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

TURBULENT STRIPE SURVIVAL IN PLANE COUETTE FLOW

Hirohito Kawajiri
Department of Mechanical Engineering,
Doshisha University, 1-3 Tatara
Miyakodani, Kyotanabe-shi, Kyoto, Japan
ctwg0532@mail4.doshisha.ac.jp

Koji Fukudome
Department of Mechanical Engineering,Tokyo University of Science, 6-3-1
Niijuku, Katsushika-ku, Tokyo, Japankfukudome@rs.tus.ac.jp

Shumpei Hara
Department of Mechanical Engineering,
Doshisha University, 1-3 Tatara
Miyakodani, Kyotanabe-shi, Kyoto, Japanshhara@mail.doshisha.ac.jp

ABSTRACT

heat exchangers (Fukudome et al., 2018; Fukuda et al., 2020).
As industrial products become smaller and smaller, the ﬂow
paths become narrower and the Reynolds number becomes
low-Reynolds number. It is difﬁcult to sustain turbulence under low-Reynolds number, and it is important to clarify thecritical Reynolds number at which turbulence can be maintained. Clariﬁcation of the mechanism of turbulent stripe inthe subcritical transition will lead to efﬁcient transport of materials.
It was reported that turbulent stripes appear for 325 ≤ Re
≤ 415 (Manneville, 2004). Here, the Reynolds number is deﬁned by the velocity of movement of both walls Uw , the ﬁlm
half-width δ and the kinematic viscosity ν . Subsequent studies (Barkley et al., 2005; Reetz et al., 2019) have reported thesame Re range of turbulent stripe appearance.
The following three reports of turbulent stripes in a plane
Poiseuille ﬂow are introduced below. Hashimoto et al. (2009)
reported in visualization experiments that the angle was 20◦ to
30◦ . Paranjape et al. (2020) succeeded in capturing the growth
process and splitting of a single turbulent stripe generated in
the subcritical transition regime by experiment. Fukudome etal. (2015) reported numerically that when the angle of the
turbulent stripe is larger than 40◦ , high-and-low-speed streaks
in the large scale are not maintained and the turbulent stripe
structure decayed. Compared to Poiseuille ﬂow, plane Couette
ﬂow generates turbulent energy only in the streamwise direction due to the constant shear deformation, and the turbulenceis strongly anisotropic. Therefore, the growth process of the
turbulent stripe is estimated to be different from that reported
for the channel ﬂow.
Plane Couette ﬂow requires an even larger area than plane
Poiseuille ﬂow. Turbulent stripes have been observed at verylarge aspect ratios. Prigent et al. (2003, 2005) simultaneously
visualized ﬁve turbulent bands within the visualization area
under the condition of δ = 3.00 mm in the plane Couette ﬂow.
Gomé et al. (2020) captured the split and decay of turbulent
stripes by extracting regions in a certain streamwise direction,
and arranging them over time in a numerical simulation using the plane Couette ﬂow. In this study, the spatio-temporalcharacteristics of turbulent stripes, such as angle against thestreamwise direction, spanwise wavelength, and the number

In this study, visualization experiments of plane Couette
ﬂow were conducted to investigate the spatio-temporal characteristics of the angle against the streamwise direction (θ ), thenumber of turbulent bands generated in the subcritical transition regime, and the survival time of turbulent stripes. The
experiment was conducted with different ﬁlm half-widths (δ ).
The appearance and decay of two cycles of turbulent stripes
were captured at visualized times of Reynolds number of 300
and 330, and named the long-life and short-life cycles in theorder of their appearance. There were differences in the angleof turbulent stripes and turbulent bands in each cycle. In thelong-life cycle, one turbulent band was characterized at a maximum θ of 40◦ to the streamwise direction for small δ , with asurvival time of t ∗ = 1620. For the large δ , one turbulent band
was characterized at a maximum θ of 35◦ , with a survival timeof t ∗ = 1960. In the short-life cycle, ﬁve turbulent bands werecharacterized at a maximum θ of 20◦ to the streamwise direction for small δ , with a survival time of t ∗ = 970. For the large
δ , one turbulent band was characterized at a maximum θ of37◦ , with a survival time of t ∗ = 1240. In the long-life cycle,
similar angles of turbulent stripes were observed for each δ .
On the other hand, a larger angle was observed for the shortlife cycle with large δ . The number of turbulent bands of eachcycle was found to be much for small δ . The survival time ofturbulent stripes depends on the angle of the turbulence stripes,
and θ = 35◦ was found to have the longest survival time at t ∗= 1960. The experimental results in this study showed that thevalue of spanwise wavelength was different for small and large
δ.

