Bridging the gap between particle-scale forces and continuum modelling of size segregation: application to bedload transport

Gravity-driven size segregation is important in mountain streams where a wide range of grain sizes are transported as bedload. More particularly, vertical size segregation is a multi-scale process that originates in interactions at the scale of particles with important morphological consequences on the reach scale. To address this issue, a volume-averaged multi-phase flow model for immersed bidisperse granular flows was developed based on an interparticle segregation force (Guillard et al. 2016) and a granular Stokesian drag force (Tripathi and Khakhar 2013). An advection-diffusion model was derived from this model yielding parametrisations for the advection and diffusion coefficients based on the interparticle interactions. This approach makes it possible to bridge the gap between grain-scale physics and continuum modelling. Both models were successfully tested against existing Discrete Element Model (DEM) simulations of size segregation in bedload transport (Chassagne et al. 2020). Through a detailed investigation of the granular forces, it is demonstrated that the observed scaling of the advection and diffusion coefficients with the inertial number can be explained by the granular drag force dependency on the viscosity. The drag coefficient was shown to be linearly dependent on the small particle concentration. The scaling relationship of the segregation force with the friction coefficient is confirmed and additional non-trivial dependencies including the inertial number and small particle concentration are identified. Lastly, adding a size ratio dependency in the segregation force perfectly reproduces the DEM results for a large range of small particle concentrations and size-ratios

Mobility of bidisperse mixtures during bedload transport

The flow of segregated bidisperse assemblies of particles is of major importance for geophysical flows and bedload transport in particular. In the present paper, the mobility of strictly bidisperse segregated particle beds is studied with a coupled fluid discrete element method (DEM). Large particles are initially placed above small ones in a three-dimensional domain inclined at a slope of 10%. A gravity-driven water free surface flow induces a downslope shear-driven granular flow of the erodible bed. It is observed that, for the same water flow conditions, the bedload transport rate is higher in the bidisperse configuration than in the monodisperse one. Depending on the Shields number and on the depth of the interface between small and large particles, different transport phenomenologies are observed, ranging from no influence of the small particles to small particles reaching the bed surface due to diffusive remixing. In cases where the small particles hardly mix with the overlying large particles and for the range of studied size ratios (r < 4), it is shown that the increased mobility is not a bottom roughness effect, that would be due to the reduction of roughness of the underlying small particles, but a granular flow effect. This effect is analyzed within the framework of the mu(I) rheology, modeling the stress to strain rate relation for dense granular flows. It is demonstrated that the buried small particles are more mobile than larger particles and play the role of a "conveyor belt" for the large particles at the surface. Based on rheological arguments, a simple predictive model is proposed for the additional transport in the bidisperse case. It reproduces quantitatively the DEM results for a large range of Shields numbers and for size ratios smaller than 4. The results of the model are used to identify four different transport regimes of bidisperse mixtures, depending on the mechanism responsible for the mobility of the small particles. A phenomenological map is proposed for bidisperse bedload transport and, more generally, for any granular flow on an erodible bed

Discrete and continuum modelling of grain size segregation during bedload transport

Grain-scale discrete element simulations of bidisperse mixtures during bedload transport are used to understand, and model, bedload transport and particle-size segregation in granular media. For an initial distribution of fine particles on top of a coarse granular bed, this paper investigates the gravity driven percolation/segregation of the fine particles down into the quasi-static part of the bed. The segregation is observed to be driven by the inertial number at the bottom of the fine particle layer, and is independent of the number of fine particles. A novel travelling wave solution for the evolving concentration distribution is constructed using the continuum particle-size segregation model of Thornton, Gray & Hogg (J. Fluid Mech., vol. 550, 2006, pp. 1–25) and Gray & Chugunov (J. Fluid Mech., vol. 569, 2006, pp. 365–398). The observed behaviour is shown to be related to a local equilibrium between the influence of the concentration and of the inertial number. The existence of the exact solution relies on the segregation flux and the diffusion coefficient having the same dependency on the inertial number. This functional dependence allows the continuum model to quantitatively reproduce the discrete simulations. These results significantly improve on our understanding of the size segregation dynamics and represent a step forward in the up-scaling process to polydisperse granular flows in the context of turbulent bedload transport

Modélisation discrète et continue du tri granulométrique : application au transport par charriage

