Transport Formula Research Articles

Bedload transport and temporal variation of non-uniform sediment in a seepage-affected alluvial channel

ABSTRACTUnderstanding bedload transport fluctuations in rivers is crucial for complementing the existing knowledge on sediment transport theory. In this contribution, we use a natural-scale laboratory flume to analyse bedload transport fluctuations in non-uniform sand under normal flow conditions. Based on the significance of downward seepage, we incorporate the seepage effect on bedload transport over a non-uniform sand bed channel. The weight of the dry material was measured, and the volumetric transport rate per unit width (bedload transport rate) was estimated. An important observation is that the bedload transport rate initially rapidly increases with time and reaches a maximum value. Based on experimental data, we propose an empirical expression to estimate temporal bedload transport. In addition, an empirical model for bedload transport is proposed by incorporating downward seepage among other variables. The performance of several existing bedload transport formulae was also taken into account by the experimental datasets.

Quantifying bed‐related suspended load in gravel bed rivers through an analysis of the bedload‐suspended load relationship

AbstractSuspended load transport can strongly impact ecosystems, dam filling and water resources. However, contrary to bedload, the use of physically based predicting equations is very challenging because of the complexity of interactions between suspended load and the river system. Through the analysis of extensive data sets, we investigated extent to which one or several river bed or flow parameters could be used as a proxy for quantifying suspended fluxes in gravel bed rivers. For this purpose, we gathered in the literature nearly 2400 instantaneous field measurements collected in 56 gravel bed rivers. Among all standard dimensionless parameters tested, the strongest correlation was observed between the suspended sediment concentration and the dimensionless bedload rate. An empirical relation between these two parameters was calibrated. Used with a reach average bedload transport formula, the approach allowed to successfully reproduce suspended fluxes measured during major flood events in seven gravel bed alpine rivers, morphodynamically active and distant from hillslope sources. These results are discussed in light of the complexity of the processes potentially influencing suspended load in a mountainous context. The approach proposed in this paper will never replace direct field measurements, which can be considered the only confident method to assess sediment fluxes in alpine streams; however, it can increment existing panel tools that help river managers to estimate even rough but not unrealistic suspended fluxes when measurements are totally absent. © 2019 John Wiley & Sons, Ltd.

Effects of Particle Properties on the Settling and Rise Velocities of Microplastics in Freshwater under Laboratory Conditions.

Microplastic (MP) contaminates terrestrial, aquatic, and atmospheric environments. Although the number of river sampling studies with regard to MP concentrations is increasing, comprehension of the predominant transport processes of MP in the watercourse is still very limited. In order to gain a better process understanding, around 500 physical experiments were conducted to shed more light on the effects of particle shape, size and density on the rise and settling velocities of MP. The determined velocities ranged between 0.39 cm/s for polyamide fibers (settling) and 31.4 cm/s for expanded polystyrene pellets (rise). Subsequently, the determined velocities were compared with formulas from sediment transport and, as there were large differences between theoretically and experimentally determined velocities, own formulas were developed to describe settling and rise velocities of MP particles with a large variety of shapes, sizes and densities. This study shows that MP differs significantly from sediment in its behavior and that a transfer of common sediment transport formulas should be treated with caution. Furthermore, the established formulas can now be used in numerical simulations to describe the settling and rising of MP more precisely.

On the Relationship between Experimental and Numerical Modelling of Gravel-Bed Channel Aggradation

This communication explores the use of numerical modelling to simulate the hydro-morphologic response of a laboratory flume subject to sediment overloading. The numerical model calibration was performed by introducing a multiplicative factor in the Meyer–Peter and Müller transport formula, in order to achieve a correspondence with the bed and water profiles recorded during a test carried out under a subcritical flow regime. The model was validated using a second subcritical test, and then run to simulate an experiment during which morphological changes made the water regime switch from subcritical to supercritical. The “relationship” between physical and numerical modelling was explored in terms of how the boundary conditions for the two approaches had to be set. Results showed that, even though the first two experiments were reproduced well, the third one could not be modeled adequately. This was explained considering that, after the switch of the flow regime, some of the boundary conditions posed into the numerical model turned out to be misplaced, while others were lacking. The numerical modelling of hydro-morphologic processes where the flow regime is trans-critical in time requires particular care in the position of the boundary conditions, accounting for the instant at which the water regime changes.

