Procedia Engineering 79 ( 2014 ) 289 – 294

Available online at www.sciencedirect.com

1877-7058 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).Selection and peer-review under responsibility of the National Tsing Hua University, Department of Power Mechanical Engineering doi: 10.1016/j.proeng.2014.06.345

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37th National Conference on Theoretical and Applied Mechanics (37th NCTAM 2013) & The 1stInternational Conference on Mechanics (1st ICM)

Influences of Prandtl number on transition of steady natural

convection flows in a square enclosure with a circular cylinder

Chuan-Chieh Liaoa, Chao-An Lina,∗aDepartment of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 30013, TAIWAN

Abstract

Numerical investigations are carried out for steady natural convection within domains with curved geometry by using an immersed-

boundary method. The method is first validated with flows induced by natural convection in the annulus between concentric circular

cylinder and square enclosure as Rayleigh number equals to 106, and the grid-function convergence tests are also examined. The

Prandtl number effect is further investigated within the range of 7.0 − 0.2 and the transient development of the flow is considered

as well. Local and average heat transfer characteristics are fully studied around the surfaces of both inner cylinder and outer

enclosure. The results show that the flow exists a transition as Prandtl number decreases to a critical value and the transition

improve the internal energy transport in the present study.c© 2013 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the National Tsing Hua University, Department of Power Mechanical Engineering.

Keywords: Prandtl number effect; transition; immersed-boundary method

1. Introduction

Buoyancy induced flow in an enclosure is frequently investigated because of its application in electronic packaging,

nuclear reactors and heat exchangers. Many studies had been reported addressing the geometric influences of the

enclosure and the immersed object. The commonly investigated cases can be heat transfer within the annulus of

horizontally concentric circular cylinders [1]-[5] or the natural convection in concentric annulus between an inner

circular object and an outer rectangular enclosure [6]-[9].

Focusing on the natural convection within the annulus of horizontally concentric circular cylinders, Cheddadi et

al.[2] showed that two different dynamic flow structures can be obtained depending on the initial conditions and the

Rayleigh number. The two flows are basic crescent-shaped ‘unicellular flow’ and a ‘bicellular flow’ with a pair of

counter-rotating eddies in the top region of the annulus. Similar results were obtained by Yoo[3], Desrayaud et al.[4]

with varying Rayleigh number for different Prandtl number and radius ratio. A critical review of buoyancy-induced

flow transitions in cylindrical configuration can be found in Angeli et al.[5]

∗ Corresponding author. Tel.: +886-3-5742602 ; fax: +886-3-5722840.

E-mail address: calin@pme.nthu.edu.tw

© 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).Selection and peer-review under responsibility of the National Tsing Hua University, Department of Power Mechanical Engineering

290 Chuan-Chieh Liao and Chao-An Lin / Procedia Engineering 79 ( 2014 ) 289 – 294

On the other hand, there were relatively less attention regarding the flow transitions within the configuration of

concentric annulus between an inner circular object and an outer rectangular enclosure. Therefore, we focus on the

transition of steady natural convection flows with varying Prandtl number in the present study. A preliminary results

of Prandtl number effect with constant Rayleigh number (106) will be obtained, and the transient development of the

flow is considered as well.

2. Numerical methodology

The schematic configuration of the system considered in the present study is shown in Fig. 1. Consider the problem

of a viscous incompressible fluid with thermal convection in a two-dimensional rectangular domain containing an

immersed massless boundary, where no-slip and isothermal boundary conditions are imposed on both boundaries.

X

Y

g

Isothermal object (T= 1.0)

L

Isothermal wall (T= 0.0)

D

Fig. 1. Computational domain and boundary conditions for mixed convection in the annulus between concentric rotating circular cylinder and

square enclosure.

The governing equations of this fluid-structure interaction system with thermal condition are

∇·u = 0 . (1)

∂u∂t+ ∇·(uu) = −∇p + Δu +

RaPr

T + fM , (2)

∂T∂t+ ∇·(uT ) =

1

PrΔT + fE , (3)

Here, x = (x, y), u(x, t) is the fluid velocity with components u and v, and p(x, t) is the fluid pressure. ∇ and Δ are the

usual Cartesian-coordinate gradient and Laplacian. Ra is the Rayleigh number, which is defined as Ra = g·β·(Ts−T0)·L3

ν·αwith the gravitational acceleration g, the thermal expansion coefficient β, the temperatures of the immersed object Ts

and the outer enclosure T0, characteristic length L, kinematic viscosity ν and thermal diffusivity α. Pr is the Prandtl

number, which is defined as Pr = να

. It is noted that the Cartesian coordinates x, time t, velocity u(x, t), temperature

T and pressure p(x, t) are normalized with L, L2/ν, ν/L, (Ts − T0) and ρν2/L2, respectively.

Note that the discrete momentum forcing fM(x, t) is applied to satisfy the no-slip condition on the immersed

boundaries, as in [10]. Also, the discrete energy forcing fE(x, t) is used to satisfy the prescribed thermal conditions

on immersed boundaries. Both momentum forcing and energy forcing are applied only at the nodes adjacent to the

immersed boundary in an Eulerian grid system.

