pouliquen_dfd07_short.ppt [read-only]

Copyright O. Pouliquen, 2007

Flow of dense granular mediaA peculiar liquid

Pierre Jop, Yoël Forterre,Mickaël PailhaMaxime Nicolas Olivier Pouliquen

IUSTI-CNRS, Université de Provence,

Marseille, France


Dry granular matter :

rigid (very stiff) particles with-non brownian motion-no cohesion-no attraction

hard spheres at T=0


Dry granular flowsDifferent flow regimes

Kinetic theory(Jenkins & Savage JFM 83Goldhirsh ARFM 02)


Different flow configurations studiedboth experimentally and numerically

GDR Midi, Eur. Phys. J 04


P

Da Cruz et al, PRE 05GdR Midi Eur. Phys. J 04

Numerical simulations (DEM)in plane shear

U

h

P

*Linear Profile

One imposes P and

Shear stress τ?Volume fraction φ?

!

˙ " =U /h

!

˙ "


Dimensional analysis:

!

I =˙ " d

P /#A single dimensionless number:

if one assumes that the system size plays no role

* I ratio between 2 times :

1/γ : time for the mean shear

d√ ρ/P: microscopic time for rearrangement

(Savage 84, Ancey et al 99)


« quasi-static » « liquid » « gas »

!

I =˙ " d

P /#

10


P

Da Cruz et al, PRE 05GdR Midi Eur. Phys. J 04

Numerical simulations (DEM)in plane shear

U

h

P

*Linear Profile

One imposes P and

Shear stress τ?Volume fraction φ?

!

˙ " =U /h

!

˙ "

!

" = µ I( )P

!

" = " I( )


!

I =˙ " d

P /#

!

" /P

P

!

" = µ I( )P

!

" = " I( )

!

"


Data from

Inclined plane exp. (Pouliquen 99)Inclined plane simulations (Baran et al 2006)Annular shear cell exp. (Sayed, Savage JFM, 84)

For spheres

Forterre, Pouliquen ARFM 08


And shear at constant volume fraction ??

!

" = f1(#)$sd

2 ˙ % 2

!

P = f2(")#sd

2 ˙ $ 2

Ι

f1

φ

φ

Bagnold Proc. R. Soc 54Lois et al PRE 07Lemaitre PRE 05Da Cruz et al PRE 05

!

I =˙ " d

P /#

!

" = µ(I)P

!

" ="(I)

Constant pressure

µ

Iφ

f2


For more complex configurations

we need a real 3D tensorial constitutive law


A 3D generalisation of the friction law : introducingan effectif viscosity

assumptions: 1) P isotropic2) and are co-linear

!

I =˙ " d

P /#!

" = µ(I)P.

!

"(I) =µ(I)P

˙ #

Flow threshold :

!

˙ " ij =#ui#x j

+#u j

#xi

!

˙ " =1

2˙ " ij ˙ " ij

!

" < µsP

!

" =1

2" ij" ij

γij

||γ||

. effectiveviscosity

ij

|| ||

!

˙ " ij

!

" ij(Savage 83, Goddard 86, Schaeffer 87,…)


Mass and momentum equations:


3 quantitative tests where 3D effects exist:

-steady flow in a rough channel-avalanches-roll wave instability

We keep the same material (glass beads .5mm) for which the friction law has been calibrated on steady uniform flows down inclined plane…

No free parameter….


1st test:flows on a heap : a full 3D problem

L = 1.5 m

W

Q

(Pierre Jop's PhD)

y/d

V(y,z)

z/d


0

1

2

3

4

5

6

7

-0,2 0 0,2 0,4 0,6 0,8 1 1,2

0

1

2

3

4

5

6

7

-0,2 0 0,2 0,4 0,6 0,8 1 1,2

-10

0

10

20

30

40

500 0,2 0,4 0,6 0,8 1

-10

0

10

20

30

40

500 0,2 0,4 0,6 0,8 1

Flow between rough lateral walls:

y/W y/W

h/d

!

Vsurf

gd

Jop et alNature 2006


Acceleration of the layer (V(t) ), and erosion (h(t) )

2 test: how an avalanche starts?(Jop et al, Phys. Fluids 07)


3D test : Long wave instability in granular flows

By Yoel Forterre (Forterre, JFM 06 )


Experimental Setup : forcing at a given frequency

Forterre and Pouliquen JFM 02

Loudspeakers

Power supply Function generator

Nozzle

Slides projector

Stabilized

alimentationf, AGBF

!

h

x

y

z

Photodiodes


-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0 2 4 6 8 10 12 14

fondamental1 Hz1.5 Hz2 Hz3 Hz4 Hz6 Hz

!

f (Hz)

(cm-1

)

fc

(glass beads, q=29°, h=5.3 mm)Dispersion relation

Growth rate Phase velocity

σ(cm-1)

f (Hz)

vφ(cm/s)

f (Hz)


Stability diagram(glass beads)

0

5

10

15

20

25

22 24 26 28 3022 24 26 28 30

25

20

15

10

5

0

! (deg.)

h/d

no flow

unstable

stable

0

0.2

0.4

0.6

0.8

1

23 24 25 26 27 28 29 30

1

0.8

0.6

0.4

0.2

0 23 24 25 26 27 28 29 30

F

unstable

stable

no flow

! (deg.)

