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l. J. Trans. Phenomena, Vo\. I, pp. 173-190Reprints available
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(fJ 1999 OPA (Overseas Publishers Association) N.V.Published by
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the Gordon and Breach Science
Publishers imprint.Printed in Malaysia.
An Axisymmetry-Breaking Instability of AxiallySymmetric Natural
Convection*
ALEXANDER YD. GELFGAP,t,PINHAS Z. BAR-YOSEPHa,ALEXANDER SOLANa
and TOMASZ A. KOWALEWSKIb
'Computational Mechanics Laboratory, Faculty of Mechanical
Engineering, Technion - Israel Institute of Technology,Technion
City, Haifa 32000, Israel; bCenter of Mechanics, Polish Academy of
Sciences,
IPPT, PAN, Swietokrzyska 21, Warsaw 00-049, Poland
(Received 25 September 1998; In final form 25 January 1999)
The three-dimensional instability of an axisymmetric natural
convection flow is investigatednumericaUy using a global spectral
Galerkin method. The linear stability problem separatesfor
different azimuthal modes. This aUowsus to reduce the problem to a
sequence of 2D-likeproblems. The formulation of the numerical
approach and several test calculations arereported. The numerical
results are successfully compared with an experiment on
naturalconvection of water in a vertical cylinder, which shows an
axisymmetry-breaking instabilitywith a high azimuthal
wavenumber.
Keywords: Axisymmetry-breaking instability, natural convection,
global Galerkin method
INTRODUCTION
Natural convection flows in axisymmetric regionsare common in
many technological processes andare widely used as experimental and
numericalmodels. When the characteristic temperature differ-ence
increases the laminar axisymmetric flow losesits stability and
becomes three-dimensional. Suchaxisymmetry-breaking bifurcations
always takeplace as a stage in the course of laminar-turbulent
transition. This study is devoted to the numericalanalysis of a
particular case when a steady axisym-metric flow becomes unstable
with respect to three-dimensional perturbations.
The most common way to study the axisymme-try-breaking
instability numerically is by the solu-tion of the full
three-dimensional unsteady problemwhere the axisymmetric solution
is taken as aninitial state (Neumann, 1990; Wanschura et al.,1996;
Verzicco and Camussi, 1997). A stability
* This paper is an extended version based on partial results
presented at the 10th International Symposium on TransportPhenomena
in Thermal Science and Engineering, Kyoto, 1997, and at the 27th
Israel Conference on Mechanical Engineering, Haifa,1998.
t Corresponding author. E-mail: [email protected].
-
174 A.Yu. GELFGAT et al.
analysis was applied analytically (Jones andMoore, 1979) or
numerically (Hardin et al., 1990;Goldstein et al., 1993) only in a
particular casewith heating from below, when the initial
axisym-metric state is a motionless fluid. However,
the27r-periodicity of axisymmetric flow allows one toreduce the 3D
stability problem to a sequence of2D-like problems. This follows
from the possibilityto expand the 27r-periodic 3D solution in a
trigono-metric Fourier series in the azimuthal direction. In
view of the orthogonality of the Fourier modes, thelinear
stability problem for each mode separatesfrom other modes. The
stability problem for eachazimuthal mode does not depend on the
azimuthalangle, and therefore it is analogous to the axisym-metric
stability problem. Such problems can bestudied numerically by the
approach which wasused by Gelfgat et al. (1996) (in the
followingreferred as G) for an axisymmetric case. Thisapproach to
the analysis of axisymmetry-breakingbifurcations of convective
flows using the globalspectral Galerkin method is discussed in the
presentpaper. The formulation of the method and severaltest
calculations are reported. It is illustrated howthe
three-dimensional stability analysis may be usedfor the explanation
of our recent experimentalresults (Kowalewski and Cybulski (1997),
in thefollowing referred to as KC).
FORMULATION OF THE PROBLEM
Consider a natural convection flow of a Boussinesqfluid in a
vertical cylindrical enclosure 0:::;r:::;1,0:::;z:::;A. The
dimensionless momentum, continu-ity and energy equations are
au au vau au v2-+u-+--+w---at ar r acp az r
ap 1(
a2u 1 au 1 &u
= - ar + Re ar2 + -;.ar + r2 acp2&u u 2 av
)+ az2- r2 - r2 acp ,
av av v av av uv-+u-+--+w-+-at ar r acp az r
1 ap 1 (&v 1 av 1 &v
= - -;.acp+ Re ar2 + -;.ar + r2 acp2&v v 2 av
)+ az2- r2 - r2 acp , (2)aw aw vaw aw-+u-+--+w-at ar r acp
az
ap 1(&w 1 aw 1 &w a2w
)= - az + Re ar2 + -;.ar + r2 acp2+ az2+ Gr(), (3)
au u 1 av aw-+-+--+-=0,ar r r acp az
(4)
a() 8() v a() a()-+u-+--+w-at ar r acp az
1(
a2() 1 a() 1 &() &()
)= Pr ar2 + -;.ar + r2 acp2+ az2 .(5)
(1)
Here (r, cp,z) are the cylindrical coordinates,v = (u, v, W)Tis
the velocity vector, p is the pressure,() is the temperature, Gr =
gj3f}.OR3/;;2 andPr = iJ!X. are the Grashof and the Prandtlnumbers,
A = H/R is the aspect ratio, g is thegravity acceleration, !3 is
the thermal expansioncoefficient, f}.O is the characteristic
temperaturedifference, iJ is the kinematic viscosity, X is
thethermal diffusivity, and Hand R are the heightand the radius of
the cavity (the overbar indicatesdimensional variables). We assume
that the wallsof the cylinder are stationary, with the
usualboundary conditions, and that there is an arbitrary(but
axisymmetric) thermal boundary condition.At this stage we leave
this condition slightlygeneral. In the following (see the section
Numer-ical Comparison with Experiment) we shall focuson specific
boundary conditions.
