Particles - Tel

swedishstreakMechanics

Feb 22, 2014 (3 years and 7 months ago)

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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 Lorraine

Ecole doctorale
RP2E


1er
Octobre
2012

Nancy, Lorraine


Laboratoire Environnement, Géomécanique & Ouvrages


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.).

3

How to model particle
-
laden flows?

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

4

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

5

Advection
-
diffusion

equations for particle

concentration.

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

6

very

different

from

tracers
!

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)

7

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)

8

Theoretical

study

of
focusing

effect

on
particle

transport in a fracture
with

periodic

corrugations.

Water

+

particles

Goal of the
thesis

«
focusing
»

9

0

h
omogeneos

distribution

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

10

11

I. Single
-
phase flow in a
thin

fracture.

I. Single
-
phase flow in a
thin

fracture.

12

Goal:

Obtain an explicit fluid velocity field for
arbitrary fracture

shapes

Method:

Asymptotic expansions


Simplified

model of a fracture

Model fracture: a
thin

2D
channel

with

«slow» corrugation.


Typical

corrugation
length

L
0
>>
typical

aperture H
0
.


𝜺
=
𝑯

𝑳




Small
parameter
:

𝑳


𝑯



=
𝚽

(
𝑿
)

𝐙
=
𝚽

(
𝑿
)

X

Z

13

Single
-
phase flow in fracture

Single
-
phase flow in fracture:

2D, incompressible,
stationary

𝝆
,
𝝂




𝒖
(
𝒙
,
𝒛
)

𝒖
=


𝒖
=


Streamfunction
:


𝑿
,

=
𝛁
×
𝚿

𝐳
=
𝝓

(
𝒙
)

𝐳
=
𝝓

(
𝒙
)

𝐳

𝐱

𝒙
=
𝑿
𝑳

,

𝒛
=

𝑯

;

𝒖
𝒙
=

𝒙


,


𝒖
𝒛
=
𝜺

𝒛


;


𝝍
=
𝜺
𝚿

;


Non
-
dimensional

variables:


𝑯
=


𝑯

𝝂
=

(

)


Reynolds
number
:

Navier
-
Stokes
equations
:

14

𝜺

𝑯



Equations of inertial lubrication theory

Navier
-
Stokes equations in non
-
dimensional variables:

Boundary conditions:

Hasegawa

and
Izuchi

(1983)

Borisov
(
1982), etc.

No slip at

the walls

15

Navier
-
Stokes
equations

in non
-
dimensional

variables

Boundary

conditions:

Small
parameter

ε



perturbative

method

No slip

at

t
he
walls

16

Equations of
inertial

lubrication

theory

𝜺

𝑯



𝜺



Generalization

of
previous

works

Hasegawa

and
Izuchi

(1983)

Borisov (1982)

Crosnier

(2002)

Present

thesis
:

full
parametrization

of the fracture
geometry
.


}

17

The cross
-
channel

variable:
𝜼

Cross
-
channel variable
:

𝜼
=


𝜼
=



𝜼
=


𝜼

𝒙

h(x)

h(x)

𝐳
=
𝝓
(
𝒙
)

𝒛

𝒙

𝒛
=
𝝓

(
𝒙
)

𝒛
=
𝝓

(
𝒙
)

𝜂
=
𝑧

𝜙
(
𝑥
)

(
𝑥
)

(
𝒙
,
𝒛
)

(
𝒙
,
𝜼
)

h
alf
-
aperture of the channel


middle
-
line profile

18

𝜙
(
𝑥
)
=
𝜙
1
𝑥
+
𝜙
2
(
𝑥
)
2


(
𝑥
)
=
𝜙
2
𝑥

𝜙
1
(
𝑥
)
2

Asymptotic

solution of 2
nd

order




0
th
:



1
st
:




2
nd
:



19

Asymptotic

solution of 2
nd

order




0
th
:



1
st
:




2
nd
:





3
rd
… etc.



v
iscous

correction

inertial

corrections

«local
cubic

law
»

20

Numerical

verification
:
mirror
-
symmetric

---

LCL flow,
2
nd

order

asymptotic
s
,




numerical

simulation

𝜺
=

.




21

Numerical

verification
: flat top
wall

𝜺
=

.




22

---

LCL flow,
2
nd

order

asymptotic
s
,




numerical

simulation

Application: corrections
to
Darcy’s

law







𝑳


Q

𝚫

=







𝛥




Darcy’s

law

𝛥


Inertial

corrections:

analytical

expression?

𝛥


Small flow rates

Larger

flow rates





-

curve



Flow rate
depends

on pressure drop:

23

Corrections to
Darcy’s

law

Pressure drop (
from

2
nd

order

asymptotic

solution):






















2
2
3
2
1
3
1
12




Q
K
K
H
Q
L
P
h
24

No
quadratic

term
!

I
n
accordance
with

Lo
Jacono

et al.

(
2005) and
many

others
.

