Indian Journal of Engineering & Materials Sciences Vo l. 9. December 2002, pp. 448-454

A combined numerical-experimental study of convection in an axisymmetric differentially heated fluid layert

Atul Srivastava & P K Panigrahi

Department of Mechanical Engineering, Indian Institute of Technology, Kanpur '1.08 016, Ind ia

Received 13 December 200 I .. accepted II October 2002

Buoyancy-driven convcction in a horizontal differenti ally heated fluid layer has bet:r. analysed nu merica ll y :lIld compared with thc experimental results obtained for a c ircular axisymmetric fluid layer, the -: ;,Ie wa ll s of the cav ity being thermally insulated and of constant radius. The cmployed Raylei gh numbers are 586 1 and 12, I L ~ and the work ing flui d air. The governing equations arc continuity equation, Navier-Stokes equation and the energy equation. Due to the axi symmelJ:ic nature of the fluid cavity, the governing equations have been solved in r-z coordinate system. A fine uniform grid o f 60 I xS I is employed. Experiments have been conducted in the fluid layer of 0 .64 m diameter and 23 mm vertica l height with the same working fluid as considered in numerical simulation. A Mach-Zehnder interferometer has been used to visuali ze the long-time convection pallerns in the fluid layer. The goal of the study is to investigate the influence of Rayleigh number on the steady (long time) thermal field, quantitative analysis o f heat transfer rates and compari son with the ex peri mental results. A good agreement has been found in qualitative as well as quantitative sense between theore tica l and ex perimental result s.

Rayleigh-Benard convection in horizon tal fluid layers is a problem of fundamental as well as practica l importance. The flow pattern associated with this configuration shows a sequence of transitions from steady laminar to unsteady flow and ultimately to turbulence. This configuration has been studied by various researchers using computational techniques as well as by experiments to understand the transition phenomenon. The experimental techniques have been strengthened by the avaibbility of optical methods to visualize the flow phenomena and computers for data storage, process ing and analysis of the images. A large amount of literature is available in the field of Ray leigh-Benard convection. Computational studies are quite abundant. Most of the available literature relates to small aspect ratio enclosures. This is because numerical calculations are easier to perform over a small domain. Analytical studies have shown that in an infinite fluid layer, the first transition from the conduction state to the steady cellular occurs at Rayleigh number of 1707.8'. This value is independent of the Prandtl number of the fluid , though it is a function of the overall geometry. When the Rayleigh number (Ra) just exceeds the critical value, hexagonal convection cells have been observed in experiments as well as computation studies2

. All

' Presentcd at the 28th National Conference on Fluid Mechanics & Fluid Power, he ld at Punjab Engineering College, Chandigarh, during 13- 15 December 2001

subsequent transitions are dependent on the Prandtl number and the geometry of the confining surfaces. Grotzbach] performed a direct numerical simulation of the Rayleigh-Benard convection probl em under both laminar and turbulent conditions. The fluid considered was air. The cavity was an infinite parallel plates channel. At Ra=4000, skewed-varicose instability was seen and at Ra=7000, the ve loc ity fi e ld was unsteady and three-dimensional.

Mukutmony and Yang4 have demonstrated the loss­of-rolls phenomenon as the Ray le igh number is increased in an intermediate aspect rati o box. They have simulated the experimenlal conditi ons of Kolodoner et aP., and investi gated the transiti on from 10 rolls to 6 rolls using Rayl eigh number in the range of 10,000 to 24,000. Mukutmony and Yang6 have numerically simulated the experiment conducted by Gollub and Benson 7. The process of fl ow reversal to a steady state at higher Rayleigh number was observed. The calculations were performed for low aspect ratio enclosure and for a fluid of Pr:=5. The Ray lei gh numbers considered were In the range of 40,000 to 120000.

Loss of rolls has also been documented by other workers. Lin et al. 8 have numerically shown thi s phenomenon for an intermediate aspect ra tio box and for a fluid of Prandtl number 3.5. Mi shra el a l .!) have experimentally studied Rayleigh-Benard convection in intermediate aspect ratio enclosure with air as the

SRIVASTAVA & PAN IGRAHI: CONVECTION IN AN AX ISYMM ETRI C FLUID LAYER 449

working tluid considering the Rayleigh numbers as 13900, 34800 and 40200. The authors concluded that at a Rayleigh number of 13900, the frin ges were steady near the boundary wall s but mild un steadiness was present in the central hori zontal layers. At hi gher Rayleigh numbers, the un stead iness was more pronounced with now switching between the two well-defined states. Apart from thi s study , the authors contributed notably in the fi eld of tomography.