Let the basic axially-symmetric steady naturalconvection flow
which corresponds to the bound-ary conditions be the solution of
the system (steady
-
AXISYMMETRY-BREAKING INSTABILITY
axisymmetric part of (1)-(5»
uau + Wauar az
= - a::+ ~A ~r~+ ~~~ + ~X - ~) ,
uaw + Wawar azap 1
(
&W 1aw &W)= - az + Re ar2 + -;. ar + az2
+ Gre,
au + u + aw = 0,ar r az
uae + Wae =~ (&e +~ ae + &e ). (9)ar az Pr ar2 r ar
az2The main purpose of this study is to find out whenthe steady
axisymmetric solution {U(r, z), W(r, z),per, z), e(r, z)} becomes
unstable with respect tothree-dimensional perturbations.
Consider infinitely small perturbations {u, v, w,P, O}of the
velocity, the pressure and the tempera-ture which depend on the
three coordinates (r, 'P,z)and time t. The linearized problem for
the perturba-tions can be defined as
au au au au au-+ U-+ W-+u-+w-at ar az ar az
ap 1 (&u 1 au 1 &u
=-ar+Re ar2+-;'ar+r2a'P2
&u u 2 av
)+ az2- r2 - r2 a'P '(10)
av av av aw Uv-+ u-+ W-+w---at ar az az r
ap 1 (&v 1av 1 &v
= - a'P+ Re ar2 +-;'ar + r2 a'P2&v v 2 au
)+ az2 - r2 + r2 a'P ' (11)
175
aw aw aw aw aw-+u-+w-+u-+w-at ar az ar az
ap 1 (a2w 1 aw 1 &w &w
)= - az + Re ar2+ -;. ar + r2 a'P2+ az2(6) + GrO, (12)
au + ~ + ~ av + aw = 0,ar r r a'P az
(13)
(7)
ao ao ao ae ae-+ U-+ w-+u-+w-at ar az ar az
1(&0 lao 1 &0 &0
)= Pr ar2 + -;. ar + r2a'P2+ az2 (14)(8)
with the boundary conditions
a. af + bill
= 0,I an r;
(15)
wherefrepresents one of the functions u, v, w or O.To complete
the formulation it is necessary to addconditions of
271"-periodicityof all the functions:
f('P + 271")= f('P). (16)
According to linear stability theory, the timedependence of the
perturbation functions {u, v, w,p, O} may be assumed as rv
exp(-\t), where -\determines the time rate of change of a
pertur-bation. The periodicity conditions (16) allow us torepresent
the solution of (10)-(15) as Fourier seriesin the azimuthal
direction. Thus, the perturbationfunctions can be represented
as
{u, v, w,p, O} = exp(-\t)
k=oo
X L {uk(r, z), vk(r, z), wk(r, z),Pk(r, z), Ok(r,z)}k=-oo
x exp(ik'P). (17)
Equations for the Fourier coefficients {Uh Vh Wh
Ph Ok} are obtained after substitution of (17) into
-
176 A.Yu. GELFGAT et al.
(10)-(16), and neglecting the higher-order terms:
8Uk 8Uk 8U 8UAUk+ U- + w - + Uk- + Wk-8r fu 8r fu
= - 8Pk+~ (&Uk +! 8Uk - k2 + 1 Uk8r Re 8r2 r 8r r2&Uk
2ik
)+ 8z2 - --;.2 Vk , (18)\ U 8Vk W 8Vk UVkAVk+ -+ ---
8r 8z r
. 1
(&Vk 1 8Vk k2 + 1= -lkpk+- -+ VkRe 8r2 r 8r r2&Vk 2ik )+
8z2 + --;.2Uk , (19)
8Wk 8Wk 8W 8WAWk+ U-+ W-+Uk-+Wk-
8r 8z 8r 8z
= - 8Pk+ ~ (&Wk + ! 8Wk - k2 Wk + 82Wk )8z Re 8r2 r 8r r2
8z2+ Gr(h, (20)
8Uk Uk ik 8Wk0-+-+-Vk+-= ,
8r r r 8z (21)
80k 80k 8e 8eAOk+U-+ W-+Uk-+Wk-8r fu 8r fu
= ~ (&Ok +! 80k - k2 Ok+ &Ok) .Pr 8r2 r 8r r2 8z2
(22)The functions {Uk,Vh Wk,Pk,Ok}are Fourier coeffi-cients which
define the eigenvector of (10)-(15) foreach eigenvalue A(k). The
integer number k in (17)plays a role of the azimuthal wavenumber.
