**J**_{ournal of }**A**_{pplied }**F**_{luid }**M**_{echanics}_{, Vol. 9, No. 3, pp. 1265-1272, 2016. }
Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645.

**Effect of Rotation on Natural Convection in Differentially **

**Heated Rotating Enclosure by Numerical Simulation **

### M. Narendra Kumar

1### G. Pundarika

2### K. Rama Narasimha

3†### and K. N. Seetharamu

4*1,3*

* Centre for Emerging Technologies, Jain University, Bangalore, Karnataka, 562112, India *
*2*

* Government Engineering College, Ramanagara, Karnataka, 571511, India *
*4*

* PES University, Bangalore,Karnataka, 560085, India *

†*Corresponding Author Email:k.ramanarasimha@gmail.com *

(Received May 6, 2014; accepted June 24, 2015)

**ABSTRACT **

Natural convection heat transfer in a two dimensional unsteady rotating differentially heated enclosure is studied numerically in this paper. The enclosure is filled with air and executes a steady counterclockwise rotation about the centre of the enclosure. A finite volume code on a staggered grid arrangement with TDMA algorithm is developed and employed to solve the governing equations subject to Boussinesq approximation. The numerical investigation is carried out for fixed Prandtl number equal to . , Rayleigh number equal to . while Taylors number vary from . . and Rotational Rayleigh number from

. . .Results reveal that there are considerable change in heat transfer rates beyond 15 rpm. The effect of rotation on the Nusselt number for a given Rayleigh number is shown in the present work which is not normally indicated and discussed in the available literature

**Key words:** Rotating enclosure; Natural convection; Coriolis force; Heat transfer.

**NOMENCLATURE **

k thermal conductivity (W/m-K) L length of the enclosure (m) Nu local Nusselt number P dimensionless pressure

p,Pm pressure and motion pressure (N/m2)
*Pr* Prandtl number

*Ra* Rayleigh Number
*Raw* rotational Rayleigh number

system

T temperature (K)
*t* time (s)
*Ta * taylor number

u, v dimensional velocity components (m/s)
*U, V* dimensionless velocity components
*X, Y* dimensionless Cartesian coordinate
x,y dimensional Cartesian coordinate system

Ω magnitude of angular rotation rate (rpm, rad/s)

α thermal diffusivity (m2/s)

τ dimensionless time

θ dimensionless temperature angular position of enclosure (o)

ρ density (kg/m3)

μ dynamic viscosity (Pas)

**Subscript **

c cold h hot f fluid

**Superscript **

- space average = Time average

**1.**

**INTRODUCTION **

The fluid flow and heat transfer characteristics of a rotating enclosure were treated in detail according to its functional and practical importance in the thermal management. The unsteady rotating

M. Na temper failure rotatin turbine experi in a w layer

increa
trend i
numbe
mercu
therma
box fi
roll ce
numbe
of a
experi
(1994)
arising
influen
rotatin
differe
Signifi
the rot
Raylei
in heat
reporte
*al*.(200
rectang
They
betwee
circula
oscilla
reduce
effects
therma
on a di
*al*.(200
Howev
natura
heated
natura
convec
flow
import
neglec
state o
to refl
fluid,
(Boub
A det
differe
especi
Coriol
same f
challen
reporte
In the
effects
enclos
walls.
charac
rate ar

arendra Kumar *e*

rature and vibr es. There had b ng ducts, for ins e blades by Lin mentally obser water layer for

for

,the Nuss se with the T is the case for er, oscillatory ury. Buhler an

al cellular conv illed with a vis ells changed ori

er. The flow an rotating squ mentally and n ). They conclu g from rotatio nce on heat tra ng results. Ker a ential heated ficant flow mo

tational Raylei igh number. A t transfer could ed by Baig 05)carried ou gular enclosur found that rot en clockwis ations, increase ations of Nuss es the average p s of Coriolis al buoyancy fo

ifferentially hea 07)

ver, there are al convection d sources in al convection i ction because

where as the tant role and th cted. In a sense, on earth, since ect the effect o is about unity nov and Golitsy

tailed 3-D num entially heated

ally with the in
lis force and c
frame. When th
nging and use
ed by Jin *et al*.

e present work s of rotation on sure, with differ The effect cteristics of flu re considered.

