Turbulence and turbulent mixing in accelerating and rotating flows

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

S.I. Abarzhi

FLASH Center, The University of Chicago

The work is partially supported byDOE/NNSA and NRL

Turbulence and turbulent mixing in accelerating and rotating fluids

occurs in a wide variety of physical phenomena, ranging from astrophysical to micro-scales, and in industrial applications

supernova explosion, thermonuclear flashes, photo-evaporated cloudsstellar convection, dynamo, mantle-lithosphere tectonicsinertial confinement fusion, plasmas, laser-material interaction magnetic fusion, target cooling by flowing liquid metal in acceleratorsZ-pinch extreme-ultra violet sources for semiconductor industry lithographic tools swirling flows in aerospace industry, accelerating boundary layers in aeronauticsgas-turbine engines, flames and fires, oil productionimpact dynamics of liquids and solids, material properties under high strain rate

The turbulent flows are inhomogeneous and highly anisotropic.Their properties are substantially non-Kolmogorov.

Grasping essentials of turbulent processesin accelerating and rotating fluids is a

fundamental problem in fluid dynamics.

•Theory: description of statistical and spectral properties of accelerating and rotating turbulent flowsis a real challenge, which requires a development of new theoretical concepts.

•Computations: are met with tremendous difficulties, asthe numerical solutions are very sensitive to the initial and boundary conditions as well as to the influence of small-scale structures

•Experiments: the flows are difficult to implement in well-controlled laboratory environment, andare very hard to quantify and measure with high accuracydue to limitations of available diagnostics and metrology.

“How to quantify these complex flows reliably?”

suggest new theoretical concept (rate of momentum loss) to describe accelerated turbulent mixing induced by the Rayleigh-Taylor instabilitydiscuss how to generalize this approach for rotating and reactive flowsdiscuss new research program (experimental facility, integrated platform)

Rayleigh-Taylor instability

P0 = 105Pa, P = ρ g hρh ~ 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

ρh

ρl

Turbulence and turbulent mixing in accelerating and rotating fluids

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 shouldbe hot and dense plasma

• For the plasma compression in the laboratory it is usedmagnetic implosionlaser implosionof DT targets

• RMI/RTI inherently occur during the implosion process

• RT turbulent mixing prevents the formation of hot spot

Nishihara, ILE, Osaka, Japan, 1994

ICF

STREAKCAMERA

QUARTZCRYSTAL1.86 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/cm2

target: 1 x 2 mm;perturbation: ~30µm, ~0.5 µm

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 et al DPP, DFD/APS 2005

Solar and Stellar ConvectionSolar surface, LMSAL, 2002

The observations indicatethe dynamics at Solar surface is governed by the convection in the interiorthe length scales of granules and super-granules depend on the vertical scalesThe simulations show Solar non-Boussinesq convection is dominated by downdraftsThe downdrafts are similar to either as huge large-scale vortices (thermal wind) or to small-scale spikes (thermal plumes or RT-spikes)

Simulations of Solar convectionCattaneo et al, 2002

Non-Boussinesq turbulent convectionThermal 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 differ asthermal wind (vortices) and thermal plumesthermal and mechanical equilibriums, or as entropy and density jumps

Sreenivasan et al 2001helium T~4K Re ~ 109, Ra ~ 1017

Boundaries play an important role.

