0 1 2 3 4 5 6 7 8

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

k (

-

0 )

Comparison of dimensionless amplitude with several models

Exp. M = 1.26Exp. M = 2.05Raptor M = 1.26Raptor M = 2.05Mikaelian M = 1.26Mikaelian M = 2.05Sadot et al. M = 1.26Sadot et al. M = 2.05Dimonte M = 1.26Dimonte M = 2.05

0 5 10 150

0.2

0.4

0.6

0.8

1

M=1.33M=2.88M=3.38

0 5 10 150

0.2

0.4

0.6

0.8

1

M=1.2M=1.5M=1.68M=3

0 5 10 150

0.2

0.4

0.6

0.8

1

M=1.14M=2.5M=5

0 5 10 150

0.2

0.4

0.6

0.8

1

M=1.2M=1.5M=1.68M=3

Experiments and Computations for Inertial Confinement Fusion-Related Shock-Driven Hydrodynamic InstabilitiesExperiments and Computations for Inertial Confinement Fusion-Related Shock-Driven Hydrodynamic InstabilitiesBradley Motl, John Niederhaus, Devesh Ranjan, Jason Oakley, Mark Anderson, and Riccardo Bonazza

Fusion Technology Institute, University of Wisconsin-MadisonHigh Energy Density Physics Summer School, San Diego, CA, July 29 - August 3, 2007

IntroductionIntroduction

Shock-bubble interactionShock-bubble interaction

Relevance to ICFRelevance to ICF

Wisconsin Shock Tube LaboratoryWisconsin Shock Tube Laboratory

In inertial confinement fusion, shock-driven hydrodynamic instab-ilities and the associated mixing impose a limit on the efficiency with which fuel material may be compressed to the densities required for fusion, reducing the obtainable fusion yield. These instabilities arise at nonuniformities on density and material interfaces, as ablative and radiatively-driven shocks pass through the material and compress it, as shown below schematically.

In the present work, the phenomenology, mechanisms, and spatial and temporal scales of shock-driven instabilities are investigated using experiments in a gas shock tube environment, along with numerical simulations. In the shock tube environment, hydrodynamic phenomena may be characterized much more precisely than at ICF conditions, due to differences in conditions listed in the table above. Further, the absence of electric and magnetic fields, phase changes, and radiation allows purely hydrodynamic effects to be studied independently. Here, geometric length scales of the deformed interfaces are measured as indications of shock-induced mixing.

Computational parameter studyComputational parameter study

ReferencesReferencesD. Ranjan, J. Niederhaus, B. Motl, M. Anderson, J. Oakley, and R. Bonazza, Experimental investigation of primary and secondary features in high Mach number shock-bubble interaction, Phys. Rev. Lett., in review (2006).

D. Ranjan, M. Anderson, J. Oakley, and R. Bonazza, Experimental investigation of a strongly shocked gas bubble, Phys. Rev. Lett., 94 (2005).

J. Niederhaus, D. Ranjan, J. Oakley, M. Anderson, and R. Bonazza, Inertial-Fusion-Related Hydrodynamic Instabilities in a Spherical Gas Bubble Accelerated by a Planar Shock Wave, Fusion Science and Technology 47, 4 (2005), p. 1160.

J. Greenough, J. Bell, and P. Colella, An adaptive multi-fluid interface capturing method for compressible flows in complex geometry, AIAA Paper 95-1718.

)( 0 p

7.2 m

2 m

Shock tube:

• Vertical orientation• 20 MPa impulsive

load capability• 25.4 cm square internal

cross section• Planar laser

imaging ports• Modular components:

– Bubble injector– Oscillating pistons

Imaging:

• Pulsed laser sheets

• CCD cameras• Mie scattering or

planar laser induced fluor-escence

• Dual exposure

Shock-bubble interaction

configuration

Soap-film bubble

Retractable injector

Richtmyer-Meshkov instability

configuration

Oscillating pistons

2D sinusoid interface

Driver(high pressure)

Diaphragm

Shockwave

propagation

Light source

Diagnostic laser sheet

Experiment

Raptor (2D)

τ = 0.00 τ = 4.66 τ = 8.79

τ = 0.00 τ = 1.54

τ = 0.00 τ = 1.53

τ = 3.98 τ = 4.16τ = 3.29

τ = 4.22τ = 3.95τ = 3.27

N2(acetone) / SF6 M = 1.26

Richtmyer-Meshkov instabilityRichtmyer-Meshkov instability

N2(smoke) / SF6

M = 2.05

AcknowledgementsAcknowledgementsThe authors would like to express sincere thanks to Jeff

Greenough (LLNL), for facilitating computations, and Paul Brooks (UW-Madison) for developing and maintaining the experimental setup. This work was supported by US DOE Grant #DE-FG52-03NA00061.

