Laboratoire Environnement, Géomécanique & Ouvrages
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 Lorraine
Ecole doctorale RP2E
1er Octobre 2012
Nancy, Lorraine
1
Particle-laden flows
Photo: NASA's Goddard Space Flight Center
Particles: air and water
pollutants, dust, sprays
and aerosols, etc…
2
Particle-laden flows through fractures
Hydrogeology:
Flows through fractures often carry particles
(sediments, organic debris etc.).
How to model particle-laden flows?
3
Two models of particles
Tracer particles:
Inertial particles:
point particles
finite size, density
advected by the fluid
different from fluid.
(+ brownian motion)
Example: dye in water
Example: sand in the air
4
Two models of particles
Tracer particles:
Inertial particles:
point particles
finite size, density
advected by the fluid
different from fluid.
(+ brownian motion)
Advection-diffusion
equations for particle
concentration.
Example: sand in the air
5
Two models of particles
Tracer particles:
Inertial particles:
point particles
finite size, density
advected by the fluid
different from fluid.
(+ brownian motion)
Particle inertia is important.
Advection-diffusion
Even weakly-inertial particles
equations for particle
are
concentration.
very different from tracers!
6
Clustering of inertial particles
Inertial particles tend to cluster in certain zones
of the flow.
rain initiation
Wilkinson & Mehlig (2006)
planet formation
Barge & Sommeria (1995)
aerosol engineering
Fernandez de la Mora (1996)
Particles in fractures: clustering can lead to redistribution
of particles across the fracture?
7
Clustering of inertial particles
Inertial particles tend to cluster in certain zones
of the flow.
rain initiation
Wilkinson & Mehlig (2006)
planet formation
Barge & Sommeria (1995)
aerosol engineering
Fernandez de la Mora (1996)
In periodic flows particle focus to a single trajectory:
Robinson (1955), Maxey&Corrsin (1986), etc.
8
Goal of the thesis
Theoretical study of focusing effect on particle
transport in a fracture with periodic corrugations.
Water
+
particles
0
homogeneos distribution
«focusing»
9
Outline of the talk
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
10
I. Single-phase flow in a thin fracture.
11
I. Single-phase flow in a thin fracture.
Goal:
Obtain an explicit fluid velocity field for
arbitrary fracture shapes
Method:
Asymptotic expansions
12
Simplified model of a fracture
Z
𝐙 = 𝚽𝟐 (𝑿)
𝑳𝟎
𝑯𝟎
𝒁 = 𝚽𝟏 (𝑿)
X
Model fracture: a thin 2D channel with «slow» corrugation.
Typical corrugation length L0 >> typical aperture H0.
Small parameter:
𝑯𝟎
𝜺=
≪𝟏
𝑳𝟎
13
Single-phase flow in fracture
Single-phase flow in fracture:
2D, incompressible, stationary
Navier-Stokes equations:
𝐳
𝒖=𝟎
𝐳 = 𝝓𝟐 (𝒙)
𝑼𝟎
𝒖(𝒙, 𝒛)
𝒖=𝟎
𝝆, 𝝂
𝐳 = 𝝓𝟏 (𝒙)
Streamfunction:
𝑼 𝑿, 𝒁 = 𝛁 × 𝚿
Non-dimensional variables:
𝑿
,
𝑳𝟎
𝚿
𝝍=𝜺 ;
𝑸
𝒙=
𝐱
𝒁
𝒛=
;
𝑯𝟎
𝑼𝒛
𝑼𝒙
;
𝒖𝒙 =
, 𝒖𝒛 = 𝜺
𝑼𝟎
𝑼𝟎
Reynolds number:
𝑼𝟎 𝑯𝟎
= 𝑶(𝟏)
𝝂
𝜺𝑹𝒆𝑯 ≪ 𝟏
𝑹𝒆𝑯 =
14
Equations of inertial lubrication theory
Navier-Stokes equations in non-dimensional variables:
Boundary conditions:
No slip at
the walls
Hasegawa and Izuchi (1983)
Borisov (1982), etc.
15
Equations of inertial lubrication theory
Navier-Stokes equations in non-dimensional variables
Boundary conditions:
𝜺≪𝟏
𝜺𝑹𝒆𝑯 ≪ 𝟏
No slip at
the walls
Small parameter ε ⇒ perturbative method
16
Generalization of previous works
Crosnier (2002)
Hasegawa and Izuchi (1983)
}
Present thesis: full parametrization
of the fracture geometry.
Borisov (1982)
17
The cross-channel variable: 𝜼
𝒛
𝜼
𝒛 = 𝝓𝟐 (𝒙)
(𝒙, 𝒛) → (𝒙, 𝜼)
h(x)
h(x)
𝜼=𝟏
𝜼=𝟎
𝐳 = 𝝓(𝒙)
𝒛 = 𝝓𝟏 (𝒙)
𝜼 = −𝟏
𝒙
Cross-channel variable
𝒙
𝑧 − 𝜙(𝑥)
: 𝜂 = ℎ(𝑥)
𝜙2 𝑥 − 𝜙1 (𝑥)
ℎ(𝑥) =
half-aperture of the channel
2
𝜙1 𝑥 + 𝜙2 (𝑥)
𝜙(𝑥) =
middle-line profile
2
18
Asymptotic solution of 2nd order
0th :
1st:
2nd:
19
Asymptotic solution of 2nd order
0th :
«local cubic law»
1st:
2nd:
inertial corrections
viscous correction
3rd… etc.
20
Numerical verification: mirror-symmetric
𝑸
𝜺 = 𝟎. 𝟏
--- LCL flow,
2nd order asymptotics,
numerical simulation
21
Numerical verification: flat top wall
𝑸
𝜺 = 𝟎. 𝟏
--- LCL flow,
2nd order asymptotics,
numerical simulation
22
Application: corrections to Darcy’s law
Flow rate depends on pressure drop:
𝑷𝟏
𝑷𝟐
𝚫𝑷 = 𝑷𝟐 − 𝑷𝟏
𝛥𝑃 𝑄 - curve
Q
𝑳∞
𝛥𝑃
Darcy’s law
𝛥𝑃
Inertial corrections:
analytical expression?
𝑄
Small flow rates
Larger flow rates
𝑄
23
Corrections to Darcy’s law
Pressure drop (from 2nd order asymptotic solution):
2


