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

THEORETICAL AND NUMERICAL ANALYSES OF UNIFORM BLOWINGAND SUCTION IN TURBULENT PLANE COUETTE FLOW

Yusuke Nabae
Department of Mechanical Engineering, Keio University,
Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522, Japan
Department of Mechanical Engineering, Tokyo University of Science,
6-3-1 Niijyuku, Katsushika-ku, 125-8585, Japan
yusuke.nabae@kﬂab.jp

Koji Fukagata
Department of Mechanical Engineering, Keio University,
Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522, Japan
fukagata@mech.keio.ac.jp

ABSTRACT

structures in the region far from the walls (Lee and Moser,
2018). Therefore, in the present study, we ﬁrst analytically investigate whether overall drag reduction and net energy saving
is possible with UB/US in turbulent plane Couette ﬂows, thenwe complement the analysis by numerical simulations.

We theoretically and numerically investigate the control effect of uniform blowing/suction (UB/US) in a turbulent plane Couette ﬂow under a constant velocity difference(CVD) condition in terms of the net energy saving. The semitheoretical upper bound of the ratio between UB/US amplitude
and the wall velocity, Vw /Uw , for the net energy saving is about
0.45% at very low Reynolds numbers, i.e., Rew ∼ 103 , while itis less than 0.05% at practically high Reynolds numbers, i.e.,
Rew ∼ 109 –1010 . According to the present simulations, however, it is found that the amplitude of Vw /Uw ≤ 0.1%, (i.e., theamplitude often used for external ﬂows) is too small to obtain
substantial control effects in this ﬂow conﬁguration, while thelarger amplitude, i.e., Vw /Uw ≥ 0.3%, results in a negative net
energy saving due to the drastic enhancement of the contribution from the blowing and suction.

THEORETICAL ANALYSIS
In this study, all variables are made dimensionless by
the ﬂuid density, ρ ∗ , the wall velocity, Uw∗ , and the channel
half-width, δ ∗ , where ∗ denotes dimensional quantities. The
subscripts 1, 2, and 3 represent the streamwise, wall-normal,
and spanwise directions, respectively. Also, ui (i = 1, 2, 3),p, and t denote the velocity components, pressure, and time,
respectively. Note that, for convenience, (x1 , x2 , x3 ) and(u1 , u2 , u3 ) are interexchangably denoted by (x, y, z) and(u, v, w), respectively. The Reynolds number is deﬁned as
Rew = δ ∗Uw∗ /ν ∗ , where ν ∗ is the kinematic viscosity. Similar to Sumitani and Kasagi (1995), the uniform blowing on thelower wall (y = 0) and the uniform suction on the upper wall(y = 2) are imposed. In the present theoretical analysis, study,
we investigate the range of the UB/US amplitude required for
net energy saving.
The net energy saving rate, S, is deﬁned as (e.g., Kasagi
et al., 2009)

INTRODUCTION
Since the skin friction drag in a turbulent ﬂow is muchhigher than that in the corresponding laminar ﬂow, it is important to establish ﬂow control methods for reduction of thefriction drag. Among various control methods, the uniform
blowing and suction (UB/US) is a simple and well-known ﬂow
control method. Sumitani and Kasagi (1995) investigated theeffect of this control in turbulent Poiseuille ﬂow. According
to their study, the turbulence is enhanced (suppressed) on theblowing (suction) side, while the drag is decreased (increased)on the blowing (suction) side. Also in a spatially developing
turbulent boundary layer ﬂow, the similar effect was conﬁrmed
at relatively low (Kametani and Fukagata, 2011) and moderate(Kametani et al., 2015) Reynolds numbers. Based on thesestudies, UB at the amplitude around 0.1% of the free-stream
velocity has also been used in a recent wind-tunnel experiment for airfoil drag reduction (e.g., Eto et al., 2019; Miuraet al., 2021). However, this control effect has not sufﬁciently
been investigated for a turbulent plane Couette ﬂow, despitethat it is more or less similar to a ﬂat-plate turbulent boundary
layer. This is because an enormous computational domain in
the streamwise direction is required to capture the large-scaleS=
Wp0 − Wp +Wa
,
Wp0

(1)

where Wp0 is the driving power in the uncontrolled case, i.e.,1
Wp0 = Uw ·
2Rew

∂U
∂U
+
∂ y y=0∂ y y=2

!


