Proceedings of the 15th International Heat Transfer Conference, IHTC-15
August 10-15, 2014, Kyoto, Japan

IHTC15-9615

A NUMERICAL AND EXPERIMENTAL STUDYON FLOW AND HEAT TRANSFER CHARACTERISTICS
OF VISCOELASTIC FLUID FLOW IN A SERPENTINE CHANNEL
Kazuya Tatsumi*, Wataru Nagasaka, Takuya Matsuo, Kazuyoshi Nakabe
Kyoto University, Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8530, Japan

ABSTRACT
Measurements of heat transfer coefficient and pressure loss penalty, and PIV measurements were carried out for
viscoelastic fluid in a serpentine channel. Numerical simulation was carried out together to investigate the flow
and heat transfer characteristics. Water solutions of polyacrylamide (PAAm) were used as viscoelastic fluids.
The PIV measurements showed that in the viscoelastic fluid case, unsteady flows and longitudinal-vortex-likesecondary flows were generated in the serpentine channel even under the low Reynolds number conditions
smaller than 2.0. As the Re and Weissenberg number Wi increased, the magnitude of the secondary flow and theflow fluctuation increased. This led to a effective heat transfer enhancement compared with the Newtonian fluidcase. Further, the influences of the fluid rheological properties on the heat transfer and pressure loss
characteristics were evaluated by changing the sucrose concentration of the viscoelastic fluids. The results
showed that a good correlation can be observed with the mean Nusselt number and the Wi rather than the Re.
This was attributed to the fact that heat transfer enhancement was mainly due to the secondary flow generated by
the normal stress differences, the magnitude of which can be represented by Wi. The computation showed that anormal stress was produced in the area particularly adjacent to the top and bottom walls, and that this would
accompany the generation of a pair of vortices. Effective heat transfer was observed at the channel sidewall
particularly at the inner sidewall near the inflection point of the serpentine channel.

KEY WORDS: Viscoelastic fluid flow, Heat transfer enhancement, Pressure loss, Low Reynolds number flow,
Serpentine channel, Square cross section
1. INTRODUCTION
Using viscoelastic fluid flow is another considerable solution to the geometrical and mechanicalmodification of the channel for heat transfer enhancement. The high shear stress generated in the channel
will cause the polymers solved in the fluid to contract and expand leading to an appearance of the elasticity
and additional normal stresses in the flow. In combination with a certain channel shape, the fluid elasticity
increases the flow instability resulting in generation of unsteady flow and enhancing the secondary flow evenin low Reynolds number regime. The influences of the rheological properties of the viscoelastic fluid on
laminar heat transfer have been keenly studied in the past several decades, particularly in the case of straightchannel. Hartnett & Kostic [1] and Chunbo & Hartnett [2] have reported from their experiment that the heattransfer coefficient in the case of viscoelastic fluid flow was higher than the Newtonian fluid case. The maincause was believed to be attributed to the secondary flow which was generated and enhanced by the normal
stress difference. Naccache et al. [3] conducted a numerical simulation on the non-Newtonian laminar heattransfer in a rectangular duct and confirmed that the heat transfer was strongly enhanced due to thissecondary flow.

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

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IHTC15-9615
In a curved channel, viscoelastic fluid flows can generate secondary flows, and also enhance the flow
instability and produce fluctuations even under low Re conditions. Thus, flow instability of viscoelastic fluid
flow in curved channel has also been studied with strong interest. Larson et al. [4] and Muller et al. [5]studied the flow instability of Taylor-Cuette flows in rotating coaxial cylinders. Shaqfeh [6] investigated themechanism of the increase in the flow instability of Taylor-Dean flows in coaxial cylinder. Further, Fan et al.
[7] investigate the viscoelastic flows in a curved duct by numerical simulations and showed that the normal
stresses due to the viscoelasticity can increase the magnitude of the secondary flow originated to thecurvature of the channel. Norouzi et al. [8, 9] have numerically investigated the heat transfer characteristics
of viscoelastic flow in a curved duct. The results showed that the first normal stress differences encourage
the secondary flow generation and enhance the heat transfer while the second normal stress differencessuppress heat transfer efficiency.
Recently, several studies considering the mixing characteristics of viscoelastic fluid in serpentine channels
have been reported. Groisman & Steinberg [10] carried out flow visualization and point measurement of theflow velocities using laser Doppler anemometer. Flow fluctuation was observed even in the order of Re ~ 10,
leading to the enhancement of the fluid mixing. Tamano et al. [11] made PIV measurements in curved
channels with circular cross-section and found a pair of vortices similar to the Taylor vortices in the timemean velocity fields. Zilz et al. [12] and Poole et al. [13] carried out numerical computation on theviscoelastic fluid in serpentine channel and discussed the flow instability. Yatou [14] numericallyinvestigated the characteristics of the separation vortices of viscoelastic fluid in a sinusoidal flow. Theirresults showed that considerable fluctuations can be obtained in such flows suggesting the possibility ofenhancing the mixing performance or, moreover, mass and heat transfer at the channel wall effectively.
To the knowledge of the authors, however, there are no experiments and numerical computations conducted
to link the viscoelastic fluid flow with heat transfer performance in serpentine channels. It is essential tounderstand the flow fluctuation and heat transfer characteristic of such flows in order to provide insights forevaluating the feasibility and designing of an effective channel shape for practical of thermal engineering.
From this background, the authors have carried out heat transfer measurements for viscoelastic fluid underisothermal wall heating conditions in the serpentine channel for low Reynolds number (Re~1) flows [15].
We have showed that heat transfer can be enhanced under such condition. In the present paper, the heattransfer coefficient and pressure loss penalty are carried out by changing the fluid properties in order toevaluate the correlations of dimensionless thermofluid parameters. Further, the velocity field was measured
using the particle image velocimetry (PIV) to investigate the flow field. Numerical simulation was also
conducted to study the flow and heat transfer characteristics using the non-linear viscoelastic model, namely
Giesekus model.