INTRODUCTION
Turbulent stripe appears in the subcritical transition
regime of a plane Couette ﬂow, which is one of the canonical ﬂows. Turbulent stripes are an intermittent structure inwhich a turbulent region with high velocity ﬂuctuation and aquasi-laminar region with low velocity ﬂuctuation are alternately arranged in the streamwise direction to form a stripelike pattern (Prigent et al., 2003). The study of the subcritical transition regime is important for low-Reynolds number

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
numerical simulation of a plane Couette ﬂow. Table 1 shows
the Reynolds number range in which turbulent stripes wereobserved and the angle of the turbulent stripes with against
the streamwise direction, with the reports of previous studies(Barkley et al., 2005; Tsukahara et al., 2010). The plane Couette ﬂow and the plane Poiseuille ﬂow are denoted by PCF and
PPF, respectively, and the angle of the turbulent stripe withrespect to the streamwise direction is deﬁned by θ . One turbulent band of θ = 20◦ ，35◦ and 37◦ was observed at Re = 330
in this study. The angle of the turbulent stripe was consistent
with θ = 21◦ in Tsukahara et al. (2010). Barkley et al. (2005)
reported θ = 37◦ at Re = 340, which is in alignment with thevalue of θ = 35◦ , 37◦ found in this study. The angle of the
turbulent stripe in the PCF and PPF of the previous study andthe value of the present study are consistent, suggesting thatthe angle is independent of the PCF and PPF. In previous studies, the value of θ tends to decrease as the Reynolds number
increases. However, in the present study, turbulent stripes of θ
= 20◦ ，35◦ and 37◦ appeared at Re = 330, suggesting that theangle of the PCF turbulent stripes is independent of the valueof the Reynolds number.
Two patterns of cycles of turbulent stripe appearance anddecay existed at Re = 330 in this study. The cycles were named
long-life cycle (Fig. 3) and short-life cycle (Fig. 4) in the orderof appearance. Turbulent stripes are guided by yellow lines.
Here, ∆ t ∗ (= Uw ∆ t/δ ) is the time elapsed since the start ofobservation, which is non-dimensionalized by the ﬁlm velocity and ﬁlm half width. At ∆ t ∗ = 7700 (Fig. 3 (a)), a turbulent
stripe with an angle of 20◦ across the entire stremwise direction appeared on the lower side of the spanwise direction. At∆ t ∗ = 8510 (Fig. 3 (b)), a turbulent stripe with an angle of35◦ appeared on the upper side of the spanwise direction at thesame time as the turbulent stripe that appeared decayed. The
turbulent stripe appearing at ∆ t ∗ = 8510 (Fig. 3 (b)) sustains,
and at ∆ t ∗ = 10500 (Fig. 3 (d)) the turbulent stripe diminished
to exist throughout the ﬂow. From the above, the survival timeof the turbulent stripe was t ∗ = 1960 with θ = 35◦ .
A space-diagram with a long-life cycle of δ = 3.39 mm
was created to capture the temporal variation of large turbulent
stripes. A row of pixels (1.10 mm wide) at approximately 80
δ upstream from the streamwise direction was combined inchronological order from left to right. The space-diagram fora long-life cycle of Re = 330 is shown in Fig. 5. The verticalaxis is spanwise and the horizontal axis is time, both of which
are non-dimensionalized. In Fig. 5, turbulent stripe appears in
the lower spanwise direction from ∆ t ∗ = 7700 to ∆ t ∗ = 8510,
when the long-life cycle is conﬁrmed, and the space-diagramshows that the structure of turbulent stripe guided by yellow
lines is sustained until ∆ t ∗ = 8510. Then, at ∆ t ∗ = 8510,
the turbulent stripe on the lower side of the spanwise direction
decays, and another turbulent stripe appears on the upper side.
The newly emerged turbulent stripe sustained its structure until
the end of the long-life cycle.
Figure 4 shows the short-life cycle. At ∆ t ∗ = 11300 (Fig.
4 (a)), one turbulent band of 37◦ appeared at the center of thevisualization area. At ∆ t ∗ = 12200 (Fig. 4 (b)) the turbulent stripe split into two guided by yellow circles. However, at∆ t ∗ = 12300 (Fig. 4 (c)), the split turbulent regions remerged
and returned to their original shape. Thereafter, the turbulent
stripe structure sustained, and at ∆ t ∗ = 12600 (Fig. 4 (d)) the
turbulent stripe diminished to exist throughout the ﬂow. The
survival time of the turbulent stripe was t ∗ = 1240 with θ =37◦ .
Comparing each cycle with δ = 3.39 mm, turbulent stripes
with longer survival times appeared in the long-life cycles.