Understanding particle size segregation is one of the great challenge in fluvial geomorphology. It is still notoriously difficult to predict sediment transport more accurately than within one order of magnitude. One of the main origin of this difficulty is particle size segregation, a granular process of particle sorting in the sediment bed. Size segregation is therefore a grain scale process impacting the morphological scale.This PhD presents a numerical study of size segregation as a granular process during bedload transport. A coupled fluid discrete element method (DEM) is used to study the infiltration of small particles in a large particle bed. This configuration, close to granular flows on erodible beds, is characterized by a particle velocity profile, a shear rate profile and an inertial number profile exponentially decreasing into the bed. It presents a particular segregation phenomenology with small particles infiltrating the bed as a travelling wave, the velocity being controlled by the inertial number at the bottom of the layer. The segregation velocity is observed dependent on the local small particle concentrations and on the size ratio. The segregation problem is also analyzed with an advection diffusion model. With advection and diffusion coefficients both proportional to the inertial number, the continuum model perfectly reproduces the dynamics observed in the DEM results.Very recently, a new segregation advection diffusion model has been derived based on particle scale forces, in particular a granular buoyancy force (or segregation force) and an inter-particle drag force. This provides new physically based parametrisations for the advection and diffusion coefficients. This new model is analysed in the bedload configuration, and reproduces qualitatively the DEM results. To improve the model, new dependencies on the inertial number and small particle concentration are proposed for the segregation and drag forces.Finally, the impact of size segregation on sediment transport is studied through the mobility of bidisperse already segregated particle beds. Large particles are placed above small ones, and it is observed that, in the same fluid and surface bed conditions, the transport rate is higher in the bidisperse configuration than in the monodisperse one. For the range of studied size ratio (r<4), it is showed that it is not a rugosity but a granular effect. This is analyzed within the framework of the mu(I) rheology and it is demonstrated that the buried small particles are more mobile than larger particles and play the role of a conveyor belt for the large particles at the surface. Based on rheological arguments, a simple predictive model for the additional transport in the bidisperse case is proposed, which reproduces quite well the DEM results for a large range of Shields numbers and for size ratios smaller than 4. The results of the model were used to identify four different transport regimes of bidisperse mixtures, depending on the mechanisms responsible for the mobility of the small particles.This work represents an important improvement in the understanding of size segregation during bedload transport and questions our understanding of bidisperse granular media, which have not been much studied. It also represents a first step in an upscaling process towards the morphological scale through continuum models.La compréhension du tri granulométrique des particules est un enjeu majeur pour l’étude des évolutions morphologiques des rivières de montagne. La prédiction des flux de transport reste difficile avec des écarts de plusieurs ordres de grandeurs entre les valeurs prédites et mesurées. L’une des raisons principales de cette difficulté est la ségrégation, phénomène granulaire de tri granulométrique des particules constituant le lit sédimentaire. La ségrégation est donc un phénomène à l’échelle du grain ayant un impact à l’échelle morphologique.Cette thèse présente une étude numérique de la ségrégation en tant que phénomène granulaire dans le cas du charriage et son impact sur le transport sédimentaire. Un modèle aux éléments discrets (DEM) couplé à un modèle fluide turbulent unidimensionnel est utilisé. A l’instant initial, des petites particules sont déposées au dessus de particules plus grosses. Le fluide s’écoule par gravité et transporte les particules du lit sédimentaire. Cette configuration, proche d’un lit érodable, est caractérisée dans la profondeur par des profils exponentiellement décroissant de la vitesse particulaire, du taux de cisaillement et du nombre inertiel et présente une phénoménologie de ségrégation particulière. Les petites particules s’infiltrent en couche, sous forme d’onde progressive, dont la vitesse est contrôlée par le nombre inertiel en bas de la couche. On observe aussi que la vitesse de ségrégation est dépendante de la concentration locale en petites particules et du ratio de taille. Le problème de ségrégation est ensuite analysé à partir d’un modèle d’advection-diffusion. Avec un coefficient d’advection proportionnel au nombre inertiel, le modèle continu reproduit parfaitement la dynamique de la phase des petites particules. Enfin on démontre que pour reproduire l’onde progressive observée dans les simulations DEM, le coefficient de diffusion doit avoir la même dépendance avec le nombre inertiel que le coefficient d’advection.Très récemment, un nouveau modèle d’advection-diffusion a été proposé dans la littérature à partir de forces inter-particulaires, notamment une force de portance (ou force de ségrégation) et de traînée, apportant de nouvelles paramétrisations physiques aux coefficients d’advection et de diffusion. Ce nouveau modèle est analysé ici dans la configuration du charriage. La dépendance en nombre inertiel, observée dans les résultats DEM, peut être retrouvée à partir de ces nouvelles paramétrisations. Pour reproduire quantitativement les simulations DEM, de nouvelles dépendances en nombre inertiel et concentration en petites particules sont proposées pour la force de ségrégation et le coefficient de traînée.Enfin, l’impact de la ségrégation sur le transport sédimentaire est étudié en s’intéressant à la mobilité d’un lit bi-disperse déjà ségrégé. Les grosses particules sont placées au dessus des petites et on observe que, pour la même contrainte fluide et pour le même état granulaire de surface, le transport est plus élevé dans le cas bi-disperse que dans le cas mono-disperse. Pour la gamme de ratio de taille étudié (r<4), on montre que l’augmentation de mobilité n’est pas un effet de rugosité mais un effet rhéologique. À partir d’une analyse dans le cadre de la rhéologie mu(I), il est démontré que les petites particules en profondeur sont plus mobiles que les grosses particules, jouant le rôle d’un tapis roulant pour les grosses particules de surface et augmentant ainsi la mobilité globale du lit sédimentaire. Basé sur des arguments rhéologiques, un modèle simple de prédiction de l’augmentation du flux sédimentaire est proposé, reproduisant correctement les résultats DEM pour une large gamme de nombre de Shields et pour des ratio de taille inférieurs à 4. Les résultats du modèle sont exploités pour identifier quatre régimes de transport différent selon les mécanismes responsables de la mobilité des petites particules

