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

GROWING TURBULENT SPOT IN PLANE COUETTE FLOW PROVIDESDISSIMILARITY BETWEEN MOMENTUM AND HEAT TRANSFERS

Koji Fukudome
Takahiro Tsukahara
Department of Mechanical Engineering
Department of Mechanical Engineering
Tokyo University of Science
Tokyo University of Science
Niijuku 6-3-1, Katsushika-ku, Tokyo 125-8585, Japan Yamazaki 2641, Noda-shi, Chiba 278-8510 Japankfukudome@rs.tus.ac.jp
tsuka@rs.tus.ac.jp

Hiroya Mamori
Makoto Yamamoto
Department of Mechanical andDepartment of Mechanical EngineeringIntelligent Systems Engineering
Tokyo University of Science
The University of Electro-Communications
Niijuku 6-3-1, Katsushika-ku, Tokyo 125-8585, Japan
Chofugaoka 1-5-1, Chofu, Tokyo, 182-8585, Japan
yamamoto@rs.tus.ac.jp
mamori@uec.ac.jp

ABSTRACTet al. (2013) performed DNS (direct numerical simulation) ofa channel flow with wall suction and blowing based on an
optimal control theory. Their streamwise-traveling wave of
blowing and suction broke the analogy in heat and momentum transfers, obtaining an effective heat transfer. Motoki
et al. (2018) determined the optimal heat transfer state by using the Euler–Lagrange equations. More recently, the effective
heat transfers were reported for a Taylor–Couette flow with thetraveling-wave control (Mamori et al., 2021), and for a turbulent flow on a permeable wall (Motoki et al., 2021).
Recently, there have been reports of effective heat transfer in spontaneously intermittent turbulent fields without any
flow control. Fukudome et al. (2018) found an effective heattransfer in the plane Couette flow at a subcritical Reynolds
number, where the flow field was accompanied by a turbulent
stripe. The turbulent stripe is a large-scale spatial intermittent pattern of coexisting laminar and turbulent regions (Prigent et al., 2002; Barkley & Tuckerman, 2007; Duguet et al.,2010; Tsukahara et al., 2014; Fukudome & Iida, 2012). The
same effective heat transfer is confirmed with helical turbulence in annular Poiseuille flows (Fukuda & Tsukahara, 2020).
More recently, we revealed that the effective heat transfer stateis also obtained by a growing turbulent spot in the plane Couette flow (Fukudome et al., 2021). However, the generation
mechanism of the dissimilarity has not been understood.
In this paper, we have performed DNS of a developing turbulent spot in the laminar plane Couette flow and its
passive-scalar thermal field, in order to discuss the dissimilarity between the momentum and heat transfers in the spatiallyintermittent turbulent regime. As a seed of the spot, we added apair of longitudinal vortices (Lundbladh & Johansson, 1991),
and the similarity between streamwise velocity and temperature fields was secured at the beginning of the simulation.
Then, the dissimilarity mechanism was analyzed by the identity equations for the skin-friction and the heat-transfer coefficients, i.e., the FIK identities (Fukagata et al., 2002; Yamamoto et al., 2013). Furthermore, the budgets of Reynolds
shear stress and turbulent heat flux are compared to identify
the generation mechanism of the dissimilarity.

Direct numerical simulation of a turbulent spot developing in a laminar plane Couette flow was performed to study thedissimilarity between the heat and momentum transfers. The
seed of the turbulent spot was a vortex pair, and the initial flow
had an absolute similarity state between heat and momentum
transfers. The turbulent spot grows gradually in the early stage
and rapidly in the latter stage. The effective heat transfer state,
which achieves larger heat transfer than momentum transport,
was confirmed in the rapidly developing process. The dissimilarity mechanism was investigated by using a FIK identity
equation. We clarified that the turbulent contributions in thefriction coefficient and the Stanton number provided the effective heat transfer state. The budgets of Reynolds shear stressand heat flux exhibited that the dissimilarity was attributed tothe differences in the pressure-strain and dissipation terms, andthe difference in the production terms leads to adverse effects
on the effective heat transfer. After the developing process,the pressure-strain term causes a decrease of the heat transfer
compared to the skin friction.

INTRODUCTION
In turbulent flows, vortical structures enhance heat andmass transfers compared to those in the laminar flow. While
this fact often leads a practical advantage for many engineering applications, the promoted momentum transport should
result in the disadvantage of increased frictional drag. It is
still challenging for us to achieve an effective heat transfer
that outweighs a momentum transfer or to intentionally control a dissimilarity between the transfers, because of a similarity between the heat and momentum transfers, as known as the
Chilton–Colburn analogy.
Some effective heat-transfer states were demonstrated byproviding a certain vortex structure or applying a flow-control
technique. Katoh et al. (2013) revealed that the spanwise vortices with cyclonic rotation generate a dissimilarity in the plane
Poiseuille flow. The dissimilarity due to anti-cyclonic rotation was also recently clarified (Kubo et al., 2021). Yamamoto1

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

20ʌįHot wall

UW

TW

y
z

y

2į

-UW

-TW
Cold wall

z

0

A

10ʌį
x

Figure 1.