P
12 Q 
2
2 Q  

1  K1  K 3  
3 
L
Hh 
   
No quadratic term!
In accordance with Lo Jacono et al. (2005) and many others.
24
Corrections to Darcy’s law
Pressure drop (from 2nd order asymptotic solution):
2


P
12 Q 
2
2 Q  

1  K1  K 3  
3 
L
Hh 
   
Geometrical factors:
𝒛
𝒉(𝒙)
𝒛 = 𝝓(𝒙)
𝒉(𝒙)
𝒙
25
Corrections to Darcy’s law
Pressure drop (from 2nd order asymptotic solution):
2


P
12 Q 
2
2 Q  

1  K1  K 3  
3 
L
Hh 
   
Geometrical factors:
𝒛
𝒉(𝒙)
𝒛 = 𝝓(𝒙)
𝒉(𝒙)
𝒙
Slope of the linear law depends on
both aperture and shape of the middle line.
26
Corrections to Darcy’s law
Pressure drop (from 2nd order asymptotic solution):
2


P
12 Q 
2
2 Q  

1  K1  K 3  
3 
L
Hh 
   
Geometrical factors:
𝒛
𝒉(𝒙)
𝒛 = 𝝓(𝒙)
𝒉(𝒙)
𝒙
Cubic correction only depends on aperture variation.
27
Numerical verification
Pressure drop
vs
Reynolds number
Darcy’s law
numerics (mirror-symmetric channel)
our asymptotic solution
numerics (channel with flat top wall)
28
II. Transport of particles in
the periodic fracture
29
Periodic channel
corrugation period
𝑯𝟎
𝑳𝟎
«focusing»
Particles: small, non-brownian, non-interacting,
passive.
Flow: asymptotic solution (leading order)
30
Particle motion equations
𝒈
Re p 
𝒂
𝑽𝒑
𝑼𝒇 (𝑿𝒑 )
aVs