of bands, as well as the survival time of the turbulent stripes
themselves, are of interest for understanding the phenomenon.
In this study, the ﬁlm half-width differs from that of Prigent et al. (2003), visualization experiments of the plane Couette ﬂow were conducted to investigate the spatio-temporalcharacteristics of the appearing turbulent stripes and theirformation and decay processes (survival time), comparing
them with previous studies on plane Couette ﬂow and plane
Poiseuille ﬂow.

EXPERIMENTAL SET-UP
Figure 1 shows a schematic diagram of the experimental
apparatus for the plane Couette ﬂow. The experimental apparatus used in this study is similar to that utilized by Couliu etal. (2015). Equipment in Fig. 1 was submerged in a water tank(1000×1660×300 mm3 ) ﬁlled with water and driven by a motor at a constant speed to form a plane Couette ﬂow. A seamless cylindrical ﬁlm was utilized to prevent disturbance to theﬂow. The ﬁlm was adjusted to be stretched evenly. To verify
the uniformity of the ﬁlm half-width, the ranged measurement(green area in Fig. 1) was divided into nine square blocks. The
ﬁlm half-width at the center of each block was measured usinga laser pointer and an acrylic cube. In this study, two experiments were conducted with different ﬁlm half-widths. In theﬁrst experiment, the mean value for the nine ﬁlm half-widths
was 3.39 mm, with a standard deviation of 0.27 mm. To estimate the kinematic viscosity, thermometers were inserted into
four corner points in the tank, and the average value of 288 K
was used. In the second experiment, the ﬁlm half-width was
2.90 mm, its standard deviation was 0.26 mm, and the watertemperature was 297 K. In this study, the Reynolds number
was lowered gradually from 450 and ﬁnally to 260. To visualize the ﬂow, ﬂake particles were inserted into the tank, andafter conﬁrming that the ﬂow became steady at each Reynolds
number, the ﬂow was captured by a camera for 5 minutes.

CHARACTERISTICS
BAND

OF

ONE

TURBULENT

Figure 2 shows the images at Reynolds numbers of 450,
330, and 280. Each image is dimensionless in the entire visualized area with ﬁlm half-widths in the streamwise direction andspanwise direction, respectively. In Fig. 2 (a), the entire ﬂow
has a light/dark pattern, so the overall ﬂow condition is turbulent. In Fig. 2 (b), turbulent stripe composed of light/dark
turbulent regions, and dark quasi-laminar regions. Turbulent
stripes can be seen at a Reynolds number of 330. Turbulent
stripe is guided by yellow lines. In Fig. 2 (c), no turbulent
stripe is observed, and the ﬂow has a dark pattern, so the overall ﬂow condition is laminar ﬂow. From Fig. 2, turbulent,turbulent stripe, and laminar ﬂow were discriminated. Basedon the turbulent stripes in Fig. 2 (b), the turbulent stripes inthis study were discriminated.
The Re range in which turbulent stripes could be visualized in this study was from 330 to 350, which is consistent
with the Re range in which turbulent stripes appeared, compared to the report by Manneville et al. (2004). The visualization movie at Re = 330 shows a coexistence of laminar andturbulent regions, and turbulent stripes appear. At Re = 320,
the entire ﬂow is in the laminar region. Therefore, the critical Reynolds number in this study is considered to exist between Re = 320 and 330. The critical Reynolds number rangeof this study is consistent with Duguet et al. (2010), which
reported a critical Reynolds number of Re = 324 ± 1 in their212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Both long-life and short-life cycles visualized at one turbulence stripe in the visualization area, and the angle was notdifferent in each cycle.

turbulent bands with θ = 0◦ and two turbulent bands with θ =20◦ , 40◦ were observed, respectively, during 114 s imaging at
Re = 300 in this study. Turbulent stripes with θ = 0◦ appeared
most frequently, and those with θ = 20◦ and 40◦ appeared at
the same frequency.

