Presentation of the discrete model, continuous framework and bedload transport configuration 

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Discussion on the Shields number definition

The Shields number θ is defined as the dimensionless fluid bed shear stress. By analogy to a free surface flow on a fixed bed, the fluid bed shear stress is classicaly taken as τb = ρf g sin(α)hw, where hw is the water depth. The volume fraction profile presented in figure 2.4a shows that there is no clear separation between the pure fluid phase and the granular bed and it is therfore difficult to measure a water depth. It is only possible to define a geometrical water depth, corresponding to a virtual depth where all particles are deposited and fully compacted at Φ = 0.61. Since water depth is difficult to measure, Revil-Baudard and Chauchat (2013) and Maurin et al. (2015) proposed to define a Shields number based on the Reynolds stress profile (see figure 2.5b) such that τb = max(Rxz). Two definitions of the Shields number are then considered. The first definition, based on hw, will be referred to as the macroscopic Shields number and the second one, based on the maximum Reynolds Stress, as the microscopic Shields number. The latter gives values of the Shields number smaller than the macroscopic one, i.e. for this configuration θ = 0.2 for the microscopic Shields number and θ = 0.3 for the macroscopic one. Indeed, due to interactions with particles, a negative feedback applies on the fluid and the Reynolds shear stress is rapidly damped to zero in the flowing layer (see figure 2.5b). This microscopic Shields number should therefore be the relevant parameter when studying phenomena in the bedload layer. However the fluid bed shear stress based on the geometrical water depth appears in the expression of the granular shear stress (see equation (2.42)). Therefore the macroscopic Shields number is as more relevant as the microscopic number when studying phenomena below the bed surface.
In practice, the microscopic definition of the Shields number is difficult to use. This definition indeed implies to be able to measure the Reynold stress profile which is far from being easy. It is experimentally possible to measure the fluid velocity fluctuations and to deduce a Reynold stress profile. In the field, with current techniques, measuring the Reynold stress is extremely difficult. The macroscopic definition based on the water depth is therefore more classical. Defining the water depth is also a subject of discussion. To measure a water depth, the bed position should be defined, which is not obvious as the transition layer between the dense bed and the pure fluid phase may reach several particle diameters. But this is more a question of definition (where is the bed position ?) than a technical difficulty.

The bedload configuration

This discussion about both definitions of the Shields number is essentially relevent for small water depths of the order of the particle diameter on steep slopes. Indeed, when the water depth is large in comparison to the particle diameter, both definitions are very close and it is therefore more convenient to use the macroscopic definition. For large water depths, the width of the transition layer becomes negligible and there is no problem of definition for the water depth.

Segregation mechanisms analysis and continuum modelling in the quasi-static regime

This chapter is dedicated to the description, analysis and modelling of size segregation during bedload transport in the quasi-static regime. As pointed out in the introduction, size segregation is still not yet well understood, in particular in configurations with complex forcing such as bedload transport. It will be analysed using the coupled fluid DEM model (chapter 2) in the framework of the dimensional analysis (aq. 1.7) presented in the introduction (section 1.2). The aim is to understand size segregation in the quasi-static part of the bed and to improve the continuum modelling parameterizations. In particular, the influence of the three main parameters, which are the size ratio r, the inertial number I and the small particle concentration φs, will be investigated through numerical experiments in the bedload configuration. The results will then be analysed in the framework of the segregation model of Thornton et al. (2006).
This work has led to the publication of a scientific article in Journal of Fluid Mechanics (Chassagne et al., 2020b).

