invited lecture - Snezhana I. Abarzhi

monkeyresultMechanics

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

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Coherence and randomness in
non
-
equilibrium turbulent
processes

Snezhana I. Abarzhi

University of Chicago

Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

27 July


07 August 2009

Second International Conference and Advanced School “Turbulent Mixing and Beyond”

is considered the last unresolved problem of classical physics.


Complexity and universality of turbulence fascinate scientists and
mathematicians and nourish the inspiration of philosophers.


Similarity, isotropy and locality are the fundamental hypotheses
advanced our understanding of the turbulent processes.


The problem still sustains the efforts applied.



Turbulent motions of
real fluids

are often characterized by



non
-
equilibrium heat transport



strong gradients of density and pressure



subjected to spatially varying and time
-
dependent acceleration




Turbulent mixing induced by the Rayleigh
-
Taylor instability is


one of generic problems in fluid dynamics.



Its comprehension can extend our knowledge beyond the limits of


idealized consideration of isotropic homogeneous flows.

Turbulence

Rayleigh
-
Taylor instability

Fluids of different densities are accelerated against

the density gradient.

A turbulent mixing of the fluids ensues with time.

Grasping essentials of the mixing process is

a fundamental problem in fluid dynamics.

RT flow is non
-
local, inhomogeneous, anisotropic and accelerated.
Its properties differ from those of the Kolmogorov turbulence.

“How to quantify these flows reliably?”

Is a primary concern for observations.

RT turbulent mixing controls



inertial confinement fusion, magnetic fusion, plasmas, laser
-
matter interaction



supernovae explosions, thermonuclear flashes, photo
-
evaporated clouds



premixed and non
-
premixed combustion (flames and fires)



mantle
-
lithosphere tectonics in geophysics



impact dynamics of liquids, oil reservoir, formation of sprays…

Rayleigh
-
Taylor instability

P
0

= 10
5
Pa,


P =
r
g h

r
h

~ 10
3

kg/m
3
,
g
~ 10
m/s
2

h

~ 10 m

Water flows out from an

overturned cup


Lord Rayleigh, 1883,

Sir G.I. Taylor 1950

r
h

r
l

Dynamics of Continuous Media

0

r


r


v
t


0

















r
S
p
t
g
v
v
v


...
2

r

r


e
t
v




e
t
z
y
x
p
p
t
z
y
x
t
z
y
x
),
,
,
,
(
,
,
,
,
,
,
,
,


r

r
v
v
Conservation laws:

Navier
-
Stokes or Euler equations

Isotropy, homogeneity, locality:

+ Boundary Conditions + Initial Conditions

0



v
,
0
,
,



r
g
const
e
const


0

r













p
t
v
v
v
Isotropic homogeneous
turbulence, Sreenivasan 1999





vL
Re
LK
L




1
tK
t


v
v
scaling invariance Kolmogorov 1941

L
v
dt
v
d
/
~
/
3
2


The Rayleigh
-
Taylor turbulent mixing

Why is it important to study?

Photo
-
evaporated molecular clouds

The fingers protrude from the wall

of a vast could of molecular hydrogen.


The gaseous tower are light
-
years long.


Inside the tower the interstellar gas

is dense enough to collapse under

its own weight, forming young stars

Stalactites?

Stalagmites?

Eagle Nebula.

Birth of a star.

Hester and Cowen, NASA, Hubble pictures, 1995

Ryutov, Remington et al, Astrophysics and Space Sciences, 2004.

Two models of magnetic support for photo
-
evaporated molecular clouds.

Supernovae

Supernovae: death of a star


type II
: RMI and RTI produce
extensive mixing of the outer and
inner layers of the progenitor star


type Ia
: RTI turbulent mixing
dominates the propagation of the
flame front and may provide
proper conditions for generation
of heavy mass element

Burrows, ESA, NASA,1994

Pair of rings of glowing gas, caused

perhaps by a high energy radiation

beam of radiation, encircle

the site of the stellar explosion.

