kLD CIRCULAR CONTAINERS I IN LOW GRAVITY I - … · I kLD CIRCULAR CONTAINERS I IN LOW GRAVITY ... ~~arcll IL Engineering Center, ... gravity force length of square side radial distance

I LMSC-HREC TR Db9782Y

¶

li NUMERICAL ANALYSIS

I OF NATURAL CONVECTION IN TWO-DIMENSIONAL SQUARE

I kLD CIRCULAR CONTAINERS I IN LOW GRAVITY I August 1980

Contract NASW-3281 (Interim Report)

Prepared for

NASA HEADQUARTERS WASHINGTON, DC 20546

I H~ntsvilk Research & Engimring Centor 48W Bmdfrrd Drive, HutrvPIe, A1 35807

I (NASA-Ck-1 b3487) N U M E h I i A i I I A ~ A L Y ~ I S u i? , ~ 8 ~ - 3 0 6 8 0 Y A T U H A L C O N V K T X G N I N l ' K 3 - ~ I i l l ; ~ i S l d N A ~ SQUAdE ANU L l d C U L A R C(iNTfr.lNELiS i n ib* k ; a A V I ' i ' Y I u t e r i n He port ( L o c k heed LILL;,L it:s .;IIC~ S ~ C L C C ! I1 i~c:las

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FOREWORD

This docurlrcnt r e p o r t s ttlc r c s u l t s of e f fo r t by pe r sonne l

of Lorkhced Miss i l ee Rr Space C o n ~ p a n y , Inc., Huntsv i l le R s -

~ ~ a r c l l IL Engineering Cen te r , f o r the National Acronau t i c s a n d

S p c t - Adnrinis t rat ion under Con t r ac t NASW - 328 1, "Manufactur - ing i ~ r Space: Flu id Dynamics Nunrcr ica l Analyeis." T h e NASA

teclrnic.al d i r e c t o r f o r t h i s con t r ac t is D r . R o b e r t F. D r e s a l e r ,

Mitnager, Advanced Technology P r o g r a m , NASA Z-{cadqtmrters,

Washington, D. C.

ACKNOWLEDGMENT

Tlrt? atrtllurs arc ploascd to e x p r c s s app rec i a t ion t o the

NASA tr*cllnici~l d i r t -c tor , Dr . Rober t F. Dressier, and the

originittor of tlrt* t-ffort, Dr . John H. Car ru t l r c r s , f o r t h e i r

intcrt8st in i\nd suppor t f o r this work. D r . D r e s s l c r l e guid-

ancr* and ~ ~ r o n i t o r i n g of o u r affor ta cont r ibu ted subs tan t ia l ly

to (JIG O L I ~ C O I I I C of thia task.

Wc arc a l s o g ra t e fu l t o Lockhecd pe r sonne l who con-

t r ibu ted in va r ioua ways t o thia e f for t , in pa r t i cu l a r , Kenneth E.

Xiqura who ~rladc? eubs tan t ia l con t r ibu t iu~r s to the nulrlc r i cn l

computat ional work.

i i

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ABSTRACT

A numerical study of natural convection in c i rcu la r cylinder and square

enclosures shows that the analytic low Rayleigh number theory of previous

investigators i s valid for Rayleigh numbers up to 1000. F o r a Rayleigh nurn-

be r of 5000, steady s ta te values of maximum fluid velocity differ by 20 per-

cent. This deviation between analytic theory and numerical resul ts increases

for higher Rayleigh numbers. In addition, the low Rayleigh number theory i s

shown to be valid for higher Rayleigh numbers for a portion of the transient

phase before significant deviation becomes apparent. It is a l so shown that

square shaped experiment configurations may be analytically approximated

with good accuracy by c i rcular cylinders of equal c r o s s sectional a r e a for

the prediction of convection velocities and flow patterns a t low Rayleigh

nttnr be r .

i i i

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- *--...---~~----*-----.- .---- --

Section

Fig urc-

1

2a

CONTENTS

FOREWORD

ACKNOWLEDGMENT

ABSTRACT

NOMENCLATURE

CIRCULAR CYLINDER ENCLOSURE

SQUARE ENCLOSURE

CONCLUSIONS

REFERENCES

LIST O F ILLUSTRATIONS

Page

i i

i i

i i i

vi

1

3

15

3 1

32

G c o l ~ ~ c t r y for Circular Cylinder Enclosure 3

Maxinlunl Velocity a s a Function of Time for Water 5 Contained in a Horizontal Circular Cylinder Enclosure a t Various Raylcigh Numbers