Global sensitivity analysis of parameter uncertainty in landscape evolution models

Abstract. The evaluation and verification of landscape evolution models (LEMs) has long been limited by a lack of suitable observational data and statistical measures which can fully capture the complexity of landscape changes. This lack of data limits the use of objective function based evaluation prolific in other modelling fields, and restricts the application of sensitivity analyses in the models and the consequent assessment of model uncertainties. To overcome this deficiency, a novel model function approach has been developed, with each model function representing an aspect of model behaviour, which allows for the application of sensitivity analyses. The model function approach is used to assess the relative sensitivity of the CAESAR-Lisflood LEM to a set of model parameters by applying the Morris method sensitivity analysis for two contrasting catchments. The test revealed that the model was most sensitive to the choice of the sediment transport formula for both catchments, and that each parameter influenced model behaviours differently, with model functions relating to internal geomorphic changes responding in a different way to those relating to the sediment yields from the catchment outlet. The model functions proved useful for providing a way of evaluating the sensitivity of LEMs in the absence of data and methods for an objective function approach.

Direct Discretization Method for the Cahn---Hilliard Equation on an Evolving Surface

We propose a simple and efficient direct discretization scheme for solving the Cahn---Hilliard (CH) equation on an evolving surface. By using a conservation law and transport formulae, we derive th...

Estimating Annual Onshore Aeolian Sand Supply from the Intertidal Beach Using an Aggregated-Scale Transport Formula

In this paper, we explore an approach for annual-scale transport prediction from the intertidal beach, in which we aggregate the surface conditions of the intertidal beach, in particular moisture content and roughness, and use hourly monitoring data of wind speed and wind direction. For our case study area (Egmond Beach, The Netherlands), we include Argus video imagery in our analysis to assess the occurrence of aeolian sand transport. With the approach described to determine a characteristic moisture content value for aeolian transport, we obtained surface moisture values of 1.2% to 3.2% for wind average and wind gust respectively, implying that we need a quite dry beach. This indicates that the main area for aeolian transport corresponds to the upper part of the intertidal source, most likely the region between mean high tide line and spring high tide line.

Numerical modelling of the flow and bed evolution of a single bore-driven swash event on a coarse sand beach

This paper examines the numerical prediction of the sediment transport and bed evolution for a single swash event on a coarse sediment beach. In these conditions bed load is the dominant mode of sediment transport. Laboratory experiments of a single bore-driven swash event are simulated numerically using a fully-coupled model that solves the system of Non Linear Shallow Water Equations and the Exner sediment conservation formula. The analysis focuses on two aspects: the optimal choice of parameters for the Meyer-Peter and Müller sediment transport formula and the model used for computing the shear stress. The methods tested for the bed shear stress are the momentum integral method and the Chezy formulation in which the friction factor is computed using two different formulae. Infiltration into the beach and its effects on the shear stress and sediment transport are also modelled. Results show that the basic Meyer-Peter and Müller sediment transport formula provides good results in the run-up. On the other hand, the sediment transport in the early stage of the backwash is overestimated. A reduction of the sediment mobility constant in the formula in the backwash marginally improves the results. However, the causes of the overestimation of the sediment transport at the early stage of the backwash is the overestimation of the shear stress, while at later stages there are several contributions that are identified, i.e. modelling of the sediment transport and infiltration. It is also suggested that the Meyer-Peter and Müller sediment transport formula might not capture the complexity of the processes involved during the backwash. The comparison of the methods for the estimate of the bed shear stress show that comparable results can be obtained using the momentum integral method and the Chezy formulation with time and space varying friction factor. The resulting bed evolution is also described. In the predicted final profile, deposition is found in the upper part of the beach and erosion in the lower part. A bed step is formed just below the position of the initial shoreline. This feature is determined by the onset of a hydraulic jump during the backwash.

