Turbulent mixing and beyond

1
Turbulent mixing and beyond
Snezhana I. Abarzhi
Many thanks to K.R. Sreenivasan (ICTP), L.
Kadanoff (U-Chicago) R. Rosner (ANL), S.I.
Anisimov (Landau Institute)
The work was partially supported by NRL (Steve
Obenschain) and DOE/NNSA
2
Turbulence

• 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
• a generic problem in fluid dynamics.
• Its comprehension can extend our knowledge
  beyond the limits of
• idealized consideration of isotropic
  homogeneous flows.

3
Rayleigh-Taylor instability
Fluids of different densities are accelerated
against the density gradient. A turbulent mixing
of the fluids ensues with time.

• 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

RT flow is non-local, inhomogeneous, anisotropic
and accelerated. Its properties differ from those
of the Kolmogorov turbulence.
Grasping essentials of the mixing process is a
fundamental problem in fluid dynamics.
How to quantify these flows reliably? Is a
primary concern for observations.
4
Rayleigh-Taylor instability
Water flows out from an overturned cup Lord
Rayleigh, 1883, Sir G.I. Taylor 1950
P0 105Pa, P r g h rh 103 kg/m3, g 10
m/s2 h 10 m
5
The Rayleigh-Taylor turbulent mixing
Why is it important to study?
6
Photo-evaporated molecular clouds
Stalactites? Stalagmites? Eagle Nebula.
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
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.
7
Supernovae
Supernovae and remnants 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
Pair of rings of glowing gas, caused perhaps by
a high energy radiation beam of radiation,
encircle the site of the stellar explosion.
Burrows, ESA, NASA,1994
8
Inertial confinement fusion

• 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
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Inertial Confinement Fusion
Nike, 4 ns pulses, 50 TW/cm3 target 1 x 2
mm perturbation 30mm, 0.5 mm
Aglitskii, Schmitt, Obenschain, et al,
DPP/APS,2004
10
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 nano-scales
Zhakhovskii, Zybin, Abarzhi, Nishihara,
Remington, DPP, DFD/APS 2005
11
Solar and Stellar Convection
Solar surface, LMSAL, 2002
Simulations of Solar convection Cattaneo et al, U
Chicago, 2002

• Observations indicate
• dynamics at Solar surface is governed by
  convection in the interior.
• Simulations show
• Solar non-Boussinesq convection is dominated by
  downdrafts which are either large-scale vortices
  (wind) or smaller-scale plumes (RT-spikes).

12
Non-Boussinesq turbulent convection
Thermal Plumes and Thermal Wind
Sparrow 1970 Libchaber et al 1990s Kadanoff et
al 1990s
Sreenivasan et al 2001 helium T4K Re 109, Ra
1017

• The non-Boussinesq convection and RT mixing may
  differ as
• thermal and mechanical equilibriums, or as
  entropy and density jumps

13
Non-premixed and premixed combustion

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

14
Turbulent mixing induced by the Rayleigh-Taylor
instability
What is known and unknown?
15
Rayleigh-Taylor evolution

• linear regime

• nonlinear regime
• light (heavy) fluid penetrates
• heavy (light) fluid in bubbles (spikes)

• turbulent mixing

• RT flow is
• characterized by
• large-scale structure
• small-scale structures
• energy transfers to
• large and small scales

16
Nonlinear Rayleigh-Taylor / Richtmyer-Meshkov
Krivets Jacobs Phys. Fluids, 2005

• large-scale dynamics
• is sensitive to the
• initial conditions
• small-scale dynamics
• is driven by shear

17
Rayleigh-Taylor turbulent mixing
Dimonte, Remington, 1998
3D perspective view (top) and along the
interface (bottom)

• internal structure of
• bubbles and spikes

18
Rayleigh-Taylor turbulent mixing
FLASH 2004 3D flow density plots
broad-band initial perturbation
small-amplitude 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.
19
Unsteady turbulent processes
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 rotating and
  reactive flows and
• for other applications

20
How to model turbulent processes in unsteady
(multiphase) flows?
Any physical process is governed by a set of
conservation laws conservation of mass,
momentum, angular momentum and energy
Kolmogorov turbulence transport of kinetic
energy isotropic, homogeneous
Accelerating flows transports of momentum
(mass), anisotropic, inhomogeneous potential and
kinetic energy
Rotating flows transports of angular
momentum momentum, mass, potential and
kinetic energy
Unsteady turbulent mixing induced by the
Rayleigh-Taylor is driven by the momentum
transport
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Modeling of RT turbulent mixing
Dynamics balance per unit mass of the rate of
momentum gain and the rate of momentum loss
These rates are the absolute values of vectors
pointed in opposite directions and parallel to
gravity.
buoyant force
rate of momentum gain
rate of potential energy gain
dissipation force
rate of momentum loss
energy dissipation rate e dimensional
Kolmogorov
L is the flow characteristic length-scale,
either horizontal l or vertical h
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Asymptotic dynamics

• characteristic length-scale is horizontal L
  l nonlinear

• characteristic length-scale is vertical L
  h turbulent

a 0.1
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Accelerated turbulent mixing

• The turbulent mixing develops
• horizontal scale grow with time l gt2
• 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.

24
Unsteady turbulent flow
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)
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Basic concept for the RT turbulent mixing

• The dynamics of momentum and energy depends on
  directions.
• There may be transports between the planar to
  vertical components.
• It may have a meaning to write the equations in
  4D for momentum-energy tensor
• and study their covariant and invariant
  properties in non-inertial frame of reference.

