J
Fluid
M
ec
h.
(2014).
uo
l.
748.
pp.
113-(42.
©Cambridge
Univer
s ity
Pre
ss
2014
doi
:
I
0. 1017
/jfm.20
14
.
163
113
Turbulent
mixing
driven
by
spherical
implosions.
Part
2.
Turbulence
statistics
M.
Lombardin
F,
D .
I.
Pullin
an
d
D.
I.
Meiron
Graduate
Aerospace
Laboratories,
California
Institute
of
Techno
logy.
Pa
sadena.
CA
91125.
USA
( Received
12
J une
2013;
revised
29
January
2014;
accepted
20
March
2014;
first
published
online
28
April
2014)
We
present
large-eddy
simulatio
ns (LES)
of
turbu
l
ent
mixing
at
a perturbed,
spher
ical
inte
rface
separating
tw
o
fluids
of
differing
densities
and
subseque
ntly
impacted
by
a
spherical
ly
imploding
shock
wave.
Th
is
paper
foc
uses
on
the
differences
between
two
fundamenta
l configu
rati
ons,
keeping
fixed
the
initial
shock
Mach
number
:::::;1.2,
the
density
rat
io
(precise
ly
IAol
:::::;
0.67)
and
the
pe rturbation
shape
(dominant
sp
herical
wavenumber
e
0
=
40
and
amplitude-to-initia
l
radius
of
3
%):
the
incident
sh
ock
travels
from
the
lighter
fluid
to
the
heavy
one,
or
inverse
l
y,
from
the
heavy
to
the
light
fluid
.
fn
Part
l (L ombardini,
M.,
Pullin
,
D .
I. &
Meiron,
D.
I.
,
J.
Fluid
Mech.,
vol.
748
,
2014,
pp.
85-112),
we
described
the
computational
problem
and
presented
result
s
on
the radially
symmetric
flow,
the
mean
flow,
and
the
growth
of
the mixing
layer.
l n
particu
lar,
it
was
shown
that
both
configurations
reach
similar
convergence
ratios
:::::;
2.
Here,
turbu
lent
mixing
is
studied
through
various
turbulence
statistics.
The
mixing
activity
is
first
measured
through
two
mixing
parameters,
the
mixing
fraction
param
et
er
e
and
the
effective
Atwood
ratio
A
..,
,
which
reach
similar
late
time
values
in
both
light
-he
avy and
heavy-light
config
urations.
The
Taylor-scale
Reynolds
numbers
attained
at
late
times
are
estimated
at
approximate
ly
2000
in
the
light-heavy
case
and
1000
in
the
heavy-light
case.
An
anaJysis
of
the
density
self-correlatio
n
b,
a
fundamenta
l quantity
in
the
study
of
variable-density
turbulenc
e,
shows
asymmetries
in
the
mixing
layer
and
non-
Boussi
nesq
effects
ge
nera
ll
y observed
in
high-
Reynolds-
number
Ray
le igh- Taylor
(RT
) turbulence.
Th
ese
traits
are
more
pronounced
in
the
light
- heavy
mi
xing
layer,
as
a
result
of
its
flow
history
, in
particular
because
of
RT
-un
stable
phases
(see
Part
1).
Another
measu
re
disti
ng
ui
sh
in
g
li
ght-
heavy
from
h
eavy-
li
ght
mixing
is
the
velocity-to-scalar
Tay
lor
mi
croscales
rati
o.
In
particular,
at
late
times,
large
r values
of
this
ratio
are
reported
in t
he
heavy
-
li
ght case.
Th
e
late-time
mixing
disp
lays
the
traits
some
of
the
traits
of
the
decaying
turbu
lence
observed
in
planar
Richtmyer
-M
eshkov
(
RM
)
flows.
On
ly
partial
isotropization
of
the
flow
(
in
the
sense
of
turb
ule
nt
kinetic
energy
(
TKE
)
and
dissipation)
is
observed
at
late
times
, the
Reynolds
n
or
mal
stresses
(and,
thu
s,
the
directional
Tayl
or
rnicroscales)
being
anisotropic
while
the
directionaJ
Kolmogorov
microscales
ap
pro
ach
isotropy.
