Migflow

Michel Henry , Simon Yans , Alexandre Pirot, James Gosselet,

supervised by Vincent Legat, & Jonathan Lambrechts

Simulate immersed granular flows ...

...with a variable volume fraction
...in complex configurations
...at a conveniant cost

Grains sorting using water jigging (M. Constant)

Submarine Avalanche of elongated grains (N. Coppin)

Solid phase

Discrete element
Lagrangian representation
Insight in contacts physics

Fluid phase

Continuous medium
Eulerian representation
Computational convenience

Fluid-grains interaction

Empirical transfer

Contact Solver

2D - Tetris Simulation

3D - Mixer Simulation

Rigid body kinematics : \[ \begin{aligned} \dot{\mathbf{q}} &= \mathbf{T}(\mathbf{q})\mathbf{v},\\ \end{aligned} \]

Rigid body dynamics : \[ \begin{aligned} \mathbf{M} \text{d} \mathbf{v} &= \mathbf{f}_{\text{e}} \text{d}t + \mathbf{f}_{\text{q}} \text{d}t + \text{d} \mathbf{p} \end{aligned} \]

\[ \begin{alignedat}{2} &\delta\mathbf{u}^+ = \delta\mathbf{u}^- + \mathbf{W} \mathbf{r}, && (\text{Contact Dynamics})\\ &g(t) \geq 0, && (\text{Impenetrability}),\\ &\rm{r}_n(t) \geq 0, && (\text{No attraction}),\\ &\rm{r}_n(t) g(t) = 0, && (\text{Signorini}),\\ &g(t) = 0 \Rightarrow \rm{r}_n(t) \geq 0, \delta\rm{u}^+_n = 0,\qquad\qquad &&(\text{No restitution}).\\ &\Vert{\mathbf{r}_t}\Vert \leq \mu \rm{r}_n, &&(\text{Coulomb's cone})\\ &\Vert{\delta \mathbf{u}^+_t}\Vert\neq 0 \Rightarrow \mathbf{r}_t = -\mu \rm{r}_n \dfrac{\delta\mathbf{u}^+_t}{\Vert{\delta\mathbf{u}^+_t}\Vert}, &&(\text{Slipping}). \end{alignedat} \]

VANS Module

Granular Rayleigh-Taylor Instability

Sedimentation of a cylinder

Fluid Equations (VANS): \[ \begin{alignedat}{2} &\boldsymbol{\nabla}\cdot(\color{red}{\varepsilon} \boldsymbol{u}) + \color{blue}{\boldsymbol{\nabla}\cdot(\phi \boldsymbol{v})} = 0,\\\\ &\partial_t\left(\rho \color{red}{\varepsilon} \boldsymbol{u}\right) + \boldsymbol{\nabla} \cdot (\color{red}{\varepsilon} \rho\boldsymbol{u} \otimes \boldsymbol{u}) = \\&\qquad-\color{red}{\varepsilon} \boldsymbol{\nabla}{p} + \boldsymbol{\nabla}\cdot (\color{red}{\varepsilon} \boldsymbol{\tau}) - \color{blue}{\phi \boldsymbol f_{\text{d}}} + \color{red}{\varepsilon}\rho\boldsymbol{g},\\ \end{alignedat} \] with $ \color{red}{\varepsilon} = \color{blue}{1-\phi}$
Grain dynamics : \[ \begin{aligned} \rho \frac{\text{d} \mathbf{v}}{\text{d} t} &= \rho \boldsymbol{g} + \mathbf{f} + \mathbf{f}_c \end{aligned} \]
Pressure-drag coupling : \[ \mathbf{f} = - \nabla p + \mathbf{f}_d \]
Di Felice and Dallavalle's laws : \[ \begin{aligned} \mathbf{f}_d = \varepsilon^{-2.8} \left( 0.77 \sqrt{\varepsilon \text{Re}} + 5.88 \right)^2 \frac{\eta}{d^2} (\boldsymbol{u} - \mathbf{v}) \\ \end{aligned} \]

Fluid-Grains transfer \[ \begin{alignedat}{6} &\int_{V} &\color{blue}{\phi} &&\tau \, &\mathrm{d}V = &&\sum_p\int_{V_p} &\tau \, &\mathrm{d}V_p \\ &\int_{V} &\color{blue}{\phi\boldsymbol{v}^h} &&\cdot \nabla \tau \, &\mathrm{d}V = &&\sum_p\int_{V_p} &\color{blue}{\mathbf{v}_p} \cdot \nabla \tau \, &\mathrm{d}V_p\\ &\int_{V} &\color{blue}{\phi\boldsymbol{f}^h} &&\tau \, &\mathrm{d}V = &&\sum_p\int_{V_p} &\color{blue}{\mathbf{f}}\, \tau \, &\mathrm{d}V_p \end{alignedat} \]

