THREE-DIMENSIONAL INSTABILITY
Laurent Joly, Jean Reinaud
Departmentof Fluid Me hani s,ENSICA
1,Pl. E. Blouin, 31056 Toulouse, Fran e
laurentensi a.fr, jeanm s.st-and.a .uk
Patri k Chassaing
Institutde Me aniquedes FluidesdeToulouse
Allee duProf. C.Soula, 31400 Toulouse, Fran e
hassainensi a.fr
ABSTRACT
Some new elements aboutthe transition to
three-dimensionalityofvariabledensitymixing
layers are proposed. The two-dimensional
a -tion of thebaro lini torqueis examinedfrom
the strain elds. It is seen to yield
drasti- ally enhan ed strain ratesin the ore region.
Then the onsequen e on the development of
three-dimensional instabilities is detailed and
theasymmetri developmentofthestreamwise
ribs is des ribed. The streamwise ir ulation
is shown to be preferentially generated on the
heavy-side of the ore and the me hanism of
the translative instability issurprinsingly
pre-served. Comparedto the onstant density
sit-uation, the ampli ation of the 3D modes is
lowerand theironset isslightlydelayed.
INTRODUCTION
The generation-destru tion of vorti ity by
the baro lini torque may substantially alter
the transition dynami s of shear ows. In the
two-dimensional strati ed mixing layer under
theBoussinesqapproximation,Staquet(1995)
des ribedsmall-s alese ondaryeddieslo ated
at thesaddlepointsbetweenlarge-s ale
stru -turesonso- alledbaro lini layers. The
three-dimensional study of su h a situation lead
Showalter et al. (1994) to stress that
baro- lini allygenerated,orenhan ed,vorti es may
possibly result in an earlier transition to
tur-bulen e. In ompressible mixing layers, Lele
(1989)orSarkar&Pantano(2000)stressedthe
onsequen es ofhigh densityratios.
The present ontribution deals with the
thepresentaddressofthese ondauthoris"S hoolof
Math-emati sand Statisti s,UniversityofStAndrews,StAndrews
baro lini e e ts beyond the Boussinesq
ap-proximation but un orrelatedto
ompressibil-ity. The baro lini torque results from the
inertial omponent of the pressure gradient
only. The vorti ity evolves within a
quasi-solenoidal velo ity eldwithoutsu eringfrom
strong dilatational e e ts that are s aled by
any relevant Ma h number. This purely
iner-tial in uen e of density variations is likely to
o ur in highReynolds numbermixing of
u-ids of di erent densities orinthermalmixing.
Su h variable-density shear-layers were
previ-ously onsidered by Davey & Roshko (1971)
who on luded that the situation where the
lighter uidisthefasterleadstoanin reasein
theampli ationrateofinstabilityos illations.
More re ently a step further was a hieved in
thenumeri alanalysis ofSoteriou &Ghoniem
(1995) on spatially evolving 2D shear layers.
They on rmed that an asymmetri
entrain-ment resulting from the baro lini torque is
responsiblefortheshiftingofthe enter ofthe
mainstru tures into thelighter uid,a
di er-ent onve tionvelo ityandamodi ed
spread-ingrate.
Inthevariable-densitymixinglayerthe
vor-ti ity is redistributed in favor of the
light-side vorti ity braid, the other being
vorti -ity depleted in a rst stage and then fed
with a vorti ity sour e of sign opposite to
the one of the initial layer, see Reinaud et
al. (1999). These two opposite-sign
vorti -ity sheets lay around the vanishing primary
stru ture ore, still guring the enter of this
two-layers system. Then the braids are
on-tinuouslystre hedwhileenrollingtowardsthis
enter in a spiral like s heme. At in nite
ondaryrollups.
In three-dimensional ows the vorti ity
dy-nami sisalsoa e tedbythevortexstret hing
me hanism that enables enstrophy to travel
amongvorti ity omponentsthrough3D
insta-bility modes. The onsequen es of the
baro- lini redistribution of spanwise vorti ity on
the development of three-dimensionnal modes
isthefo uspointofthepresentpaper. The
in-terferen ewiththepairingpro essandfurther
subharmoni semergen e isnotyet onsidered.
