Soutenance de thèse

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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 & Ouvr 1

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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 - PowerPoint PPT Presentation

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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…

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Particle-laden flows through

fracturesHydrogeology:

Flows through fractures often carry particles

(sediments, organic debris etc.).

How to model particle-laden flows?

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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!

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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

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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

09

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

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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 /

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Focusing/sedimentation diagram

)(crz

z

Case C:

C

hh JJ /

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Focusing/sedimentation diagram

)(crz

z

Case D:

D

hh JJ /

𝜺

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• 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:

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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

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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:

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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

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• 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

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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