Approaches to Turbulence in High-Energy-Density Experiments R. Paul Drake University of Michigan

Approaches to Turbulence in Approaches to Turbulence in High-Energy-Density ExperimentsHigh-Energy-Density Experiments

R. Paul Drake R. Paul Drake University of MichiganUniversity of Michigan

2007 TMBW, TriesteWork supported by the U.S. Department of Energy under grants DE-FG52-07NA28058, DE-FG52-04NA00064, by the Naval Research Laboratory

under contract NRL N00173-06-1-G906, and by other grants and contracts

2007 August Turbulent Mixing and Beyond

Who contributed; where we are going

• Key collaborators (others listed later)– Michigan: C.C. Kuranz, E.C. Harding, M. Grosskopf – LLNL: J.F. Hansen, H.F. Robey, B.A. Remington, A. Miles – Rochester: J. Knauer – Florida State: T. Plewa – NRL: Yefim Aglitskiy

• Outline– How lasers accomplish hydrodynamic experiments – Shock-driven Richtmyer Meshkov– Turbulence?– Blast-wave-driven instabilities & why not yet turbulent – Steepness of shear appears to matter (blast-waves, jets) – An experiment to create a steep shear layer– An experiment having a steep shear layer and turbulence– Some speculation regarding requirements for quick turbulence


High-Energy-Density Physics

• The study of systems in which the pressure exceeds 1 Mbar (= 0.1 Tpascal = 1012 dynes/cm2), and of the methods by which such systems are produced.

• This also matters for astrophysics

• Direct reasons – pressures > 1 Mbar are important in planets – High-temperature dense matter is important in stars

• Indirect reasons– Dynamics of Mach >> 2, very high Re systems

– T >> 1 eV (104 K) and/or = (8p/B2) >> 1 systems

– Radiation hydrodynamic systems

• We work in my group with high Mach number hydrodynamic and radiative hydrodynamic systems


The Omega laser is our major tool at present

Target chamber at Omega laser

There are lots of kJ, ns lasers around the world

Omega is the most capable for experiments.

Omega is at the Laboratory for Laser Energetics, affiliated with the University of Rochester.

Its primary mission is inertial fusion research.


Here is what such lasers do to a material

• The laser is absorbed at less than 1% of solid density

From Drake, High-Energy-Density Physics, Springer (2006)

Rad xport, high-v plas Hydro, rad hydro


Shock waves establish the regime of an experiment


The Mach number in these experiments is effectively infinite

• Sound speeds are ~ 1 km/sec – Exact value depends on upstream “preheating”

• Shock velocities are ~ 50 km/sec

• Mach number terms in shock jump relations are fundamentally present as 1/M2

• Implications – Density jump is ~ (– 1)/(– 1) – Post-shock temperature is ~

– Here , Z are post-shock values • A complication is that they are temperature-dependent.

kBT =

Amp

1+ Z( )

2 −1( )

+1( )2

us2


Here is a drawing of a typical target for hydrodynamics experiments on lasers

• Precision structure inside a shock tube

Experiment design: Carolyn Kuranz


Ungated imaging provides improved resolution in such experiments (side-on schematic)

4 backlighter beams

Delayed 10-30 ns X-ray photons

X-ray filmTargetTitanium/

Scandium backlit pinhole

20 mm Au shield protects film from

overexposure

Drive beams t=0

Credit: Carolyn Kuranz


This shows images of actual targets built at Michigan

Acrylic cone

Gold cone

Laser-driven surface

Side view

1 m

m

Targets: Mike Grosskopf, Donna Marion, Robb Gillespie, UROP team

2007 August Turbulent Mixing and Beyond XTVS YTVS

Alignment in Omega Chamber


Main target positioner

Backlighter stalkBacklighter positioner

Acrylic shield

What a shot looks like in the chamber…


We have greatly improved the resolution and signal-to-noise in the data

Also, the UM-led team was the first group to get simultaneous orthogonal physics data

Mid-1990’sdata

Recent data and analysis: Carolyn Kuranz

Aug. 2005 Dec. 2006

Grid squares have 43 µm openings

Eggcrate + 424 µm sine wave, 10 to 20 µm tapered pinhole, 25 ns

Eggcrate mode only, 20 to 50 µm stepped pinhole, 17 ns


Theoretical considerations for these experiments

• One likes to imagine that these systems are described by the Euler equations

• But do these accurately describe the physical system?

ρ∂ v

∂ t

+ v ⋅ ∇ v ⎛

⎝

⎞

⎠= −∇ p

∂ ρ

∂ t

+ ∇ ⋅ ρ v( ) = 0

∂ p

∂ t

− γp

ρ

∂ ρ

∂ t

+ v ⋅ ∇ p − γp

ρ

v ⋅ ∇ ρ = 0

,

,

,

and

ρ = density v = velocity p = pressure = constant (adiabatic

index)


Is it sufficient to use only hydrodynamics?