The experimental eviden e, e.g. Bernal
& Roshko (1986), the stability analysis of
Pierrehumbert & Widnall (1982) and
Cor- os et Lin (1984), and the dire t simulations,
e.g. Rogers & Moser (1992), all onverge
to-ward a similar route to three-dimensionality
leading to streamwise vorti es lying in the
braid region as a result of both an
instabil-ity lo ated in the Kelvin-Helmoltz (KH)
bil-low alled translative instability(TI) and one
lo ated in the vorti ity-depleted braid
here-after alled shear instability (SI). Knio &
Ghoniem(1992) ontributedtothe rst
analy-sisofthese o-workingme hanismsunder
non-symmetri vorti ity onditions resulting from
a weak baro lini torque. They fo ussed on
symmetrylosses and a knowledged foruneven
intensi ation and weakening of the
stream-wise vorti ity.
As stated by these authors the baro lini
torque is responsiblefor su h a di erent
two-dimensional stru ture that the results on the
spanwise stability of Stuart vorti es or even
the KH billoware irrelevant to the 3D
stabil-itypropertiesofthevariable-densitysituation.
Thoughthis asedemandsa urrently
unavail-able stability study, a step further has been
attemptedinintensifyingthedensityvariation
and re ning the rosswise des ription of the
layer in order to get a full baro lini torque
e e t, i.e. opposite-sign vorti ity sheets as in
Reinaudet al. (1999). The ore being
vorti -ity depleted in favor of surrounding vorti ity
ups, the translative instability me hanism is
expe tedtoweaken leavingvorti ity ups
sub-mitted to the braid instability. This s enario
is examinedinthelastpart ofthepaper.
PRELIMINARIES
The baro lini torque
Thebaro lini e e tistheadditionaltorque
that isfelt when aninhomogeneous mass eld
id two-dimensional in ompressible ows, the
baro lini torqueistheonlysour eofvorti ity
variationalong theparti le path, asstatedby
the orrespondingvorti ityequation :
d t ! = 1 2 rP r (1) where d t = t
+ur is thematerial
deriva-tive. In this simpli ed situation the pressure
gradient is dire tly onne ted to the material
a elerationa=du=dtsu hthatthebaro lini
torque bis given by:
b=ar(ln) (2)
Hen e the interpretation of the baro lini
torqueasbeingofinertial nature. Thissimple
s heme is the one that a ts primarily on the
spanwise rollers of the KH instability leading
rstly to a strong asymmetry of the
vorti -ity eld, and further on yielding the
on en-tration of the ir ulation on thinning
vorti -ity sheets of opposite signs, see gure 1. In
three-dimensions the vorti ity budget is
om-plementedwiththevortexstret hingtermand
in real ows vis osity a ts at the di usion of
vorti ity gradients. At the zero Ma h number
limit, dilatation results from density di usion
only. Thedivergen eofthevelo ity eldbeing
s=ruand%standingfor(ln),the
ontinu-ity equation turns into an adve tion-di usion
equation : d t %=D%= s (3) Provided=p=+u 2
=2isthespe i pressure
head,themomentumequation anbere asted
inthefollowing form:
t u=u! r p r%+u (4)
The numeri al pro edure
Theseequationsaresolvedforthetemporal
mixing layer with the streamwise (x) velo ity
pro leinthe rosswise (z)dire tion:
u x =Uerf( 1=2 z Æ 0 ! ) (5)
Streamwise and spanwise (y) periodi
ondi-tions are imposed within a lassi al
pseudo-spe tral approa h on a mesh ounting up to
19296264 nodes yielding Reynolds
num-bers, based on the initial vorti ity thi kness,
of 500. The density ratio of the upper to the
x
z
(a)
x
z
(b)
x
z
(c)
Figure 1: Spanwise vorti ity !y ontours of the
two-dimensional variable density mixing-layerat t = 12: (a)
Re = UÆ 0
!
= = 3000 (b) Re = 500. The density ratio
S = upper = lower
=3. ( ) ontoursofthestraininthe
senseofCaul eldandKerswell(2000): ontourin rement
isU=3Æ 0
!
. Shadedregionwhererotationprevailsoverstrain
a ordingtoCK.
to promotetheKHand thespanwise
instabili-ties. Thetwo-dimensionalmodeisperturbated
with the eigenfun tions of the most unstable
mode 2D =2= x =0:48 p =Æ 0 ! given bythe
onstant density linear stability analysis. In
the variable density ase, a global onve tion
velo ity, u
, opposite to the group velo ity
of the 2D perturbation, is added to the
ve-lo ity pro le (5) in order to keep the main
stru ture entered in the temporally evolving
0
5
10
15
20
10
0
10
1
t
Γ
−
/ Γ
0
Constant density − Re=500
Density ratio 3 − Re=500
Density ratio 3 − Re=3000
Figure2:Thein reaseofthe ir ulation asso iatedwith
baro lini ally enhan ed negativeregions of vorti ity,
non-dimensionedbytheinitial ir ulation
0 = 2U. as y =0:5 x
, losetothemostunstable
wave-lengthgivenbythestabilityanalysisofStuart
vorti es by Pierrehumber & Widnall (1982).
