# Discrete simulations of an armoured sediment bed during bedload transport

Rémi Chassagne, Raphaël Maurin, Julien Chauchat, Philippe Frey

► To cite this version:

Rémi Chassagne, Raphaël Maurin, Julien Chauchat, Philippe Frey. Discrete simulations of an armoured sediment bed during bedload transport. EGU General Assembly 2020, May 2020, Vienna (Online), Austria. 10.5194/egusphere-egu2020-9367. hal-03135479

HAL Id: hal-03135479

https://hal.science/hal-03135479v1

Submitted on 1 Nov 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

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

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

# Discrete simulations of an armoured sediment bed during bedload transport

Rémi Chassagne, Raphaël Maurin, Julien Chauchat and Philippe Frey

May 5 2020

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

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

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

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

UNIVERSITÉ
Grenoble
Alpes

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

<figure><img src="image_10.png" /></figure>
<figure><img src="image_11.png" /></figure>

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

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

# Introduction - Context

During bedload transport :

- Necessity to estimate sediment flux

- Poorly sorted sediment → size sorting

- Impact on the sediment flux. Which one ? Are we able to characterise them ?

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

<figure><img src="image_15.png" /><figcaption>Pictures of armouring on the field resulting from size-segregation</figcaption></figure>
<figure><img src="image_16.png" /></figure>

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

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

# Numerical Model

## Discrete Element Method (DEM) coupled with a 1D turbulent fluid model (Maurin et al. 2015, 2016) :

- DEM (code YADE) : Lagrangian method based on contact between particles

- Fluid : 1D vertical turbulent fluid flow based on mixing length closure

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

## Granular phase, for each particle $p$ :

$$m^p \frac{d^2 \vec{x}^p}{dt^2} = \vec{f}_c^p + \vec{f}_g^p + \vec{f}_f^p$$

$$\mathcal{T}^p \frac{d\vec{\omega}^p}{dt} = \vec{T} = \vec{x}_c \times \vec{f}_c^p$$

## Fluid phase :

$$\rho^f \epsilon \frac{\partial \langle u_x \rangle^f}{\partial t} = \frac{\partial S_{xz}^f}{\partial z} - \frac{\partial R_{xz}^f}{\partial z} + \rho^f \epsilon g_x - n \langle f_D \rangle$$

$\vec{f}_f^p$ : fluid forces over particles

- $\vec{f}_b^p$ : Buoyancy force

- $\vec{f}_D^p$ : Drag force

$$\vec{f}_D^p = \frac{1}{2} \rho^f \frac{\pi d^2}{4} C_D \left\| \vec{u}^f - \vec{v}^p \right\| \left( \vec{u}^f - \vec{v}^p \right)$$
<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

# Numerical Model

Discrete Element Method (DEM) coupled with a 1D turbulent fluid model (Maurin et al. 2015, 2016) :

- DEM (code YADE) : Lagrangian method based on contact between particles

- Fluid : 1D vertical turbulent fluid flow based on mixing length closure

<figure><img src="image_23.png" /></figure>

Granular phase, for each particle $p$ :

$$m^p \frac{d^2 \vec{x}^p}{dt^2} = \vec{f}_c^p + \vec{f}_g^p + \vec{f}_f^p$$

$$\mathcal{T}^p \frac{d\vec{\omega}^p}{dt} = \vec{T} = \vec{x}_c \times \vec{f}_c^p$$

Fluid phase :

$$\rho^f \epsilon \frac{\partial \langle u_x \rangle^f}{\partial t} = \frac{\partial S_{xz}^f}{\partial z} - \frac{\partial R_{xz}^f}{\partial z} + \rho^f \epsilon g_x - n \langle f_D \rangle$$

Reynolds shear stress :

$$R_{xz}^f = \rho^f \nu^t \frac{d \langle u_x \rangle^f}{dz}$$

$\nu^t$ : turbulent viscosity computed with a mixing length
<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

## Numerical setup

<figure><img src="image_27.png" /></figure>

Geometrical parameters :

