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.
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