Thevalue k = 0 corresponds to the axisymmetricperturbation.
It is seen that the linear stability problem can besolved
separately for each value of the azimuthalwavenumber k. This allows
us to replace the fullthree-dimensional stability problem by a
series ofaxisymmetric problems for different azimuthalwavenumbers
k.
The main problem of numerical solution ofthe system (18)-(22) is
caused by the terms
proportional to 1/,1, which lead to a non-integrable
discontinuity at the axis of the cylinderr = O. Note that this
discontinuity is an artifactintroduced by the use of polar
coordinates in the(r,cp) plane. However, this artificial
discontinuitycan be easily avoided. Note, that the azimuthalangle
cpis not defined at r = O.This means that anon-zero value of each
function can be assigned atr = 0 to one ofthe Fourier modes, while
all the othermodes can be put equal to zero at the axis. Hence,
itfollows for Eqs. (20) and (22), that non-zero valuesof the axial
velocity and the temperature should beassigned to the axisymmetric
mode k = O. To do thesame for Eqs. (18) and (19), one can express
theterms (2ikuk/r2) and (2ikvk/,1) from the continuityequation (21)
as
2ik Uk = - 2ik (8Uk + ik Vk + 8Wk ) ,r2 r 8r r 8z2ik Vk= _~(8Uk
+ Uk + 8Wk )r2 r 8r r 8z
(23)
and substitute (23) into (18) and (19), which gives
8Uk 8Uk 8U 8UAUk+ U- + W - + Uk- + Wk-
8r 8z 8r 8z
= - 8Pk + ~(
&Uk + ~8Uk - k2 - 1 Uk8r Re 8r2 r 8r r2
&Uk 2 8Wk
)+- 8 2+-- 8 'z r z (24)\ U 8Vk W 8Vk UVkAVk+ -+ ---8r 8z r
. 1
(&Vk 1 8Vk k2 - 1= -lkpk+- -+--+-VkRe 8r2 r 8r r2
+ &Vk - 2ik 8Uk - 2ik 8Wk)
.8z2 r 8r r 8z
(25)
It is easy to see now, that terms proportional tol/r2 disappear
at k = 1. This gives us a possibilityto assign non-zero values of U
and v at r = 0 to theFourier components corresponding to k =
::I:1.Finally, we obtain the following restrictions for
-
AXISYMMETRY-BREAKING INSTABILITY
values of the Fourier modes at r =0:
Uo= 0, Vo= 0, 00 =I0, Wo=I0,U:l:1=I0, V:l:1=I0, W:l:1;= 0:1:1=
0,
Uk = Vk = Wk = Ok = 0, for Ikl > 1.
(26a)
(26b)
NUMERICAL METHOD
The axisymmetric problem (6)-(9) together withthe
three-dimensional linear stability problem (18)-(22) are solved
using the spectral Galerkin method,as described in detail in G.
Here we shall outline
some the main steps.The system of basis functions of the
Galerkin
method is divided into axisymmetric and asym-metric subsystems.
This allows us to extract theaxisymmetric problem for the basis
flow as aseparate part and then consider only a three-dimensional
stability problem. Furthermore, itfollows from the continuity
equation (21), thatamong the three systems of basis functions for
UbVk and Wk only two will be linearly independent.Taking this into
account, the resulting Galerkinexpansion of the velocity can be
written asfollows:
M, Mz
v= LLAijUij(r,z)i=1 j=1
+k~ { t~[BtVij(r,z) + et Wij(r,z)]}
x exp(ik
-
178 A.Yu. GELFGAT et al.
Therefore, there is no need to determine the pressureif a
solenoidal basis, satisfying no-throughflowboundary conditions, is
used for the global Galerkinmethod. All terms containing Fourier
modes of thepressure Pk vanish after projection of the Eqs.
(18)-(22) on the solenoidal bases (28)-(30).
For the temperature (or other transported scalarproperty) the
Galerkin expansion can be written asfollows:
B=G(r,z)
+ k~{q(k,r) ttnttailTi+/(r)
x ~ 8jm1J+m(~) } exp(ikrp),
q(k, r) = ikr, if k i- 0; q(O,r) = 1.
(34)
The coefficients ail, and 8jmare used to satisfy thehomogeneous
boundary conditions (15). The func-
tion G(r, z) is used t,o satisfy non-homogeneousboundary
conditions for the temperature also ex-pressed as a series of the
Chebyshev polynomials
N, N,
(Z
)G(r,z) = L LgijT;(r)TjA .i=O j=O
(34*)
This approach was used in G to analyze theaxisymmetric
instabilities of a basic rotating flow.Here we use a similar
technique for the non-axisymmetric instability of a basic
non-rotatingflow. Further details of the numerical solution
follow the same steps as in G. The results are givenin the
following.
Test Calculations
The first test case considered was the Rayleigh-Benard
instability of motionless fluid in a cylinderheated from below.
