Bedload Transport Model Research Articles

Bedform response to flow variability

ABSTRACTLaboratory observations and computational results for the response of bedform fields to rapid variations in discharge are compared and discussed. The simple case considered here begins with a relatively low discharge over a flat bed on which bedforms are initiated, followed by a short high‐flow period with double the original discharge, during which the morphology of the bedforms adjusts, followed in turn by a relatively long period of the original low discharge. For the grain size and hydraulic conditions selected, the Froude number remains subcritical during the experiment, and sediment moves predominantly as bedload. Observations show rapid development of quasi‐two‐dimensional bedforms during the initial period of low flow with increasing wavelength and height over the initial low‐flow period. When the flow increases, the bedforms rapidly increase in wavelength and height, as expected from other empirical results. When the flow decreases back to the original discharge, the height of the bedforms quickly decreases in response, but the wavelength decreases much more slowly. Computational results using an unsteady two‐dimensional flow model coupled to a disequilibrium bedload transport model for the same conditions simulate the formation and initial growth of the bedforms fairly accurately and also predict an increase in dimensions during the high‐flow period. However, the computational model predicts a much slower rate of wavelength increase, and also performs less accurately during the final low‐flow period, where the wavelength remains essentially constant, rather than decreasing. In addition, the numerical results show less variability in bedform wavelength and height than the measured values; the bedform shape is also somewhat different. Based on observations, these discrepancies may result from the simplified model for sediment particle step lengths used in the computational approach. Experiments show that the particle step length varies spatially and temporally over the bedforms during the evolution process. Assuming a constant value for the step length neglects the role of flow alterations in the bedload sediment‐transport process, which appears to result in predicted bedform wavelength changes smaller than those observed. However, observations also suggest that three‐dimensional effects play at least some role in the decrease of bedform wavelength, so incorporating better models for particle hop lengths alone may not be sufficient to improve model predictions. Published in 2011. This article is a US Government work and is in the public domain in the USA.

Initial stages of erosion and bed form development in a turbulent flow around a cylindrical pier

[1] Bed load transport and erosion in fine sediment beds are mainly driven by the dynamics of the near-bed turbulent flow. In situations when the shear stress is not sufficiently high to produce significant transport, the presence of an obstacle can initiate erosion and trigger the development of bed forms, which are produced by the emergence of the turbulent horseshoe vortex (THV) system. We develop a numerical model to investigate the initial stages of erosion and the development of ripples produced by the THV system in the vicinity of a surface-mounted cylindrical pier. The flow is simulated using the detached eddy simulation approach, which has been shown to accurately resolve most of the turbulent stresses produced by the THV. To compute the erosion, the Exner equation is coupled to a new bed load transport model that directly incorporates the effect of the instantaneous flow field on sediment transport. The morphodynamic model is integrated simultaneously with the flow equations using an arbitrary Lagrangian-Eulerian method for moving boundaries. Even though the time rate of scour is slower compared to the observations, the computed results exhibit essentially all the dynamics of erosion, including the emergence of ripples reported in the experiments of Dargahi (1990). The bed forms show similar velocities as reported in the experiments and are shown to be statistically similar to ripples measured in laboratory experiments and in nature. To our knowledge, this is the first three-dimensional simulation to capture the ripple dynamics that evolve naturally from the nonlinear interactions between the flow and the bed.

Non-uniformity and layering in sediment transport modelling 2: river application

The objectives of this research are to verify a numerical sediment transport model based on field data from a reach of the Austrian Danube River, to identify the influence of non-uniform bed-load transport modelling and the selected bed schematization on the simulation results and to investigate the cause and the numerical reproducibility of spatio-temporal variations found in field data. Simulation results based on variants of the transport equation by Meyer-Peter and Müller were compared to basket sampler and radio tracer stone observations. The best agreement was obtained for a non-uniform formulation with an exchange layer, while the uniform formulation of the transport equation underestimated the transport rates for discharges lower than the mean flow since it predicts a later onset of bed-load transport. The consideration of an exchange layer in the numerical code led to the prediction of bed-load sheets due to sorting waves, allowing the model to account for the spatio-temporal variability of field data.