Inertial confinement fusion

17.6 MeV
+
+
+
He
n
D
T
neutron
proton


For the nuclear fusion


reaction, the DT fuel should


be hot and dense plasma





For the plasma compression


in the laboratory it is used



magnetic implosion



laser implosion of DT targets





RMI/RTI inherently occur


during the implosion process




RT turbulent mixing


prevents the formation


of hot spot

Nishihara, ILE, Osaka, Japan, 1994

Inertial Confinement Fusion

STREAK

CAMERA

QUARTZ

CRYSTAL

1.86 keV

imaging



2D IMAGE

MAIN

LASER BEAMS

Time

Magnification x20

BACKLIGHTER

LASER BEAMS

BACKLIGHTER

TARGET
Si

RIPPLED

CH TARGET

Rayleigh
-
Taylor

Imprint

Richtmyer
-
Meshkov

Feedout

Aglitskii, Schmitt, Obenschain, et al, DPP/APS,2004

Nike, 4 ns pulses, ~ 50 TW/cm
3

target: 1 x 2 mm;

perturbation: ~30
m
m, ~0.5
m
m

dislocations
phase
inversion
(a)
dislocations
phase
inversion
(a)
(b)
molten nuclei
incipient spall
(b)
molten nuclei
incipient spall
Impact dynamics in liquids and solids

MD simulations of the Richtmyer
-
Meshkov instability: a shock refracts

though the liquid
-
liquid (up) and solid
-
solid (down) interfaces

Zhakhovskii, Zybin, Abarzhi, etal, DPP, DFD/APS, APS/SCCM 2005, 2006

~4 10
6

LJ atoms (2005), ~7 10
6

LJ atoms (2007), nm, ps

~0.2
m
m & ~ps, for ~2 10
8

atoms on 1.6 10
5

CPU, IBM BG/L, 48


96 hrs


Solar and Stellar Convection

Solar surface, LMSAL, 2003


Observations indicate:

dynamics at Solar surface is governed by convection in the interior.


Simulations show:

Solar non
-
Boussinesq convection is dominated by downdrafts; these are
either large
-
scale vortices (wind) or smaller
-
scale plumes (“RT
-
spikes”).

Simulations of Solar convection

Cattaneo et al, U Chicago, 2002

Non
-
Boussinesq turbulent convection

Thermal Plumes and Thermal Wind

heating

cooling

liquid

instabilities

instabilities

Sparrow 1970

Libchaber et al 1990s

Kadanoff et al 1990s


The non
-
Boussinesq convection and RT mixing may differ as


thermal and mechanical equilibriums, or as entropy and density jumps

Sreenivasan et al 2001

helium T~4K

Re ~ 10
9
, Ra ~ 10
17

Sprays and Atomization

The dispersion of a liquid volume by a

gas steam occurs in



spume droplets over the ocean



pharmaceutical sprays



propellant atomization in combustors

Marmottant and Villermaux, JFM 2004



The Kelvin
-
Helmholtz instability results


in a primary destabilization of a jet.




The Rayleigh
-
Taylor instability causes


the transverse destabilization of the jets


and determines the drop size distribution.

Non
-
premixed and premixed combustion

FLAMES



Landau
-
Darrieus (LD) +


RT in Hele
-
Shaw cells

linear nonlinear

1.0 mm x 200 mm,

0.17 mm/s, Atwood ~ 10
-
3

Ronney, 2000



The distribution of vorticity is the key difference between the LD and RT

Tieszen et al, 2004

FIRES



shear
-
driven KH



buoyancy
-
driven RT

hydrogen and methane

1 m base

Oil production

The process of oil recovery:

water is pumped into reservoir

to force oil to the surface

The Rayleigh
-
Taylor instability

develops at the oil
-
water

Interface.

The mixing imposes strong

limitations on the amount of

the extracted oil

Glimm and Tryggvason 1980

Technology and Communications

Non
-
equilibrium turbulent processes:


Rayleigh
-
Taylor Turbulent Mixing


What is known and unknown?

