Surface curvature and secondary vortices

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in dense shallow granular flows

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C. Gadal, C.G. Johnson, J.M.N.T Gray

The University of Manchester

Dense granular flow down an inclined halfpipe

Glass beads, \(d \sim 145 \mu\text{m}\), \(\theta = 26^{\circ}\)

Upward curvature of the flow surface

Glass beads, \(d \sim 145 \mu\text{m}\), \(\theta = 26^{\circ}\)

Can not be obtained with standard \(\mu(I)\) rheology! What’s missing?

Normal stress differences

\[\sigma = \underbrace{\begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \\ \end{bmatrix}}_{\text{total stress}} = \underbrace{\begin{bmatrix} \sigma_{xx} & 0 & 0\\ 0 & \sigma_{yy} & 0 \\ 0 & 0 & \sigma_{zz} \\ \end{bmatrix}}_{\text{normal stresses}} + \underbrace{\begin{bmatrix} 0 & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & 0 & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & 0 \\ \end{bmatrix}}_{\text{shear stresses}}\]

• Simple (first-order) rheology (newtonian): \(\sigma_{ii} = p\)

Complex fluids: \(\sigma_{xx} \neq \sigma_{yy} \neq \sigma_{zz}\)    Normal stress differences

   Often induced by anisotropie at the microscale

Normal stress differences

Complex fluids: \(\sigma_{xx} \neq \sigma_{yy} \neq \sigma_{zz}\)    Normal stress differences

\(\mathcal{N}_{1} = \sigma_{xx} - \sigma_{zz}\)    tension along streamlines

Self pouring polymer (Royal Society of Chemistry)

\(\mathcal{N}_{2} = \sigma_{zz} - \sigma_{yy}\)    compression perp. streamlines

Surface curvature in flowing dense suspensions (Couturier et al. 2011)

Second normal stress difference in granular flows

Grain scale simulations

Sketch of stress controlled simulations
(Srivastava et al. 2020)

Measured scaled second normal stress difference
(Srivastava et al. 2020)

second-order granular rheology   

\(\sigma = -p \left(I_{3} + \displaystyle\frac{\mu(I)}{\lVert\dot{\gamma}\rVert}\dot{\gamma} + \displaystyle\frac{N_{2}(I)}{\lVert\dot{\gamma}\rVert^{2}}\left[\dot{\gamma}^{2} - \displaystyle\frac{1}{3}\text{tr}\left(\dot{\gamma}^{2}\right)\right] \right)\)

Summary of the mathematical modelling

Summary of the mathematical modelling

Assumptions

• steady, incompressible, uniform in \(x\)
• a single average grain size, \(d\)

• shallow \(\epsilon = H/W \ll 1\):

\(\begin{aligned} \mathbf{u} & = u^{(0)} \mathbf{e}_{x} + \epsilon \mathbf{u}^{(1)} + \epsilon^{2} \mathbf{u}^{(2)} + ... \\ p & = p^{(0)} + \epsilon p^{(1)} + \epsilon^{2} p^{(2)} + ... \\ & \qquad \vdots \end{aligned}\)

Mass conservation

\(\nabla \cdot \boldsymbol{u} = 0\)

Momentum conservation

\(\rho (\boldsymbol{u} \cdot \nabla)\boldsymbol{u} = \nabla \cdot \boldsymbol{\sigma} + \rho\boldsymbol{g}\)

Second-order granular rheology

\(\boldsymbol{\sigma} = -p \left(\mathbf{I}_{3} + \displaystyle\frac{\mu(I)}{\lVert\boldsymbol{\dot{\gamma}}\rVert}\boldsymbol{\dot{\gamma}} + \displaystyle\frac{N_{2}(I)}{\lVert\boldsymbol{\dot{\gamma}}\rVert^{2}}\left[\boldsymbol{\dot{\gamma}}^{2} - \displaystyle\frac{1}{3}\text{tr}\left(\boldsymbol{\dot{\gamma}}^{2}\right)\right] \right)\)

\(\boldsymbol{\dot{\gamma}} = \left(\nabla\boldsymbol{u} + \nabla\boldsymbol{u}^{T}\right)\) and \(I = \displaystyle\frac{\lVert\dot{\gamma}\rVert d}{\sqrt{p/\rho_{\text{s}}}}\)

Boundary conditions

• no slip at the base: \(\boldsymbol{u}(b) = \boldsymbol{0}\)
• kinematic boundary condition: \(\boldsymbol{u}(s) \cdot \boldsymbol{n}_{s} = 0\)
• dynamic boundary condition: \(\boldsymbol{\sigma}(s) \cdot \boldsymbol{n}_{s} = \boldsymbol{0}\)

Assymptotic solution: first non-zero quantities

How to verify this solution?

Leading order

• \(s^{(0)}\): \(\displaystyle\frac{\text{d}}{\text{d}y}s^{(0)} = \displaystyle\frac{5 N_{2}}{5 N_{2} + 4}\displaystyle\frac{\text{d}}{\text{d}y}b\)

• \(u^{(0)}\): juxtaposition of Bagnold profiles

• \(p^{(0)}\): hydrostatic

Next orders

Secondary vortices

Validation using grain scale simulations (DEM)

Access to particle position, velocity, contact stresses (\(N_{2}\))    statistical averages

Validation using grain scale simulations (DEM)

Good prediction of the flow structure

Validation using grain scale simulations (DEM)

Excellent prediction of the surface shape

Back to the lab: inference of \(N_{2}(I)\)

• glass beads, \(d \sim 150\) microns
• slope angle, \(\theta\)    control the inertial number, \(I\)
• mass flux, \(Q\)    only scales up or down the velocity/flow depth

\(\displaystyle\frac{\text{d}}{\text{d}y}s^{(0)} = \displaystyle\frac{5 N_{2}}{5 N_{2} + 4}\displaystyle\frac{\text{d}}{\text{d}y}b\)

Back to the lab: inference of \(N_{2}(I)\)

In a nutshell

• second normal stress difference    surface curvature and secondary flows in dense granular flows

• mathematical model able to predict this quantitatively    inference from experiments of \(N_{2}(I)\)

C. Gadal, C. J. Johnson C. J. and J. M. N. T. Gray. “Surface curvature and secondary vortices in dense shallow granular flows.” Submitted to Journal of Fluid Mechanics

Bidisperse dense granular flow down an inclined halfpipe

Bidisperse glass beads, green \(d_{\text{g}} \sim 225 \mu\text{m}\), white \(d_{\text{w}} \sim 100 \mu\text{m}\), \(\theta = 26^{\circ}\)

Lateral segregation, stripes

Bidisperse glass beads, green \(d_{\text{g}} \sim 225 \mu\text{m}\), white \(d_{\text{w}} \sim 100 \mu\text{m}\), \(\theta = 26^{\circ}\)

Segregation in convecting granular flows
Amber Pearse
Tuesday - 16h40