Comparison with other resultsfor a stationary cylinder (Hardin et
aI., 1990) andfor a cylinder rotating around its axis (Jones
andMoore, 1979; Goldstein et al., 1993) showed thatthe calculated
critical Rayleigh number is correctup to the fifth digit with the
use of 10 x 10
basis functions in the r- and z-directions. Details
may be found in Gelfgat and Tanasawa (1993).However, these tests
are not sufficient, because theconvective terms of the momentum
equation vanishin the case of motionless initial state.
The next test case considered was the onset of the
secondary, oscillatory instability of the axisym-metric
Rayleigh-Benard convective flow. A cylin-der with isothermal top
and bottom and perfectlyinsulated lateral wall was considered. For
aspectratio equal to 1, the axisymmetry-breaking bifurca-tion sets
in as a transition to steady 3D flow with theazimuthal number k =
2. An illustration of the con-vergence of the critical Rayleigh
number (Ra =GrPr) and a comparison with recent results ofWanschura
et al. (1996) are shown in Table I.
A hysteresis of Racr at k = 2 was found byWanschura et al.
(1996) for Pr = 1. With theincrease of Ra the axisymmetric flow
becomesunstable with respect to asymmetric perturbationsat a
certain value Rai~) and then, with furtherincrease of the Rayleigh
number, it becomes stableat a larger value Rai;) > Rai~). This
result was usedas another test, and was extended further: the
thirdvalue Rai~) > Rai;) at which the steady axisym-metric flow
becomes finally unstable was alsocalculated. The convergence of all
three criticalRayleigh numbers is shown in Table 11.
TABLE I Critical Rayleigh number for theazimuthal mode k =2
TABLE 11 First, second, and third criticalRayleigh numbers for
Pr= I, A = I, k=2
N, x Nz R~:) R~;) R~;)
10 x 1020x2030 x 30
Wanschura et al. (1996)
300430043004
3016
259242594525945
784178427842
7900
N, x Nz Pr=0.02 Pr=1
6x6 2493.74 300310 x 10 2493.72 300420 x 20 2493.72 3004
Wanschura et al. (1996) 2463 3016
Neumann (1990) 2525
-
AXISYMMETRY-BREAKING INSTABILITY
Further tests of convergence of Racr and Wcrwere made for A = 1,
Pr = 0.02 and 1.0,and modewavenumbers k = 0, 1,2,3,4, and 5,with
numbers ofbasis functions running frorp. 6 x 6, 8 x 8, . . ., to28
x 28, 30 x 30. For all parameter values exceptPr = 0.02, k = 0, the
values of Racrandwcrconvergedto four or five significant digits
from 14 x 14functions on. For Pr = 0.02,k = 0
convergencewasslightly slower and was reached from 28 x 28functions
on. The final converged results areshown in Tables III(a)-(c).
Apart from the issueof convergence testing, it can be seen that the
valueof Racr for k = 2 is lower than that for other k, forboth Pr =
1and Pr = 0.02, i.e., the k = 2 mode is themost unstable.
Furthermore, it is interesting toobserve (Table III(c)) that at low
Pr the value ofRacr is quite strongly dependent on Pr.
EXPERIMENT
The onset of convec;tion and the stability ofan initially
isothermal fluid in regular cavities
TABLE III Critical Rayleigh numbers for var-ious azimuthal modes
k, and various Prandtlnumbers using 30 x 30 basis functions
*Compare with Neumann (1990) Raer=4100 andWanschura et al.
(1996) Racr = 4224.
179
instantaneously cooled from above have beenextensively
investigated for water, both with andwithout phase change (see KC).
A sketch of theexperimental setup is shown in Fig. 1. A
cylinder(37.1 mm inner diameter by 41 mm inner height)filled with
water was immersed in a thermostatic
water bath held at a hot temperature and was closedon its top by
a metal plate held at a cold temperature.The walls of the cylinder
(side and bottom) weremade of 2.1 mm thick glass. Experiments
werecarried out at bath temperatures in the rangeOhot= 10-25°C and
top plate temperaturecold,ranging from slightly below the bath
temperature,viz., Ocold= 20°C, down to below the freezing
point,viz., Ocold= -10°C. In all situations the
thermalstratification resulted in a free convective flow, andfor
below-freezing lid temperature an ice frontformed and grew downward
from the top. A steadystate flow configuration consists of a single
colddownward jet along the cavity axis and a reverseupward flow
along the side wall.
Observations were made mainly by careful anddetailed
measurements of the temperatures andtracks of liquid crystal
tracers, at various verticaland horizontal cross-sections. The
color change ofthe tracers convected by the flow allowed us to
detectvariations of the thermal field as small as O.I°C,providing a
direct indication of the stability andstructure of the investigated
flow field. Details of theexperimental procedure and extensive
results are
./metal plate at 9=-it" i Geoid
FIGURE 1 Sketch of the experimental setup. Glass cylinderwith a
cooled lid immersed in a hot bath.