291 Chuan-Chieh Liao and Chao-An Lin / Procedia Engineering 79 ( 2014 ) 289 – 294

The numerical procedure used herein consists of a finite-volume method discretized in Cartesian coordinates with

a staggered-grid arrangement of dependent variables. This procedure is based on the integration of the transport

equations over arbitrary control volumes, leading to the conservation of mass and the balance of momentum and

any scalar flow property over each such volume. Spatial derivatives are approximated using second-order centered

differencing, and a fractional-step (projection) method implemented with a combination of the Adams–Bashforth and

Crank–Nicolson methods for advective and diffusive terms, respectively, is used for temporal discretization. Here, the

four-step time-advancement formulation[11,12] is employed for the momentum equations due to its storage economy

and straightforward implementation of boundary conditions. Further details of the immersed-boundary method used

here for dealing with heat transfer problems are given in Liao and Lin[8,13].

3. Result and discussion

3.1. Grid independence test

Flows induced by natural convection in the annulus between concentric circular cylinder and square enclosure

have been reported in recent years and are adopted here to validate the capability of the present method. Consider a

horizontal concentric annulus between a heated circular inner cylinder and a square outer enclosure. Here, as shown

in Fig. 1, no-slip and isothermal boundary conditions are imposed on both boundaries. The temperature difference

between the cylinder and the outer walls introduces a temperature gradient in the fluid, and the consequent density

difference induces a convective fluid motion. Air is used to be the working fluid and the Prandtl number is fixed at

0.71. The enclosure length and cylinder diameter ratios (L/D) equals to 2.5.

Grid-function convergence tests are first conducted using four uniform grids of 200×200, 300×300, 400×400 and

600×600 cells, when Ra=106. The local and surface-averaged Nusselt numbers are defined as

h = −k∂T∂n|wall, and NuLocal =

hSk, (4)

h =1

2π

∫ 2π

0

hdθ, and NuMean =hSk. (5)

where n is the normal direction along the boundary and the characteristic length S is defined as the perimeter of the

circular cylinder and square enclosure. Table 1 shows the comparisons of NuMean for different uniform grids with

those of Moukalled and Acharya [6] and Shu et al. [7]. All simulated surface-averaged Nusselt number are consistent

with literature data and the difference by using the coarsr (200×200, 300×300, 400×400) and finest grids (600×600) are

about 0.46%, 0.02% and 0.03%. Therefore, the uniform grid resolution of 300×300 cells is adopted in the following

simulated cases. It is clear that the present predictions are consistent with literature data, and the capability of the

present method for dealing with natural convection in the annulus is ascertained.

Table 1. Comparisons of the simulated surface-averaged Nusselt number for different uniform grids with those of Moukalled and Acharya [6] and

Shu et al. [7] for grid-function convergence tests.

200 × 200 300 × 300 400 × 400 600 × 600 Ref.[6] Ref.[7]

NuMean 17.688 17.765 17.775 17.769 18.748 17.796

3.2. Influence of the Prandtl number (Pr) for natural convection in the annulus

Since the present study explores the prandtl number effects on steady natural convection flows, thus varying Pris adopted here. The enclosure length and cylinder diameter ratios (L/D) equals to 2.5 and the Rayleigh number is

fixed to 106. Figure 2 shows the isotherms and streamlines for different Pr within the range of 7.0 − 0.2 where all the

contour values are normalized and range from 0.0 to 1.0.

When Pr = 7.0, one pair of symmetric circulation can be observed in the streamlines and one plume appears on

the top of the inner cylinder in the isotherms, as depicted in Fig.2(a). All contours are similar as Pr decreases from

292 Chuan-Chieh Liao and Chao-An Lin / Procedia Engineering 79 ( 2014 ) 289 – 294

7.0 to 0.4 and an apparent change can be observed when Pr further decreases. As shown in Fig.2(d)(e), the plume

appeared on the top of the inner cylinder is divided into two plumes on both sides of the cylinder, and two pairs of

counter-rotating vortices can be observed in the streamlines. Because all observed circulation were symmetric with

respect to the y-axis, the results can be divided into two sections of ‘unicellular flow’ [7.0 ≥ Pr ≥ 0.4] and ‘bicellular

flow’ [Pr = 0.3, 0.2], according to on the flow structures in a half domain.

Streamlines Streamlines

Isotherms Isotherms

(a) Pr= 7.0 (e) Pr= 0.2(d) Pr= 0.3(c) Pr= 0.4(b) Pr= 1.0

Isotherms

Streamlines

Isotherms Isotherms

Streamlines Streamlines

Fig. 2. Isotherms (up) and streamlines (down) for Ra = 106: (a)Pr = 7.0; (b)Pr = 1.0; (c)Pr = 0.4; (d)Pr = 0.3; (e)Pr = 0.2.