=

1
Uw+

2
. (2)

Also, Wp and Wa are the driving power and actuation powerin the controlled case, respectively. According to the overall
energy balance (e.g., Fukagata et al. 2009), the summation of1

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

non-effective range

effective range

Figure 1. Relationship between the upper bound of the UB/US amplitude, Vwc /Uw , and the Reynolds number, Rew , according tothe obtained theoretical formula (Eq. (10)). Blue and red represent the effective and non-effective ranges of the UB/US amplitude,
respectively.

these values, i.e., Wp +Wa , can be computed using the dissipation rate, i.e.,
Wp +Wa =1
V

The drag coefﬁcient in the uncontrolled ﬂow, C f 0 , is computed
as

Z


2
Si j S ji dV,
Re
V
w

(3)

Cf0 =

where Si j and V are the strain-rate tensor and the volume of
the computational box, respectively.
In the present study, the effective range of the UB/US amplitude is derived under a hypothesis that the lower bound of
Wp + Wa is the value corresponding
to the laminar ﬂow with
UB/US, i.e., Wp +Wa lam :
Wp +Wa lam ≤ Wp +Wa ,

Cf0 =

,

(9)

where G is the empirical coefﬁcient, and we use the value proposed by Pirozzoli et al. (2014), i.e., G = 0.424. Substituting
Eqs. (2), (6), (8), and (9) into (5), we can obtain the following
inequality,

Vw

0.212
exp (2Vw Rew ) + 1≤.
exp (2Vw Rew ) − 1 (ln Rew )2

(10)

(5)
Solving Eq. (10) in the range of Vw > 0, we can obtain theeffective range of Vw , i.e., 0 < Vw < Vwc , where Vwc denotes theupper bound of the UB/US amplitude having the possibility of
net energy saving.
Figure 1 shows the relationship between Vwc /Uw and Rew .
According to this ﬁgure, Vwc /Uw decreases as the Reynolds
number increases, and it is less than 0.05% at signiﬁcantly
high Reynolds numbers, i.e., Rew ∼ 109 –1010 . Even at very
low Reynolds numbers, i.e., Rew ≈ 1000, Vwc /Uw is about
0.45%. Although the UB/US can relaminarize the turbulent
Couette ﬂow, this semi-theoretical analysis suggests that the
net energy saving is possible only when the UB/US amplitude
is very small.

(6)

where Vw denotes the blowing or suction velocity amplitude(Vw > 0). Note that the laminar solution with UB/US can analytically be derived asexp (Vw Rew y) − 1
Ulam = 2− 1.
exp (2Vw Rew ) − 1

G(ln Rew )2(4)

which suggests that the net energy saving is possible
only
when this range exists. Using Eq. (3), Wp +Wa lam is computed asZ
12 ∂ Ulam
Wp +Wa lam =·dV
V V Rew∂yexp (2Vw Rew ) + 1= Vw,
exp (2Vw Rew ) − 1(8)where τw0 denotes the wall shear stress in the uncontrolled
ﬂow. Also, using the empirical formula proposed by Robertson (1959), C f 0 is expressed aswhere the subscript ”lam” denotes quantities in the corresponding laminar ﬂow with UB/US. In the situation where thepositive net energy saving rate is achieved, Wp + Wa should
also be less than the pumping power in the uncontrolled case,i.e.,
Wp +Wa lam ≤ Wp +Wa ≤ Wp0 ,



∗
2τw0
1 2
=
2
,
Uw+
ρ ∗Uw∗ 2

DIRECT NUMERICAL SIMULATIONAs a complementary investigation, we perform a direct
numerical simulation (DNS) of a fully developed turbulent(7)212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1.

Computational conditions for the validation and veriﬁcation.