These flows are simulated numerically by integrating the 3D Euler equations using a piecewise-linear 2nd-order Godunov method with adaptive mesh refinement (AMR). The code is called Raptor, and was developed at LLNL and LBL (see Greenough, et al.). A computational parameter study was performed for shock-bubble interactions, with 14 scenarios at 1.14 ≤ M ≤ 5 and -0.76 ≤ A ≤ 0.61.

= 1.3 = 4.1 = 7.7 = 11.5

8 cm

= 23.6

= 1.3 = 4.0 = 7.7 = 11.6 = 23.8

5 cm

= 45.6 = 46.3

8 cm

= 63.5 = 71.6 = 63.5 = 69.5

M = 1.4 M = 2.08 M = 2.95 = 65.7 = 63.5 = 71.6

f1

Rut /1

The results above show the growth of the mixed region within the shocked bubble as a function of dimensionless time, with M and A as parameters. This is quantified as , where is the mean volume fraction of the ambient gas within the bubble region. These data show that the intensity and extent of mixing increases with increasing values of the Atwood number A, though Mach number effects may be scaled out using the post-shock flow speed.

(b) (d)(c)(a)

The evolution of the Richtmyer-Meshkov instability is shown schematically above: (a) initial configuration just prior to shock, (b) linear growth regime, (c) start of nonlinear growth, (d) appearance of mushroom structures, and (e) turbulent mixing.

Below are images generated using planar laser diagnostics (PLIF or Mie scattering) from shock tube experiments studying the Richtmyer-Meshkov instability for a 2D sinusoidal interface between nitrogen (with flow tracer) above and SF6 below.

(e)

Instability evolution

Mr

Mt

Mi

(+) (-)

p

He bubble in air: A = -0.76 Ar bubble in N2: A = 0.18

Kr bubble in air: A = 0.49 R12 bubble in air: A = 0.61

The evolution of a helium bubble of initial radius R during and after acceleration by a planar shock wave in air is shown schematically above: (a) initial configuration; (b) compression and rotation induced by shock passage; (c) deformation and vortex ring formation.

(b) (c)(a)

p

(+)(-)

tk 0

Mi

Shock-driven hydrodynamic instabilities are present in accelerated inhomogeneous flows. Vorticity is deposited on density interfaces by baroclinicity , causing interfaces to become unstable and deform. The geometric features of the deformed interfaces and mixing zones in two particular shock-driven flows are studied here: the Richtmyer-Meshkov instability of a planar interface with a small-amplitude sinusoidal perturbation, and the interaction of a planar shock wave with a discrete spherical bubble.

t = 13 s t = 35 s t = 70 s t = 113 s t = 230 s

Bubble distortion

t = 430 s t = 721 s t = 725 s

Top: experimental shock tube images obtained using laser light scattered at the bubble midplane for a helium bubble shocked at M = 2.95. Bottom: results from 3D Eulerian AMR simulations: density (left) and vorticity magnitude (right) at bubble midplane.

Below: late-time experimental and numerical images showing multiple vortex rings and complex structure. Left: helium volume fraction ( f ); right:

vorticity magnitude ().

RtWt /

Late-time images for varying Mach number.

Radiative heating

Ablation Compression Thermo-nuclear burn

Fusion Technology InstituteUW-Madison

21

21

A

Results from simulations with Raptor (2D) are also shown above, At left, the dimension-less amplitude of the interface (excluding wall effects) from simulations and experiments at various shock strengths (M) is plotted on a dimensionless timescale, with analytical model predictions.

= post-shock particle velocity

Wt = transmitted shock wave speed

R = initial bubble radius

k = perturbationwavenumber

tk 0

ICF Shock tube

L scale 10-6 m 10-2 m

T scale 10-9 s 10-6 s

scale 103 kg/m3 100 kg/m3

M ~30 ≤ 5

A [-1,1] [-1,1]

0 20 40 600

1

2

3

4M=1.45 Expt.M=2.08 Expt.M=2.95 Expt.

RtWt /

h

)2/( RhLh

h / (

2R)

f

100

10-6

[logscale]

max

-max

0

1u

B

B

dV

dVf

1

B = bubble-fluid region (f ≠ 0)

f = bubble-fluid volume fraction