Growth Rates of the Deceleration-Phase
Rayleigh–Taylor Instability
V. LOBATCHEV, M. UMANSKY, and R. BETTI
University of Rochester
Laboratory for Laser Energetics
Growth Rates of the Deceleration-Phase Rayleigh–Taylor Instability
V. Lobatchev, M. Umansky, and R. Betti
Laboratory for Laser Energetics, U. of Rochester
The evolution of inner-surface perturbations during the deceleration phase of a planar foil and a spherical
shell is studied by using high-resolution, two-dimensional Eulerian simulations, and the results are compared
with a theoretical model for the growth rates. The planar simulations concern the deceleration of a foil
compressing a low-density shocked material against a rigid wall. The two-dimensional spherical simulations
concern direct-drive NIF capsules. The output of the one-dimensional code LILAC at the beginning of the
deceleration phase with an imposed inner-surface perturbation is used as the input of the two-dimensional
simulations. It is found that the simulated growth rates are significantly lower than their classical values. In
addition to the finite density-gradient scale length, further stabilization is provided by mass ablation off the
inner surface of the shell. This work was supported by the U.S. Department of Energy Office of Inertial Confinement
Fusion under Cooperative Agreement No. DE-FC03-92SF19460.
The deceleration-phase instability occurs
in the final phase of the implosion
Outer
surface
Shell
DT
Inner
surface
Laser
TC5357
g
During the deceleration
phase, the inner surface
of the shell is RT unstable
and its perturbations grow
exponentially.
What is investigated?
• Mass ablation
• Density-gradient scale length
Important in the RT instability
• Inner-surface acceleration
• Reduction of the deceleration-phase Rayleigh–Taylor instability
• Both planar and spherical numerical simulations were carried
out to study all of the above.
TC5546
Eulerian code was developed to simulate
the deceleration-phase instability
• Standard gas dynamic equations
• Heat transfer
• Local a-particle deposition
• To speed up the heat transfer calculation,
the simulations were done with an Eulerian code.
TC5547
Impulsive deceleration by a series of shocks
is followed by continuous deceleration
g (mm/ns2)(× 1000)
10
Stagnation
8
6
4
Impulsive
deceleration
Continuous
deceleration
2
–0.4
–0.3
–0.2
–0.1
0.0
Time (ns)
• Why does the deceleration behave in this manner?
• What do we call the deceleration phase?
TC5548
• Growth rate?
The decelerating foil problem provides the basic
understanding of the deceleration-phase instability
Wall
Foil
U
Shock
Gas
Shocked
gas
Density
profile
x
· The shock reflected from the wall slows down the foil,
which in turn compresses the gas and decelerates.
· The 1-D problem can be solved analytically leading to a
clear understanding of the relevant physics issues.
TC5391
The foil is slowed down both impulsively and continuously
Evolution of pressure and density
Wall
Reflected
shock
20 Mbar
Density
1 gr/cm3
Pressure
Foil
Arbitrary units
Wall
Pressure
Wall
Wall
Arbitrary units
Pressure
Density
Density
Pressure
0
TC5392
Density
30 0
Distance (mm)
30
Distance (mm)
What physical mechanism is responsible
for the mass ablation at the shell’s surface?
• The shell is much colder
than the hot spot.
• The heat flux leaving the
hot spot is deposited on
the shell surface.
• Expansion of the heated
material causes the mass
ablation from the shell into
the hot-spot.
T
Heated
layer
Heat
flux
Cold
shell
Hot spot
Ablation
R
TC5394a
r
Spherical simulations were performed to determine
the growth rates during the deceleration phase
• 1-D LILAC output for a typical NIF target was used for the initial
density, pressure, and velocity distribution.
• A 1-D Eulerian simulation is performed until the beginning
of the continuous deceleration phase.
• The 2-D pertubations of the shell’s inner surface are introduced
~200 ps before the stagnation begins.
• A Fourier transform is used to determine the amplitude
of each mode.
• The growth rates are determined by fitting the time history
of each mode.
TC5570
High-resolution simulations are required
to resolve short-wavelength modes with k > 1/Lm
-1
1 drI
F
Lm = min G
x H r dxJK
Lm (mm)
1.4
1.2
1.0
0.8
0.6
0.4
10
20
30
40
Number of points/mm
• Lm obtained in 1-D numerical simulations depends on resolution.
TC5571
• Lm does not decrease dignificantly when the grid refinement
is Dx < 0.05 mm.