Bedload transport in shallow water models: Why splitting (may) fail, how hyperbolicity (can) help

In this paper, we are concerned with sediment transport models consisting of a shallow water system coupled with the so called Exner equation to describe the evolution of the topography. We show that, for some bedload transport models like the well-known Meyer-Peter and Müller model, the system is hyperbolic and, thus, linearly stable, only under some constraint on the velocity. In practical situations, this condition is hopefully fulfilled. Numerical approximations of such system are often based on a splitting method, solving first shallow water equation on a time step and, updating afterwards the topography. It is shown that this strategy can create spurious/unphysical oscillations which are related to the study of hyperbolicity. Using an upper bound of the largest eigenvector may improve the results although the instabilities cannot be always avoided, e.g. in supercritical regions.

Lagrangian model of bed-load transport in turbulent junction flows

Motivated by the need to gain fundamental insights into the mechanisms of bed-load sediment transport in turbulent junction flows, we carry out a computational study of Lagrangian dynamics of inertial particles initially placed on the bed upstream of a surface-mounted circular cylinder in a rectangular open channel (Dargahi, J. Hydraul. Engng, vol. 116, 1990, pp. 1197–1214). The flow field at Re = 39000 is simulated using the detached eddy simulation (DES) approach (Spalart et al., In Advances in DNS/LES, ed. C. Liu & Z. Liu, 1997, Greyden), which has already been shown to accurately resolve most of the turbulent stresses produced by the low-frequency, bimodal fluctuations of the turbulent horseshoe vortex (Paik et al., J. Hydraul. Engng, vol. 131, 1990, pp. 441–456; Escauriaza & Sotiropoulos, Flow Turbul. Combust., 2010, in press). The trajectory and momentum equations for the sediment particles are integrated numerically simultaneously with the flow governing equations assuming one-way coupling and neglecting particle-to-particle interactions (dilute flow) but taking into account bed–particle interactions and the effects of the instantaneous hydrodynamic forces induced by the resolved fluctuations of the coherent vortical structures. The computed results show that, in accordance with the simulated clear-water scour condition (i.e. the magnitude of the particle stresses is near the threshold of motion), the transport of sediment grains is highly intermittent and exhibits essentially all the characteristics of bed-load sediment transport observed in laboratory and field experiments. Groups of sediment grains are dislodged from the bed simultaneously in seemingly random bursting events and begin to move, saltating or sliding along the bed. Furthermore, particles that are not entrained into the bed-load layer are found to form streaks aligned with near-wall vortices around the cylinder. The global transport of particles is studied by performing a statistical analysis of the bed-load flux to reveal scale-invariance of the process and multifractality of particle transport as the overall effect of the coherent structures of the flow. A major finding of this work is that a relatively simple Lagrangian model coupled with a coherent-structure resolving simulation of the turbulent flow is able to reproduce the sediment dynamics observed in multiple experiments performed under similar conditions, and provide fundamental information on the initiation of motion and the multifractal nature of bed-load transport processes. The results also motivate the development of new Eulerian bed-load transport models that consider unsteady conditions and incorporate the intermittency of the unresolved scales of sediment motion.

A one-dimensional bedload transport model for steep slopes

A sediment routing model called SETRAC has been developed to simulate bedload transport in torrents and mountain streams. SETRAC is the acronym for Sediment Transport in Alpine Catchments. Flow resistance is modelled using an approach proposed by Smart and Jäggi, and bedload transport is predicted with an equation by Rickenmann. Form roughness losses are accounted for by an empirical approach which is based on flow resistance observations in mountain streams. It is possible to compute fractional bedload transport taking grain sorting effects into account. Flow hydrographs are routed through the channel network by using the kinematic wave approach. An application of the model is discussed for an extreme flood event of August 2005 in the Chiene catchment, Switzerland, which mobilized 120,000 m3 of bedload along the mountain stream. The simulation results underline the importance of form resistance when modelling bedload transport in steep and rough channels.

Fractional dispersion in a sand bed river

Random walk models of fluvial bed load transport use probability distributions to describe the distance a grain travels during an episode of transport and the time it rests after deposition. These models typically employ probability distributions with finite first and second moments, reflecting an underlying assumption that all the factors that influence sediment transport tend to combine in such a way that the length of a step or the duration of a rest can be characterized by a mean value surrounded by a specific amount of variability. The observation that many transport systems exhibit apparent scale‐dependent behavior and non‐Fickian dispersion suggests that this assumption is not always valid. We revisit a nearly 50 year old tracer experiment in which the tracer plume exhibits the hallmarks of dispersive transport described by a step length distribution with a divergent second moment and no characteristic dispersive size. The governing equation of this type of random walk contains fractional‐order derivatives. We use the data from the experiment to test two versions of a fractional‐order model of dispersive fluvial bed load transport. The first version uses a heavy‐tailed particle step length distribution with a divergent second moment to reproduce the anomalously high fraction of tracer mass observed in the downstream tail of the spatial distribution. The second version adds a feature that partitions mass into a detectable mobile phase and an undetectable, immobile phase. This two‐phase transport model predicts other features observed in the data: a decrease in the amount of detected tracer mass over the course of the experiment and enhanced particle retention near the source. The fractional‐order models match the observed plume shape and growth rates better than prior attempts with classical models.