In the present study , numeri ca l simulation has been carri ed out for two Rayleigh numbers namely 5861 and 12, 124. Since these values are significa ntl y lower than the cri tical Ray leigh number for turbulent flow (::::; 50,000 for a square cav ity of intermediate aspect rati o)~, the study is in the laminar reg ime. The governing di fferential equati ons for buoyancy-driven fl ow in an ax isy mmetri c cavity have been numeri ca ll y so lved. The grid used is 60l x8 1 in rand ? direc ti ons, respectively. Ex periments have been conducted in a fluid layer of 64 cm diameter and 2.3 cm verti cal hei ght. A Mach-Zehnder interferometer has been used to visualize the convec ti on patterns in the fluid layer. The numeri ca ll y generated roll patterns, streamlines and the heat transfer rates in terms of the Nusselt numbers at the top and the bottom wall s have been reported and compared with the experimental res ults.

Mathematical Model The axi symmetri c form of co ntinuity, Navier­

Stokes and energy eq uations have been numeri ca ll y solved in the present study . The flow is taken to be incompressibl<:, with buoyancy accounted for via the Boussi nesq approx i mati on. The convecti ve terms of the momentum equa ti ons are di screti zed using the 2nd

order upwi nd scheme while central difference discretization has b~en appl ied to the di ffusi ve te rms. The Ilow equati ons have been solved using the stream function-vorticity approach. The matri x in version is carri ed out by the Gauss-Siedel method. The solutions for velocity and temperature have been obtained by marching in time wi th initial transiti ons neg lected. Due to the axtsymmetric nature of the fluid cavity , the governing eq uations have been so lved in r --;

coord inate sys tem. The grid on whi ch the results have been obtai ned has 60 I x8 1 cell s in the r and :: directions respectivel y. The radial data obtained from the nu merical simulation have been transformed onto a rectangular grid , which is of the same dimensions as the dimensions of the window Llsed the

ex perimental tes t cell. The rectangular grid has also been divided into 81 x60 I grid points.

Governing Equations The governing equations for the present numeri cal

study are continuity eq uati on, Navier-Stokes equati ons and the energy equati on, which in general form are:

COl/lilluily

all It av - +- + - =0 a,. ,. oz. . .. ( I )

r-M OIl/el/ IUIII

[au au au ] P - + lI - +V - = al ar az

ap (aCII -- + f..I --0 ar ar - ... (2)

:: -M (J ill el/ 11/11/

... (3)

Dlergy

+--+--l aT a "2 T ] r ar az"2

. . . (4)

NOIl-dimClIsiollalizatioll

Various non-dimensional parameters used in the numeri cal simulation are:

Reference lelllperuillre- The temperature has been T T

non-dimensionalised as e ::: ,. 7~, - T,

The temperature is defined in sLl ch a way that it va ri es from zero at the cold wal to unity at the hot V\ all.

Referel/ ce leI/gIlt-The height of the cavity, It, plays a major role in the !low LIS he temperat ure variat ions

450 INDI AN 1. ENG. M ATER. SCI., DECEMB ER 2002

and thus density vari ations responsible for buoyancy force due to gravity exi st in the vertical direction . Therefore, the height of the enclosure is considered as the reference length for non-dimensionali zation .

Refe rence veloci(y-The non-dimensionali zed velo-

city is taken as V = ~ II

Refm" ce pm.",re·· P ~ p( * )' The non-dimensional parameter responsible for the

buoyancy induced flows in the cav ity namely the Grashoff number is defin ed as:

C r = Raj PI'

where Ra is defined as :

g{J (T" - 7~. ) h 1 Ra = -'------"--------'--

V Pr =­

ex

va

W - IV - T Formulation

The u and v veloc iti es have been defin ed in terms of strea m functions cp as:

l Olfl II = --

r OZ l olfl

V=--r or

Accordingly, the governing equati ons can be transformed to the strea m fun ction, vorti city and energy equat ions as:

Streall/ Function Equation

Vo rticity Equation

aw aw (lw u 0 2W I aw -+ {( - + v---w=--+- -at ar 0::. r dr 2 I' dr

a2w I Ra aT +-7---' w+--aZ. - r- Pr ar

. . . (5)

. . . (6)

Energy Equation

Boundary conditions

The boundary conditi ons fo r stream functi ons, vorticity and temperature for the circul ar cavity can be deri ved by using the fac t that the veloc ity sa ti sfies the no-slip condition, the hori zontal surfaces are isothermal , and the side wall s are insulated. These can be expressed mathemati cally as:

1 a2cp Top Wall - T = 0 ; IJf = 0 ; and, w = - -; 0::' 2

I a2cp Bottom Wall - T = I ; IJf = 0 ; and, w = ---­

r O~ 2

. aT Side Wall s - If! = 0 ; - = 0 ; and W = 0 ar Initial conditions

The following initi al conditi ons have been applied:

IJf = 0 ; W = 0 ; and, T = linear

Experimental Laser interfero metri c technique has been empl oyed

to visuali ze the flow field when subjected to a temperature gradient between its top and the bottom surfaces. A Mach-Zehnder interferometer is used to collect the line-of-sight projec ti ons of the temperature inside a Ray leigh-Benard set-up. Since the cavity (as di scussed earl ier) is circular in pl an, hence one is led to an ax isy mmetric confi gurati on. Fluid employed is air with Pr=0 .7 1. Ex periments were cond ucted in a tluid layer of 64 cm diameter and 2.3 cm verti ca l height. The side wa ll s were kept insulated with temperature gradient applied in the vertica l direct ion. All ex peri ments have been performed to record the long- time convecti on patterns in the test cav ity. Thi" has been referred to as the steady state thermal field in the fl uid layer. The ex periments are run fo r a period of 11-1 2 h to achi eve thi s cond ition . Long ti me convecti on patterns are coll ected to ensure that steady state prevail s in the fluid layer with the initi al transients eliminated. At steady state, th ree to four interferograms have been recorded in a gap of 10- 15

SRI V ;\STAV;\ & PANIGRAHI: CONVECTION IN AN AX ISYMMETRIC FLUID LAYER 45 1

min . The ex periments have been performed at two

different ang les namely 0° and 45° by ro tating the test cell. These steady state inte rferograms are processed and analysed to ex tract the quantitati ve informatio n. The thermal fi e ld in the tluid layer is three dimensional, despite the axisy mmetric geometry. Hence the inte rferog rams are to be inte rpreted as path integrals o f the refrac ti ve index fi e ld.

Results and Discussion Nussclt numher calculation

T he Nusselt number representing the dimensio nl ess wa ll heat transfe r rates is de fined as:

Nil = - Ii aT I T - T a v =O.II

I I C Y .. . (8)

In the present work, the local and average Nu sse lt nurnbers al the top and the bo tto m wa ll s have been reported. The average Nu ssel t number fo r each of the plates has a lso been compared with the experimenta l corre lati on reported by Gebhart el al.lO. For a ir, the corre lation is g iven by:

[ 1708] [( R )1/3 l NII = 1+1 .44 \--- + _a_ - I

J Ra 5830 . .. (9)

Numcrical resuits

T he results di scussed here perta in to the steady state solut ions o f the process. The numerica ll y

i ~ r;; ~~ ~ ~ ~

:::::-- /=

~~ @] ~I ® ~ @ m I \'- ~ ~ Q '-~

Fig. I- SlI-ealll lincs fo r full cavity (Ra=586 1)

Fi g. 2- lso therllls fo r fu ll cavity (Ra=586 1 )

Fi g. 3-Strea l11l ill eS fo r fu ll ca vity (Ra= 12, 124)

de termined streamlines and isotherms in the tluid layer for the full cavity for Ra=586 1 are shown in Fi gs I and 2.

A comparab le plot for Ra= 12, 124 is shown in Figs 3 and 4. In both the fi gures, the formatio n of ro ll s is quite c lear. The number of roll s is less than the aspect ratio of the tluid layer and decreases with an increase in the Ray le igh number. Thi s loss-of-roll pheno meno n is quite similar to that seen in a rectangul ar cavity (Ko lodner el al.\

To compare with the ex perimental results, the numeri call y generated radi a l data have been transformed onto a rectangular grid. The grid has the dimensions of the windo w used in the experiments fo r capturing the in terferometri c projections. Figs 5 and 6 show the numericall y generated contours fro m the projections calculated a lo ng the length o f the cavity for Ra=586 l and Ra=1 2, 124 , respecti ve ly . For clarity , o nly a few ro ll s are shown. The formatio n o f ro ll s simil ar to those inte rferograms captured experimenta lly (sho wn in Fig. 7) is c learly seen in the cav ity. In an ax isy mmetric geometry, an equ iva lent result is not ava ilable in the publi shed literature till date to the bes t of the knowledge of the authors.