g

l

~

h

r
h

r
l

Rayleigh
-
Taylor evolution




l
t
h
~


nonlinear regime


light (heavy) fluid penetrates


heavy (light) fluid in bubbles (spikes)




t
h
h
exp
~
0


linear regime





,
~
l
h
l
h
g
r

r
r

r
l



turbulent mixing




h
l
h
gt
h
r
r

r
2
~

RT flow is


characterized by:



large
-
scale structure



small
-
scale structures



energy transfers to


large and small scales

max
~
l
l
Krivets & Jacobs

Phys. Fluids, 2005

5
.
0
27
.
1


A
M
Nonlinear Rayleigh
-
Taylor / Richtmyer
-
Meshkov

ms
t
cm
~
6
~
l


large
-
scale dynamics


is sensitive to the


initial conditions



small
-
scale dynamics


is driven by shear





l
h
l
h
A
r

r
r

r

Nonlinear Rayleigh
-
Taylor


Density plots in horizontal planes

He, Chen, Doolen, 1999, Lattice Boltzman method

ky
kx
cos
cos
~
5
.
0

A
1054
Re

Rayleigh
-
Taylor turbulent mixing

Dimonte, Remington, 1998



cm
L
cm
L
g
g
A
z
y
x
8
.
8

,
3
.
7

,
73

,
2
.
0
0






max
max
~

,
~
l

l
O
mm
5
4
10
Re
,
10


We
3D perspective view (top)

and

along the interface (bottom)



internal structure of


bubbles and spikes

Rayleigh
-
Taylor turbulent mixing

broad
-
band

initial

perturbation

The flow is sensitive to the horizontal boundaries of the fluid tank,

is much less sensitive to the vertical boundaries,

and retains the memory of the initial conditions.

small
-
amplitude
initial

perturbation

FLASH 2004

3D flow

density plots





l
h
l
h
gt
h
r

r
r

r


2
~
Quantification of Rayleigh
-
Taylor flows


For nearly two decades, the observations were focused on



the diagnostics of the vertical scale, readily available for measurements



the ascertainment of the “universality” law



Several folds scatter in the observed values of


indicates a need in new approaches for


understanding the non
-
equilibrium mixing process

2
~
gt
h
2
gt
l
The growth of the horizontal scales


was considered as a primary mixing mechanism

2
~
~
~
gt
h
l

with “unique” constant



Significant efforts and large resources were involved.

Conservation laws

no mass flux


momentum


no mass sources

0
0
0
0
0


















z
l
z
h
l
h
l
h
p
p
v
v
n
v
n
v
Theory of Rayleigh
-
Taylor Instability

0

r


r


v
t


0

















r
S
p
t
g
v
v
v


...
2

r

r


e
t
v
initial conditions






symmetry



max
~
l
l
1883: Rayleigh

1950
th
: Fermi & von Newman, Layzer, Garabedian, Birkhoff;

1990
th
: Anisimov, Mikaelian, Tanveer, Inogamov,

Wouchuk, Nishihara, Glimm,
Hazak, Matsuoka, Velikovich, Abarzhi, ...


Solution of nonlinear PDEs



Physica Scripta T132, 2008

Singular and non
-
local aspects of the interface evolution cause
significant difficulties for theoretical studies of RTI/RMI.

Group
-
theory based approach

Abarzhi
1990th



passive active

scale separation

group theory







l
h
l
h
l
h
Ψ
v






l

l
~


RTI/RMI nonlinear dynamics

is essentially multi
-
scale:


amplitude
h


and wavelength
l

contribute independently

g

l

~

h

r
h

r
l

RTI:

const
g
dt
dh
~
)
/
(
3
2
l
l


l


f
v
dt
dh

RMI:

curvature





The nonlinear dynamics is hard to quantify reliably (power
-
laws).

Our phenomenological model

identifies


the new invariant, scaling and spectral properties of


the accelerated turbulent mixing

accounts for


the multi
-
scale and anisotropic character of the flow dynamics


randomness of the mixing process

discusses


how to generalize this approach for other flows/applications

Non
-
equilibrium turbulent processes

How to model non
-
equilibrium turbulent processes

(in unsteady multiphase flows)?

