12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

MODIFICATION OF TAYLOR-COUETTE TURBULENCE IN AN
ASYMPTOTIC ULTIMATE REGIME BY THE VISCOELASTICITY OF
DILUTE SURFACTANT SOLUTIONTaisei Hayama
Yasufumi Horimoto
Faculty of Engineering
Graduate School of Science and Technology
Hokkaido University
Tokyo University of Science
060-8628, N13 W8, Sapporo, Hokkaido, Japan
horimoto@eng.hokudai.ac.jp

Homare Okuyama
Graduate school of Engineering
Hokkaido University

the radius and angular velocity of the inner (outer) cylinder，and ν is the kinematic viscosity of the ﬂuid. d = ro − ri is thewidth of the gap between the cylinders. This is an great advantage from an experimental view point because controlling therotations by motors is easy.
In recent years, quite high Reynolds-number TaylorCouette turbulence has been investigated by experimentallyand numerically where ﬂow has an analogy of turbulent
Rayleigh-Bénard convection with a high Rayleigh number
with respect to the angular momentum transport [see a recent
review by Grossmann et al. (2016) and references therein].
That is, a power law

ABSTRACT
We experimentally investigate the modiﬁcation of thewall frictional drag and mean ﬂow structures in high Reynoldsnumber Taylor-Couette turbulence due to the viscoelasticity
of a dilute surfactant solution. The examined turbulence isin an asymptotic ultimate regime realized by utilizing a verylarge facility with co- and counter-rotation of the inner andouter cylinders. We experimentally observe the drag reduction for the counter-rotation cases in a higher Reynolds number
regime for the ﬁrst time while, for the co-rotation, a remarkable drag enhancement as reported by recent numerical studies
for the pure inner cylinder rotation [Liu & Khomami, Phys.
Rev. Lett., (2013); Song et al., J. Fluid Mech., (2019)]. Thisdrag modiﬁcation is well understood by deﬁning the Reynolds
number based on the bulk shear rate and the viscous timescale, which suggests the the bulk shear rate normalized by
the relaxation time of the viscoelasticity (i.e. the Weissenberg
number) is an important parameter of the drag modiﬁcation.
For the mean ﬂow structure, we ﬁnd a non-monotonic proﬁleof azimuthal velocity and a constant angular velocity region(i.e. quasi solid-body rotational ﬂow) in a bulk region. The
angular velocity also shows a clear correlation with the dragmodiﬁcation.

Nuω ∼ Taβ(2)holds in the angular momentum transport (Eckhardt et al.,2007). Here, Ta is the Taylor number

Ta =
=

1
4



1+η
√
2 η

4

(ro + ri )2 (ωi − ωo )2 d 2
ν2(1 + η )6(Rei − η Reo )2
64η 4

(3)

INTRODUCTION
Taylor-Couette ﬂow, ﬂow between two concentric cylinder independently rotating, is a one of fundamental ﬂow systems. Utilizing this ﬂow, one can study the dynamics of ﬂuid
ﬂow including ﬂow instability, turbulence transition and fully
developed turbulence [see a famous ﬂow diagram in Andereck
et al. (1986)]. This ﬂow is controlled only by two dimensionless parameters, namely the Reynolds numbers

Rei =r i ωi dνand Reo =r o ωo dνand Nuω is the angular Nusselt numberJω
Nu = ω
Jlam

ωi − ωoωJlam= 2ν ri2 ro2 2
ro − ri2!
(4)

