparisons between measured and simulated laboratory spatial breach formations confirm the basic model assumptions.

Net beach change in the swash zone: A numerical investigation

A range of bed-load sediment transport formulae are used to run fully coupled, morphodynamic simulations of one [1] swash cycle on an erodible plane beach. A system comprising shallow water equations and Exner equation is solved, in which sediment transport rate q is either dependent only on depth averaged velocity (u), or on u and water depth (h). The results are in agreement with equivalent uncoupled results [2] in that all sediment transport formulae considered applied to the event of [1] yield net erosion in the whole of the swash. Consistent with [3], however, full coupling yields significantly less erosion for all the q=q(u) formulae compared to the equivalent uncoupled results. It is shown that differences between uncoupled and coupled approaches (for most formulae) accumulate over the course of the swash event. The main reason for the reduced net erosion is the smaller maximum inundation. It is also shown that including a dependence on h in the bed load sediment transport formula for fully coupled simulations can result in net deposition in the upper swash.Bed shear stress described by a Chezy law is further included in fully coupled simulations to examine net beach change. Much reduced maximum inundation and net offshore sediment transport are predicted both for q=q(u) and q=q(h,u). It is shown that although the net sediment flux at the base of the swash under one [1] swash event is still offshore, deposition in the middle or upper swash may be predicted when bed shear stress is included, particularly when the drag coefficient in the backwash is reduced compared to that in the uprush, consistent with some in-situ measurements. The implication is that bed shear stress must be included not just to obtain correct quantitative beach change, but also to obtain correct qualitative beach behaviour.

Particle path characteristics at the large gravel‐bed river Danube: results from a tracer study and numerical modelling

ABSTRACTA tracer study performed on a 3 km long reach of the Danube River in Austria is presented. Forty artificial stones of three different sizes (intermediate b‐axis: 25 mm, 40 mm, 70 mm) were produced and a coded radio acoustic transmitter was implanted. The measurement system had to be improved to be applicable to large rivers with water depths up to 12 m. The positions of the stones were observed approximately once a week, depending on hydrology, over a period of at least one year by radio‐tracking from a boat, including a 15 year flood event. Transport paths and velocities, as well as the incipient motion of bedload transport, could be monitored for the first time on a large gravel‐bed river. The particle paths were found to be mostly bankline‐parallel, even though the stones passed a 30° river bend. The median of the transverse particle displacement was found to be 4% of the longitudinal displacement. Calculations considering both transverse slope and transverse flow velocities showed transverse transport to be 6·6% of the longitudinal transport indicating that marginal lateral transport is mainly influenced by morphology. A three‐dimensional (3D) numerical model using a stochastic particle tracing approach was validated with the data, indicating that the observed positions are well reproduced by the model. Within the observation period, 74% of all stones passed the reach. With more than 1000 detections, particle transport could be characterized by a mean travel velocity of about 10 m per day (variable for the different grain sizes); single tracer stones were transported up to 1000 m during a single flood event. Size‐selective behaviour could be shown and the incipient motion of the large 70 mm gravel was detected at lower discharges than predicted by commonly used uniform bedload transport formulae. Copyright © 2012 John Wiley & Sons, Ltd.

Testing bedload transport equations with consideration of time scales

ABSTRACTBedload transport is known to be a highly fluctuating temporal phenomenon, even under constant (mean) flow conditions, as a consequence of stochasticity, bedform migration, grain sorting, hysteresis, or sediment supply limitation. Because bedload transport formulas usually refer to a single mean transport value for a given flow condition, one can expect that prediction accuracy (when compared to measurements) will depend on the amplitude and duration of fluctuations, which in turn depend on the time scale used for observations. This paper aims to identify how the time scale considered can affect bedload prediction. This was done by testing 16 common bedload transport formulas with four data sets corresponding to different measurement period durations: (i) highly fluctuating (quasi‐)instantaneous field measurements; (ii) volumes accumulated at the event scale on two small alpine gravel‐bed rivers, potentially affected by seasonal fluctuations; (iii) volumes accumulated at the interannual scale in a meandering gravel bed river, thought to be weakly subject to fluctuations; (iv) time‐integrated flume measurements with nearly uniform sediments. The tests confirmed that the longer the measurement period, the better the precision of the formula's prediction interval. They also demonstrate several consequential limitations. Most threshold formulas are no longer valid when the flow condition is below two times the threshold condition for the largest elements' motion on the bed surface (considering D84). In such conditions, equations either predict zero transport, or largely overestimate the real transport, especially when D84 is high. There is a need for new sediment data collected with highly reliable techniques such as recording slot bedload samplers to further investigate this topic. Copyright © 2012 John Wiley & Sons, Ltd.