Laminar Turbulent

Schematic of target plane Couette flow.

Figure 2. Cross-sectional view of initial disturbance. Red andblue iso-surfaces show vortices identified by the second invariant of the deformation tensor II + = 0.1 with clockwise and anticlockwise rotation, respectively. The in-plane velocity vector iscolored by its magnitude.

Figure 3. Temporal development of the turbulent spot. Red and blue iso-surfaces represent vortical structures identified by the second
invariant of the deformation tensor II + = 0.05 with clockwise and anti-clockwise rotation in streamwise direction, respectively.

NUMERICAL SIMULATION& Johansson (1991). The streamwise velocity and temperature distributions are identical with the laminar value, i.e.,
u/Uw = y/δ and θ /Tw = y/δ , indicating that the initial flow
field satisfies the complete similar state between heat and momentum transfers. The stream function Ψ of an initial disturbance is denoted as

The flow configuration including the thermal boundary
condition is illustrated in Fig. 1, where the periodic boundary
conditions are imposed in the streamwise (x) and spanwise (z)directions. The computational domain is 20πδ × 2δ × 10πδand the number of grid points is 256 × 65 × 256, in x, y, and z.
The top and bottom walls move at speeds of +UW and −UW ,respectively, and their temperatures are +TW and −TW , respectively.
The governing equations are the incompressible continuity, Navier–Stokes, and energy equations. The temperatureis treated as a passive scalar. For spatial discretization, thespectral method (Kim et al., 1987; Fukudome et al., 2018)is adopted with Fourier series in x and z, and the Chebyshev
polynomial expansion in the wall-normal (y) direction. The
collocation grid is employed to compute the nonlinear terms inphysical space and the grid has 1.5 times finer resolution to remove aliasing errors. As the time integration, the second-order
Adams–Bashforth and Crank–Nicolson schemes are adopted
for the nonlinear and viscous terms, respectively. Here, the explicit Euler scheme is adopted for the first step of the nonlinear
terms to compute from the initial flow field.
The Reynolds and Prandtl numbers are Re = 450 andunity, respectively, where Re is based on UW and δ . The time
step size is ∆t = 0.01δ /UW . Hereafter, the parameters are normalized by UW , δ , and/or TW .
As the initial flow field, we mimic a transient jet fromthe wall as a single disturbance, introduced by Lundbladh


2
2
2
Ψ = A 1 − y2 ze(−x −y ) ,(1)

where the wall-normal and spanwise velocities can be derived
by v = ∂Ψ /∂ z and w = −∂Ψ /∂ y. Here, the parameter A represents the strength of initial disturbance or the peak value ofthe local wall-normal velocity at the mid of the vortex pair.
Figure 2 shows a cross-sectional flow field of the initial disturbance for A = 1.2. A pair of counter vortices can be confirmedin the channel center, and the maximum wall-normal velocity
at the midpoint of the vortices corresponds to A = 1.2.

RESULTS AND DISCUSSION
Figure 3 shows the temporal evolution of the disturbancefor A = 1.2. At t = 100 (Fig. 3(c)), very-extended longitudinalvortical structures composes a spot structure, in which turbulent motions are rather weak yet. Small-scale vortices can be
confirmed at t = 150 (Fig. 3(d)), then the turbulent spot rapidly
grows and it reaches the edge of the computational domain at
t = 300 (Fig. 3(g)). As reported in the following, our main2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
1

2St 㸫 Cf100 [%]
Cf

6

0.8

0.01

4

Γt

2St, Cf

0.012

0.6
0.008

2St
Cf

0.004
0

100

Figure 4.

0

200

300

400

0.2

500

t

-2
0

0
600

Figure 6.

Temporal evolution of St, C f , and Γt .

100

200

300

400

500 t 600

Temporal evolution of the dissimilarity factor.

0.012

0.3
2St, Cf

u'i rms, θ'rms

0.4

Γt

0.006

2

u'rms
v'rms
w'rms
θ'rms

0.2

0.008

0.1

0
0

0.01

Cf
Cf (FIK:Turbulent)
2St
2St (FIK:Turbulent)

0.006
0

100

200

300

400

100

200

300

400

500 t 600

500 t 600

Figure 7. Temporal evolution of the turbulent contribution of
FIK identity of St and C f .