Two examples on the property of the noise in the systems of equations of fluid mechanics

In this manuscript, we discuss two examples of the results on the systems of partial differential equations in fluid mechanics obtained via stochastic analysis, for which deterministic case has no counterpart. Firstly, it is well-known that for a deterministic system of equations in fluid mechanics, a lack of diffusion creates a significant obstacle in proving its global well-posedness for a sufficiently small initial data. Nevertheless, it may be proven that for the Hall-magnetohydrodynamics system forced by L e ́ vy noise, such a result may be obtained even with zero viscous diffusion. Secondly, we may extend the classical Kelvin’s conservation of circulation flow, as well as the Cauchy’s vorticity transport formula, on the Euler equations to show that the solution to the damped Euler equations enjoys the exact same identities except with an exponential decay in time. Such identities are known to completely fail once we add diffusion and consider the damped Navier–Stokes equations. Nevertheless, discovering a stochastic Lagrangian formulation, we are able to show that such identities indeed do continue to hold informally on average, i.e. via a mathematical expectation, over the random ensemble at earlier time. The stochastic Lagrangian formulation and these identities have also been obtained for the solution to the Boussinesq system. These results are rigorously proven in Mohan and Yamazaki (0000) and Yamazaki (0000). In this manuscript we will not pursue complete proofs, which were in particular mostly inspired by Kim (2011) and Iyer (2006), but rather elaborate on motivations, difficulty and remaining open problems for future works.

Direct Discretization Method for the Cahn–Hilliard Equation on an Evolving Surface

We propose a simple and efficient direct discretization scheme for solving the Cahn–Hilliard (CH) equation on an evolving surface. By using a conservation law and transport formulae, we derive the CH equation on evolving surfaces. An evolving surface is discretized using an unstructured triangular mesh. The discrete CH equation is defined on the surface mesh and its dual surface polygonal tessellation. The evolving triangular surfaces are then realized by moving the surface nodes according to a given velocity field. The proposed scheme is based on the Crank–Nicolson scheme and a linearly stabilized splitting scheme. The scheme is second-order accurate, with respect to both space and time. The resulting system of discrete equations is easy to implement, and is solved by using an efficient biconjugate gradient stabilized method. Several numerical experiments are presented to demonstrate the performance and effectiveness of the proposed numerical scheme.

Bedform migration in a mixed sand and cohesive clay intertidal environment and implications for bed material transport predictions

Many coastal and estuarine environments are dominated by mixtures of non-cohesive sand and cohesive mud. The migration rate of bedforms, such as ripples and dunes, in these environments is important in determining bed material transport rates to inform and assess numerical models of sediment transport and geomorphology. However, these models tend to ignore parameters describing the physical and biological cohesion (resulting from clay and extracellular polymeric substances, EPS) in natural mixed sediment, largely because of a scarcity of relevant laboratory and field data. To address this gap in knowledge, data were collected on intertidal flats over a spring-neap cycle to determine the bed material transport rates of bedforms in biologically-active mixed sand-mud. Bed cohesive composition changed from below 2 vol% up to 5.4 vol% cohesive clay, as the tide progressed from spring towards neap. The amount of EPS in the bed sediment was found to vary linearly with the clay content. Using multiple linear regression, the transport rate was found to depend on the Shields stress parameter and the bed cohesive clay content. The transport rates decreased with increasing cohesive clay and EPS content, when these contents were below 2.8 vol% and 0.05 wt%, respectively. Above these limits, bedform migration and bed material transport was not detectable by the instruments in the study area. These limits are consistent with recently conducted sand-clay and sand-EPS laboratory experiments on bedform development. This work has important implications for the circumstances under which existing sand-only bedform migration transport formulae may be applied in a mixed sand-clay environment, particularly as 2.8 vol% cohesive clay is well within the commonly adopted definition of “clean sand”.