Itsshapeisthestreamwiseinvariantrod(STI)
used in the three-dimensional stability
analy-sis of Rogers & Mosers (1992). As in Knio
& Ghoniem (1992), the hara teristi s ales
andinitial onditionsarethesamebetweenthe
onstant and variabledensitysimulations.
THE STRAIN RATE OF THE 2D MIXING
LAYER
Sin e the baro lini torque is strongly
de-pendent on the density gradients it is also
sensitive to the Reynolds number. Figures
1(a) and (b) illustrate the stru ture of the
two-dimensionalvorti ity eldofmixing-layers
with densityratio of 3 and Reynoldsnumbers
(basedontheinitialvorti itythi kness)of500
and 3000, respe tively.
Though the growth of the primary mode
energy is only weakly a e ted by the
baro- lini torque (not shown), it is striking how
the amounts of negative and positive
ir ula-tion asso iated with that vorti ity sour e is
anythingbutnegligeableagainsttheinitial
ir- ulation intheperiod. It isstraightforwardto
show that the latter is
0
= 2U where
is the period length. Figure 2 illustrates the
departure of the negative ir ulation from
that value. Sin e the ir ulation is an inv
ari-ant of the problem, a orresponding positive
ir ulation is generated on heavy-side braids,
seeReinaudetal. 2000. Again,thehigherthe
Reynoldsnumberthe more signi ative is the
departure fromthe onstant densitysituation.
0
5
10
15
20
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t
γ
Constant density − Re=500
Density ratio 3 − Re=500
Density ratio 3 − Re=3000
Figure3:Theevolutionofthestrain normaltothedensity
gradientatthesaddlepointbetweenthemain2Dstru tures.
0
5
10
15
20
10
0
10
1
t
S
max
Constant density − Re=500
Density ratio 3 − Re=500
Density ratio 3 − Re=3000
Figure4: Theevolutionof themaximumstrainrateSmax
(positiveeigenvalueoftensorD)overtheentiredomain.
2D strain eld is needed. The strain tensor
D = (ru+ru t
)=2 is proje ted there in the
dire tionoftheunitaryve tor nnormaltothe
lo aldensitygradient onformlyto,e.g. Ottino
(1989), : =D:nn. Figure 3shows how the
evolution of the stain rate at the saddlepoint
region isweakly a e ted by theredistribution
of vorti ity in the main stru ture.Thisresults
from the onservation of the overall
ir ula-tion withintheperiod,see Cor os& Sherman
(1984). But this global invarian e statement
does not extend to lo al amounts of spanwise
ir ulation. From gure1( )itis learthatthe
strain eld looses the symetry of the onstant
density roll-up. Compared to the ontours of
togetherwiththerotation-dominatedshaded
regionbyCaul eld&Kerswell(2000),itturns
out that the upperbraid (heavy uid side) is
the region of maximum strain rate and
mini-mum rotation. Figure 4 on rms that mu h
higher levels of strain are produ ed there
un-der the a tion of the baro lini torque. This
isretainedasamaindi eren eexpe tedto
af-Name up = ` R e N x u x = 0 y PS / 500 128 0. 0.023 VD 3 500 192 0.28 0.023
Table1: .Globalparametersofthepassives alarand
baro- lini allymodi edthree-dimensionalsimulations
in3DmixinglayersthroughtheSIme hanism.
THE STRUCTURE OF THE 3D
SHEAR-LAYER
Two simulationsareanalysedinthispaper,
one solving the passive s alar (PS) equations
and the other (VD) with fullvariable density
e e ts. The parameters of the PS and VD
ases arereported in table 1. Throughoutthe
paper,timeisnormalizedby =Æ 0
!
=U,
vorti -ity by the initial maximum 2U=Æ 0
!
and strain
by 1= . Unless quoted, the vorti ity ontours
in rement is always U=Æ 0
!
starting from zero.
The positive vorti ity ontours are sket hed
bysolidlinesand negative ontoursbydashed
ones. The ti marks along the spatial
oordi-nates aredistributedevery Æ 0
! .
The spanwisevorti ity
The stru ture of the spanwise vorti ity
ross-se tionisderiveddire tlyfromthefolded
distributionestablishedin 2D.In boththe rib
plane andthe"o -rib"one, spanwise vorti ity
isredistributedinthinsheetsofalternatesigns.