A
spectral
analysis
is
developed
for
the
general
study
of
statistically
isotropic
turbule
nt
fields
on
a spherical
surface,
and app
lied
to
the
present
flow.
Th
e resu
ltin
g
angular
power
spectra
show
t
he
deve
lopment
of
an
inertial
subrange
approaching
a
Kolm
ogorov-like
-5/3
power
law
at
high
wavenumbers, similarly
to
the
scali
ng
obtained
in
planar
geometry.
It confirms
the
fi
ndin
gs
of
T homas
&
Kares
(Ph
ys.
R
ev.
Leu.,
vo
l.
I
09
, 2012,
075004
)
at
hi ghe
r convergence
ratios
and
ind
i
ca
t
es
that
the
turbu
lent
scales
do
not
seem
to
feel
the
effect
of
the
spherical
mixing-layer
curvature.
Key
words:
compressible
turbulence,
shock
waves,
tu
rbul
ent
mixing
t
Ema
il
address
for
correspondence:
111anu1:I
<!1
1
1.:allt'ch
~Ju
l l 4
.
\1.
Lomhardini,
D.
I.
Pullin
and
D.
I.
Meiron
1.
Introduction
Molecular
mixing
as
a
consequence
of
st1
rnng
by
fluid
motion
is
a
process
of
fundamental
impo
rtance
in
a
myriad
of
a pplications.
Variable-density
mixing,
i.e.
mixing
between
fluids
of
differing
microscopic
densities,
which
is
enco
un
tered
for
instance
in
geophysical
and
astrophysica
l flows,
combustion
and
fluidized
beds,
is
often
driven
by
acce
leration,
e.g.
gravity.
Wh
en
the
local
density
gradient
and
the
pr
essure
gradient
ge
nerated
by
the
accele
ration
field
are
misaligned,
barodinic
vorticity
is
generated
and
perturbations
of
the
initia
l
de
nsity
layer
can
grow
non
linearly
and
lead
to
turbulent
mixing.
In
a constant
acceleratio
n environment,
this
is
known
as
the
Rayleigh-Tayl
o r (RT
) instability
(Taylor
19'0
).
Its
impulsive
analogue,
e.g.
when
a density
interface
is acce
lerated
by
the
passage
of
a shock
wav
e,
is refe
rred
to
as
the
R ich
tmy
e
r-M
eshkov
(
RM
) instability
(R ichtmyer
1960
;
M
eshk
ov
1969
).
Unlike
RT
flows,
which
are
unstable
only
wh
e n the
density
gradient
is in
the
op
posite
direction
to
the
acceleration,
i.e
.
Vp
·
V
p
<
0
(e.g.
heavy
fluid
atop
light
fluid
in
a gra
vita
tional
field),
RM
perturbations
grow
whether
the
in
cident
sh
ock
wave
propagates
from
a
light
to
a heavy
fluid
(
Vp
•
V
p
<
0)
or
from
a heavy
to
a light
fluid
(
Vp
·
V
p
>
0
).
Th
e
study
of
barocli
nic
inst
ab
ilities
in
curved
geometries,
in
parc
icu
l
ar
whe
n
the
mean
isopycnic
and
isobaric
cylindrical
/sphe
rical
surfaces
are
concentric
to
each
othe
r,
is relevant
to
prob
le ms
sp
anning
a
wide
ran
ge
of
scales,
from
supernovae
collapse
(Joggerst
, Almgren
&
Woosley
2010
)
to
explosive
detonation
(Balakrishna
n
&
Me
non
201
I )
and
inertial
confi
n
eme
nt
fusion
(
ICF
) (
We i
se
r-She
rri
ll
et
al.
:2008
).