Evaluating spatial variation of suspended sediment rating curves in the middle Yellow River basin, China

AbstractAbstractThe suspended sediment load in the middle Yellow River basin (YRB) cannot be well predicted by capacity‐based transport formulas because a large fraction of suspended sediment load is composed of wash load. This study evaluated the spatial variations of sediment rating curves (SRCs) in the middle YRB. Both power and linear SRCs were used to fit daily flow and suspended sediment concentration (SSC) historical data at 49 gauging stations throughout the middle YRB. The spatial variation in regression coefficients was investigated, and the relationship between regression coefficients and the physical characteristics of watersheds was discussed. The results indicate that SRC regression coefficients vary with drainage area and basin slope, but their responses to these parameters are remarkably different in watersheds with different underlying surfaces, which indicates the significance of sediment availability, erodibility, and grain size distribution. For power SRCs representing sediment transport in unsaturated flows, the regression coefficients are more closely correlated with the drainage area in loess regions and with the basin slope in rock mountain regions. For linear SRCs representing sediment transport in saturated flows, saturated SSCs vary with coarse (particle size > 0.05 mm) and fine (particle size < 0.01 mm) fractions in suspended sediment. The maximum saturated SSC among the different gauging stations is associated with the optimal grain size composition of suspended sediment, which has been proposed for loess regions in previous studies. This study provides theoretical support for estimating the regression parameters for sediment transport modelling, especially in ungauged basins.

Revisiting slope influence in turbulent bedload transport: consequences for vertical flow structure and transport rate scaling

Gravity-driven turbulent bedload transport has been extensively studied over the past century in regard to its importance for Earth surface processes such as natural riverbed morphological evolution. In the present contribution, the influence of the longitudinal channel inclination angle on gravity-driven turbulent bedload transport is studied in an idealised framework considering steady and uniform flow conditions. From an analytical analysis based on the two-phase continuous equations, it is shown that: (i) the classical slope correction of the critical Shields number is based on an erroneous formulation of the buoyancy force, (ii) the influence of the slope is not restricted to the critical Shields number but affects the whole transport formula and (iii) pressure-driven and gravity-driven turbulent bedload transport are not equivalent from the slope influence standpoint. Analysing further the granular flow driving mechanisms, the longitudinal slope is shown to not only influence the fluid bed shear stress and the resistance of the granular bed, but also to affect the fluid flow inside the granular bed – responsible for the transition from bedload transport to debris flow. The relative influence of these coupled mechanisms allows us to understand the evolution of the vertical structure of the granular flow and to predict the transport rate scaling law as a function of a rescaled Shields number. The theoretical analysis is validated with coupled fluid–discrete element simulations of idealised gravity-driven turbulent bedload transport, performed over a wide range of Shields number values, density ratios and channel inclination angles. In particular, all the data are shown to collapse onto a master curve when considering the sediment transport rate as a function of the proposed rescaled Shields number.

Efficient analytical implementation of the DOT Riemann solver for the de Saint Venant–Exner morphodynamic model

