effects of rotation on natural convection cooling by l.f. jin a, k.w. tou a, c.p. tso b, (1)

8/2/2019 Effects of Rotation on Natural Convection Cooling by L.F. Jin a, K.W. Tou a, C.P. Tso b, (1)

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Effects of rotation on natural convection coolingfrom three rows of heat sources in a rectangular cavity

L.F. Jin a, K.W. Tou a, C.P. Tso b,*

a School of Mechanical and Production Engineering, Nanyang Technological University, 50 Nanyang Avenue,

Singapore 639798, Singaporeb Faculty of Engineering and Technology, Multimedia University, Jalan Ayer Keroh Lama, 75450 Melaka, Malaysia

Received 24 August 2004; received in revised form 15 April 2005

Abstract

Two-dimensional unsteady numerical studies are made on an air-filled enclosure rotated about its horizontal axis withan array of three rows of heat sources on one of the walls, revealing three physically realizable phenomena, namely, uni-periodic oscillation, multi-periodic oscillation and chaotic oscillation. The evolutionary process of flow field and naturalconvection characteristics from stationary to rotating situation is studied. Rotation results in an imbalance betweenclockwise and counter-clockwise circulations, increases heat transfer in the worst scenario, reduces the oscillation of Nus-selt number, and improves or reduces mean performance in each cycle. The optimal distribution of heaters in rotatingfluid is close to the results in the stationary situation if they have same dominated circulation direction.

2005 Elsevier Ltd. All rights reserved.

Keywords: Rotating cavity; Discrete heat sources; Natural convection; Electronics cooling

1. Introduction

Electronics cooling under natural convection hasbeen studied by many [16], the common model beingthe discrete heat sources in a rectangular enclosure. Withadvanced technology, thermal management in the un-

steady or rotating condition may be encountered in suchsituations as rotary machines, guided missiles and space-based manufacturing processes. Bobco [7] introducedthe application of natural convection in the design of avented Gallileo mission descent module parachutinginto the Jupiter atmosphere. It was reported [8] that

the failure caused by temperature and vibrationamounts to 75% of all failures. There had been manystudies on flow in rotating ducts, for instance in relationto cooling of turbine blades [9]. However, there are lim-ited studies in unsteady natural convection cooling fromdiscrete heat sources in rotation. The situation with nat-

ural convection is very different from forced convectionbecause the latter is pressure-driven flow where the cen-trifugal force plays an important role and the thermalbuoyancy effect is neglected. In a sense, there is no abso-lute stationary state on earth, since Taylor number, nor-mally used to reflect the effect of Coriolis force in therotating fluid, is about unity at the equatorial sea level[10]. This is the motivation of the present investigationon the effects of rotation on the periodical behavior ofnatural convection cooling from three rows of heat

0017-9310/$ - see front matter 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2005.04.013

* Corresponding author. Tel.: +606 252 3075; fax: +606 2316552.

E-mail address: [email protected] (C.P. Tso).

International Journal of Heat and Mass Transfer 48 (2005) 39823994

www.elsevier.com/locate/ijhmt
mailto:[email protected]:[email protected]


2/13

sources in a rectangular cavity, extending our earlierwork on the stationary cavity [6,11,12].

Although a detailed 3-D numerical computation forthis discrete heat sources condition is formidable, espe-cially with the influence of the buoyancy force, Coriolisforce and centrifugal force all within the same framewhen the cavity is rotated, it is still challenging and use-ful to study a 2-D model. For the present geometry itwill be shown that the results are close to 3-D results.Comparison will be made with reported work on uni-form wall temperature condition [13] and physical expla-nations of results given wherever possible. An attemptwill also be made to obtain an optimal heater spacing

design criterion.

2. Numerical methodology and validation

2.1. Governing equations and boundary conditions

The geometry and boundary conditions are shown inFig. 1. The enclosure, filled with air, executes a steadyuniform counterclockwise angular velocity about a hor-izontal axis. (The rotation axis is vertical to the XOYplane.) Three discrete rows of heat source are flushed-

mounted onto one of the side walls, while the opposite

wall acts as a uniform temperature cold surface. Theremaining walls are insulated from the surroundings.This model can be tracked to studies in stationary cases[26,11,12,14].

