Coherence and randomness in nonequilibrium turbulent processes

1
Second International Conference and Advanced
School Turbulent Mixing and Beyond
Coherence and randomness in non-equilibrium
turbulent processes
Abdus Salam International Centre for Theoretical
Physics, Trieste, Italy 27 July 07 August 2009
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
• one of generic problems 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
Dynamics of Continuous Media
Conservation laws Navier-Stokes or Euler
equations
Boundary Conditions Initial Conditions
Isotropy, homogeneity, locality
scaling invariance Kolmogorov 1941
Isotropic homogeneous turbulence, Sreenivasan 1999
6
The Rayleigh-Taylor turbulent mixing
Why is it important to study?
7
Photo-evaporated molecular clouds
Stalactites? Stalagmites? Eagle Nebula. Birth of
a star.
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.
8
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
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
9
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
10
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
11
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
4 106 LJ atoms (2005), 7 106 LJ atoms
(2007), nm, ps 0.2mm ps, for 2 108 atoms on
1.6 105 CPU, IBM BG/L, 48 96 hrs
Zhakhovskii, Zybin, Abarzhi, etal, DPP, DFD/APS,
APS/SCCM 2005, 2006
12
Solar and Stellar Convection
Solar surface, LMSAL, 2003
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 these are either large-scale vortices
  (wind) or smaller-scale plumes (RT-spikes).

13
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

14
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

• 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.

Marmottant and Villermaux, JFM 2004
15
Non-premixed and premixed combustion

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

16
Oil production
17
Technology and Communications
18
Non-equilibrium turbulent processes Rayleigh-Tay
lor Turbulent Mixing
What is known and unknown?
19
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

20
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

21
Nonlinear Rayleigh-Taylor
Density plots in horizontal planes He, Chen,
Doolen, 1999, Lattice Boltzman method
22
Rayleigh-Taylor turbulent mixing
Dimonte, Remington, 1998
3D perspective view (top) and along the
interface (bottom)

• internal structure of
• bubbles and spikes

23
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.
24
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

25
Theory of Rayleigh-Taylor Instability
Conservation laws
no mass flux momentum no mass sources
1883 Rayleigh 1950th Fermi von Newman,
Layzer, Garabedian, Birkhoff 1990th 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.
26
Group-theory based approach
curvature z

• RTI/RMI nonlinear dynamics is essentially
  multi-scale
• amplitude h and wavelength l contribute
  independently

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

27
Non-equilibrium 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 other
  flows/applications

28
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
29
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
30
Asymptotic dynamics

• characteristic length-scale is horizontal L
  l nonlinear

• characteristic length-scale is vertical L
  h turbulent

a 0.1
31
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.

32
Non-equilibrium 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)
33
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.

34
Invariant properties of RT turbulent mixing
RT turbulent mixing
Kolmogorov turbulence
35
Scaling properties of RT turbulent mixing
RT turbulent mixing
Kolmogorov turbulence
transport of momentum
transport of energy

• similarly dissipative scale, surface tension

36
Spectral properties of RT mixing flow
What is the set of orthogonal functions?
Scaling, invariant, spectral properties depart
from classical scenario
37
Time-dependent acceleration, turbulent diffusion
The transport of scalars (temperature or
molecular diffusion) decreases the buoyant force
and changes the mixing properties
38
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

39
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.

40
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

with M. Cadjan, S. Fedotov, Phys Letters A, 2007
41
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 sensitive
  parameter

42
Statistical properties of RT mixing
probability density function at distinct moments
of time
43
Statistical properties of RT mixing

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

44
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
45
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.
46
RT mixing coherence and randomness
Turbulent mixing is disordered. However, it is
more ordered compared to isotropic turbulence
Is a solid body acceleration an asymptotic
state of RT flows?
Group theory approach Abarzhi etal
1990th 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
Tight control over the experimental conditions is
required.
47
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.

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

• Energy is conjugated with time, momentum is
  conjugated with 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
  mass, momentum and energy
• in non-equilibrium turbulent flows.

48
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

49
Conclusions
Theory

• 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,

Results

• 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.

Works in progress

• 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.