2. WORKING FLUIDS
An aqueous solution of 500ppm polyacrylamide (PAAm, Mw = 1.8107), 1wt% NaCl, and sucrose was usedas viscoelastic fluids. Sucrose was mixed in the fluid to mainly increase the viscosity of solution, and NaCl
was added as a counter ion to prevent the shear thinning effects and the aggregation of the polymers. The
sucrose concentration was changed for three conditions, which were 57.0, 60.0 and 64.4 wt%, in order toevaluate the influences of the fluid property. The solutions were used for the experiment within 72 hours
after the preparation. An aqueous solution of 64.4wt% sucrose only was used as Newtonian fluids tocompare with the viscoelastic fluids.
The apparent viscosity of PAAm solutions is a function of shear rate and temperature showing the relation of
power law, P= K J n 1 [16]. J is the flow shear rate, n is constant, and K is a coefficient related to thetemperature. The viscosity Pand relaxation time O were measured by a rheometer (ANTON PAAR,
MCR301) before each experiment was conducted. As one example, Table 1 summarizes the representativevalues of . n, O, and Prandtl number, Pr for the solutions. These values are the results of the measurement
at fluid temperature of 20ǀC. P and Pr of sucrose solutions at 20ǀC are 0.15Pas and 1030, respectively.
The values of K and O at the reference temperature in experiments are calculated using K-G equation (Mirsa& Varshni [17]), which is the function for temperature dependence of the viscosity of sucrose solutions. The
viscosity is a function of shear rate and temperature which can be written as Eq. (1).

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IHTC15-9615
Table 1 Properties of the solutions (fluid temperature of 20qC)
solution

PAAm [ppm]Sucrose [wt%]P [Pas]

O [s]

Pr

SucrosePAAm(64)
PAAm(60)
PAAm(57)


500
500
500

64.4
64.4
60.0
57.0

0.15
0.37
0.16
0.09


4.0
3.5
2.5

1030
2340
1030
650

Table 2 Constants for viscosity and normal stress difference in the case of PAAm solutions
solution
PAAm(64)
PAAm(60)
PAAm(57)

Km

n

0.47
0.25
0.16

0.88
0.85
0.82

E

H
3

-3.110-3.6103-3.8103

EN

KN1
5

7.9107.9105
7.9105

3

0.090
0.045
0.018

-0.810-1.7103-3.2103

O [s]

Pr

4.0
3.5
2.5

2340
1030
650

(a) N1 vs shear rate(b) N1 vs temperature
Fig. 1 Apparatus of heat transfer experiment for the serpentine channel.

log10 P Jm , Tm

D Jm E
Tm  273

H
Tm  273(1)

2

Shear rate is calculated as J =4Um/Dh . Um is the channel mean velocity and Dh is the hydraulic diameter ofthe channel. E and H are constants which were obtained by the measurement carried out under the conditionof Tm=20qC. These two values were not largely influenced by the temperature, and was used as constants forother temperature conditions. As for D which is dependent with the shear rate, the value was obtained using
Eq. (2), in which the values of Tm=20qC was also used.

D Jm

·
§
E
H
¸
log10 K m, 20Jmn 1  ¨¨

2 ¸
T
273

T
273

20
¹
© 20

(2)

Similarly, the empirical equations to derive the normal stress differences are defined as Eqs. (3) and (4).

log10 N1 Jm , Tm

D N Jm

D N Jm ENTm  273



HN

Tm  273

2

·
§ EN
HN
¸
log10 K N 1, 20Jmn 1  ¨¨

2 ¸
T
273

T
273

20
¹
© 20

(3)

(4)

Figure 1 shows the relation between N1 and shear rate J , and between N1 and temperature T. The
approximated curve of Eqs. (3) and (4) agrees fairly well with the measurements. The constants applied toEqs. (1)(4) are summarized in Table 2.

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IHTC15-9615
Table 3 Experimental condition of heat transfer measurement.
solution

Re

Gz

Wi

ND

SucrosePAAm(64)
PAAm(60)
PAAm(57)

0.72.5
0.22.6
0.45.1
0.55.8

434
340
538
439


11137
101368125

0.22.2
0.11.5
0.22.9
0.33.5

Fig. 2 Apparatus of heat transfer experiment for the serpentine channel.
The specific heat capacity at constant pressure cp, the thermal conductivity k, and the density U of thesolutions were referred to the values measured by Frank et al. [18] and Werner et al. [19].