In
these
implosion/explo
sio
n-drive
n
flows,
density
inhomogeneities
are
not
on
ly
RM
unstable,
but
also
radially
accelerated/decelerated
and
the
refore
subj
ect
to
RT
in
sta
bilities
as
they
geometrically
contract
or expand,
as
discussed
in
Pa
rt
1
(Lo mbardini,
Pullin
&
Meiron
2014
).
We
have
also
seen
that
a spherical
interface
initia
ll
y
impacted
by
a
converging
shock
is
res
hocked
multipl
e
times,
whether
the
interface
is
initially
processed
in
a light-to-heavy
fashion
or
vice
versa.
The
turbulent
mi
xing
observed
in
RM
flows
with
reshock
has
bee
n studied
in
light
-he
avy
and
h
eavy-light
configurations
in planar
geometry
(Lombardini
et
al.
2011
),
but
little
work
h as
been
done
in
spheric
al
geometry,
fo r two-dimensiona
l
(2
0 )
ax
isymmetric
flows
(Glimm
el
al.
2002
)
or
fully
three-dimensional
(30 )
flows
(Youn
gs
&
William
s
2008
).
Th
o
ma
s
&
Kares
(2012
)
init
iali
zed
a
30
simulation
in a
n
oc
tant
from
a
2D
flow:
first,
a
20
axisymmetric
problem
with
azimuthal
perturbations
in
a
quadrant
was
run
up
to
afte
r the
fir
st
reshock;
at
that
poinc
the
so
lut
ion
was
rotated
into
a
30
octant
and
co
ntinu
ed
co
late
times.
You
ngs
&
Williams
(2008
)
and
Th
o mas
&
Kares
( 2012
)
u
se
d
num
er
icall
y
diffusive
sche
mes
co
captu
re the
shock
and
model
the
turbulent
dis
sipation.
l n
Part
1
we
de
sc
ribed
a different
approach
to
that
of
Youngs
&
Williams
(2008
)
to
simu
late
the
30
turbulent
mixing
driven
by
sp
herical
implosion
s . We
performed
Cartes
ian-
gr
id
based
large-ed
dy s
imul
ations
(LES
)
of
the
two-component
flow
usin
g
an
explicit
subg
rid-
scale
(SGS
)
model
and
a
low-
num
e rical
dissipation
a
dv
ectio
n
s
cheme
appli
ed
in
the
turbulent
but
smoo
th
regio
ns.
Th
e
computational
domain
co
nsidered
is e
ith
er
an
oc
tant
of
a
sp
here o
r a
full
sphere.
Th
e
initial
converging
s hock
is
ge
nerated
as
a
se
lf-similar
, radially
sy
mmetric
soluti
on
of
the
Eul
er
equation.
Spherical
harm
o nic
s
are
used
to
construct
the
pre-shock
interfacial
perturbations.
We
conti
nu
e
her
e
the
investigation
of
two
cano
nical
configuration
s:
the
Light
fluid
enclosed
by
the sphe
rical
interface,
or vice
versa,
as
summarized
in
table
1.
While
P
art
l
ha
s
focused
on
the
importance
of
understanding
the
mean
flow
fo
r
the
stud
y
of
the
mixing-layer
growth,
va
ri
ous
turbulence
stati
st
i
cs
are
pre
se
nt
ed
here.
In
§
2,
we
analyse
the
mixing
through
various
measures,
with
a particular
focus
on
variable-de
nsity
effects
through
the
density
self-cor
relation
b
defi
n
ed
below.
Some
Turhulent
mixing
driven
by
spherical
implosion
s.
Part
2
(a )
Air-?
SF
6
A
o~
0.667
Ms
0
=
1.2
(b)
Sf
6
.-
air
A
o~
- 0.66
7
Ms
0
~
1
.2
22
115
TABLF.
I . Table
of
runs
for
the
light
-h
eavy
and
heavy-light
configurations
considered
indicacing
pre-shock
Atwood
rat
ios
and
incident
shock
Mach
number
at
impact.
c h
aracte
ri
stics
of
the
turbulence
such
as
Ta
y l
or
and
Ko lm
ogorov
microscales,
based
on
turbulent
kinetic
energy
(
TKE
) and
dissipation,
are
discussed
in
§
3.
