J.
Fluid
Mech.
(2014)
,
vol
.
748
.
pp.
85
11
2.
©
Cambndge
University
P
re
ss
2014
doi
:
I 0.10
I
7/jfm.2014
.
161
85
Turbulent
mixing
driven
by
spherical
implosions.
Part
1.
Flow
description
and
mixing-layer
growth
M.
L
ombardi
ni
-'-,
D .
I.
Pullin
an
d
D.
I.
M
e
iron
Graduate
Aerospace
Lab
o ratories,
California
Institute
of Technology
, Pasadena,
CA
91125,
USA
( Received
12
June
2013;
revised
29
Janu
ary
2014;
accepted
20
March
2014
;
fir
st
published
online
28
April
2014)
We
pre
se
nt
l
arge-eddy
si mulations
(
LES)
of
turbulent
mixing
at
a
perturbed,
spherical
interface
se
parating
two
fluids
of
differing
densities
a
nd
subseque
ntly
impacted
by
a
s
ph
erically
imp
l
odi
ng
shoc
k wave.
This
paper
focuses
on
the
differences
between
two
fundamental
configurations,
keeping
fixed
the
initial
s
hock
Mach
number
~1.2,
the
density
ratio
(precisely
I
Ao
l
~
0.67
)
and
the
perturbation
shape
(dominant
sp
herical
wavenumber
.f
0
=
40
and
ampl
itud
e-to
-initial
radius
of
3 %):
the
incident
s
hock
travels
from
the
ligh
ter
fluid
to
the
heavy
fluid
or,
inversely,
from
the
heavy
to
the
l
ight
fluid.
After
describing
the
computational
problem
we
present
results
on
the
radially
symmetric
flow,
the
mean
flow
,
and
the
growth
of
the
mi
xing
lay
er.
Turbulent
statistics
are
developed
in
Part
2
(Lomba
rdin
i,
M.,
Pullin,
D .
I.
&
M
eiron,
D.
I.
J.
Fluid
Mech.,
vol.
748,
2014,
pp.
113-142).
A
wave-diagram
analysis
of
the
radially
symmetric
flow
highli
gh
ts
that
the
light
-
heavy
mixing
layer
is
processed
by
consecutive
reshocks,
and
not
by
reverberating
rarefaction
waves
as
is
usually
observe
d
in
plana
r
geometry.
Less
surpr
i
sing
ly,
reshocks
process
the
heavy
- li
ght
mixing
layer
as
in
the
planar
case.
In
both
configurations,
the
incident
imploding
shock
and
the
reshocks
induce
R
ichtmyer-Meshkov
(RM
) instabilities
at
the
density
layer.
H
owever
,
we
obse
rve
differences
in
the
mixing-layer
growth
because
the
RM
instability
occ
urr
ences,
Rayl
eigh-
Tay
lor
(R
T)
uns
table
scenarios
(due
to
the
radially
accelerated
motion
of
the
layer)
and
phase
inversion
events
are
different.
A
small-am
plitude
stability
analysis
al
ong
the
lin
es
of
Be
ll
(L
os
Alamos
Scientific
Laboratory
Report,
LA-132
L,
1951
)
and
Plesset
(J.
Appl.
Phys.,
vol.
25,
1954,
pp.
96-98)
help
s quantify
th e
effects
of
the
mean
flow
on
the
mixing-layer
growth
by
decoupling
the
effects
of
R
T/RM
instabilities
from
Be
ll
- Pl
essel
effects
associated
with
geometric
co
nvergence
and
compressibility
for
arbitrary
convergence
rat
i
os.
The
analysis
indicates
that
baroclinic
instabilities
are
the
dominant
effect,
conside
ring
the
low
convergence
ratio
(~2)
and
rather
high
ce
>
10)
m
ode
numb
ers
considered.
Ke
y
word
s :
compressible
turbulence,
shock
waves,
rurbulent
mixing
1.
In
trod
ucti
on
Th
e
contemporary
study
of
accelerated
density
inhomogeneities
began
with
the
small
-amp
l
itude
analysis
of
a
perturbed
interface
by
Taylor
(
19:'0
).
Thi
s
constant
acceleration
envi
rn
n mcnt
is
calle
d
the
Ray
l
eigh-
Tay
l
or
( RT
)
instability.
Taylor
t
Email
address
for
correspondence:
m.muel(a'calcech
l"du
,
86
M.
Lomhardini,
D.
I.