3. EXPERIMENTAL METHODS
3.1 Heat Transfer and Pressure Loss Measurements Two channels of different shapes were used in
the heat transfer experiment. One was a serpentine channel and the other was a straight one. Figure 2 shows
the schematic diagram and dimensions of the serpentine channel. The serpentine channel had a square crosssection of 5㽢5 mm, and consisted of semicircle parts connected periodically in the streamwise direction.
The inner and outer radii of the semicircle part were Ri = 5mm and Ro = 10mm, respectively. As shown in
Fig. 2, the maximum of the curve unit number, N, was 10.㻌 Therefore, the total length along the serpentine
channel, L, was 470mm. The straight channel was designed to have the same channel length of 470mm.
Water of constant temperature was supplied to the copper block of the test section to heat the channel wall
isothermally. Working fluid was supplied from the pressurized tank to the channel, and the mass of the fluid
flowing from the outlet was measured by an electronic balance to measure the mass flow rate, m . The bulk
mean temperatures of the fluid at the inlet and outlet of the test section, Tb,i and Tb,o, respectively, wereobtained using the fluid temperatures measured at each location and the fully developed velocity profile of asquare channel. A single pair of sheath thermocouples was used to measure the inlet fluid temperature while
the outlet fluid temperature was measured by seven thermocouples located evenly in the height direction at
the center of the channel width direction. The wall temperature, Tw, was a value of the arithmetic-averaging

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IHTC15-9615
temperatures measured at seven locations of the copper block surface. Average heat transfer coefficient, hm,
was thus calculated by applying the bulk mean temperature and mass flow rate to Eq. (5).

hmm c pAsu ln

Tw  Tb ,i(5)

Tw  Tb ,o

As is heat transfer area. Pressure loss, 'P, was measured by two pressure gauges installed to the inlet andoutlet of the test section as shown in Fig. 2.
The average Nusselt number was defined as Num = hmDh/k. A logarithmic mean temperature of Tb,i and Tb,o
was used as the reference temperature to estimate the fluid properties in heat transfer measurement. The
Reynolds number, Re, Prandtl number, Pr, Graetz number, Gz, Weissenberg number, Wi, and Dean number,
ND, are defined as Eq. (6). The values of these dimensionless parameters are summarized in Tables 1 and 3.
The Re condition was varied from 0.2 to 5.8.

Re

UU m Dh, Pr
P0

c p P0k, Gz

Dh
RePr , WiL

4U m O
,
Dh

ND

Dh
Re
Ro  Ri(6)

3.2 PIV Measurements PIV measurements were carried out using the channel made of transparent acrylic,
which had the same shape and dimension with those of the channel employed in the heat transfer experiment.
Measurement was made for the velocity fields at two cross sections. One was the xy plane located at thecenter of channel height (z = 0mm). The other was the streamwise cross section (rz plane) at T = 78o.
These measurements were carried out at the 2nd curve unit (N = 2) of the streamwise position.
During the measurement, the influences of the curvature and the refraction index were checked bycomparing the velocity distributions with theoretical values in the case of laminar flow for Newtonian fluids.
The results showed that these two effects were less than several %. Therefore, no correction is applied to thepresent images and results.
The PIV system consisted of a double pulse Nd:YAG laser and a high resolution CCD camera. The
scattering images of the tracer particles were captured by the CCD camera. The spatial resolution of theimage was 15Pm/pixel, and the recording frame rate was 5fps. The total number of pair of images measured
was 600, which was 120s converted to the measuring time. Glass beads (Glass hollow sphere, nominal
diameter =10Pm, relative density = 1.1) were used for tracer particles. PIV analysis software (LaVision,
Davis 6) was used to measure the velocity data from the images.
The Reynolds number and Dean number for the flow in the PIV measurement were Re=3.9 and ND=3.1 inthe case of sucrose solutions. In the PAAm solution case, the PIV measurement was carried out under three
conditions of Re=0.5, 1.2 and 2.2, in which the Dean number was ND=0.4, 1.0 and 1.6, and the Weissenberg
number was Wi=98, 211 and 333, respectively.

4. NUMERICAL METHODS
4.1 Numerical Model and Procedure Figure 3 shows the computational domain and the applied cylindricalcoordinates. Finite-Volume Method (FVM) was adopted in the present numerical computation. In thecomputation, continuity and Navier-Stokes equations were solved in three-dimensional, time-dependent andincompressible forms. For the temperature fields, the energy conservation equation of the fluid was solved.
No-slip condition was employed at the channel wall for the flow fields. For the thermal boundary conditions of
the channel walls, isothermal wall condition was applied.
The serpentine channel considered in the computation was the one in which the single curve unit illustrated inFig. 3 was connected periodically in streamwise direction. For numerical simplicity, the serpentine channel isassumed to be considerably long and the region to be calculated is located at a position with a sufficiently large
distance from the inlet and outlet. Therefore, periodic boundary conditions were applied to the flow and thermal
fields. The scheme proposed by Patankar et al. [20] was applied in the present computation to solve the flow
fields having periodic boundaries in the streamwise direction.
For the model of the viscoelastic fluid, the Giesekus model [21] was applied to constitutive equation.

5

IHTC15-9615
Table 4 Fluid properties applied to the computation.

U [kg/m ]cp [J/(kgK)]k [W/(mK)]

Ps [Pas]

P0 [Pas]

O [s]

1.30 u 103

2.74 u 103

0.448

0.10

0.53

3.20

3

y
R o =2 H

R i= H
T
O z

x

H
H

Fig. 3 Computational domain.
W  OW  D

O
W :W
Pa

P a J

(7)

W and J are the stress tensor and strain rate tensor, respectively. Pa is the additional viscosity which isdefined by Eq. (8) using the zero-shear-rate viscosity, P0, and Newtonian-solvent viscosity, Ps.