Corrections to
Darcy’s

law

Pressure drop (
from

2
nd

order

asymptotic

solution):






















2
2
3
2
1
3
1
12




Q
K
K
H
Q
L
P
h

(
𝒙
)


(
𝒙
)

𝒛
=
𝝓
(
𝒙
)

𝒙

𝒛

Geometrical

factors
:

25

Corrections to
Darcy’s

law

Pressure drop (
from

2
nd

order

asymptotic

solution):






















2
2
3
2
1
3
1
12




Q
K
K
H
Q
L
P
h

(
𝒙
)


(
𝒙
)

𝒛
=
𝝓
(
𝒙
)

𝒙

𝒛

Geometrical

factors
:


Slope

of the
linear

law

depends

on


both

aperture and
shape

of the middle line.

26

Corrections to
Darcy’s

law

Pressure drop (
from

2
nd

order

asymptotic

solution):






















2
2
3
2
1
3
1
12




Q
K
K
H
Q
L
P
h
Geometrical

factors
:

Cubic

correction
only

depends

on aperture variation.

27


(
𝒙
)


(
𝒙
)

𝒛
=
𝝓
(
𝒙
)

𝒙

𝒛

Numerical

verification

Darcy’s

law

o
ur

asymptotic

solution

n
umerics

(
mirror
-
symmetric

channel
)

n
umerics

(
channel

with

flat top
wall
)

Pressure drop

v
s

Reynolds
number

28

II. Transport of
particles

in

the
periodic

fracture

29

Periodic

channel

Particles
:
small
, non
-
brownian
, non
-
interacting
,
passive.

𝑳


𝑯


c
orrugation
period

Flow
:
asymptotic

solution (
leading

order
)

«
focusing
»

30

Particle

motion
equations



(
𝑿
𝒑
)




𝒑

1
Re



s
p
aV

𝑠
=

𝑝


𝑓

Particle

dynamics
:


from

Stokes
equations


around

the
particle


Maxey
-
Riley
equations

g
m
F
dt
V
d
m
p
H
p
p









𝒔

Maxey

and Riley (1983)

Gatignol

(1983)

31

Maxey
-
Riley
equations
:





6
1


10
2


6
6


)
(
0
2
2
2
ds
U
a
V
U
ds
d
s
t
U
a
U
Dt
D
dt
V
d
m
U
a
V
U
a
g
m
m
Dt
U
D
m
dt
V
d
m
t
f
p
f
f
f
p
f
f
p
f
f
p
f
f
p
p


































































d
rag force

f
luid

pressure gradient +
gravity

added

mass

Basset’s


m
emory
term

Particle

motion
equations

32

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

33



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
)

34

Heavy

particles

Light

particles

Goal:

Find

conditions for
particle

focusing

d
epending

on
channel

geometry

and flow rate
.

Method
:


Poincaré
map

+

a
symptotic

motion
equations

for

weakly
-
inertial

particles

35

Simplified

Maxey
-
Riley
equations

f
p
f





2
2
R
Density

contrast
:

Particle

response

time:

2
0
9
Re
2









H
a
R
H


36

Simplified

Maxey
-
Riley
equations

For
weakly
-
inertial

particles
:



2
/
3
1
)
(
)
(


O
x
v
x
u
x
p
p
f
p









particle

inertia

+
weight

f
luid

velocity

f
p
f





2
2
R
Density

contrast
:

Particle

response

time:

2
0
9
Re
2









H
a
R
H


37

1


f
rom

Maxey
-
Riley
equations

Maxey

(1987)

Poincaré
map

for
weakly
-
inertial

particles

𝜼
𝒌

𝜼
𝒌
+


𝜼


𝜼


𝜂

= rescaled cross
-
channel variable z

𝜂
𝑘
=
𝜂
(
𝑡
𝑘
)

a
fter

k
periods

𝜂
𝑘
+
1
=
𝜂
𝑘
+
𝑓
(
𝜂
𝑘
)

f
rom

simplified

Maxey
-
Riley

equations

38

1


Poincaré
map

for
weakly
-
inertial

particles

𝜼
𝒌

𝜼
𝒌
+


𝜼


𝜼


𝜂
𝑘
+
1
=
𝜂
𝑘
+
𝑓
(
𝜂
𝑘
)

Stable
fixed

point:

𝑓
𝜂

=
0
;



1
<
𝑓′
𝜂

<
0
.

Focusing
!

39

Poincaré
map
:

Particles converge

to the
streamline


𝜂
𝑥
,
𝑧
=
𝜂























z
h
h
G
J
J
R
f
2
2
1
8
9
)
(
'
~
1
2
3
)
(









Analytical

expression for the Poincaré
map

Poincaré
map

for the LCL flow
:

Gravity

number

Channel
geometry

Fluid
/
particle


d
ensity

ratio

1
2
3
1
2
3


R
R
l
ighter


than

fluid

h
eavier


than

fluid

2
1
2
2
'
'
h
h
h
J





2
'
h
h
h
J

Fr
U
L
g
G
z
z
1
2
0
0


40






















z
h
h
G
J
J
R
f
2
2
1
8
9
)
(
'
~
1
2
3
)
(









Analytical

expression for the Poincaré
map

Poincaré
map

for the LCL flow
:

Gravity

number

Channel
geometry

Fluid
/
particle


d
ensity

ratio

1
2
3
1
2
3


R
R
l
ighter


than

fluid

h
eavier


than

fluid

2
1
2
2
'
'
h
h
h
J





2
'
h
h
h
J

Fr
U
L
g
G
z
z
1
2
0
0


41

𝑓
𝜂

=
0
;



1
<
𝑓′
𝜂

<
0
.

