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Numerical and laboratory experiments of sidewall heating thermohaline convection

32 pages
Numeri al and laboratory experiments of sidewall heating thermohaline onve tion C. Sabbah, R. Pasquetti, R. Peyret Lab. J.A. Dieudonné, UNSA, UMR CNRS 6621, Ni e, Fran e V. Levitsky, Y.D. Chashe hkin Institute for Problems in Me hani s, Mos ow, Russia Abstra t Spe tral al ulations of the ow whi h develops near a heated verti al wall in a uid stably stratied by a salinity gradient are arried out. The ase of weak strati a- tion and heating due to a onstant heat ux is onsidered. The numeri al results are ompared to experimental ones. Both on qualitative and quantitative arguments, a satisfa tory agreement is obtained. The ell formation pro ess is dis ussed by fo us- ing on the o urren e of ounter-rotative ells and on the instability phenomenon of the simultaneous reation of ells. The high sensitivity of su h phenomena to slight disturban es at the initial time is pointed out. 1 Introdu tion Thermohaline experiments were mainly stimulated by the dis overy of layered stru tures in the o ean or in the atmosphere, where thi k homogeneous layers are separated by thin high gradient interfa es (see e.g. [1?). It is now re ognized that one me hanism whi h an lead to the formation of su h sharp interfa es is double diusive onve tion, i.

  • tion

  • uid refra tive

  • ux density

  • al ulations whi

  • uid

  • multi-domain method

  • uniform heat

  • ounter-rotating ells

  • initial onditions

  • numeri al


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pap
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[19,20
are
,21].
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ectral


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to
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n
on
temp
erheating
the
v
ratio
o

the
o
that
the
(4)
they
with
y:
wing

kno
Na
for
b
the
is
unit
the
v
ximation.

within
v
using
ector,
what
when
to
for
salinit
the
and
time,
temp

,
for
ariations
the
the
v
t
elo
,

linearly
y
and
,
term
are
,
for
y
the
buo
deviation
the
of
the
the
y
pressure
the
from
y
the
the
h
viscosit
ydrostatic
resp
one,
ely
assumptions
The
and
(
h

resp
are
ectiv
v
ely
)
for
Lewis
the
densit
deviations
(ii)
of
y
the
and
temp
)
erature
um
their
ers
(5)
the
that


of
of
uid,
as
the

yleigh
is
um
v
er
t
the
the
and
d-
buo
but,

the
equations
ysical
are
oin

of
the
really
w.
for
or
t
presen
e
problem
mains
are
.
b
w
ort-
er,
to
is
equations
a
vier-Stok
p

to
y
a
erned
v
go
t
w
length
uid
the
(i)
and
Th
to
appro
the
the
umerical
delled
b
and
salinit
Note
y
the

hoice
are
tration
from

their
length
mean

initial
enien
v
for
alues
mo
and
elling,

from
done
ph
our
p
of
t
n
view,
and
justied
erimen
slot
results.
yp
These
do-
quan
only
tities
Ho
ha
ev
v
it
e
alw
b
ys
een
ossible
made
use
dimensionless

b
enien
y
reference
using
for
the

follo
then
wing
discuss
reference
n
v
results
alues:
y
transfers
other
This
lengths.
,
is
salinit
is

in
for
analysis
the
the
material
umerical
deriv
exp
ativ
tal
e.
5V.∇S
IP I xI,J
k¯J y ΩI,J
k k¯Ω 1≤ k ≤ K ΩI,J
2 k k¯ ¯k = 1,...,K V IP (Ω ) p T S IP (Ω )I,JI,J
kΔT −σ T =f in ΩT T I,J
∗ kΔS−LeV .∇S−σ S=f in ΩS S I,J
kΔV −σ V −∇p=f (T,S) in ΩV V I,J
k¯∇.V =0 in ΩI,J
+ BoundaryConditions.
k = 1,...,K−1
[[V ]] = [[T ]] = [[S]] =0k k k
[[∂ V ]] = [[∂ T ]] = [[∂ S]] = [[p]] =0y x k y k y k k
k k+1¯ ¯[·] I = Ω ∩Ωk k
elds.
The
partial
dieren
domain
w
,
tial
e
equations
uit
are
es
appro
a
ximated

with
n
the
whic
Cheb
ximation.
yshev
for

bined

sp
metho
of
d
y
using
v
a
equation,
strong
or
form
dams-Bashforth
ulation.
the
The
order
general
Discr
Stok
in
es
(6)
problem

is
the
solv
-
ed
use
with
on
the
this
same
an
v
(9)
ector
w
space
alue
is
follo
used
for
for
im-
the
for
the
a
v
ectiv
elo
are

kw
y
y

The
onen
where
ts
the
and
eing
for
h
the
parallel
pressure.
is
Finally

,

with
e:
and
etization
y

salinit
an
the

for
time-step
the

space
implicitly
of
(8)
the
w
p
treated
olynomials
the
of
small
maxim
lead
um
b
degree
the
equation
large
.
with
e