P0

Pa  Ps(8) shown in Eq. (7) indicates the upper convected derivative. In order to satisfy the principle of material
objectivity, the upper convected derivative was applied to the unsteady term of the equation.
In the curve channels, the second normal stress difference shows a considerable influence on the vortex structure
generated in the streamwise cross section. Therefore, in order to take the anisotropic motions of the polymers
into account, the Giesekus model, which includes the non-linear term W : W , is employed in this study. D is set
as D=0.5 in the computation. The parameter and condition of the computation are described in Section 4.2.
These values were applied to the computation on the basis of the measurements. It should be noted that thetemperature effects on the properties are not considered in the computation, and the value of therepresentative temperature is employed.
Further, in prior to the simulation for serpentine channel, computation employing the upper convected
derivative Maxwell and Phan-Thien-Tannar (PPT) models were carried out for simple shear flows to validate
the models for predicting the characteristics of the viscosity and first order normal stress differences inrelation to the shear rate. The results of these two models were compared with the Giesekus model. It
appeared that the Maxwell and PTT models showed a better performance to predict the viscosity distribution.
On the other hand, for the N1 distribution, Giesekus model predicted the characteristic well. As previously
mentioned, the generation of the secondary flow is considered to be one of the major factors for heat transfer
enhancement. Since the normal stresses influence these flow significantly, although the power law model
was used for presenting the measured fluid property, we have used the Giesekus model for computation.
To discretize the Navier-Stokes and energy conservation equations, QUICK (Quadratic Upstream Interpolation
for Convective Kinematics) method is employed as the difference scheme for the convective terms, and 4thorder central-difference scheme was employed for the diffusion terms. These higher order schemes were appliedin the present code to reduce the discretization errors, particularly numerical diffusion errors. Semi-Implicit
Method for Pressure-Linked Equations (SIMPLE) algorithm was applied to solve the pressure correction. To
solve the discretized algebraic equations, a linear-iterative method using the modified Tri-Diagonal Matrix
Algorithm (TDMA) combined with an Alternating Direction Implicit (ADI) method was employed.

4.2 Numerical Conditions The properties of the fluid were adjusted to those measured during theexperiment. The fluid density U, specific heat capacity cp, thermal conductivity k are shown in Table 4. Further,Ps, P0 and the relaxation time O, are also shown in the table. The definition of the Reynolds number Re,
Weissenberg Wi, and Dean number ND are the same shown in Eq. (6). And Re, Wi, and ND were set
respectively as Re=0.63, Wi=205, and ND=0.13 in the computation.

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IHTC15-9615
100

8

㻌

Num

12

PAAm(64): serpentine
PAAm(64): straightSucrose: serpentine
Sucrose: straight
Numerical [22]

f

16

10

4
0

1

2

PAAm(64): serpentine
PAAm(64): straightSucrose: serpentine
Sucrose: straight
Modified value of 16 /Re*

3

1

0.5

5

Re

Re*

(a) average Nusselt number, Num(b) friction factor, f

Fig. 4 Relation between Reynolds number Re and (a) average Nusselt number Num and, (b) friction
factor f : comparison of serpentine and straight channels. Results of PAAm solution of 64wt%, sucrose
solution, and numerical results for Newtonian fluid laminar flow in straight channel are shown.

Num

12

SucrosePAAm(64)
PAAm(60)
PAAm(57)
Numerical [22]

16

8
4
0

PAAm(64)
PAAm(60)
PAAm(57)

12

Num

16

8
4

1

2

3

4

5

0

6

20

40

Re

60

80

100

120

140

Wi

(a) Num vs Reynolds number(b) Num vs Weissenberg number

Fig. 5 Relation between (a) Reynolds number Re and average Nusselt number Num and, (b) Weissenberg number
Wi and Num: comparison of PAAm(57) ~ (64) cases showing PAAm fluids with different sucrose concentrations.
Results of sucrose solution and numerical of Newtonian fluid is also shown for comparison.

5. RESULTS AND DISCUSSION
5.1 Heat Transfer and Pressure Loss Measurements Figure 4-(a) shows the relation between the
average Nusselt number Num and Reynolds number Re. Num of the PAAm solutions (viscoelastic fluids) inthe cases of the serpentine and straight channels are denoted by □ and ×, respectively. The sucrose cases(Newtonian fluids) are depicted by ○ and 㸩. The Nusselt number shown by the solid line are the values ofnumerical simulation of steady Newtonian fluid flow in a square duct where the developing region of thethermal boundary layer is taken into account [22]. Further, the influences of the temperature distribution
formed in the channel cross-section on the fluid properties are also taken into consideration [23].
As Re increases, Num of the sucrose case slightly increases due to the influence of the entrance region of thethermal boundary layer. The result agrees excellently well with the numerical values showing the validity ofthe present measurement. In the sucrose case, Num of the serpentine channel is nearly the same as those of thestraight channel case. This is because the Dean number in the present flow condition is small (ND ~ 1) and,thus, no noticeable secondary flow due to the centrifugal force is generated. The values of PAAm case of theserpentine channel agree with the numerical ones of the Newtonian fluid at Re < 0.4. However, as Re

7

2

2

1

1

1

0

0

-1

-1
-2

-1

0

x/Dh

1

2

(a) uT / U m (sucrose case)

0

-1

-2

0 0.5 1 1.5 2-2-2

0 0.5 1 1.5 2

-2

-1

0

x/Dh

1

2

(c) uT / U m (PAAm, Re=0.5 case)

1

1

1

y/D h

2

0

-2

-2

-0.5-0.3-0.1 0.1 0.3 0.5

-2

-1

0

x/Dh

1

2

(b) u r / U m (sucrose case)