k Raer "'er
(a) Pr= I0 28469 0I 4202* 02 3004 03 23851 11.0434 17610 05 17
392 0
(b)Pr=0.020 17442 247.1I 2662.82 02 2493.72 03 3313.9 04 4908.06
05 7406.92 0
(c)k=20.19 2625 00.2 2493.72 00.21 2242 0
-
180 A.Yu. GELFGAT et al.
given elsewhere (see KC, also Kowalewski et al.,1998). Insofar
as the present study is concerned, theinteresting observation was
that at a certain set ofparameters the temperature field measured
inthe fluid in a horizontal cross-section slightly belowthe top
(z=0.9A) split into a pattern of 16-181wedge-like sectors extending
radially from the axis tothe circumference (Fig. 2). Thus, despite
the cylin-drical geometry the flow underneath the lid becamedivided
into a regular pattern of radial structures.Clearly, the basic
axisymmetric flow split into a non-axisymmetric (but quite regular)
flow with this highwavenumber. These structures appeared for
purewater convection for temperature differences /}.()=()hot -
8cold exceeding 5°c. The corresponding
Grashof number is Gr=2.46 x 104. At largertemperature difference
(/}.()> lO°e) the flow struc-ture became unstable and the
vertical "cold jet"started to bounce. The previously regular
"star-like"horizontal structure of 16-18 spikes became dis-turbed,
their number and length varied in time.
It is worth noting, that the observed flow pat-tern remains when
the phase change takes place
FIGURE 2 Temperature distribution visualized by liquidcrystals.
Color image taken at the horizontal cross-sectionz=0.9A, Gr~2.5 x
104.
(freezing of water for ()co1d= -lO°e). The char-acteristic
star-like grooves were well visible in theice surface growing under
the lid.
NUMERICAL COMPARISON WITHEXPERIMENT
In view of the experimental observation, the fur-ther thrust of
the present analytical study was toanalyze numerically the
splitting of axisymmetry ofthis flow.
A secondary, but non-trivial, problem arose inconnection with
the definition of the thermal
boundary conditions: In a naive, first-sight descrip-tion, the
system appears to be defined as isothermalcold top and isothermal
hot sidewall and bottom.However, as pointed out by KC, the
conductionthrough the glass wall is finite (i.e., the inner
wallsurface is neither isothermal nor perfectly insulated)and there
are significant temperature gradients nearthe upper corner, where
the top plate meets thecylinder wall. To account for this effect,
twodifferent approaches were taken for the definitionof the thermal
boundary conditions:
(a) the boundary conditions at the inner wallswere assumed to
be:
o() = - Bi( () - 1) at z = 0;oz ()=O, at z=A; (35)
~~ = Ri (() - [1 - (~r]) at r = 1, (36)
and no-slip conditions for the velocity on allboundaries. Here
()= (iJ - iJcold)/(iJhot - iJcold), Biis a semi-artificial Biot
number, and the powerfunction (z/Ar with the artificial exponent n
isadded to smoothen the temperature boundaryconditions at the top
edge of the cylinder;
(b) the compound problem of axisymmetricconvection in the
cylinder with finite conductionthrough the walls was computed by a
finite-volumemethod (the isothermal conditions were assumed
I The shape of the wedges was not perfectly uniform. A simple
count gave 17. The correct number could be either 17 or the
nearesteven numbers 16 or 18.
-
AXISYMMETRY-BREAKING INSTABILITY
on the outside of the walls). Then the resultingtemperature
distribution on the inner walls wastaken as boundary condition for
the non-axisym-
metric stability analysis. .The details of the two approaches
and the results
are described below.
"Artificial" Ri and n
As stated above, the thermal boundary condition atthe inner
walls was approximated by assuming (35)and (36). With these
boundary conditions, thefunction G(r, z) in (34) was chosen as
G(r,z) = [1- (~) n]
[
Ri 2
] (z)
n
(Z
)+ l+-r - 1--2 + Ri A A . (37)
The Biot number Bi and the exponent n depend onheat transfer
between the metal plate, the thermo-static bath, and the entjre
enclosure. Unfortunately,
2
1.5
.,.,b";<..p 1
~
0.5
0
0 5
181
there is not enough experimental data to determineaccurate
values of these parameters. Therefore, Riand n were varied with the
goal to find whetherthere exists a most dominant 3D
perturbationwhich is divided into 18 similar parts in theazimuthal
direction.
A coarse estimate of the Biot number may beobtained from the
balance of the heat flux at the
inner and outer boundary of the cylindrical wall.Assuming that
the heat transfer coefficient from thewall to the outer water bath
is about 103W/m2Kand that the heat conductivities of glass and
waterare 1.02 and 0.566 WImK, respectively, the estimateis
Ri>:::,10. In the following calculations the Biotnumber was
varied from 0.5 to 20.
Preliminary calculations were done with theexponent n = 20 in
(37). (All computational resultspresented from here on are for A =
HIR = 2.2 andPr = 8.0, which correspond to the parameters ofthe
experiments.) The dependence of the criticalGrashof number on the
azimuthal number k for
different values of Ri is shown in Fig. 3. At Bi = 20
10
k
15 20
FIGURE 3 Critical Grashof number Grcr corresponding to different
azimuthal wavenumbers k for n =20 in (36).
