Progress with high-resolution AMR wetted-foam simulations.

Progress with high-resolution AMR wetted-foam simulations.

• Two issues are central: the role of density fluctuations at the ablation surface, shock speed.

• The new material tracking routines show a short mixing length.

• Simulations modeling the CH ablator show agreement with Rankine-Hugoniot jump conditions.

DDI 3.3: High-gain wetted-foam target design

A single fiber is subject to the Richtmyer-Meshkov and Kelvin-Helmholtz instabilities

• The primary instability is Richtmyer-Meshkov, which generates a pair of vortices as the shock passes the fiber.

A lone fiber is “destroyed” in ~13 ps, or about 3 fiber-crossing times.

Time (ps)

Ce

nte

ro

fm

ass

po

sitio

n(

m)

0 10 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

singlefiber9 / singlefiber9_com.lpkTime (ps)

Ce

nte

ro

fm

ass

po

sitio

n(

m)

0 10 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

singlefiber9 / singlefiber9_com.lpk

Fiber-crossing time

Time (ps)

%fib

er

ma

ssw

ithin

on

efib

er

rad

ius

0 10 200

25

50

75

100

singlefiber9 / singlefiber9_massfrac.lpk

• A characteristic hydrodynamic time scale is the shock-crossing time tc

of the fiber.

• The fiber is accelerated to the speed of the DT in about ~2tc, or ~8 ps.

• 75% of the fiber mass lies outside its original boundaries after ~3tc, or ~13 ps.

The fiber destruction time depends on the ratio of fiber density to fluid density

• For a larger density ratio: – the Atwood number is higher and the Richtmyer-Meshkov instability

is increased– The velocity shear between the fiber and DT is greater, resulting in

greater Kelvin-Helmholtz instability

Time (ps)

%fib

er

ma

ssw

ithin

on

efib

er

rad

ius

0 10 200

25

50

75

100

4:140:1

singlefiber9 / singlefiber9_massfrac.lpk

The fiber destruction time depends onthe fiber : DT density ratio40:1 density ratio

Identification of the CH as a second material type provides a measure of mixing

Tagging a single fiber as a third material shows the degree of mixing

• Any cell with over 10 mg/cc of the “tagged” material is colored red.

Fourier decomposition of the tracer mass fraction shows a mixing length of ~1.3 m

• The average e-folding distance for decay of the mass-fraction fluctuations is ~1.3 m.

Mix region depth (m)

Fo

uri

er

de

com

po

sitio

no

ffib

er

ma

ssfr

act

ion

(%)

0 1 2 3 410-2

10-1

100

101

n=1n=2n=3n=4n=5n=6

150.2.long.8, 150.2.long.a, 150.2.lr.6

Shocks reflected from the fibers raise the pressure, elevating the post-shock pressure

• The higher pressure results in an elevated shock speed relative to a shock in a uniform field of the same average density, with the same inflow pressure.

x (m)

p(M

ba

r)

0 10 20 30

0.5

1

1.5

2

2.5

3

3.5

4

mixregionreflected

shocks

sho

ck

x (m)

p(M

ba

r)

0 10 20 30

0.5

1

1.5

2

2.5

3

3.5

4

150.2.long.6; 150.2.long.6_fr72_p.lpk

homogeneous densityfield, 3-Mbar shock

x (m)

(g

/cc)

0 10 20 30

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 mixregion

un-shocked

fibers

sho

ck

x (m)

(g

/cc)

0 10 20 30

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 mixregion

un-shocked

fibers

sho

ck

150.2.long.6; 150.2.long.6_fr72_rho.lpk

reflectedshocks

homogeneous densityfield, 3-Mbar shock

When the CH ablator is included, the Rankine-Hugoniot jump conditions are satisfied

• These targets will be fabricated with a thin plastic overcoat.• The post-shock conditions are the same as in the average case with the

same pusher.• On average the Rankine-Hugoniot conditions are obeyed, and the shock

speeds are the same.• An average treatment of density, as in LILAC, is accurate.

x (m)

De

nsi

ty(g

/cc)

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

CH

foam

rarefactionwave

shock

homogeneousfiber-resolved

x (m)

p(M

ba

r)

0 2 4 60

2

4

6

8

10

12

14

16

18homogeneous

inhomogeneous

rarefactionwave

shock

CH foam

Time (ps)

Sh

ock

po

sitio

n(

m)

0 20 40 60 80

0.5

1

1.5

2

2.5

3

3.5

4

4.5 homogeneousinhomogeneous

The fiber destruction time depends on the ratio of the fiber density to the fluid density

40:1

4:1

4 ps 8 ps 12 ps

The fiber-resolved simulations behave, on average, like the equivalent 1-D simulation

x (m)

de

nsi

ty(g

/cc)

0 2 4 6 8 100

0.25

0.5

0.75

1

1.25

1.5

1.75

2

inhomogeneoushomogeneous

x (m)

pre

ssu

re(M

ba

r)

0 2 4 6 8 100

1

2

3

4

5

x (m)

inte

rna

len

erg

y(M

ba

r)

0 2 4 6 8 100

1

2

3

4

5

6

7

x (m)

velo

city

(m

/ns)

0 2 4 6 8 100

5

10

15

20

25

30

35

Shocks reflected from the foam fibers elevate the post-shock pressure.

• The main shock is partially reflected off the foam fibers.• The reflected shocks make their way though the mix region, eventually

crossing the ablation surface and entering the corona.• Conservation of mass requires the density in the mix region match the

post-shock speed.• Since ~ log(p / 5/3), the post-shock adiabat is higher by p / p ~ ??.

x (m)

(g

/cc)

0 10 20 30

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 mixregion

reflectedshocks

un-shocked

fibers

sho

ck

x (m)

(g

/cc)

0 10 20 30

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 mixregion

reflectedshocks

un-shocked

fibers

sho

ck

150.2.long.6; 150.2.long.6_fr72_rho.lpk

x (m)

p(M

ba

r)

0 10 20 30

0.5

1

1.5

2

2.5

3

3.5

4

mixregionreflected

shocks

sho

ck

x (m)

p(M

ba

r)

0 10 20 30

0.5

1

1.5

2

2.5

3

3.5

4

150.2.long.6; 150.2.long.6_fr72_p.lpk

The fiber destruction time depends on the ratio of fiber density to fluid density

• For a larger density ratio: – the Atwood number is lower and the Richtmyer-Meshkov instability

is increased– The velocity shear between the fiber and DT is greater, resulting in

greater Kelvin-Helmholtz instability

Time (ps)

%fib

er

ma

ssw

ithin

on

efib

er

rad

ius

0 10 200

25

50

75

100

4:140:1

singlefiber9 / singlefiber9_massfrac.lpk

The fiber destruction time depends onthe fiber : DT density ratio

Artificial viscosity is modeled in BEARCLAW by splitting the contact discontinuity

• The Riemann problem at a cell boundary is solved with three waves: shock, rarefaction (collapsed to a midpoint line) and contact discontinuity (CD).

• Eulerian codes are subject to the growth of noise due to discretization.• These are eliminated in BEARCLAW by splitting the CD from a sharp

transition to a smooth transitional region.• For appropriate values of the artificial viscosity, the shock speed is not

affected.

Time (ps)

Sh

ock

po

sitio

n(

m)

0 2 4 60

1

2f=0.95f=0.1

150.2.long.g_150.2.long.f_xs.plt