(horizon

tal
salinit
axis)
.
and
is
yp
salinit
in
olation,
ection-diusion-t
order
(v
y

terms
axis),
(11)
adv
implicitly
steady
linear
a
Euler
(ii)
dierence
erature,
a
temp
discretized
the
deriv
set
in
of
d
the
v
grid-p
es
oin
at
ts
Stok
of
b
the
sub
sub-domain

the

for
to
equation

(
hnique
Helmholtz
osition
an
domain
(i)

ely
v

Along
)
ac
and
in
and
Discr
time-cycle
(7)
h
pro

iterativ
at
of
the
the
sub-set

of
but,
the
the
inner
t
p
the
oin
strongly
ts,
term
at
Considering

.
h
a
time-cycle
explicit
one
in
solv
as
es
term
the
if
follo
time-steps
wing
ery
problem:
to
F
ould
or
er
e
um
solv
Lewis
to
of
used
v
is
the
d
(10)
metho
the
ectral
wing
sp
tin
,
y
nd
ts:
pseudo-
y
in
the
a
F
domain,
plicitly
sub
treated
h
h

y
In
the
osed.

imp
extrap
are
A


transmission
using
and
b
natural
explicitly
,
e
and
v
,

tial
and
in
treated
essen
terms
the
The
domains,
appro
sub
ard
the

een
nite
w

et
using
b
b
terfaces
are

ativ
h
time
as
time:
:
etization
in
-
the
(12)
t
elo
pressure
the
and
problem
y
is

jump

the

terface
asso
a
generalized
A

metho
(iii)
pro
a
al
to
2.3
in
6σ f
∗V
T S V
Δ−σS
IP −IPN N
[p] = 0k
0.5(Ra )ϕ
8Pr = 7 Le = 100 Ra = 3.46× 10 R =ϕ ρ
1.66 K = 34 I = 250
5J = 70 6×10
x
τ ≈ 0.2s
t = 30mnF
0.014K
imp
ose
space
with

the

transmission


,
These
y

nal
matrices,
a
whic
divided
h
wing
are


from
in
ulate
preliminary
w

initial
asso
ha

of
for
v
the
in

v
ond-
n
ing
solv
homogeneous
t
problem
the
the
h
in
transform
terface
heater
v
v
alues
v
to
the
the
the
jumps.
Finally
Then,
men
the
o
full
whic
problem
alues
is
are
solv
b
ed
in
with
tal
the
done

the
v
at
alues
homogeneous
of
problem
matrices
and
,
.

use
and
of
inu-
(7),
at
out
the
ts.
in
elo
terfaces.
oin
In
w

units
h
from
sub-domain,
(S1
(i)
p
the
and
equation
ulations
for
temp
the
is
temp
to
erature
w
is
e
solv
that
ed
oid
with
v
a
dimensionless


metho
reference
d,
time
(ii)

for
ectiv
the
m
salinit

y
but
equation
.
one
exp
uses
Calculations
the
b

the
Conju-
alues
gate

Residual
b
(see
unkno
e.g.
alues
[33])
ed
iterativ
is
e
olation.
pro
using

o
with
at
the
or
Helmholtz
w
op
elo
erator
domains,
using
v
y

b
in
determined
i.e.
as
in
preconditioner,
grid-p
and

(iii)
used
the
to
generalized
mesh
Stok
near
es
the
problem
esp
is
ph
solv
time-step
ed

in
.
a
sim

S2)
w
b
a
till
y

,

using
These
a
from
P
for
oisson
for
equation
temp
for
disturb
the
amplitude
pressure.
expressions
The
,
pressure
e
b
v
oundary
to
v
tion
alue
to
is
v

to
b
small
y
alues
using
the
again
time-step,
an
h


matrix
the

v
hnique,
for
yielding
and
a
elo
div
y
ergence-free
resp
v
ely
elo
and

ultiplied
y
y
eld.
to
Nev
taking
ertheless,
terfaces,
due
the
to
3
the
and
use
erimen
of
results
the
ha
are
e
alues
een
v
with
terface
follo
in
v

of
the

t
dimensionless
yp
um
e
ers:
appro
wns
ximation,
the
the
of
pressure
v
is
with
not
,
unique
rst
and
(6)-(10)
aected
The
with
extrap
the
a
so-called
time-cycles,
spurious
previous
mo
w
des
the
of
y
pressure.
F
Consequen
the
tly
discretization
,
e
the

transmission
v

sub
on
with
the
alues
pressure,
the
Then,
and

is
oundary
equation
b

the
sub-domain,
and
ab
terms
the
forcing
,
,

the
oin
t
A
to
ordinate
no
is
are
in
for
,
As

ected,
the
disturb
p

ts
yields
the
in
where
etter
o
t
is
the
ecially
erimen
In
data
ysical
y
the
imp
is
osing
The
them
the
w
result
eakly
T
,
o
using
ulations
as
and
trial
ha
functions
e
a
een
set
erformed,
of
the
p
time
olynomials
terms
orthogonal
ts
to

the
.
pressure
sim
space
dier
k
the
ernel.

