Proceedings of the Asian Conference on Thermal Sciences 2017, 1st ACTS
March 26-30, 2017, Jeju Island, Korea

ACTS-P00321

LOCAL HEAT TRANSFER CHARACTERISTICS AND VORTEX STRUCTURES
OF VISCOELASTIC FLUIDS IN SERPENTINE CHANNELS
K. Tatsumi *, N. Shinotsuka, R. Kimura, R. Kuriyama and K. Nakabe
Kyoto University, Kyotodaigaku-katsura, Nishikyo-ku, Kyoto 615-8540, Japan
*

Presenting and Corresponding Author: tatsumi@me.kyoto-u.ac.jp

ABSTRACTIn this study, we focus on the flow instability and the generation of the strong unsteady flow in serpentine channels
under very low Reynolds number conditions of Re~1. The local heat transfer, pressure loss and flow characteristicsof the viscoelastic fluid in the transition and developed flows through a serpentine channel were measured anddescribed. Water solution of polyacrylamide with sucrose was used as viscoelastic fluid. The results showed that theflow becomes unsteady and longitudinal vortices-like secondary flows were generated due to the normal stressdifferences produced by the viscoelasticity of the fluid located near the top and bottom walls. This enhanced the heattransfer performance markedly. The vortex structures changed depending on the streamwise position andWeissenberg number Wi showing three transition stages: a pair of moderate vortices, a single vortex with a drift ofmain flow toward the wall, and a pair of strong vortices. These vortices affected the local heat transfer characteristics
at the top/bottom walls and sidewalls, streamwise distributions, and their unsteady behavior. We showed that thesebehaviors and characteristics could be correlated by the combination of Wi and Re.
KEYWORDS: Viscoelastic fluid, Serpentine channel, Heat transfer, Convection, Low Reynolds number flow,
Unsteady flow, Vortex structure

1. INTRODUCTION
Viscoelastic fluid produces interesting and effective flow structures and features in various channels andconditions. One of these effects is the increase of the flow instability which induces the generation of unsteady flows
under low Reynolds number conditions. The viscoelasticity can also generate additional stresses due to the normal
stress differences which leads to the generation of secondary flows. Such unsteady flow and vortices can enhance
the fluid mixing and heat transfer at the channel walls [1, 2]. Intensive work has been made to study these effects
with various channel shapes and under wide range of flow conditions [3-5]. In our pervious works, we have been
focusing on the flow and heat transfer characteristics of viscoelastic fluid in serpentine channel (schematic is shownin Fig. 1). We have confirmed that the average heat transfer performance can be enhanced due to the generation of
longitudinal vortex like secondary flows and its unsteady behavior [6, 7].
In this study, measurements of flow visualization and the local heat transfer coefficient are carried out. The
developing and developed regions of the flow and heat transfer were studied by measuring the heat transfer
coefficient at different streamwise positions and flow rates. Further, the viscoelastic properties of the fluid werechanged to discuss the appropriate dimensionless number to correlated the heat transfer and pressure loss
characteristics. From these results, the relationship between the flow structure and heat transfer characteristics are
discussed to understand the mechanism of heat transfer enhancement.

2. EXPERIMENTAL METHODS AND CONDITIONS
Figure 1 shows the experimental apparatus and the serpentine channel of the present study. The serpentine
channel had a square cross section 5 mm on a side, and consisted of semicircle parts connected periodically in thestreamwise direction. The number of the curve units, N, was 30. Constant heat flux conditions were applied to the1

Fig. 1 The channel and test section of the heat transfer and flow visualization measurements.
channel walls. Two test sections, one with the top and bottom walls heated, and the other with sidewalls heated wereused to measure the heat transfer coefficient of each wall. Stainless steel sheets were attached to the wall to apply
constant heat flux while the local wall temperature was measured to obtain the heat transfer coefficients.
The unheated walls were made transparent to perform the flow visualization through the walls simultaneouslywith the heat transfer measurement. The streak lines were visualized by recording the image of the ink dye supplied
to the flow through a needle set in the upstream region of the channel. The flow pressure loss was measured by twopressure gauges installed to the inlet and outlet of the test section, respectively.
Aqueous solution of 500ppm polyacrylamide (PAAm, MW: 1.8×107) was used as viscoelastic fluids. To stable
the fluid properties, 1wt% NaCl was added. Sucrose was mixed with the concentration of 57.0, 60.0 and 64.4wt% toincrease the viscosity and relaxation time of the fluid. Aqueous solution of 64.4wt% sucrose only (Newtonian fluids)
was also measured for comparison. The fluid viscosity and relaxation time were measured by a rheometer inadvance of each experiment. The flow rate was changed and the related flow conditions ranged as follows based on
the dimensionless number: Reynolds number, Re=0.2-5.8, Graetz number, Gz=3-40, Weissenberg number, Wi=8137, and Dean number, ND=0.1-3.5.