- 3D bi-periodic domain

- Slope : 10%

- $d_l/d_s = 2$ (6 mm / 3 mm)

- $H = 8d_l$

- Shields Number : $\theta = \frac{\tau_f}{(\rho^p - \rho^f)gd_l}$

<figure><img src="image_28.png" /><figcaption>(a) $N_l = 2$</figcaption></figure>

<caption>(b) $N_l = 4$</caption>

<caption>(c) Monodisperse</caption>
<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /></figure>

Simulations

<figure><img src="image_32.png" /></figure>

■ $v_x^p$: exponential decrease with depth

<figure><img src="image_33.png" /></figure>

■ Total concentration: $\phi = \phi_s + \phi_l$

Transport rate:

$$Q_s = \int_z \phi v_x^p dz$$

$$Q_s^* = \frac{Q_s}{\sqrt{(\rho^p / \rho^f - 1)g\cos(\alpha)d^3}}, \quad \bar{d}: \text{mean surface diameter}$$
<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /></figure>

<figure><img src="image_36.png" /></figure>

Transport. Comparison Monodisperse and $N_l = 2$

<figure><img src="image_37.png" /></figure>

Comparison Mono, $N_l = 2$ :

* $Q_s^{*\text{mono}} = 15.77\Theta^{1.88}$

* $Q_s^{*N2} = 21.59\Theta^{1.88}$

* Transport 37% more efficient
<figure><img src="image_38.png" /></figure>

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

Transport. Comparison Monodisperse and $N_l = 2$

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

Comparison Mono, $N_l = 2$:

■ $Q_s^{*mono} = 15.77\Theta^{1.88}$

■ $Q_s^{*N2} = 21.59\Theta^{1.88}$

■ Transport 37% more efficient

■ Local transport : $q_s^i = \phi_s^i v_x^i$

■ Both small and large particles are transported
<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.png" /></figure>

<figure><img src="image_45.png" /></figure>

Transport. Comparison Monodisperse and $N_l = 2$

<figure><img src="image_46.png" /></figure>

Comparison Mono, $N_l = 2$:

■ $Q_s^{*mono} = 15.77\Theta^{1.88}$

■ $Q_s^{*N2} = 21.59\Theta^{1.88}$

■ Transport 37% more efficient

■ Local transport : $q_s^i = \phi^i v_x^i$

■ Both small and large particles transported
<figure><img src="image_47.png" /></figure>

<figure><img src="image_48.png" /></figure>

<figure><img src="image_49.png" /></figure>

Transport. Comparison Monodisperse and $N_l = 4$

<figure><img src="image_50.png" /></figure>

Comparison Mono, $N_l = 4$ :

- Small shield : no effect

- Big shield : big effect

- $\Rightarrow$ depth of interface small/large important
<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /></figure>

<figure><img src="image_53.png" /></figure>

Transport. Comparison Monodisperse and $N_l = 4$

<figure><img src="image_54.png" /></figure>

Comparison Mono, $N_l = 4$ :

- Small shield : no effect

- Big shield : big effect

- $\Rightarrow$ depth of interface small/large important

- Local transport : $q_s^i = \phi_s^i v_x^i$

- Necessity to transport small particles

- Not a rugosity effect
<figure><img src="image_55.png" /></figure>

<figure><img src="image_56.png" /></figure>

<figure><img src="image_57.png" /></figure>

Transport. Comparison Monodisperse and $N_l = 4$

<figure><img src="image_58.png" /></figure>

Comparison Mono, $N_l = 4$ :

- Small shield : no effect

- Big shield : big effect

- $\Rightarrow$ depth of interface small/large important

- Local transport : $q_s^i = \phi_s^i v_x^i$

- Necessity to transport small particles

- Not a rugosity effect
# Conclusion

## Summary :

- Model : reproduce an increase of mobility in bidisperse case

- Small particles need to be transported

- Small and large particles take part of the increase

## Why are small particles sometimes transported ?

- Fluid effect : fluid shear stress sufficient to transport small particles ?

- Granular effect : how, what effect ?