Instability Theory of Sand Ripples Formed by Turbulent Shear Flows

AbstractA theory of turbulent shear flow over a sand bed is developed, addressing the instability principle of the fluid-granular bed interface leading to the formation of ripples. The Reynolds-averaged Navier-Stokes (RANS) equations and the time-averaged continuity equation are analyzed using a 1/7-power law of the time-averaged streamwise velocity and treating the curvilinear streamlines by the Boussinesq approximation. The integration of the RANS equations leads to a governing dynamical equation of flow over a mobile bed. A near-bed flow layer of 3.5 times the ripple height is considered being affected by the ripples. The dynamical equation of the mobile sand bed is based on the Exner’s sediment continuity equation in conjunction with the Meyer-Peter and Muller bed-load transport formula as modified to account for the effect of local bed slope attributable to bed forms. The coupled dynamical equations are then analyzed to estimate the parameters for the instability that results in the formation of ripp...

Variability of wave-induced ripple migration in wave-flume experiments and its implications for sediment transport

A thorough discussion of results from laboratory experiments with regular waves sheds light on the gap that lies between the sediment transport associated with ripple migration and the performance of a standard bedload transport formula in terms of bed shear concept. It is found that the extent of deviations of the bedload transport formula by Ribberink (1998) from the measured rate of sediment transport associated with ripple migration becomes systematically apparent under conditions of increasing settling time factor Ω s (= η/( w 0 T); η is the ripple height, w 0 the settling velocity and T the wave period). Re-examination of previous two field studies demonstrates a further reinforcement for phase-lag argument addressed in this paper.

Application of Image Processing in Unsteady Flows to Investigate Bed Load Transport

Image processing is a promising technique to investigate bed load characteristics and its movement which has been applied in steady flow conditions for the last few decades. In this study, it is aimed to perform experiments in unsteady flows through generating a hydrograph to apply this technique and calculate the bed load transport. The sediment motion is recorded by a video recorder. The video records are analyzed by image processing techniques to determine the number and area of active grains moving at any instant as well as the average velocity of the grains. A bed load transport formula is introduced particularly used by the digital video analysis in which the time variation of the bed load due to the hydrograph could be precisely determined. It is revealed that the bed load determined at two sections of the flume is in accord and the bed load – time curve has a fluctuating character.

A comparison between flume and field bed load transport data and consequences for surface‐based bed load transport prediction

The ability of simple equations to predict bed load transport with limited knowledge of the bed surface material was investigated. This was done using a data set consisting of 7,636 bed load transport values from the flume (1,317 data) and from 84 river reaches (6,319 field data). It was possible to collapse field and flume data by correcting the ratio between the Shields number and its critical value with a very simple hiding function proposed as a power law of the D84/D50 ratio. In so doing, a surface‐based bed load transport formula was proposed. It was successfully tested on an independent data set (comprising sand and gravel bed rivers with slope in the range 0.0002–0.08), with 86% of the values predicted within a precision of 1 order of magnitude. Moreover, the formula reproduced the low transport rates well, contrary to the usual surface‐based formulas also tested, and is particularly well suited for estimating low transport rates associated with near‐bankfull flow discharge. This new formula is neither time consuming (no fractionwise calculation) nor data consuming (the required parameters are the flow discharge, the active width, the slope, and the surface grain diameters D50 and D84).

Surface-Based Fractional Transport Predictor: Deterministic or Stochastic

Stochastic bed load transport formulas for nonuniform sediment exist, but most of them do not account for the composition of surface material to predict fractional transport rate. This study transformed a surface-based bed-load transport predictor to a stochastic one by approximating the fluctuation of bed-shear stress with a standard log-normal distribution. The deterministic predictor underpredicts fractional transport rate at low values of bed-shear stress and Reynolds number. The modified stochastic predictor predicts fractional transport rate more accurately and converges to the deterministic one at high shear stresses.