Velocity Deviation a s a Function of Time for Water 6 Cont-aincd in a klorizonbl Circular Cylinder Enclosure a t Various Rayleigh Numbers

Variation of Maximum Equilibrium Velocity with Rayleigh 7 Number for Water Contained in a Horizontal Cylinder

Conlparison of Velocity Distributions Along Horizontal 9 Radius a s a Function of Time fo r Rayleigh Numbers of 1 and 100,000

GIM Code Velocity Vector Plots a t Steady State for Circular 10 Cylinder a t Various Rayleigh Numbers

GIM Code Velocity Contour Plots a t Steady State for Circular 11 Cylinder a t Various Rayleigh Numbers

LOCKHLLD. HUN1SVILI.E RESEARCH & ENGINEERING CENTER

" . , . . " ..-...-.-. .. .- ..

Figure

7 GIM Code Temperature Contour Plots a t Steady State f o r Circular Cylinder a t Vrtrious Rayleigh Numbers 12

Comparison of GIM Code Temperature Contour and Velocity 13 Vector Variation with Time for Circular Cylinder a t Rayleigh Number of 100,000

Geometry for Square Enclosure

Variation of Maximum Equilibrium Velucity with Rayleigh Number for Water Contained in Square Enclosure

LOCAP Velocity Vector Plots a t Steady State for Square Enclosure a t Various Rayleigh Numbers

LOCAP Streamline Plots a t Steady State fo r Square En- c losure a t Various Rayleigh Numbers

LOCAP Absolute Velocity Contour Plots at Steady State fo r Square Enclosure a t Various Rayleigh Numbers

LOCAP Temperature Contour Plots at Steady State fo r Square Enclosure a t Various Rayleigh Numbers

GIM Code Velocity Vector Plots a t Steady State fo r Square Enclosure a t Various Rayleigh Numbers

GIM Code Velocity Contour Plots a t Steady State fo r Square Enclosure a t Various Rayleigh Numbers

GIM Code Temperature Contour Plots a t Steady State fo r Square Enclosure a t Various Rayleigh Numbers

Maximum Velocity a s a Function of Time fo r Water Con- tained in a Square Enclosure a t Various Rayleigh Numbers

Vclocity Distribution a s a Function of Time fo r Water Con- tained in a Square Enclosure with Rayleigh Number = 1

Comparison of GIM Code Temperature and Velocity Vector 28 Variation with Time for Square Enclosure a t Rayleigh Num- ber of 100,000

Variation of Fluid Velocity with Angular Position Along 3 0 Circular Arcs in Liquid Filled Square Container

v

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Symbol

d

6

L

r

R

Ra

NOMENCLATURE

Description

cylinder diameter = 2 R

gravity force

length of square side

radial distance

cylinder radius

Rayleigh number = , cylinder va

- - 8, square am

temperature

initial mid-point temperature

temperature difference across circular cylinder o r square enclosure

time

dimensionless time I vt /~ ' , cylinder 2

V~/ ' (L /Z) , sqnare

velocity

dimensionless velocity = lL8' v, cylinder g p ~ ~ d 2

rectangular coordinates (Figs. 1 and 9)

thermal diffusivity

volumetric coefficient of thermal expansion , ,

; 1

i' v i

f f. LOCKHEED HUNTSVILLE RESEARPY 6 ENGINEERING CENTFR

*a*., . U - W i C a * r W l i i *..,,- - ,.-.* wV...-- V I P * ,I,U I . .? .a7.. lf .U- "ll)--.*C.-. .,.-*- ""- .. . . - -- . - we...... -..<.- r

polar angle (Fig. 1 )

kinematic viscosity

stream function

dimensionless stream function = lZBv JI, cylinder gB AT d3

vii

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LMSC-HREC TR 0697821

1. INTRODUCTION

The NASA Materials Processing in Space program includes a number

of experiments to be performed using the Fluids Experiment System (FES)

facility aboard the Spacelab vehicle in orbit. These experiments will investi-

gate various fluid phenomena of interest to materials processing in space

applications. . Fluid convection induced by residual gravity forces under

orbital conditions is of interest to a number of possible materials process-

ing applications.

