# invited lecture - Snezhana I. Abarzhi

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.