• General Fluid Energy Equation:

€

∂∂t

ρε +ρu2

2+ ER

⎛

⎝ ⎜

⎞

⎠ ⎟+∇ ⋅ ρu ε +

u2

2

⎛

⎝ ⎜

⎞

⎠ ⎟+ pu

⎡

⎣ ⎢

⎤

⎦ ⎥=

−J • E+ Fother • u−∇ • FR + pR + ER( )u+Q − σ v • u( )[ ]

Material Energy Flux m

€

m

Pe

€

m

Re

€

m

PeRad

=

τ rad

τ hydro

Γm

Smalleror Hydro-like

€

ν ei

ωpe

~ 1 Typ. small

Or Ideal MHD


Typical parameters in these experiments

• U ~ 10 km/s Range 1 to < 1,000 • Driving scale for turbulence ~ 100 µm Range 10 µm to 1 mm• Kinematic viscosity ν ~ 10-5 m2/s Range 10-6 to 10-4

• Re ~ 105 Range 104 to 107

• Kolmogorov scale ~ 0.02 µm

• Viscosity ~ Diffusion (Schmidt # ~1). RT & RM typically quenched on scales of 1 to a few µm (Robey 2004)

• Pe and Perad are both >>> 1 • The plasmas are well localized (fluid models are OK)

See: D.D. Ryutov, et al. ApJ 518, 821 (1999); ApJ Suppl. 127, 465 (2000); Phys. Plasmas 8, 1804 (2001)


Consider what we mean by “turbulence”

• Is turbulence – Growth of structures beyond the nonlinear saturation of existing

modes? – The development of strong mixing as indicated by supra-linear growth

of a mixing layer in time? – The appearance of an inertial range in the fluctuation spectrum? – Something else?

• Different communities use different definitions

• For turbulence corresponding to strong mixing – Dimotakis (2000) argued that the necessary condition was a sufficient

separation of the Taylor microscale and the dissipation scale


Does the large Re lead to “turbulence”?

• HEDP experiments typically have Re of 104 to 106

The Dimotakis picture. Credit: Zhou 2003


The experimental Re is > 105, so these systems should be turbulent?

• Let’s look at Richtmyer-Meshkov

• Practical experiments are “heavy to light”



At high Mach number, RM does not become turbulent by almost any definition

• Hard experiments – Difficult to sustain the shock

• Glendinning et al. (Phys. Plas., 2003) – The spikes rapidly overtake the

shock, distorting it and limiting their development

– Initial amplitude 5%

• Sensible result:– Incompressible Meyer-Blewett

amplitude growth (d/dt)

– Can exceed the shock separation velocity

data

simulation

d / dt =A*ui

* ko + ko*( ) / 2

u

s−ui

* → −12

ui*


Do we see turbulence in scaled hydrodynamic experiments relevant to supernovae?

• SN 1987A– A core-collapse supernova – Early high-Z x-ray lines with large

Doppler shifts– Early glow from radioactive heating– The issue is the post-core-collapse

explosive behavior

• In 19 years of simulations– Only one (Kifonidis, 2006) makes

fast enough high-Z material – 3D simulations coupling all the

interfaces where initial conditions matter are not feasible

– Experiments will be able to address this fully within a decade

– We now address a single interface

SN1987A, WFPC2, Hubble

Kifonidis, 2003


This type of system involves a blast-wave-driven interface

• Laser ablation drives shock wave for 1 ns

• Front surface rarefaction overtakes shock wave by 2 ns, forming planar blast wave

• Blast wave crosses interface, followed by deceleration and rarefaction



Boundary conditions in time & space matter for scaling

Interface velocity vs timePressure and density

star

lab

Conclusion: scaling is possible


Experimental image of aturbulent flow at Re = 3x104

Numerical image of an unstable, though non-turbulent flow at Re(sim) ~ 103; Re = 1010

Kelvi

n-Hel

mholtz

roll-

up

Lamin

ar fl

ow

All simulations have too much numerical viscosity to produce the observed turbulent state.

But it is an open question whether the flows in supernovae actually become turbulent

There is a turbulence angle to these experiments and supernovae

Van Dyke, Album of Fluid Motion

Kifonidis et al., 2003


t = 8 ns t = 12 ns t = 14 ns

• Data from a 2D single-mode perturbation having an initial perturbation = 50 µm, a = 2.5 µm, ka = 0.3

•The perturbation growth is well into the non-linear regime and Re > 3 x 104 by 7 ns, but the system does not appear to become turbulent within 14 (or 20) ns

shock shock shock

spikes

bubbles

Our Rayleigh-Taylor experiments produce time-dependent, high-Reynolds-number systems.