0 Bi=O.5
0 0
0 Bi=!
0. Bi=2
0 00
x Bi=20 0
0
0 0
0
0 00
0 0 0 00 0
00
0
0 0
. . . .0
0. 0 .x 0 0
0 . . . . .. . x xX
.X X
X X * x xx
x x x
-
182 A.Yu. GELFGAT et al.
the minimum Grcr(k)corresponds to the dominantazimuthal mode
with k = 5. With the decrease of Rithe number of the dominant
azimuthal mode
grows. At Ri = 2 there is an abrupt decrease to thedominant
azimuthal mode k = 7 from the modek = 6 (Fig. 3). With further
decrease of the Biotnumber this abrupt decrease becomes larger
andoccurs at larger k. At Ri = 0.5 the most unstableazimuthal mode
is k = 9.
The axisymmetric convective flow for Ri = 0.5and n = 20 is shown
in Fig. 4 for the critical Grashofnumber corresponding to k = 9.
Figure 5 shows thecorresponding dominant three-dimensional
pertur-bation of the temperature whose azimuthal period is27rj9.
Figures 6 and 7 illustrate the same but forRi = 20 and k = 5. Note,
that the most unstableperturbation (Figs. 5 and 7) consists of a
pair ofantisymmetric patterns which are separated by aplane cp=
const. The perturbation of the three-dimensional velocity is
similar.
A comparison of the dominant perturbation(Figs. 5 and 7) with
the distribution of the tem-
r=1 r=O
FIGURE 4 Streamlines and isotherms of 'the axisymmetricflow
Bi=O.5, n=20, Gr=Grcr=40,900.
perature in the mean axisymmetric flow (Figs. 4and 6) allows us
to make some conclusions aboutthe nature of the instability. It is
seen that anun stably stratified fluid layer is always locatednear
the upper cold plate. The depth of the layerdepends on the Biot
number and on the smoothingof the temperature at the upper edge
(the expo-nent n). The maximal absolute values of theperturbation
of the temperature are also locatednear the upper plate. The
patterns of the pertur-bation of the vector potential of velocity
(w(r) andw(z),not shown in the paper) also look similar. This
r=1
FIGURE 5 Isosurfaces of the 3D perturbation of the tem-perature.
Bi=O.5, n=20, k=9, Gr=Grcr=40,900.
I I 0.05
I 0,0006el I -0.0006-0.054Jl)1
I
-
AXISYMMETRY-BREAKING INSTABILITY 183
r=1 r=O r=1
FIGURE 6 Streamlines and isotherms of the axisymmetricflow
Bi=20, n=20, Gr=Grcr=4600.
allows us to conjecture that the observed instabilityis caused
by a Rayleigh-Benard mechanism inthe un stably stratified fluid
layer.
Maximal values of the perturbation appear onsurfaces which have
almost rectangular crosssection at the cylindrical wall (Figs. 5
and 7). Thesize of these "rectangles" grows with the growth ofthe
depths of the stratified layers. This means thatfor thinner layers
the size of the characteristicpatterns of the most unstable
perturbation will besmaller and the corresponding critical
azimuthalnumber will be larger (the length of each "rectangle"in
the circumferential direction is 7r/k).
The depth of the unstably stratified fluid layerstrongly depends
on the heat transfer conditions inthe vicinity of the upper edge of
the cylinder. Quan-titative comparison with the experiment is
hardlypossible without better approximation of theseconditions in
the calculations (see below, FiniteWall Conduction Analysis).
However, in the frame-work of the present numerical model it is
possible tocontrol the depth of the stratified layer by varyingthe
exponent n in (37), which corresponds to
0.3
0.01
0.001
-0.001
-0.01
-0.3
FIGURE 7 Isosurfaces of the 3D perturbation of the tem-perature.
Bi=20, n=20, k=5, Gr= Grcr=4600.
different smoothings of the discontinuity of thetemperature at
the upper edge of the cylinder.
It was found that for the values of the Biot
number Ri= 10 or 20 the instability sets in with theazimuthal
number k = 9 if one assigns n = 34 in (37).This is illustrated in
Figs. 8-10 for Ri = 20. It is seen(Fig. 8) that the critical
azimuthal number growswith the growth of the exponent n, which
corre-sponds to the thinning of the stratified layer. It
isinteresting to compare flow patterns with the samek but different
Ri and n. Thus, compare the resultsfor Ri = 0.5,n = 20, Gr=
40900(Figs. 4 and 5) andfor Ri = 20, n= 34, Gr= 6770(Figs. 9 and
10),bothof which correspond to a dominant instabilitywith k = 9.
The depths of the stratified layers inFigs. 4 and 9 are almost
equal, which leads to theonset of instability with the same
azimuthal numberand with similar perturbations (compare Figs. 5
-
184
5
A.Yu. GELFGAT et al.
4
0
0 5 10
k
15 20
FIGURE 8 Critical Grashof number Grcrcorresponding to different
azimuthal numbers k for Hi= 20.
r=O
FIGURE 9 Streamlines and isotherms of the axisymmetricflow
Hi=20, n=34, k=9, Gr=Grcr=6770.
r=l
and 10). Note, that for larger n the criticalazimuthal number is
larger (k = 11 for n = 40,Fig. 8). Obviously, there should be some
limit ofthe critical k when n tends to infinity.