The ontour maps given in gure 5, taken at
t = 8, are still simple. In the "o -rib" base
plane, utting the enter of the upper
span-wisemushroomstru ture,two ounter-rotative
thinvorti itylayers arebrought loserto ea h
other than in the 2D ase, lo ally de ning a
jet ow of light uid towards the heavy side.
In the rib plane, the sket h is quite similar,
though no more symetri , to theone given by
Rogers &Mosers (1992) g. 19(a). The
anal-ysis of subsequent spanwise vorti ity maps is
more diÆ ultdueto the omplex stru tureof
the ore region as seen from the streamwise
vorti ity ontoursof gure 8.
The ribvorti es
Streamwise vorti ity is ollapsing into rib
vorti es as in the onstant density ase. At
t = 8, gure 6 indi ates that the streamwise
stru ture is growing more rapidlyon its right
sidelying above themain stru ture. This an
be learly asso iated with the favourable
ef-fe t of theadditional strain in that region, as
mentioned inthe 2D analysis. A weak region
x
z
ω
y
x
z
ω
y
Figure 5: Contours of spanwise vorti ity !y of the
three-dimensionalvariable densitymixing-layerat t=8. Top :
BasePlane rossse tion,bottom: RibPlane rossse tion.
of the still a tive translative instability
me h-anism. At t=12themain ontributionto the
streamwise ir ulation omes from the
domi-nantribvorti es,see gure7. Theme hanism
a tingonspanwisevorti ityin2D,namelythe
baro lini sour e on the light side and sink
on theheavy side, is re overed in the
stream-wisedire tion. From gure8 (right views)the
rib vorti es are developing with a ompanion
ounter-rotative layer on theheavy side. This
asso iation yields again a higher entrainment
of light uid into the heavy medium as seen
from thedensityhalf maps.
Analysis of the growthof 3D modes
The spatial stru ture of the rib vorti es
is seen to be a e ted by the modi ed strain
eld and by the streamwise baro lini torque.
The sensitivity of the ampli ation rate of
the three-dimensionalmodes to these hanges
is dis ussed. Figure 9 gives the time
evo-lution of the energy ontent of the pure
two-dimensional mode A , orresponding to
x
z
ω
x
x
z
ω
x
Figure6: Contours streamwisevorti ity!x att=8(top)
andt=12(bottom).Contourin rementsisU=Æ 0 ! ,ti marks areatÆ 0 ! .
the streamwise and spanwise wavenumbers
(k x ;k y ) = (2= x
;0), and of the umulated
three-dimensional modes , i.e. those whi h
satisfy k
y
6= 0. In this semilog plots the
slopes of linear portions are equivalent to the
exponential ampli ation rates of the small
perturbations. Both the PS and VD ases
undergo a two-stage evolution for A
3D , the
rst one onne ted withthegrowing2Dmode
and the se ond one spe i of the
three-dimentionalization of the ow. The VD
ex-hibits a delayed transition at t = 12 (9 for
PS) and a lower ampli ation rate
3D VD =
3D PS
=2. This rst on lusionstandsforthe
linear response of the layer to the presently
smallperturbations. The non-linearregime as
well as the higher Reynolds number domains
have to befurtherexplored.
REFERENCES
5
10
15
20
x
t
Γ
x
5
10
15
20
x
t
Γ
x
Figure 7: Time evolution of the streamwise ir ulation
x
(x;t). Top : passive s alar, bottom: variable density.
Me h.,vol. 170, pp. 499.
Cor os, G.M., and Sherman. F.S., 1984,
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S howalter, D.G., Van Atta, C.W. and
Lasheras,J.C., 1994,J. FluidMe h., vol. 281,
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Phys. FluidsA,vol. 7(8), pp. 2036.
Staquet, C., 1995, J.Fluid Me h.,vol. 296,
pp. 73.
y
z
ρ
y
z
ω
x
y
z
ρ
y
z
ω
x
Figure8:Contoursofdensityandstreamwisevorti ityofthe
three-dimensional variable density mixing-layerat t=12.
Contour in rementsforthe density(left)are=6. Top:
Mid-braid rossse tion,bottom: Coreplane rossse tion.
0
5
10
15
20
10
0
10
1
10
2
t
A
3D
*
,
A
10
*
σ
2D
σ
3D−VD
σ
3D−PS
VD
PS
Figure 9: Time evolution of the normalized energy inall
three-dimensional modes A
3D
and in the two-dimensional
oneA
10 .
