Proceedings of CHT-21
ICHMT International Symposium on Advances in Computational Heat Transfer
August 15 - 19, 2021, Rio de Janeiro, Brazil
CHT-21-270

Viscoelastic Fluid and Dean Flow Effects on Flow and Heat Transfer Characteristics ofSerpentine Channel
Kazuya Tatsumi *,§, Yousuke Tanaka*, Reiko Kuriyama* and Kazuyoshi Nakabe**

Department of Mechanical Engineering and Science, Kyoto University, Kyoto, Japan§Correspondence author. Fax: +81 75 383 3608 Email: tatsumi@me.kyoto-u.ac.jp

ABSTRACT Viscoelastic fluid can increase the heat transfer coefficient of the low Reynolds numberflow in the serpentine channel by generating unsteady and secondary flows. A pair of longitudinalvortices which is similar in form to Dean vortices is generated mainly due to the first normal stressdifferences. These vortices carry the low temperature fluid toward the channel sidewalls and increase theheat transfer coefficient. However, each contribution of the viscoelastic effect and the centrifugal force(Dean vortices) on the secondary flow and heat transfer characteristics are not clearly investigated anddistinguished, especially in the case of serpentine channel. In this study, we conducted three-dimensional
numerical computation on viscoelastic fluid in serpentine channel using Oldroyd model. The effects of
the Reynolds number (Dean number) and Weissenberg number on the flow structure and heat transfer
coefficients were evaluated. In the results, the viscoelastic flow and Dean flow gave certain similar
aspects. However, the viscoelastic fluid generated stronger secondary flows and producing higher heattransfer coefficient for high Weissenberg number conditions. Further, we showed that the first normal
stress difference plays an important role to characterize the flow, and good agreement can be obtained
between the measurement and computation by introducing the dimensionless normal stress difference.
INTRODUCTION
Viscoelastic fluid is a non-Newtonian fluid that exhibits elastic properties in addition to viscosity. The
elasticity will produce normal stresses in the flow, when shear and stress are applied to the fluid. This
force changes the main flow structure, generates secondary flows, and increases the flow instability. The
unsteady and secondary flows enhance the fluid mixing and heat transfer, and therefore, studies onviscoelastic fluids in various types of channels have been made by many researchers. Among these,channel with curvature is known to produce longitudinal vortex-like secondary flow similar to Dean
vortex due to the normal stress differences of the viscoelastic fluid [e.g., Joo et al. 1994, Norouzi et al.
2009, Helin et al. 2009]. These vortices also appears in the serpentine channel, and furthermore, theinflection point of the curvatures increases the strength of the radial flow and the flow instability[Burghelea et al. 2004, Tatsumi et al., 2011, Zilz et al. 2012]. Thus, the heat transfer coefficient increases
especially at the outer sidewalls of the serpentine channel [Tatsumi et al., 2014, Abed et al. 2016, Tatsumi
et al. 2019]. The flow in these channels is a mixture of the viscoelastic fluid flow and the Dean vortices.
However, the difference between the contribution of the two flow on the heat transfer characteristics,
and moreover, the combined effects are not clearly evaluated. In this study, we conducted a threedimensional computation on flow and temperature fields of the viscoelastic fluid in serpentine channeland discussed the effect of the Dean vortices and the viscoelastic fluid on the flow and heat transfer
characteristics varying the Reynolds number and Weissenberg number. We will first present theTable 1 Fluid properties and conditions of the computation and experiment
Viscosity Relaxation
0 [Pas] time  [s]
0.53
0.056
0.28
3.73

Computation
Measurement

ReND

WiPr

N1*

0.5-28
1.0-2.1

0.28-16
0.57-1.2

0.5-6
6.0

3.2×103
3.2×103

6.5-234
6.4-260

20yHNumHxO z
R o =2 H

Calculation
Experiment

16
1284

Ri=H010Figure 1 Computational domain.

100N*1Figure 2 Nusselt number and dimensionlessnormal stress differences N1*.

(a) z/H=0.5(b)  = −60(c)  = 0(d)  = 60
Figure 3 Contours of the streamwise velocity u and velocity vectors on the r-z and r- planes.

(a) Case 01O02(b) Case 01O04(c) Case 01O06
Figure 4 Contour of the dimensionless temperature  on the r-z plane at  = −60.
comparison of the numerical results with the measurements for validation. The dimensionless value ofthe first normal stress difference is introduced here to adjust the conditions of the computation andmeasurement. The effects of the Dean flow and fluid viscoelasticity on the flow and heat transfer
coefficients will, then, be presented; the flow structure and Nusselt number distribution along the curved
channel is compared, and how the combined effect can increase the heat transfer performance are shown.
METHODS
Numerical Procedure Unsteady and three-dimensional form of the continuous equation, Navier-Stokes
equation, and energy conservation equation are solved in the present computation. These equations arediscretized by applying the QUICK scheme to the convection terms and 4th order central difference
scheme to the diffusion terms. The pressure term was solved using the SIMPLE algorism. Theviscoelasticity of the fluid is applied to the computation by employing the Oldroyd-B model as theconstitutive equation and adding the additional stress term to the Navier-Stokes equation. Oldroyd-B