The
the
details
erature:
of
S2
the
initial
algorithm
erature
are
randomly
giv
ed
en
an
in
equal
[28
of


The
b
e
whereas
enforced
disturbances
strongly
used
.
S1.
The
exp
dicult
the
y
ed
has
S2
b
results
een
b
o
agreemen
v
with
er-
exp

tal
b
7oS = 5( / )oo
Tf
ϕm
Λ Tf
ϕ Pr Le Ram ϕ

o −2Λ(m) T ( C) ϕ (W.m ) Pr Le Ra Rf m ϕ ρ
8107.8 18 71.9 7.15 99.3 3.23×10 1.72
8108.6 19 74.2 7.08 100 3.51×10 1.57
8103.2 20 69.4 7 100 3.46×10 1.66
α
α αϕ
α
ϕ
o −4 −1α α(18 C) = 1.8510 K
o −4 −1 o −4 −1α(19 C) = 1.9610 K α(20 C) = 2.0610 K
),
Lewis
salinit

mean
(
er,
eing
1
b
thermal
),
for
Ra
only
yleigh

(
of
er
(
y
h
la
ha
)
is
n
in
um
so
b
ma
ers
of
and
er
buo
(at
y
the

ers
ratio
t
(
on
er
erature.
upp

)
since
for
w.
the
b
exp
that
erimen

ts
n
and
v
for
balanced
the
.
sim
used
ulations.
heat
Exp.
ysical
the
estimate
,
sit
y
used
salinit
uid
kground
Stratication

from
eak

w
,
a
dep
with
oth
done
and
een
w
b
e
e
v
v
w
ha
salinit
ts
ery
erimen
w
exp
ma
oratory
observ
lab
equation
of
app
set
the
A
exp
disturbances.
for
initial

of
in

of
E1
b
the
y
study
in
to
v
order
that
in
T
5,
densit

the
in
and
S2
uid
of
v
those
),
to
to

ux
e
the
b
(
will
erature
S1
for
ulation

sim
b
the

for
the
obtained
expansion
results

The
T

whic
this
strongly
in
ends
used
b
are
the
so
y
2:
temp
E2
Here
where
e
1
v
able
simply
T
its
in
alues
wn
fresh
sho
ater,
as
the

y
the
v
for
lo
used
Ho
those
ev
to
it
equal
y
ely
e
ximativ
ed,
appro
the
ers
(5),
b
ts).
um
ears
n
through
dimensionless
pro
the
erimen
of
,
alues
that
v
the
yield
umerical
h
a
whic
hange
S1,S2
the
E2),
alue
and
the
(E1
y
them
e
of
b
o
a
w
hange
t
during
on
The
T
alues
e
y
able
are
fresh
in
w
able
Note
are:
for
ux
sak
mean
of
for
y
).
same
parameters
are
ph
for
the
v
alues
with
the
without
,
the
ater
(
(
Prandtl
y
stands
den-
latter
heat
T
),
go
lo
from
erature
to
temp
of
and
the
temp
dieren
),
t
w
dimensionless
length
n

um
b
1
men
in
(
W
en
giv
.
also
that
are
the
ulations
e
sim

umerical
,
n
notations
the
used
for
the

alues
used
or
alues
dimension,
v
dimension
the
eing
reference
tioned
alues
the
listed

y
o
in
easily
the
ones
determination
others,
the
main
The
v
main
are
dicult
in
fo
able
and
8−12 −1 −2L L κ κ L κ LT TT
5 −6 −1 −5 −10.12m ≈ 10 s 1.1810 m.s 0.9910 s
−1 −1 −1κ Lϕλ HΛ βT
−7 2 −1 o1.4210 m .s 14.1K ≈ 5 /oo
ω ψ T S
t ={t = 0.0048,t = 0.006,t =1 2 3
0.012,t (= t ) = 0.018} t = {8,10,20,30mn}4 F
∂ ρx
tF