CHARACTERISTICS
BANDS

Here, the turbulent fraction for the experimental resultsfor each δ were determined and compared with those reportedby Duguet et al. (2010). In this study, the turbulent fraction,
which indicates the percentage of turbulent areas within thevisualized area, was deﬁned by the percentage of white areas (turbulent areas) in the image after morphology processing after binarization of the image. The turbulent fraction was
followed for 120 s, and remained constant at each Reynolds
number, regardless of the appearance or absence of turbulent
stripes. Figure 8 shows the intermittency calculated by time
averaging at 120 s for each Reynolds number of squares (largeδ ), and diamond (small δ ). The turbulent fraction reported byDuguet et al. (2010) by direct numerical simulation is shownin circles. Duguet et al. (2010) obtained the turbulent fraction from the percentage of regions with turbulence generation
above a certain threshold, which is strictly different from theturbulent fraction deﬁned in this study. The measured turbulent fraction of large δ is almost identical to that reported byDuguet et al. (2010). The trend in this study of decreasing the
turbulent fraction from Re = 380 as the Reynolds number is decreased from Re = 450 is consistent with the decreasing rate of
turbulent fraction (Duguet et al., 2010). Numerical simulation
has shown that the turbulent fraction of the plane Couette ﬂow
has two types of slopes: one is a gradual decrease in value as
the Reynolds number is lowered from turbulent fraction, andthe other is a sharp decrease in turbulent fraction from a certain Reynolds number. This experiment of each δ was able to
conﬁrm both a gradual and a sharp decrease in the turbulent
fraction. The difference between the maximum and minimum
values of the turbulent fraction (∆ Re) and the critical Reynolds
number (Recr ) were obtained by applying the sigmoid function
to the turbulent fraction values of Duguet et al. (2010) andpresent study. For large δ , ∆ Re was 0.20, for small δ , it was
0.18, and for Duguet et al. (2010), it was 0.22. For large δ ,Recr was 0.67, for small δ , it was 0.63, and for Duguet et al.
(2010), it was 0.60. For small δ , the increase in the turbulent
fraction rate occuered earlier than in the other two results, butboth ∆ Re and Recr showed similar values. there is no difference between the experimental and numerical results for the
turbulent fraction.

