-7 

Semiannular Enclosure 

P 

NASA Technical Memorandum 10201 1 

1 

Lin-Wen Wang 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 

IChain-Nan Yung 

University of Toledo 

Toledo, Ohio 

An-Ti Chai 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 

and 

Nasser Rashidnia 

Sverdrup Technology, Inc. 

Cleveland, Ohio 

NASA Lewis Research Center Group 

Prepared for the 

1989 National Heat Transfer Conference 

sponsored by the American Society of Mechanical Engineers 

Philadelphia, Pennsylvania, August 6-9, 1989 

(NASA-TM-102011) kUMFRTCAL STUGIFS dF N99-28666 CONVECTIVE HtAT TRANSFER TN AN TWLINEU 

SEMIANNULAR ENCLOSURE (NASA- Lewis 

'# Research Center) 9 p CSCL ?ZA uncl as 

I G3/29 023171d 
Numerical Studies of Convective Heat Transfer 

in an Inclined Semiannular Enclosure 

Lin-Wen Wang* 

NASA Lewis Research Center 

Cleveland, Ohio 44 135 

An-Ti Chai 

NASA Lewis Research Center 

Cleveland, Ohio 44135 

Abstract 

Natural convection heat transfer in a two-dimensional differentially heated semiannular enclosure was studied. The enclosure 

is isothermally heated and cooled at the inner and outer walls, 

respectively. A commercial software based on the SIMPLER 

algorithm was used to simulate the velocity and temperature 

profiles. Various parameters that affect the momentum and heat 

transfer processes were examined. These parameters include 

the Rayleigh number, Prandtl number, radius ratio, and the 

angle of inclination. The study covers a flow regime extending 

from conduction-dominated to convection-dominated flow. The 

computed results of heat transfer are presented as a function 

of flow parameters and geometric factors. It is found that the 

heat-transfer rate attains a minimum when the enclosure is tilted 

about +50° with respect to the gravitational direction. 

Introduction 

Natural convection in a differentially heated cavity is an 

important characteristic in material science experiments, such 

as the growth of single crystals from melts, the solidification 

of castings, and the purification of materials. The study is also 

valuable to applications such as thermal energy storage, building energy use, nuclear reactor safety, cooling of electronic 

components, and analysis of smoke and fire spread in buildings. 

The classical problems of natural convection in an annular 

enclosure with differentially heated surfaces have been the 

*National Research Council-NASA Research Associate. 

?NASA Resident Research Associate. 

Chain-Nan Yungt 

University of Toledo 

Toledo, Ohio 43606 

Nasser Rashidnia 

Sverdrup Technology, Inc. 

NASA Lewis Research Center Group 

Cleveland, Ohio 44 135 

subject of numerous investigations. For example, Kuehn and 

Goldstein (refs. 1 and 2) conducted experimental and numerical 

analyses to determine the heat-transfer rate for Rayleigh 

numbers to lo5. However, little is known regarding the free 

convection in an inclined semiannular enclosure. Recently, 

heat transfer in partially divided enclosures has received 

attention because of its application in the design of energy 

efficient buildings, electronic cooling systems, and materials 

processing. It is the purpose of this research to analyze flow 

and heat-transfer behavior in a semiannular enclosure. 

The basic flow structure in a rectangular enclosure can be 

classified into three types, each governed by a different heattransfer process. The first type of flow is induced by heating 

the enclosure from the top surface, sometimes called the heatedfrom-top flow phenomenon. Since the lighter fluid is always 

in the upper portion of the enclosure, there is no convective 

flow inside the cell. The second flow phenomenon is caused 

by a temperature gradient across the enclosure. The direction 

of heat transfer is perpendicular to the gravitational direction. 

In this situation free convection always exists (ref. 3). In the 

third type, a heated-from-below flow phenomenon, the heat 

is transferred in a direction parallel but opposite to gravity. 

The flow phenomenon in this case is closely related to the 

Bernard cell problems (ref. 4). 

In a semiannular enclosure, all the three types of heat transfer may exist simultaneously and interact with each other. As 

a result, the flow pattern is more complicated than that in a 

rectangular enclosure. 