Fig . 4--lso therl11s fo r full cavity (Ra= 12, 124)

Fig. S--Numeri ca ll y generated conto urs from the projection da ta (Na= 586 1)

Fig. 6-Numerica ll y genera ted contours from the projec tion data (Ra=1 2.1 24)

-l-S2 INDI AN J. ENe;. MATER. SCI. DECEMBER 2002

Ev idence reported in the present work shows that the axisy mmetric ro ll s arc poss ibl y in the form of conce ntri c rings, the schematic ror whi ch has been sho\Vn in Fig. ~. For H{/= 12, 124, the ro ll s are well ­defined through a clear displacement or the contollrs. The con tours in Fig. 6 have been show n along with a shad i ng scheme, a darker background i ndi ca ti ng higher gradi ent s. As ex pected, the temperature gradients nca r the wa ll s are hi ghes t. The shaded reg ions near the wa ll s arc relatively strai ght, similar to strai ght rri nges in the ex peri mental i nterfcrog rams. These are al so coincident with the sites o r the hi ghest convccti ve r1uid ve loc ities.

Comparision with the ex perimental results

Interrerograms co llec ted ex perimen tall y were proce~sed and analysed to get quantitat ive data in te rm: of temperature di stri but ion in the flu id layer, hea t transfer rates de termined by local and average Nussc\ t numbers at the top and bottom walls.

Usi ng the nu merica ll y determined projection data, the vari ati on o r the width -averaged temperature pro ril e with respect to the vertical coordin ate has been plll llCd in Figs l) and 10. Ex pcrimentall y obtained \V idth-averaged tcmperature pro fil e is also shown for the two view angles (0° and 45°) and for both the Rayleigh numbers in the sam e fi gure. A good agreement can be seen between the nu meri ca l simulat ion and the ex perimental results. A sli ght difference can be at tri buted to the mild un steadiness

Fig. 7- Expcri l1lc ntally o bta ined illlerfCrogralll

I :Ig. X-Scilematic of thc rol ls in thc circ ular cavity

and lateral movement of the fringes present in the fluid layer during the experiments that was not re rl ected in the simulation. The in verted S-shaped curve in both the cases reveal s the buoya ncy clri ven convec ti ve fl ow. The slopes o r the indi vidual curves in ncar the walls (v=O, I) arc quite close to each other. It refl ec ts the fact that the net heat transfer across the cav ity at steady state from the hot surface to the cold does not depend on the view angle. In th is respect, the co nvec ti on in th e fluid layer is axi sy mmetri c. Thi s also refl ects the energy ba lance at the steady state.

The numeri cally determ ined local Nussel t numbers within a ce ll at the hot and the cold wall s are present ed in Figs II and 12 ror both the Ray leigh numbers. The ave rage Nus';e lt number for the full cav ity at !?a=586 I and !?a=! 2, 124 numbers have been ca lculated to be 2.06 I and 2.53 I, respective ly.

Ex perimentall y, the local and average heat transfer rates at the boundi ng wa ll s have also been cal culated in lerms of local Nusselt number defined by Eq . (8). Fig. 13 show the va riati on of the local Nusselt number at the top and the bottom wall s of the cav ity at !?a=586 I and !?a= 12, I 24 for 0° and 45° projection

0 .8

0 .6 £

>-0.4

0 .2

0 0

- , .... " ,

Numerical ' . \ 0 ° . ,

., ., '. , ,

0.2 0 .4 0 .6 0.8

e

r:i g . l).·-Widlh-ave ragcd tClllpcrature var iati o ll (Na=5X6 1)

0 .8

0 .6 .£:

>-0.4

0 .2

0 0 0.2 0.4 0 .6 0 .8

e

Fig. IO--Wiclth-avcragccl tcmperaturc va ri atio ll (Ra= 12.124)

SRIVASTAVA & PAN IGRAH I: CONVECTION IN AN AXISYMM ETRIC FLUID LA YER 453

~ ---1 3 . lower NU 2 - ' --, ~= 1 ._. _ . _ . . _._ . _'

Distance scaled by the cavity height

rig. \ l- Nul1lc rica ll y obta ined local Nusselt number va riation, (Rfl=586 1)

NU ;~ __ J Distance scaled by the cavity height

Fig. 12- Numerically obtai ned loca l Nusse ll numher va riation, (/?a= 12,124)

angles . For 0° projection angle, the individual wall ­averaged Nusselt numbers are 2.07 and 1.996 at the cold and hot surfaces respectively . For the 45° projection angle, these are 1.9 16 and 1.893 respectively. The average values of the Nusselt numbers, as calcul ated from both the projecti on angles, match well. This reveal s the axi symmetri c nature of the convection in the tluidlayer at Ra=5 86 1.