The non-inertial reference frame XOY is shown inFig. 1. Boussinesq approximation is widely used in nat-ural convection [36,913,1518] and the justificationfor rotating fluids can be found in [15,16]. Then theterms representing the thermal and rotational buoyan-cies and Coriolis force are, respectively, equal toq0 g

*bT Tc, q0bT TcX

*

X*

r*

and 2q01bT TcX

*

V*

. Using L, a/L, L2/a, a2q0/L2 and q00L/k

as scales for length, velocity, time, pressure and temper-

ature, respectively, the dimensionless variables are de-fined as

X x

L; Y

y

L; U

uL

a; V

vL

a;

s ta

L2; P

pmL2

a2q0; h

T Tcq00L=k

;

Rax bX2L5q00

kam; Ra

gbL4q00

kam;

Ta 4X2L4

m2; Pr

m

a;

AW W

L

; AH H

L

;

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

1

Nomenclature

AH, AW aspect ratio along x- and y-directiond, d0 distance between side heater and center

heater, m

g* gravitational acceleration vector, m2/sg magnitude of gravitational acceleration,

m2/sH the side of the enclosure, mk thermal conductivity, W/mKL heater length, mL1, L2 enclosure heater location, mn rotation number for one periodic cycleNu Local Nusselt numberNu; Nu space averaged and time and space averaged

Nusselt numberNui Nusselt number for heater of i row

Numax, Numin maximum and minimum Nusseltnumberp, pm pressure and motion pressure, PaP dimensionless motion pressurePr Prandtl numberq00 heat flux, W/m2

Ra Rayleigh numberRa* modified Rayleigh numberRaw rotation Rayleigh numbert time, s

T temperature, KTa Taylor numberTc temperature of the cold wall, K

u, v dimensional velocity components, m/sU, V dimensionless velocity componentsV!

velocity vector, m/sW enclosure width, mx, y dimensional Cartesian coordinate system, mX, Y dimensionless Cartesian coordinate system

Greek symbols

a thermal diffusivity, m2/sb thermal expansion coefficient, 1/Kh dimensionless temperaturem kinematic viscosity, m2/s

q0 density at the mean temperature, kg/m3

/ angular position of cavity, /1, /2, /3 angular position of the first, second, and

third inflexion, s dimensionless times0 dimensionless time for one rotationDs dimensionless time step (variable)X*

angular rotation rate vector, rad/sX magnitude of angular rotation rate, rad/s

L.F. Jin et al. / International Journal of Heat and Mass Transfer 48 (2005) 39823994 3983


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where the motion pressure is defined as

Pm p1

2q0X

2x2 1

2q0X

2y2 q0gx sinXt

q0gycosXt. 2

As mentioned by Vanyo [17], in a rotating Euleriancoordinate system, the centripetal term can be includedin the pressure term and disappears from the typicalrotating fluid computation. With Boussinesq approxi-mation, the main part of the centrifugal force term iscombined with the pressure term and others are causedby density change and centrifugal buoyancy. In mostcases, if the change of density is small and the rotationspeed is low (x2r/g) ( 1, the effect of this term can beneglected completely [13], and this is verified in thepresent studies.

With a selected coolant, Prandtl number (Pr) is as-

sumed fixed and there are two physical variables, heatflux and rotation speed, which are reflected by the mod-ified Rayleigh number (Ra*) and Taylor number (Ta),respectively. Due to the rotation Rayleigh number(Raw) not being an independent parameter but depen-dent on Ra, Pr, and Ta, we will not discuss Raw exceptin specifying it explicitly. The Coriolis buoyancy force isneglected, due to jb(T Tc)j ( 1 in present studies.Then the dimensionless governing equations and bound-ary conditions are as the following:

Continuity equation:

oU

oX

oV

oY

0. 3

Momentum equations:

oU

os U

oU

oX V

oU

oY

oP

oX Pr

o2U

oX2o

2U

oY2 Ta0.5PrV|fflfflfflfflffl{zfflfflfflfflffl}Coriolis force term

RaxPrXh|fflfflfflfflffl{zfflfflfflfflffl}Centrifugal force term

RaPrh sin0.5Ta0.5Prs|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Buoyancy force term

