Post on 05-Jan-2016
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Soutenance de thèse
Transport, dépôt et relargage de particules inertielles dans une fracture à rugosité périodique
T. Nizkaya
Directeur de thèse: M. Buès
Co-directeur de thèse: J.-R. Angilella,
LAEGO, Université de LorraineEcole doctorale RP2E
1er Octobre 2012Nancy, Lorraine
Laboratoire Environnement, Géomécanique & Ouvrages
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Particle-laden flows
Photo: NASA's Goddard Space Flight Center
Particles: air and water pollutants, dust, sprays and aerosols, etc…
3
Particle-laden flows through
fracturesHydrogeology:
Flows through fractures often carry particles
(sediments, organic debris etc.).
How to model particle-laden flows?
4
Two models of particles
Tracer particles:
point particles
advected by the fluid
(+ brownian motion)
Example: dye in water
Inertial particles:
finite size, density
different from fluid.
Example: sand in the air
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Two models of particles
Tracer particles:
point particles
advected by the fluid
(+ brownian motion)
Inertial particles:
finite size, density
different from fluid.
Example: sand in the air
Advection-diffusion
equations for particle
concentration.
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Two models of particles
Tracer particles:
point particles
advected by the fluid
(+ brownian motion)
Advection-diffusion
equations for particle
concentration.
Inertial particles:
finite size, density
different from fluid.
Particle inertia is important.
Even weakly-inertial particles
are
very different from tracers!
7
Clustering of inertial particles
Inertial particles tend to cluster in certain zones
of the flow.
Particles in fractures: clustering can lead to redistribution
of particles across the fracture?
rain initiation Wilkinson & Mehlig (2006)
planet formation Barge & Sommeria (1995)
aerosol engineering Fernandez de la Mora (1996)
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Clustering of inertial particles
Inertial particles tend to cluster in certain zones
of the flow.
In periodic flows particle focus to a single trajectory:
Robinson (1955), Maxey&Corrsin (1986), etc.
rain initiation Wilkinson & Mehlig (2006)
planet formation Barge & Sommeria (1995)
aerosol engineering Fernandez de la Mora (1996)
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Theoretical study of focusing effect on particle
transport in a fracture with periodic corrugations.
Water +
particles
Goal of the thesis
«focusing»
0
homogeneos distribution
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I. Single-phase flow in a model fracture
II. Focusing of inertial particles in the fracture
III. Influence of lift force on particle focusing
IV.Conclusion and perspectives
Outline of the talk
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I. Single-phase flow in a thin fracture.
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I. Single-phase flow in a thin fracture.
Goal:
Obtain an explicit fluid velocity field for
arbitrary fracture shapes
Method:
Asymptotic expansions
13
Simplified model of a fracture
Model fracture: a thin 2D channel with «slow» corrugation.
Typical corrugation length L0 >> typical aperture H0.
𝜺=𝑯𝟎
𝑳𝟎≪𝟏Small parameter:
𝑳𝟎
𝑯𝟎 𝒁=𝚽𝟏(𝑿 )
𝐙=𝚽𝟐(𝑿)
X
Z
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Single-phase flow in fracture
Single-phase flow in fracture:
2D, incompressible, stationary
𝝆 ,𝝂𝑼 𝟎
��(𝒙 ,𝒛 )
��=𝟎
��=𝟎
Streamfunction:𝑼 (𝑿 ,𝒁 )=𝛁×𝚿
𝐳=𝝓𝟐(𝒙)
𝐳=𝝓𝟏(𝒙)
𝐳
𝐱
𝒙=𝑿𝑳𝟎,𝒛=𝒁𝑯𝟎
;
𝒖𝒙=𝑼 𝒙
𝑼 𝟎
,𝒖𝒛=𝜺𝑼 𝒛
𝑼 𝟎
;𝝍=𝜺𝚿𝑸;
Non-dimensional variables:
𝑹𝒆𝑯=𝑼 𝟎𝑯𝟎
𝝂=𝑶(𝟏)
Reynolds number:
Navier-Stokes equations:
𝜺𝑹𝒆𝑯 ≪𝟏
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Equations of inertial lubrication theory
Navier-Stokes equations in non-dimensional variables:
Boundary conditions:
Hasegawa and Izuchi (1983)
Borisov (1982), etc.