OF

FIVE

TURBULENT

Long-life and short-life cycles were also observed at Re =300 for δ = 2.90 mm. Figure 6 shows long-life cycle from theappearance of the turbulent stripe to its decay at Re = 300. Turbulent stripes are guided by yellow lines. At ∆ t ∗ = 162 (Fig. 6(a)), a turbulent stripe with θ = 40◦ appeared downstream, andat ∆ t ∗ = 616 (Fig. 6 (b)), a turbulent stripe with θ = 40◦ was
visualized, which was opposite to the angle of turbulent stripe
in Fig. 6 (a). At ∆ t ∗ = 681 (Fig. 6 (c)), the turbulent stripe
in Fig. 6 (b) was mixed with the upstream turbulent region
and decayed. Then, at ∆ t ∗ = 1780, the turbulent stripe surviving since the beginning of the observation decayed, and at ∆t ∗ = 1910 (Fig. 6 (d)), the turbulent stripe diminished to exist
throughout the visualized ﬂow. From the above, the survival
time of the turbulent stripe was t ∗ = 819 with θ = 20◦ and t ∗ =1620 with θ = 40◦ .
Figure 7 shows short-life cycle from the appearance ofthe turbulent stripe to its decay at Re = 300. At ∆ t ∗ = 2110 in
Fig. 7 (a), two turbulent bands with θ = 0◦ and one turbulent
band with θ = 20◦ appeared downstream. One turbulent bandwith θ = 0◦ ﬂowed upstream and decayed, while the other one
mixed with the surrounding turbulent region. Three turbulent
bands with θ = 0◦ appeared downstream and upstream of the
spanwise direction at ∆ t ∗ = 2760 in Fig. 7 (c), and ﬁve turbulent bands were visualized in the entire ﬂow. This was thelargest number of turbulent bands visualized in the ﬂow at Re= 300 of ∆ t ∗ = 2760 in this study. At ∆ t ∗ = 2890 in Fig.
7 (d), two turbulent bands with θ = 0◦ decayed. At ∆ t ∗ =3080, two turbulent bands with θ = 20◦ decayed, and at ∆ t ∗= 3690, turbulent stripes diminished to exist in the entire ﬂow
as in Fig. 7 (d). From the above, the survival time of the turbulent stripe was t ∗ = 970 with θ = 20◦ . The case of the decayof the turbulent stripe with θ = 0◦ at t ∗ = 162. Fukudome etal. (2013) reported that collision between high-and-low-speed
streaks does not occur when the turbulent region exists parallelto the streamwise direction. Onishi et al. (2010) reported thatthe collision between high-and-low-speed streaks is essential
for the generation and maintenance of turbulent stripes, which
may be the reason why the θ = 0◦ of turbulent stripe observedin this study decayed in a short time.
Comparing the two cycles for δ = 2.90 mm, the survival
time of turbulent stripe of θ = 40◦ in long-life cycle was t ∗ =1620, whereas the survival time of θ = 20◦ turbulent stripe inshort-life cycle was t ∗ = 970. Duguet et al. (2010) reported
that an angle close to θ = 40◦ was most likely to appear in theearly stage of turbulent stripe development, which was consistent with this experiment. The survival time of the turbulent
stripe at θ = 40◦ was the longest, suggesting that the energy
cycle of the turbulent stripe was stably supplied at θ = 40◦ .

The long-life and short-life cycles are compared for eachﬁlm half-width δ = 2.90 and 3.39 mm. The angle and numberof turbulent bands, appearing in the long-life cycle, was 40◦
for δ = 2.90 mm, with two turbulent bands visualized. And
35◦ for δ = 3.39 mm, with one turbulent band was visualized.
While relatively similar angles could be identiﬁed, the numberof turbulent bands in the entire ﬂow was much for δ = 2.90
mm. The angle and number of turbulent bands appearing inthe short-life cycle were 0◦ for δ = 2.90 mm, and ﬁve turbulent
bands were visualized at 20◦ . At 37◦ with δ = 3.39 mm, a
maximum of one turbulent band was visualized throughout the
ﬂow. Regarding the angles of the turbulent stripes observed in
the short-life cycle, the angle at δ = 3.39 mm was about twiceas large as the angle appearing at δ = 2.90 mm. As with thelong-life cycles, the number of turbulent bands, that appearedin the entire ﬂow, was much for δ = 2.90 mm.

RELATIONSHIP BETWEEN ASPECT RATIOAND THE NUMBER OF TURBULENT BANDS
The Re range, in which the turbulent stripe was visualizedat δ = 2.90 mm, was within 300 and 350. It is a little lower
than the Re range at δ = 3.39 mm in this study and that reportedby Manneville (2004). However, since the standard deviationof δ in this study are 0.26 mm, the Re range where the turbulent stripe appears are consistent with previous study. Five