; 4

oV

os U

oV

oX V

oV

oY

oP

oY Pr

o2V

oX2o

2V

oY2

Ta0.5PrU|fflfflfflfflffl{zfflfflfflfflffl}

Coriolis force term

RaxPrYh|fflfflfflfflffl{zfflfflfflfflffl}Centrifugal force term

RaPrh cos0.5Ta0.5Prs|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Buoyancy force term

. 5

Energy equation:

oh

os Uoh

oX Voh

oY o

2h

oX2 o

2h

oY2 . 6

Boundary conditions:

At X AH=2 : U V 0;oh

oX 0;

At X AH=2 : U V 0;oh

oX 0;

At Y AW=2 : U V 0;oh

oY 1 for the iso-flux regions heaters;

oh

oY 0 for the insulated regions;

At Y AW=2; U V 0; h 0.

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

7

Fig. 1. Physical and geometrical arrangement.

3984 L.F. Jin et al. / International Journal of Heat and Mass Transfer 48 (2005) 39823994


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To examine the time evolution of the velocity andtemperature fields, results for local, space-averaged

and time-space-averaged Nusselt number Nu;Nu;Nucan be evaluated from

NuYj/u 1

hY/u q00L

K TY Tc/u;Nui

/u

q00LR

LK TY Tc dX

/u

; i 1; 2; 3;

Nu/u

X3i1

q00LRLK TY Tc dX

/u

" #,3;

Nui

R2np0

Nui d/

2npaveraged over one cycle;

Nu X3i1

Nui

!=3.

9>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>; 8

2.2. Solution method

For this unsteady problem, first-order implicitscheme is used for the unsteady term and the othernumerical procedures of finite volume method are thesame as reported previously [6,12]. The steady-state result for each step is obtained by two conver-gence criteria: (1) SMAX6 1 1010, and (2)MAXhn1i;j h

n

i;j 6 1 105. SIMPLE arithmetic is

adopted to solve the coupled velocity and pressure. Inthe iteration procedure, the motion pressure correctionequations are used to derive the new velocity from theprevious calculated velocity. SMAX is the largest abso-lute value of the mass source used in the motion pres-sure correction equations and SMAX 6 1010 is used asthe velocity convergence criterion. The numerical oscil-lations required to reach the final state varies from 10to 200 rotations. The Ds, s0 are the dimensionless timestep in numerical simulations and dimensionless timerequired for one rotation circulation, respectively. Inpresent studies, Ds equals 1/72s0.

Some simulations may require up to one week of

computational time with a Pentium IV PC. For example,if we use 20 90 mesh grids, it may take only one minuteto get one convergence result for the steady problem.Uniform grid is used in the x-direction, while a non-uni-form grid distribution with the characteristic of finemesh at the two sides and coarse mesh in the middlezone is used in the y-direction of the computation do-main, in order to capture the surface heat transfer with-out bringing about too many grid points. The meshlengths are increased gradually from the finer mesh, inthe ratio 1:1.1, to match smoothly with the coarse mesh.However, for the present unsteady problem, 72 time

steps are used in each rotation circulation and 175

circulations are necessary to ensure that the final peri-odic condition has been reached, as shown in Fig. 4(c).Then the total computation time is estimated to be 1 72 175 = 12600 min = 8.75 days. If 360 time stepsare used in each rotation circulation that is necessaryin high Taylor number, it will take more than one month

of computation time. Therefore, only 2-D numericalsimulations are adopted and the work is confined tostudying the periodic field in natural convection.

2.3. Numerical validations

In view of the complex flow to be simulated, stringenttests are conducted. First, computations are carried outfor cases at various stationary (X = 0) orientationsshown in Fig. 11 in [12]. Without edge effects, the resultsof 2-D model are close to those of 3-D model. Therefore,2-D model can be used without losing generality in the

stationary case. Next, tests are performed for a rotatingcavity of square cross-section with constant temperatureat the hot wall and cold wall. Compared with forcedconvection, velocity gradient of natural convection islow since normally the maximum velocity magnitude isabout 102101 m/s and the flow not of the boundarylayer type. However, the temperature gradient is relativehigh and the periodic temperature fields are easier to bemeasured successfully than flow fields. Therefore, vali-dation of the program and most discussion are basedon temperature results. As shown in Fig. 2, at differentorientations and rotation speeds, close agreement is ob-tained when compared with results of Hamady et al. [13]who used the same governing equation, and this givesconfidence to our simulation results.