3. RESULTS AND DISCUSSION
Figure 2 shows the Nusselt number Num distributions in relation to Re and Wi. Num averaged in the streamwise
region of N=21-30 is shown in the figure. At Re<0.3 or Wi<20, Num of PAAm cases matches with the solid line
showing that the flow structure is similar to that of the steady state laminar flow of Newtonian fluid. As Re increases
Num increases. However, the Re at which Num starts to increases in the cases of PAAm(54), PAAm(60) andPAAm(67) do not correspond with each other. On the other hand, a good correlation among these three distributions
are observed with Wi shown in Fig. 2(b). This indicates that the heat transfer characteristics are attributed more to
the vortices generated by the normal stress differences of the viscoelastic fluid than the fluid dynamic instability
characteristic.
Figure 3 shows Num in the cases when the top-bottom walls and sidewalls were heated. Nu shown here are theaveraged values of the sections dividing the channel into three with each part composed of 1-10th, 11-20th and 2130th curve units. The Re in which Num starts to increase are of a same value in the three regions. This shows that theflow becomes unsteady and vortex is generated nearly at the same time through the channel. However, thedistribution does not increase monotonically and shows a characteristic pattern in relation to the streamwise position
and Wi particularly in the sidewall heated case. Taking the visualization results into account, the heat transfer
characteristics can be roughly divided into several regions which are shown in Fig. 3 (b) as (A), (B), (C) and (D).

2

20

Average Nusselt number: Num

Average Nusselt number: Num

25
PAAm(64)
PAAm(60)
PAAm(57)
SucroseNewtonian fluid (numerical)

20
15

15

PAAm(64)
PAAm(60)
PAAm(57)
Newtonian fluid (numerical)

10

10
5
0
0.1

1

10

5

0

10

100

Reynolds number: Re

Weissenberg number: Wi(a) Num vs Re(b) Num vs Wi
Fig. 2 Average Nusselt number Num in relation to Reynolds number Re and Weissenberg number Wi. Num isaveraged in the region of N=21-30, and PAAm(57), PAAm(60) and PAAm(64) are the cases of PAAm soluton withsucrose 57.0, 60.0 and 64.4wt% concentrations, respectively.

20

Average Nusselt number: Num

Average Nusselt number: Num

25
PAAm(64)_N1-10_tb
PAAm(64)_N11-20_tb
PAAm(64)_N21-30_tb
Newtonian fluid

15
10
5
0

(A)
10

(B)

(C)

(D)
100

Weissenberg number: Wi

25
20

PAAm(64)_N1-10_s
PAAm(64)_N11-20_s
PAAm(64)_N21-30_s
Newtonian fluid

15
10
5

(A)
0

10

(B)

(C)

(D)
100

Weissenberg number: Wi(a) Wi vs Num (top-bottom walls)(b) Wi vs Num (sidewalls)

Fig. 3 Locally averaged Nusselt number Num distributions of top-bottom and sidewalls in the case of PAAm(64).
Num is averaged in three regions, N=1-10, 11-20 and 21-30, respectively.
Figure 4 shows the local Nusselt number Nux distributions of the top-bottom walls and sidewalls averaged inseveral Wi regions. These regions roughly correspond to those shown in Fig. 3 (b). Figure 5 shows the time
distributions of Nux at the top-bottom walls of N=16, and the flow visualization performed simultaneously at thesame location. The period (b)-(f) shown by the dash lines in Fig. 5 (a) corresponds to those of Figs. 5 (b)-(f).
As shown in Fig. 4, Nux begins to increase from the downstream showing that the secondary flow generation andunsteady flow starts from the downstream side. Larger Nux is observed in the sidewall compared to the top-bottom
walls. This is due to the generation of a pair of longitudinal vortices like secondary flow which carries the lowtemperature fluid located at the channel center toward the outer wall of the curvature and increases Nu there. Thiscan be confirmed by comparing the Nux and flow structure shown in Fig 5 particularly in the cases of (b) and (e).
The error bar shown in Fig. (4) increases with Wi. This error bar is the deviation of the instantaneous Nux fromthe averaged one. Therefore, it presents the fluctuation of the heat transfer coefficient and the flow and not themeasurement error. As shown in Fig. 5, the direction of the pair of vortices changes related to the height position of
the flow core region: as the region of high velocity moves in the height direction, the normal stress changes its
amplitude in the region adjacent to the top and bottom walls leading to the change of the vortices direction. The
instantaneous Nux changes and fluctuates over time due to this change of the flow structure.
Thus we can explain the heat transfer and flow characteristics of regions (A)-(D) as follows.
In region (A), the flow structure is nearly the same with the Newtonian fluid flow. In region (B), as mention inabove, secondary flow is observed and the flow becomes unsteady. Num of the sidewalls increases while Num slightly
decreases in the case of top-bottom walls heated. Further, a noticeable peak appears in the N21-30 region of the
sidewall heated case.