This report describes numerical fluid dynamics analyses of simplified

geometries to investigate fluid convection behavior under various conditions

of thermal gradients and gravitational loading. These combined effects a r e

indicated by the magnitude of the Rayleigh number (see Nomenclature).

The particular configurations investigated in this study were two-

dimensional circular cylinder and square enclosures with horizontally im-

posed initial temperature gradients. The fluid was water in both cases.

Batchelor [Ref. I ) investigated rectangular enclosures using non-numerical

analytical methods for the limiting case of low Rayleigh number (Ra --, 0 )

and steady state conditions. Weinbaum (Ref. 2) made a similar study for the

circular cylinder. Dr. Robert F. Dressler (Ref. 3) of NASA Headquarters has

recently developed a non-numerical transient solution for the cylinder con-

figuration.

The purpnsc of the numerical study described herein is to establish

thc range of Rayleigh numbers for which the low Rayleigh number theories

of Batchelor, Weinbaum and Dressler may be considered valid. We used

the Lockheed developed LOCAP (Eckheed - Convection Analyzer Program) - - and GIM (General Interpolant Method) codes to perform these numerical

LOCKHEtD HUNTSVILLE RESEARCH & ENGINEERING CENTER - M III -- .----I---- -.--- -I-I----- -.. --- - - --.-- -

c.ot~rputalions. Tllc W C A P code is a two-dimensional code for analyzing

convection in rectangular enclosures, and the GIM code is a general purpose

fluid mechanics code for al l kinds of geometries and flow conditions. The

LOCAP computations were performed on the NASA-MSFC Univac 1108 com-

puter, and the GIM code on the much faster and more efficient NASA-Langley

STAR- 100 vector computer.

2

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LMSC-HREC TR 0697821

2. CIRCUIAR CYLINDER ENCLOSURE

r 1 Numerical computations were made for natural convection within the

circular cylinder configuration shown in Fig. 1.

Cold Hot Side - X

Fig. 1 - Geometry for Circular Cylinder Encloeure

The initial temperature distribution w Le based on a horizontal gradient in

the x direction, with the boundary pointe held constant in time;

T(R, t) = To + AT/^) c0.8

3

LOCKHEED HUNTSVILLE RESEARCH & ENGINFFQ iNG CENTER

The geometry module of the GIM code divided the c i rcular region into

an a r r a y of 20 x 20 a r e a elements with 21 x 21 nodal points. The proces s

treated the boundary as a four-sided figure, each side being a ci rcular a r c .

By use of stretching ful~ctions, each a r e a element became a quadrilateral,

with curvilinear sides, and with the nodal points located a t i t s four corner:.

F rom the discussion of the computer geaerated velocity vector plots, i t will

become c lear where the nodal points were placed.

We obtained transient flowfield conlputations over a range of Rayleigh

numbers f rom 1 to 100,000. Figure 2a shows a t ime history of maximum

computed flowfield velocities for various Rayleigh numbers. Both velocity

and time a r e dimensionless a s described in the nomenclature.

Up through Ra = 1000, the values of Tmax closely follow the same curve

through steady state. This curve i s a lso close to the theoretical t r a n ~ i e n t of

Dress le r (Ref. 3). Sorl?e deviation f r o m the low Ra results begins to occur a t - t = 0.1 for Ra = 5000 and 10,000. The steady state values for these cases

were extrapolated f rom the derivatives of ;max, since the numerical calcula-

tions converged slowly.

F o r higher Ra the deviations begin a t ea r l ie r t imes. Thus, for Ra = 100,000,

the low R a curve is followed until? = 0.06 where Tmax peaks and then falls to a

lower steady state value. Figure 2b shows a time history of deviation f r o m max the low R a values. Fo r Ra = 5000, the deviation is l ess than I ! yo throughout the

transient phase, while for Ra = 10,000 this level i s reached halfway through the

transient. Fo r the higher Rayleigh numbers, 20% deviation is reached about one-

third through the tranaient period.

Figure 3 shows Tmax a t steady state a s a function of Ra. The computed

results closcly approach the analytic value for low Ra, showing that the theory

i s valid up to Ra = 1000. The range of values 5000 < Ra < 50,000 may be viewed

a s a "transitiontt region. l o r l a rger Ra, the trend in Tmax approaches the (Rn)-f

dependence given by Weinbaum (Ref . 2).