A frictional–collisional model for bedload transport based on kinetic theory of granular flows: discrete and continuum approaches

In this work, the modelling of collisional bedload transport is investigated with a focus on the continuum modelling of the granular flow. For this purpose, a frictional–collisional approach, combining a Coulomb model with the kinetic theory of granular flows, is developed. The methodology is based on a comparison with coupled fluid–discrete simulations that the classical kinetic theory model fails to reproduce. This inaccuracy may be explained by the assumptions of negligible interparticle friction and the absence of a saltation model in the continuum approach. In order to provide guidelines for the modelling, the fluctuating energy balance is computed in the discrete simulations and systematically compared with the kinetic theory laws. Interparticle friction is shown to affect the radial distribution function and to increase the energy dissipation, in agreement with previous observations. In addition, a saltation regime is identified, leading to departure from the viscosity and pseudo-thermal diffusivity laws of the kinetic theory in the dilute regime. Based on these observations, modifications to account for interparticle friction are included in the two-fluid model, and the kinetic theory is coupled with a saltation model. The results show that for inelastic frictional particles, interparticle friction controls energy dissipation, and the macroscopic behaviour of the granular flow does not depend on the microscopic particle properties. The proposed model reproduces the $\mu (I)$ rheology in the dense regime of the granular flow. Finally, the model is evaluated with experiments, showing significant improvements concerning the granular flow modelling

Vertical size-segregation in bedload sediment transport : from grain scale to continuum models

International audienceBedload sediment transport (transport of particles by a flowing fluid along the bed by rolling, sliding and/or saltating) has major consequences for public safety, water resources and environmental sustainabilty. In mountains, steep slopes drive intense transport of a wide range of grain sizes implying size sorting or segregation largely responsible for our limited ability to predict sediment flux and river morphology. Concerning size sorting, most studies have concentrated on the spontaneous percolation of fine grains into immobile gravels. However when the substrate is moving, the segregation process is different as statistically void openings permit downward percolation of larger particles, a process also known as 'kinetic sieving'. In order to gain understanding of this process, bedload transport numerical experiments of two-size particle mixtures were carried out, using a coupled Eulerian-Lagrangian fluid-discrete element model developed at Irstea and validated with experiments (Maurin et al. 2015, 2016). It is composed of a 3D discrete element model (based on the open source code Yade), describing each individual particle, coupled with a one dimensional Reynolds average Navier Stokes model (Chauchat 2017). A 3D 10% steep domain (angle of 5.71 •) consisting at initial time of a given number of layers of 4 mm spherical particles deposited on top of a 6 mm particle bed, were submitted to a turbulent, hydraulically rough and supercritical water flow and let evolved with time. Shields numbers (dimensionless water shear stress) of 0.1 and 0.3 were considered. For a given Shields number, the elevation of the center of mass of the infiltrated fine particles has been shown to remarkably follow the same logarithmic decrease with time, whatever the number of fine layers initially deposited. This decrease is steeper for a higher Shields number. These numerical experiments were also analysed in the framewok of a continuum theoretical model for the segregation of binary mixtures based on a kinematic approach (Thornton et al. 2006). Modelling bedload size sorting at the particle scale and upscaling in continuum models should improve our knowledge of sediment transport and river morphology