Section
4
d
eta
il
s
h
ow
to
perfo
rm
a
s pectral
analysis
of
turbu
le nt
fields
on
a
sp
herical
s urface
,
which
is u
sed
co
represent
power
spectra
of
the
late-time
turbu
l
ent
mix
i
ng
as
well
as
to
build
the
initial
perturbation
field
presented
in
Part
I.
2.
Mixing
s
tatistic
s
In
th is
section,
we
study
the
evolurion
of
some
important
stat
1st1cs
in
var
iable-
dens
it
y
flows
:
the
mixing
quan
tities
e
(t)
and
A
,.
(t)
,
and
the
density
self-correlatio
n
b(r.
t).
Th
ese
three
quantities
are
computed
from
spherica
l
surface
ave
r
ages
defined
below
. We
recal!_
that
fluctuations
from
a s
urf
ace
ave
r
age
(Q )
and
from
a
Fa
vre-like
surface
average
Q
=
(
pQ
}/(
p
),
where
p
is
the
density
of
the
mixture
, are
given
by
Q
'(
r.
e.
</J:
1)
=
Q (r.
e.
</J:
t )
-
(Q}(r.
t) .
O:
' (r,
e.
</J
;
l)
=
Q (r.
e.
</J:
t)
-
Q (r ,
l ) .
(2. l
a,b
)
We
in sist
chat
these
are
s
urf
ace-averaged
statistics
performed
on
LES
data
obtained
by
s
olving
the
Favre-filtered
Navier-Stokes
equations.
Fo
r clarity,
we
have
omitted
bars
and
tildes
that
usually
denote
the
filtering
operations.
2.1.
Mi
xing
quantities
A
and
A
,,
Consider
that
the
amount
of
mix
ed fluid
results
from
the passive
ch
em
ica l equi
librium
between
light
a
nd
heavy
fluids.
T
he
mass
fraction
of
product
is
[I
-
(1/1)
( r ,
t )]
(1/t)
(r,
t)
[ l -
(l/t
)
(re.
t
)]
(l/t
)
(r
n
l)
'
with
(
l/t
)(
r
,,
t)
= 0.5,
and
the
mixing-layer
width
8
at
time
1,
defined
by
8(1)
=
1oc
4 (
1/t
}
( 1 -
(
l/t
))
dr,
(2.2)
(2.3
)
can
then
be
int
erp
ret
ed
as
a
product
thickness
that
would
res
ult
if
the
entrai
ned
fluids
were
perfectly
mixed
in
the
directions
()
a
nd
</J.
Using
the
scalar
field
Y
=
21/t
-
I,
the
rat
io
J
oo
(I
-
( Y
2})
dr
Y(t
)
8 (
t)=
~
=--,
lo
(I
-
( Y
)2)
dr
8(t)
(
2.4
)
cha
ra
cte
ri
zes
the
re lativ
e
amount
of
molecularly
mixed
fluid
within
the
mixing
layer,
i.e.
the
toca
l
chemica
l
product
formed
(Ji'J)
re lative
to
the
maximum
chemical
L
16
M. Lomhardini,
D. l
Pullin
and
D.
!.
Meiron
produce
or
product
tha
t would
be
formed
if
all
ent
rain
ed
fluid
were
completely
mixed
within
the
layer-centre
sp
herical
surface
(8
),
as
defined
by
Youngs
(I
h>-i)
.
Here
,
molecular
mixing
is
a
surrogate
for
a
chemical
reaction.
Anocher
mixing
variable
is the effective
Atwood
ratio
A e,
which
is
defined
as
the
tu rbulent
density
intensity
(p '
2
)
1
1
2
/(
p } evaluated
at
the
layer-centre
surfac
e:
V(p
'
2
}
( r
(.