The difference in the angle and number of turbulent bands
is considered to be caused by the ﬁlm half-width. Table 2
compares the values of this study with those reported from experiments in which turbulent stripes were visualized at differ-3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
regime with helical turbulence in annular ﬂow. International Journal of Heat and Fluid Flow 82, 108555.
Fukudome, K. & Iida, O. 2013 Effect of large-scale structure generated in low-reynolds-number poiseuille ﬂow on
longitudinal vortices. Transactions of the Japan Society of
Mechanical Engineers. Series B 79 (807), 2352–2362 (inJapanese).
Fukudome, K. & Ogami, Y. 2015 On the scale and angular dependence of large-scale intermittent structures inlowreynolds-number interplanar ﬂows. RIMS 1944, 15–23(in Japanese).
Fukudome, K., Tsukahara, T. & Ogami, Y. 2018 Heat and momentum transfer of turbulent stripe in transitional-regime
plane couette ﬂow. International Journal of Advances inEngineering Sciences and Applied Mathematics 10 (4),291–298.
Gomé, S., Tuckerman, L, S. & Barkley, D. 2020 Statisticaltransition to turbulence in plane channel ﬂow. Physical Review Fluids 5 (8), 083905.
Hashimoto, S., Hasobe, A., Tsukahara, T., Kawaguchi, Y. &Kawamura, H. 2009 An experimental study on turbulentstripe structure in transitional channel ﬂow. In ICHMT DIGITAL LIBRARY ONLINE. Begel House Inc.
Manneville, P. 2004 Spots and turbulent domains in a modelof transitional plane couette ﬂow. Theoretical and Computational Fluid Dynamics 18 (2-4), 169–181.
Ohnishi, K., Tsukahara, T. & Kawaguchi, Y. 2011 Turbulence budget in transitional plane couette ﬂow with turbulent stripe. In Journal of Physics: Conference Series, , vol.
318, pp. 32–33.
Paranjape, C. S., Duguet, Y. & Hof, B. 2020 Oblique stripe
solutions of channel ﬂow. Journal of Fluid Mechanics 897,
A7.
Prigent, A. & Dauchot, O. 2005 Transition to versus from turbulence in subcritical couette ﬂows. In IUTAM Symposiumon Laminar-Turbulent Transition and Finite Amplitude Solutions, pp. 195–219.
Prigent, A., Grégoire, G., Chaté, H. & Dauchot, O. 2003 Longwavelength modulation of turbulent shear ﬂows. Physica D:
Nonlinear Phenomena 174 (1-4), 100–113.
Reetz, F., Kreilos, T. & Schneider, T. M. 2019 Exact invariant solution reveals the origin of self-organized obliqueturbulent-laminar stripes. Nature Communications 10 (1),
1–6.
Tsukahara, T., Kawaguchi, Y., Kawamura, H., Tillmark, N.
& Alfredsson, P. H. 2010 Turbulence stripe in transitional channel ﬂow with/without system rotation. In Seventh IUTAM Symposium on Laminar-Turbulent Transition,
pp. 421–426.
Yimprasert, S., Kvick, M., Alfredsson, P, H. & Matsubara, M.
2021 Flow visualization and skin friction determination intransitional channel ﬂow. Experiments in Fluids 62 (2), 1–
16.

ent ﬁlm half-widths. Comparison is made in terms of visiblearea, ﬁlm half-width, number of turbulent bands, and spanwise wavelength (λz ); Yimprasert et al. (2021) reported λz /δ= 188 and one turbulent band at δ = 3.50. This study identiﬁed one turbulent band with δ = 3.50 of λz /δ = 31.7, andﬁve turbulent bands with δ = 2.90 of λz /δ = 24. Prigent et al.
(2005) reported λz /δ = 21.1 and ﬁve turbulent bands at δ =1.50. Experiments conducted in this study with different ﬁlm
half-widths resulted in larger number of turbulent bands andsmaller values of spanwise wavelength for the small δ . It isdifﬁcult to say that it cannot be organized by δ because thestandard deviation of the ﬁlm half-width is 0.26 mm for smallδ and 0.27 mm for large δ . The same order of spanwise wavelengths can also be assumed from the report by Duguet et al.
(2013) that non-dimensionalized the spanwise wavelengths byδ . On the other hand, Tsukahara et al. (2010) reported that thevalue of spanwise wavelength decreases as the value of Reτ
increases in PPF and PCF. From this study and Tsukahara etal.’s report, spanwise wavelengths are considered to be organized on a viscous scale. Thus, future works need inclusion of
discussion about the inﬂuence of δ on turbulent bands.

CONCLUSION
In this study, the spatio-temporal variation of turbulent
stripes in the plane Couette ﬂow was investigated. The ﬁndings
are summarized as follows.
• Two cycles of turbulent stripe appearance and decay were
ﬁlmed in the transition region of a plane Couette ﬂow,
and differences in the angle (θ ) and number of turbulent
bands existed in each cycle. The survival time of turbulent
stripes was found to be longer in the ﬁrst long-life cyclethat appeared than in the second short-life cycle.
• In the short-life cycle at Reynolds number of 300, a maximum of ﬁve turbulence bands were visualized in the ﬂow,
and the survival time of turbulent stripe with θ = 40◦ was
the longest compared to stripes at other angles at t ∗ =1620, which was non-dimensionalized by the ﬁlm velocity and ﬁlm half-width (δ ). At Re = 330, the survival timeof turbulent stripe with θ = 35◦ was the longest in thisstudy. Based on the results of turbulent stripe with survival time, the angle, at which turbulent stripes are stablein plane Couette ﬂow experiments, are considered to be
the range is 35◦ and 40◦ .
• The experimental results in this study showed that thevalue of spanwise wavelength was different for small andlarge δ .