We have also carried out experimental studies usingthe setup in [12] and changed the program code to con-sider those experimental conditions, and they will bereported separately. Here, a brief comparison with theexperimental results is presented in Fig. 3, where theheater aspect ratio equals 6. The comparison shows sim-ilar trends and the maximum deviation between them isless than 15%. These show that 2-D model can be validin applications when the heater aspect ratio is high andthe effect of rotation is relatively low. Finally, some grid-

independence tests are conducted. The differences inNui; Nu predicted from the original grid (20 90) andfiner grid (40 120) are within 1% during the entire peri-odic oscillation, with differences in the last temperaturefield kept within 2%. Different time steps Ds up to (1/360)s0 are used to ensure convergence when Ta is higherthan 103. Through these tests, the present numericalmethod is considered suitable for the present problem.

3. Results and discussion

The study fields are shown in Table 1.



5/13

3.1. Oscillation phenomena

Under a non-inertial frame, the driving buoyancy

force rotates in the enclosure and results in an oscillation

phenomenon. When the time for one cycle equals thetime for finishing one rotation, it is uni-rotation periodicoscillation (n = 1), otherwise it is multi-rotation oscilla-tion (1 < n < 1).

Typical result for uni-rotation periodic oscillationproblem is shown in Fig. 4(a), where after about fiverotations (s = 5s0), the final periodic oscillation patternis obtained. To ensure correct results, additional compu-tation was carried out beyond five rotations. Fig. 4(b)shows the Nu evolution for multi-rotating periodic oscil-lation. With the same Ra* and increased Ta, chaoticoscillation occurs as shown in Fig. 4(c) where even after175 rotations, there is still no sign of periodic oscillation.At the special values of flow parameters, the flow mayhave different features, i.e., periodic oscillating or cha-otic oscillating flows, similar to observations reportedin a numerical study of combined free and forcedconvection in a rotating curved duct [18].

3.2. Flow and temperature fields with rotation

Due to the close relationship between stationary andlow rotation case, the discussion will begin with the sta-tionary case first. In the vertical orientation, / = 90(270), the enclosure is filled with strong uni-cellularclockwise-circulation (counter-clockwise-circulation).In the orientations that / is close to 0 or 180, thereare different multi-cellular flow patterns shown in Table2 of [12], with both clockwise and counter-clockwisevortices present, and this can be treated as the transitionstage between clockwise-circulation and counter-clock-

wise-circulations. In particular, in the orientation that

Fig. 2. Comparison of computed isotherms with experiment results of [13] shown in the upper figures (Ra = 3 105, Pr = 0.7):(a) / = 0, X = 17.5 rpm; (b) / = 270, X = 15.5 rpm; (c) / = 180, X = 17.5 rpm.

101

102

103

1

1.5

2

2.5

3

3.5

4

4.5

5

4.39X103

9.56x103

2.22x104

3.79x104

5.12x104

6.30x104

Ta

Nu

3.13x103

1.57x104

4.70x104

7.83x104

Experimental results

Numerical resultsRa*

Ra*

Fig. 3. Brief comparison between numerical and experimentalresults.

Table 1Summary of model parameters

Ra* Ta Pr AW AH

103107 0104 0.7 1, 2 7.5

Note: Heater length, L = 12.7 mm.



6/13

/ equals 0 or 180, the strengths of the two circulations

equal each other and the flow and temperature fields are

symmetric. In the whole cycle (/ = 0360), the flow andtemperature fields are in reversed symmetry with respectto 180. From scale analysis [12], the clockwise or coun-ter-clockwise circulation is dominated by buoyancyforce and friction force.