No slip atthe walls
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Navier-Stokes equations in non-dimensional variables
Boundary conditions:
Small parameter ε perturbative method
No slip atthe walls
Equations of inertial lubrication theory
𝜺𝑹𝒆𝑯 ≪𝟏𝜺≪𝟏
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Generalization of previous works
Hasegawa and Izuchi (1983)
Borisov (1982)
Crosnier (2002)
Present thesis: full parametrization
of the fracture geometry.
}
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The cross-channel variable:
Cross-channel variable :
𝜼=𝟏
𝜼=−𝟏
𝜼=𝟎
𝜼
𝒙
h(x)
h(x)𝐳=𝝓 (𝒙)
𝒛
𝒙
𝒛=𝝓𝟐(𝒙)
𝒛=𝝓𝟏(𝒙)
𝜂=𝑧−𝜙 (𝑥)h(𝑥)
(𝒙 ,𝒛 )→(𝒙 ,𝜼)
half-aperture of the channel
middle-line profile𝜙(𝑥 )=𝜙1 (𝑥 )+𝜙2(𝑥)
2
h (𝑥)=𝜙2 (𝑥 )−𝜙1(𝑥)
2
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Asymptotic solution of 2nd order
0th :
1st: 2nd:
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Asymptotic solution of 2nd order
0th :
1st: 2nd:
3rd… etc.
viscous correctioninertial corrections
«local cubic law»
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Numerical verification: mirror-symmetric
--- LCL flow, 2nd order asymptotics,
numerical simulation
𝜺=𝟎 .𝟏
𝑸
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Numerical verification: flat top wall
𝜺=𝟎 .𝟏
𝑸
--- LCL flow, 2nd order asymptotics,
numerical simulation
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Application: corrections to Darcy’s law
𝑷 𝟐𝑷 𝟏
𝑳∞
Q
𝚫𝑷=𝑷 𝟐−𝑷𝟏
Darcy’s law𝛥 𝑃
Inertial corrections:analytical expression?
𝛥 𝑃
Small flow rates Larger flow rates𝑄 𝑄
- curve
Flow rate depends on pressure drop:
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Corrections to Darcy’s law
Pressure drop (from 2nd order asymptotic solution):
22
32
131
12
Q
KKH
Q
L
P
h
No quadratic term!In accordance with Lo Jacono et al. (2005) and many others.
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Corrections to Darcy’s law
Pressure drop (from 2nd order asymptotic solution):
22
32
131
12
Q
KKH
Q
L
P
h
𝒉(𝒙 )
𝒉(𝒙 )𝒛=𝝓(𝒙 )
𝒙
𝒛Geometrical factors:
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Corrections to Darcy’s law
Pressure drop (from 2nd order asymptotic solution):
22
32
131
12
Q
KKH
Q
L
P
h
𝒉(𝒙 )
𝒉(𝒙 )𝒛=𝝓(𝒙 )
𝒙
𝒛Geometrical factors:
Slope of the linear law depends on
both aperture and shape of the middle line.
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Corrections to Darcy’s law
Pressure drop (from 2nd order asymptotic solution):
22
32
131
12
Q
KKH
Q
L
P
h
Geometrical factors:
Cubic correction only depends on aperture variation.
𝒉(𝒙 )
𝒉(𝒙 )𝒛=𝝓(𝒙 )
𝒙
𝒛
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Numerical verification
Darcy’s law
our asymptotic solution
numerics (mirror-symmetric channel)
numerics (channel with flat top wall)
Pressure dropvs
Reynolds number
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II. Transport of particles in
the periodic fracture
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Periodic channel
Particles: small, non-brownian, non-interacting, passive.