Any transport process is governed by a set of conservation laws:

conservation of mass, momentum, and energy

Kolmogorov turbulence


transport of kinetic energy

isotropic, homogeneous:

Non
-
equilibrium flows



transports of momentum (mass)

anisotropic, inhomogeneous:


potential and kinetic energy

Unsteady turbulent mixing induced by the Rayleigh
-
Taylor

is driven by the momentum transport

Modeling of RT turbulent mixing

Dynamics:

balance per unit mass of the rate of momentum gain



and the rate of momentum loss

L

is the flow characteristic length
-
scale
,
either horizontal

l

or vertical

h


g
r
r

m
~

rate of momentum gain

v


m
~
~
r
r


g
v
~
rate of potential energy gain

buoyant force

energy dissipation rate




dimensional & Kolmogorov

L
v
C
3


rate of momentum loss

v


m
dissipation force

,
v
dt
dh

m

m

~
dt
dv
These rates are the absolute values of vectors pointed in opposite
directions and parallel to gravity.

Asymptotic dynamics

2

~






~
2
t
g
h
t
g
v


characteristic length
-
scale is vertical


L ~ h


turbulent



r
r


m
g
a
1




t
g
a
a
2
1
r
r



r
r

m
g
~


t
g
a
2
~
r
r


a ~
0.1

l
l
g
t
h
g
v
~



~


characteristic length
-
scale is horizontal

L ~
l


nonlinear



2
1
2
3
~
l
r
r

g


2
1
2
3
~
~
l
r
r

g
r
r

m
g
~
r
r

m
g


2
2
t
g
h
a
r
r


The turbulent mixing develops:




horizontal scale grow with time
l
~ gt
2




vertical scale
h


dominates the flow and is regarded as


the integral,
cumulative

scale for energy dissipation.




the dissipation occurs in small
-
scale structures produced by shear


at the fluid interface.

Accelerated turbulent mixing


In the turbulent mixing flow:




length scale and velocity are time
-
dependent




kinetic and potential energy both change


changes in potential energy are due to buoyancy


changes in kinetic energy are due to dissipation




momentum gains and losses



~
2
~
,
~
gt
L
t
g
v
m
~
m
Non
-
equilibrium turbulent flow



rates of momentum gain and momentum loss are scale and time invariant

L
v
2
~
m
r
r

m
g
~


rate of change of potential energy gain and


of dissipation of kinetic energy are time
-
dependent

r
r


vg
~
L
v
3
~

P

remains time
-

and scale
-
invariant


for time
-
dependent and spatially
-
varying acceleration,


as long as potential energy is a similarity function


on coordinate and time (by analogy with virial theorem)



ratio between the rates is the characteristics of the flow




m
m

P
~
~
Basic concept for the RT turbulent mixing



The dynamics of momentum and energy depends on directions.



There are transports between the planar to vertical components.



4D momentum
-
energy tensor equations should be considered, and


their covariant/invariant properties should be studied in non
-
inertial frame of reference.


RT turbulent mixing is non
-
inertial (non
-
Galilean),

anisotropic, non
-
local and inhomogeneous.

l
v
L
v
l
2
2
~
~
m


The flow invariant is the rate of momentum loss



We consider some consequences of time and scale


invariance of the rate of momentum loss in the


direction of gravity.


Kolmogorov turbulence is inertial (Galilean
-
invariant),

isotropic, local and homogeneous.



Energy dissipation rate is the basic invariant,


determines the scaling properties of the turbulent flow.

l
v
L
v
l
3
3
~
~

Invariant properties of RT turbulent mixing

Kolmogorov turbulence

RT turbulent mixing



v
v





helicity







2
2
3
~
~
~
L
v
vL
L
v
v
L
v



energy dissipation rate

energy transport and inertial interval

time
-

and scale
-
invariant

time
-
dependent

transport of momentum

not a diagnostic parameter

time
-

and scale
-
invariant



enstrophy



2
2
v






rate of momentum loss

l
v
L
v
l
2
2
~
~
m
time
-

and scale
-
invariant

time
-
dependent

not
-
Galilean invariant

~g
, time
-

and scale
-
inv

Scaling properties of RT turbulent mixing

Kolmogorov turbulence

RT turbulent mixing

l
v
L
v
l
2
2
~
~
m
transport of energy

l
v
L
v
l
3
3
~
~

transport of momentum



scaling with Reynolds





3
2
~
~
Re
t
g
vL


2
/
3
Re
~
Re
L
l
l


const
vL
~
~
Re



3
/
4
Re
~
Re
L
l
l


2
/
1
~
L
l
v
v
l


3
/
1
~
L
l
v
v
l


local scaling

more ordered





3
/
1
2
3
/
1
2
~
~
g
l

m



viscous scale



4
/
1
3
~


l
mode of fastest growth



similarly: dissipative scale, surface tension

Spectral properties of RT mixing flow



spectrum of kinetic energy (velocity)