*et al.* /**JAFM**_{, V}

ration amounts
been many stu
stance in relatio
n *et al.* (2001)
rved the subcr
r

. In addition elt number w Taylor number

air. Besides, a convection i nd Oertel (198 vection in a rota scous fluid and entation with in nd heat transfe uare enclosur numerically by uded that the on may have ansfer when co and Lin (1996, rotating cu difications are igh number is

significant incr
d be achieved d
*et al*.(2001,
ut numerical

re with discre tation results in se and co es heat transfer selt number. performance in force, centrifu rce are segrega ated square enc

limited studie cooling from rotation. The is very differe

the latter is e centrifugal f he thermal buo , there is no abs Taylor number of Coriolis forc y at the equat

yn 1995).

merical compu source conditio nfluence of the entrifugal forc he enclosure is r

ful to study a (2005).

k, it is intende n natural convec rentially heated of rotational uid flow as we

Vol. 9, No. 3, p

s to 75% of a udies on flow i on to cooling o . Rossby (1969 ritical instabilit

and in an ai
n, for water a
was found t
r. The opposit
at a large Taylo
is preferred i
82) studied th
ating rectangula
d found that th
ncreasing Taylo
er characteristic
e are studie
y Hamady *et a*
Coriolis forc
e a remarkabl
ompared to no
1997) studied
ubic enclosure
obtained whe
greater than th
rease or decreas
due to rotation a
2006). Jin *e*
studies on
te heat source
n an imbalanc
ounter-clockwis
and reduces th
It improves o
each cycle. Th
fugal force an
ated numericall
closure by Tso* e*

es on unstead m differentiall situation wit ent from force pressure-drive force plays a oyancy effect i solute stationar r, normally use e in the rotatin torial sea leve

utation for thi on is formidable buoyancy force e all within th rotated, it is sti a 2-D model a

ed to study th ction in a squar d top and bottom

speeds on th ell heat transfe

pp. 1265-1272,
all
in
of
9)
ty
ir
at
to
te
or
in
he
ar
he
or
cs
ed
*al*.
ce
le
on
a
e.
en
he
se
as
*et *
a
e.
ce
se
he
or
he
nd
ly
*et *
dy
ly
th
ed
en
an
is
ry
ed
ng
el.
is
e,
e,
he
ll
as
he
re
m
he
er

**2.**

The sch
under in
time
stationar
enclosur
clockwis
surface
temperat
subjecte
walls a
surround
position
the enc
rotation
the Bou
1982; K
density
both the
terms r
buoyanc
equal to
and
developm
**Fig. 1.**The con are

*u*

*v*

*x*

*y*

_{} 2

*u*

*u*

*t*

*g*

_{} (

*v*

*u*

*t*

*g*

*y*

_{} 2016.

**MATHEMA**

**FORMULA**

hematic diagra
nvestigation is d
, the enclosu ry and isoth re executes a se rotation ab at ⁄ is ture and d to uniform are thermally dings. The in

of the enclosu closure is sim and the therm ussinesq approx Ker and Lee 199 variation with

body force and representing th cies and Corio

ment can be de

**Schematic rep**
**s**
ntinuity, mome
0
1
( )sin(

2 ( )

*o*
*c*
*c*
*u* *u*
*v*
*x* *y*

*T* *T* *t*

*v* *T* *T*

_{} _{ }
1
( )cos(

) 2 2 (

*o*
*c*
*v* *v*
*v*
*x* *y*
*T* *T*

*y* *u* *T*

_{} _{ }

**ATICAL **

**ATION **

am of the phy depicted in Fig ure and inside hermal at . A a steady unif bout a horizon subjected the surface temperature well insulate n Fig. 1 define ure. Thus the a multaneously d mal buoyancies ximation, (Buhl