,
I)
A e(t)
=
.
(p )
(r
..
.
l )
(2.5)
Th
e
quan
ti
ties
8
and
Ae
complement
statistics
ba
se
d
solel
y
on
( Y)
(e.g.
o)
that
cannot
disting
ui sh
between
fluid
locally
mixe
d
for
example
at
a
fraction
Y (x ,
t)
=
0
an
d
unmix
ed
fluid
in
equ
al
proporti
o ns
in
a
particular
spherical
surface
for
which
(
Y)
(r
,
t)
=
0.
According
to
the
d efinition
s
of
8
and
Ae,
co
mpl
etely
mix
ed
fluid
(i.e.
homogeneity
across
the
lay
er-centr
e s
ph
eric
al
surface)
is c
haract
er
ized
by
8
=
l
a
nd
Ae/ A
=
0,
whereas
A
=
0
and
Ae/ A
=
l
cor
responds
co
com
plet
e
segrega
tion
(i.e
. immiscible
case).
Figur
e
I
compares
the
evo
luti
o n
of
th e maximum
chemica
l product
8(1)
with
that
of
the
total
chemical
product
.9>(t).
We note
that
, in
the lig ht- heavy
case,
.
?JJ
increases
by
a
factor
of
app
roximately
5
in
the
time
stretch
0 .6
~
6u
t/
R
o~
l.
I
following
the
first
reshock
and
an
RT
-unstable
window
, wherea
s
8
ri
ses
by
an
even
larger
factor
(approximately
6-fold)
in
that
window.
T his
suggests
not
o nly
an
intense
mixing
activity
, but
an
eve
n
more
im
portant
g r
owt
h
of larg
er,
entr
ained
eddies
which
are
res ponsib
le
fo r an
actual
decline
o f
e
immediately
following
the
reshock-induced
phase
reversal
(6
ut
/ R
0
~
0.6
)
until
6ur
/ R
0
~
0.8
(figure
2a
).
Fro m
then,
molecu
lar
mixi
ng s
usca
in
ed
by
RT
potential
energy
bec
o m
es
predominant
and
e
starts
inc
reasi
ng
whil
e
A
.,
dec
reases.
Ac
late
times
,
the
mixing
s lows down
as
the
mix
ing
layer
k
eeps
g rowi
ng
.
Ultimately
,
e
and
A
e/
A
reach
th e value
s
~
o.8
and
0.6,
respectively.
Tn
the
heavy
-
li
g ht
configuration,
the
increase
of
e
and
Ae
from
6ut
/ R
0
,2:
0.12
apparent
in
fig
ure
2(h)
confirms
that
the
main
contributing
factor
to
the
mixing
activity
is
the
RT
un
sta
ble
pha
se,
w h
en
the
inte
rface
decelerate
s inw
ard
prior
co
the
first
reshock
(see
figur
e
14b
of
Part
1).
The
two
gases
mix
at
the
fi
nal lev
e ls
E:>
~
0.8
and
A
e/
A
~
0.6
comparable
to
the
values
obtai
n
ed
in
th e light
- heavy
configuration,
even
tho ugh
the
turbulent
mixing
follows
from
a
differe
nt
fl
ow
history.
For
comparison,
the
planar
RT-un
stab
le
mixing
simulations
o f Cook,
Cabot
&
Mill
er
(2 004
),
which
al
so
e mpl
oyed
the
same gas
combination
and
an
initial
pe
rturbati
o n
spectrum
of
the
G au
ss
ian
type but
with
different
pe
ak
wav
en
umb
er
and
standard
deviation, achieved
8
~
0.78
and
A
e/
A
~
0.48.
2.2.
Density
self-correlation
b
We
have
see
n
in
appendix
-\
of
Par
t I that
the
mixing
lay
er grows
asy
mmetrica
lly,
with
the
sp
ik
es
and
bubbles
rising
at
different
mean
radial
ve
l
oc
ities.