Pullin
and
D.
l
Meiron
showed
that
an
interface
se
parating
two
fluids
of
differing
density
is
unstable
depending
o n
whether
the
acceleration
is
directed
from
the
lighter
co
the
heavier
fluid.
Wh
e n
the
perturbation
amp
litude-lo-wavelength
ratio
approaches
one,
the
growth
becomes
nonlinear,
secondary
Ke lvin
-
Helmholtz
instabilities
appear
along
the
fingering
structures
of
the
distorced
interface
and
mixing
may
eventually
occur
at
sma
ll
scales
(e.g.
Cook,
Cabot
&
Miller
200.f
).
The
impulsive-acceleration
analogue
of
the
RT
instability
, also
referred
to
as
the
Richtmyer-Me
shkov
(
RM
) instability
(Ri
chtmyer
l
960
;
Meshkov
1969
),
is
produced
when
a
density
interfac
e is subjected
to
a
shock
wave.
Unlike
the
RT
instability
, the
occurrence
of
the
RM
instability
does
not
depend
upon
the
direction
of
impulsive
acceleration;
the
interface
is
unstab
le
whether
the
shock
travels
from
the
light
to
the
heavy
fluid
or
from
the
heavy
to
the
light
fluid.
The
RM
instability
is
of
fundamental
importance
in
understanding
the
physics
of
inhomogeneous,
compressib
l
e,
turbulent
flows.
In
particular,
the
transition
to
turbul
e
nt
mixing
generated
by
a
single
shock
wave
bas
been
the
subject
of
rec
ent
inv
est
igations,
both
experimentally
(Motl
er
al.
2009
;
Orlic
z
el
al.
2009
)
and
numerically
(
Thornber
et
al.
20
IO
;
Lombardini,
Pullin
&
Meiron
2012
).
While
most
research
on
both
barocl
inic
instabililies
bas
focused
on
planar
geometry,
little
wo rk
has
been
done
o n
imploding/exploding
flows
where
density
inhomogeneities
are
radially
accelerated/decelerated,
espec
ially
on
the
turbulent
mixing
generated.
H owever,
in
a variety
of
applications,
such
as
inertial
confinement
fusion
(
ICF
)
(Lindi
1998
; Wei
ser-Sherrill
et
al.
2008
),
supernova
collapse
(Jun,
Jones
&
Norma
1996
;
Joggerst,
Almgren
&
Woo
sley
20
I 0),
explosive
detonati
on
(Balakrishnan
&
Menon
20
I I ),
underwater collapsing
bubbles
(Lin,
Storey
&
Szeri
'2002
)
and
drop
impact
(
Kr
ec
hetnikov
&
H
omsy
2009
),
these
hydrodynamic
instabilities
take
place
in
curved
geometry.
In
contrast
to
planar
geometry
where
only
RM
grow
th
is
expected
to
occur,
converging/diverging
shock-accelerated
interfaces
can
be
RT
unstable
as
they
geometrically
contract
o r expand.
The
early-time
growth
of
these
instab
ilities
has
been
investigated
in
cyli
ndrical
(
Bell
1951
;
Mikaelian
2005
;
Yu
&
Livescu
2008
;
Lombardini
&
Pullin
2009
)
and
spherical
geometries
(
Bell
195
I ;
Ples
set
I
95.+
;
Mika
elia
n
1990
;
Kumar
, H
ornung
&
Sturtevant
200
3;
Mankbadi
&
Balachandar
2
01
2
).
Th
e stability
analysis
by
Krech
et
nikov
(2009
)
actually
unifie
s
some
of
the
work
previously
cited
by
uncovering
the
interrelation
between
the
RT
and
RM
instabilities
and
the
general
effect
of
interfacial
curvature.
Late-time
mixing
pa
st the
reshock
event,
which
occurs
when
an
initially
converging
shock
wave
impact
s
the
interface
a
second
time
on
its
way
out
after
reflecting
off
of
the
origin,
has
been
studied
for
two-dimensional
(
2D)
flows
in
purely
azimuthal,
cylindrical
geometrics
(
Zhang
&
Graham
1998
;
Hosseini
&
Takayama
200
")
and
(
axisymmetric
)
sp
herica
l geometries
(Glimm
er al.
2002
;
Th
o
ma
s
&
Kares
201
2).
There
are
even
fewer
sLudies
that
consider
fully
three-dimensional
(
3D)
flows
whe
re
vortex
stretching
can
now
generate
a
wide
range
of
turbulent
scales,
all
the
way
down
to
Kolmogorov
viscous
scales.
