ELSEVIER Physica D 97 (1996) 180-184

PHYSICA D

Convection in cylindrical containers at supercritical Rayleigh numbers *

U. Mti l ler 1 Forschungszentrum Karlsruhe Gmbh, lnstitut fiir Angewandte Thermo- und Fluiddynamik, Postfach 3640,

D-76021 Karlsruhe, Germany

Abstract

An experimental study is described of various cellular convection pattems in cylindrical containers at supercritical Rayleigh numbers. The fluid containers are heated from below and as the Rayleigh number is slowly increased the convection patterns undergo a series of bifiarcations to nonaxisymmetric configurations. The form of convection depends on both the Rayleigh number and aspect ratio of the container. The observations are discussed in the light of current understanding of pattern-forming phenomena.

1. Introduct ion

In an early paper Stork and Miiller [13] reported

investigations on Benard convection in annular and

cylindrical containers of small aspect ratio. A some-

what surprising result of these experiments in compar-

ison with observations of Koschmieder and Pallas [7]

was that axisymmetric convection patterns were gen-

erally not stable in cylindrical containers of small as-

pect ratio even at Rayleigh numbers very close to the

critical value. For cylindrical boxes "ring roll" patterns

were observed for aspect ratios 1.8 < d / h <_ 3.2 with

d the diameter and h the height of the layer. For larger

aspect ratios nonaxisymmetric convection patterns in

form of straight or bent rolls, nearly parallel to the box

diameter were observed. Since these early observa-

tions, many experiments on pattern formation and on

transition to weak turbulent flow have been carried out

* Dedicated to Professor Fritz Busse on the occasion of his 60th birthday.

1 Tel: ++49(7247) 82 3450; fax:++49(7247)82 4837; e-mail: ulrich.mueller@ iatf.fzk.de.

in cylindrical containers by several groups and inves-

tigators for example by Kirchards et al. [6], Croquette

et al. [2] and Steinberg et al. [12]. A summary of the

relevant phenomena can be found in the survey arti-

cles of Pocheau [11] and Cross and Hohenberg [3].

Most of these experiments were aimed at exploring

the effect of distant side walls on the cellular pattern

and slow variation of motion with time as essential

ingredients for the transition to spacial and temporal

turbulence. The aspect ratios of the cylindrical boxes

were therefore mostly chosen in a range d / h > 10.

There are only a few experimental investigations

in small aspect ratio boxes with d~ h < 3. They refer

either to purely axisymmetric flows for one particular

aspect ratios only and slightly supercritical conditions

such as in [10], or to axi- and nonaxisymmetric flows

at highly supercritical conditions such as in [4,8].

Here we present some observations obtained from

convection tests in cylindrical boxes of aspect ra-

tios 1.6 < d~ h < 3. The observed pattern develop-

ment may be considered a supplement to other flow

pattern investigations in different aspect ratios and

0167-2789/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S0167-2789(96)00078-4

U. Miiller/Physica D 97 (1996) 180-184 181

2R = 16mm 2R = 22ram

Fig. 1. Flow pattern development at supercritical Rayleigh numbers d/h = 1.6 and 2.2.

Rayleigh-number ranges. The experiments have been carded out by Stork [15] and some of his unpublished

photo material and test data have been reevaluated. The experimental apparatus and the tracer technique

for flow visualization used by Stork has been described in his thesis [13] and by Stork and Miiller [14] and

is only briefly outlined here. The axisymmetric boxes

are formed by a heated copper plate at the bottom, a cooled glass plate at the top and a PVC plate, 10mm thick with a cylindrical borehole of 14 and up to 30 mm in diameter, respectively in the center. The PVC plate was sandwiched between the plates and the cavity was filled with silicon oil M 100 with a Prandtl number of

1000. Aluminum flakes were used as a tracer material.

The layer was heated slowly from isothermal condi- tions to critical conditions during a period of typically

6-10 diffusion times based on the layer height and be- yond critical conditions in a similarly slow transient

with time intervals of several hours with no tempera- ture change at the heating or cooling plate.

2. Results

The computational results of Charlson and Sani [1] for the critical Rayleigh numbers of different modes

182 U. Miiller/Physica D'97 (1996) 180-184

2R= 2 6 m m 2R = 30ram

Fig. 2. Flow pattern development at supercritical Rayleigh numbers d/h = 2.6 and 3.0.

Fig. 3. Schematic flow pattern of photographs of Figs. l(e), i(f), 2(e) and 2(f); (® @ ®) symbols of downflow, (63 63 63) symbols of primary upflow, mode effect; (OOO) converging lines of secondary upflow due to Eckmann pumping.

in axisymmetric containers indicate that an easy and

clean experimental generation of steady axisymmet-

ric and nonaxisymmetric modes can be expected only

up to diameter to height ratios of about 3.5. We argue

that the eigenvalues of the first modes differ in cer-

tain wave number ranges sufficiently from each other,

only in this range, for the formation of a distinct flow

pattern in an experiment. To support this conjecture

U. Miiller/Physica D 97 (1996) 180-184 183

experiments performed in containers of aspect ratios d / h = 1.4; 1.6; 1.8; 2.0; 2.2; 2.6; 3.0 were evaluated.