 1
𝑉𝑠 = 𝑉𝑝 − 𝑈𝑓
−𝑽𝒔
Particle dynamics:
mp

dV p
dt


 FH  m p g
from Stokes equations
around the particle
Maxey-Riley equations
Gatignol (1983)
Maxey and Riley (1983)
31
Particle motion equations
Maxey-Riley equations:
mp

dV p
drag force
added mass
Basset’s
memory term
dt
fluid pressure
gradient + gravity

 mf
DU f

 (m p  m f ) g 
Dt
 a2  

 6a U f  V p  U f  
6



2
 
m f  dV p D  
a

U f  U f   


2  dt
Dt 
10
 
t
 a2  
1 d 
U f  V p  U f ds

6
t  s ds 

0
32
Typical long-time behaviors
(numerics - LCL flow, no gravity)
Heavy
particles
Q
Light
particles
Q
Heavy particles can focus
to a single trajectory (or not!)
depending on channel geometry.
Q
33
Typical long-time behaviors
(numerics - LCL flow, with gravity)
𝒈
𝒈
Light
particles
Heavy
particles
Low Q
High Q
Focusing persists in presence of gravity,
if the flow rate Q is high enough
(permanent suspension)
34
Goal:
Find conditions for particle focusing
depending on channel geometry and flow rate.
Method:
Poincaré map
+
asymptotic motion equations for
weakly-inertial particles
35
Simplified Maxey-Riley equations
Particle response time:
2 Re H

9R
 a 


 H0 
2
Density contrast:
2 f
R
2 p   f
36
Simplified Maxey-Riley equations
Particle response time:
2 Re H  a 



9R  H 0 
For weakly-inertial particles:
Density contrast:
2 f
R
2 p   f
  1
 

 
 
x p  u f ( x p )  v1( x p )  O  3 / 2
fluid velocity
2
Maxey (1987)
particle inertia + weight
from Maxey-Riley equations
37
Poincaré map for weakly-inertial particles
𝜼𝟎
𝜼𝟏
𝜼𝒌
𝜼𝒌+𝟏
𝜂 = rescaled cross-channel variable z
𝜂𝑘 = 𝜂(𝑡𝑘 ) after k periods
𝜂𝑘+1 = 𝜂𝑘 + 𝑓(𝜂𝑘 )
from simplified
Maxey-Riley
  1
equations
38
Poincaré map for weakly-inertial particles
𝜼𝟎
𝜼𝟏
𝜼𝒌
𝜼𝒌+𝟏
Poincaré map: 𝜂𝑘+1 = 𝜂𝑘 + 𝑓(𝜂𝑘 )
Stable fixed point:
𝑓 𝜂∞ = 0;
−1 < 𝑓′ 𝜂∞ < 0.
Particles converge
to the streamline
Focusing! 𝜂 𝑥, 𝑧 = 𝜂∞
39
Analytical expression for the Poincaré map
Poincaré map for the LCL flow:
2
 3R
   9

2


f ( )   
 1 ~
J


J


1

G

h
h
z 
 2
  ' ( )  8


Fluid/particle
density ratio
heavier
3R
1
than fluid
2
3R
lighter
1
than fluid
2
Channel geometry

Gravity number
J h  h' h
2
J h   '2
2
h
  '1
2
g z L0
1
Gz 

2
U0
Fr
h
40
Analytical expression for the Poincaré map
Poincaré map for the LCL flow:
2
 3R
   9

2


f ( )   
 1 ~
J


J


1

G

h
h
z 
 2
  ' ( )  8


Fluid/particle
density ratio
heavier
3R
1
than fluid
2
3R
lighter
1
than fluid
2
Channel geometry

Gravity number
J h  h' h
2
J h   '2
𝑓 𝜂∞ = 0;
−1 < 𝑓′ 𝜂∞ < 0.
2
h
  '1
2
g z L0
1
Gz 

2
U0
Fr
h
Attractor
position 𝜂∞
41
Focusing/sedimentation diagram
Rescaled gravity:
8 GZ
z 
9 h' 2
h
Corrugation
asymmetry factor:
 '2 h   '1
2

 (  ) (analytical expression)
cr
z
z
2
h
2
h' h
  J h / J h
42
Focusing/sedimentation diagram
Light particles
 ( )
cr
z
  
Heavy particles
A
Case A: 𝛽 = 0, 𝛾𝑧 = −0.217

  J h / J h
43
Focusing/sedimentation diagram
 ( )
cr
z
z
B
Case B: 𝛽 = 0, 𝛾𝑧 = −0.5