This
is
not
o
nly
a
purely
geometric
effec
t, but
also
a general
f
ea
ture
of
mixing
at
high
A,
as
o
ppo
se
d
to
Boussi
nesq
flows
in
which
de
nsiti
es
are cl
ose
in
value
and
the
mixing
l
ayer
remains
symmetrical
around
the
centre
lin
e.
[n
variab
le-de
nsity,
multicomponent
fl
ows
at
vast
ly
different
densities,
the d
iffusive
ma
ss
flux
stro ngly
depends
on
the
ma
ss
frac
tio
ns
and
the
dens
ity
field,
which
is
it
self
a function
of
the
mass
fractions.
Thi
s l
eads
to
nonlinear
phe
nomena
that
are
not
see
n
in
co
nstant-den
sity
mixing.
For
in
sta
nce
, Livescu
&
Risto
r
ce
lli
(2009
)
co
mm
e
nt
how,
in
va
riabl
e-density
flows,
light
fluid
mixing
in
to
heavy
fluid
operates differ
entl
y
than
heavy
int
o
light
mixing,
and
how
this
int
roduces
a new
source
of
skewness
of
the proba
bility
density
function.
A
general
featu
re
of
mixing
at
high
A
is
the
tight
coupling
between
the
density
and
velocity
fields
that
require
s
a
careful
st
udy
of
turbulent
stati
st
ics
such
as
the
Turhu
le
nt
m
ixi
ng
dr
ive
n b
y s
ph
eric
al
i
mpl
osio
n
s.
Pa
rt
2
(a )
0.40
.--
--r---.--""""""T""-
---.--.....----.
0.40
0.
35
0.
30
0.
25
0
~
0.20
~
0. 15
0.
10
0
(b)
0.40
0.
35
0.
30
0.25
0
~
0.20
~
0.
15
0.
10
0.
05
0
0.35
/\
:
~
!
~
i
~
! \
.
..:
0.30
0 .
25
0
0 .20
~
'<>
0.
15
0 .
10
~~
...
~
..
~.:
0.05
~
..........
~~~
..........
~~
......................
~....----....
a
0.5
1.0
I .S
2.0
2.5
3.0
c
0.20
~
'<>
0.
15
0.
10
0 .05
..........
.............
......_._
.............
~
.........._~........._~......_~....._.............,o
0.2
0.4
0.6
0.8
1.0
1.2
1.4
(
!J.u
/
R
o)
t
117
FIG
U R E
I .
Total
molecular
ffilxmg
f!/J
(solid
lin
e)
and
maximum
molecular
mixing
8
(dashed)
versus
t:
(a )
air-+
SF
6
and
(b
)
SF
6
-+
air.
no rmali
ze
d
lurbulenl
mass
flux
a,
in
the
ith
dir
e ction
and
th e Favre
Reynolds
stress
component
s
Ru,
especially
the
TKE
,x;,
defined
re
sp
ectively
as
(2.6a
,b,c)
Whi
le
the
de ns ity
variance
(p
12
}
mediates
the turbul
e nt
ma
ss
flux
in
Bou
ssine
sq
flows
,
in
variabl
e
-d
e ns ity
mixing
(Livescu
&
Ristorcelli
2007
)
thi
s
role
is played
by
the
de nsity
-specific
vo lume
correlation,
or
densit
y
se lf-c
o rrelati
on,
b
=
- (p ' v
'}
.
with
v
=
- .
(2.7
)
p
Th
e
quamity
b
aff
ec
ts the
production
of
radial
turbul
e nt
ma
ss
flux
ar
t
hr
o ugh
the
term
b
a(p }
/a
r
(s
ee
appendix
..\..2
),
and
ar
se ts the e
ner
g y c
onv
ers ion
rare
through
the
producti
o n
te rm
ar
a
(p)
j
ar
in
the
surface-av
erag
ed
TKE
equali
o n (s
ee
appendix
A
3).
Figur
e
1
depict
s the
radial
profile
of
b
at
three
diff
erent
late
time
s,
t
=
2tR
t:
sl,
3tR
esl
and
41R
e
~
i.
