Turbulent mixing and beyond

Many thanks to: K.R. Sreenivasan (ICTP), L. Kadanoff (U-Chicago) R. Rosner (ANL), S.I. Anisimov (Landau Institute)

Snezhana I. Abarzhi

The work was partially supported byNRL (Steve Obenschain) and DOE/NNSA

Turbulenceis 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.

Rayleigh-Taylor instabilityFluids of different densities are accelerated against

the density gradient.A turbulent mixing of the fluids ensues with time.

Grasping essentials of the mixing process isa 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

P0 = 105Pa, P = g hh ~ 103 kg/m3, g ~ 10 m/s2

h ~ 10 m

Water flows out from an overturned cup

Lord Rayleigh, 1883, Sir G.I. Taylor 1950

h

l

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.

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 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

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 spotNishihara, ILE, Osaka, Japan, 1994

Inertial Confinement Fusion

STREAKCAMERA

QUARTZCRYSTAL

1.86 keVimaging

2D IMAGE

MAIN LASER BEAMS

Tim

e

Magnification x20Magnification x20

BACKLIGHTERLASER BEAMS

BACKLIGHTERTARGET Si

RIPPLEDCH TARGET

Rayleigh-TaylorImprintRichtmyer-MeshkovFeedout

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

Nike, 4 ns pulses, ~ 50 TW/cm3

target: 1 x 2 mm;perturbation: ~30m, ~0.5 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 refractsthough the liquid-liquid (up) and solid-solid (down) interfaces; nano-scales

Zhakhovskii, Zybin, Abarzhi, Nishihara, Remington, DPP, DFD/APS 2005

Solar and Stellar Convection

Solar surface, LMSAL, 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”).

Simulations of Solar convectionCattaneo et al, U Chicago, 2002

Non-Boussinesq turbulent convection

Thermal Plumes and Thermal Wind

heating

cooling

liquid

instabilities

instabilities

Sparrow 1970 Libchaber et al 1990sKadanoff 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 2001helium T~4K Re ~ 109, Ra ~ 1017

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

Turbulent mixing induced bythe Rayleigh-Taylor instability

What is known and unknown?

g

~h

h

l

Rayleigh-Taylor evolution

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

thh exp~ 0• linear regime

,~ lhlh g

• turbulent mixing

hlhgth 2~

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

max~

Krivets & JacobsPhys. Fluids, 2005

5.027.1 AM

Nonlinear Rayleigh-Taylor / Richtmyer-Meshkov

mstcm ~6~

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

Rayleigh-Taylor turbulent mixing

Dimonte, Remington, 1998

cmLcmLggA zyx 8.8 ,3.7 ,73 ,2.0 0

maxmax ~ ,~ Omm

54 10Re,10 We

3D perspective view (top) andalong the interface (bottom)

• internal structure of bubbles and spikes

Rayleigh-Taylor turbulent mixing

broad-bandinitial

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 20043D flowdensity plots

Our phenomenological model

identifiesthe new invariant, scaling and spectral properties of

the accelerated turbulent mixingaccounts for

the multi-scale and anisotropic character of the flow dynamicsrandomness of the mixing process

discusseshow to generalize this approach for rotating and reactive flows andfor other applications

Unsteady turbulent processes

How to model turbulent processesin 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 energyisotropic, 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

Modeling of RT turbulent mixing

Dynamics: balance per unit mass of the rate of momentum gainand the rate of momentum loss

L is the flow characteristic length-scale, either horizontal or vertical h

g

~

rate of momentum gain v ~~

gv~rate of potential energy gain

buoyant force

energy dissipation rate dimensional & Kolmogorov

LvC 3

rate of momentum loss v dissipation force

,vdtdh ~

dt

dv

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

Asymptotic dynamics

2 ~ ~ 2tghtgv

• characteristic length-scale is vertical L ~ hturbulent

ga1

tgaa 21

g~

tga 2~ a ~ 0.1

gthgv ~ ~

• characteristic length-scale is horizontal L ~ nonlinear

2123~ g 2123~~ g

g~ g

22 tgha

The turbulent mixing develops:

• horizontal scale grow with time ~ 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.

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~,~ gtLtgv

~

Unsteady turbulent flow

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

Lv2~ g~

• rates of gain of potential energy gain and dissipation of kinetic energy are time-dependent

vg~ Lv3~

remains time- and scale-invariantfor 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 characteristic value of the flow

~~

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.