In the present work, a numerical study of two-dimensional 

free convection in a semiannular enclosure is carried out. The 

computation uses the software, microCOMPACT, for velocity and temperature predictions. Because of the limited capa1 
bility of a personal computer, the grid nodes used are relatively 

coarse, and a highly accurate solution cannot be expected. 

However, it was believed that the study would help the understanding of the transport processes in inclined semiannular 

enclosures. 

Formulations and Numerical Approach 

The geometry of a semiannular enclosure is illustrated in 

figure 1. The radii of the enclosure for the inner and outer 

cylinder are Rin and Rout, respectively. The inner wall is kept 

at a higher temperature Tin and the outer wall, at a lower temperature Tout. The symmetrical planes at 19 = 0 and n are 

insulated. By applying the Boussinesq approximation for free 

convection flow, the governing equations for the steady-state 

velocity and temperature distributions in this geometry can be 

written in a vector form as 

Continuity 

v*v=o 

Momentum 

I ~(v* V)V= - VP+ F -pV x(V x V) (2) 

Energy 

k( V * V)T pcP ( V V) T (3) 

where F, the body force, is given by 

Based on these equations and the consideration of the 

geometry, the dimensionless quantities, Rayleigh number (Ra), 

Prandtl number (Pr), and radius ratio (RR), and the angle of 

inclination d are found to be important in the present study. 

These quantities are defined as 

gPATL3 Ra = ~ 

VCY 

V 

a Pr = - 

where L and ATare the differences of the radius and temperature, respectively, between the outer and inner cylinder, g is 

the gravitational acceleration, 0 is the thermal expansion 

coefficient, and CY is the thermal diffusivity of the fluid. The 

physical properties, p, v, k and cp are, respectively, the fluid 

density, kinematic viscosity, thermal conductivity, and specific 

heat. 

The analyses were conducted over various flow regions and 

geometrical conditions: The Rayleigh number varied from 

lo3 to lo5, the diameter ratio from 1.5 to 2.5, and inclination 

angles from -90" to +90". The results of velocity and 

temperature profiles were examined. The heat-transfer rate, 

expressed as the Nusselt number for the inner cylinder, is 

presented as a function of the Rayleigh number, the radius 

ratio, and the tilt angle. 

The computation uses a control volume method and a staggered grid to discretize the equations and employs the SIMPLER 

procedure for the predictions and corrections of the velocity 

field. A description of the SIMPLER algorithm can be found 

in the book (ref. 5) by Patankar, who also developed, based on 

this method, a computer code for two-dimensional elliptic flow 

problems. This computer software, called microCOMPACT 

(ref. 6), can be run on a personal computer. The present work 

takes advantage of this program to simulate the flow in a 

semiannular enclosure. 

Results and Discussions 

Because of the limited worlung space in a personal computer, most calculations used a 20 by 20 grid; however, some 

of them are based on a grid of 12 by 14 (12 in circumferential 

direction and 14 in radial direction). The Nusselt numbers 

obtained from these two mesh systems differ by 2 percent. 

The predicted temperature distribution, streamline, and 

velocity vector are presented in figures 2 to 4, for a flow 

condition of Pr = 0.7, Ra = 0.7 X lo5, and RR = 2.0 and for 

various inclinations (or tilt angles). When the enclosure is 

placed in a horizontal position, the tilt angle is either -90" 

or +90°, and the thermal gradient produces a pair of convective streams. These circulated streams are symmetric with the 

central axis; they move upward in a path parallel to the axis 

and return along the solid surface to join the rising fluid in 

the lower portion of the test cell. Although the flow patterns 

are similar, the heat transfer that induces the flow is different. 