respectively. J ω = T /2πρ L is the angular velocity current
ω is its laminar
across the gap between the cylinders and Jlam
value (ρ : the density of the ﬂuid). The regime where the scaling (2) holds is called an asymptotic ultimate regime. Typically, it appears for a sufﬁciently high Taylor-number regime,
Ta ≳ 109 (Grossmann et al., 2016). Incidentally, while β istheoretically predicted to be 1/2 as well as turbulent RayleighBénard convection, it has been shown that β ≈ 0.39 due to the(1)based on the rotations of the ﬂow for ﬁxed geometrical parameters, the radius ratio η = ri /ro and the aspect ratio Γ = L/d(L is the height of the cylinders). Here, ri (ro ) and ωi (ωo ) are1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
logarithmic correction from many previous studies (Kraichnan, 1962; Doering & Gibbson, 1995; Eckhardt et al., 2007).
In this study, we report the impact of the viscoelastic
property of the ﬂuid on Taylor-Couette turbulence in the ultimate regime. More concretely, we experimentally investigate
the modiﬁcation of the frictional drag and ﬂow structure by theaddition of a small amount of surfactant to turbulence of water.
It is well known that dilute surfactant solution has remarkableviscoelasticity and show large drag reduction compared withturbulence of water due to the suppression of small-scale turbulent eddies (Zakin et al., 1998; Li et al., 2012). However,
although this drag reducing effect is very attractive in termsof ﬂuid tranport, turbulence of a viscoelastic ﬂuid is hard tounderstand because of its peculiar phenomena (Lumley, 1969,
1973;Tabor & de Gennes, 1986; White & Mungal, 2008; Xi,
2019). Hence, researches to elucidate the physical mechanismof the drag reduction has widely conducted for fundamental
ﬂow systems such as pipe ﬂow, channel ﬂow, boundary layer,
etc. In this context, Taylor-Couette ﬂow is a good candidate
because this closed ﬂow system is useful to sustain turbulence
generated by strong shear and the wall frictional drag can bedirectly assessed from torque on the cylinders.
Nevertheless, a little suprisingly, previous studies of turbulent Taylor-Couette ﬂow of viscoelastic ﬂows is limited for
relatively low Reynolds number where ﬂow instability andearly transitions to turbulence are concerned (e.g. Larson et al.,
1990; Avgousti & Beris, 1993; Thomas et al., 2009; Dutcher& Muller, 2013). What is worse, direct numerical simulations(DNS) for Rei = O(103 ) and Reo = 0 shows that the viscoelasticity triggered a pronounced drag enhancement rather thandrag reduction (Liu & Khomami, 2013; Song et al., 2019).
Note that their DNS did not reach the ultimate regime [seeEq. (3)]. In other words, the modiﬁcation of Taylor-Couette
turbulence in the ultimate regimes by the viscoelasticity has
not been revealed yet.
In the present study, for the ﬁrst time, we conducted aseries of expriments both for co- and counter-rotation of thecylinders to answer the following question by utilizing a large
experimental facility: can the drag reduction occur in the ultimate turbulence of Taylor-Couette ﬂow? We also shows themodiﬁcation of the ﬂow structure obtained by mesurements ofvelocity ﬁeld and discuss the modiﬁcations of the ability ofangular momentum transport.

variable on its subscript. We also estimate the angular Nusselt number Nuω through ⟨T ⟩t by using J ω = T /(2πρ L) to
examine the angular momentum transport.
As mentioned above, the outer cylinder and its top surface
are made of transparent acrylic resin for ﬂow visualization. To
discuss the statistics in turbulent ﬂow, we measure the turbulent ﬁeld by particle tracking velocimetry (PTV) with the nearest neighbour algorithm. As a tracer particles for PTV, we use
silver-coated hollow glass beads (10 µm diameter). A planer
laser sheet is in the plane at mid-height of the inner cylinderand a high speed camera above the facility records the motionof the tracer particles visualized by the laser sheet.
We examine a dilute aqueous solution of a cationic surfactant, cetyltrimethylammonium chloride (CTAC. 320.00 g/mol)
as a viscoelastic ﬂuid as well as water as a Newtonian ﬂuid.
With counterions to reduce the charge on the surface of surfactant micellar structures, the structure can grow and the CTAC
solution shows remarkable viscoelasticity due to the existent ofvery large micellar network structure (Li et al., 2012). As thecounterion, we use sodium salicylate (NaSal. 160.11 g/mol).
The mass concentrations of CTAC and NaSal are both 100
ppm: the molar ratio of NaSal to CTAC is 2 where the micellar networks are stably exist in the solution. In this study,
we estimate the physical quantities of the CTAC solution by
those of water at the same temperature because measurementsof them are quite difﬁcult for such dilute solution.