Flow and temperature fields at the onset of rotation

are shown in Figs. 5 and 6. There are two uneven vorticesin Fig. 5(a) as a result of transition from counter-clockwise to clockwise circulation, with clockwisemotion appearing stronger. And in the temperature field,the convective heat transfer resulted from clockwisemotion can be seen clearly. Therefore, a higher Nu isexpected compared to the stationary case where thermalconduction dominates. In Fig. 5(b) (/ = 45), eventhough the clockwise motion dominates the flow, thetemperature field shows hierarchical structure and resultsin an expected local minimum Nu, shown in Fig. 7 for thecircle line ofTa = 103. However, it is different from con-

duction domination where Nu is close to unity. FromFig. 5(c) to Fig. 6(a), clock-wise motion dominates theflow, but in Fig. 6(b) (/ = 225), the flow enters the tran-sition stage again. In the core region, counter-clockwisecirculation appears and becomes increasingly stronger.The clockwise circulation falls back to the corner regionsand becomes weakened until obliterated finally. In thisorientation or near it, hierarchical structure re-appearsand brings about the second local minimum Nusseltnumber. In the transition stage, there are more vorticesin the cavity that cause difficulties in numerical simula-tions. After the transition stage, the counter-clockwisemotion dominates the flow as shown in Fig. 6(d), but itis not as strong as the earlier dominant clockwise motion.The flow and temperature field for the phase angle of360 is the same as that for 0, thus completing a peri-odic process. The rotation-induced symmetry-breakingphenomenon is mainly caused by the inertial force. Withthe increase in rotation and hence inertial force, thecounter-clockwise circulation will disappear at last, asdiscussed in the next section.

3.3. Effects of rotation to heat transfer

For the stationary case, as shown by the mean Nus-

selt number results with zero Ta in Figs. 7 and 8(a), threephenomena can be seen: (1) There are three local peaksin one period. The two side-peak values are respondingto the strongest points of clockwise and counter-clock-wise motions while the middle value is the result of Ray-leighBenard convection. (2) Corresponding to the threelocal peaks, there are three inflexion points and the an-gles at the inflexion points are shown in Table 2 as /1,/2, /3. (3) The minimum Nu of 1.76 is closer to unitydue to conduction heat transfer. The effects due to rota-tion are summarized in Table 2.

For the stationary and low rotation cases, thermal

buoyancy force causes two or more local peaks in Nu,

Number of rotations

Nu

0 1 2 3 4 5 6 7 8 9 102

3

4

5

6

7

8

Ra*=1x105, Ta=1x10

3, Pr=0.7,AW=1

Number of rotations

Number of rotations

Nu

0 6 12 18 24 30 362.5

3

3.5

4

4.5

5

5.5

Nu

Ra*=1x10

5

, Ta=1x10

4

, Pr=0.7, AW=2

2.5

3

3.5

4

36343230

0 25 50 75 100 125 150 1752.5

3

3.5

4

4.5

5

5.5

Ra*=1x105, Ta=1x10

5, Pr=0.7, AW=2

a

b

c

Fig. 4. Three physically realizable oscillations: (a) uni-rotationperiodic oscillation; (b) multi-rotation periodic oscillation; (c)chaotic oscillation.



7/13

and the flow consists of clockwise circulation, counter-

clockwise circulation and two transition stages between

them. With the increase in rotation speed, the clock-wise

circulation is enlarged and tends to cover the other

Table 2Effects of rotation (Ra* = 1 105, Pr = 0.7)

Ta No. of peak Nu Numax Numin DNu (NumaxNumin) /1 /2 /3

0 3 4.520 1.760 2.76 $0 $160 $200103 2 4.139 2.660 1.479 $30 $225104 2 3.586 2.549 1.037 $85 $305105 1 3.577 3.065 0.512 $135

Fig. 5. First part of flow and temperature fields with rotation (Ra* = 1 105, Ta = 1 103, Pr = 0.7, AW= 1).



8/13

circulations. As a result, except for the first local peakNu, the others become weakened and disappeared whenthe rotation speed is high enough. In Fig. 7, it can beseen that with increasing rotation and Ta, the swing ofNu is reduced and the minimum Nu is improved (in-creased) gradually. The continuous line (Ta = 105) inFig. 7 shows that the flow is dominated by clockwise cir-culation. Correspondingly, there is only one peak andvalley for Nu. This shows that the inertial force playsan important role in rotating fluids, similar to earlier

observations in [13].

The Fig. 8 shows that the heat transfer behavior ofrow 1 and row 3 are not in reversed symmetry and theybecome asymmetrical gradually with the increase ofrotation speed and Taylor number, resulting from theevolutionary process in the flow and temperature fields.