! (deg.)

h/d

! (deg.)

F


Full 3D linear stability analysis:

No fit parameter…


Dispersion relationForterre JFM 06


Instability thresholdForterre JFM 06


The simple visco-plastic approach:

captures the viscous nature of granular flows and givesquantitative predictions in 3D geometries

… serious limits !!!But …

!

" =µ(I)P

#

!

" ij =#$ ij

??? Other configuration ?????? other material ???

-Flow threshold

- Quasi-static regime


Shear bands in quasi-static flow not capturedLimits in the quasi-static limit when I-> 0

1.0

0.8

0.6

0.4

0.2

0.0121086420

V/Vw

y/d

« narrow shear band » « wide shear bands »

Depken et al PRE 06

(C .Cawthorn APS DFD 07, Forterre Pouliquen ARFM 08)

Howell et al PRL 99Mueth et al Nature 00Bocquet et al PRE 02…


flow threshold not well captured

Present model

-no hysteresis

-no finite size effect

Pouliquen, Phys. Fluids 99Daerr, Douady Nature 99Ecke, Borzsonyi APS DFD 07

h

θ


link with the microstructure?

Role of the fluctuations ?

Role of the correlations ?

Link with other glassy systems?

Aranson and Tsimring PRE,01, Louge Phys. Fluids 03, Josserand et al 06

Mills et al 99, 00, 04 Jenkins and Chevoir 01,

Pouliquen et al 01,

Ertas and Halsey 03,

Lemaitre 02

Jenkins Phys. Fluids 06,…

Radjai and Roux PRL 02

5 10 15 20 25 30

5

10

15

20

25

30

Pouliquen PRL 04


Conclusion:

The visco-plastic model gives good predictions when the granular material flows in the inertial regimebut fails close to quasi-static regime

Perspectives:

Link with the microstructure

More complexe materials:

cohesion? Rognon et al PRE 06

role of the interstitial fluid ? Cassar et al Phys. Fluids 06


Changing time scales…by putting the granular material in water

Laser

P.C

DV

recorder

!P

d=112 µm

glass beads


Dry

Immersed


A naive idea :

fluid only plays a role by changing the time scale of rearrangements

I = γ tmicroτ = P µ(I) with

viscous :

dry :

!

tmicro

=d

P /"!

tmicro =" f

#P


µ=tgθ

Idry Ivisc


Suggestion for immersed granular media:

τ = P µ(I)

!

I =" ˙ #

$P

!

" ="(I)

!

" = f1(#)$ ˙ %

!

P = f2(")# ˙ $

Similar to rheology of dense suspensions….

Morris and Boulay 99Ovarlez et al 06…

Or


A simple experiment:

Loose sample Dense sample


Experimental setup(Mickael Pailha PhD)

Fastcamera

Inclinationangle

Lasersheet

Pressurehose

Granularmaterialthickness

Pressuresensor

1m

20cm

7cm


14

12

10

8

6

4

2

0

-2

u(mm/s)

403020100

t(s)

Free surface Velocity starting at different initial volume fraction


Hand waving argument(Iverson Rev. Geo. 97, Huppert 04)

Φ⇒Fluid expelled⇒Pfluid ⇒Pgrain ⇒Friction ⇒Less friction betweengrains

Loose case Dense case

Φ⇒Fluid sucked⇒Pfluid ⇒Pgrain ⇒Friction ⇒higher friction betweengrains


Rheology of Immersed granularflows

Volume fraction variationat small deformations

Two-phase flowsequations

Soilsmechanics

fluidmechanics

This work

We need to put together

Copyright O. Pouliquen, 2007Predictions of a simple model:

14

12

10

8

6

4

2

0

-2

u(mm/s)

403020100

t(s)

-10

-5

0

P(Pa)

403020100

t(s)

-10

-5

0

P(Pa)

403020100

t(s)

14

12

10

8

6

4

2

0

-2

u(mm/s)

403020100

t(s)

Velocity Pressureexperiments:


Conclusions

A visco-plastic description of granular flows is a first steptowards a granular rheology (quantitative predictions in3D)

⇒Other configuration (Pb of simulation of apeculiar visco-plastic rheologie…)

⇒Transition to the quasi-static regimeJamming transition

⇒ transition to the kinetic gaseous regime

⇒Link with the micro-structure

The granular rheology gives a new point of view for immersed granular media: link with suspension rheology ?


Thanks to

Yoël Forterre Pierre Jop, Maxime Nicolas Cyril Cassar Mickael Pailha Prabhu Nott Jeff Morris Neil Balmforth Bruno Andreotti Olivier Dauchot

pouliquen_dfd07_short.ppt [read-only]

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