• t ≈ t1
∂ ρ t≈tx 2

times
t
dieren
simultane
steep
the
the
temp
at
ariables.
ed,
of
y
v
displa
sho
ely
e
ectiv
together
resp

are
the
y
of
salinit
t
the
nal
and
temp
erature
the
temp
phenomena
the

,
egin

Afterw
the
on
,

y
A

of
v
ell
the
a
of
enlargemen
isolines
y
The
,
S2.
plexit
ulation
y
sim
the
the
the
with
,
obtained
b
results
forms
the
the
w
ectiv
sho
its
5
then
(
app
-
ones
2
This
Fig.

esults
tral
r
are
al
the

ation
3.1
Just
y
observ
salinit
earance,
erature
horizon
temp
from

able
y
of

and
v
at
in
time
ph
order
ysical
the
units).
of
In
and
Fig.
with
6
ts
is
y
giv
terfaces
en,
v
at
Essen
the
follo
same
v
times,
observ
the
up-o
y
the

distortion
elo
y
elds,

whic

h
form
ma
rst
y
ottom
b
the
e
new
view
in
ed
the
as
h
the
gro

a
tures

pro
ation
vided
the
b
and
y
the
the


t
Sc
,
hlieren

system.
a

o
that
efore,
these
streamlines
gures
as
only
the
visualize
the
the
eld,

strip
e
a
part
heater.
of
Main
the
es

ts
domain:
the
elsewhere
erature
the
salinit
uid
elds
is
the
nearly
time
at
length
rest.
in
Ho
to
w
w
ev

er,
y
one
the
observ
erature
es,
salinit
a
elds,
w
the
a
gradien
y
of
from
salinit
v
at
a
in
of
of


e
ectiv
ed

along
tially
heater.
the
Then
wing

ha
is
e
b
een
the
ed:
of
An
doubling
w

along

heater,

a
elops
of
to
salinit
set
isolines,
t
with
o
v
in
e

b
later,
to
to
at
set
extremities,
three
at

b
a
and
system
at
of
top.
dissipativ
ards
e

gra
ear
vit

y
of
w
rst
a
whic
v
go
es
in
whic
wing.
h
is
propagate
phase
in

to
essive
the
e
undisturb
of
ed
In
uid.

It
part
is
near
in
heater,
teresting
streamlines
that
v
the
v

t
has
dieren

for
this
the
w
ous
a
e
v
of
e-phenomenon
set
whic

h

w
b
as

analyzed
the
earlier,
are
b
ed
oth
w
on
as
its
app
the-
in
oretical
alues
and
reference
exp
of
erimen
tal
tal
es,
asp
w
ects
y
[34,35,32
the

A
Fig.
time
7
2
giv
the

pattern
v

ectiv
b
e
observ

all
and
the
T
the
heat
the
fron
width
t,

horizon
y
tal
mean
strip
a
es
pro
asso
:

h
with
dev
a
in
mo
a
dulation
of
along
w
the
small
v
ternal

and,

in
These
a
strip
of
es
9∂ ρx


δTm
h lm m
15cm hm
∂ ρ lx m
t =t1
ha
3αgδThaRa = , h =αδTΛa
νκT
Ka
h K =h /ha a m a
δT h l Km m m a
Ra
Ka
Ra
4αδT Ra ∝ (αδT)
ards
its

whic
the


tral
w
part.
the
egins
the
Within
a

sensitivit
h
yleigh

its
and
the
in
the
ternal
t

t
the
the
salt
the

since
tration
ed
tends
measured
to
o
b
[12
ecome
y
homogeneous.
ratio
This
t

to

for
h
,
ne
results

the
limited
the
b
orse
y
is
sti
ternal
salinit
to
y

gra-
e
dien
e:
ts
er
at
v
the

dieren
particle
t
y
outer
e
and
the
inner

in
,
terfaces.
heigh
3.2
oten
Comp
h
arisons
.
of
the
the
E1

S2.
al

and
ecially
exp
.
erimental
go
data
erimen
Bey
of
ond
the
the
with
go
.
o
erimen
d
heat
agreemen
when
t
um
of
results
the
n
exp
then
erimen
the
tal
v
and
ha
n

umerical
tities,
o
the
w
um
pattern
on
ev

olutions,
and
esp
heigh
ecially
of
with
erheated
rst
reac
the
neutral
formation
lev
of
at

v
v
t
ectiv
onding
e
erio

the
at
eep
the
to
ex-
the
tremities
of
of
the
the
rising
heater
out,
and
es
then
mean
with
ables
the
giv
sim
obtained
ultaneous
erimen

E2
of
sim
a
t
set
one
of
elds.

visible
v
,
ectiv
pro
e
n

ear
quan
d
titativ
the
e
E1,

heigh
ha

v
smaller
e
than
also
erimen
b

een
gro


out.
the
Th
E2,
us,
the
w
densit
e
esp
ha

v
yleigh
e
er

This
the
the
ev
of
olution
b
of
pro
the
heater
mean
of
temp
lo
erature
Moreo
rise
er,
b
e
rst
v

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