In Fig. 9, the multi-rotation periodic oscillation isshown. It should be pointed out this has the same Ra*

and Ta as the indented line in Fig. 7, and the only differ-ence is the value of the aspect ratio, AW. With increasingrotation speed, chaotic oscillation, as shown in Fig. 4(c)

will take place.

Fig. 6. Second part of flow and temperature fields with rotation (Ra* = 1 105, Ta = 1 103, Pr = 0.7, AW= 1).



9/13

3.4. Effects of heat flux to heat transfer

The effects of heat flux to heat transfer can be ob-tained with a fixed Ta and different Ra*, as shown inFigs. 10 and 11. When Ra* is as low as 103, conductionheat transfer with characteristics of low and flat Nu isseen. When Ra* is increased to 104, the fluctuation ofNu indicates the onset of convection and there is onlyone peak Nu. Further increase in Ra* shows more oscil-lations in Nu and two or three local peaks Nu appear inFig. 10. These evolution phenomena seem like a reverse

process as discussed earlier on Fig. 7 and Table 3, show-ing that the process is affected by the combined effect ofrotation and heat flux.

3.5. Effects of rotation to time-and-space averaged heat

transfer

Fig. 12 shows that rotation delays the onset of con-vection, agreeing with the findings of Fernando andSmith [19]. This is especially clear when Ta is as highas 104. As a result of continuity, rotation reduces heattransfer performance in the low rotation speed range

and increases it in the relatively high speed ranges. Thesummary of correlations ofNu as a function of Ra*, atdifferent Ta, is shown in Table 3.

3.6. Optimal distribution of discrete heat sources

A design process requires the evaluation of a numberof available options to determine the most appropriatecase that meets the primary design requirement, e.g.,maximum mean heat transfer, or minimum peak tem-perature in the system. For natural convection, con-strains come not only from computational challenges,

but also design and manufacture requirements. An at-

tempt is now given to determine the optimal distribution

of the discrete heat sources using the present model.

Nu

0 45 90 135 180 225 270 315 3601

1.5

2

2.5

3

3.5

4

4.5

5

()

1

Ta=1x104

Ta=1x105

Ta=1x103

Ta=0

Fig. 7. Effects of Taylor number (Ra* = 1 105, Pr = 0.7,AW= 1).

Nu

0 45 90 135 180 225 270 315 361

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Nu

Nu

Nu

()

3

2

1

Nu

0 45 90 135 180 225 270 315 3601

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Nu

Nu

Nu

Nu

()

()

3

2

1

0 45 90 135 180 225 270 315 3601

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Nu

Nu

Nu

3

2

1

a

b

c

Fig. 8. Mean Nusselt number for each row of heater at differentTaylor number (Ra* = 1 105, Pr = 0.7, AW= 1): (a) Ta = 0,(b) Ta = 1 103, (c) Ta = 1 104.



10/13

For the facility of discussion, two new dimensionvariables, d and d0, related to geometrical configuration

are shown respectively in Figs. 13 and 15. Firstly, calcu-lations are carried out when the middle heater is fixed atthe center line of the wall and the locations of the twoside heaters are changed symmetrically. From Fig. 13,even though minimization is the developmental trendof electronics packaging, greater distance between theside heater and middle heater will result in better heattransfer performance, except when the side heaters aretoo close to the corner where stagnant fluid is deleteriousto heat transfer. Fig. 14 shows that the optimal locationwill change with Ra*, as well as Ta. Computations arealso carried out with fixed side heaters and free shifting

of the center heater, as shown in Fig. 15. The optimal

Nu

0 90 180 270 360 450 540 630 7201

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Nu

Nu

Nu

3

2

1

()

Fig. 9. Mean Nusselt number for each row of heater for multi-rotation oscillation (Ra* = 1 105, Ta = 1 104, Pr = 0.7,AW= 2).

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Nu

0 45 90 135 180 225 270 315 3601

2

3

4

5

6

7

8

Ra*=1x103

Ra*=1x104

Ra*=5x104

Ra*=1x105

Ra*=1x106

+

()

+ + + +

Fig. 10. Effects of modified Rayleigh number (Ta = 1 104,Pr = 0.7, AW= 1).