Case

Reτ

Lx × Ly × Lz

Nx × Ny × Nz

∆x+

∆y+

∆z+

Case 1180.6

8 π × 2 × 4π

256 × 96 × 256

17.7

0.9–6.0

8.8

Case 2181.2

8 π × 2 × 4π

512 × 96 × 512

8.8

0.9–6.0

4.4

Case 3178.9

16π × 2 × 4π

512 × 96 × 256

17.7

0.9–6.0

8.8

Case 4180.2

32π × 2 × 4π

1024 × 96 × 256

17.7

0.9–6.0

8.8

Case 5179.8

64π × 2 × 4π

2048 × 96 × 256

17.7

0.9–6.0

8.8

Avsarkisov et al. (2014)180

20π × 2 × 6π

1296 × 251 × 768

13.1

0.92–5.9

6.6

100

140

(a) 30(b) 1.2

25

1

20

0.8

15

0.6

10

0.4

5

0.2

0

100

101

0

102

0

20

40

60

80

120

160

180

Figure 2. Mean velocity and the Reynolds shear stress (RSS) of the uncontrolled ﬂow in the different computational conditions: (a)mean velocity proﬁle; (b) RSS proﬁle.

plane Couette ﬂow controlled using the UB/US. The governing
equations are the incompressible continuity and Navier-Stokes
equations, i.e.,

∂ ui
= 0,
∂ xi

The velocity amplitude of the UB/US is varied as 0.05%,
0.1%, 0.3%, 0.5%, and 1.0% of the wall velocity, Uw .
The present DNS code is developed based on the DNS
code developed by Fukagata et al. (2006). The governing
equations are spatially discretized by means of the energyconserving fourth-order ﬁnit difference scheme (Morinishi
et al., 1998) in the streamwise and spanwise directions andsecond-order ﬁnite difference scheme in the wall-normal
direction. The time integration is performed by meansof the low-storage, third-order Runge-Kutta/Crank-Nicolson
scheme (Spalart et al., 1991) with the high-order SMAC-likevelocity-pressure coupling scheme (Dukowicz and Dvinsky,1992).
First, we validate and verify the present DNS code. Table 1 shows the computational conditions for the validationand veriﬁcation of the present simulation. Here, Li , Ni , and ∆xi
denote the length of the computational box, the number of thegrid points, and the grid spacing, respectively, in the i-th direction. Figure 2 shows the mean velocity proﬁle and Reynolds
shear stress (RSS) of the uncontrolled ﬂow in all cases. As
shown in Fig. 2, these statistics in all cases are in good agreement with the reference results by Avsarkisov et al. (2014).
Figure 3 shows the comparison of the mean velocity proﬁlein the relaminarization case, i.e., Vw /Uw = 1.0% (explainedlater). As can be observed in Fig. 3, the dependence of thecomputational domain can hardly be observed. Therefore, it isfound that the computational condition of Case 1 is sufﬁcientfor the present simulation.
Figure 4 shows the comparison of the mean velocity pro-(11)∂ ui u j1 ∂ 2 ui
∂ ui
∂p
=−
−
+
.
∂t
∂xj
∂ xi Rew ∂ x j ∂ x j(12)

The Reynolds number is set to Rew = 3200, which corresponds
to the friction Reynolds number of Reτ ≃ 180 in the uncontrolled ﬂow.
The periodic boundary condition is applied in the streamwise (x) and spanwise (z) directions. The boundary conditionsof the streamwise and spanwise velocities on the walls are applied as u = Uw , w = 0
(upper wall),(13)u = −Uw , w = 0 (lower wall).

Also, the boundary condition of the wall-normal velocity on
the wall in the uncontrolled and controlled cases as follows: v(x, 0, z) = v(x, 2, z) = 0 (uncontrolled),
(14)v(x, 0, z) = v(x, 2, z) = Vw (UB/US).

3

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

2
1.5
1
0.5
0

Figure 3.

-1

-0.5

0

0.5

1

Mean velocity proﬁles of the controlled ﬂow (Vw /Uw = 1.0%) in the different computational conditions.

(a) 2(b) 1.2
1

1.5
0.8
0.6

1

0.4

0.5
0.2

0

0

-1

-0.5

0

0.5

1

0

50

100

150

200

250

300

350

Figure 4. Mean velocity proﬁles and RSS for different UB/US amplitudes, Vw /Uw , obtained by DNS: (a) mean velocity proﬁle; (b)RSS.
Table 2.

Net energy saving rate, S, for different UB/US amplitudes, Vw /Uw , computed from DNS results.