0

x/Dh

1

2

0

-1

-1

-1

-1

(e) uT / U m (PAAm, Re=1.2 case)20

0 0.5 1 1.5 2

-2

2

y/D h

y/D h

y/D h

2

y/D h

y/D h

IHTC15-9615-2-0.5-0.3-0.1 0.1 0.3 0.5

-2

-1

0

x/Dh

1

2

(d) u r / U m (PAAm, Re=0.5 case)-0.5-0.3-0.1 0.1 0.3 0.5

-2

-1

0

x/Dh

1

2

(f) u r / U m (PAAm, Re=1.2 case)Fig. 6 The distributions of the time mean velocities at x-y plane of z = 0 in the cases of sucrose solution(Re =3.9, ND=3.2), and PAAm solutions (cases of Re =0.5, Wi=98.4, ND=0.4 and Re =1.2, Wi=189,
ND=1.0) measured at the second curvature of the serpentine.
increases, Num of PAAm solution increases markedly and shows higher value compared with the sucrose
case. This result shows that heat transfer can be enhanced by using the viscoelastic fluid and serpentine
channel.
In the case of straight channel, Num of PAAm case increases as Re increases, and takes larger values than the
numerical results. However, the values are much smaller than the case of serpentine channel. As described in
the work of Hartnett & Kostic [1], secondary flow can be generated in the straight channel due to the squarecross-section and the viscoelasticity of the fluid, and enhanced the heat transfer in the channel. However, theflow structures in the straight and serpentine channels are different from each other due to the difference ofthe channel curvature. As will be discussed in Sections 5.2 and 5.3, a pair of strong longitudinal vortex-likesecondary flow was generated in the serpentine channel case. Further, due to the increase of the flow
instability at the curved part, and the effect of the inflection point at which the curvature changes its direction,
the flow becomes unsteady. These differences of the flow characteristics led to higher heat transfer
enhancement in the serpentine channel case compared to the straight one.
Figure 4-(b) shows the relation between the Fanning’s friction factor, f = Dh'P/2UUm2L, and modified
Reynolds number, Re*. Re* is defined by Eq. (5).

Re*

UU m2n Dhn
8n1 K B  A / nn(5)

A and B are the geometrical constants. The value n is referred to the one used to express P as a function of J .
Both values of the serpentine and straight channels of PAAm and sucrose cases are plotted in Fig. 4-(b). The
solid line drawn in Fig. 4-(b) is the friction factors of steady flow in a square duct theoretically obtained by
Kozicki et al. [24]. The influences of the temperature distribution are considered for the theoretical values inthe same way as applied to the Nusselt number.

8

IHTC15-9615
Um
0.5

z/D h

0.25

0

-0.25

-0.5
2.0

(a) Re=0.5

1.5

r/D h

(b) Re=1.2

1.0(c) Re=2.2

Fig. 7 Velocity vectors of the PIV measurement on the cross-sectional plane (y-z plane) at T= 78q in
the cases of Re =0.5, 1.2 and 2.2.

(a) u 'T 2 / U m (Re=0.5)(c) u 'T 2 / U m (Re=1.2)(b) u ' r 2 / U m (Re=0.5)(d) u ' r 2 / U m (Re=1.2)

Fig. 8 The distributions of the fluctuation intensities of the velocities at xy plane of z=0 in the cases ofsucrose solution, and PAAm solutions (Re =0.5 and 1.2).
In the case of sucrose solution, f of the straight and the serpentine channels agree well with the theoreticalone (solid line). This shows that the flow in the sucrose solution case is steady and no considerable
secondary flows exist even in the serpentine channel case. In the PAAm case, the values of f also agree wellwith the line at Re* < 0.4. However, as Re* increases, the experimental values are located above the line.
Therefore, other factors in addition to the shear thinning effect exist in the serpentine channel case toincrease f. This is believed to be due to the transition of the flow from steady state to unsteady one, and thegeneration of secondary flows as mentioned previously.
In comparison with PAAm case of the straight and serpentine channels, the values of the straight channel aresmaller than those of the serpentine channel. The values are, however, larger than the theoretical ones.
Therefore, even in the straight channel case, secondary flows with a magnitude smaller than the serpentine
channel case can be generated. This is consistent with the results of the Num distribution.

9

IHTC15-9615
Figure 5-(a) shows the influences of the sucrose concentration of the PAAm solution on the relationshipbetween Num and Re in the case of serpentine channel. Since the viscosity largely differs among these threesolutions with different sucrose concentrations, the Num distributions show different tendency against Re.
Namely, the Reynolds number at which Num starts to increase, decreases with an increase in sucrose
concentration. Thus, Num cannot be correlated simply with the Reynolds number only, which implies that theflow and heat transfer characteristics of the PAAm solution could rely on its viscoelastic properties.
Figure 5-(b) shows the relation of Weissenberg number Wi and Num. One can see that a good agreement isobserved among the three different concentration cases. Weissenberg number is well known to represent the
influence of the fluid viscoelasticity on the flow characteristics, particularly the normal stress differences.
Although not shown here, the relation between Wi and the first normal stress difference N1=W11W22
measured by using the rheometer for each PAAm solution showed that N1 starts to increase at Wi ~ 30, thevalue of which would correspond to the one at which Num starts to increase in Figures 4-(a) and 5-(a).
Further, N1 of the PAAm(64), (60) and (54) solutions showed similar values particularly in the area of Wi <30. These results showed that Wi has a correlation with N1, and indicted that the normal stress, which
increases as Wi increases, is one of the main reasons for triggering the unsteady flows, generating thesecondary flows, and enhancing the heat transfer.
As Wi increases, the dispersion of Num increases in Fig. 5-(b). The reason for this is mainly considered to be
twofold. The first reason is the dispersion of N1 which becomes significant as Wi increases. The second oneis the influences of the increase of the Reynolds number. As Re increases, the viscous diffusion decreases
and the vortices generated in the channel are convected downstream. Under such situation, the normal stress,
i.e. Weissenberg number, might not be the only dominant factor to govern the flow structure in theserpentine channel.