Continuous modeling of grain size segregation in bedload transport

International audienceRivers carry sediments having a wide grain size distribution, ranging from a few hundreds microns to meters. This leads to grain size segregation mechanism that can have huge consequences on morphological evolution. Accurate comprehension and modeling of this mechanism with continuous equations is a key step to upscale segregation in sediment transport models.Thornton et al. (2006) developed continuous equations for bidisperse segregation in the context of the mixture theory. Based on the momentum balance of small particles, a simple advection-diffusion equation for the volumetric concentration of small particles was derived. This equation enables to explicit the advection term, that tends to segregate the different particle sizes and the diffusive term, that tends to remix the particles. However, this approach does not immediately provide the physical characteristics of the granular flow in the advection and diffusion terms.Recently, Guillard et al. (2016) showed, using a Discrete Element Method (DEM), that the segregation force on a large intruder in a bath of small particles, can be seen as a buoyancy force proportional to the pressure. In addition, Tripathi and Khakhar (2011) showed that a large particle rising in a pool of small grains experiences a Stokesian drag force proportional to the granular viscosity.These new results enable to infer a force balance for a single coarse particle in bedload transport. Solving this force balance showed that the large particle rises with the accurate dynamics, meaning that this force balance is relevant to model grain-size segregation.Based on these new forces, a continuous multi-class model has been developed to generalize to the segregation of a collection of large particles. The concentration and the segregation velocity of the small particles have been compared with coupled-fluid DEM bedload transport simulations from Chassagne et al. (2020) and show that the accurate dynamics of segregation can be modeled using this continuous model.Based on this continuum multi-class model, a similar advection-diffusion equation as Thornton et al. (2006) has been obtained. The latter appears to provide the physical origin of the advection and diffusion terms by linking them to the parameters of the flow

Vertical grain size sorting in bedload transport on steep slopes with a coupled fluid-discrete element model

In order to study vertical grain size sorting in bedload sediment transport, numerical experiments of two-size particle mixtures were carried out, using a validated coupled fluid-discrete element model developed at Irstea. A 3D 10% steep domain, consisting at initial time of a given number of layers of 4 mm particles deposited on top of a coarser 6 mm particle bed, was sheared by a turbulent and supercritical fluid flow (Shields numbers of 0.1 and 0.3). The elevation of the centre of mass of the infiltrated fine particles is observed to follow the same logarithmic decrease with time, whatever the initial number of fine layers. This decrease is steeper for a higher Shields number. The main result is that this typical behaviour is related at first order to the particle shear rate depth profile

Discrete element simulations and continuous modeling of vertical size-segregation in bedload transport

International audienceDiscrete element simulations and continuous modeling of vertical size-segregation in bedload transport Chassagne, Maurin, Chauchat, Frey Bedload sediment transport has major consequences for public safety, water resources and environmental sustainability. In mountains, steep slopes drive intense transport of a wide range of grain sizes implying size sorting or segregation largely responsible for our limited ability to predict sediment flux and river morphology. Concerning size sorting, when the substrate is moving, statistically void openings allows fine particles to percolate into the bed, a process called 'kinetic sieving'. To better understand this process, bedload transport numerical experiments of two-size particle mixtures were carried out, using a coupled Eulerian-Lagrangian fluid-discrete element model (DEM) developed at Irstea (Maurin et al. 2015, 2016, Chauchat 2017). A 3D 10% steep domain consisting at initial time of a given number of layers of 4 mm spherical particles deposited on top of a 6 mm particles bed, were submitted to a turbulent, hydraulically rough and supercritical water flow and let evolved with time. Shields numbers of 0.1 and 0.3 were considered. For a given Shields number, the elevation of the center of mass of the infiltrating fine particles has been shown to remarkably follow the same logarithmic decrease with time, whatever the number of fine particles. In addition, the profile of concentration of fine particles is a Gaussian like function with constant width during the percolation into the bed : the entire layer percolates at the same velocity. Based on this observation, it has been showed that the segregation velocity is driven by the shear rate at the bottom of the layer. These numerical experiments were also analyzed in the framework of a continuum theoretical model for the segregation of binary mixtures based on a kinematic approach (Thornton et al. 2006). The continuum model shows very good agreement with the DEM simulations (comparable characteristic time of segregation, exact segregation velocity) and reproduces the same properties (Gaussian like concentration profile, constant width of the fine layer). Chauchat J. 2017. A comprehensive two-phase flow model for unidirectional sheet-flows. Journal of Hydraulic. Maurin R, Chauchat J, Chareyre B, Frey P. 2015. A minimal coupled fluid-discrete element model for bedload transport. Physics of Fluids 27(11): 113302. Maurin R, Chauchat J, Frey P. 2016. Dense granular flow rheology in turbulent bedload transport. Journal of Fluid Mechanics 804: 490-512. Thornton AR, Gray J, Hogg AJ. 2006. A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. Journal of Fluid Mechanics 550: 1-25

Vertical grain size sorting in bedload transport on steep slopes with a coupled fluid-discrete element model

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