LMSC-HREC TR D697821


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d

The low Ra theory i s a l s o valid f o r higher Ra during s o m e ini t ial pa r t d I of the transient . A s noted e a r l i e r , f o r Ra = 100,000 th is ini t ial s t age last

until; = 0.06. Th i s i s fu r the r indicated by compar ison of the radia l d i s t r i -

bution of velocities for Ra = 1 and 100,000 shown in Fig. 4. Not only the

maximum velocity, but a l s o the en t i r e velocity field c o m p a r e s well up t o - t = 0.06. Note that the computed distr ibution approaches the low Ra theo-

r e t i ca l s teady s t a t e distribution. Note a l s o that the e a r l y velocity d is t r ibu-

t ions a r c essent ia l ly l inear f o r a considerable d is tance away f r o m the cen te r ,

thus forming a solid body rotation inner c o r e that gradually d iminishes with

tinrc.

GIM code generated velocity vec to r plots r evea l that tb? flow a t low

Raleigh n\imbcr remains unicellular f r o m initial t r ans ien t through s teady

sh tc . The velocity vector plot shown in Fig. 5 f o r Ha = 1 a t s teady s t a t e is

seen to be nearly perfectly c i r cu la r . This plot i s essent ia l ly duplicated

throughout the t rans ient phase fo r a l l Rayleigh numbers up through Ra = 1,000.

A slight distort ion in the c i r c u l a r flow pattern i s seen fo r Ra = 10,000, and a

dist inct breakup into n ~ u l t i p l e ce l l s i s noted f o r Ra = 130,000. Th i s depar tu re

f r o m c i r c u l a r flow is a l s o indicated by the velocity contour plots in Fig . 6.

Notc thc skcwness that has developed in the s t i l l unicel lular contours f o r

Ila = 10,000. This skewness i s apparently c a r r i e d over into the mult iple c e l l

c-ontours for Ra = 100,000.

l lca t t r ans fe r a t low Raylcigh numbers i s domina tcd by conduction, and

thc 1-ffcct of increasing Rayleigh number is to inc rease the d is tor t ian of the

temperature? cc3ntours f r o m the ini t ial conduction profile. This ef fec t i s shown

in the steady s ta te t e n ~ p e r a t u r e contour plots in Fig. 7. Note tha t f o r Ra = 1

the contours a r e near ly s t r a igh t l ines , corresponding to a nea r ly pure conduction

t cmpera tu re field. The d is tor t ion in the t cmpera tu re field a t h igher Rayleigh

n i ~ ~ n b e r s leads to a n unstable condition that r e su l t s in depar tu re f r o m c i r c u l a r

flow and, fo r sufficiently high Rayleigh numbers , eventual breakup into multiple

ce l l s . This is c lea r ly indicated in the compar ison of t empera tu re contour and

vclocity vcctor plots shown in Fig. 8 f o r var ious t imes with Ra = 100,000. At .7..

t 1 0.1, the t cmpera tu re field is grea t ly d is tor ted f r o m the ini t ial pure conduction


LMSG-HREC TR D6978L1

9

LOCKHEED - HUNTSVILLE RESEARCH h ENGINEERING CENTER

LOCKHEED

10

HUNTSVILLE RESEARCH & ENGINEERING CENTER

LMSC-HREC TR ~ 6 9 7 8 2 1

LOCKHEED - HUNlSVlLLE RESEARCH 6 ENGINEERING CENTER

LOCKHEEO - HUNTSVlLl E RESEARCH 6 ENGINEERING CENTER

7-

LMSC-HREC TR ~ 6 9 7 8 2 1

13


-....-, .---.--. a

field, with the interior contours nearly horizontal. The flow i s s t i l l unto IC

cellular, but it has become somewhat skewed. By t = 0.2, the flow has

broken up into two cells , and this has caused the temperature contours to B r loop past the horizontal and to further dis tor t the temperature field. By -

,;; t = 0.3, a third cel l has developed and this has tended to shift the tempera-

tu re contours back toward a more stable configuration. i

'. 4 LOCKHEED - HUNTSVILLE RESEARCH 6, ENGINEERING CENTER w

. ; 111* - ------.-.---- 1: , A

, , *r*r. +-- ---. " , ~ - I. -"-?."we*---

LMSC-HREC TR ~ 6 9 7 8 2 1 3 ; 1

3. SQUARE ENCLOSURE

Fig. 9 - Geometry for Square Enclosure

The square enclosure configuration is illustrated in Fig.