96 and Jin *et al*
temperature is
d centrifugal fo
he thermal a
olis force are,

,

. The scribed as follo

**presentation of**
**system. **

entum and ene

###

###

2 2 )*c*

*p*

*u*

*x*

*x*

*t* *T* *T*

*v*
_{} _{}
_{}

###

###

2 2 ) ( )*o*

*c*

*c*

*p*

*v*

*y*

*x*

*t* *T* *T*

*T* *T* *u*

_{} _{}
_{}
ysical system
g.1. Initially at
e air are both

At ,the form counter ntal axis. The

to uniform at ⁄ ,

. The lateral ed from the es the angular air flow inside driven by the

. By adopting
ler and Oertel
*l*. 2005) linear
considered in
rce terms. The
and rotational

respectively,

Ω Ω

resulting flow ows:

**f the physical **

2 2

2 2 (4)

*T* *T* *T* *T* *T*

*u* *v*

*t* *x* *y* *x* *y*

_{} _{} _{} _{} _{} _{}

_{} _{}

Boundary Conditions are

0 0

2

0 0

2

(5) 0

2

0 2

*c*

*h*

*L* *T*

*At x* *, u* *v* *, *
*x*

*L* *T*

*At x* *, u* *v* *, *
*x*
*L*

*At y* *, u* *v* *, T* *T*

*L*

*At y* *, u* *v* *, , T* *T*

_{ }

_{}

_{}

_{}

_{}

With the motion pressure defined as

2 2 2 2

1 1

sin( ) cos( )

2 2

*m*

*P* _{} *x* *y* *gx* *t* *gy* *t* _{}

(6) As mentioned by Vanyo (1993) in a rotating Eulerian coordinate system the centrifugal term can be included in the pressure term and disappears from the typical rotating fluid computation. With Boussinesq approximation, the main part of the centrifugal force term is combined with the pressure term and others are caused by density change and centrifugal buoyancy. The governing equations can be converted to non-dimensional forms using the non-dimensional parameters as indicated in (7).

###

###

###

###

2

2 2

3

2 4 2 4

2

Pr

4

*m*

*h* *c*

*c*

*h* *c*

*h* *c*

*w*

*x* *y* *uL* *vL* *tα* *P L*

*X* *Y* *U* *V* *τ* *P*

*L* *L* *α* *α* *L* *ρα*

*g* *T* *T* *L*

*T - T*

*θ* * Ra*

*T - T*

*T* *T* *L* *L*

*Ra* * Ta*

_{}

_{}

(7) The Coriolis buoyancy force is neglected as

| | in the present study [Jin *et al*.
(2005)]. The dimensionless governing equations
and boundary conditions are written as

follows:-Continuity equation:

0 (8)

*U* *V*

*X* *Y*

_{} _{}

Momentum equations:

###

###

###

###

###

###

###

###

2 2

2 2

0.5

0.5

Pr

sin 0.5 (9)

*w*

*U* *U* *U* *P* *U* *U*

*U* *V* *Pr*

*X* *Y* *X* *X* *Y*

*Ta* *V* *Ra Pr X*

*RaPr* *Ta Pr*

_{} _{} _{ } _{} _{} _{} _{}

_{} _{}

###

###

###

###

###

###

###

###

2 2

2 2

0.5

0.5

Pr

cos 0.5 (10)

*w*

*V* *V* *V* *P* *V* *V*

*U* *V* *Pr*

*X* *Y* *Y* *X* *Y*

*Ta* *U* *Ra Pr X*

*RaPr* *Ta Pr*

_{} _{} _{ } _{} _{} _{} _{}

_{} _{}

Energy equation:

2 2

2 2 (11)

*U* *V*

*X* *Y* *X* *Y*

_{} _{} _{} _{}

Boundary Conditions:

0.5 0 0

0.5 0 0 (12)

0.5 0 0.5

0.5 0 0.5
*θ*

*At X* *, U* *V* *, *

*X*
*θ*
*At X* *, U* *V* *, *

*X*
*At Y* *, U* *V* *, *
*At Y* *, U* *V* *, *

_{ }

_{}

_{}

_{}

_{}

The above formulation clearly shows that the flow
to be examined is governed by dimensionless
parameters, namely Prandtl number, Rayleigh
number and Taylor number and rotational Rayleigh
number. The rotational Rayleigh number reflects
the effect of the rotational buoyancy force which
depends on the other non-dimensional parameters.
The rotational buoyancy becomes important when
the rotational speed is high or the enclosure
dimensions are too large, viz is whenΩ L is much
larger than g (Lin *et al*. 1996).