The minimum density-gradient scale length is ~1mm
3.0
Lm (mm)
2.5
Simulation
2.0
1.5
1.0
Formula
0.5
0.0
–0.15 –0.10 –0.05 0.00
0.05 0.10
Time (ns)
TC5572
-1
1 ¶ rI
F
Lm = min G
r H r ¶x JK
52
mi Phs atf I
F
Lm = 6.8 Rhs G
H 2rshellThsa0, tfJK
Ablation velocities obtained from both theory
and simulation are of the order of 20 mm/ns
30
Va (mm/ns)
25
Simulation
20
15
10
Formula
5
0
–0.15 –0.10 –0.05
0.00
0.05 0.10
Time (ns)
mm
VaF I = 1.2
H ns K
TC5573
·
103
a f
F g IJ
Rhs am mfrshell G
H cm3 K
T 5 2 keV
hs
Growth rates of the deceleration-phase instability
for the planar foil are reduced by mass ablation
35
Growth rate (ns–1)
g=
30
Without
ablation
kg
1 + kLm
Theory
25
With
ablation
20
15
10
5
Simulation
1
2
3
K = 2p/l (mm–1)
TC5366a
4
5
6
Results of 2-D spherical simulations of a NIF-like
capsule confirm the importance of mass ablation
40
g =
g(l) (ns-1)
30
20
g = 0.9
10
kg
1+ k Lm
kg
- 1.4 k Va
1+ k Lm
Simulations
0
0
20
40
60
80
l-mode
TC5574
• The cutoff is observed at l = 90 in the instability spectrum.
All modes with l > 90 are stable.
Conclusions
• Two stages of the deceleration-phase instability are observed:
deceleration by a series of shocks and continuous deceleration.
• Mass ablation through the inner surface and finite densitygradient scale length are the main stabilizing factors during
the RT growth.
• The significant reduction in the RT instability growth rate
is observed due to the mass ablation. The spectrum
of the deceleration-phase instability exhibits a cutoff
for mode numbers of about l = 90.
• For l~50, the calculated growth rate is less by a factor
of 3 than the conventional estimate, neglecting ablation.
TC5575
Planar Model
Planar simulations reproduce the behavior
of ICF capsule implosions
· Hot-spot temperature and radius and peak shell density have
Shell
1.2
0.8
0.4
0.0
TC5359
Wall
Shock wave
100 200 300 400 500 600
X (mm)
Stagnation
Density
(g/cm3)
30
25
20
15
10
5
0
Initial distribution
Temperature
(keV)
Temperature
(keV)
Density
(g/cm3)
the same order of magnitude in planar and spherical cases.
250
200
150
100
50
0
2.0
1.5
1.0
0.5
0.0
5
10
X (mm)
15
20
The density-gradient scale length is small
· Balance of heat flux to the shell and internal energy flux
leaving the shell
b g dTdrsh » k bTshg Tsh r1
d r sh
sh dr
pVa » - k Tsh
· The density-gradient scale length is found using the formula
for the ablation velocity:
M
r
Lm = 1 d
r dr
-1
min
· For NIF: Lm ~ 1 mm
TC5409
b g
F
GH
I5 2
JK
M k Tsh
T
» 1.6 i
= 8R hot-spot shell
r shVa
T0
The ablation velocity is determined from the energy balance
temperature
rshell
shell
Va
density
F 3 pbVb + pbVb I = k Spitz bTb g dTb
H2
K
dRhs
Internal
energy flux
pdV work rate
Vb
hot spot
r
Rhs
Energy flux balance
2 5
2
F
I
r
· Hot-spot temperature profile: Ths = T0 G 1 GH Rhs2 JJK
·
· Use the EOS: pb Vb = 2rb Vb Tb / Mi = 2mTb / Mi
·
· Ablation velocity: Va =
TC5408
m
&
r shell
= 0.2
b g
Mi k Spitz T0
r shell Rhot spot
The deceleration-phase instability consists
of a RM phase and a RT phase
· Linear RM instability is observed when shocks are coming.
· Exponential RT instability is seen during the continuous
deceleration stage.
Perturbation (mm)
0.25
0.20
Exponential
growth
0.15
0.10
Linear growth
0.05
0.00
–0.05
1.3
TC5363
1.4
1.5
Time (ns)
1.6
1.7
1.8
Finite density-gradient scale length and mass
ablation have the largest stabilizing effect
· Simple models include only d2h/dt2 = kgh,
compressibility, ablation, and Lm.
l = 10 mm
Lm
1.0
h (mm)
Compressibility
d2h
dt2
0.1
Prediction
= kgh
Simulation
1.75
TC5365
Ablation
1.80
Time (ns)
1.85
1.90