5.2 Velocity Fields (PIV Measurements) Figures 6 (a)(f) show the distributions of the streamwise(circumferential, uT) and spanwise (radial, ur) components of the normalized time-mean velocities in the xyplane at z = 0mm. In Fig. 6, (a) and (b) show the results of the sucrose case of Re=3.9, and (c)(f) show thoseof PAAm case (viscoelastic fluid) at Re=0.5 and 1.2
In the sucrose case, the flow is laminar and a parabolic shape of the uT distribution is observed along thestreamwise direction. Focusing on the spanwise position at which the maximum value of uT takes place, it islocated relatively closer to the inner wall. As mentioned in Section 5.1, the Dean number of the flow is small
and no conspicuous effect of the centrifugal force appears. On the other hand, in the serpentine channel case,the flow will tend to flow through the channel with the shortest distance in the streamwise direction. Thisfeature can be seen in the spanwise velocity ur distribution also, the value of which is not zero in the area of
the inflection point (T = s90o) where the curvature of the curve switches its direction. This is one of thecharacteristics of the serpentine channel flow. However, it is limited only in a small area and does notproduce any large fluctuation or secondary flows.
On the other hand, in the PAAm case, the position at which the streamwise velocities uT becomes maximum
is located near the outer wall of the curve. And in the downstream of the inflection point T = 90o ~ 75o),the distributions of uT varies, and ur shows positive values. These results indicate that the flow flows fromthe outer side of one curve to the outer side of the next curve. Further, as Reynolds number increases from
Re = 0.5 to 1.2, the deviation of the flow in the spanwise direction and the values of ur increase.
Fig. 7 shows the time-mean distribution of velocity vectors in the streamwise cross section of T = 78o planein the PAAm case. This is approximately the location at which ur takes its maximum value.
A large longitudinal-vortex-like secondary flow can be found in the whole cross section in the case of (a) Re= 0.5. However, in the case of (b) Re = 1.2 and (c) Re =2.2, a pair of vortices is observed in the figure. And
in the center of the channel height where the two vortices merge, flow with large velocity heading toward theouter wall is generated.
Fan et al. [7] reported that for viscoelastic flow in a curved duct, circumferential normal stresses, WTT, areproduced with particularly larger values found in the area adjacent to the top and bottom walls of the channel.
This normal stress drives the flow located close to both walls toward the inner direction of the curve, andenhances the generation of the secondary flows.

10

IHTC15-9615

y/H

3

Nu

0 2 4 6 8 101214161820

flow

2

1

0

-1

0

x/H

1

2

3

4

5

-1

-2

(a) top (bottom) wall
2

2
1

Nu

0 4 8 12 16 20 24 28 32 36 40

z/H

z/H

Nu
flow

0 4 8 12 16 20 24 28 32 36 40

flow

0

0
-90

1

0

90

180

T>@q(b) sidewalls (outer to inner walls)

270

-90

0

90

180

T>@q(c) sidewalls (inner to outer walls)

270

Fig. 9 Nusselt number distributions of top (bottom) wall and sidewalls (Re=0.63 and Wi=205).
Considering their results and those of the present numerical simulation which will be discussed shortly, theflow and vortex observed in Fig. 7-(a) are believed to be in transition state. Namely, due to the above normal
stresses, a single vortex rotating in the clockwise or anti-clockwise directions is generated. As Re increases,
the centrifugal force and WTT, which is strongly related to the shear rate, increases. This enhances thegeneration of the pair of vortices, and thus, two vortices appear in Figs. 7-(b) and (c). These vortices will
increase the radial velocity and accompany larger deviation of the main flow toward the outer side of thecurve as shown in Figs. 6-(e) and (f).
Another reason why strong vortex-like secondary flows appears in the downstream of the inflection point isbecause radial velocities increase in this region due to the main flow flowing from the outside of one curveto the outside of the next curve mentioned previously. The radial velocity increases and enlarges the apparent
vortex-like secondary flow in this region.
Figure 8 shows the distributions of the normalized fluctuation intensities, u '2 / U m , where u ' =u  Um , of
PAAm case at Re 0.5 and 1.2. Although not shown here, the fluctuation intensities of sucrose solutions at
Re=3.9 were nearly zero because the flow is steady. uT '2 / U m and u r '2 / U m of PAAm case are large especiallyin the downstream of the inflection point, the position at which the strong secondary flows exist. Further,larger fluctuation intensities are observed in the Re = 1.2 case compared to the Re =0.5 case. These features
are believed to be due to the vortex structure shown in Fig. 7.
In the case of Re=1.2 and 2.2 shown in Figs. 7-(b) and (c), a pair of vortices is generated and the radial
velocity of the flow located at the center of the channel height is large. If the flow and the vortices are
perturbed, the apparent flow fluctuation intensities will increase as the locations of the vortices varies. On theother hand, in the Re = 0.5 case, the influences of a vortex on the fluctuation intensities in the area located at
the center of the channel height should be small: as the radial velocity is small at the center of the vortex, the11

IHTC15-9615

0

1.6

r/H

1.8

2

1

(a) T= 90q
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

0.2

z/H

1.4

1.6

r/H

1.8

2

1

0.35 uT,m

0

-0.2

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

0.2
0

1.6

r/H

(e) T= 0q

1.8

2

1

1.4

1.6

r/H

1.8

2

0.35 uT,m

1.4

1.6

r/H

1.8

2

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

0.2
0

1

(f)T=q

1.2

1.4

1.6

r/H

1.8

0.35 uT,mTT [Cq]

