<figure><img src="image_1.png" /></figure>

## Introduction

* Bedload has been investigated mostly from the fluid perspective as if the bed were immobile

<figure><img src="image_2.png" /></figure>

Sediment transport= bedload (rolling, sliding, saltation)+suspension

* Yet grain-grain interactions are important (Frey & Church 2009, 2011)

* Warrants:

- Use of discrete element models (DEM)

- Upscaling in continuum models

* Objective of this poster

- report globally on an ongoing research effort on sediment transport multi-scale modelling within the framework of the ANR project SegSed 'Size segregation in sediment transport'

- Draw materials from PhDs by R. Maurin, R. Chassagne and H. Rousseau

<figure><img src="image_3.png" /></figure>

Philosophy and workflow of the SegSed project
Tasks reported in this poster are circled in red

## Coupled fluid discrete element model

<figure><img src="image_4.png" /></figure>

Integral Mixing length taking into account $\phi$

<figure><img src="image_5.png" /></figure>

Experimental bedload dataset (Frey 2014): narrow flume, Slope 10%, glass beads 6mm, used to validate the fluid discrete model (cumulative image of 20 frames)

<figure><img src="image_6.png" /></figure>

Depth profiles of the streamwise solid velocity $v_x^P$ (m/s), solid volume fraction $\phi$, and sediment transport rate density $q_s$ (m/s). Comparison with experiments by Frey 2014
Maurin et al. PoF 2015

## Granular rheology $\mu(l)$ for upscaling

<figure><img src="image_7.png" /></figure>

Maurin et al. JFM 2016

- Phenomenological constants $\mu_s, \mu_2, l_0$ derived from DEM

- Fed into a two phase continuous (Euler-Euler) model (Revil-Baudard and Chauchat 2013; Chauchat 2018)

$$ 
\begin{aligned}
0 &= \frac{\partial \tau_{xz}^p}{\partial z} + \rho^p \phi g \sin \alpha + n \langle f_x \rangle^p, \\
0 &= -\frac{\partial P^p}{\partial z} + \rho^p \phi g \cos \alpha + n \langle f_z \rangle^p
\end{aligned}
$$

**Solid phase equation**
Fluid component is the same as in the fluid-DEM model

$$ P^p = \frac{1}{l^2} \rho_p d^2 \left( \frac{d\langle v_x^p \rangle^s}{dz} \right)^2 $$

$$ \tau^p = \frac{\mu(l)}{l^2} \rho_p d^2 \left( \frac{d\langle v_x^p \rangle^s}{dz} \right)^2 $$

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

Solid depth profile comparison between the two-phase continuous model (Euler–Euler) and the coupled fluid–discrete-element model (Euler–Lagrange)

## Multi-Phase Model

$$ 
\begin{aligned}
\rho^p \left[ \frac{\partial \Phi^s w^s}{\partial t} + \frac{\partial \Phi^s w^s w^s}{\partial z} \right] &= -\frac{\partial p^s}{\partial z} - \Phi^s \frac{\partial p^f}{\partial z} - \rho^p g \cos \theta + n_s f_{drag}^{f \to s} - n_l f_{drag}^p - n_l f_{seg}^p \\
n_l f_{drag}^p &= \frac{6c \Phi^l \eta^p}{d_l^2} (w^s - w^l) \quad \text{and} \quad n_l f_{seg} = - \Phi^l \mathcal{F}(\mu) \frac{\partial p^m}{\partial z}
\end{aligned}
$$

<figure><img src="image_10.png" /></figure>

Forces on a single large particle

<figure><img src="image_11.png" /></figure>

Rousseau et al. JFM submitted

<figure><img src="image_12.png" /></figure>

$$ \frac{\partial \phi^s}{\partial t} + \frac{\partial}{\partial z}(\phi^l \phi^s S_r) = \frac{\partial}{\partial z}(D \frac{\partial \phi^s}{\partial z}) $$

$$ S_r = \mathcal{F}(\mu) \frac{\rho^p d_l \sqrt{d_l g} \partial p^m}{18 \eta^p \partial z} \quad \text{and} \quad D = \frac{\Phi^s}{\Phi} \rho^p d_l \sqrt{d_l g} \frac{\partial p^m}{\partial z} $$