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Nu

0 45 90 135 180 225 270 315 3601

2

3

4

5

6

7

8

Ra*=1x103

Ra*=1x104

Ra*=5x104

Ra*=1x105

Ra*=1x106

+

1

()

+ ++ +

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Nu

0 45 90 135 180 225 270 315 3601

2

3

4

5

6

7

8

Ra*=1x103

Ra*=1x104

Ra*=5x104

Ra*=1x105

Ra*=1x106

+

()

()

2

Nu3

+ + + +

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

0 45 90 135 180 225 270 315 3601

2

3

4

5

6

7

8

9

Ra*=1x103

Ra*=1x104

Ra*=5x104

Ra*=1x105

Ra*=1x106

++ + + +

a

b

c

Fig. 11. Mean Nusselt number for each row of heater atdifferent modified Rayleigh numbers (Ra* = 1 104, Pr = 0.7,AW= 1): (a) row 1, (b) row 2, (c) row 3.



11/13

distribution is not uniform and the heaters are pushedtoward the starting cold current of the dominatedclock-wise circulation, close to the description by Silvaet al. [14] in the stationary situation where the enclosureis dominated by clock-wise circulation. With rotation,the distribution is affected by balancing of clock-wise

and counter-clock-wise circulations. The optimal distri-

Table 3Summary of correlations ofNu at different Ta

AW= 1Ta = 0 Nu 0.3691Ra0.2028 103 6 Ra*6 107

Ta = 102 Nu 0.3367Ra0.2083 103 6 Ra*6 107

Ta = 103 Nu 0.2996Ra0.2139 103 6 Ra*6 107

Ta = 104 Nu 0.1649Ra0.2505 104 6 Ra*6 107AW= 2

Ta = 0 Nu 0.3983Ra0.1930 102 6 Ra*6 106

Ta = 102 Nu 0.3184Ra0.2098 102 6 Ra*6 107

Ta = 103 Nu 0.2281Ra0.2341 103 6 Ra*6 107

Ta = 104 Nu 0.1694Ra0.2506 103 6 Ra*6 107

+ +

+

+

+

+

+

Ra*

Nu

102

103

104

105

106

107

1

2

3

4

5

6

7

89

101112

Ta=0

Ta=1x102

Ta=1x103

Ta=1x104

+

AW

=1

+

+

+

+

+

+

+

Ra*

Nu

102

103

104

105

106

107

1

2

3

4

5

6

789

10

Ta=0

Ta=1x102

Ta=1x103

Ta=1x104

+

A = 2W

a

b

Fig. 12. Effects ofRa* and Ta on the overall Nusselt number intwo aspect ratios.

X

X XX

X

X

XX X X

X

X+

+

+

+

+

+

+ + + +

+

+

Nu

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4Nu

Nu

Nu

Nu

X

+

1

2

3

d/L

Nu( )max

( )Numaxi

dL Heater 3

Fig. 13. Results for varying the location of side heaters

(Ra* = 1 105, Ta = 1 104, Pr = 0.7, AW= 1).

+

+

+

+

X

X

X

X

Ra*

Nu

103

104

105

106

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

d/L = 0.25

d/L = 1.9167

d/L = 2.0833

d/L = 2.25

+

X

Nu

Ta=1x104, Pr=0.7, AW=1

++

+

Ta10

-110

010

110

210

310

42

2.5

3

3.5

4

4.5

5

d/L = 2.25

d/L = 2.0833

d/L = 0.25

+

Ra*=1x105, Pr=0.7, Aw=1

a

b

Fig. 14. Effects of Ra*

(a) and Ta (b) to optimal distribution.



12/13

bution of heaters in rotating fluid is close to the results inthe stationary situation if they have the same dominantcirculation.

4. Conclusions

The following conclusions can be drawn from thepresent studies:

(1) Three physically realizable phenomena, uni-rota-tion periodic oscillation, multi-rotation periodicoscillation and chaotic oscillation are identifiedfor rotating discrete heat sources in a cavity.