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Invariant properties of the RT turbulent mixing
RT turbulent mixing
Kolmogorov turbulence
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Scaling properties of the RT turbulent mixing
RT turbulent mixing
Kolmogorov turbulence
transport of momentum
transport of energy

• similarly dissipative scale, surface tension

28
Spectral properties of RT mixing flow
What is the set of orthogonal functions?
These properties have not been diagnosed.
The difference with Kolmogorov and/or
Obukhov-Bolgiano is substantial.
29
Time-dependent acceleration, turbulent diffusion
The transport of scalars (temperature or
molecular diffusion) decreases the buoyant force
and changes the mixing properties
with A. Gorobets, K.R. Sreenivasan, Phys Fluids
2005
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Asymptotic solutions and invariants
buoyancy g dr/r vs time t
dimensionless units

• Buoyancy g dr/r vanishes asymptotically with
  time.
• Parameter P is time- and scale-invariant
  value, and
• the flow characteristics

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Randomness of the mixing process
Some qualitative features of RT mixing are
repeatable from one observation to another. As
any turbulent process, Rayleigh-Taylor turbulent
mixing has essentially noisy character. Kolmogor
ov turbulence RT turbulent mixing Noisiness
reflects the random character of the dissipation
process. velocity fluctuates velocity and
length scales fluctuate energy dissipation rate
is invariant energy dissipation rate grows with
time

• 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 value, and
  fluctuates about its mean.

• Observations focus on the diagnostics of
  integral scale and
• do not provide necessary information on
  dissipation statistics.

with M. Cadjan, S. Fedotov, Phys Letters A, 2007
32
Stochastic model of RT mixing
Dissipation process is random. Rate of momentum
loss fluctuates
If

• Fluctuations
• do not change the time-dependence, h gt2
• influence the pre-factor (h /gt2)
• long tails re-scale the mean significantly

33
Statistical properties of RT mixing
lt P gt
sustained acceleration
lt a gt
t/t
uniform distribution log-normal distributions

• The value of a h /g(dr/r)t2 is a very
  sensitive parameter

34
Statistical properties of RT mixing
probability density function at distinct moments
of time
P(P)
p(a)
sustained acceleration log-normal distribution
a
P
The rate of momentum loss is
statistically steady
35
Statistical properties of RT mixing
lt P gt
time-dependent acceleration turbulent
diffusion uniform and log-normal distribution
lt a gt
t/t

• The value of a h /g(dr/r)t2 is very
  sensitive parameter
• Asymptotically, its statistical properties are
  very sensitive
• to noise and retain a time-dependence.
• The length-scale is not well-defined

36
Statistical properties of RT mixing
probability density function at distinct moments
of time
p(a)
P(P)
time-dependent acceleration turbulent
diffusion log-normal distribution
a
P

• The ratio between the momentum rates is
• statistically steady for any type of
  acceleration
• a robust parameter to diagnose

37
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.
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RT mixing between order and disorder ?
Turbulent mixing is disordered. However, it is
more ordered compared to isotropic turbulence
Is a solid body acceleration being the
asymptotic state of RT unsteady flow?
Group theory approach, Abarzhi et al 1990s In
RT flow, the coherent structures with hexagonal
symmetry are the most stable and isotropic.
Self-organization may potentially occur. How to
impose proper initial perturbation? Faraday
waves (Faraday, Levinsen, Gollub) can be a
solution
This imposes very high requirements on the
precision and accuracy in the experiments.
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Diagnostics of unsteady turbulent processes
Basic invariant, scaling and spectral properties
of accelerated mixing differ from those in the
classical Kolmogorov turbulence.

• In Kolmogorov turbulence, energy dissipation
  rate is statistic invariant,
• rate of momentum loss is not a diagnostic
  parameter.

• In unsteady turbulent flow, the rate of momentum
  loss is the basic invariant,
• whereas energy dissipation rate is
  time-dependent.

• Energy is complimentary to time, momentum is
  complimentary to space.

• In classical turbulence, the signal is one (few)
  point measurement
• with detailed temporal statistics

• Spatial distributions of the turbulent flow
  quantities should be
• diagnosed for capturing the transports of
  momentum and energy
• in non-Kolmogorov turbulence

40
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.

• a new research platform is attempting to launch
  for studies of turbulence
• in unsteady turbulent flows
• leverage existing technology, the unique
  experimental facility,
• holographic data storage http//en.wikipedia.o
  rg/wiki/HDSS

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Conclusions
The model

• We suggested a phenomenological model to
  describe the
• unsteady turbulent mixing induced by
  Rayleigh-Taylor instability.
• The model describes the invariant, scaling and
  spectral properties of the flow.
• The model considers the effects of randomness,
  turbulent diffusion,

The results

• Unsteady turbulent flow is driven by the
  transport of momentum,
• whereas isotropic turbulence is driven by
  energy transport.
• The invariant, scaling, spectral properties and
  statistical properties of the
• accelerating mixing flow differ from those in
  Kolmogorov turbulence.
• The rate of momentum loss is the basic invariant
  of the accelerated flow,
• the energy dissipation rate is time-dependent.
• The ratio between the rates of momentum loss and
  gain is time and scale-
• invariant, for sustained and/or time-dependent
  acceleration

Works in progress

• The model can be applied for rotating,
  compressible and reactive flows
• The results of the model can be applied for a
  design of experiments and
• for numerical modeling (sub-grid-scale models)
• Rigorous theory is on the way. New experiments
  are attempting to launch.