RT turbulent mixing is non-inertial (accelerated),anisotropic, non-local and inhomogeneous.

lvLv l22 ~~• 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.

lvLv l33 ~~

Invariant properties of the RT turbulent mixing

Kolmogorov turbulence RT turbulent mixing

vv • helicity

223 ~~~ LvvLLvvLv• energy dissipation rate

energy transport and inertial interval

time- and scale-invariant time-dependent

transport of momentumnot a diagnostic parameter time- and scale-invariant

• enstrophy 22 v

• rate of momentum loss

lvLv l22 ~~

time- and scale-invariant time-dependent

not-Galilean invariant ~g, time- and scale-inv

Scaling properties of the RT turbulent mixing

Kolmogorov turbulence RT turbulent mixing

lvLv l22 ~~

transport of energy

lvLv l33 ~~

transport of momentum

• scaling with Reynolds

32~~Re tgvL

2/3Re~Re Lll

constvL ~~Re

3/4Re~Re Lll

2/1~ Llvvl 3/1~ Llvvl

• local scaling

more ordered

3/123/12 ~~ gl

• viscous scale

4/13~ lmode of fastest growth

• similarly: dissipative scale, surface tension

Spectral properties of RT mixing flow

These properties have not been diagnosed.

spectrum of kinetic energy (velocity)

23/23/2

3/23/53/2 ~~~~ l

kk

vlk

dkkdkkE

kinetic energy =

Kolmogorov turbulence:

• spectrum of kinetic energy

kinetic energy = 22 ~~~~ lkk

vlk

dkkdkkE

• spectrum of momentum

momentum = lkk

vlk

dkkdkkM eeeee ~~~~ 2/12/1

2/12/32/1

RT turbulent mixing:

The difference with Kolmogorov and/or Obukhov-Bolgiano is substantial.

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

TT ~

,vdtdh ,

2

hv

Cgdtdv 2

hv

Cdtd

t

dynamical system

Rate of temperature change is

T~We assume

T 2T

22 ~~ TLvLTvL Landau & Lifshits

asymptotic solution

t 0 1exp~h 022 ln~ hgtgth

with A. Gorobets, K.R. Sreenivasan, Phys Fluids 2005

Asymptotic solutions and invariants

Buoyancy g vanishes asymptotically with time. Parameter is time- and scale-invariant value, and the flow characteristics

buoyancy g vs time t vs time t ~

0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

/

t / 0

Ct=3

Ct=0~

dimensionless units

0 5 10 150

0.05

0.1

0.15

0.2

0.25

/

t / 0

Ct=3

Ct=0

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.

Kolmogorov turbulence RT turbulent mixing

Noisiness reflects the random character of the dissipation process.

velocity fluctuates velocity and length scales fluctuateenergy dissipation rate is invariant energy dissipation rate grows with time

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

• 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.

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

Stochastic model of RT mixing

Dissipation process is random. Rate of momentum loss fluctuates

normallog is Cp

2ln

2 2

C

Cdt

C

dCdW

CC

Fluctuations do not change the time-dependence, h ~gt2 influence the pre-factor (h /gt^2) long tails re-scale the mean significantly

vdtdh d~ddv dtg~d dthvC 2dM

is stochastic process, characterized bytime-scale and stationary distributionC

tC Cp

0C

Cp is non-symmetric: Cmean mode maxC std

If

Statistical properties of RT mixing

The value of ah /gt2 is a very sensitive parameter

a

t

uniform distribution

log-normal distributions

maxCC 0C3max0 C

1C maxCC 0 05.2

sustainedacceleration

probability density function at distinct moments of time

Statistical properties of RT mixing

The rate of momentum loss is statistically steady

~

05.2

P

a

~p(a) sustained

acceleration

log-normaldistribution

1C

100,50,1t

Statistical properties of RT mixing

The value of ah /gt2 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

a

t

time-dependentacceleration

turbulent diffusion

uniform and log-normaldistribution

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

~

P

~p(a)

a

time-dependentacceleration

turbulent diffusion

log-normaldistribution

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 (sustained acceleration) or

parameter (time-dependent acceleration)

To monitor the momentum transport, spatial distributions of the flow quantities should be diagnosed.

RT mixing: between order and disorder ?

Turbulent mixing is “disordered.” However, it is more ordered compared to isotropic turbulence

2/1~ Llvvl 3/1~ Llvvl

unsteady turbulent flowKolmogorov turbulence

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

Is a “solid body acceleration” being the asymptotic state of RT unsteady flow?

This imposes very high requirements on the precision and accuracy in the experiments.

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.

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

In unsteady turbulent 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 momentum and energy in non-Kolmogorov turbulence

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.

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.org/wiki/HDSS

Conclusions

• 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 model

The results

• 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.

Works in progress

• 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