These
small
scales
are
curre
ntly
unresolvable
at
the
R
eynolds
numbers
involved,
and
subgrid
modelling
is
therefore
necessary.
The
large-eddy
sim
ulati
ons
(
LES)
of
Lombardini
&
Deiterding
(20
I
0)
considered
the
effect
of
both
azimuthal
and
axial
perturbations
on
the
turbulent
mixing
driven
by
cylindrical
implosions,
while
Youngs
&
Williams
(2008
)
performed
monotone
integrated
LES
(
MILES)
of
turbulent
mixing
in
sp
herical
implosions.
Thomas
&
Kar
es
(
2012
)
also
considered
30
mixing
using
a
G
odu
n
ov
so
lver
for
the
hydrodynamic
part
of
the
radiation-hydrodynamic
code
RAGE
Gittin
gs
el al.
(2008
).
H owever,
their
simulation
was
initialized
from
a
20
axisymmetric
flow
in
a
quadrant
then
mapped
to
an
octant
after
the
passage
of
lhe
first
reshock
when
the
turbulent
mixing
is
intensified.
Turbulent
mixing
driven
hy
spherical
implosions
.
Part
I
87
Youngs
&
Williams
(
200~)
calculated
che
mixing
in
a
spherical
sector
using
a
spher
ical
polar
mesh.
The
implosion
was
driven
by
applying
a
varying
pressure
at
an
outer
boundary
which
moves
inward
in
a Lagrangian
manner,
the
applied
pressure
being
constant
unLil
the
boundary
reaches
the
interface
at
rest.
The
initial
interface
perturbation
height
was
given
by
a superposition
of
normal
modes
in
the
polar
and
azimuthal
directions
()
and
¢,
with
a prescribed
radial
power
spectrum,
similar
to
what
is
usually
done
for
planar
interfaces.
The
numerical
method
provided
the
necessary
artificial
dissipation
to
capture
the
shocks
and
model
the
subgrid
turbulent
activity.
We
are proposing
here
a different
approach
to
simu
la
te
the
turbulent
mixing
driven
by
spherical
implosions.
Following
Lombardini
&
Deiterding
(2010
),
we
perform
Cartesian-grid-based
LES
of
the
two-component
flow
using
an
explicit
subgrid-scale
(SGS)
model
and
a
low-numerical
dissipation
advection
scheme
activated
in
the
turbulent
regions
(§
2
).
As
explained
in
§
3,
the
computational
domain
considered
is
either
an
octant
of
a sphere
or
a
full
sphere;
the
initial
converging
shock
is
set
up
as
a
self-similar,
radially
symmetric
solution
of
the
Euler
equation.
An
analysis
using
spherical
harmonics,
which
is
detailed
in
Part
2
(Lombardini,
Pullin
&
Meiron
2014
),
is
used
to
construct
the
pre-shock
interfacial
perturbations.
We
investigate
two
canonical
configurations:
the
light
fluid
enclosed
by
the
spherical
interface,
or
vice
versa.
Various
results
are
exposed
in
subsequent
sections,
focusing
on
the
fundamental
differences
between
the
two
configurations,
and
hi ghlight
in g
the
unique
aspects
of
the
spherical
geometry
compared
with
the
planar
one.
ln
§
.+
we
analyse
the
main
wave
interactions
through
a
study
of
the
unperturbed,
radially
symmetric
flow.
The
30
pertu
rbed
flow
is
illustrated
in
§
) ,
with
an
emphasis
on
the
mean
flow.
The
growth
of
the
mixing
layer
is
finally
interpreted
in
§
6
using
conclusions
from
the
radially
symmetric
flow
and
from
a linear
stability
analysis.
2.
Computational
approach
To
be
able
to
computationally
represent
the
large
dynamical
range
of
scales
produced
in
such
compressible,
high-
Reynolds-number
environments
,
we
consider,
as
described
in
Lombardini
et
al.
(
201
I),
the
two-component
Favre-filtered
Navier-Stokes
eq
uati
ons
governing
the
transport
of
the
filtered
density
p,
momentum
pu;
and
total
energy
£
of
the
mixture,
and
the
filtered
heavy-fluid
partial
density
pl/I,
with
1/1
the
heavy-fluid
mass
fraction.
We
explicitly
model
the
subgrid
terms
representing
unresolvable
flow
features,
i.e.
below
a
fixed
cutoff
scale
taken
here
as
the
finest
gri
d
si
ze
L\
in
the
co
mput
ational
domain.