The
bimodal
character
of
the
light-heavy
b-profil
e
(figure
}
a)
persists
as
the
spik
es
and
bubbles
continue
to
r.ransport
partially
mixed
gases
from
the
outer
118
M . Lomhard
ini, D. I
Pullin
and
D.
f
Mei
ro
n
0.7
5
lit
'\
./'\
/ •
i'.,
•
•
.I•
"
/,:
~·
··- ..
..
.:'>,
~
.....
(-j
0.50
/~·
i
~
0.5
0
~
~
....
...
0.25
0.
25
0
0.5
1.0
l.
5
2.0
2.5
3
.~
(b)
l.00
1.00
(\
0.75
!
•
..........
.
...
~
...
.
0.
75
~
f
~
e
0.50
r
0.50
-
"
~
!!
•
0.
25
0.2
5
.........................
~
.............
~
.......................
........_~
......._~_._.~
......
()
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
(
!!..u
/
Ro
)t
FIGURE
2.
Ratio
of
mixed
to
entrained
fluid
e
(solid
line)
and
effective
Atwood
number
ratio
Ae/A
(dashed
) versus
t :
(
a)
air
~
SF
6
and
(b)
SF
6
~air.
regions
of
the
mixing
layer.
As
the
mixing
zone
grows,
it
becomes
more
asymmetric
,
with
a stronger
peak
on
the
heavy
side,
as
confirm
ed
by
a positive
peak
of
the
ob
/
ot
-
profile
on
the
heavy
side
( not
shown
here)
.
The
he avy
- light
b-profile
evolves
more
into
a
single
peak
almost
aligned
with
the
layer
centre
as
observed
in
planar
RT
mixing
(Livescu
et al.
2009
),
but
stiU
witb
a slight
asymmetry
towards
the
heavy
side.
The
study
of
b
encompasses
the
previous
analysis
of
A
,,
that
was
a measure
of
the
density
fluctuat
ions
at
the
layer
centre:
the
Boussinesq
approximation
can
be
seen
as
the
leading
order
of
the
Taylor
expansion
of
b
for
small
density
fluctuations
€p
=
(p
12
)
1
1
2
/
(
p)
«
1 (the
value
f p
=
0.05
is
usually
taken
to
define
the
limit
be
low
which
the
Boussinesq
approximation
applies):
(2.8)
whe
re
the
odd
terms
characterize
the
skewness
of
the
density
profile.
Differences
between
b
and
the
normalized
density
variance
€~
indicate
non-Boussinesq
effects
already
anticipated
by
the
asymm
etric
growth
of
the
mixing
Layer
.
The
departure
Turbul
e
nt
mi
x ing
driv
en
hy
sph
erical
implo
sio
n
s.
Part
2
119
(a
)
0.25
I=
:!.tR_.I
0.20
t
=
31R
t<I
1=41R
t.<I
--
0. 15
b
'
'
0.
10
'
0.05
'
'
0
- 0.4
- 0.3
-0
.2
- 0. I
0
0.1
0.2
0
.3
0.4
(b)
0.3
0
0.25
0.20
/
_,
'
'
\
I
.
i
•
b
0. 15
.
\\
0.10
0.05
0
-
0.4
- 0
.3
- 0.2
- 0. 1
0
0.1
0.2
0
.3
0.4
(r - rc) / Ro
FIGUR
E
3.
Radial
h-profile
(
in
the
frame
of
th
e moving
layer
) a
l thre
e
diff
erent
lac
e times
:
(
a)
air-+
SF
6
and
(b)
SF
6
-+
air
.
from
the
Boussinesq
approximation
is
largest
at
the
edges
of
the
layer,
consistent
with
figure
...+
.