In the -90" enclosure a high temperature region (red) is 

located in the.vertica1 axis (fig. 2(a)); the fluid rises solely 

because of the buoyancy force. On the other hand, in the +90° 

enclosure the motion along the symmetrical plane is due to 

the fluid in that region being pushed upward by the heavier 

fluid around the outer cylinder, which tends to descend to the 

bottom of the enclosure. This phenomenon can also be interpreted from the temperature and velocity distributions of 

figures 2(e) and 3(e). A large part of the semiannulus remains 

at low temperature, and the fluid in the region is almost 

stagnant. In fact, a comparison of figures 3(a) and (e) shows 

that the velocity along the centerline in a +90° enclosure is 

much smaller than that in a -90" enclosure. The slightly 

asymmetric temperature and streamline patterns in these two 

cases (with respect to the central axis) result from the coarse, 

12 by 14 grid used in calculations. Using the 20 by 20 grid 

eliminates the problem. 

2 
As the angle of inclination increases, the heat-transfer 

direction shifts from upward to downward. Consequently, the 

influence of buoyant convection becomes smaller. This effect 

changes the heat-transfer pattern from convection dominant 

to conduction dominant. Thus, the heat-transfer rate was 

expected to decrease as the angle varied. The isotherms and 

streamlines for a semiannulus inclined -50°, O", and +50° 

(parts (b) to (d) of figs. 2 and 3) show that the cold region 

expands as the angle changes from -50" to +50". For the 

+50° case, stratification is strong; the fluid in the lower 

portion of the semiannulus is relatively stagnant and stays cold. 

The temperature and velocity patterns are presented for a 

wide range of flow parameters, in an accompanying video tape. 

The variables for each picture are listed in table I. 

The heat-transfer rates, in terms of Nusselt number, for 

various radius ratios, are presented as a function of the inclination angle in figure 5. The Nusselt number obtained in this 

study varies from 2.30 to 6.47 depending upon the geometrical 

conditions and the Rayleigh number. These values are the same 

order of magnitude as computed by Kumar (ref. 7) for horizontal annuli. The Nusselt number increases with increasing 

diameter ratio. However, when the enclosure is tilted in a 

positive direction, the Nusselt number decreases and reaches 

a minimum at angles around 30" to 50", depending on the 

radius ratio. Further increasing the tilt angle causes the Nusselt 

number to increase by a small value due to an increase of the 

convective flow. 

For a fixed radius ratio (RR = 2.0), the changes in heat 

transfer for the two Rayleigh numbers vary with the angle of 

inclination (fig. 6). The heat-transfer rate increase is 

proportional to the Rayleigh number. 

i 

Summary of Results 

and heat-transfer processes in a differentially heated semiannular enclosure. The calculations are performed on a personal 

computer and use a relatively coarse grid. Although, very 

accurate results were not expected, the predicted velocity and 

temperature patterns are good enough to explain the physical 

phenomena inside the enclosure. The computed results indicate 

that the heat-transfer rate increases with increasing Rayleigh 

number and radius ratio and varies strongly with the angle 

of inclination. 

References 

1. Kuehn, T.H.; and Goldstein, R.J.: An Experimental and Theoretical Study 

of Natural Convection in the Annulus Between Horizontal Concentric 

Cylinders. I. Fluid Mech., vol. 74, pt. 4, 1976, pp. 695-719. 

2. Kuehn, T.H.; and Goldstein, R.J.: A Parametric Study of Prandtl Number 

and Diameter Ratio Effects on Natural Convection Heat Transfer in 

Horizontal Cylindrical Annuli. J. Heat Transfer, vol. 102, 1980, pp. 

3. Kamotani, Y.; Wang, L.W.; and Ostrach, S.: Experiments on Natural 

Convection Heat Transfer in Low Aspect Ratio Enclosures. AIAA J., 

vol. 21, no. 2, 1983, pp. 290-294. 

4. Wang, L.W.; Chai, A.T.; and Sun, D.J.: Convective Flows in Enclosures 

with Vertical Temperature or Concentration Gradients. AIAA Paper 

89-0069, 1989 (also NASA TM-101373). 

5. Patankar, S.V.: Numerical Heat Transfer and Fluid Flow. Hemisphere, 

New York, 1980. 

6. Reference Manual for "MicroCOMPACT," Version 1.1: A GeneralPurpose Computer Program for Two-Dimensional Fluid Flow and Heat 

Transfer. Innovative Research, Inc., Maple Grove, MN, 1988. 