model is a fundamental model derived by also considering the behaviour of the viscoelasticity inmolecular level. We used this model in order to simplify the problem which is discussed and comparewith the measurements. The constitutive equation was discretized by applying 1st order upwind scheme
to the convection term, and 2nd order central difference scheme to other derivatives.
Computational domain and conditions Fig. 1 (a) shows the computational domain. The serpentine
channel consists of two semi-circular parts with curvature of opposite direction. The channel has a squarecross section of 5 mm (=H) on side, and the inner and outer radii of the semicircle part are Ri/H = 1 andRo/H = 2, respectively. Non-slip and constant temperature conditions are applied to the channel walls.
Periodic boundary condition is applied to the streamwise boundaries. The fluid properties anddimensionless parameters of the computation are shown in Table 1. Grid dependency test was carriedout using grid density of 42×124×42, 82×239×82 and 164×479×164. The variation of the average heattransfer coefficient among these conditions were 12.3% and 2.95%, and of the pressure loss were 2.71%
and 0.974%, respectively. Therefore, we chose the grid density of 82×239×82 in the main computation.
RESULTS and DISCUSSON
Comparison with measurement The average Nusselt number Num is compared with the authors’
measurement [Tatsumi et al., 2014] for validation. The condition of the computation and measurement
are shown in Table 1. The Reynolds number Re of the experiment and computation are similar while theWeissenberg number Wi differs largely. This is due to the difference in the relaxation time . Therefore,
we will introduce another dimensionless parameter related to the normal stress. The term expressed as(-rr)/r, which appears in the equation of motion of the radial direction as the normal stress difference
N1, plays an important role in the generation of the secondary flow by producing force in the radial
direction near the top and bottom walls and driving the fluid toward the central direction of the curvature.
We can derive the following dimensionless parameter by normalizing the equation of motion applying
H and the streamwise mean velocity Um as the characteristic length and velocity.
𝜌𝐻 2

𝑁1∗ = 𝜇2 𝑁1(1)0

We will rearrange the relationship of the Num with the Reynolds number or Weissenberg number to thethe dimensionless first normal stress difference N1* as shown in Fig. 2. The results of the computationand measurement show a reasonable agreement, which indicates the validity of the present computationand that N1* can be an important index to represent the flow and heat transfer characteristics. Difference
between the computation and measurement is found in the region of large N1*. This can be attributed tothe shear-thinning effect of the fluid in the experiment, which decreases the normal stress in the nearwall region significantly compared to the effect on the Reynolds number and viscosity in Eq. (1).
We now show some of the computational results which represent the viscoelastic effect on the flow andtemperature field of the serpentine channel. Fig. 3 shows the streamwise velocity u distributions on ther- plane of z/H=0.5, and r-z planes of =−60°, 0° and 60° in the case of high Weissenberg number andReynolds number condition; Wi=5.0 (N1*=908) and Re=28. Fig. 4 shows the dimensionless temperature
T*=(T-Tw)/(Tb-Tw) at =60° for the cases of Wi=2.0, 4.0 and 6.0 where Re is constant as Re=1.0.
In Fig. 2, Num is constant at low N1* showing the steady state and small secondary flow. As N1* increases,
Num increases due to the generation of pair of longitudinal vortex like secondary flows. Generation ofthese vortices can be observed clearly in the velocity distribution shown in Fig. 3 (b) and (d). The flowcore region of low temperature is carried toward the outer sidewall, and enhance the heat transfer
especially at the outer wall. The secondary flow is generated by the normal stress difference which givesa large value near the top and bottom wall due to the high shear stress of the flow. Therefore, the strengthof the secondary flow increases as Wi (or N1*) increases, and the low temperature region largely moves
in the spanwise and height directions of the channel as shown in Fig. 4.

Similar trend was observed in the Dean flow case (Newtonian fluid); Nu increased at the outer wall forlarge Dean number flow. However, the Dean vortices developed in the curve channel and converges tothat of circular channel. Therefore, the heat transfer coefficient gradually increased at the outer wallalong the channel. On the other hand, in viscoelastic fluid flows, the location of the high velocity region
not only affected the centrifugal force, but also the normal stress difference mentioned above. The force
generated at the outer wall and the top-bottom walls locally generates a secondary flow which moves thefluid of high velocity first toward the top-bottom walls and then back to the center of the cross-section.
This produced the redevelopment effect in the thermal boundary layer. Further, a strong flow crossing
the channel in the spanwise direction in the downstream region of the inflection point of the serpentine
channel (=−60°) was generated in the viscoelastic fluid case. Thus, the viscoelasticity enhanced the heattransfer by increasing the strength of the secondary flow, and also by keeping the low temperature fluid
region sweep the channel walls. Detail of these effects in relation to the Wi, Re, and N1* will be shownin the presentation, especially focusing on the difference of the viscoelastic fluid and Dean flow.
CONCLUSION
The flow and heat transfer characteristics of viscoelastic fluid in serpentine channel was numericallyinvestigated in this study. We have shown that the velocity distribution and the average Nusselt number
agrees well with the measurements especially when considering the dimensionless normal stressdifferences. Further, the independent and combined effects of the Dean vortex and the fluid
viscoelasticity will be evaluated in the presentation by mapping the characteristics against the Reynolds
number (Dean number) and Weissenberg number. Viscoelastic effect can increase the strength of thesecondary flow, and furthermore, enhance the movement of the flow core region which leads to theredevelopment of the thermal boundary layer and further increase of the overall heat transfer coefficient.
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