RESULTS & DISCUSSION
Drag Modiﬁcation
In this study, we the rotational condition of the cylinders
by using the angular velocity ratio deﬁned asa=−

We use a quite large Taylor-Couette facility to sustain
very high Reynolds-number turbulence. It consists of two
cylinders independently rotated by stepper motors. The radius and height of the inner cylinder are 150 mm and 990 mm
while those of the outer cylinder are 205 mm and 1014 mm,
respectively. Thus, the radius ratio and the aspect ratio of the
facility are η = 0.732 and Γ = 18. The outer cylinder and its
top surface are made of acrylic resin for ﬂuid visualization andvelocity measurement (explained below).The end surfaces ofeach cylinder rotate together with the cylinders.
By using a torquemeter set on the shaft of the inner cylinder, the measured torque T on the inner cylinder can be converted into the wall shear stress. Based on this, we estimate
the frictional drag by the drag coefﬁcient deﬁned as⟨T ⟩t.
πρ (ri ωi − ro ωo )2 ri2 L(6)Note that a < 0 and a > 0 respectively represent the co- andcounter-rotation of the cylinder. Fig. 1 shows examples of
the torque measurement; the wall frictional eoefﬁcient C f asa function of the inner Reynolds number Rei for a = −0.33, 0,
and 0.33. It is clear that while the drag enhancement occur fora = −0.33, 0 [Fig. 1(a,b)], we can observe the speciﬁc switchof the drag modiﬁcation from the enhancement to the reduction for a = 0.33 [Fig. 1(c)]. That is, the drag enhancement
occurs for a sufﬁciently high Rei -region (Rei ≳ 4 × 104 ).
Here, it is worth mentioning that Fig. 1 shows an answer
to the question in the introduction. More concretely, the dragreduction in turbulent Taylor-Couette ﬂow by the viscosity canbe expected for a high Reynolds-number regime. To conﬁrm
this, we conduct the torque measurement for a wide range ofa; −0.5 ≤ a ≤ 0.5. The results are summarized in Fig. 2(a).
In this ﬁgure, the drag modiﬁcation is quantiﬁed by the dragreduction fraction

EXPERIMENTAL SETUP & METHOD

Cf =

ωo
.
ωi

DR =

C f ,w −C f ,s
× 100.
C f ,w(7)

Here, C f ,w and C f ,s are the drag coefﬁcient for water and thesrufactant (CTAC) solution, respectively.
First of all, from Fig. 2(a), we notice that DR is larger
for larger a, and the clear drag reduction is observed for thecases of counter rotation (a > 0) in a higher Rei -regime up
to 50%. On the other hands, for a ≤ 0, only drag enhancement (DR < 0) appears and it is more remarkable than the(5)Here, the bracket ⟨·⟩ represents average with respect to the212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
0.04

(b)

(a)

(c)

0.02

Cf

0.01
0.008
0.004
0.002
0.001
104

105 104

105 104

Rei

105

Rei

Rei

Figure 1. Wall frictional coefﬁcient C f as a function of the inner cylinder Reynolds number Rei for (a) co-rotation (a = −0.33), (b)pure inner cylinder rotation (a = 0), and (c) counter-rotation (a = 0.33). Black and red dots represent the cases for water and surfactant
solution (100 ppm), respectively.

100

(a)

(b)

0
−0.5
−0.4
−0.33
−0.2
−0.1
0
0.1
0.2
0.33
0.4
0.5

−100
−200

DR

−300
−400
−500
−600
−700
−800
50000

10000050000

Rei

100000150000Re

Figure 2. Drag reduction ratio DR as a fuction of (a) Rei and (b) Re for −0.5 ≤ a ≤ 0.5. The colors correspond the values of a asshown in (b).
0.5

above drag reduction. However, the Rei -dependence that DR
increases with Rei is common for all value of a. Hence, evenfor co-rotation (a < 0), the drag reduction can be expected ina much more high Rei -regime.

0.4

β

0.3

Next, we consider what parameter is suitable for the understanding of the drag modiﬁcation. Recalling the deﬁnition (6) of a, we notice that a characteristic shear rate γ̇ =(ri ωi − ro ωo )/d is higher for larger a. To make it clear, wedeﬁne another Reynolds number as(Ri ωi − Ro ωo )dRe == Rei − Reo .
ν

0.2
0.1
0

0.6 0.4 0.2

0

0.2

0.4

0.6

a

Figure 3. Scaling index β in the scaling law (2) of angularmomentum transfer as a function of the angular velocity ratioa. Black and red dots represent the cases for water and surfactant solution (100 ppm), respectively.

(8)

Note that Re−1 indicates the ratio of a time scale of the ﬂowγ̇ −1 to the viscous time scale d 2 /ν . We replot DR as a function of Re in Fig. 2(b). It is obvious that DR seems to be wellcollapsed on a curve as an envelop. Hence, this shows thatthe ﬂow time scale γ̇ −1 is an important parameter for the dragmodiﬁcation in viscoelastic Taylor-Couette turbulence. This
results is reasonable by noting that a dilute surfactant solution has single (or sometimes a few) relaxation time τ of theviscoelasticity (Zakin et al., 1998; Li et al., 2012). That is,
the Reynolds number Rec , which is Rec ∈ [5000, 7000] for our
experimental setup, where the drag enhancement turns to thedrag modiﬁcation is given a function of the Weissenberg number We = τ γ̇ .