𝑳𝟎
𝑯𝟎
corrugation period
Flow: asymptotic solution (leading order)
«focusing»
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Particle motion equations
𝑼 𝒇 ( ��𝒑)
𝒂𝑽 𝒑
1Re s
p
aV
𝑉 𝑠=𝑉 𝑝−�� 𝑓
Particle dynamics: from Stokes equations around the particle
Maxey-Riley equations
gmFdt
Vdm pH
pp
��
−𝑽 𝒔
Maxey and Riley (1983)
Gatignol (1983)
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Maxey-Riley equations:
6
1
102
66
)(
0
2
2
2
dsUa
VUds
d
st
Ua
UDt
D
dt
Vdm
Ua
VUa
gmmDt
UDm
dt
Vdm
t
fpf
ffpf
fpf
fpf
fp
p
drag force
fluid pressure gradient + gravity
added mass
Basset’s memory term
Particle motion equations
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Typical long-time behaviors (numerics - LCL flow, no gravity)
Heavy particles
Light particles
Heavy particles can focus
to a single trajectory (or not!)
depending on channel geometry.
Q
Q
Q
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��
Focusing persists in presence of gravity,
if the flow rate Q is high enough
(permanent suspension)
��
Low Q High Q
Typical long-time behaviors (numerics - LCL flow, with gravity)
Heavy particles
Light particles
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Goal:
Find conditions for particle focusing
depending on channel geometry and flow rate.
Method:
Poincaré map+
asymptotic motion equations for weakly-inertial particles
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Simplified Maxey-Riley equations
fp
f
2
2R
Density contrast:Particle response time:
2
09
Re2
H
a
RH
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Simplified Maxey-Riley equations
For weakly-inertial particles:
2/31 )()( Oxvxux ppfp
particle inertia + weightfluid velocity
fp
f
2
2R
Density contrast:Particle response time:
2
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Re2
H
a
RH
1
from Maxey-Riley equations
Maxey (1987)
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Poincaré map for weakly-inertial particles
𝜼𝒌 𝜼𝒌+𝟏𝜼𝟏𝜼𝟎
= rescaled cross-channel variable z
𝜂𝑘=𝜂(𝑡𝑘) after k periods
)from simplifiedMaxey-Rileyequations 1
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Poincaré map for weakly-inertial particles
𝜼𝒌 𝜼𝒌+𝟏𝜼𝟏𝜼𝟎
)
Stable fixed point:
Focusing!
Poincaré map:
Particles converge to the streamline
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zhh GJJR
f22 1
8
9
)('~12
3)(
Analytical expression for the Poincaré map
Poincaré map for the LCL flow:
Gravity numberChannel geometryFluid/particle density ratio
12
3
12
3
R
R
lighter than fluid
heavier than fluid
2
1
2
2 ''hhhJ
2'hh hJ
FrU
LgG zz
120
0
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zhh GJJR
f22 1
8
9
)('~12
3)(
Analytical expression for the Poincaré map
Poincaré map for the LCL flow:
Gravity numberChannel geometryFluid/particle density ratio
12
3
12
3
R
R
lighter than fluid
heavier than fluid
2
1
2
2 ''hhhJ
2'hh hJ
FrU
LgG zz
120
0
Attractorposition
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Focusing/sedimentation diagram
)(crz
z
Rescaled gravity:
Corrugation asymmetry factor:
2
2
1
2
2
'
''
h
hh
h
(analytical expression)
hh JJ /
2'9
8
h
Zz
h
G
43
Focusing/sedimentation diagram
)(crz
Case A:
A
hh JJ /
Heavy particles
Light particles
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Focusing/sedimentation diagram
)(crz
z
Case B:
B
hh JJ /
45
Focusing/sedimentation diagram
)(crz
z
Case C:
C
hh JJ /
46
Focusing/sedimentation diagram
)(crz
z
Case D:
D
hh JJ /
𝜺
• Percentage of deposited particles
• Maximal deposition length
• Focusing rate
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Using the Poincaré map we can calculate:
Other applications of Poincaré map
Verified numerically Ok
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Influence of channel geometry
on transport properties
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𝒉(𝒙 )
𝒉(𝒙 )𝒛=𝝓(𝒙 )
𝒙
𝒛
𝑸 𝒛=𝝓𝟏(𝒙)
2
1
2
2 ''hhhJ
2'hh hJ 2
'h
J 23 1h
h
Shape factors of the channel
Shape factors:
𝒛=𝝓𝟐(𝒙)
«apparent»aperture
aperturevariation
middle linecorrugation
difference betweenwall corrugations
l
h xh
dxxa
la
03
22
)(
)(1
Aperture-weighted norm:
50
Pressure drop curve:
Single phase flow: geometry influence
22
32
131
12
Q
KKH
Q
L
P
h
3
303
h
HH h
Slope of the linear law:
Inertial correction:
2
1
2
2 ''hhhJ
2'hh hJ 2
'h
J
23 1h
h Shape factors:
Weak dependence on channel shape!