Within the framework of the de Saint Venant equations coupled with the Exner equation for morphodynamic evolution, this work presents a new efficient implementation of the Dumbser-Osher-Toro (DOT) scheme for non-conservative problems. The DOT path-conservative scheme is a robust upwind method based on a complete Riemann solver, but it has the drawback of requiring expensive numerical computations. Indeed, to compute the non-linear time evolution in each time step, the DOT scheme requires numerical computation of the flux matrix eigenstructure (the totality of eigenvalues and eigenvectors) several times at each cell edge. In this work, an analytical and compact formulation of the eigenstructure for the de Saint Venant–Exner (dSVE) model is introduced and tested in terms of numerical efficiency and stability. Using the original DOT and PRICE-C (a very efficient FORCE-type method) as reference methods, we present a convergence analysis (error against CPU time) to study the performance of the DOT method with our new analytical implementation of eigenstructure calculations (A-DOT). In particular, the numerical performance of the three methods is tested in three test cases: a movable bed Riemann problem with analytical solution; a problem with smooth analytical solution; a test in which the water flow is characterised by subcritical and supercritical regions. For a given target error, the A-DOT method is always the most efficient choice. Finally, two experimental data sets and different transport formulae are considered to test the A-DOT model in more practical case studies.

Generalized network transport and Euler-Hille formula

In this article we consider asymptotic properties of network flow models with fast transport along the edges and explore their connection with an operator version of the Euler formula for the exponential function. This connection, combined with the theory of the regular convergence of semigroups, allows for proving that for fast transport along the edges and slow rate of redistribution of the flow at the nodes, the network flow semigroup (or its suitable projection) can be approximated by a finite dimensional dynamical system related to the boundary conditions at the nodes of the network. The novelty of our results lies in considering more general boundary operators than that allowed for in previous papers.

SELECTION OF APPROPRIATE TRANSPORT FORMULAS FOR ESTIMATION OF TRANSPORT AND DEPOSITION RATES AT BABREKA CANAL

Siltation is a significant factor, affecting the efficient operation and maintenance of the irrigation canals which influence the hydraulic behavior of the canals and economic benefits of irrigation. In last 25 years a tendency of silting up at the irrigation canals in Bulgaria is observed due to the following reasons: decreased area under irrigation, fluctuation in supply, non-regime section, ”berming” of the canal, economic, crime and safety situation. The large number of available bed load and permissible velocity formulas and the significant differences between obtained results enforce selection of appropriate sediment transport formulas in each particular case. In this study discharge and velocity approaches are used for estimation of sediment transport and deposition rates at Babreka Canal, Malka Vitska Irrigation Project in Bulgaria. Seven equations for prediction total bed load transport rates and five equations for minimum permissible velocity are compared. The results show substantial differences in performance. Two formulas for total bed load and two formulas for minimum permissible velocity are selected as appropriate for the studied canal.

Perhitungan Kecepatan Sedimentasi Melalui Pendekatan Usle dan Pengukuran Kandungan Tanah dalam Air Sungai yang Masuk ke dalam Waduk Sermo

Dam is one of the essential man-made buildings which was developed to various beneficialy purposes such as irrigation, fresh water supply, flooding and drying control, and also tourism. Due to those mentioned functions, condition of the dam has to be maintained continuously especially from sedimentation. Sedimentation is being a serious threat for the continuity of dam services. Protecting dam from sedimentation is not only managing dam area, but also we have to care all watershed area, because the sedimentation is caused by erotion on the area. In order to know the rate of the sedimentation within the dam, measurement and prediction must be done. One of methods to predict and measure the sedimentation is by analyzing geospatial data using USLE Approach, and analyzing suspended sediment in river’s water toward to the dam. By those both analysis, rate of sedimentation happened within the dam can be calculated. This paper try to use geospatial (GIS) based analysis to estimate the sedimentation rate using USLE approach within Ngrancah Watershed. The USLE Formula requires four types of maps, they are soil type, slope, land cover, and rain erosivity maps. Each of maps is classified to the specific standards, then will be analized by overlaying to another map. Another method will be used to estimate these dimentation rate is the suspended sediment measurement. This method was determined using sediment transport formula. The data used are samples of river water ing Ngerancah Watershed that was flow toward to the Sermo Dam, and daily volume of inflow water. Those resulted values, then were compared each other. Based on the calculation, sedimentation rate resulted using USLE Approach is 276.100,917 m3 per year or 8,675 mm thickness per year. While the calculation based on the measurement of suspended sediment in river’s water is 270.206,363 m3 per year or 8,490 mm thickness per year. The difference value between the both methods is 5894,555 m3 per year or 0,185 mm thickness per year. Based on the watershed monitoring guidelines published by The Ministry of Forestry of Indoensia, the Sermo Dam sedimentation rate is categorized in poor class, because the sedimentation rate is exceeding 5 mm per year, as the safe limit of dam sedimentation rate.