The fluid motion is displayed using the stream function obtained from the velocity components and and the temperature fields using the isotherms. In addition to the time evaluation of velocity and temperature fields, results for the local, spaced averaged and time space averaged Nusselt number on the heated or cooled wall are important in thermal design and can be evaluated from

0 5 0.5

0.5 2

0

(13)

1 2

*Y* *.*

*n*

*Nu*
*Y*

*Nu* *NudX*

*Nu* *Nud*

*n*

_{}

_{}

###

###

**3. SOLUTION **

**METHODOLOGY **

line-by-M. Na

line enforc at eac proced equati the pr conver the su variab been e the obtain chosen Code develo convec

**4.**

The an is perf

.

explici indepe as

**4.1 G**

Unifor enclos examin density from indepe consid Consid compu chosen

**Fig. 2**
**size **

As a v the iso Hamad shown

arendra Kumar *e*

application o cing under-rela

ch time step dure, the m ons are used to revious calcul rgence of the m of the absol bles between t evaluated. Whe

convergence ned. The conv

n as in th based on oped indigeno ction problem

**. RESULTS**

nalysis in the p formed in the fo

. , . , a

is not disc itly. This is d endent paramet well as on

**Grid Testing **

rm grid distrib sure. The eff

ned in order t y. Appropriate

endence. After derable change dering the a utational time n for the presen

**2. Grid indepen**

**for** .

validation, the
otherms compa
dy *et al. *(1994
n in Fig. 3.

*et al.* /**JAFM**_{, V}

of the TDM axation to ensu

considered. I motion pressu o derive the new

ated velocity. sequential ite lute differences two successive en this summat criterion, co vergence crite his study. A de Finite Volum ously to solv in enclosure w

**S AND DISC**

present numeric ollowing dimen

. ,

and .

cussed except due to the

ter but depend
*Ta*. Other pa

**and Code Va**

bution is used fect of grid to select the a grid densities are consider r grid in the average N accuracy requ

involved, a nt study as exhib

**ndency study **

, .

.

present results are well with t 4) and with. Jin

Vol. 9, No. 3, p

MA algorithm ure convergenc In the iteratio ure correctio w velocity from

To check th erative solution s of the solutio e iterations ha tion falls below onvergence i erion has bee edicated Fortra me Method i ve the natura with rotation.

**CUSSION **

cal investigatio nsionless groups

.

.

t to specify not being a ding on an arameter isΩ

**alidation **

d for the whol resolution wa appropriate gri among a rang red to study gri ds, there is n Nusselt number uired and th grid i bited in Fig. 2.

*Nu***versus grid**

of the code fo
that obtained b
n* et al.*(2005) a

pp. 1265-1272,

m, ce on on m he n, on as w is en an is al

on s:

it an nd

le as id ge id no r. he is

**d **

or by as

**Fig. 3.**
**pres**
**(left)**
**(b)**

Fig. 4. isotherm the expe the two with present experim

**Fig. 4.**
**prese**

** (b)**

Fig. 5 a dimensio length o different compare figures s with th (1994).

**Fig. 5. **
**temp**

2016.

**. Comprasion **
**sent work (mid**

**) and with Jin **
* °*,

(a) and (b) ms of the presen

erimental work angular position

*°* and
case very clos
mental work.

**. Comprasion **
**ent work with**

**) ** * °*,

and 6 show the onless temper of the enclosure

t positions edwith that of H

show good agr e experimenta

**Comprasion o**
**erature along **

. , .

**of computed I**
**ddle) with Ham**

**et al. (right) (a**
. ,

shows the h
nt work matche
of Hamady et
n of the enclosu
*°* . The iso
sely match wit

**of computed I**
**Hamady et al.**

. ,

e variation of t rature along e of the present

of the
Hamady *et al.*
reement of the
al work of H

**of computed d**
**dimensiomless**

**sotherms of **
**mady et al. **

**a)** **°**** _{ , }**
.
.

half sectioned ed with that of

al. (1994) for ure considered otherms of the th that of the

**Isotherms of **

**.(a)** **°**

. .

the computed
dimensionless
t work at two
enclosureand
(1994). These
present work
amady *et al*.