REFERENCES
Barkley, D. & Tuckerman, L. S. 2005 Turbulent-laminarpatterns in plane couette ﬂow. In IUTAM Symposium on
Laminar-Turbulent Transition and Finite Amplitude Solutions, pp. 107–127.
Couliou, M. & Monchaux, R. 2015 Large-scale ﬂows in transitional plane couette ﬂow: a key ingredient of the spot
growth mechanism. Physics of Fluids 27 (3), 034101.
Duguet, Y., Monokrousos, A., Brandt, L. & Henningson, D. S.
2013 Minimal transition thresholds in plane couette ﬂow.
Physics of Fluids 25 (8), 084103.
Duguet, Y., Schlatter, P. & Henningson, D. S. 2010 Formationof turbulent patterns near the onset of transition in plane
couette ﬂow. Journal of Fluid Mechanics 650, 119–129.
Fukuda, T. & Tsukahara, T. 2020 Heat transfer of transitional412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Table 1: Reynolds number at which the turbu-

lent stripe appeared and the angle of the turbulent stripe with respect to the ﬂow direction (θ )were compared.

Barkley et al. (2005)

Flow
PCF

Tsukahara et al. (2010)

PCF
PPF

Figure 1: Schematic view of the experimentset-up. Flow is left to right. The captured
area by the camera is the green. Experiments
were conducted with different ﬁlm half-widths(large δ , small δ ).

Present

PCF

Re
340
394
350
375
268
293
330 - 350θ (deg)
37
25
27
27
21
21
20 - 37

Figure 2: Flow visualization of plane Couette ﬂow. (a)All captured area is turbulent at Re = 450; (b)Visualization of

turbulent stripes guided by yellow at Re = 330; (c)All captured area is laminar ﬂow at Re = 280. The captured area is of
600 mm × 700 mm.

Figure 4:

Short-life cycle of turbulent stripesthat appear and then decay at Re = 330.
(a)∆ t ∗ = 11300; (b)∆ t ∗ = 12200; (c)∆ t ∗ = 12300;(d)∆ t ∗ = 12600.

Figure 3:Long-life cycle of turbulent stripesthat appear and then decay at Re = 330.
(a)∆ t ∗ = 7700; (b)∆ t ∗ = 8510; (c)∆ t ∗ = 9430;(d)∆ t ∗ = 10500.

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 5: Space-diagram of the long-life cycle at Re = 330. The white dashed line shows that the decay and turbulent

stripes occur almost simultaneously.

Figure 6: Long-life cycle of turbulent stripes

Figure 7: Short-life cycle of turbulent stripesthat appear and then decay at Re = 300.
(a)∆ t ∗ = 162; (b)∆ t ∗ = 616; (c)∆ t ∗ = 681;(d)∆ t ∗ = 1910.

that appear and then decay at Re = 300.
(a)∆ t ∗ = 2110; (b)∆ t ∗ = 2370; (c)∆ t ∗ = 2760;(d)∆ t ∗ = 2890.

100

Table 2: Differences in the spanwise wave-

Ft (%)

80

length (λz ), and number of bands of turbulent
stripes across the ﬁlm half-width.

60
40
20
0
250

Present Exp. large
Present Exp. small
Duguet (2010)

300

350

400

450

Re

Yimprasert et al.
(2021)
Present study
Prigent et al.
(2005)

Figure 8: Turbulent fraction at time average ofeach Reynolds number.

6

δ
3.50

bands
1

λz /δ
188

x/δ × z/δ
1330 × 166

Flow
PPF

3.39
2.90
1.50

1
5
5

31.7
24.0
21.1

177 × 206
207 × 241385 × 169

PCF
PCF