2
3
/
2
3
/
2
3
/
2
3
/
5
3
/
2
~
~
~
~
l
k
k
v
l
k
dk
k
dk
k
E








kinetic energy =

Kolmogorov turbulence:



spectrum of kinetic energy

kinetic energy =



2
2
~
~
~
~
l
k
k
v
l
k
dk
k
dk
k
E
m
m
m







spectrum of momentum

momentum =





l
k
k
v
l
k
dk
k
dk
k
M
e
e
e
e
e
~
~
~
~
2
/
1
2
/
1
2
/
1
2
/
3
2
/
1
m
m
m





RT turbulent mixing:

Scaling, invariant, spectral properties depart from classical scenario

What is the set of orthogonal functions?

Time
-
dependent acceleration, turbulent diffusion

The transport of scalars (temperature or molecular diffusion)

decreases the buoyant force and changes the mixing properties

T
T

r
r
~
,
v
dt
dh

,
2
h
v
C
g
dt
dv



2




h
v
C
dt
d
t
r
r


dynamical system

Rate of temperature change is

T
~
r
We assume

T



2
T












2
2
~
~
T
L
v
L
T
vL



Landau & Lifshits

asymptotic solution



t
0





1
exp
~
h


0
2
2
ln
~
h
gt
gt
h
Asymptotic solutions and invariants



Buoyancy
g

r/r

vanishes asymptotically with time.



Parameter
P

is time
-

and scale
-
invariant value, and


the flow characteristics

buoyancy

g
r/r

vs time
t

vs time
t

m
m

P
~
0
5
10
15
0
0.05
0.1
0.15
0.2
0.25
0.3

m
/
m
t /

0

C
t
=3
C
t
=0
~
dimensionless units

0
5
10
15
0
0.05
0.1
0.15
0.2
0.25

r
/
r
t /

0
C
t
=3
C
t
=0
Randomness of the mixing process

Some features of RT mixing are repeatable from one observation to another.

As any turbulent process, Rayleigh
-
Taylor turbulent mixing has noisy character.




Kolmogorov turbulence



RT turbulent mixing


velocity fluctuates




velocity and length scales fluctuate

energy dissipation rate is invariant

energy dissipation rate grows with time


statistically steady




statistically unsteady



We account for the random character of the dissipation process in RT flow,


incorporating the fact that the rate of momentum loss is



time
-

and scale
-
invariant that fluctuates about its mean.



t
v
V
v



0




t
v
t
V
v



0




1
0


t
V
t
v
a
t
V
~
0


1
0


V
t
v
b
t
v
~

a
b

Stochastic model of RT mixing

Dissipation process is random. Rate of momentum loss fluctuates



normal
log

is


C
p















2
ln
2
2
C
C
dt
C
dC
dW
C
C

Fluctuations




do not change the time
-
dependence,
h ~

gt
2




influence the pre
-
factor
(
h /gt^
2)



long tails re
-
scale the mean significantly

vdt
dh





d
~
d
dv
dt
g


~
d


dt
h
v
C
2
dM

is stochastic process, characterized by

time
-
scale

and stationary distribution

C



t
C


C
p
0

C


C
p
is non
-
symmetric:

C
mean

mode

max
C

std

If

with M. Cadjan, S. Fedotov, Phys Letters A, 2007

Statistical properties of RT mixing



The value of
a


h /g
r/r
t
2

is a sensitive parameter

 P 


a


t
/

uniform distribution


log
-
normal distributions

max
C
C

0


C
3
max
0
C


1




C
max
C
C

0



0
5
.
2



sustained

acceleration

probability density function at distinct moments of time

Statistical properties of RT mixing


The rate of momentum loss is




statistically steady

m
m

P
~
0
5
.
2



P

P
P

a

~

p
(
a
)


sustained

acceleration


log
-
normal

distribution

1




C
100
,
50
,
1


t
Statistical properties of RT mixing



The value of
a


h /g
r/r
t
2

is a sensitive parameter



Asymptotically, its statistical properties retain time
-
dependency.