(2) For the stationary and low rotation cases, thermalbuoyancy force causes two or more local peakNusselt numbers and the flow consists of clock-wise circulation, counter-clockwise circulationand two transition stages between them. Withthe increase of rotation speed, the clock-wise cir-culation is enlarged and tends to cover the other

circulations. As a result, except for the first localpeak Nu, the others become weakened and disap-peared when the rotation speed is high enough.This process is determined by the relative effectsof thermal flux and rotation.

(3) Rotation reduces oscillation in Nusselt number,improves heat transfer in the weak stages butmay reduce or increase mean heat transferperformance.

(4) Heat transfer behavior for the two side rows ofheaters is in reversed symmetry in the stationarycase (shown in Fig. 8(a)), but becomes asymmetri-

cal gradually with the increase of rotation speed.

(5) The correlations between Ra*, Ta, and Nu aregiven in Table 3 for the present conditions.

(6) The optimal distribution of heaters in rotatingfluid is close to the results in the stationary situa-tion if they have same dominated clock-wise orcounter-clock-wise circulation.

Future work could include effects of aspect ratio, dif-ferent coolants, experimental investigations consideringcomplex geometrical structure and conjugate heat trans-fer, or three-dimensional simulations.

References

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[2] M.S. Polentini, S. Ramadhyani, F.P. Incropera, Singlephase thermosyphon cooling of an array of discrete heatsources in a rectangular cavity, Int. J. Heat Mass Transfer36 (1993) 39833996.

[3] T.J. Heindel, S. Ramadhyani, F.P. Incropera, Laminarnatural convection in a discretely heated cavity: I. Assess-ment of three-dimensional effects, J. Heat Transfer 117(1995) 902909.

[4] T.J. Heindel, F.P. Incropera, S. Ramadhyani, Laminarnatural convection in a discretely heated cavity: Compari-sons of Experimental and theoretical results, J. HeatTransfer 117 (1995) 910917.

[5] F.P. Incropera, Liquid Cooling of Electronic Devices bySingle-phase Convection, in: Wiley series in thermalmanagement of microelectronic and electronic systems,

1999, pp. 90124.[6] S.K.W. Tou, C.P. Tso, X. Zhang, 3-D numerical analysis

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[7] R.P. Bobco, Free convection in enclosures exposed tocompressive heating, Heat Transfer Therm. Control:Progr. Astronaut. Aeronaut. 78 (1981) 487515.

[8] L.T. Yeh, Review of heat transfer technologies in elec-tronic equipment, J. Electron. Packag. 117 (1995) 333339.

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[10] B.M. Boubnov, G.S. Golitsyn, Convection in RotatingFluids, Kluwer Academic Publishers, 1995.

[11] S.K.W. Tou, X.F. Zhang, Three-dimensional numericalsimulation of natural convection in an inclined liquid-filledenclosure with an array of discrete heaters, Int. J. HeatMass Transfer 46 (2003) 127138.

[12] C.P. Tso, L.F. Jin, S.K.W. Tou, X.F. Zhang, Flow patternevolution in natural convection cooling from an array ofdiscrete heat sources in a rectangular cavity at variousorientations, Int. J. Heat Mass Transfer 47 (2004) 40614073.

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Transfer 116 (1994) 136143.

X

X

X X XX X

XX

++

+

+

+

++

++

Nu

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

Nu

Nu

Nu

Nu

X

+

1

2

3

d'/L

Nu( )max

( )Nu maxi

Heater 1 Heater 3d'L

Cold tide Hot currentCold wall

Fig. 15. Results for varying the location of center heaters(Ra* = 1 105, Ta = 1 104, Pr = 0.7, AW= 1).



13/13

[14] A.K. Silva, S. Lorente, A. Bejan, Optimal distribution ofdiscrete heat sources on a wall with natural convection, Int.J. Heat Mass Transfer 47 (2004) 203214.

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[18] T.L. Yang, L.Q. Wang, Bifurcation and stability ofcombined free and forced convection in rotating curvedducts of square cross-section, Int. J. Heat Mass Transfer46 (2003) 613629.

[19] H.J.S. Fernando, D.C. Smith IV, Vortex structures ingeophysical convection, Eur. J. Mech. B/Fluids 20 (2001)437470.


effects of rotation on natural convection cooling by l.f. jin a, k.w. tou a, c.p. tso b, (1)

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