  Jh / J h
44
Focusing/sedimentation diagram
 ( )
cr
z
C
z
Case C: 𝛽 = −3, 𝛾𝑧 = 0

  Jh / J h
45
Focusing/sedimentation diagram
  
z
 ( )
cr
z
𝜺
D
Case D: 𝛽 = −3, 𝛾𝑧 = −0.5
  Jh / J h
46
Other applications of Poincaré map
Using the Poincaré map we can calculate:
•
Percentage of deposited particles
•
Maximal deposition length
•
Focusing rate
Verified numerically  Ok
47
Influence of channel geometry
on transport properties
48
Shape factors of the channel
Aperture-weighted
norm:
ah
2
1
l
l
𝒛
𝒛 = 𝝓𝟐 (𝒙)
𝒉(𝒙)
𝒛 = 𝝓(𝒙)
2
a ( x)dx
0 h3 ( x)
𝒉(𝒙)
𝑸
𝒛 = 𝝓𝟏 (𝒙)
𝒙
Shape factors:
h
3
 1h
2
J h  h' h
2
«apparent» aperture
aperture
variation
J   ' h
2
J h   '2 h   '1
2
2
h
middle line difference between
corrugation wall corrugations
49
Single phase flow: geometry influence
Pressure drop curve:
2


P
12 Q 
2
2 Q  

1  K1  K 3  
3 
L
Hh 
   
Slope of the linear law:
H 03
H  3
h
3
h
Inertial correction:
h
3
Shape
factors:
 1h
2
J h  h' h J    ' h
2
2
J h   '2 h   '1
2
Weak dependence on channel shape!
2
h
50
Particle transport: geometry influence
Particle Poincaré map:
P ( )
f ( )   3R / 2  1 ~
 ' ( )
2
9
2
J h  Jh    1  Gz
P ( ) 
8


3
Shape
factors:
 1h
2
Particle behavior depends
h
strongly on the difference
J h  h' h J    ' h
2
2
J h   '2 h   '1
2
in wall corrugations!
2
h
51
Particle transport: geometry influence
flat
J flat


J
h
h
J hflat  J hmirror
mirror
J flat

J
h
h
Example: channel with flat top wall
and mirror-symmetric channel.
Equivalent for single phase flow
but different for particles.
J mirror
0
h
52
IV. The effect of lift force on
particle focusing
53
Motion equations with lift
 
U f (X )
Particle motion equations:
mp

dV p
dt
 mf

DU f
Dt

 6a (U f

m f  dV p


2  dt

 (m p  m f ) g 

 Vp ) 

DU f 
 + Lift force

Dt 

FL

VP
Lift in simple
shear flow
(Saffman, 1956)
Lift appears when particle leads or lags the fluid.
No formula for lift in a general flow…
«Generalization» of Saffman’s lift:

 
2
1/ 2
FL  6.46a (  f )   U f
1/ 2



((U f  V p )   0 )
54
Lift force induced by gravity
Gravity in the direction of the flow (vertical channel):
Heavy particles lead 
Light particles lag
pushed to the walls
 pushed to the center
Effect opposite to focusing!
Poincaré map with lift calculated analytically
55
Lift force induced by gravity
G
Two attracting
streamlines
(theory)
Effect opposite to focusing!
Poincaré map with lift shows splitting of the attractor.
56
Lift force induced by particle inertia
No gravity

FL
lead
lag

FL
 k 1
Lift-induced chaos at finite response times
Particles lead or lag because of their proper inertia.
The direction of lift changes many times.
57
Lift force induced by particle inertia
No gravity

FL
lead
lag

FL
 k 1
Lift-induced chaos at finite response times
Particles lead or lag because of their proper inertia.
The direction of lift changes many times.
Effect on focusing?
Poincaré map does not work here…
58
Lift-induced chaos at finite response times
Period doubling cascade
Chaos!
k
 (response time)
Feigenbaum constants:
59
IV. Conclusion
60
Conclusions: single-phase flow
• 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.
61
Conclusions: particle transport
• 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.
62
Conclusions: lift effect
• 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.
63
Perspectives
• 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
64
Perspectives
• 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.
Thank you for attention!
65