Stability of bedforms in laminar flows with free surface: from bars to ripples

The present paper is devoted to the formation of sand patterns by laminar flows. It focuses on the rhomboid beach pattern, formed during the backswash. A recent bedload transport model, based on a moving-grains balance, is generalized in three dimensions for viscous flows. The water flow is modelled by the full incompressible Navier–Stokes equations with a free surface. A linear stability analysis then shows the simultaneous existence of two distinct instabilities, namely ripples and bars. The comparison of the bar instability characteristics with laboratory rhomboid patterns indicates that the latter could result from the nonlinear evolution of unstable bars. This result, together with the sensibility of the stability analysis with respect to the parameters of the transport law, suggests that the rhomboid pattern could help improving sediment transport models, so critical to geomorphologists.

Modeling Bedload Transport of Juvenile Bivalves: Predicted Changes in Distribution and Scale of Postlarval Dispersal

The potential consequences of bedload transport of postlarvae for patterns of distribution of marine invertebrates were explored by developing a bedload transport model for juvenile bivalves in a small estuary in New Jersey, USA. A simple numerical model of tidal current hydrodynamics was developed based on field measurements of shear stresses near the bottom. Burrowing behavior of bivalves was incorporated into the model of bedload transport by using estimates of entrainment rates of Gemma gemma and Mya arenaria in a laboratory flume, and jump lengths of the bivalves were estimated by methods previously developed for noncohesive particles. Based on the flood domination and strong gradient of shear stresses in the Navesink estuary, our model predicted that juvenile bivalves would accumulate in the center of the estuary, traveling up to several kilometers over 30 days. Field distributions of juvenile bivalves were consistent with the model predictions for other species of bivalves but not for G. gemma, for which field distributions of both 500-μm individuals were concentrated in the eastern end of the estuary. Differences between the bedload model and G. gemma distributions suggest that spatial variation in burrowing behavior or biological interactions are playing an important role in maintaining distribution patterns of this species in spite of high levels of bedload transport. This modeling approach is applicable to other juvenile benthic invertebrates that disperse as bedload and is a useful model against which to compare field observations of rates of transport and patterns of distribution and abundance.

A Three-Dimensional Coupled Fluid-Sediment Interaction Model With Bed-Load/Suspended-Load Transport for Scour Analysis Around a Fixed Structure

The predictive capability of a three-dimensional (3D) numerical model for sediment transport and resulting scour around a structure is investigated in this study. Starting with the bed-load and suspended-load sediment transport (reference) model developed by Takahashi et al. (2000, “Modeling Sediment Transport Due to Tsunamis With Exchange Rate Between Bed Load Layer and Suspended Load Layer,” Proceedings of the 27th International Conference on Coastal Engineering, ASCE, Sydney, Australia, pp. 1508–1519), we first introduce an extension to incorporate Nielsen’s modified Shields parameter to account for the effects of infiltration/exfiltration flow velocity across the fluid-sand interface on the sediment transport (the modified Shields-parameter model). We then propose a new model to include the influence of the effective stress to account for the stress fluctuations inside the surface layer of the sand bed (the effective-stress model). The three analytical models are incorporated into a 3D numerical solver developed by Nakamura et al. (2008, “Tsunami Scour Around a Square Structure,” Coast. Eng. Japan, 50(2), pp. 209–246) to analyze the dynamics of fluid-sediment interaction and scour. Their solver is composed of two modules, namely, a finite-difference numerical wave tank and a finite-element coupled sand-skeleton pore-water module. The predictive capability of the three alternative coupled models is calibrated against hydraulic experiments on sediment transport and resulting scour around a fixed rigid structure due to the run-up of a single large wave in terms of the sediment transport process and the final scour profile after the wave run-up. It is found that, among the three models considered, the proposed effective-stress model most accurately predicts the scouring process around the seaward corner of the structure. The results also reveal that the deposition and erosion patterns predicted using the effective-stress model are in good agreement with measured results, while a scour hole at the seaward corner of the structure cannot be always predicted by the other two models.