0.4

2

TT [Cq]

0.4

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

0.2
0

-0.2

-0.4
1.2

1

(d)T= 30q

-0.2

-0.4
1.4

0

-0.4
1.2

TT [Cq]

-0.2

-0.4

0.2

(c)T= 60q

0.4

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

-0.2

-0.4
1.2

TT [Cq]

0.4

1.2

0

(b)T=78q

z/H

0.35 uT,m0.2

TT [Cq]

0.4

z/H

1.4

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

-0.2

-0.4
1.2

0.35 uT,mTT [Cq]

0.4

z/H

0.2

-0.2

-0.4

1

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

z/H

z/H

0

-0.2

0.35 uT,mTT [Cq]

0.4

z/H

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

0.2

1

0.35 uT,mTT [Cq]z/H0.35 uT,m
0.4

-0.4
1.2

1.4

1.6

r/H

1.8

(g)T= 60q

2

1

1.2

1.4

1.6

r/H

1.8

2

(h)T= 0q

Fig. 10 Velocity vectors and dimensionless temperature contours at the cross-sectional plane (yz plane).
influence of the perturbation of the vortex on the fluctuation intensities is small. It is expected, however, thatthe fluctuation intensities in the area adjacent to the top and bottom walls should be large.
Since Wi increases as Re increases, the good correlation of Num and Wi shown in Fig. 5-(b) and thediscussion made above leads to the conclusion that as Wi increases, a single vortex is first generated. When
Wi is further increased, the flow develops to a pair of vortex-like secondary flow due to the normal stress.
The strong radial flow will convect the low temperature fluid located at the channel core to the sidewalls andenhance the heat transfer markedly. In addition to this, the deviation of the main flow to the outer side of thechannel curve, and the flow fluctuation increases the Num. Further discussion will be made for this matter in
the next section on the basis of the numerical results.

5.3 Numerical Simulation Results Figure 9 shows the local Nusselt number Nu distributions of (a) the top(bottom) wall and, (b) and (c) the sidewalls of one periodical unit. (b) shows the streamwise distribution startingfrom the outer sidewall of the first semi-circle curve to the inner sidewall of the 2nd semi-circle curve. (c) shows
the distribution of the opposite side (counter side) which starts from the inner sidewall to the outer sidewall.
In Fig. 9-(a), an area of large Nu is observed near the inner wall of the curve in the upstream region. Nu
gradually decreases in the downstream region, and is nearly unity at T > 0q. On the other hand, Nu increase in
the area near the outer wall of the curve as T increases downstream of T = 0q. This tendency can be observed
also in the Nu distributions of the sidewalls shown in (b) and (c). Further, a much greater value is obtained at the
sidewalls particularly in the area near the inflection point (T= 90q, 90q and0q).
Figure 10 shows the velocity vectors and dimensionless temperature TT = (TTw / TbTw) contour maps of thecross sectional planes. The streamwise locations of the cross sections are T= 90q, 78q, 60q, 30q,
0q,30q,60q and90q, starting from the inlet and ending at the outlet of the first semi-circle curve.
The velocity vectors show that a strong secondary flow of a pair of longitudinal vortices is generated in thechannel. It should be noted that such secondary flow was not observed clearly in computation of the Newtonian
fluid case since the Dean number is small as described in Section 5.1. Considering the stress distributions
obtained by the computation, which are not shown here, the secondary flows are attributed to the production of
the normal stress WTT mainly. Due to the high shear stress generated near the channel walls, the first order normal
stress differences increases markedly in that area. This normal stress difference will influence the flow fields as asource term with the order of (WrrWTT) /r . Since it is WTT which markedly increases, this will produce a driving
force to the fluid in the direction toward the center of the curvature. Hence, a pair of vortices with flow heading
toward the outer sidewall near the center of the cross section, the vortices similar to the one generated due to theeffect of the centrifugal force, can be formed.

12

IHTC15-9615

30

Nu (top wall)
Nu (in o out)
Nu (out o in)