The question is why these experiments did not enter a turbulent state

• One answer is time dependence

• For turbulence corresponding to strong mixing – Dimotakis (2000) argued that the necessary condition was a sufficient

separation of the Taylor microscale and the dissipation scale

– Zhou et al. (2003) and Robey et al. (2003) argued that sufficient time was also needed

• The role of large Re is to allow diffusive laminar boundary layers to become large enough to separate driving scales from dissipation scales

• This also takes time

– Miles et al. (2004, 2005) argued from simulations that interacting structures could produce turbulent conditions, but not soon enough to be observed in the Omega experiments


The Robey/Zhou picture. Credit: Robey 2003

Here is the Robey/Zhou picture

• The point is– It can take longer to

establish the needed boundary layers than it does to reach the threshold value of Re

– Zhou et al. 2003 show that this picture applies successfully to numerous cases


This picture is consistent but not universal

• Consistency:– Dimotakis requires that the Liepmann-Taylor scale exceed the inner

viscous scale, or 5Re-1/2 > 50Re-3/4

– Robey/Zhou requires the viscous boundary layer, 5(νt)1/2 > 50Re-3/4

– These are consistent if t = /U

• For typical turbulence /U = l /u, a large-eddy turnover time, so both these arguments are consistent with development of an inertial range and strong mixing in one eddy turnover time

• I will refer to this as “quick turbulence”, distinct from e.g. long-term RT

• Issues– Perhaps /U ≠ l /u. – Sometimes velocity shear is gentle …


The path to quick turbulence at shear layers begins with Kelvin Helmholtz (KH)

• Experiments and some simulations do not show Kelvin-Helmholtz growth along Rayleigh-Taylor spikes

• We certainly do not see it, especially in our improved 3D experiments

• However, simulations of our experiments show that the velocity shear is too shallow to allow KH growth


High-energy-density jet experiments produce supersonic jets

• They create a significant bow shock

– Surrounds the jet with a cocoon of shocked material

– Weakens the velocity shear

• These experiments see a lot of structure at the head of the jet, but with many possible causes

4.0 mm dia.

700 µm

125 µm

300 µm dia. hole

0.1 g/cc

RF foam

Titanium

4 µm CH ablator

Target dimensions

Laser Ablation


Astro & HEDP Jets show less Kelvin Helmoltzthan laboratory jets in gasses

Van Dyke, Album of Fluid Motion (1982)

Turbulent jetin gas

HH34 AstroJet

Burrows, Hester, Morse, WFPC2 Hubble

Z-pinch Jet

Lebedev et al.Ap J

Laser-drivenJet

Foster et al.Ap J

The difference is most likely in Lu.


This motivates HEDP experiments to study shear flow effects directly

Target Cross Sectional Views

Expanding fluid bubble Gold “knife-edge”

t =4nst=0 t >4ns

CH ablatorAl driver

1st fluid

Cold foam

Shocked foam2nd fluid

Drive Beams

KH here?

Credit: Eric Harding


Initial experiments show we can get data

• The edge that clips the flow is too close to laser-driven material,

• Complex laser-heating above the rippled surface greatly complicates results

• The next experiments will have an improved design

Experiments at NRL

Credit: Eric Harding


One experiment having shear at a boundary definitely produces turbulence

• … but not in a way that lets one diagnose details

• The experiment involves blast-wave-driven mass stripping from a sphere

• Early experiments used Cu in plastic; recent experiments use Al in foam


This has found application

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

• Experimental results used to help interpret Chandra data from the Puppis A supernova remnant

• Well-scaled experiments have deep credibility• Una Hwang et al., Astrophys. J. (2005)


Observations of the Al/foam case continued until mass stripping had destroyed the cloud

• Hansen et al., ApJ 2007


The stripping is clearly turbulent, consistent with the necessary conditions

• The turbulent model is based on “Spalding’s law of the wall”, Spalding (1961)

• ParametersRe ~ 105 to 106

U ~ 10 km/s

ν ~ 10-5 to 10-6 m2/s ~ Sphere ~ 60 µm radius

/U ~ 6 ns ~ rollup time (data)

• Robey/Zhou time is ~ 1 ns

RemainingMass (µg)

Time (ns)

Hansen et al., ApJ 2007


The HEDP results taken together suggest a condition on steepness of shear

• The Dimotakis and Robey/Zhou models have turbulence begin when fluctuations in the inertial range exist primarily within viscous diffusion layers

• Something must establish these fluctuations, almost certainly beginning with Kelvin Helmholtz (KH), followed by some secondary instability

• In the standard model from Chandrasekhar, a linear velocity scale length H creates a KH threshold of kth ~ 1.4/H