A three-dimensional distinct element model for bed-load transport

A three-dimensional model is developed to simulate bed-load transport using Distinct Element Method (DEM). The model describes the bed-load transport as dynamically interdependent motions of individual sediment particles under the flow action. A steady logarithmic velocity distribution over the flow depth is considered to represent fluid motion, while turbulent intensities are also included within the closure model. The present model accounts for both fluid–particle and particle–particle interactions, which are the predominant micro-mechanisms of bed-load transport under low and high tractive forces, respectively. Characteristic features of bed-load transport, previously reported by experimental observations, are numerically reproduced. The concept of vertical momentum transfer is exploited to describe features of concurrently present of saltation and sheet-flow layers. It is concluded that the vertical motion of particles is not significantly active in sheet-flow layer and the particle momentum is preserved. Consequently the interparticle collisions is the predominant mechanism of bed-load transport at high fluid tractive forces.

Numerical Simulation of Contraction Scour in an Open Laboratory Channel

A finite-volume computer code developed at the Institute for Hydromechanics, University of Karlsruhe, has been used to calculate the flow and sediment transport in a laboratory channel with constriction and movable bed. The flow is calculated by solving the fully three dimensional Reynolds-averaged Navier-Stokes equations with k−ε turbulence model. The bed deformation is obtained from an overall mass-balance equation for sediment transport and the bed-load transport is simulated with a nonequilibrium model. The calculated results for flow and scour development in the laboratory channel are compared with experimental measurements. The sensitivity of the simulated results to the nonequilibrian adaptation-length parameter in the nonequilibrium bed-load transport model is investigated systematically, which represents the main contribution of this paper.

An Integrated Hydrodynamic-Sedimentologic-Morphologic Model for the Evolution of Alluvial Channels Cross-Sections

A practical implementation of an integrated hydrodynamic-sedimentologic-morphologic model for the evolution of the cross section of an alluvial channel is presented. It is shown that the driving hydrodynamics can be solved through a series of concatenated 1D hydrodynamic models, which reduces the mathematical complexity to its minimum. A mechanistic model for bed load transport is presented, which can be used readily by practitioners. As part of the morphologic model, a sliding algorithm is proposed to limit the lateral slope which, in particular, efficiently deals with non-cohesive bank erosion. The obtained integrated model is validated by comparing its results with experimental data and previous calculations for evolving channel cross sections. The comparison shows good agreement between these data and the numerical results. To illustrate the model capabilities, numerical experiments are performed on open channels with different radii of curvature to attain stabilized cross-section forms.

Estimation of Obstructing Effect on Longshore Bedload Transport Caused by River Mouth Terrace

The wave and current-induced bedload transport model by Nakamura (2006) is applied for evaluating the obstruction to the longshore transport of bedload with the terrace geographical feature at river mouth. Numerical results show that the terrace feature has the obstructing effect on the longshore transport of bed-load. The obstructing effect is caused by two mechanisms. One is, in case of non-wave breaking on the terrace, the longshore transport decreases by the opposite directional transport caused by the cell circulation at the downstream edge of the terrace. Another is, in case of wave breaking at the terrace edge and edge slope, the longshore transport changes the direction offshoreward at the upstream edge of the terrace. In both cases, the obstructing effect grows by the terrace width wide.

A physically-based bedload transport model developed for 3-D reach-scale cellular modelling

A bedload transport model is outlined which aims to provide a more physically-based description of bed surface and bedload grain-size distributions (GSDs) than existing empirical bedload models, but is simple enough to be incorporated in a reduced-complexity modelling framework. One motivation for this is the increasing need for 3-D models of decadal, reach-scale channel and floodplain evolution for use in river management. The computational demands of simulation at these scales mean that the appropriate models are necessarily of the reduced-complexity type. The bedload model is derived through analysis of sediment beds created using a discrete element model. These beds are “eroded” using a probabilistic entrainment algorithm which predicts the bedload GSD. There are two parts to the bedload model. The first identifies a functional relationship between the applied shear stress and the fraction of the bed undergoing active transport, while the second uses this fraction, in conjunction with the surface GSD, to produce a bedload GSD. The model produces bedload coarsening as shear stress increases from a value of one to a value of five times a reference shear stress. This model is tested in a series of numerical experiments, applied using a simple cellular model of a straight flume, which is run in both recirculation and feed modes. In both modes, the equilibrium GSDs of the bedload and surface are consistent with expected flume behaviour, suggesting that the model provides a reasonable reproduction of sediment entrainment, transport and sorting.