Figure 5. Temporal evolution of RMS values of velocity andtemperature fluctuations.

ponents of Reynolds stresses, resulting in the energy transferfrom u0 component to the other components. Meanwhile, thev0rms and w0rms are almost constant for 0 < t < 100, and gradually increase for 100 < t < 350, implying the spatial growth of“actually turbulent” region with fine-scale eddies.
Let us define the dissimilarity factor (2St −C f )/C f , considering the Chilton–Colburn analogy with Pr = 1. Its positive value represents the efficient heat transfer, while its negative does the dominance of frictional drag. The time series of its percentage is plotted in Figure 6. In the periodof spatial growth for the spot (100 < t < 350), we obtained
positive values. In particular, the factor rapidly increases for
175 < t < 250, and then it turns to decrease. Therefore, theeffective heat transfer state (2St −C f > 0) is confirmed for the
rapidly growing process of the turbulent spot, until the spotcovers the entire domain. When the entire flow field is in aturbulent state (t > 400), the factor is negative, which is a tendency known in the literature (Fukudome et al., 2018).
Each of C f and 2St can be decomposed into laminar andturbulent contributions by using FIK identity (Fukagata et al.,
2002; Yamamoto et al., 2013). Here, due to the temporal
growth of localized turbulence, the unsteady and asymmetricterms (denoted by Cuv and Cvθ ) cannot be ignored, as follows:

focus is on this spatial-growth period (t = 150–300) of the turbulent spot, where an effective heat transfer has been detected.
Figure 4 shows time series of the friction coefficient C f(= u2τ /UW2 ), the Stanton number St (= h/C p ρUW ), and the intermittent factor Γt , where uτ , h, C p , and ρ are the friction
velocity, the heat-transfer coefficient, the specific heat, and thedensity, respectively. The definition of Γt is the ratio of the turbulent area against the entire channel area, where the turbulent
area is identified based on the dissipation rate. The point at
which the local dissipation rate averaged in y exceeds half of
the average dissipation rate for the entire channel in the instantaneous flow field was determined to be a turbulent region. As
shown in Fig. 4, all values gradually increase for 0 < t < 150
and they rapidly grow for t > 150, where the turbulent spot
grows spatially and with generating fine-scale eddies inside it.
The growth rate of each coefficient during this rapid development process is the largest and constant value. Eventually, they
converge at t ' 350 because the spot size reaches the finite
computational domain. Therefore, the turbulent spot grows
through the two stages before reaching the domain size: thegradual expansion at the early stage and the rapid expansion
at the latter stage. This two-stage growth corresponds to theexperiment observation by Tillmark & Alfredsson (1992).
Figure 5 shows the time evolution of the turbulent intensities of velocities and temperature. The fluctuating component
is defined by the deviation from the spatially-averaged value
at each time instance. Then, the root-mean-square values inthe entire channel are obtained irrespective of y. The figure0exhibits that the u0rms and θrms
increase linearly with time. It
0
is interesting to note that θrms
keeps larger value than u0rms .
This is cased by the redistribution between the normal com-1
2+
−u0 v0 dy +Cuv ,Re−1|{z} |{z}

Z

C f ave =Laminar

Turbulent

Z 1
22Stave =+−v0 θ 0 dy +Cvθ ,|RePr{z } | −1 {z}Laminar3(2)

Turbulent

(3)

Gain

Gain

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

0.002

0.0001
0

0

Loss

Loss

-0.0001-0.0020-0.0002100 200 300
Unsteady(LHS)
P12

ε12

Gain

Figure 8.
stress.

400 500 t 600-p'dv'/dx

0-p'du'/dy

0
Loss

400

500 t 600ε2θ-ε12Φ2θ-Φ12D h−v0 θ 0 i(5)= hP2θ i + hε2θ i + hΦ2θ i ,Dt∂Θ∂V
P2θ = v0 v0+ u0 θ 0
∂y
∂y
!


1
1
∂ v0 ∂ θ 0 ∂ v0 ∂ θ 0 ∂ v0 ∂ θ 0
ε2θ =
+
+
+
Re RePr
∂x ∂x
∂y ∂y
∂z ∂z
 0
∂θ
Φ2θ = −p0
∂y

0.002

-0.002100 200 300
Unsteady(LHS)P2θ

400 500 t 600-p'dθ'/dx

where P12 and P2θ are production terms, ε12 and ε2θ are dissipation terms, Φ12 and Φ2θ are pressure-strain terms, and p ispressure.
Figures 8 and 9 represent temporal evolution of the budget terms for Reynolds shear stress and turbulent heat flux, respectively. The production and pressure-strain terms becomelarger than the unsteady and dissipation terms. The production terms show similar time evolution; however, P12 marks aslightly larger value than P2θ . The pressure-strain terms marknegative values, and the term of −p0 ∂ u0 /∂ x shows a smaller
value than −p0 ∂ θ 0 /∂ x in spite of considerable similarity between u0 and θ 0 . The summation of both pressure-strain terms
for the Reynolds shear stress is comparable to the pressurestrain term of turbulent heat flux.
Figure 10 represents the differences in each budget term
between the shear component of Reynolds stresses and the turbulent heat flux. For the gradually-growth period (t < 150),
the dissipation and the pressure-strain term take positive andnegative values, respectively. For the spatial-growth period(t = 150–300), the dissipation and the production term show
positive and negative values, respectively. The pressure-strain
term oscillates and the net value shows positive for the spatialgrowth period. On the other hand, after the developing process
for t > 350 (i.e., fully-turbulent state), the pressure-strain term
leads to a decrease in heat transfer compared to the friction,
while the production and dissipation show the adverse effect ofthe dissimilarity where the friction outweighs the heat transfer.