Angular Momentum Transport
To discuss the modiﬁcation of angular momentum transport, we show the scaling exponent β in Eq. (2) for the various
values of a. The examined Reynolds-number regime shown
above corresponds to Ta ≳ 109 , which is the ultimate regimes,
and we conﬁrm the scaling law (2) indeed holds both for waterand the CTAC solution (ﬁgures are omitted).
For water, β reaches around 0.4 as well known by many
previous studies for a ≳ 0 (Grossmann et al, 2016). Incidentally, accoding to the discussion of Gils et al. (2012), β has the
maximum value around the bisection line between two neutral lines in Rei –Reo space, the pure outer cylinder rotation3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 20221

heuθ itθ

0.8

(a)

(b)

(c)

(d)

(e)

(f)

0.6
0.4
0.2
0
1

he
uθ itθ

0.8
0.6
0.4
0.2
0
0

0.2

0.4

0.6

0.8

1 0

0.2

0.4

re

0.6

0.8

1 0

0.2

0.4

re

0.6

0.8

1

re

Figure 4. Radial proﬁle of mean azimuthal velocity proﬁle ⟨euθ ⟩t θ for (a-c) Rei = 4.3 × 104 and (d-f) Rei = 7.0 × 104 . Black andred dots represent the cases for Newtonian ﬂuid (water) and viscoelastic ﬂuid (the CTAC solution), respectively. (a,d) Counter-rotation(a = 0.33), (b,e) pure inner cylinder rotation (a = 0), and (c,f) co-rotation (a = −0.33).
0.5

0.3

DR

he
ωbulk itθ

0.4

0.2
0.1
0
0

50000

100000150000

100
0
−100
−200
−300
−400
−500
−600
−700
−800

0

0.1

ebulk ⟩t θ
Figure 5. Re-dependence of the angular velocity ⟨ωin the quasi SBR region for a = ±0.5, ±0.33, ±0.1, and 0.
Colors are the same as those shown in Fig. 2.

Finally, we discuss the modiﬁcation of mean ﬂow structure. Figure 4 shows the radial proﬁle of mean azimuthal velocity ⟨uθ ⟩t θ . Here, the velocity and radial position is normalized as

and

ueθ =

0.4

0.5

We notice a remarkable modiﬁcation of the proﬁles for the
drag enhancement cases (a = −0.33 and 0). That is, quite
surprisingly, the velocity gradient is nonmonotonic; it turns
positive in a bulk region while it is negative near the wall. Furthermore, in the bulk region, the velocity proﬁle seems to belinear, which implies that there is a ﬂow structure rotating as
if it is solid body between the cylinders. We call this the quasi
solid body rotational (SBR) region. It is worth mentioning that
although the previous DNS showing the drag enhancement in
the viscoelastic turbulent Taylor-Couette ﬂow reported a similar nonmonotonic proﬁle for Rei = 5000 (Liu % Khomami2013) and Rei = 3000 (Song et al. 2019), they did not showsuch quasi SBR region probably because the Reynolds number
is not so high and they treat the pure inner cylinder rotation.
From the results of PTV, we calculate the proﬁles of theangular velocity by using ⟨ω ⟩t θ = ⟨uθ ⟩t θ /r and normalizing it
asMean Velocity Modiﬁcationr − rod

0.3

ebulk ⟩t θ in
Figure 6. Relation between the angular velocity ⟨ωthe quasi SBR region and the drag reduction rate DR for a =±0.5, ±0.33, ±0.1, and 0. Rei = 7.1 × 104 only for a = ±0.5
and Rei = 7.0 × 104 for the others. Colors are the same as
those shown in Fig. 2.