An Equilibrium Transport Formula for Modeling Sedimentation of Dredged Channels

The proposed formula for modeling sedimentation is based on a concept of hydrodynamic equilibrium in sediment transport. Hydrodynamic equilibrium occurs when the sediment transport is in a form of grain jumps with the distance traveled dx smaller than the one established in the numerical modeling. A first-order upwind scheme FDM is applied. It provides the solution to equations describing changes in bathymetry in time and space coupled with simultaneous changes of bed sediment distribution. A clear-cut division of sediment transport into those during wave crest and wave trough is proposed. The results of bathymetry modeling coincided satisfactorily with the laboratory data. Moreover, the longer the experiments run, the more the results of measurements are compliant with the calculations. A certain time is needed for sediment transport to reach the hydrodynamic equilibrium, especially when the suspended load is significant. It is reached after going through the initial stages of bed level changes coupled with simultaneous changes of grain size distribution. The application of the model to field conditions shows its ability to predict sedimentation of dredged channels. Furthermore, the calculations show a good agreement with measured bed grain size distributions of sediment samples taken from the sites within the navigational channel of Łeba Port in Poland.

Numerical modelling of downstream migrating antidunes

AbstractA numerical model is presented that compute the geometrical dimensions and movement of downstream migrating antidunes. The model solves the Navier–Stokes equations together with the k‐epsilon turbulence model to find the water flow field over the bedforms. A two‐dimensional width‐averaged grid is used. The bed elevation changes are computed by solving the convection–diffusion equation for suspended sediments and bedload, together with the Engelund–Hansen sediment transport formula. The free surface is computed with an algorithm based on water continuity in the surface cells. Non‐orthogonal adaptive grids were used, moving vertically with the computed location of the bed and the free water surface. The numerical model was tested on data from a physical model study where regular downstream migrating antidunes had been observed. The numerical model started out with a flat bed and the trains of antidunes formed over time. Many of the physical processes observed in earlier studies were replicated by the numerical model. Four dune parameters were computed in the current tests: The antidune wavelength, height and celerity, together with the average water depth. The antidune wavelengths were best predicted with an accuracy of 3 to 8% compared with the measurements. The antidune heights were computed with a deviation of 11 to 25% compared with an empirical formula. The water depths over the antidunes were predicted with an accuracy of 3 to 9% related to the measured values. The average antidune celerity was the parameter with largest deviation: For the coarsest grid it was overpredicted with 37%. Copyright © 2017 John Wiley & Sons, Ltd.

Generalizing Study Results: A Potential Outcomes Perspective.

Great care is taken in epidemiologic studies to ensure the internal validity of causal effect estimates; however, external validity has received considerably less attention. When the study sample is not a random sample of the target population, the sample average treatment effect, even if internally valid, cannot usually be expected to equal the average treatment effect in the target population. The utility of an effect estimate for planning purposes and decision making will depend on the degree of departure from the true causal effect in the target population due to problems with both internal and external validity. Herein, we review concepts from recent literature on generalizability, one facet of external validity, using the potential outcomes framework. Identification conditions sufficient for external validity closely parallel identification conditions for internal validity, namely conditional exchangeability; positivity; the same distributions of the versions of treatment; no interference; and no measurement error. We also require correct model specification. Under these conditions, we discuss how a version of direct standardization (the g-formula, adjustment formula, or transport formula) or inverse probability weighting can be used to generalize a causal effect from a study sample to a well-defined target population, and demonstrate their application in an illustrative example.