Attractor

p
osition
𝜂


Focusing
/
sedimentation

diagram

)
(


cr
z
z

Rescaled

gravity
:

Corrugation

asymmetry

factor:

2
2
1
2
2
'
'
'
h
h
h
h






(
analytical

expression)

h
h
J
J
/



2
'
9
8
h
Z
z
h
G



42

Focusing
/
sedimentation

diagram

)
(


cr
z
Case A:

=
0
,


𝑧
=

0
.
217

A



h
h
J
J
/



43





Heavy
particles

Light
particles

Focusing
/
sedimentation

diagram

)
(


cr
z
z

Case B:

=
0
,


𝑧
=

0
.
5



B

h
h
J
J
/



44

Focusing
/
sedimentation

diagram

)
(


cr
z
z

Case C:

=

3
,


𝑧
=
0

C

h
h
J
J
/





45

Focusing
/
sedimentation

diagram

)
(


cr
z
z

Case D:

=

3
,


𝑧
=

0
.
5

D

h
h
J
J
/



46





𝜺


Percentage

of
deposited

particles


Maximal
deposition

length


Focusing

rate

47

Using

the Poincaré
map

we

can

calculate
:

Other

applications of Poincaré
map

Verified

numerically



Ok


Influence of channel geometry

on transport properties

48


(
𝒙
)


(
𝒙
)

𝒛
=
𝝓
(
𝒙
)

𝒙

𝒛



𝒛
=
𝝓


(
𝒙
)

2
1
2
2
'
'
h
h
h
J





2
'
h
h
h
J

2
'
h
J



2
3
1
h
h


Shape
factors

of the
channel


Shape
factors
:

49

𝒛
=
𝝓


(
𝒙
)

«apparent»

aperture



a
perture

variation


m
iddle line

corrugation


difference

between

wall

corrugations






l
h
x
h
dx
x
a
l
a
0
3
2
2
)
(
)
(
1
Aperture
-
weighted


norm
:

Pressure drop
curve
:

Single phase flow:
geometry

influence





















2
2
3
2
1
3
1
12




Q
K
K
H
Q
L
P
h
3
3
0
3


h
H
H
h
Slope

of the
linear

law
:

Inertial

correction:

50

2
1
2
2
'
'
h
h
h
J





2
'
h
h
h
J

2
'
h
J



2
3
1
h
h



Shape


factors
:

Weak

dependence

on
channel

shape
!

Particle

Poincaré
map
:

Particle

transport:
geometry

influence

51



)
(
'
~
)
(
1
2
/
3
)
(





P
R
f






z
h
h
G
J
J
P




2
2
1
8
9
)
(





Particle

behavior

depends


strongly

on the
difference


in
wall

corrugations!

2
1
2
2
'
'
h
h
h
J





2
'
h
h
h
J

2
'
h
J



2
3
1
h
h



Shape


factors
:

Particle

transport:
geometry

influence

52

Example
:
channel

with

flat top
wall

and
mirror
-
symmetric

channel
.

Equivalent for single phase flow


but
different

for
particles
.

mirror
h
flat
h
J
J

mirror
h
flat
h
J
J



0

mirror
h
J

flat
h
flat
h
J
J



IV. The
effect

of lift force on

particle

focusing

53





2


)
(
6


)
(

















Dt
U
D
dt
V
d
m
V
U
a
g
m
m
Dt
U
D
m
dt
V
d
m
f
p
f
p
f
f
p
f
f
p
p









Particle

motion
equations
:

Motion
equations

with

lift

+ Lift force



)
)
((
)
(
46
.
6
0
2
/
1
2
/
1
2












p
f
f
f
L
V
U
U
a
F

(
Saffman
, 1956)

«
Generalization
» of
Saffman’s

lift:



L
F



P
V



)
(
X
U
f


Lift
appears

when

particle

leads or
lags

the
fluid
.

54

Lift in simple

shear

flow

No formula for lift in a
general

flow…

Lift force
induced

by
gravity


55

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

Lift force
induced

by
gravity


56


Effect

opposite to
focusing
!


Two

attracting



streamlines


(
theory
)


Poincaré
map

with

lift shows
splitting

of the
attractor
.


G

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.

57

No
gravity



L
F



1

k



L
F

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.

58

No
gravity

Lift
-
induced

chaos
at

finite

response

times

lead

lag



1

k



L
F



L
F

Lift
-
induced

chaos
at

finite

response

times

k


(
response

time)

Chaos!

Period

doubling

cascade

Feigenbaum

constants:

59

IV. Conclusion

60



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


61


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


62


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


63


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

64









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

65