(a → ∞) and the Rayleigh criterion (a = −η 2 ) for the centrifugal instability. The bisection line corresponds to a value abis of
the angular velocity ratio. For the present setup (η = 0.732),
abis = 0.371, which is show that the result for water is reasonable for the previous study.
On the other hand, for the CTAC solution, β show a quite
different trend. That is, β is drastically reduced for counterrotation (a > 0) and keep around 0.1 over the examined range
of a. This shows that the angular momentum transport is also
reduced by the viscoelasticity as well as the wall frictional drag(shown in Fig. 2).

e
r=

0.2

he
ωbulk itθReuθ − ro ωo.
r i ωi − r o ωo

eθ =
ω

(9)

4

⟨ω ⟩t θ − ωo.
ωi − ωo(10)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
ebulk ⟩t θ of the bulk region as
We deﬁne the angular velocity ⟨ωthe average in 0.2 ≤ er ≤ 0.8. Figure 5 show the Re-dependence
ebulk ⟩t θ in the quasi SBR regeion forof the angular velocity ⟨ωa = ±0.5, ±0.33, ±0.1, and 0 [colors are the same as shownin Fig. (2)]. Due to the resolution of the angular velocity on
the present set up, the inner cylinder Reynolds number for a =±0.5 (Rei = 7.1 × 104 ) is slightly different from the others(Rei = 7.0 × 104 ).
It is obvious that there is a positive correlation between Re
ebulk ⟩t θ . We can understand this result by recalling thatand ⟨ω
the deﬁnition of Re: a higher Re means larger shear rate γ̇ in abulk region. That is, the viscoelasticity has a larger effect andthe angular velocity in quasi SBR region becomes larger. Herethis result might be confusing in the comparison with that inFig. 4 but notice that the difference of the normalization of velocity [Eq. (9)] and angular velocity [Eq. (10)]. Furthermore,
ebulk ⟩t θ
as expected the Re-dependence of DR [Fig. 2(b)] and ⟨ωebulk ⟩t θ as
(Fig. 5), DR show the positive correlation with ⟨ω
shown in Fig. 6 implying that the quasi SBR region is important for the drag modiﬁcation [colors are the same as shown in
Fig. (2)].

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CONCLUSION

Li, F.-C., Yu, B., Wei, J.-J., and Kawaguchi, Y., 2012,”Turbulent Drag Reduction by Surfactant Additives”, Wiley,
New York.

To elucidate the modiﬁcation of high Reynolds-number
Taylor-Couette turbulence by the viscoelasticy of a dilute surfactant (CTAC) solution, we investigate the wall fcirtion andturbulent velocity ﬁeld by utilizing a quite large experimental
facility. We examined the turbulence both for the co-cotation(a < 0) and counter-rotation (a > 0) of the two cylinders.
We ovserve the drag reduction up to DR = 50% for thecounter-rotation for the ﬁrst time while the drag enhancement
DR < 0 for the co-rotation and the pure inner cylinder rotation (a = 0) as well as the previous DNS (Liu & Khomami,
2013; Song et al., 2019) (Fig. 2). This drag modiﬁcation canbe uniﬁedly represented by a Reynolds number Re indicating
the shear on the bulk region [Eq. (8)]. The viscoelasticity of
the CTAC solution also modiﬁes the angular momentum transport in Taylor-Couette turbulence in the ultimate regime where
the scaling law (2) holds. The scaling exponent is reduced andalmost the same even for the counter-rotation as that for the
co-rotation (Fig. 3).
Concerning the mean ﬂow structures, a nonmonotonic velocity proﬁle and the quasi solid body rotational region in abulk appear for the CTAC solution as shown in Fig. 4. Suchquasi SBR regime seems to be exist for a sufﬁciently higher
Reynold-number regime. The magnitude of the angular velocity of the quasi SBR region is larger for higher Re and has
a positive correlation with DR, which implys that the occurrence of this quasi SBR region is important to discuss the dragreduction in Taylor-Couette turbulence in the ultimate regime.
In the conference, we will present the more detailed results about the ﬂow structures and discuss the sustainaningmechanism of the quasi SBR region by showing the resultsfor other Reynolds-number regimes.

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Song, J., Teng, H., Liu, N., Ding, H., Lu, X., andKhomami, B., 2019, ”The correspondence between drag enhancement and vortical structures in turbulent Taylor–Couetteﬂows with polymer additives: a study of curvature dependence”, Journal of Fluid Mechanics, Vol. 881, pp. 602-616.
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pp. 353-382.
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ACKNOWLEDGEMENT
This study was partially supported by JSPS Trant-in-Aid
for Early-Careere Scientists (19K14893).

Xi, L., 2019, ”Turbulent drag reduction by polymer additives: fundamentals and recent advances”, Physics of Fluid,Vol. 31, 121302.

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