ε2θFigure 9.
flux.

300

Figure 10. Time evolution of the differences between shear
component of Reynolds stresses and turbulent heat flux budget
terms.

Time evolution of budget for the Reynolds shear0

100 200
Unsteady
P2θ-P12

Time evolution of budget for the turbulent heat

where the upper bar shows spatial average in x and z. The laminar contributions are constant at 2/Re in both equations, since
the present Pr = 1. Figure 7 shows the summations of the laminar and turbulent contributions of C f and 2St: see the solidlines in the figure. Again, we focus on the spatial-growth period at t = 150–300. As the term of 2St is larger than the termof C f , we conjecture that the turbulent contribution produces
the effective heat transfer state, i.e., the deviation betweens
shear component of Reynolds stresses and turbulent heat flux
directly affect the effective hat transfer state.
Finlay, we discuss the transport equations of the Reynolds
shear stress −u0 v0 and the turbulent heart flux −u0 θ 0 . By averaging in the entire channel (i.e., averaging not only in x andz, but also in y), represented by angle blankets hi, the diffusion
terms are discarded. The transport equations are denoted as
D h−u0 v0 i
= hP12 i + hε12 i + hΦ12 i ,Dt∂U∂V
P12 = v0 v0+ u0 v0∂y∂y∂ u0 ∂ v0 ∂ u0 ∂ v0 ∂ u0 ∂ v0
+
+
∂x ∂x
∂y ∂y
∂z ∂z
 0

0
∂v
∂u
Φ12 = −p0
+
∂x
∂y
2
ε12 =
Re

(4)

!

CONCLUSION
We performed a direct numerical simulation of a spectral
method to study the dissimilarity between momentum and heattransfers for a developing turbulent spot generated by the pairvortices. We obtained the following conclusions.

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
turbulent spots in plane couette flow. Journal of Fluid Mechanics 229, 499–516.
Mamori, H., Fukudome, K., Ogino, K., Fukushima, N. & Yamamoto, M. 2021 Heat transfer enhancement and torquereduction by traveling wave-like blowing and suction in turbulent taylor-couette flow. Journal of Thermal Science andTechnology 16 (1), JTST0003.
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Mechanics 835, 1157–1198.
Motoki, S., Tsugawa, K., Shimizu, M. & Kawahara, G. 2021
The ultimate state of turbulent permeable-channel flow.
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within turbulent shear flows. Physical review letters 89 (1),014501.
Tillmark, N. & Alfredsson, P. H. 1992 Experiments on transition in plane couette flow. Journal of Fluid Mechanics 235,
89–102.
Tsukahara, T., Kawaguchi, Y. & Kawamura, H. 2014 An experimental study on turbulent-stripe structure in transitional
channel flow. arXiv preprint arXiv:1406.1378 .
Yamamoto, A., Hasegawa, Y. & Kasagi, N. 2013 Optimal control of dissimilar heat and momentum transfer in a fully developed turbulent channel flow. Journal of Fluid Mechanics
733, 189–220.

The obtained turbulent spot grows via two stages: gradually and rapidly growth periods. For the gradual-growth period, the longitudinal vortices extending in the flow direction
dominate. For the rapid growth period, which is called also
as the spatial-growth period, fine vortices are generated in
the turbulent spot and exhibits a linear spatial growth of the
spot. In this second stage, an effective heat transfer state isconfirmed. The FIK identity revealed that the dissimilarity
is mainly caused by the turbulent contributions owing to the
Reynolds shear stress and turbulent heat flux. In addition, thedifference between the Reynolds shear stress and the turbulent heat flux is found in their dissipation and pressure-strain
terms, and their production terms show the adverse effect on
the effective heat transfer. For the fully-developed turbulent
state, effective heat transfer is prevented by the pressure-strain
terms.

ACKNOWLEDGEMENT
K.F. acknowledges the support of the Grant-in-Aid for
Early-Career Scientists (No. 20K14671), and T.T. acknowledges the supports of the Grant-in-Aid for Scientific Research (B) (No. 19H02071) by Ministry of Education, Culture,
Sports, Science and Technology of Japan.

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