25

15

㻌

NuT

20

10
5
0
-90 -60 -30

0

30

60

90 120 150 180 210 240 270

T [q]
Fig. 11 The distributions of the average Nusselt number of each cross-sectional position. (in o out) isfor the sidewall located at the side which starts from the inner wall and connects to the outer wall, andvice versa.
Table 5 Average Nusselt numbers of the top (bottom), sidewalls and overall surfaces.
Top (bottom) wall
Inner wall
Outer wall
Overall
Newtonian fluid
2.95
3.55
2.61
2.93
Viscoelastic fluid
4.66
10.21
8.54
6.88
At the inflection point (a) T= 90q, a strong cross flow heading toward the inner wall is observed. However, due
to the above normal stresses, the flow gradually weakens as Tincreases and in the area of T> 60q, a cross-flow
heading toward the outer wall is generated. As T further increases, the magnitude of the vortices increases. And
at (h) T= 90q, the strength of the secondary flow becomes maximum. Since the curvature changes its direction at
this inflection point (T= 90q), the flow tends to flow from the outer wall to the inner wall at the beginning of thenext semi-circle curve. Further, the flow in the serpentine channel has the feature that the main flow tends toflow from the inner wall of the first curve to the inner wall of the next curve which could be observed even in the
Newtonian fluid (sucrose) case. This flow will produce a cross-flow which enhances the secondary flow in theviscoelastic fluid case, too. These characteristics agree well with the one observed in the PIV measurement
discussed in Section 5.2.
The temperature distributions show that the position of the core of the low temperature fluid changes in the
spanwise direction as the flow flows along the channel. At (a) T= 90q, the low temperature fluid is located at
the area near the inner wall and is convected to the outer wall by the secondary flow mentioned previously. By
this motion of the fluids, Nu in the area adjacent to the inner wall gradually decreases at T< 0q, and Nu in thearea near the outer wall increases at T> 0q in the first curve of the channel. Near the inflection point, Nu
increases markedly at the outer sidewall as the flow impinges and sweeps the wall. Further, as the lowtemperature fluids flow toward the top and bottom walls along the sidewall, Nu in this area increase as shown in
Fig. 9-(a).
The distributions of NuT, which is the average value of Nu at the streamwise cross section of angle T, is shownin Fig. 11. The pattern of the NuT distribution (out o in) is the same as the one of NuT distribution (in o out),
and only the phase is shifted by 180q, the feature of which was observed in Figs. 9-(b) and (c) also. The averagevalue of the channel walls are summarized in Table 5. The tendencies previously mentioned can be observed in
the distributions of NuT. Considering the NuT of top and sidewalls, the overall of NuT appears to take a large
value in the area near T= 90q while at T = 0q the values become small. Therefore, an effective heat transfer
enhancement is obtained near the inflection point.
For the Num shown in Table 5, it should be noted first that in the case of Newtonian fluid, Num of the inner andouter walls are not equal, and the inner wall shows a greater value. This is due to the fact that in the serpentine
channel, the velocity profile is not fully symmetric and the main flow tends to flow through a shorter passage,closer to the inner wall of the curve. This heat transfer characteristic of the sidewalls is an opposite one

13

IHTC15-9615
compared to the one when the heat transfer is influenced by the Dean flow. In the viscoelastic fluid case, Num isnearly two times larger at the sidewalls compared with the top (bottom) wall. Further, Num shows a greater value
at the inner wall than the outer wall. Therefore, although a pair of vortices is generated and a radial flow heading
toward the outer sidewall appears in the curved channel, the effective heat transfer is actually obtained at the
inner sidewall owing to the geometry of the serpentine channel.
When the Reynolds number was increased in the computation, the magnitude of the secondary flow and thenumber of the vortices changed. This influenced the temperature distribution in the cross section dramatically:
the fluctuation amplitude and the number of the cores of the low temperature region moving in the spanwise
direction increased as Re increased. This is related to the switching of the curvature in the serpentine channel.
Further, although a very small perturbation was observed in the flow, unsteady flow was not observed in thecomputation even if the Re, Wi and other parameters were varied for several conditions. These two factors will
largely influence the relationship between the Re (or Wi) and Nu, and further improvement and discussion on thecomputation should be made as future works.

6. CONCLUSIONS
The flow and heat transfer characteristics of viscoelastic fluid in the serpentine channel at Re~1 wasinvestigated experimentally and numerically.
The average Nusselt number Num increased in the case of viscoelastic fluid (PAAm solution) in theserpentine channel compared with the case of PAAm solution in the straight channel and the case of sucrose
solution (Newtonian fluid) in the serpentine channel.
Investigating the influences of the fluid properties on the heat transfer performances by changing the sucrose
concentration of the PAAm solution, it was difficult to obtain a good correlation between the Reynolds
number and Num. In this case, a good correlation was achieved by using the Weissenberg number instead of
Re. This was due to the fact that Wi represents the magnitude of the normal stress difference of the fluid, andthat a longitudinal-vortex-like secondary flow was generated by these normal stresses and enhanced the heattransfer at the channel walls.
PIV measurements showed that at low Re condition, a single longitudinal vortex was generated, and as Re
increased it developed to a pair of vortices in the case of the viscoelastic fluid flow in the serpentine channel.
This was due to the increase of the normal stress produced near the top and bottom walls of the channel as
Re increased. The vortices produced a deviation of the main flow toward the outer sidewall, and the variationand fluctuation of the vortices’ shape and position, which was observed particularly in the area near the
inflection point, led to the strong unsteady flow generation in the channel.
The results obtained by the numerical computation confirmed the above conclusions achieved by theexperiment. A pair of longitudinal-vortices-like secondary flow was generated in the channel owing to thenormal stress differences which drove the fluid located close the top and bottom wall toward the center of thechannel curve. The secondary flow convected the lower temperature fluids in the spanwise direction which
effectively enhance the heat transfer at the sidewalls particularly the inner side.

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

NOMENCLATURE
Dh
f
Gz
k
L
ND
Nu
NuT

Num
Re
Ri
Ro
r, T, zTTbhydraulic diameter(m)
Fanning friction factor(-)
Graetz number(-)
thermal conductivity( J/Km )length of the serpentine channel (m)Dean number(-)
Nusselt number(-)
cross sectional average Nu(-)

14overall Nu
Reynolds number
inner radius

(-)
(-)

outer radius

(m)
(m)

cylindrical coordinates
temperaturebulk mean temperature(m, q, m)
(Cq)
(Cq)

IHTC15-9615
Tw
TT
Um
x, y, z
Wi

D
O

P
Pa
Ps
U

wall temperature(Cq)
dimensionless temperature(-)
mean velocity(m/s)
Cartesian coordinates(m)
Weissenberg number(-)
coefficient for nonlinearity rate ( - )relaxation time(t)zero-shear-rate viscosity
additional viscosity
viscosity of solvent
density(Pas)(Pas)(Pas)(kg/m3)

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