Numerical study on wave-induced filtration flow across the beach face and its effects on swash zone sediment transport

A numerical model, coupling an analysis of beach groundwater flow with an analysis of swash wave motion over a uniform slope, is presented. Model calculations are performed to investigate the variations of swash-induced filtration flows across the beach face for different input parameters. Swash zone sediment transport under the influence of such filtration flow across the beach face is investigated through modification of effective weight of sediment particle and modification of swash boundary layer thickness. These effects are quantified based on a bed load transport model with a modified Shields parameter.

Machine Learning Approach to Modeling Sediment Transport

Inaccuracies of sediment transport models largely originate from our limitation to describe the process in precise mathematical terms. Machine learning (ML) is an alternative approach to reduce the inaccuracies of sedimentation models. It utilizes available domain knowledge for selecting the input and output variables for the ML models and uses modern regression techniques to fit the measured data. Two ML methods, artificial neural networks and model trees, are adopted to model bed-load and total-load transport using the measured data. The bed-load transport models are compared with the models due to Bagnold, Einstein, Parker et al., and van Rijn. The total-load transport models are compared with the models due to Ackers and White, Bagnold, Engelund and Hansen, and van Rijn. With the chosen data sets on bed-load and total-load transport the ML models provided better accuracy than the existing ones.

A bed‐load transport model for rough turbulent open‐channel flows on plane beds

AbstractData from flume studies are used to develop a model for predicting bed‐load transport rates in rough turbulent two‐dimensional open‐channel flows moving well sorted non‐cohesive sediments over plane mobile beds. The object is not to predict transport rates in natural channel flows but rather to provide a standard against which measured bed‐load transport rates influenced by factors such as bed forms, bed armouring, or limited sediment availability may be compared in order to assess the impact of these factors on bed‐load transport rates. The model is based on a revised version of Bagnold's basic energy equation ibsb = ebω, where ib is the immersed bed‐load transport rate, ω is flow power per unit area, eb is the efficiency coefficient, and sb is the stress coefficient defined as the ratio of the tangential bed shear stress caused by grain collisions and fluid drag to the immersed weight of the bed load. Expressions are developed for sb and eb in terms of G, a normalized measure of sediment transport stage, and these expressions are substituted into the revised energy equation to obtain the bed‐load transport equation ib = ω G 3·4. This equation applies regardless of the mode of bed‐load transport (i.e. saltation or sheet flow) and reduces to ib = ω where G approaches 1 in the sheet‐flow regime. That ib = ω does not mean that all the available power is dissipated in transporting the bed load. Rather, it reflects the fact that ib is a transport rate that must be multiplied by sb to become a work rate before it can be compared with ω. It follows that the proportion of ω that is dissipated in the transport of bed load is ibsb/ω, which is approximately 0·6 when ib = ω. It is suggested that this remarkably high transport efficiency is achieved in sheet flow (1) because the ratio of grain‐to‐grain to grain‐to‐bed collisions increases with bed shear stress, and (2) because on average much more momentum is lost in a grain‐to‐bed collision than in a grain‐to‐grain one. Copyright © 2006 John Wiley & Sons, Ltd.

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.

Bed-load transport model based on fractional size distribution

Two models based on the fractional size distribution approach, used in conjunction with the excess discharge theory, have been developed by using bed-load data collected from the Roaring River (Rocky Mountain National Park, Colorado) during the summer of 1995. The first model is based on the critical discharge value of individual fractional (IF) sizes, IF model (for log-normal and nonlog-normal size distribution modes), while the other is based on critical discharge value for total (combined) sizes, total fractional (TF) sizes model (for log-normal and nonlog-normal size distribution modes). The performance of the log-normal size distribution based models was tested with data from the Roaring River, Rich Creek, and Fourmile Creek (three Colorado streams), whereas the performance of the nonlog-normal size distribution based models was tested using Pitzbach River data. The performance of the models was also tested by comparing their results with the Inpasihardjo fractional size distribution based model. For all tests, the TF model performed better for both the log-normal and nonlog-normal grain size distributions.Key words: fractional size, critical discharge, IF model, TF model, discharge theory, Roaring River.