## Size segregation – Kinetic sieving

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

$D_l=6mm D_s=4mm S=10\% N_s=2$ small grain layers

<figure><img src="image_15.png" /></figure>

Time evolution of particle concentration in fine particles $\phi$

Experimental segregation pattern (Frey et al. 2020)

<figure><img src="image_16.png" /></figure>

Streamwise velocity profile and time evolution of center of mass

Mixture model (Thornton et al. 2006)

$$ \frac{\partial \phi}{\partial t} - \frac{\partial}{\partial z}(S_{r0} e^{z/c} \phi(1-\phi)) = \frac{\partial}{\partial z} \left( D(z) \frac{\partial \phi}{\partial z} \right) $$

$$ \text{DEM: } dz_c/dt = S_{r0} I_b^{0.81} $$

$$ \text{Model: } w_p = S_{r0} I^{0.81} (1 - \phi) $$

$$ D(z) = D_0 I^{0.81} = D_0 e^{z/c} $$

Diffusion must have the same exponential dependency as segregation

<figure><img src="image_17.png" /></figure>

Chassagne et al. JFM In press

<figure><img src="image_18.png" /></figure>

Center of mass velocity vs. $l$ at the bottom of the layer for varied numbers of layers $N_s$

<figure><img src="image_19.png" /></figure>

Concentration profiles (DEM vs. mixture model) almost superimposed

## Conclusions and outlook

* Coupled fluid-discrete element model validated on experiments to:

- gain access to grain scale physics (impossible to measure in the laboratory)

- Confirm relevancy of the $\mu(l)$ rheology in bedload transport

- Study segregation dynamics

- Provide parametrizations for the grain-scale physics in continuum models

- Upscaling for applications at larger scales

* Future work:

- Improve particle-particle interaction forces

- drag (e.g. effect of small particle concentration)

- segregation (e.g. scaling with size ratio)

- Need of experimental data not only numerical DEM results

## Contact

Philippe Frey
Irstea, UR ETNA
BP 76, 38402 St-Martin-d'Hères
France
philippe.frey@inrae.fr
www.irstea.fr/en/frey

## Acknowledgements

This research was supported by Irstea (now INRAE, formerly Cemagref), by ANR (the French national research agency) project ANR-16-CE01-0005 SegSed, the labex OSUG@2020, the French Institut National des Sciences de l'Univers programs EC2CO-BIOHEFECT and EC2CO-LEFE MODSED,

## References

Chauchat, J., 2018. A comprehensive two-phase flow model for unidirectional sheet-flows. *Journal of Hydraulic Research* 56, 15-28

Frey P, Church M. 2009. How river beds move. *Science* 325(5947): 1509-1510.

Frey P, Church M. 2011. Bedload: A granular phenomenon. *Earth Surface Processes and Landforms* 36(1): 58-69.

Frey P. 2014. Particle velocity and concentration profiles in bedload experiments on a steep slope. *Earth Surface Processes and Landforms* 39(5): 646-655.

Guillard, F., Forterre, Y., Pouliquen, O., 2016. Scaling laws for segregation forces in dense sheared granular flows. *Journal of Fluid Mechanics* 807, R1

Jackson, R., 2000. The dynamics of fluidized particles. Cambridge University Press.

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.

Revil-Baudard, T., Chauchat, J., 2013. A two-phase model for sheet flow regime based on dense granular flow rheology.

*Journal of Geophysical Research-Oceans* 118, 619-634, doi:10.1029/2012jc008306

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

Frey P, Lafaye de Micheaux H, Bel C, Maurin R, Rorsman K, Martin T, Ducottet C. 2020. Experiments on grain size segregation in bedload transport on a steep slope. *Advances in Water Resources* 136: 103478.

Chassagne, R., Maurin, R., Chauchat, J., Gray, J. M. N. T., Frey, P., in press. Discrete and continuum modelling of grain size segregation during bedload transport. *Journal of Fluid Mechanics*

Rousseau, H., Chassagne, R., Chauchat, J., Maurin, R., Frey, P., subm. Bridging the gap between particle-scale forces and continuum modelling of size segregation: application to bedload transport. *Journal of Fluid Mechanics*