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 KHbuoyancy-driven RT

hydrogen and methane

1 m base

Accelerated turbulent mixing induced bythe Rayleigh-Taylor instability

What is known and what is unknown?

g

λ

~h

ρh

ρl

Rayleigh-Taylor evolution

( )τλ th ~• nonlinear regimelight (heavy) fluid penetratesheavy (light) fluid in bubbles (spikes)

( )τthh exp~ 0• linear regime( ) ( ),~ lhlh g ρ−ρρ+ρλτ

• turbulent mixing ( ) hlhgth ρρ−ρ2~

RT flow is characterized by: large-scale structuresmall-scale structuresenergy transfers to large and small scales

max~ λλ

Rayleigh-Taylor turbulent mixing

Dimonte, Remington, 1998

( ) cmLcmLggA zyx 8.8 ,3.7 ,73 ,2.0 0 ====

( )maxmax ~ ,~ λΛλ Omm

54 10Re,10 <<We

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

• internal structure ofbubbles and spikes

Rayleigh-Taylor turbulent mixing

broad-bandinitial

perturbation

The mixing flow is sensitive to the horizontal boundaries of the fluid tank (computational domain) though much less so to the vertical,

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 mixing

accounts forthe anisotropic, inhomogeneous and random character of the turbulent process

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

Turbulent and turbulent mixingin accelerating and rotating fluids

Modeling of turbulent and turbulent mixingin accelerating and rotating fluids

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 momentummomentum, mass,potential and kinetic energy

Accelerating turbulent mixing induced by the Rayleigh-Taylor is driven by the momentum transport

Modeling of RT turbulent mixingDynamics: 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 λ or vertical h

ρδρ

=µ~

rate of momentum gain vε=µ ~~

ρδρ=ε gv~rate of potential energy gain

buoyant force

energy dissipation rate εdimensional & Kolmogorov

LvC 3=ε

rate of momentum loss vε=µ dissipation force

,vdtdh

= µ−µ= ~dtdv

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

Asymptotic dynamics

2~ ~ gthgtv

• characteristic length-scale is vertical L ~ h turbulent

( ) ρδρ−=µ ga1

( )( ) tgaa 21 ρδρ−=ε

ρδρ=µ g~

( ) tga 2~ ρδρ=ε

The mixing develops:

• 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 interface.

a ~ 0.1

Accelerated mixing flow• rates of momentum gain and momentum loss are scale and time invariant

hv2~µρδρ=µ g~

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

ρδρ=ε vg~ hv3~ε

Π is time- and scale-invariant for time-dependent and spatially-varying acceleration, as long as potential energy is a similarity function of the coordinate and time

• ratio between the rates is the characteristic value of the flow

εε=µµ=Π ~~ Fr~Π

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 tensorand study their covariant and invariant properties in non-inertial frame of reference.

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

lvLv l22 ~~µ• 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 thedirection of gravity.

Kolmogorov turbulence is Galilean-invariant,isotropic, local and homogeneous.

• Energy dissipation rate is the basic invariant, determines the scaling properties of the turbulent flow.

lvLv l33 ~~ε

Invariant properties of the RT turbulent mixingKolmogorov turbulence RT turbulent mixing

( )vv ×∇⋅• helicity

( ) ( )( )223 ~~~ LvvLLvvLvε• energy dissipation rate

energy transport and inertial intervaltime- and scale-invariant time-dependent

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

• enstrophy( )22 v×∇=ω

• rate of momentum loss

lvLv l22 ~~µ

time- and scale-invariant time-dependent

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

Scaling properties of the RT turbulent mixingKolmogorov turbulence RT turbulent mixing

lvLv l22 ~~µ

transport of energy

lvLv l33 ~~ε

transport of momentum

• scaling with Reynolds

( ) νν 32~~Re tgvL( ) 2/3Re~Re Lll

( ) constvL ~~Re ν

( ) 3/4Re~Re Lll

( ) 2/1~ Llvvl( ) 3/1~ Llvvl

• local scaling

more ordered

( ) ( ) 3/123/12 ~~ gl νµν

• viscous scale

( ) 4/13~ ενlmode of fastest growth

• similarly: dissipative scale, surface tension

Spectral properties of RT mixing flow

• spectrum of kinetic energy

kinetic energy = ( ) 22 ~~~~ lkk

vlk

dkkdkkE µµ

µ∫∫∞

−∞

• spectrum of momentum

momentum = ( ) ( ) lkk

vlk

dkkdkkM eeeee ~~~~ 2/12/1

2/12/32/1 µ

µµ∫∫

∞−

∞

These properties have not been diagnosed.