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Particle Poincaré map:
Particle transport: geometry influence
)('~)(
12/3)( P
Rf
zhh GJJP 22 1
8
9)(
Particle behavior depends
strongly on the difference
in wall corrugations! 2
1
2
2 ''hhhJ
2'hh hJ 2
'h
J
23 1h
h Shape factors:
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Particle transport: geometry influence
Example: channel with flat top wall
and mirror-symmetric channel.
Equivalent for single phase flow
but different for particles.
mirrorh
flath JJ
mirrorh
flath JJ
0mirrorhJ
flath
flath JJ
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IV. The effect of lift force on
particle focusing
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2
)(6
)(
Dt
UD
dt
Vdm
VUa
gmmDt
UDm
dt
Vdm
fpf
pf
fpf
fp
p
Particle motion equations:
Motion equations with lift
+ Lift force
))(()(46.6 0
2/12/12
pfffL VUUaF
(Saffman, 1956)
«Generalization» of Saffman’s lift:
LF
PV
)(XU f
Lift appears when particle leads or lags the fluid.
Lift in simple shear flow
No formula for lift in a general flow…
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Lift force induced by gravity
Heavy particles lead pushed to the walls
Light particles lag pushed to the center
Effect opposite to focusing!
Gravity in the direction of the flow (vertical channel):
Poincaré map with lift calculated analytically
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Lift force induced by gravity
Effect opposite to focusing!
Two attracting
streamlines
(theory)
Poincaré map with lift shows splitting of the attractor.
G
57
Lift-induced chaos at finite response times
lead lag
Lift force induced by particle inertia
Particles lead or lag because of their proper inertia.
The direction of lift changes many times.
No gravity
LF
1k
LF
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Lift-induced chaos at finite response times
Lift force induced by particle inertia
Effect on focusing? Poincaré map does not work here…
Particles lead or lag because of their proper inertia.
The direction of lift changes many times.
No gravity
Lift-induced chaos at finite response times
lead lag
1k
LF
LF
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Lift-induced chaos at finite response times
k
(response time)
Chaos!
Period doubling cascade
Feigenbaum constants:
60
IV. Conclusion
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• A new asymptotic solution of Navier-Stokes equations is
obtained for thin channels.
• This solution generalizes previous results to arbitrary wall
shapes.
• Inertial corrections to Darcy’s law are calculated analytically
as functions of channel geometrical parameters.
Conclusions: single-phase flow
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• Particles transported in a periodic channel can focus to
an attracting streamline which depends on channel
geometry.
• This attractor persists in presence of gravity, if the flow
rate is high enough.
• The full focusing/sedimentation diagram for particles
in periodic channels has been obtained analytically, using
Poincaré map technique.
Conclusions: particle transport
63
• Lift has been taken into account in form of a classical
generalization of Saffman (1965).
• In presence of gravity (vertical channel), lift causes
attractor splitting: two attracting streamlines are visible.
• In the absence of gravity, lift causes a period-doubling
cascade leading to chaotic particle dynamics.
Conclusions: lift effect
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• Particles in a non-periodic (disordered) fracture
Do particles still cluster? How to quantify the clustering?
• Collisions
Does focusing increases collision rates?
• Brownian particles with inertia
Maxey-Riley equations with noise?
• Experimental verification
Experimental setup is under construction at LAEGO
Perspectives
65
Thank you for attention!
• Particles in a non-periodic (disordered) fracture
Do particles still cluster? How to quantify the clustering?
• Collisions
Does focusing increases collision rates?
• Brownian particles with inertia
Maxey-Riley equations with noise?
• Experimental verification
Experimental setup is under construction at LAEGO.
Perspectives