The length
-
scale is not well
-
defined

 P 


a


t
/

time
-
dependent

acceleration


turbulent diffusion


uniform and

log
-
normal

distribution

probability density function at distinct moments of time

Statistical properties of RT mixing


The ratio between the momentum rates is





statistically steady for any type of acceleration



a robust parameter to diagnose

m
m

P
~
P
P

~

p
(
a
)


P

a

time
-
dependent

acceleration


turbulent diffusion


log
-
normal

distribution

Is there a true “alpha”?

Our results show that the growth
-
rate parameter alpha is significant

not because it is “deterministic” or “universal,”

but because the value of this parameter is rather small.


Found in many experiments and simulations,

the small alpha implies that in RT flows

almost all energy induced by the buoyant force dissipates,

and a slight misbalance between the rates of momentum loss and gain

is sufficient for the mixing development.


Monitoring the momentum transport is important

for grasping the essentials of the mixing process.


To characterize this transport, one can choose the

rate of momentum loss
m

(sustained acceleration) or

parameter
P

(time
-
dependent acceleration)


To monitor the momentum transport
,

spatial distributions of the flow quantities should be diagnosed
.

RT mixing: coherence and randomness

Turbulent mixing is “disordered.”

However, it is more ordered compared to isotropic turbulence



2
/
1
~
L
l
v
v
l


3
/
1
~
L
l
v
v
l
non
-
equilibrium RT flow

Kolmogorov turbulence

Group theory approach:






Abarzhi etal 1990
th

In RT flows, coherent structures with hexagonal symmetry are

the most stable and isotropic. Self
-
organization may potentially occur.


Laminarization of accelerated flows is known in fluid dynamics.


How to impose proper initial perturbation?


Faraday waves (Faraday, Levinsen, Gollub) can be a solution

Is a “solid body acceleration” an asymptotic state of RT flows?

Tight control over the experimental conditions is required.

Diagnostics of non
-
equilibrium turbulent processes

Basic invariant, scaling, spectral and statistical properties of

non
-
equilibrium turbulent flows depart from classical scenario.



In Kolmogorov turbulence, energy dissipation rate is statistic invariant,


rate of momentum loss is not a diagnostic parameter.



Energy is conjugated with time, momentum is conjugated with space.



In accelerated RT flow, the rate of momentum loss is the


basic invariant, whereas energy dissipation rate is time
-
dependent.



Spatial distributions of the turbulent flow quantities should be


diagnosed for capturing the transports of mass, momentum and energy


in non
-
equilibrium turbulent flows.



In classical turbulence, the signal is one (few) point measurement


with detailed temporal statistics

Verification and Validation



Metrological tools available currently for fluid dynamics community


do not allow experimentalists to perform a


detailed quantitative comparison with simulations and theory:


qualitative observations, indirect measurements, short dynamic range …



The situation is not totally hopeless.



Recent advances in high
-
tech industry unable the
principal

opportunities


to perform the high accuracy measurements of turbulent flow quantities,


with high spatial and temporal resolution, over a large dynamic range,


with high data rate acquisition.



some of potential approaches are being discussed at TMB
-
2009


holographic data storage technology

Conclusions



We suggested a phenomenological model to describe the


non
-
equilibrium turbulent mixing induced by Rayleigh
-
Taylor instability.



The model describes the invariant, scaling and spectral properties of the flow, and


considers the effects of randomness, turbulent diffusion, …

Theory:

Results:



The model can be potentially applied for other flows



The results can be applied for a design of experiments and



for numerical modeling (sub
-
grid
-
scale models)



Rigorous theory is being developed. New experiments are attempting to launch.

Works in progress:



Non
-
equilibrium turbulent flows are driven by transports of mass, momentum



and energy, whereas isotropic turbulence is driven by energy transport.



The invariant, scaling, spectral properties and statistical properties of the


non
-
equilibrium turbulent flows depart from classical scenario.



In RT mixing flow, the rate of momentum loss is the basic invariant,




the energy dissipation rate is time
-
dependent.



The ratio between the rates of momentum loss and gain is time and scale
-



invariant and statistically steady, for sustained and/or time
-
dependent acceleration.