Particle Centroid Method (PCM)

Fast
Loss of accuracy
Needs of stability
Available in 2D-3D

Overlap Centroid Method (OCM)

Expensive
Exact geometry integration
Stable even at low void fraction
Available in 2D

Unresolved scale : $h > 3d$

Semi-resolved scale : $h \sim d $

Resolved scale : $ 10h < d $

Particle Centroid Method (PCM) & Continuity Equation

Overlap Centroid Method & Volume Conservation

Particle Centroid Method (PCM) & Continuity Equation

Overlap Centroid Method & Volume Conservation

Heat Transfer Module

Transfer by conduction

Vapour enables fluidisation

At the grain scale : \[ \rho c \frac{\text{d} T_p}{\text{d} t} = q^c + q \]

Conduction from the Contacts Network

Convection from correlation

At the grain scale : \[ \rho c \frac{\text{d} \mathrm{T}_p}{\text{d} t} = \mathrm{q}^c + \mathrm{q} \]

Conduction Contacts Network

Convection from correlation

At the fluid scale,

\[ \partial_t \left( \varepsilon \rho c T \right) + \boldsymbol{\nabla} \cdot \left( \varepsilon \rho c \boldsymbol{u} T \right) = \boldsymbol{\nabla} \cdot \left( \varepsilon k \boldsymbol{\nabla} T \right) - \phi q \]

Boussinesq approximation: \[ \begin{alignedat}{2} &\boldsymbol{\nabla}\cdot(\varepsilon \boldsymbol{u}) + \boldsymbol{\nabla}\cdot(\phi \boldsymbol{v}) = 0,\\\\ &\partial_t\left(\rho \varepsilon \boldsymbol{u}\right) + \boldsymbol{\nabla} \cdot (\varepsilon \rho\boldsymbol{u} \otimes \boldsymbol{u}) = \\&\qquad-\varepsilon \boldsymbol{\nabla}{p} + \boldsymbol{\nabla}\cdot (\varepsilon \boldsymbol{\tau}) - \phi \boldsymbol f_{\text{d}} + \varepsilon\rho\boldsymbol{g}\left( 1 - \beta \left( T - T_R \right) \right), \end{alignedat} \]

At the grain scale : \[ \rho c \frac{\text{d} \mathrm{T}_p}{\text{d} t} = \mathrm{q}^c + \mathrm{q} \]

Conduction Contacts Network

Convection from correlation

At the fluid scale,

\[ \partial_t \left( \varepsilon \rho c T \right) + \boldsymbol{\nabla} \cdot \left( \varepsilon \rho c \boldsymbol{u} T \right) = \boldsymbol{\nabla} \cdot \left( \varepsilon k \boldsymbol{\nabla} T \right) - \phi q \]

Boussinesq approximation: \[ \begin{alignedat}{2} &\boldsymbol{\nabla}\cdot(\varepsilon \boldsymbol{u}) + \boldsymbol{\nabla}\cdot(\phi \boldsymbol{v}) = 0,\\\\ &\partial_t\left(\rho \varepsilon \boldsymbol{u}\right) + \boldsymbol{\nabla} \cdot (\varepsilon \rho\boldsymbol{u} \otimes \boldsymbol{u}) = \\&\qquad-\varepsilon \boldsymbol{\nabla}{p} + \boldsymbol{\nabla}\cdot (\varepsilon \boldsymbol{\tau}) - \phi \boldsymbol f_{\text{d}} + \varepsilon\rho\boldsymbol{g}\left( 1 - \beta \left( T - T_R \right) \right), \end{alignedat} \]

Local transfers model by the dimensionless numbers : \[ \mathrm{C_d} = \frac{\mathrm{f_d}}{\frac{1}{2} \rho \Vert \boldsymbol{u} - \mathbf{v} \Vert^2}\frac{V}{A}, \qquad \mathrm{Nu} = \frac{\mathrm{q} d}{k (T - \mathrm{T}_p)} \frac{V}{S} \]

The Reynolds analogy to estimate the Nusselt number: \[ \mathrm{Nu} \approx f(\mathrm{Re}) \frac{\mathrm{C_d Re}}{12} \mathrm{Pr}^{\alpha} \]

Which is extended to an assembly : $\mathrm{Re} \rightarrow \varepsilon \mathrm{Re}$ \[ \mathrm{Nu} \approx g(\varepsilon, \mathrm{Re}) f(\varepsilon \mathrm{Re}) \frac{\mathrm{C_d(\varepsilon\mathrm{Re}) \varepsilon \mathrm{Re}}}{12} \mathrm{Pr}^{\alpha} \]