spectrum of kinetic energy (velocity)

( ) ( ) 23/23/2

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

kkvl

kdkkdkkE ε

εε∫∫

∞−

∞kinetic energy =

Kolmogorov turbulence:

RT turbulent mixing:

Time-dependent acceleration, turbulent diffusionThe transport of scalars (such as temperature or molecular diffusion) decreases the buoyant force and changes the mixing properties

TTδρδρ ~

,vdtdh

= ,2

hvCg

dtdv

−Θ= 2Θ−=Θ

hvC

dtd

t ρδρ=Θ

dynamical system

Rate of temperature change is

T~ρWe assume

Tϕ ( )2T∇χ=ϕ

( )( ) ( )( )22 ~~ TLvLTvL δδϕLandau & Lifshits

asymptotic solution

∞→t 0→Θ ( )Θ1exp~h ( )022 ln~ hgtgth

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

Asymptotic solutions and invariants

µµ ~Buoyancy g δρ/ρ vanishes asymptotically with time.The ratio is time and scale invariant value, and this is the flow characteristics

buoyancy g δρ/ρ vs time t vs time tµµ ~

0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

µ /

µ

t / τ0

Ct=3Ct=0~

dimensionless units

0 5 10 150

0.05

0.1

0.15

0.2

0.25

δρ/ ρ

t / τ0

Ct=3Ct=0

Randomness of the mixing processThe Rayleigh-Taylor turbulent mixing has a noisy character.

• No detailed measurements are available from the observations.• Rigorous analysis of statistical properties of the governing equations is

a real challenge and a subject for future studies.

• This randomness is caused by the retained memory of the (unresolved) initial conditions and the influence of shear-driven small-scalestructures, whose dynamics is governed by the density ratio.

• The rate of momentum loss is determined by random dissipative processes and fluctuates about its mean.

• We consider the influence of these fluctuations on the mixing process and show that the mixing growth-rate is very sensitive to the stochastic effects

with M. Cadjun, S. Fedotov, L. Kadanoff, in preparation

Stochastic modeling

( ) ( )( )220 2lnlnexp

21

σ−−πσ

= CCC

Cp ( )2exp 20 σ= CC

hvC 2=µThe value of is statistically steady.

dynamical system

,dtvdh = ,2

dthvCgdtdv −

ρδρ

= ⎟⎟⎠

⎞⎜⎜⎝

⎛ σ−

τ+=

τσ

2ln2 2

CCdt

CdCdW

CC

the fluctuations intensity is σ the fluctuations time-scale is τc

In the asymptotic solution h = a (δρ/ρ) gt2

the fluctuations do not change the exponent but may change the pre-factor: long tails re-scale the mean

Its distribution is in general non-symmetric.

Statistical properties of RT mixing

The value of a = h /gt2 is very sensitive diagnostic parameter.Its statistical properties change significantly with timeTo prove the universality of scaling h = a(δρ/ρ)gt2 as an empirical law, at least 303=27,000 data points are required

position h/gt2 probability density function h/gt2

vs time tµµ ~probability density function for

Statistical properties of accelerated flows

The ratio between the rates of momentumis statistically steady for any type of accelerationis robust diagnostic parameter

sustained acceleration time-dependent acceleration

gLv2~~µµ

Application and generalization of the model

Inhomogeneous, accelerated, anisotropic turbulent flows

Boundary layers (accelerated and decelerated)

Turbulent combustion in gravity field (premixed and non-premixed)

Compressible turbulent flows

Turbulence and turbulent mixing in rotating fluids

Large-eddy simulations and sub-grid models for accelerated flow

Covariant, invariant and scaling dynamics ofnon-inertial, non-Galilean turbulent flows

4D for momentum-energy tensor transports analogy with general relativity theorytwin paradox ☺

Diagnostics of accelerating and rotating fluidsBasic invariant, scaling and spectral properties of accelerating

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 related to time, momentum is related to space.