Effect of Coarse Surface Layer on Bed-Load Transport

Existing bed-load transport formulas may overestimate the transport rate in mountain rivers by two orders of magnitude or more. Recently published field data sets provide an opportunity to take a fresh look at the bed-load transport relationship and it is hypothesized that the overestimate is due to a failure to account for the effect of a coarse surface layer of bed material inhibiting the release of fine subsurface material. Bed-load transport is determined as gs=aρ(q−qc) where q=water discharge per unit width; qc=critical value for initiation of bed material movement; ρ=water density; and a=coefficient. The gs∕q relationship is typically piecewise linear, characterized by two transport phases with, respectively, low and high rates of change. Twenty-one flume and 25 field data sets were used to quantify the relationship for Phase 2. The flume data confirm the dependence of a on S1.5, where S=channel slope, in agreement with earlier studies. The field data additionally show that a varies inversely with the degree of bed armoring, given by the ratio of surface to subsurface bed material size. The finding is consistent with the hypothesis and suggests the need to account for the bed material supply limitation in the bed-load transport formula. However, the available data are not entirely sufficient to rule out an alternative dependency, or codependency, on flow resistance. The critical conditions for initiation of Phase 2 transport are also quantified as a function of bed material size and channel slope. The resulting set of equations allows a more accurate estimation of Phase 2 bed-load transport rates. However, the equations are empirical and should be restricted for use within the range of conditions used in their development, to determine mean rather than instantaneous transport rates and to determine bulk transport rates, not transport by size fraction.

3-D NUMERICAL SIMULATION OF FLOW AND CLEAR WATER SCOUR BY INTERACTION BETWEEN BRIDGE PIERS

In this paper, the flow and local scour variation a round single pier and introducing interaction effect between bridge piers were studie d using 3D flow model. The model used a finite-volume method to solve the non-transi ent Navier-Stokes equations for three dimensions on a general non-orthogonal grid. The e k turbulence model is used to solve the Reynolds-stress term. The numeric al model solves the sediment continuity equation in conjunction with van Rijn’s bed-load sediment transport formula to simulate the bed evolution. The 3D flow model was verified through experimental study in the non cohesive bed material in an experimental flume. The different causes of local scour around the pier wer e simulated well, such as bow flow, down flow, horseshoe vortex, pressure variation and lee-wake vortex. It was found from this research study that the local scour depth by interaction process between bridge piers depends on the Froude number, the dist ance between piers and the diameter of piers. The maximum scour depth for double piers is higher than that for single pier. Furthermore, the effect of pier shape on the scour process was studied and it was found that the maximum scour depth for circu lar pier is less than that for rectangular one for both single and double piers ca ses. The results show good agreement between simulation and experimental results. Also, empirical equation was developed from the experimental data for computing the maximum scour depth due to the interaction between bridge piers.

Unified View of Sediment Transport by Currents and Waves. I: Initiation of Motion, Bed Roughness, and Bed-Load Transport

Attention is given to the properties of sediment beds over the full range of conditions (silts to gravel), in particular the effect of fine silt on the bed composition and on initiation of motion (critical conditions) is discussed. High-quality bed-load transport data sets are identified and analyzed, showing that the bed-load transport in the sand range is related to velocity to power 2.5. The bed-load transport is not much affected by particle size. The prediction of bed roughness is addressed and the prediction of bed-load transport in steady river flow is extended to coastal flow applying an intrawave approach. Simplified bed-load transport formulas are presented, which can be used to obtain a quick estimate of bed-load transport in river and coastal flows. It is shown that the sediment transport of fine silts to coarse sand can be described in a unified model framework using fairly simple expressions. The proposed model is fully predictive in the sense that only the basic hydrodynamic parameters (depth, current velocity, wave height, wave period, etc.) and the basic sediment characteristics (d10, d50, d90, water temperature, and salinity) need to be known. The prediction of the effective bed roughness is an integral part of the model.

VALIDATION OF A NUMERICAL MODEL FOR FLOW AND BEDFORM DYNAMICS

An explorative study has been carried out within the scope of this paper in order to validate a morphodynamic numerical model with the assessment of model sensitivity to some factors and parameters. The flow model component is a vertical two-dimensional with non-hydrostatic, unsteady free surface flow condition. The flow model has been coupled with sediment transport models. Two different sediment transport approaches have been used, namely Ashida & Michiue's bedload transport formula and a stochastic pick up deposition formulation for non-equilibrium sediment transport proposed by Nakagawa & Tsujimoto. The model performance has been tasted for different turbulence closures, namely a zero-equation, a standard k-e and a non-linear k-e models, in the context of morphodynamic simulation. Model performance has been evaluated for the prediction of temporal variation of flow-depth and boundary shear stress induced by bedform evolution based on experimental measurements.