**imensionless **
**s length for **

**Fig .6. Com**
**tempera**
**for**

Close agreem results of Ham governing eq the present sim

**4.2 Effect **

**Temperatu**

Fig. 7 (a) to F Fig. 7 (b) to F different rota figures repres where the hot left vertical p positionwhere relatively the column for wall occupy t the last colu isothermal wa position of th

For a fixed

Ω ,

dominating fo Coriolis force the Taylor nu magnitude of a function o buoyancy is 7(a) shows m function. The near isotherm the core as

, the s formation of magnitude c isotherms in F stratification convective m Coriolis force their magnitu thermal buoy disruption tak thermal and decrease of h Taylor numb order. Further

Ω

change in the rolls as evid

**mprasion of com**
**ature along dim**

. ,

ment is obtaine
mady *et al*. (19
quations and th

mulation result

**of Rotati**

**re Fields**

Fig 10(a) repre Fig. 10 (b) show

tional speed. T sents the po t isothermal wa position, the se e the hot iso e bottom horizo position w the relatively ri umn for

all occupy relat he enclosure.

. , the thermal orce compared es due to the fa umber is smalle f Rayleigh numb

of Taylor num a function of multi cellular e isotherms sho mal walls with

s evident fro stream function f multiple ro ompared to th Fig. 8 (b) depic in the core motion in the co

es are stronger ude is quite s yancy. Stable s kes place by o d Coriolis for

heat transfer, a ber and Raylei r increase in ro

andΩ e pattern format dent in Fig. 9

**mputed dimen**
**mensiomless len**

.
**. **

ed when compa 994), who used his gives confi ts.

**ion on Flo**

sents the stream ws isotherm cou The first column osition of the e all occupy relat econd column othermal wall ontal position, where the hot is ght vertical pos positionwhere tively the top h

.

buoyancy to smaller mag act that the mag er when compar ber. As Corioli mber and the

Rayleigh num roll pattern o ow thin bound stable stratific om Fig. 7(b). n [Fig. 8(a)] aga olls of slightl he previous c ct beginning of signifying inc re of the enclos near the walls small compare tratification of opposing intera rces which re as the magnitud igh number is otation correspo

, itresults in tion of stream

(a) and 10 (a

**nsionless **
**ngth **

ared with the same idence to

**ow and **

mline and unters for n in these enclosure, tively the for occupy

the third sothermal sition and e the hot horizontal

and at is the gnitude of gnitude of red to the is force is thermal mber, Fig.

of stream ary layer cation of . AtΩ ain shows ly larger ase. The f unstable crease in

sure. The s but still ed to the f the core

action of esults in de of the

of same onding to n drastic functions a). These

f t r C t t c I o e s d c s

**4**

F

Ω

l c a s s e s

figures show f towards the iso rolls are prima Coriolis forces the streamline the magnitude compared to th Isotherms in Fi of spiral shap enclosure wh stratification of dominant force cold wall whic shaped isotherm

**Fig. 7. (a) Stre**
**contours for **

**4.3 Effect of**

For low rota

Ω , th

local peaks in N circulation. Th and tends to co seen that with suppressed by evident from F speed to Ω

formation of c othermal wall o

arily formed d on the fluid wh from the centr of the Taylor n he magnitude ig. 9 (a) and10 ped structure hich leads t f the core. Cor e from centre of ch leads to the ms.