• The KH growth rate to a factor of two is

/ (kxΔU ) =12

1−A2 1−kkth

⎡

⎣⎢

⎤

⎦⎥


A crude estimate of the impact of distributed shear

• Take the linear scale length of the velocity transition to be

• Integrating in time, the linear number of e-foldings is

• Assume 10 e-foldings needed and find minimum wavelength with this much growth

• Can’t quickly populate the turbulent spectrum for smaller wavelengths

H =Ho + νt

Re

3k1−A2 1−

k1.4

Ho

⎛

⎝⎜⎞

⎠⎟

3

The KH curve must be below the inner viscous scale for KH driven turbulence

Steeper shear layers are required at higher Re


Conclusions

• High-energy lasers readily produce high-Mach-number, ionized, high-Reynolds-number flows.

• These flows may develop turbulence, but the nature of turbulent onset is typically important.

• Richtmyer-Meshkov development in such systems can be constrained by interaction with the shock.

• Driving Rayleigh Taylor long enough to see turbulence develop is challenging.

• The challenge in shear flows is to produce steep enough shear. – A fundamental experiment is in progress – Mass stripping of a shocked sphere was clearly turbulent

• The development of turbulence from shear layers in an eddy turnover time may involve a steepness constraint

• The next page shows the extended list of collaborators


This work and this talk have involved contributions from many individuals

• My students and post docs at Michigan • Bruce Remington, Harry Robey, Dmitri Ryutov, Adam Frank, Kent Estabrook,

Sasha Velikovich, Riccardo Betti

• The present and past Omega NLUF collaborators

• Numerous Michigan collaborators at LLNL & NRL

• High Energy Density Laboratory Physicists around the world

Harry Robey Dave Arnett Dick McCray Robert Rosner LLNL University of Arizona University of Colorado University of Chicago Ted Perry Romain Teyssier Jim Knauer Bruce FryxellLLNL CEA Saclay, France LLE; Univ. of Rochester University of Chicago Dimitri Ryutov James Glimm James Stone Alexei Khokhlov LLNL SUNY-Stony Brook Univ. of Maryland Naval Research LaboratoryJave Kane Omar Hurricane John Grove James CarrollLLNL LLNL LANL Eastern Michigan Univ.Adam Frank Kim Budil Keisuke Shigemori Riccardo Betti University of Rochester LLNL Osaka University; LLNL University of Rochester, MIT Mary Jane Graham Christof Litwin Gail Glendinning Grant Bazan West Point University of Chicago LLNL LLNL Serge Bouquet Michel Koenig Aaron Miles CEA Bruyeres LULI Univ. Maryland


Next consider systems driven by flowing plasma

• Ejecta-driven systems – Rarefactions drive

nearly steady shocks– Supernova remnants – Experiments– Rarefactions often

evolve into blast waves

A rarefaction can produce flowing plasma that can drive instabilities


Supernova remnants produce the instability driven by plasma flow in simulation, …

• 1D profile and 2D simulation

Chevalier, et al. ApJ 392, 118 (1992)


.. in observation, and in lab experiment

Supernova Remnant E0102- 72 from Radio to X- Ray Credit: X- ray (NASA/C XC/ SAO); optical (NASA/HST); radio: (ATNF/ ATCA) http://antwrp.gsfc.nasa.gov/apod/ap00 0414.html

Blast-wave driven labresult

RemnantE0102


But we do understand how to scale hydro. An important example: strongly shocked systems

• Consider two hydrodynamic systems driven by strong shocks

• Suppose their density structure is spatially identical:

• Suppose they are caused to evolve by a strong shock of speed vi

• The two systems will evolve identically on normalized time scales t/i, with

ρi = Ci H(r/hi)

Constant

Shape function

Normalizing dimension

i = hi / vi Example hi (cm) vi i

(km/s)SN 1987A 9x1010 = ~ 1million 2000 450 s kmLab 0.0053 = 1/2 human 13 4 ns

hair

System 1 System 2

Details: D.D. Ryutov, et al., Ap.J. 518, 821 (1999)


In other lab experiments, the instabilities moved material all the way to the shock13 ns

17 ns13 ns

21 ns17 ns

modulated

planar21 ns

Drake, Phys. Plasmas 2004


We are now observing the role of complex initial conditions in spike penetration

Interferogram of complex surface on component provided by GA (analysis: Kai Ravariere)

Preliminary data on mix layer thickness

Data and analysis: Carolyn Kuranz


We collaborate with simulation groups to evaluate our results and validate codes

• Work with the FLASH Center (Chicago), to include 3D adaptive modeling, has now begun

Preliminary FLASH simulation at 30 ns of recent experiments

Left: Single eggcrate mode. Right: Two-mode system.

Approaches to Turbulence in High-Energy-Density Experiments R. Paul Drake University of Michigan

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