Figure 3 further shows the distributions of local Nusselt numbers (NuLocal) along the hot surface of the circular

cylinder and the cold surface of the enclosure at different Pr. In general, there exist a locally maximum value on

the top wall of the outer enclosure (i.e. θ = 90o) and four minimum values on the corners of the enclosure (i.e.

θ = 45o, 135o, 225o and 315o) for unicellular flows (Pr = 7.0, 1.0 and 0.4). The locations of minimum value along

the inner cylinder and maximum value along the outer enclosure are affected by the plume. As Pr decreases to 0.3,

the plume starts dividing and the position of maximum NuLocal along the outer enclosure migrates from one local

maximum at θ = 90o to two local maxima at θ = 60o and 120o. The influence of the divided plumes on the inner

cylinder is significant where the position of minimum NuLocal changes from one local minimum at θ = 90o to two

local minima at θ = 30o and 150o, as shown in Fig.3(a).

To further understand the nature of the bicellular flow, the transient development of the flows are investigated

for Pr = 0.3. Figure 4(a) shows the evolution of surface-averaged Nusselt number in time and Figs. 4(b)-(g) are

the transient development of isotherms and streamlines contours at five instants. Initially, the temperature difference

between the two walls introduces a temperature gradient in the fluid, and the consequent density difference induces

an upward plume in isotherms and one pair of symmetric circulation can be observed in the streamlines. As time goes

by, the strong convection flow moves outward along the outer surface and the thermal plume starts separating. When

time equals to 0.083s, two divided plumes establish and a small pair of counter rotating eddies can be observed in the

top area, as depicted in Fig.4(d). The divided plumes move along the inner surface continuously, which induces the

eddies in the top area become larger. Finally, the flow with two separated plumes and two pairs of counter-rotating

vortices is created, as shown in Fig.4(g).

The surface-averaged Nusselt number for different Prandtl numbers are depicted in Fig.5 where both unicellular

flow and bicellular flow are included. In general, the NuMean becomes smaller as the Pr decreases from 7.0 to

0.4. However, an obvious increase can be observed as Pr decreases to 0.35, where the flow structure changes from

unicellular flow to bicellular flow. The corresponding NuMean is 18.664, whereas the NuMean is 17.539 when the Pr

293 Chuan-Chieh Liao and Chao-An Lin / Procedia Engineering 79 ( 2014 ) 289 – 294

θ

Nu L

ocal

0 30 60 90 120 150 180 210 240 270 300 330 3600

10

20

30

40

50

60

70

80

90

Pr= 7.0Pr= 1.0Pr= 0.4Pr= 0.3Pr= 0.2

(b)

θ= 270

θ= 90

θ= 0θ= 180

θ= 270

θ= 90

θ= 0θ= 180

θ

Nu L

ocal

0 30 60 90 120 150 180 210 240 270 300 330 3600

5

10

15

20

25

30

35

40

Pr= 7.0Pr= 1.0Pr= 0.4Pr= 0.3Pr= 0.2

(a)

Fig. 3. Steady distributions of local Nusselt number along the surface of the (a) inner cylinder and (b) outer enclosure for different Pr(7.0 − 0.2)

when Ra = 106.

t*

Nu M

ean

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.416

17

18

19

20

(f)

(a)

(b)

(c)

(d)

(e)

(b) t*= 0.017s (f) steady(e) t*= 0.117s(d) t*= 0.083s(c) t*= 0.05s

Fig. 4. (a): Instantaneous distributions of surface-averaged Nusselt number, and (b)-(g): transient development of isotherms and streamlines when

Ra = 106, Pr = 0.3.

equals to 0.4. In other words, a 6.4% increase of the NuMean can be obtained as the Pr decreases to a critical number,

which indicating that the flow transition improve the internal energy transport in the present study.

4. Conclusion

Numerical investigations are carried out for steady natural convection within domains with curved geometry by

using an immersed-boundary method. The method was first validated with flows induced by natural convection in the

annulus between concentric circular cylinder and square enclosure as Rayleigh number equals to 106, and the grid-

function convergence tests were also examined. The Prandtl number effect is further investigated within the range

of 7.0 − 0.2 and the transient development of the flow was considered as well. In general, the steady flow can be

294 Chuan-Chieh Liao and Chao-An Lin / Procedia Engineering 79 ( 2014 ) 289 – 294

: bicellular flow

: unicellular flow

log (Pr)

Nu M

ean

10-1 100 10117

17.5

18

18.5

19

Fig. 5. Comparisons of surface-averaged Nusselt number for different Prandtl numbers when Ra = 106.

categorized as unicellular flow or bicellular flow as Pr decreases from 7.0. The results show that the flow exists a

transition from unicellular flow to bicellular flow as Pr further decreases to 0.35, where a 6.4% increase of the NuMean

can be obtained, indicating that the flow transition improve the internal energy transport in the present study.

Acknowledgements

The authors gratefully acknowledge the support by the National Science Council of Taiwan and Low carbon energy

research center of National Tsing Hua University, and the computational facilities provided by the Taiwan National

Center for High-Performance Computing.

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