3

30
Top wallBottom wall
Sucrose(Numerical)

25
20

10

5
0

5
10
15
20
25
Streamwise position: x /L

030

Local Nusselt number: Nux

Local Nusselt number: Nux

Top wallBottom wall
Sucrose(Numerical)

5
0

(b)
0

(c) (d)
100

(e)
200

(f)

300

t [s]

(a) Time distributions of Nux

20

15

15

10

10

5
0

5
10
15
20
25
Streamwise position: x /L

5
0

30

(c) Wi=2945 (top-bottom)

0

5
10
15
20
25
Streamwise position: x /L30(b) t=18s(d) Wi=2945 (sidewalls)

30

30
Top wallBottom wall
Sucrose(Numerical)

25

Local Nusselt number: Nux

Local Nusselt number: Nux

30

Outer wall
Inner wall
Sucrose(Numerical)

25

20

Outer wall
Inner wall
Sucrose(Numerical)

25

20

20

15

(c) t=95s

15

10

10

5
0

5
10
15
20
25
Streamwise position: x /L

5
0

30

(e) Wi=5085 (top-bottom)
30

Top wallBottom wall
Sucrose(Numerical)

25

0

30

(d) t=115s

Outer wall
Inner wall
Sucrose(Numerical)

25

20

5
10
15
20
25
Streamwise position: x /L(f) Wi=5085 (sidewalls)
30

Local Nusselt number: Nux

Local Nusselt number: Nux

5
10
15
20
25
Streamwise position: x /L

30

25

20

15

15

10

(e) t=271s

10

5
0

10

0

(b) Wi=729 (sidewalls)

30

0

15

5

(a) Wi=729 (top-bottom)

0

20

15

10

N=15(Upper wall)
N=15(Lower wall)

25

20

15

0

30

Outer wall
Inner wall
Sucrose(Numerical)

25

Local Nusselt number: Nux

Local Nusselt number: Nux

Local Nusselt number: Nux

30

0

5
10
15
20
25
Streamwise position: x /L30(g) Wi=100150 (top-bottom)

5
0

0

5
10
15
20
25
Streamwise position: x /L30(h) Wi=100150 (sidewalls)(f) t=321sFig. 4 Streamwise local Nusselt number distributions at the Fig. 5 Simultaneous measurement of Nux andtop-bottom walls and sidewalls in the Wi four regions (A)- visualization. (a) Time distributions of Nux and (b)(D) shown in Fig. 3.
side views of the streaklines both measured at
N=16.
In this Wi region, a pair of longitudinal vortices like secondary flows are generated in the channel. Thissecondary flow generation is attributed to the normal stress difference of the viscoelastic fluid, and convects the fluid
located at the channel center toward the outer wall while the fluid located adjacent to the outer wall is driven toward
the top-bottom walls. Therefore, the Num at the sidewalls is enhanced while the Num at the top-bottom walls decrease.
These secondary flow is fully developed at the downstream side of the channel, and, thus, a marked increase of Num
is observed in Fig. 3 (b).
In region (C), these pair of vortices develop to a single vortex. The vortices in this region becomes unstable, andone of the two vortices increases its size and occupy a large portion of the cross section. Further, the main flow islocated near the center of this larger secondary flow, and drifts toward one side of the top-bottom walls increasing
the Num there. As mentioned above, the direction of the vortices changes depending on the location of the flow core.

4In region (D), a symmetric pair of vortices appears again from the downstream of the channel. As Wi increases,
the normal stress described in the discussion of region (B) increases. This makes the pair of vortices become morestable, and thus the single vortex like secondary flow and the drift of the main flow in the height direction decay.
Thus, the phenomena observed in region (C) moves upstream to the N11-20 region, the region at which we can findthe fluctuation of Nux at the top-bottom walls increasing. On the other hand, in the N21-30 region, the sidewall heattransfer increases markedly and the fluctuation decreases compared to those of top-bottom walls. The fluctuation at
the sidewalls, however, still shows a large value due to the change of the direction of the pair of vortices.

4. CONCLUSIONSHeat transfer coefficient, flow visualization and pressure loss measurement were conducted for viscoelastic fluid inserpentine channel under low Reynolds number conditions of Re~1. The average Nusselt number Nu increased as
the Re and Weissenberg number Wi increased showing a three times higher heat transfer performance compared to
Newtonian fluid. A better correlation was observed between the Wi number and Nu number indicating that thevortices generation and unsteady flow which increase the Nu was mainly attributed to the normal stress of theviscoelastic fluid. The local Nu and flow structure varied in the streamwise location, top-bottom and sidewall, andWi conditions. This was mainly due to the difference of vortex structure: a pair of longitudinal vortices, a single
vortex with the position of the core region – high flow velocity region – changing in the height direction, and a pairof strong longitudinal vortices changing their rotational directions.

ACKNOWLEDGMENT
This work was partially supported by JSPS KAKENHI Grant Number 24360080 and 25630066.

NOMENCLATURE
D
Um




hydraulic diametermean flow velocity
relaxation time
fluid density

(m)
(m/s)
(s)
(kg/m3)ReWi

Reynolds number, UmD/
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