Pouliquen_DFD07_short.ppt [Read-Only]
Transcript of 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
Copyright O. Pouliquen, 2007
Dry granular matter :
rigid (very stiff) particles with-non brownian motion-no cohesion-no attraction
hard spheres at T=0
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Copyright O. Pouliquen, 2007
Dry granular flowsDifferent flow regimes
Kinetic theory(Jenkins & Savage JFM 83Goldhirsh ARFM 02)
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Different flow configurations studiedboth experimentally and numerically
GDR Midi, Eur. Phys. J 04
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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
!
˙ "
Copyright O. Pouliquen, 2007
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)
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« quasi-static » « liquid » « gas »
!
I =˙ " d
P /#
10
Copyright O. Pouliquen, 2007
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( )
Copyright O. Pouliquen, 2007
!
I =˙ " d
P /#
!
" /P
P
!
" = µ I( )P
!
" = " I( )
!
"
Copyright O. Pouliquen, 2007
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
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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
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For more complex configurations
we need a real 3D tensorial constitutive law
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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,…)
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Mass and momentum equations:
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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….
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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
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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
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Acceleration of the layer (V(t) ), and erosion (h(t) )
2 test: how an avalanche starts?(Jop et al, Phys. Fluids 07)
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3D test : Long wave instability in granular flows
By Yoel Forterre (Forterre, JFM 06 )
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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
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-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)
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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
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Full 3D linear stability analysis:
No fit parameter…
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Dispersion relationForterre JFM 06
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Instability thresholdForterre JFM 06
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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
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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…
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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
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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
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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
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Changing time scales…by putting the granular material in water
Laser
P.C
DV
recorder
!P
d=112 µm
glass beads
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Dry
Immersed
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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
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µ=tgθ
Idry Ivisc
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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
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A simple experiment:
Loose sample Dense sample
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Experimental setup(Mickael Pailha PhD)
Fastcamera
Inclinationangle
Lasersheet
Pressurehose
Granularmaterialthickness
Pressuresensor
1m
20cm
7cm
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14
12
10
8
6
4
2
0
-2
u(mm/s)
403020100
t(s)
Free surface Velocity starting at different initial volume fraction
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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
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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:
Copyright O. Pouliquen, 2007
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 ?
Copyright O. Pouliquen, 2007
Thanks to
Yoël Forterre Pierre Jop, Maxime Nicolas Cyril Cassar Mickael Pailha Prabhu Nott Jeff Morris Neil Balmforth Bruno Andreotti Olivier Dauchot