**eamline contou**
. ,

**f Rotations to**

ation cases w hermal buoyan Nu and the flow he clockwise ci over the other h low rotation,

the thermal Fig. 11. With th

and

counter rotatin f the enclosure due to the act hich offsets the re of the enclos number is large of Rayleigh n (b) shows gen in the core to strong u riolis forces ar f hot wall towa e formation of

**urs and (b) Iso**
.
.

**o Heat Trans**

with Ω ncy force cau w consists of clo irculation is en

circulation. It the swing of buoyancy wh he increase in r

Ω ,t

ng rolls . These tion of e eye of

sure, as er when number. neration of the unstable

re more ards the f spiral

**otherm **

**sfer **

and uses no ockwise

M. Na

buoyan Nu an counte betwee rise an therma Corres three i points and al minim therma corresp the do summa

**Fig .8**
**cont**

**4.4 E**

**Avera**

The t compu Fig. 1 time a progre numbe is dom

arendra Kumar *e*

ncy force caus nd the flow con er clockwise ci en them. The p nd fall of Nu is al buoyancy sponding to th inflexion points

are shown in T lmost cyclic h ma correspond

al buoyancy is pond to the du minating force. arized in Table

**8. (a) Streamlin**
**ours for**

**Effects of Rot**

**aged Heat Tr**

transient behav uted as a funct

2. Initially, th and varies per esses this behav er attains steadi minated by the

, there is disr

*et al.* /**JAFM**_{, V}

ses two or mor nsists of clockw irculation and t possible source due to alternat y and Co e three local p s and the angle Table 1.Fig. 11 heat transfer at

to the time s dominant wh uration when C . The effects du 1.

**ne contours an**
. ,

**tations to Tim**

**ransfer **

vior of the N tion of rotation he heat transfer riodically with vior disappears iness. At Ω

e thermal buo ruption of he

Vol. 9, No. 3, p

e local peaks i wise circulation transition stage

of almost cycli te domination o oriolis forces peaks, there ar e at the inflexio shows unstead the walls. Th interval whe hile the maxim Coriolis force i ue to rotation ar

**nd (b) Isotherm**
.

**me and Spac**

Nusselt numbe n is exhibited i r is sensitive t time. As tim and the Nusse

, the flow oyancy. At Ω

eat transfer b

pp. 1265-1272,

in n, es ic of s. re on dy he en ma is re

**m **

**ce **

er in to me lt w

by

opposing which

by Cori boundar and henc heat tran range an ranges a

**Fig. 9. (**
**contou**

**5.**

The foll present s

1. For dom sup tran 2. For tran rota of h 3. For loca hea

2016.

g interaction o results in de and , th iolis force and ry layers but by ce there is rise nsfer performan nd increases it as shown in Fig

**(a) Streamline**

**urs for** .

**CONCLUS**

lowing conclus studies

rΩ an

minated by th presses the nsfer is by therm

rΩ ,

nsfer by opposin ational buoyanc heat transfer.

r Ω a

ation of the loc at transfer are s

f thermal and
ecrease of *N*

he heat transfe d occurs not y the spiral sh

of *Nu*. Thus ro
nce in the low
in the relative
. 13.

** contours and **

. , .

.

**SION **

sions can be dr

nd Ω

he thermal b oscillation in mal buoyancy e there is disrup ng interaction o cy which resul

and Ω cal maximum sensitive to rota

Coriolis force

*Nu*. For Ω

er is governed through the aped structure otation reduces rotation speed ely high speed

**(b) Isotherm **

rawn from the

,the flow is
buoyancy and
n*Nu*and heat

effect.

ption of heat of thermal and lts in decrease

4. The loca to move rotated a 5. At high

Ω

dominati forces an of Nuss correspo buoyancy correspo is the dom 6. At high

Ω

Nusselt dominati buoyancy

**Fig. 10. (a) S**
**contours fo**

**Tab**

al maximum and to higher angle at higher rotatio h rotational sp

and Ω ion of Coriolis nd results in al elt number w nd to the time y is dominan nd to the durat minating force. h rotational sp ), there is number which ion of Coriolis

y effect.

**Streamline cont**

**r ** . ,

**ble 1 Effect of r**

d minimum poi es when the enc onal speed.

peed correspon

,there is s and thermal b lmost periodic with time. The e interval when nt while the tion when Corio

. peeds (Ω

significant in h is primarily s forces and r

**tours and (b) I**
.
.

.

**rotation ( **

ints tends closure is

nding to alternate buoyancy variation e minima

n thermal maxima olis force

and crease in y due to

rotational

**Isotherm **

B

B

. ,

**Fig. 11. Va**
**angular p**

**Fig. 12. Var**
**number **

**Fig. 13. Var**
**number**

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