At Ra= 12, 124, fo r 0° projection angle, the indi vidual wall-averaged Nusselt numbers are 2.48 and 2.38 at the cold and the hot surfaces, res pecti vely. The corresponding values fo r 45° projecti ons are 2.3 1 and 2.26 at the cold and th e hot surfaces respectively. For Ra= 12, 124, the difference in the va lu es of the hea t trans fer rates for the two projection ang les may be attribu ted to the mil d unsteadiness present in tile flow fie ld. Th is can be the reason for the loss of ax isy mmetry in strictly quantitative terms. These values of the heat transfer rates, detennined ex perimentall y and numerically , are In good agreement with the va lues given by experimental correlation of Gebhart el 0 1.10 The average Nusselt number as determined from the correlation is 2.081 at Ra=586 I and 2.5 1 at Ra= 12, 124 with an uncertainty lc vC'l of ±20/){ .

The loca l Nusselt number in Fig. 13 has been plotted as a function of the dimensionless distance within a rol l. Regions of hi gh heat transfer on one wall correlate well with those of low heat transfer on the other. This trend brings out the skewness of the streamline pattern In the indi vidual rolls. An

' ~ ......... 1 3 lower

, ; ~ ,: ~· ~-:" " ~l ~2 ~_ .•.. ---

~ -- - ------------ -~pp~;------ --.-- 4~

Distance scaled by the cav~y height

Fig . i}---Ex pcrimenta lly obta ined local Nussellnul1lber variati on

examination of the adjacent cell s would show that the ro ll s have opposing inclinat ion.

Two factors that can additi onall y lead to a di screpancy between the ex periments and the numerical simulation are: (a) the average Nusselt number in the experiment is calculated over only a si ngle roll , and (b) unsteadi ness in the experiments that is not reflected in the simulation .

The results obtained from the numerical simulation support the formation of the concentri c roll s as predicted from the ex perimental observation. Ex perimentall y, the fringe patterns as shown in Fi g. 7 have been recorded in a direction parall el to the diameter of the tes t cell. Appearance of ci rcul ation ro ll s can be proposed for the two Ray leigh numbers. Thus, the viewing axi s parallel to the light beam is at ri ght angles to that of the circul at ion pattern within the rolls (Fig. 8). Owing to the component of the radial velocity normal to the viewing direct ion, add itional cell-like structures can be ex pected in the ex peri mentally obtai ned intelf erometric projections. A typical interferogram of the circular fluid layer is shown in Fig. 7. The tlow cell in the interferogram, distinct from those in Fig. 8 can be identifi ed as fo llows. The cell boundari es wi II pass through the locations where the fringe slope is zero. This procedure is meaningful because one can visualize hot fl uid ri sing along one boundary, on the right side in Fig. 7, and descend along the other. Hence the cell in Fig. 8 can be considered as primary and that in Fig. 7 as secondary.

Conclusions The present investi gati ons are focused on the study

of Rayleigh-Benard convection for an axisymmetric

454 INDIAN J. ENG. MATER. SCI., DECEMBER 2002

cavity both experimentally and numerically which has not been reported till now. The published literature on thi s topic shows that this s tudy is limited to rectangular and square cavities only till date. This paper presents the numerical study of Rayleigh­Benard convection in a circular, differentially heated large aspect ratio fluid layer for two Rayleigh numbers, 5861 and 12,124 with air as the working fluid. The numerical results are compared with the experimentally obtained quantitative results. For both the values of the Rayleigh numbers, the numerically generated streamlines and the isotherms in the fluid layer for the full cavity show the clear formation of rolls. The loss-of-rolls phenomenon has also been observed with an increase in the Ray le igh number. The numerically generated contou rs from the projections calculated along the length of the cavity also show the formation of rolls s imilar to those in the i nterferograms obtai ned ex peri mentall y. Wall heat Iransfe r rates calculated in terms o f Nusselt numbers show good agreement between the numerical simu lation and the experiments and compare well with the correlation proposed by Gebhart et at. 10. An acceptable agreement In the wid th-averaged temperature profiles between the numerical simulation and the experimental values is also observed.

Nomenclature C; Accelaration due to gravit y h Hcight of the cavity

II Refractive index of the Iluid Nu Nusselt number Pr Prandtl number of the Iluid Na Rayleigh number T Temperature 7:, Reference telnperature r, Z Cylindrical co-ordi nates a Thermal diffusivity of the Iluid f3 Coefficient of volume expansion A Wavelength of the lascr v Kinematic viscoc ity of the Iluid p Density of fluid

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