Eq. (12) in the paper of Charlson and Sani [1] was

used to characterize the vertical component of the ve- locity of the different modes. In appropriate form the

equation is written as

Wren = C Jn(amr) cos n ~ F (z),

where am represent the series of the transcendental

algebraic equation

amRJn+l (amR) -I- Jn(amR) = O.

For aspect ratios d~ h = 1.4 and 1.6 a single roll with its axis in line with the diameter of the box was

observed at the onset of convection as can be seen

from Fig. l(a). This figure shows a tracer photo of a single roll aligned with the box diameter. This con-

figuration corresponds to the mode reference numbers m = 1, n = 1. At only slightly supercritical temper-

ature differences the roll is split in the center of the box by a zone of rising fluid along a diameter, forming an angle of lzr with the axis of the original convec-

tion rolls. Distinct downward and upward flow, respec- tively, is found close to the side walls perpendicular to the roll axis. At the two ends of the convection rolls

the fluid spirals inwards to the roll axes driven by an Eckmann pumping effect in the wall boundary layer at supercritical Rayleigh numbers. A return spiraling

occurs further along the roll axis towards the center of

the box, where it rises again forming the rising zone

perpendicular to the roll axis. Although these details were not observed in the experiments, the phenomena

have been simulated by several authors numerically for

cellular convection in rectangular boxes among others by Mallinson and de Vahl Davis [9] and Kirchards [5]. Therefore we conjecture that the roll splitting perpen-

dicular to the vortex axis and at supercrifical Rayleigh numbers is caused by Eckmann pumping near rigid walls. This is shown in Fig. l(c) and (e) for d / h =

1.6. For aspect ratio values larger than d / h = 1.6 ax- isymmetric modes with m = 1, n = 0 were observed at the onset of convection corresponding to a single toms. However, these tori became unstable to nonax- isymmetric perturbations at higher Rayleigh numbers. This can be seen in the photographs of Figs. l(d),

2(c) and 2(d). The experiments showed that zones of

upward flow are characterized by narrow dark lines while regions of predominant downward flow can in

general be recognized by dark lines which are fainter

but broader. With this in mind Figs, l(d), 2(c) and (d)

with d / h = 2.2; 2.6; 3.0 can be described by mode reference numbers m = 1, n = 2; m = 1, n = 3 and

m = 1, n = 4, respectively. At even higher Rayleigh numbers this azimuthally structured cellular flow be-

comes even more complex, because the secondary vor- tices with axes in the radial direction interact with

the side walls giving rise to Eckmann pumping in the boundary layers. The resulting flow pattern because

of this interaction between buoyancy induced spiral

flow and the coriolis forces in the wall region can be seen in the photographs of Figs. l(f), 2(e) and 2(f). For n = 2, 3, 4 four, six and eight counterrotating vor-

tices with radially oriented axes are formed (see pho-

tographs of Figs. l(d), 2(c) and 2(d)). Each of these

vortices induces an inward spiraling flow at the side

walls, which gives rise to spiraling return flows near

the vortex axes and together with the spiraling flow coming from the center of the box lead to flow welling

up along cord lines connecting centers of down flow near the side walls. The outlined explanation for the pattern should be considered a conjecture, which may

be confirmed by direct numerical simulation in the future. An illustration of the flow pattern is given in

Fig. 3 by a sketch showing lines of upward and down- ward flow.

References

[1] G.S. Charlson and R.L. Sani, Int. J. Heat Mass Transfer 14 (1971) 2157.

[2] V. Croquette, M. Mory and F. Schosseler, J. Physique 44 (1983) 293.

[3] M.C. Cross and P.C. Hohenberg, Rev. Modern Phys. 65 (1993).

[4] R.S. Figliola, Phys. Fluids 29 (1986) 2028. [5] K.R. Krichards, Thesis, Universit~it Karlsruhe (1988). [6] K.R. Kirchards, U. Miiller, H. Oertel and J. Zierep, Acta

Mech. 40 (1981) 181. [7] E.L. Koschmider and S.G. Pallas, Int. J. Heat Mass

Transfer 17 (1974) 991. [8] S.E Liang, A. Vidal and A. Acrivos, J. Fluid Mech. 36

(1969) 239.

184 U. Miiller/Physica D 97 (1996) 180-184

[9] G.D. Mallinson and G. de Vahl Davis, J. Fluid Mech. 83 (1977) 1.

[10] W.T. Mitchell and J.A. Quinn, AIChE J. (1966) 1116. [11] A. Pocheau, J. Phys. France 49 (1988) 1127. [12] V. Steinberg, G. Ahlers and D.S. Cannel, Phys. Scripta 32

(1985) 534.

[13] K. Stork, Thesis, Universit~it Karlsruhe (1974) [14] K. Stork and U. Mtiller, J. Fluid Mech. 71 (1975) 231 [15] R. Stork, 1972.