' Y

Hot Side - X Cold Sidc! -

I

Numerical computations were made with the LOCAP code using a 10 x 10 a r r a y

of square elemcnts and the GIM code using a 20 x 20 a r r ay . The LOCAP code

was used to get initial resul ts because i t had existing capability for considering

buoyal-cy effects. The GIM code was used la te r after modifications were made

to the program to include buoyancy. As in the c i rcu la r cylinder, the initial

temperature distribution was based on a horizontal gradient in the x direction.

Calculations were made using the LOCAP code fo r both adiabatic ( ze ro normal

temperature gradient) a.nd constant temperature boundary conditions on the top

and bottom surfaces. The side surface temperatures were held constant in

both cases . The GIM code calculations were for constant temperature bound-

a r i e s only.

t

i

LOCKHEEO - HUNTSVILLE RESEARCH & ENGINEERING CENTER I

" ,.---.-- -,-.----- -

LMSC-MREC TR ~ 6 9 7 8 2 1

Shown in Fig. 10 a r e the computed maximum flow velocities a s a func-

tion of Raylpigh number after the flow has reached an equilibrium condition,

The theoretical low Rayleigh number limiting value, based on the f irst order

stream function in Ref. 1, is shown for comparison. Good agreement is noted

between the theoretical low Rayleigh number limit and the computed results.

The LOCAP results indicate significant deviation from the low Rayleigh nurn-

ber results starting a t about Ra = 1000. The GIM code results follow the same

general trend, but the deviation occurs a t highex Rayleigh number. Note that

the adiabatic boundary produces the greatest deviation from the low Rayleigh

number results in the LOCAP data.

LOCAP generated plots of velocity vectors, streamlines and absolute

velocities a re shown in Figs. 11, 12 and 13. Note the skewness that develops

in the unicellular flow patterns as Rayleigh number increases. At Ra = 100,000,

the flow appears to be nearly broken up into two cells. LOCAP plots of tem-

perature contours a r e shown in Fig. 14. Note that the adiabatic boundary

permits a greater distortion in the temperature contours than the constant

temperature boundary. This probably accounts for the greater deviation

from the low Rayleigh number results shown in Fig. 10 for the adiabatic

boundary cases.

CIM code generated velocity vector, absolute velocity and temperature

contour plots a r e shown in Figs. 15, 16 and 17 for constant temperature bound-

aries. The GIM code results generally confirm the LOCAP data. The GIM

code, however, predicts a definite breakup into three convection cells a t

Ra = 100,000, whereas the LOCAP code noted only an apparent incipient

double cell pattern a t the same Rayleigh number.

The differences between the LOCAP and GIM results may be attributed

not only to differences in the LOCAP and CIM computational techniques, but

also to the fact that the GIM computations were based on finer grid of nodal

points. The finer grid for the GIM computations was made possible by the

greater speed and efficiency of the STAR- 100 vector computer.


17


" .

2 E .: : t g -1a;,-. c, Id 5 4a 2 k: c E a ~d o s o P; U H h


LOCKHEED - HUNTSVILLE RESEARCH & ENGINLERING CENTER

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LOCKHECD HUNTSVILLE RESEARCH 6 ENGINEERING CENTE R


2 1


* . " - - --- - .-. - ".- - ..- " -- - . . - .. - . -

I dRIGINAL PAGE IS OF POOR QUALITY


23

LOCKHEED. HUNTSVILLE RESEARCH 6 ENGINEERING CENTER


LOCKHEED - HUNTSVILLE RESEARCH 4 ENGINCERING CENTER

The variation in maximum velocity a s a function of time is shown in

Fig. 18 for the GIM code data. The LOCAP code did not provide realistic

rcsults during the transient phase for the lower Rayleigh numbers. The

GIM code rcsults in Fig. 18 exhibit basically the same trends noted earlier

in Fig. 2 for the circular cylinder. Again, the theoretical low Rayleigh num-

ber limiting value, based on the first order stream function, + in Ref. 1, is 1' shown for comparison.