Vw /Uw

0.05%

0.1%

0.3%

0.5%

1.0%

S

3.9%

1.9%

−20.3%−60.9%−216.0%than the upper bound at Rew = 3200, i.e., Vwc /Uw ≈ 0.33%
shown in Fig. 1.
In order to investigate the detailed effect of UB/US
control, we investigate different contributions to the wall
shear stress using the Fukagata-Iwamoto-Kasagi (FIK) identity (Fukagata et al., 2002). The FIK identity in the turbulent
plane Couette ﬂow with UB/US control under CVD conditionis expressed as

ﬁle and RSS in the uncontrolled and controlled cases. Notethat circle denotes the laminar solution of Vw /Uw = 1.0% in
Eq. (7). As can be observed in Fig. 4(a), the mean velocity
proﬁle shifts upward as the UB/US amplitude increases. The
proﬁle in the case of Vw /Uw = 1.0% is in perfect agreementwith the laminar solution of Eq. (7). In other words, in thecase of Vw /Uw = 1.0%, the ﬂow is relaminalized by UB/US.
According to Fig. 4(b), the RSS in the case of Vw /Uw = 1.0%,
i.e., the relaminarization case, is perfectly suppressed. In othercases (not relaminarization), although the RSS are enhanced in
the region near the wall of the suction side, those in the otherregion are suppressed compared with the uncontrolled case.
The net energy saving rates in all cases are shown in Table 2.
Although the small net energy saving effect can be observedfor Vw /Uw ≤ 0.1%, the net energy saving effect cannot be obtained for Vw /Uw > 0.1% despite the relaminarization. Thisis because the UB/US amplitude of Vw /Uw = 1.0% is greater

τw =

1
1
+
Rew 2
|{z} |
τwl

Z 2
0


−u′ v′ dy −VwUb ,| {z }
{z
}
bs
τwr

(15)

τw

where ()l , ()r , and ()bs denote the contribution from the laminar ﬂow, the RSS, and the blowing and suction, respectively.

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

0.012
0.01
0.008
0.006
0.004
0.002
0

NC

0.05%

0.1%

0.3%

0.5%

1.0%

Figure 5. Contribution to the wall shear stress, τw , from each term of Eq. (15) in the uncontrolled and controlled cases. Black, darkgray, and light gray represent the contribution from the laminar ﬂow, the RSS, and the blowing and suction, respectively.

REFERENCES

Also, Ub denotes the bulk-mean velocity. Figure 5 shows
the contribution from each term in right-hand-side (RHS) of
Eq. (15). As shown in Fig. 5, the contribution from the blowing and suction increases and the contribution from the RSS
decreases as the UB/US amplitude increases. In the relaminarization case, i.e., Vw /Uw = 1.0%, there is no contribution fromthe RSS, and that from the blowing and suction is drasticallyenhanced. In the cases of the smaller UB/US amplitude, i.e.,
Vw /Uw ≤ 0.1%, the contribution from the RSS is nearly unchanged, and there is little contribution from the blowing andsuction, so that the net energy saving effect is hardly obtained.
On the other hand, in the cases of larger UB/US amplitude,i.e., Vw /Uw ≥ 0.3%, the amount of decrease of the contribution from the RSS is smaller than that of increase of the contribution from the blowing and suction, so that the total dragincreases. Large increase of the total drag leads to large negative net energy saving rate.

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CONCLUSIONS
We theoretically and numerically investigate the effect of
the uniform blowing and suction control in turbulent plane
Couette ﬂow under the constant velocity difference condition.
According to the present semi-analytical solution, the upperbound of the UB/US amplitude decreases as the Reynolds
number decreases, and it is less than 0.05% of the wall velocity
at signiﬁcantly high Reynolds numbers, i.e., Rew ∼ 109 –1010 .
The results of a direct numerical simulation (DNS) shows thatthe net energy saving can be achieved only when the blowing
amplitude is very small, Vw /Uw ≤ 0.1%, and we cannot obtainthe net energy saving for larger amplitudes even if the ﬂow
is relaminarized. According to the analysis of the FIK identity, in the cases of larger UB/US amplitude, the drag is drastically increased due to the large contribution from the blowing
and suction beyond the amount of decrease of the contribution
from the Reynolds shear stress.

ACKNOWLEDGMENTS
Authors are grateful to Drs S. Obi, K. Ando, and T.
Kawata (Keio University) for fruitful discussions. This work
was supported through JSPS KAKENHI grant No. 18H03758
and 21H05007 by Japan Society for the Promotion of Science.

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