DEM simulations of segregation dynamics

Numerical setup

The numerical setup is presented in figure 3.1. In the following, subscripts l and s denote quantities for large and small particles respectively. Initially, large particles of diameter dl = 6 mm and small particles of diameter ds = 4 mm, defining a size ratio of r = 1.5, are deposited by gravity over a rough fixed bed made of large particles. The particle and fluid densities are fixed respectively to ρp = 2500 kg m-3 and ρf = 1000 kg m-3. The size of the 3D domain is 30dl × 30dl in the horizontal plane in order to have converged average values (Maurin et al., 2015) and is periodic in the streamwise and spanwise directions. The number of particles of each class is assimilated into a number of layers, Nl and Ns. The number of layers represents in terms of particle diameters the height that would be occupied by the particles if the volume fraction was exactly Φ = 0.61, the random close packing observed in the simulation. Equivalently, the volume occupied by large particles (resp. small particles) is 0.61 × 30dl × 30dl × Nldl (resp. 0.61 × 30dl × 30dl × Nsds). Therefore, fixing Nl or Ns fixes the number of particles of each class. The height of the bed at rest is defined by H = 1/2dl + Nldl + Nsds, where the 1/2dl term accounts for the fixed particles at the bottom of the domain, and H is fixed to 10.5dl in all the simulations. The number of layers of fine particles Ns varies from 0.01 (only a few small particles) up to 2 layers (corresponding to figure 3.1), while Nl changes accordingly in order to keep H = 10.5dl. The bed slope is fixed to 10% (5.7◦), representative of mountain streams, and the water depth is set to hw = 2.6dl. It corresponds to a macroscopic Shields number of θ = ρf ghw sin(α)/[(ρp − ρf )gdl] = 0.17 and to a microscopic Shields number of θ = max(Rxzf/[(ρp − ρf )gdl]) ∼ 0.12 − 0.13.
At the beginning of each simulation, the fluid flows by gravity and sets particles into motion. A first transient phase takes place, during which fluid and particles are accelerating. Along this period, segregation is very fast and at the end of the transient phase, the small particles have already infiltrated into the first layers of the bed. It is therefore difficult to study size segregation in the bedload layer. It would require numerical tricks that are not easy to set up in order to keep small particles at the surface during the transient phase. For these reasons, in this chapter, the study focuses on the dynamics of segregation once the system is at transport equilibrium and small particles have reached the quasi-static region.
The horizontal averaged volume fraction per unit granular volume of small and large particles are defined as where Φs (resp. Φl) is the volume fraction of small (resp. large) particles defined per unit mixture volume as in the previous chapter.
The large particle diameter dl has been chosen as a reference length scale. Indeed we assume that the small particles are few enough not to perturb the bulk granular flow. The segregation mechanisms are interpreted as a response to the mean flow assumed to be a monodisperse granular flow of large particles. In the following, the tildes are dropped for sake of clarity.



Simulations have been performed for different numbers of layers of small particles Ns. Figure 3.2a shows the temporal evolution of fine particle concentration profiles for the case Ns = 2, corresponding to two layers of small particles deposited on top of a bed made of large particles. At the beginning, the small particles, indicated by the high small particle concentration regions in green, infiltrate rapidly into the first few layers of large particles. As small particles infiltrate downward, large particles rise to the surface. The DEM simulations exhibit a two-layer structure, with small particles sandwiched between two layers of large particles. While infiltrating, the thickness of the small particle layer gets slightly larger. Profiles of concentration for different times are presented in figure 3.2b. The concentration profiles exhibit a Gaussian-like shape. After the transient phase, neither the maximal value (see figure 3.2c) nor the width of the profiles evolve in time, suggesting that the small particles infiltrate the bed as a layer having a constant thickness and being just convected downward by segregation inside large particles.
The vertical time and volume-averaged streamwise velocity profile of the particle mixture is plotted in figure 3.3a for all the simulations. All the curves are superimposed meaning that the response of the granular medium to the fluid flow forcing is not modified by the number of small particles. The granular forcing is therefore the same for all the simulations independently of the number of small particles at the initial condition. Between z = 3 and 8 approximately, the velocity profiles are linear in the semi-logarithmic plot, indicating that the velocity is exponentionally decreasing in the bed. This is characteristic of a quasi-static granular flow (Komatsu et al., 2001), or creeping flow (Houssais et al., 2015; Ferdowsi et al., 2017). Due to the presence of a fixed layer of particles, the velocity vanishes at the bottom of the domain. It is interesting to note that the entire bed is in motion, even if the velocity can be very low at the bottom of the quasi-static region. The time averaged solid volume fraction of the mixture is plotted in figure 3.3b for all the simulations. The quasi-static region is characterized by the bed at random close packing (Φ ∼ 0.61). Above z ∼ 8, the volume fraction decreases to zero, corresponding to a transition zone between a quasi-static region and a pure fluid phase. Note that due to this decompaction of the bed at the surface, the total height of the bed is slightly larger than 10.5dl. For the case Ns = 2, the velocity profile is slightly modified as well as the volume fraction profile with a small increase of volume fraction around z ∼ 5. This very small increase originates from mixture effects and the volume fraction can exceed the random close packing (Cumberland and Crawford, 1987). The modifications on the velocity and volume fraction profiles are however small enough to consider that the small particles do not perturb the mean monodisperse flow. This supports the assumption that the segregation mechanisms can be considered in this configuration as a response to the mean monodisperse granular flow.
Since the small particles infiltrate the bed as a layer, the center of mass of the small particles, zc, is a representative position of the entire layer. Figure 3.3c shows the temporal evolution of this position for all the simulations, in a semi-logarithmic plot. After the initial transient phase, the curves become linear, meaning that zc is a logarithmic function of time In equation (3.5), the coefficient a corresponds to the absolute slope of the curve and characterises the segregation velocity (dzc/dt = −a/t). Whatever the number of small particles in the simulation, figure 3.3c shows that all the curves are parallel to each other, meaning that a is independent of the number of layers of small particles, Ns. Therefore the segregation velocity dzc/dt is also independent of the number of layers of small particles.
In the following, these simulations will be analysed using the dimensional analysis presented in the introduction (equation (1.7)) with the aim to confirm the dependence of the segregation flux on the inertial number I and on the local concentration φs.