Tn
deed,
even
though
Ep
is small
towards
the
edges
(typically
<
0.1
at
the
edge
s, and
>0.3
at
the
interior
),
the
skewness
of
the
density
profil
e
is positive
on
the
light
side
and
negative
on
the
heavy
side,
all
odd
terms
in
(2
~)
thus
being
of
the
s
ame
sign
away
from
the
layer
ce ntre.
We
further
observe
from
figure
.f
that
b
has
larger
magnitude
than
€;
on
the
heavy
side,
and
smaJJer
magnitude
on
the
light
side.
Therefore,
at
fixed
Atwood
ratio,
the
Boussinesq
equa
tio ns
wou
ld
lead
to
sma
ll
er
e nergy
conversion
rate
on
the
heavy
side
of
the
layer,
compared
with
the
more
general
variable-density
equations
.
Th
e
surface-averaged
equation
governing
the
evolution
o f
h(r
,
t)
is
giv
e n
by
(
see
appendix
\ I)
ab+
ii,.
ab
=
2
a,
ab
_
2
(I
+
h )
a,
8 (p }
+
(p }
~
(r
2
(p'v'
u
~)
)
81
ar
ar
(p)
ar
r 2
or
(p
)
~
1h
tJ
(h
m
(h
rr
n
[
( u)
a
]
+ 2 (p )
(u
V·
u
)-~
ar
(? (u
,)).
(
2.9
)
(/
dV
)
120
(a)
"'
~
"'
-
~
(b)
""~
<.:
-
~
M .
Lomhardini.
D.
!.
Pullin
and
D.
!.
Meiro
n
1.50
1.25
1.00
0.75
0.50
0.25
0
-0.4
2.5
2.0
1.5
- 0.3
- 0 .2
-
0.1
0
I=
2tR~<I
I=
3tR,tl
1 =
41R
nl
--
0. 1
0.2
Cl.3
0.4
o~
~~~...._
~...._~...._~...._~...._~....___.
-0.4
-0
.3
-0.2
- 0.
t
0.1
0.2
0.3
0.4
(r
-re)/
Ro
FIGURE
4.
Rat
io
of
b
to
E;
across
the
layer,
at
three
different
late
times:
(a)
air~
SF
6
and
(b)
SF
6
--+
air.
Studying
its
right-hand
side,
as
we
ll
as
those
of
the
a ;
and
.%
equations
(see
appe
ndices
\.2
and
A
1),
is
relevant
to
the
characte
ri
za
ci
on
of
the
turbulent
mixing
and
to
seco
nd-order
moment
Reynolds-averaged
Navier-Stokes
(R
ANS)
modelling
in
variable-density
flows.
Figure
5
measures
the
relative
norm
of
each
term (across
the
mixing
layer)
as
a function
of
time,
depicting
that
term
(h
IV),
i.e.
the
production
of
b
by
specific
volume-dilatatio
n
correlations,
is
the
dominant
term
at
late
times.
In
che
lighc
- heavy
case,
(b
IV)
actually
accouncs
for
more
than
80
%
of
the
right-hand
side
across
the
whole
layer,
the
oth er terms
cancelling
each
other
o
ut
due
to
opposite
signs.
The
late-time
approximate
b-equation
is
therefore
ab
[
(v)
a
]
-::::
2 (p)
(v
V·u
)---(
r
2
(u
,))
.
ai
r
2
ar
(
2.
LO)
In
che
heavy-light
case,
term
(b
TV)
is
still
dominant.
When
accounting
for
possible
cancellat
i
on
of
terms
with
opposite
signs,
we
obtain
the
same
equacion
as
(
2
I
0)
near
the
edges
of
the
layer.
As
noted
by
L ivescu
et
al.
(200lJ)
for
planar
RT
mixing,
we
find
presencly
chat
(b
II) almost
balances
(b
TV)
at
the
l
ayer
interior,
as
they
have
close
magn
itu
de
but
oppos
ite sign:
0::::
-
(l
+
b)
a,
<J{p)
+
(p)
[(v
V.
u ) -
(r
v2}
:r
(r2
(u,}
)]
.
(p}
ar
"
(2 .11)