Th
e
SGS
repr
ese
ntati
on
is
essentially
an
extension
of
the
stretched-vortex
model
of
Mi
sra
&
Pull
in
( 1997
) to
compressible,
mu
l ticomponent
flows,
in
which
the
small-sca
le mixing
is
descr
ib
ed
by
the
st
irring
of
a passive
scalar
transported
under
the
flow
of
the
axisymmetric
vortex
Pullin
(
WOO
).
Shock
waves
are
captu
red
numerically
using
a weighted
essent
ia lly
non-oscillatory
(WENO)
scheme
and,
in
that
sense,
the
shock
thickness
is
the
only
physical
scale
in
the
problem
that
is
neither
resolved
computationally
nor
modelled
physically.
Away
from
shocks,
the
numerically
diffusive
WENO
scheme
reverts
dynamically
to
a
low-numerical
dissipation,
tuned
centre-difference
(TCD)
sc
h
eme
opt
im
al
for
the
accurate
computation
of
regions
of
turbulent
mixing
where
the
SGS
model
activates
itself
H ill
&
Pull
in
(200.f
).
The
switching
of
advection
schemes
is
performed
using
a
robust
shock
sensor
Lombard
ini
(2008
).
To
reduce
nonlinear
instabilities,
e.g.
of
the
aliasing
type,
we
avoid
any
filtering
or
numerical
damping
techniques
which
could
alter
the
computation
of
high-Reynolds-number
rurbulence,
and
prefer
to
rewrite
the
TCD
discretization
of
the
convective
terms
in
the
momentum,
energy
88
M.
Lomhardini
, D.
f
Pullin
and
D.
I.
A1
eiron
and
scalar
equations
in
an
energy-con
serving
(skew-symmetric
)
form
Honein
&
Moin
(
200~
).
Te mporal
stability
is
achiev
ed
by
the
use
of
the
optimal
third-order
strong-stability-preserving
Run
gc-
Kutt
a
time-stepping
scheme
of
Gottlieb,
Shu
&
Tadmor
( 200
I ).
In
the
spherically
conver
ging
RM
flow
presently
studied,
it
is
computationally
advantageous
to
dynamically
refine
shock
waves
and
geometrically
contracting/
expanding
turbulent
regions.
This
is
undertaken
using
the
block-
structured,
adaptive
mesh
refinement
( AM
R) algorithm
of
Be rger
&
Colella
(
1989
)
as
implemented
by
Deiterding
(2005
).
The
above
numerical
method
is
formulated
on
uniform
Cartesian
grids
and
is
effectively
applied
to
each
s
ubgr
id
of
the
AMR
hierarchy.
Thanks
to
a
refinement
criterion
based
on
the
local
density
and
mass
fraction
gradients
, we
e nforce
that
the
evolution
of
the
various
shocks
and
the
entire
mixing
zone
are
solved
discretely
on
adaptive
grids
refined
to
the
maximum
level
(i.e.
with
grid
cell
of
size
L1),
at
all
tim
es
in
the
simulation.
Owing
to
the
radial
symmetry
of
the
problem,
we
investigate
various
statistics
on
spherical
surface
s o
f radius
r.
Cartesian
AMR
data
are
first
interp
o lated
over
a regular
Gaussian
grid,
i.e
.
al
the
points
of
spherical
coordinates
(
rr
2rr)
(
r.
B;.
</>
;)
=
r,
i
No
.j
N
ib
.
i
=
0
..
.
..
N
8
(r)
,
j
=
0,
..
. ,
N
¢(
r)
-
I ,
(2.1
)
where
N
0
(r)
=
N
tb
(r)
/ 2
=
rn:r/L1
l
This
ensures
that
the
finest
resolved
sca
les
are
retained
during
the
interpolation
process.
Surface-interpolated
quantities
Q(r,
()i,
</ij;
L)
will
be
used
for
spectral
statistics
in
§
4
of
Part
2,
or
for
the
discrete
evaluation
of
spherical-surface
averages
formally
defined
as
(Q
}(
r,
t )
= -
1
J[
Q(r
,
(},
¢;
l )
d.Q
,
4n
J!
n
(2.2)
where
d.Q
=
sin
(}
d()
d¢
is
the
local
infinitesimal
solid
angle.
Fl
uctuations
from
(Q}
and
from
Favre-like
surface
averages
Q
=
(pQ
)/
(p
),
where
p
is
the
density
of
the
mixture,
are
given
by
Q' (r.
(),
</J:
t )
=
Q(r,
B.