The results presented so far show that (a) theanalysis of
non-axisymmetric instability by thepresent Galerkin and
mode-separation approachyields meaningful results and (b) for the
naturalconvection problem considered here, the
axisym-metry-breaking instability appears to be closelyrelated to
the thickness of the thermally stratifiedlayer under the lid,
nearly regardless of the specificvalues of parameters which led to
the formation ofthat particular thickness.
To further investigate if the k= 16-18 circum-ferential
splitting which was observed in the ex-periment could be obtained
analytically, thecomputations were run for a variety of values ofRi
and n. Representative results for Grcr(k) forRi = 10,n= 50,60,70
are shown in Fig. 11.
00 n=20
'" n=30
x n=34 0
0 n=40 0
x 0 x
0 0 0 x0 '""
'"0 x
'"'" 0 00 0 '"0 xx x 0 0x "
'" 0x x x '" 0x'" '" '"
'" '" x '" 000
A '"0
00 0 0 0
0 0
'i0 3-
t;...
2
-
&~.
."'..
~tV.
AXISYMMETRY-BREAKING INSTABILITY
0.03
0.001
-0.001
-0.03
FIGURE 10 Isosurfaces of the 3D perturbation of the
tem-perature. Si = 20, n = 34, k = 9, Gr = Grcr= 6770.
The isolines of the main flow and of the
perturbation are of the same nature as Figs. 4-7,9-10 above, but
more clustered near the top cover,as could be expected for the
higher value of n. Thebehavior for the three values of n is
analogous, withsome shift in the characteristic values. The
common
result is that a local minimum Gr er appears at k = 13(n=50),
k=14 (n = 60), k=16 (n=70), but aglobal minimum Grer appears near k
= '" 4. Thequestion then arises why was the splitting observed
185
at k = 16-18, rather than at k = '" 4. The resolutionof this
question we hope to find in further moredetailed experiments.
Possibly, due to some tran-sient effect, the instability leads to
the k=16-18mode without first exhibiting the k = '" 4
mode(perturbations growth rates at Gr = 4 x 104 fork = 4 and 16 are
of the same order of magnitude'" 101).Alternatively, the k = 16-18
mode could bethe result of non-linear interaction oflower k
modes,since the value of Gr = 2.46 x 104in the
experimentwassignificantlyabove the criticalvaluesof Grcr' Inthat
case the observed k = 16-18 could be the non-
linear interaction of k = 7, 8, 9,10, etc.
Finite Wall Conduction Analysis
The preceding approximate description of thethermal boundary
condition ("artificial" Ri and n)indicated that the stability
results are, indeed,sensitive to the details. Therefore in the
second
approach the temperature at the inner wall wascomputed
numerically taking into account finiteconduction in the wall. (The
detailed temperaturedistribution at the inner wall was not
available
experimentally. It is hoped that in future experi-ments this
will be estimated, although the fineresolution will be difficult.)
The numerical studywas done in the following way: First, the
coupledaxisymmetric problem of convective flow inside thecontainer
and heat conduction through its wallswas solved using the
finite-volume method. Thenthe calculated profiles of the
temperature at theinner surface of the side wall and the bottom
were
applied as the boundary conditions for the Galerkinmethod, such
that boundary conditions for thetemperature became
0 = Bbottom (r ) (38)
(39)
at z = 0;
0 = 0, at z = A; 0= Bwall(Z)atr= 1,
and no-slip conditions for the velocity wereimposed on all
boundaries. Streamlines and iso-therms of the flow calculated at Gr
= 104 are
illustrated in Fig. 12. Note that the temperaturefield extends
into the walls, which are indicated by
-
186
1.2
1.0
or>
:::, 0.8';:(......"
r,:,
0.6
0.4
0.2
0
A.Yu. GELFGAT et al.
5 10
k
15 20
FIGURE 11 Critical Grashof number Grcr corresponding to
different azimuthal numbers k for Bi= 10.
FIGURE 12 Streamlines (right) and isotherms (left) of
thecalculated convective flow. Pr = 8, Gr= 104.
the straight lines. The corresponding profiles ofOwalland
°bottom are shown in Fig. 13. Similarcomputations (not shown) were
carried out forother values of Gr. The Biot number at the inner
wall was computed from the numerical solution and
was found to be approximately Bi ~ 17 at thebottom and Bi ~ 18at
the side wall.
With the basejlow temperature distribution estab-lished, the
stability study was carried out for threedifferent boundary
conditions for the perturbationofthe temperature 0 imposed on the
side wall of thecylindrical container. One assumption was
thevanishing of the perturbation of the temperature:
0 = 0 at r = 1. (40)
The two other assumptions were:
00 = BiO at r = 1,or (41)
where Bi = 0 corresponds to the vanishing of theperturbation of
the heat flux on the wall and Bi = 18corresponds to the calculated
value of Bi.
The calculated values of the critical Grashof
number Grcr for different azimuthal wavenumbersk are shown in
Fig. 14.The general trend is the same
xx 0 n=50
x /1n=60
/1-
/1X n=70
x x
x/1 0
0 0x
x X/1 X
0X /1
11
0/1
/10
/1
0
0
0
0
0
g 0 0 0
-
AXISYMMETRY-BREAKING INSTABILITY 187
0.88
0.96 0.75
-
188 A.YU. GELFGAT et al.
as for the other model, shown in Figs. 3 and 5. Theassumed
boundary conditions have some effect onthe details of the results,
but without altering thetrend2. It is seen that, for all three
assumptionsregarding the boundary condition at the wall, theminimal
values of the critical Grashof numbers
correspond to k = 7, 8 and 9 and are located closeto Gr = 104.