The velocity distribution along a horizontal bibector is shown in Fig. 19

as a function of time for R a = 1. The low Rayleigh number theoretical equi-

lrbrium distribution is shown for comparison. The traversing of the theo-

retical equilibrium curve over the GIM code results near equilibrium for

x/(L/z) > 0.7 is obviously not realistic. The solution for the f irst order

stream function given by Batchelor in Ref. 1 is actually an approximate solu-

tion intended to provide a simple algebraic equation form for the stream

function. Hence, exact agreement is not expected.

A conlparison of temperature contour and velocity vector plots at

various times is shown in Fig. 20 for Ra = 100,000. A strong correlation is

indicated between distortion of the temperature field and departure from

synlnletrical unicellular flow, as was shown earlier i12 Fig. 8 for the

circular cylinder.

A strong similarity has been noted throughout this section between the

square enclosure and the previously discussed circular cylinder results.

This suggests the possibility of using the simpler, more analytically defined

circular cylinder theory to predict convection velocities and flow patterns

in actual experiment configurations, which a r e more likely to be square than

circular. In order to match velocities in a model circular cylinder configura-

tion to an experimental square enclosure, we set the theoretical equilibrium

velocities for the two configurations equal and find the ratio of the circle

diameter to square side (page 29):

LOCKHEED. HUNISVILLE RESEARCH & ENGlNtERlNG CENTER

. . . .

LOCUHEED - HUNTSVlLLE RESEARCH & ENGINEERING CENTER

Fig

. 19

- V

elo

city

Dis

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as a

Fu

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of

Tim

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Wa

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1

LMSC-HREC TR Dbo7821

I . . - - - - - - - - - - - e m . .


m u I I I III L M

LMSC-HREC TR D69782l

Compare this ratio to the ratio 1.1284 that yields equal areas for the circle

and square. This somewhat fortuitous result clearly implies that an experi-

mental square enclosure may be closely modeled by a circular cylinder of

equal cross sectional area. The validity of this modeling is additionally

supported by the rather remarkable degree of radial symmetry that exists

in the low Rayleigh number velocity field in the square enclosure. This is

dieplaycd in the angular variation of velocity shown in Fig. 21 for various

radial distances. These velocities were determined from the first order

stream function in Ref. 1.

29

LOCKHEEO - HUNTSVILLE RESEARCH 6 tNGlNEERlNG CENTER

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(Fr

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Ref. 1

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4. CONCLUSIONS

The primary conclusion to be drawn f r o m this study i e that the low

Rayleigh number theorv for convection in c i rcular cylinder and square box

enclosures i s valid throughout the transient phase to equilibrium fo r Rayleigh

numbers up to a t leas t 1000. The GIM code resul ts , which we assume to be

most valid, show a deviation of 20% from the low Rayleigh number resul ts a t

Ra = 5,000. Additionally, for higher Rayleigh numbers, the low Rayleigh

number theory is valid f o r some portion af the transient, depending on the

value of the Rayleigh number, before significant deviation becomes apparent.

In the case of both c i rcular cylinder and square box enclosures a t Ra = 100,000

the valid portion of the transient phase extends up to?= 0.06, which i s approxi-

mately 20410 of the time required to reach equilibrium.

Another conclusion drawn f rom the comparison of the c i rcular cylinder

and square box enclosure resul ts i s that experimental square shaped con-

ta iners can be analytically modeled ra ther closely by c i rcu la r ~ y l i n . ~ e r s of

equal c r o s s sectional a r e a for the prediction of convection velocities and flow

patterns a t low Rayleigh number.

3 1

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. +-- .-.-.- *------.-- . .-

I

'I-

4 d -

I

LMSC-HREC TR D(9 ; 7821

REFERENCES

1 . Batchelor, G. W . , "Heat Transfer by F r e e Convection Across a Closed Cavity Between Vertical Boundaries at Different Temperatures," Quart. Appl. Math., Vol. 12, No. 3, 1954, p. 209.

2. W einbaurn, S., "Natural Convection in a Horizontal Circular Cylinder , I 1

J. Fluid Mech., Vol. 18, 1964, p. 409.

3. Dress ler , R . F., NASA Headquarters, Washington, D. C. (to be published).

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.. ... .

Documents

kLD CIRCULAR CONTAINERS I IN LOW GRAVITY I - … · I kLD CIRCULAR CONTAINERS I IN LOW GRAVITY ... ~~arcll IL Engineering Center, ... gravity force length of square side radial distance