</J;
t)
-
(Q}(r , /),
Q''( r ,
()
,
</J
;
t)
=
Q(r
,
()
,
</J
;
l)
-
Q(r,
t).
(2.3a
,b)
3.
Problem
set-up
3.1
.
Geometry
We
consider
two
comp
utational
domains:
an
entire
sphe
rical
volume
"f/,
for
the
production
of
spectra
l statistics
taken
within
the
mixing-layer
cent
re,
as
well
as
an
oc
tant
of
a
sphere
t;
18
,
which
is
less
computationally
expensive,
for
the
extensive
processing
of
various
surface
averages
(simply
replace
4n
by
n / 2
in
(2
2)):
"f/
-
{Cr
,
(}
,
</J
)l
r
u
11
~
r
~
r
ex
t•
O
~
()
~
n.
0
~
¢
<
2n
},
Yi
1s
-
{
(r,
(},
¢
)1
r
;m
~
r
~
r
e
.ct•
0
~
(}
~
n /
2.
0
~
¢
~
rr
/
2}
.
(
3.la
)
(3.
lb
)
A
20
view
of
'Y!
18
is
depicted
in
figure
I.
f
n
both
domains,
an
in terior
spherical
reflective
wall
of
radius
r ;
n1
is
used
to
regularize
the
centre,
while
inflow/outflow
Turhulent
mixing
driven
hy
spherical
implosions.
Part
I
---
I
!
Refl
ecti
ng
!/free
-s
li
p
wall
I
....
....
....
Outer
nuid
I
nner
flu
id
un
shocked
· 1·
Perturbed
spherical
i
ncerfa
ce
at
rest
Outtlow/intlow
spherica
l boundary
',,/
'
'
'
\
\
\
lncident
\
spheri
ca
lly
CO(lvcrging
/shock
\
\
\
Retleccing
\
\
\
YI
Rctl
ecting
frt:e
-s lip
~
~~~~~~~:·-·
-
·
-
·
-
·
-
·
-
·
-
·
·
-·-·-
'""'"•
\'"
':
-
-
· -
· -
·-·
-
· _J
z
rin
, x
R
0
R,(t
)
r,..,
89
FI
GURE
1.
Planar
cut
z
=
0
of
the
simulation
in
an
octant:
an
incident
convergi
ng shock
of
Mach
number
M s
(l)
and
position
r =
R
s(l).
1
~
1
0
,
impacts
at
t=to
a perturbed,
spherical
interface
at
mean
position
r
=
R
0
separating
two
quiescent
fluids
of
different,
unifonn
densities
p
1
0
and
P
2ri
.
boundary
conditions
are
prescribed
on
an
exterior
sp
herical
boundary
of
radius
r exs·
A
ghost
fluid
approach
(
Fedkiw
et
al.
1999
)
is
utilized
to
numerically
incorporate
the
non-Cartesian
boundary
co
nditions
arising
at
the
interior
and
exte
rior
boundaries.
F
or
practical
reasons,
we
choose
r
,,,,
=
Ro/
40
and
r
exi
=
(
rr.
/2)
R
0
,
where
Ro
is
the
initial
position
of
the
unperturbed
interface.
The
region
around
rm,
is
refined
to
th e
maximum
level
at
all
times.
The
base
grid
resolu
t ion
is
128
3
for
"f/
and
64
3
for
"fl'.
18
,
on
which
two
additional
levels
of
refi
n
ement
can
be
applied,
each
wit
h
a refinement
fac
t
or
of
two
in
each
Cartesian
direction
. As
a
resu
lt,
the
finest
reso
lution
(if
the
enti
re
domain
were
re solve
d
by
the
hig
h
est
level
of
refinement
in
each
direction)
is
5 12
3
for
'Y
a
nd
256
3
for
"ft
18
,
the
finest
g rid
s i
ze
bei
ng
th
en
~
:::::::
0.006R
0
for
both
domains.
F
or
11
18
,
it
is
assumed
that
the
shock-wave/boundary-laye
r interaction
does
not
play
a
dominant
role
in
the
growth
of
the
turbulent
mixing
zone
a
nd
free-slip
boundary
conditions
can
be
applied
al
the
reflective
walls
x
=
0,
y
=
0
and
z
=
0.
3.2.