7. Kumar, R.: Study of Natural Convection in Horizontal Annuli. Intern. 

J. Heat Mass Transfer, vol. 31, no. 6, pp. 1137-1148, 1988. 

768-770. 

The two-dimensional Navier-Stokes and energy equations are 

numerically solved to study the free convective momentum 

3 
AD I ABAT I C 

Figure 1 .--Semiannular enclosure. Lift angle, 0". 

4 
(c) 0'. (d) +50'. 

(e) +goo. 

Figure 2.--Isotherm plots. Pr = 0.7; Ra = 0.7~ 10'; and RR = 2.0. 

5 
QR1GIN.rtb PAGE 

COLOR PHOTOGRAPH 

(b) -50'. 

tc) oo. (d) +50°. 

(e) +goo. 

Figure 3.--Streamlines. Pr = 0.7; Ra = 0.7X 16; and RR = 2.0. 

6 
ORIGINAL. PAGE 

COLOR PHOTOGRAPH 

3- 

0 

6 0 

I 

d 

W 

v, co 

Z'I j; 

I 

0 

0 0 - 

0 

RAD I US 

RATIO. 

RR 

0 2.5 

0 2.0 

A 1.5 

0 

A 

A A 

A 

2 

-90 

1 

-60 -30 0 30 60 90 

ANGLE OF INCLINATION, deg 

Figure 5 .-Comparison of heat-transfer rates for various radius ratios and 

inclination angles. Pr = 0.7 and Ra = 0.7 x lo5. 

(b) -50'. 

0 

0 

RAYLEI GH 

NUMBERS, 

0.7~10~ 

0 0.7x1Oq 

Ra 

0 

0 

OO 00 

tc) +900. 

Figure 4.-Velocity vectors. Pr = 0.7; Ra = 0.7X lo5; and RR = 2.0. 
N/\sA National Aeronautics and Report Documentation Page 

1. Report No. 

NASA TM-102011 

2. Government Accession No. 

7. Author(s) 

Lin-Wen Wang, Chain-Nan Yung, An-Ti Chai, and Nasser Rashidnia 

,7. Key Words (Suggested by Author(s)) 

9. Performing Organization Name and Address 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135-3191 

2. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D.C. 20546-0001 

18. Distribution Statement 

3. Recipient's Catalog No. 

9. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No of pages 

Unclassified Unclassified 8 

5. Report Date 

22. Price' 

A02 

6. Performing Organization Code 

8. Performing Organization Report No. 

E-4723 

10. Work Unit No. 

674-25-05-08 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical Memorandum 

14. Sponsoring Agency Code 

5. Supplementary Notes 

Prepared for the 1989 National Heat Transfer Conference sponsored by the American Society of Mechanical 

Engineers, Philadelphia, Pennsylvania, August 6-9, 1989. Lin-Wen Wang, National Research Council-NASA 

Research Associate; Chain-Nan Yung, University of Toledo, Toledo, Ohio 43606 and NASA Resident Research 

Associate; An-Ti Chai, NASA Lewis Research Center; and Nasser Rashidnia, Sverdrup Technology, Inc., NASA 

Lewis Research Center Group, Cleveland, Ohio 44135. 

6. Abstract 

Natural convection heat transfer in a two-dimensional differentially heated semiannular enclosure is studied. The 

enclosure is isothermally heated and cooled at the inner and outer walls, respectively. A commercial software 

based on the SIMPLER algorithm was used to simulate the velocity and temperature profiles. Various parameters 

that affect the momentum and heat transfer processes were examined. These parameters include the Rayleigh 

number, Prandtl number, radius ratio, and the angle of inclination. The study covers a flow regime extending 

from conduction-dominated to convection-dominated flow. The computed results of heat transfer are presented as 

a function of flow parameter and geometric factors. It is found that the heat transfer rate attains a minimum when 

the enclosure is tilted about +50° with respect to the gravitational direction. 

Natural convection 

Thermal convection 

Unclassified - Unlimited 

Subject Category 29 

~ 

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