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

Spatial properties of the accelerated mixing (in situvelocity density fluctuations, local acceleration, integral dynamics, etc are essential to monitor.

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

Experiments and Validation•• Current Status:Current Status:

Metrological tools currently employed in fluid dynamics experiments do not allow direct quantitative comparison with simulations and theory:

• qualitative observations and indirect measurements• integral quantities• limited spatial and temporal resolutions, small dynamic range• low data acquisition rate and poor statistics…

•• Future Directions:Future Directions:Recent advances in high-tech industry enable the principal opportunities

to measure the turbulent flow quantities with high accuracy, high spatial and temporal resolutions, large dynamic range, and good statistics at high data acquisition rates.

model experiments allowing direct comparison with theory and simulationsnew research platform is being launched for studies of turbulence in accelerating, rotating, and reactive flows (S. Orlov, Stanford)high performance and low cost: leverage existing state-of-the-art technology, unique facility, developed for high-end data storage and holography (HDSS).http://en.wikipedia.org/wiki/HDSS

Kodak C7 1000 fps CCD

photopolymer disk/chamber with rotating fluids

high speed spindle

Optical shaft encoder

IBM FLC1000 fps SLM

Double Fouriertransform imaging

Pulsed Nd:YAG laser (not shown)

Experimental capabilitiesBackground:Background:

• $52M 6-year long DARPA HDSS program

• Final deliverable:1 Gb/s holographic disk storage system (Stanford)

• 10 Gb/s optical data rate fastest optical storagedevice ever built

• State-of-the-art technology in optics,mechanics, electronics

Capabilities:Capabilities:• Rotation rate 0-10,000 RPM, Max g: 12,000g, Reynolds >107

• Timing accuracy <10 ns, position accuracy < 1.5 µm • Chamber size ~Ø40 × 30 cm, spatial resolution < 10µm• Frame rate 103 (upto 104) fps, frame size upto 2 Mpixels

HDSS, rotating (7,500 RPM) disk Ø is 165 mm

Experiments in rotating and accelerating fluids

Computer control

Synchronization electronics

high speed digital camera

precision lasershaft encoder

high speedair-bearing spindle

laser sheet forvisualization

rotatingchamber

Pulsed Nd:YAG laser

particles forflow

visualization

imaging data

beam shaping optics

high resolution imaging optics

high precision (25 nm) translation stage

visualization volume

Computer control

Synchronization electronics

high speed digital camera

precision lasershaft encoder

high speedair-bearing spindle

rotatingchamber

Pulsed Nd:YAG laser

particles forflow

visualization

imaging data

interferometer fordensity fluctuations

measurements

high resolution imaging optics

high precision (25 nm) translation stage

illuminated volume

Schematics of PIV for velocimetry (left) and interferometry for density (right)

particles for flowvisualization

precision lasershaft encoder

high-speedair-bearing spindle

accelerating rotation

toroidal chamberwith rectangularcrossection

D

Schematic of toroidal chamber for studying accelerating flows

Conclusions

• We suggested a phenomenological model to describe the accelerated Rayleigh-Taylor turbulent mixing.

• The model describes the invariant, scaling and spectral properties of the flow• The model considers the effects of randomness and diffusion…

The model

The results

• The model can be generalized for rotating and reactive flowsand can account for the transports of mass and angular momentum

• Spatial and temporal distributions of turbulent flow quantitiesshould be monitored

• New integrated research platform is being launched.

Works in progress

• The rate of momentum loss is the basic invariant of the accelerated flowthe energy dissipation rate is time-dependent.

• The invariant, scaling, spectral properties and statistical properties of the accelerating mixing flow differ from those in Kolmogorov turbulence.