Converging
shock
T he
flow
behind
the
spherically
converging
s h
ock
is
initialized
wit
h
an
approximate
so
lu
tion
of
the
radially
symmetric
Eu
l
er
equations
Chisnell
(
1998
),
which
gives
(p,
u,,
p),
normalized
by
the
density,
radial
velocity
a
nd
pressure
i
mm
ediately
be
hind
90
M
Lomhardini
, D. l
Pullin
and
D. I. Meiron
(a)
Air
~
SF
"
Ao
::::::
0.
6
67
Ms
0
=
1.2
(b )
SF
6
~air
A
::::::
- 0.
667
Ms
0
::::::
l.222
TA
BLE
I .
Tabl
e
of run
s
fo
r the
light
- heavy
and
heavy-light
configuration
s considered
indicating
pr
e-s
hock
Atwoo
d rati
os
and
incident
shock
Mach
number
at
impact
t
=
t
0
.
the
shock,
respectively
, as
a
function
of
Che
similarity
variable
r/ R
5
(1),
where
the
shock
positi
o n
R
5
(t )
is characteriz
ed
by
a similarity
expon
ent
n ( y )
Gud
e rley
(
1942
).
For
a shock
travelling
in
air,
y
~
1.40
gives
n
~
0.7
l
7,
while
in
SF
6
,
y
~
1.09
and
n
~
0.799.
The
self-similar
structure
has
been
pre
liminary
confirmed
by
simulatio
ns
of
a
single
converging
shock.
In
particular,
n
has
been
computed
before
and
after
reflection
at
the
centr
e
Lombardini
&
Pullin
(2009
).
In
itializing
the
shock
using
Chisnell
' s self-similar
solution
not
o nly
avoids
spurious
wav
es
that
would
appear
if
setting
up
the
shock
as
a
R iemann
problem
solution
for
the
strictly
axisymmetric
shock-imp
losion
process,
but
also
leaves
the
shock
thickness
as
the
onl y
intrin
sic
length
scal
e . Chisnell'
s solution
is
also
used
as
a
time-dependent
inflow
boundary
conditions
at
r
=
r
,ni,
until
the
exploding
reshock
exits
the
computational
domain
,
after
which
ze
ro-gradient
boundary
conditions
are
applied.
We
define
an
initial
Atwood
ratio
Ao
=
(P
io
-
P
io
) / (Pi
u
+
P2
0
),
where
p
10
and
P
io
denote
the
initially
uniform
densities
of
the
pure
fluid
(i.e.
as
the
mass
fraction
1"
=
0
or
I )
respectiv
e ly
contained
within
the
sphere
of
radius
Ro
and
exterior
to
thi
s
sphere
.
For
an
air
-
SF
6
gas
combination
with
equal
init
ial
temperatures,
Ao
is
simply
given
by
the
ratio
of
molecular
weights:
JA
ol
~
0.667.
The
converging
shock
is
said
to
impact
a
density
interface
in
a
light
- heavy
fashion
when
Ao
>
0
and
in
a
heavy-light
fashion
when
Ao
<
0.
In
the
light
- heavy
configuration,
the
shock
is
positioned
behind
the
slightly
perturbed
interface
such
that
its Mach
number
at
impact
is
M
50
=
1.2
.
The
initial
perturbation
shape
being
fixed
, the
impact
Mach
number
in
the
heavy
- light
shock
interaction
is
chosen
such
that
the
perturbation
grows
with
che
same
small-amplitude
growth
rate
as
in
the
light-heavy
refraction,
i.e.
matching
k(aoA
o
+a
t At )
tiu,
where
tiu
is the
change
in
radial
velocity
due
to
shock
refraction,
a
0
is
the
pre-shock
perturbation
amplitude
and
'+'
denot
es
post-shock
quantities
.
Vandenboomgaerde,
MUg
l
er
&
Gauthier
(
199~
)
have
shown
that
this
adjustment
of
Richtmy
er
's
original
formula
app
li
es
welJ
to
both
ligh
t- heavy
and
h
eavy-light
config
urati
ons.
Thi
s
gives
a
Mach
number
Ms
0
~
I .222
at
the
heavy-light
impact.
L
ight-heavy
and
heavy
- light
co
nfigurations
are
summarized
in
table
I .
3.3.
Initial
interfacial
perturbation
We
describe
here
how
to
build
an
ini
tiaJly
isotrop
ic
perturbation
at
a
spherical
density
interface
of
radius
R
0
(i.e.
no
preferred
directions
when
moving
a l
ong
the
spherical
surface)
, with
the
idea
that
the
turbu
lent
mixing
induced
by
the
passage
of
concentric
pressure
waves
would
remain
somehow
isotropic,
and
could
then
be
ana
ly
sed
ass
uming
statistica
l isotropy
on
a sphere.