Table of contents :

1 Introduction 
1.1 Context
1.2 Size segregation in granular media
1.3 Size segregation modelling
1.3.1 Particle scale modelling
1.3.2 Advection-diffusion model for size segregation
1.4 Objectives and scope
2 Presentation of the discrete model, continuous framework and bedload transport configuration 
2.1 A Coupled fluid-DEM model
2.1.1 Granular phase
2.1.2 Fluid phase
2.1.3 Averaging procedure
2.1.4 Granular stresses
2.1.5 Extension to bidisperse configurations
2.2 The bedload configuration
2.2.1 Numerical setup
2.2.2 Description of the bedload configuration
2.2.3 Granular rheology in bedload transport
2.2.4 Momentum balance
2.2.5 Discussion on the Shields number definition
3 Segregation mechanisms analysis and continuum modelling 
3.1 DEM simulations of segregation dynamics
3.1.1 Numerical setup
3.1.2 Results
3.1.3 Dependence on the inertial number
3.1.4 Bottom controlled segregation
3.1.5 Size ratio influence
3.2 Discussion in the framework of continuum modelling
3.2.1 Local mechanism interpretation
3.2.2 Diffusion and upscaling in a continuum framework
3.3 Conclusion
4 From particle-scale to continuum modelling of size segregation 
4.1 Advection-diffusion segregation model based on particle-scale forces
4.2 Numerical resolution strategy
4.2.1 Formulation of the finite volume method
4.2.2 Estimation of the advective flux with Godunov scheme
4.2.3 Estimation of the diffusive flux
4.2.4 Boundary condition
4.2.5 Algorithm implementation
4.3 Results and Discussion
4.3.1 Results
4.3.2 Interpretation
4.3.3 Missing dependencies in particle scale forces
4.3.4 Influence of the size ratio
4.4 Conclusion
5 Sediment mobility of bi-disperse mixtures during bedload transport 
5.1 Numerical setup
5.2 Enhanced mobility due to bidispersity
5.3 Interpretation as a granular process
5.4 A predictive model for the additional transport
5.5 Discussion and conclusion
6 Conclusions and perspectives 
6.1 Conclusions
6.2 Perspectives
6.2.1 Interpretation of size segregation in laboratory experiments
6.2.2 Extension to other configurations
6.2.3 Rise of a large intruder
6.2.4 Diffusive remixing


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