An example of isolines of the pertur-bation of the temperature at
the horizontal cross-section z = 0.9A (corresponding to the
locationof the photograph in Fig. 2) is shown in Fig. 15.
The pattern of the perturbation in this cross-section contains 8
pairs of maxima and minima(total 16 regions) and looks similar to
the experi-mental pattern of isotherms (Fig. 2). However,
theagreement with the experiment is not complete,because the dark
areas in Fig. 2 correspond to theminima of the temperature.
Therefore one should
FIGURE 15 Isolines of perturbation of temperature at
thecross-section z = 0.9A. Case of vanishing temperature
pertur-bation at side wall. Grcr= 1.01 x 104,k = 8.
expect the existence of 16, 18 pairs of maxima andminima in the
perturbation of the temperature.
The disagreement of the experimental andnumerical results can be
explained if one com-pares the Grashof number corresponding to Fig.
2(Gr = 2.46 x 104) with the calculated criticalGrashof number
(Grcr~ 104). The experimentalstudy was carried out at more than
100% super-criticality, where non-linear interaction of thedominant
modes of the perturbation cannot beneglected. Thus, the 17 minima
of the temperature,seen in Fig. 2, can be a result of
non-linearinteraction of modes with k = 8 and 9, or k = 7and 10,
whose critical Grashof numbers have closevalues. On the other hand,
modes with k = 16, 18also become unstable at Gr::::,j2.5 x 104, and
canbecome dominant at certain conditions.
CONCLUSIONS
It was shown that the global spectral approachmay be
successfully applied to numerical studiesof axisymmetry-breaking
instabilities. Using thisapproach, one can consider a linearized
2-Dstability problem for each circumferential modeseparately,
instead of CPU-time-consuming, time-dependent calculations. The
proposed spectralapproach was validated by comparison with
theresults of direct numerical simulation. (Thisapproach is also
used to analyze the stability ofrotating flows, which will be
reported elsewhere.)
The use of the global spectral approach allowedus to obtain a
qualitative explanation of the recentlyreported experimental
results (KC) for naturalconvection in a cylindrical container, in
which aninstability with a relatively high azimuthal number(k = 16
-;- 18) was observed. The spectral Galerkinanalysis presented here
reproduces such instabilitiesand provides details of the flow and
temperature
2Our experiments also included an investigation in which a
slightly insulated ring of 4 mm height (adhesive tape) was wrapped
on theoutside of the cylinder just below the cold cover, thus
smoothing the temperature discontinuity. We have reproduced this
situation in thecomputations (not shown). The general trend of the
instability is stiU the same, with some differences in detail.
Further comparison ofexperiment and computation for this situation
will be meaningful only when detailed measurements of the
temperature in the transitionregion become available.
-
AXISYMMETRY-BREAKING INSTABILITY
field. In the absence of detailed experimental dataon the
temperature distribution in the critical regionnear the upper
corner, various approximationswere assumed, all leading to. a
splitting of axisym-metry with high to very high
circumferentialwavenumber. The non-simple thermal
boundaryconditions of the experiments were approximatedin several
ways, including a hybrid numericalapproach, in which the base flow
was computedtaking into account wall conduction, and the
resultswere used to define thermal boundary conditionsfor the 3-D
stability problem.
The numerical results support the conjecture ofKC that the
instabilityisof Rayleigh- Benard type,generated by the thermal
stratification near theupper corner of the cylinder. The analysis
predicts ahigh azimuthal number close to that observed
inexperiment. A more precise quantitative compar-ison with the
experimental results would requirebetter resolution of the
experimental heat transferconditions. This may suggest the line of
futureexperimental work.
Acknowledgments
This work was supported by the Israel ScienceFoundation under
Grant 110/96, by the Centerfor Absorption in Science, Israel
Ministry ofImmigrant Absorption (to A. Gelfgat), by theFund for the
Promotion of Research at Technion
(to P. Bar-Yoseph), by the Y. Winograd Chair ofFluid Mechanics
and Heat Transfer, and by theIsrael High Performance Computer Unit.
Thefourth author would like to acknowledge the con-tribution ofW.
Hiller and C. Soeller from Max Plank
Institute for Fluid Mechanics Research, G6ttingen,with whom the
experimental study was initiated.
NOMENCLATURE
A aspect ratio (height/radius) ofthe cylinderBiot numberBi
18'
GrHPrRRa
Tj(x)
Grashof number
height of the cylindrical cavityPrandtl number
radius of the cylindrical cavityRayleigh numberChebyshev
polynomial of the first
kind
Chebyshev polynomial of thesecond kind
gravity accelerationazimuthal number
pressurecylindrical coordinatestime
velocity vector in cylindricalcoordinate system
vector potential of velocitythermal expansion
coefficientkinematic viscositythermal diffusivitytemperature
Uj(X)
gk
pr, cp, Zt
v=(u, v, w)
\11
(311
x()
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