We
define
by
r
-
~
0
(0,
</>
)
=
0
the
perturbed
surface
where
light
(air)
and
heavy
(
SF
6
)
fluids
have
equal
mass
frac
tions
. The
heavy-fluid
mass
fraction
field
1"
(r ,
e.
<f>;
t)
initially
takes
the
form
o f a hyperbolic
tangent
profile
centred
at
~
0
(
0
,
</>
)
and
with
characteristic
thickness
L
i/to
=
0.002R
0
•
This
is
large
enough
to
ensure
the
resolution
Turbulent
mixing
driven
hy
sphe
ri
ca
l implosions
. Part
1
of
che
diffuse
interface
given
by
1/t(
r.8,</J;t
0
)
=
-
1
--
tanh
.
1{
IAo
l
[r
-
~0(8
,
</J
)
]}
2
Ao
L
l/lo
~
o
(O
,
</J)
=
Ro -
a
o!
f(
Ro.
0,
</J
)I
.
91
(3 .2a
)
(3.2b
)
where
f(
R
0
,
f),
</J)
is a
perturbation
field.
Our
exper
ience
with
RM
sim
ulati
o ns has
m
ot
ivated
th e u
se
of
the
ab
solute
valu
e
off
in
(3
2/?
):
the
singularity
at
the
zeros
of
[fl
cor
r
espo
nd
s to
a spread
in
the
wavenumber
space
that
encourages
earlier
nonlinear
growth.
In
our
early
planar
RM
investigations,
we
had
also
found
that
the
absolute
value
m
ode
ls well the
regularity
of
the
membrane
used
in the
shocktu
be
experiments
of
Vetter
&
Sturtevant
( I
99'i
).
In order to
bu
ild
a homogeneous,
isotropic
perturbation
fie
ld,
f
is
dec
ompose
d u
sing t
he
spherical
harmonic
s basis
as
:x:
t
f(
R
o,
0,
</J
)
=
L L
ftmY
ltn(
8 ,
</J)
,
(3.3
)
e
=0
111
=- e
with
coefficients
cos
(
2nw
~)
.ftm
=
J (
2e
+
l)C
r
,
JL-
J=-t
cos(2nw{)
2
(3.4)
where
~
are
r
an
d
omly
generated
numbers
in
[O.
I],
and
Ct
are
given
by
1
1
[
(
e
-
eo)
2
]
C
e=
exp
-
2
.
4
(2£
0
+
I )
CJ
o
../2n
2CJ
0
(3 .5)
The
pertu
rbation
is
therefo
re
r
ep
resented
by
a
wavepacket
of
modes
with
random
phase
and
so-ca
ll
ed
'ang
ul
ar
power
spectr
um
'
Ct .
We r
ef
er
to
§
4
of
Part
2
fo
r the
mathematical
derivation
off.
Tn
(
1.5
),
we
have
chose
n a narrow
Gaussia
n profile
with
domi
nant
s
ph
e rica
l wavenumber
£
0
=
40
and
variance
CJ
o
=
l
0
/
30.
Fo r
an
entire
s ph
ere
of
radiu
s
R
0
,
the
choice
for
e
0
would
r
ough
ly correspond
to
40
wavelengths
in
both
ort
h
ogon
al
dir
ect
i
ons
of
a plane
of
extent
2-J2n
R
0
x
2-J2n
R
0
( I 0
wavelengths
for
the
planar
an
alogue
of
an
oc
tant
),
i.e. an
equivalent
radial
wav
en
umber
Ko
=
i
0
/ R
0
;
CJo
is
suc
h
thal
the
width
of
the
Gau
ss
ian,
l o
-
3ao
;S
e
:S
lo
+
3cr
0
,
is
we
ll
within
the
sp
ace
of
reso
lved
spherica
l wavenumbers.
I n
( l
'i ),
the
n
orma
lizati
on
co
nstant
in
front
guarantees
t
hat
the variance
of
aof
matches
the
int
erface
disp
lac
e me
nt
va
rianc
e
of
a
si ngl
e-mode,
planar
corrugacion
of
maximum
amplitude
a
0
and
wavenumber
ko
=
t
0
/(-J2R
0
)
in
both
orthogonal
dir
ec
tio
ns
of
that
plane.
We
fix
a
0
=
0.03
R
0
,
i.e.
an
initial
amp
litude-t
o-wavelength
ratio
of
appr
oxim
ately
20
%,
larg
e e
nough
to
precipitate
the
nonlinear
development
o f the
perturbation.