Some considerations about the simulation of breach channel erosion on landslide dams

The analysis of the flood hazard related to the areas downstream of landslide dams is one of the most interesting aspects of studying the formation and the failure of natural dams. The BREACH code [14], simulating the collapse of earthen dams, both man-made and naturally formed by a landslide, was chosen in order to analyse the case of the Valderchia landslide (central Italy). The bed-load transport formula used in BREACH (Meyer-Peter and Muller, modified by Smart [27]) is based on flume experiments with well-sorted sediments. Such a methodology probably makes this equation not very suitable for describing the sediment transport peculiar to a landslide body presenting a very poor material sorting. The Schoklitsch [26] formula was implemented into the programme as an alternative to the Smart equation. However, because the landslide deposits may often have a strongly bimodal grain–size frequency curve, the percentile D 50 (the typical granulometric parameter requested by bed-load sediment transport formulas) can sometimes correspond to one of the grain-size classes which are really present to a lesser degree. To consider this phenomenon, the BREACH programme (version 7/88-1) was implemented with a new procedure that calculates two granulometric curves, one for each mode of the original distribution, and evaluates transport of the landslide material separately. Results of the analysis show that the model is very sensitive to the bed-load equation and that the procedure implemented to consider the eventual bimodal distribution of the dam material simulates the armouring phenomenon (which can stop the erosion of the dam during the overtopping phase).

Testing bedload transport formulae using morphologic transport estimates and field data: lower Fraser River, British Columbia

AbstractMorphologic transport estimates available for a 65‐km stretch of Fraser River over the period 1952–1999 provide a unique opportunity to evaluate the performance of bedload transport formulae for a large river over decadal time scales. Formulae tested in this paper include the original and rational versions of the Bagnold formula, the Meyer‐Peter and Muller formula and a stream power correlation. The generalized approach adopted herein does not account for spatial variability in flow, bed structure and channel morphology. However, river managers and engineers, as well as those studying rivers within the context of long‐term landscape change, may find this approach satisfactory as it has minimal data requirements and provides a level of process specification that may be commensurable with longer time scales. Hydraulic geometry equations for width and depth are defined using morphologic maps based on aerial photography and bathymetric survey data. Comparison of transport predictions with bedload transport measurements completed at Mission indicates that the original Bagnold formula most closely approximates the main trends in the field data. Sensitivity analyses are conducted to evaluate the impact of inaccuracies in input variables width, depth, slope and grain size on transport predictions. The formulae differ in their sensitivity to input variables and between reaches. Average annual bedload transport predictions for the four formulae show that they vary between each other as well as from the morphologic transport estimates. The original Bagnold and Meyer‐Peter and Muller formulae provide the best transport predictions, although the former underestimates while the latter overestimates transport rates. Based on our findings, an error margin of up to an order of magnitude can be expected when adopting generalized approaches for the prediction of bedload transport. Copyright © 2005 John Wiley & Sons, Ltd.

Transport solide par charriage sous une interaction houle-courant

ABSTRACT A bed-load transport formula was investigated for steady flows, oscillatory flows and combined steady and oscillatory flows. 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 function 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. A specific study on the effects of sediment phase-lag on net sediment transport was carried out. A good correlation was found between the bed-load model and the measurements for colinear oscillatory and steady flows.

Sand grain threshold, in relation to bed ‘stress history’: an experimental study

AbstractBesides particle size, density and shape, the erodibility of a sediment bed depends also upon the exposure to prethreshold velocities in the overlying flow. Such flow effectively rearranges the grains (at and below the bed surface), causing them to become more resistant to subsequent erosion. The effects of the ‘stress history’, leading up to the critical condition for sediment movement, are investigated for unidirectional flows generated in a recirculating laboratory flume. The sediment beds investigated consisted of cohesionless quartz sand grains, with mean grain diameters of 0·194 mm (fine sand), 0·387 mm (medium sand) and 0·774 mm (coarse sand), with narrow particle‐size distributions. The critical (threshold) shear velocity (target value) for the three beds was established, within 2·5 min of increasing the flow from zero velocity. The subsequent experiments were performed under prethreshold velocities at 70% (for 5, 10, 20, 40 and 80 min exposure duration), 80% (for 5, 10, 20, 40 and 80 min exposure duration), 90 and 95% (for 5, 10, 20, 40, 80 and 120 min exposure duration) of the target value. Following exposure to these different prethreshold conditions, the flow was increased then to reach actual critical conditions, within a period of 2·5 min. The critical condition for the initiation of sediment movement was established using visual observation (supplemented by video recordings), according to the Yalin criterion. The results show that if the exposure duration to prethreshold velocities remains constant, then the critical shear velocity increases with increasing prethreshold velocity. Likewise, if the prethreshold velocity remains constant, then the critical shear velocity increases with increasing exposure duration. In some circumstances, the critical shear velocity was found to increase by as much as 27%. An empirical formula is proposed to account for the exposure correction to be applied to the critical shear velocities of sand‐sized sediment beds; this is prior to their inclusion into bedload transport formulae, for an improved prediction of the magnitude and nature of transport.

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.