Th
e
initial
perturbation
shape
obtai
n
ed
is
depicted
in
figure
2.
Instea
d
o f
emp
loying
a G auss
ian
power
s pectrum
, which
decays
ex
po nentially
at
hi
g h
waven
u mbers
,
num
e
ri
ca
l
studies
of
planar
R
M/
RT
flows
hav
e often
co ns idered
a broadband
co mbination
of
modes
sa
ti
sfying
a power-law
d
ecay
ing
spectrum
ex
K-q,
q
>
0
(e.g.
Th
ombe
r
et
al.
20
0).
Thi
s perturbation
s hap
e
is o
f practical
importance
as
it
is representative
of
the
mea
sur
ed
surfac
e
finish
of
an
I
CF
capsule
Barn
es
et a
l.
( 2002
).
F
or
s imp
licity,
we
presently
foc
us o
n a
narrowband
Gau
ss
ian
spectrum
and
92
M. Lomhardini.
D.
l
Pullin
and
D.
[
Meiron
FrGURE
2.
(Colour
online)
Tnitial
isosurface
of
equal
mass
fractions.
The
statistically
isotropic
initial
perturbati
on
field
has
a characteristic
amplitude
a
0
=
0.03R
0
and
a shape
given
by
a Gaussian
angular
power
spectrum
with
dominant
spherical
wavenumber
£
0
=
40
and
variance
a
0
=
£
0
/30.
leave
for
future
work
the
study
of
the
influence
of
multimode
initia
l perturbations
on
the
growth
of
a spherica
l mixing
layer.
Using
a
decom
position
in
terms
of
a
sphericaJ
harmo
n
ics
basis
allows
one
to:
(i)
avoid
the
po
le
singu
larity
()
=
O;
(ii)
keep
a
homogeneo
us c
haracteristic
wave
n
umber
Ko
along
the
(0,
¢)-man
ifold;
(i
ii
) inject
wavenumbers
around
Ko
that
can
slig
ht
ly
alter
the
symmetry
as
well
as
stimulate
nonlinear
mode
coupling
and
he
n
ce
the
mixing.
Another
way
to
des
ign
a
per
turbation
that
satisfies
(i)
and
(ii),
bu
l
tha
t i
s not
str
ic
tl
y
speak
ing
sta
tistically
isot
ropic,
wo
uld
be
to
co
nside r for
examp
le
f
ex
sin (k
0
8)
sin[ko
sin(O)ef>
],
i.e.
a
single-mode
perturbation
with
constant
wavenumber
ko
in
the
directions
()
and
ef>.
However,
this
perturbation
function
wou
ld
not
work
well
for
the
whole
sphere
because
it
is
n
ot
periodic
in the
azimuthal
direction.
3.4.
Summary
of
initial
conditions
Th
e vector
of
conserved
quantities
(p.
p u .
£ ,
pl/t)
for
a
mixture
o f two
ideal
gases
is
initialized
in
the
simu
l
at
i
on
j ust before
impact
(i.e.
t
=
t
0
-
),
using
the fo
ll
owi
ng choice
of
(p,
u
,p,
i/f).
Th
e couple
(u ,
p)
is
equa
l to
(0 , p
0
)
ah
ea
d
of
the s
h
oc
k
r
<
R
5
(l
0
-
),
an
d
g iven
by
the
press
ure
and
rad
ia l ve
l
oc
ity
of
Chisne
ll
( 1998
) for
r;;:::
R
sCto
- ).
T he
field
i/f
is
given
by
O
'2.h
)
and,
ahead
of
the
shock
r
<
R
5
(1
0
),
p
is
the
n
directly
derived
from
1/1
assuming
continuity
of
pressure
and
temperature
across
the interface
.
ln
particu
lar,
ac
p
=
min(p
10
,
p-,_
,)
as
1/1
=
0
(pure
light
fluid
),
and
p
=
max
(p
10
•
P
i.,
)
as
1/1
= 1
(pu
re heavy
flui
d). F
or
r
~
R
5
(t
0
-
),
p
is
specified
followi
ng Chisnell
(
1998
).
4 .
The
radi
a lly
s
ymm
et
ric
flow
ln
this
sect
ion,
we
inspect
the
radia
ll
y
symmetr
ic
flow,
i.e.
when
the
interface
is
un
pe
rt
urbed.
An
(r.
t)
wave
diagram
is
obtained
from
one-dimensiona
l sim ulatio
ns