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

FEEDBACK CONTROL EFFECT IN TURBULENT CHANNEL FLOW WITH
A BUMP BY MEANS OF DIRECT NUMERICAL SIMULATION

Yusuke OKOCHI
Department of Mechanical Engineering
Keio University
Hiyoshi 3-14-1, Kohoku-ku, Yokohama, Japany.okochi@keio.jp

Yusuke NABAE
Department of Mechanical Engineering
Keio University
Hiyoshi 3-14-1, Kohoku-ku, Yokohama, Japan
yusuke.nabae@kflab.jp

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

ABSTRACTa bump has a crucial importance toward practical implementation of feedback control.
Based on the aforementioned background, the objectiveof the present study is to investigate the effect of feedback control in a turbulent channel flow with a bump by means of DNS.
We apply the opposition control as a representative example offeedback control and investigate its control effect. Then, twodifferent bump heights are considered to investigate the dependence of the drag reduction effect on the bump height. Additionally, the control method is applied at two different bulk
Reynolds numbers to investigate the Reynolds number dependence of the control in this flow geometry.

We investigate the effect of feedback control in a turbulent channel flow with a bump by means of direct numericalsimulation (DNS). In this study, the opposition control proposed by Choi et al. (1994), which is the simplest feedback
control method, is applied and the detection plane is set at
about 15 wall units from the wall. Two bulk Reynolds numbers, Reb = 5600 and 12600, and two different bump heights,
h+ ≈ 20 and 40, are considered. According to the DNS results,
although the pressure drag is hardly changed, the total drag isdecreased due to reduction of the friction drag in both casesof h+ ≈ 20 and 40. In addition, the control effect decreases
as the bump height increases due to enhancement of the flow
separation. The control effect decreases also as the Reynolds
number increases.

NUMERICAL METHODS
Coordinate system
In order to enable the simulation in the orthogonal coordinate system, the coordinate transformation proposed by Kang
& Choi (2000) is applied as follows:

INTRODUCTION
Although reduction of flow drag is of great importance forenergy savings, there are few practical flow control methods
for friction drag reduction. Therefore, development of practical flow control methods has been strongly desired.
The opposition control proposed by Choi et al. (1994) is
the simplest feedback control scheme, which suited to study
the effect of attenuation of quasi-streamwise vortices in the region near the wall. In this control method, the control input on
the walls is determined based on the velocity in the detection
plane. Choi et al. (1994) achieved 25% drag reduction in a relatively low-Reynolds-number turbulent plane channel flow bymeans of direct numerical simulation (DNS). Hammond et al.
(1998) optimized the location of the detection plane in the opposition control, and the drag reduction rate was found to be
maximum when the detection plane was placed about 15 wall
units above the wall, i.e., y+ ≈ 15.
On the other hand, the effect of feedback control in a turbulent channel flow with a bump has been still unknown due
to its complexity, e.g., flow separation. However, investigationof the feedback control effect in a turbulent channel flow withx = ξ1 , y = ξ2 (1 + η ) + ηl , z = ξ3 ,(1)

where ξ1 ,ξ2 and ξ3 denote the streamwise, wall-normal andspanwise directions in the computational space, respectively.
Also, η = (ηu − ηl )/2 with ηu and ηl denoting the displacement of the upper and lower walls. Following Banchetti et al.
(2020), who studied the effect of streamwise-travelling wavesof spanwise velocity over a bump, ηu and ηl are defined asηu = 0,

(2)

" 
" 
 #
 #
ξ1 − b 2
ξ1 − b 2′
ηl = a exp −+ a exp −
,
c
c′

(3)

where a, a′ , c and c′ are the parameters to determine the configuration of the bump. In this study, these parameters are
set to (a, a′ , c, c′ ) = (0.0505, 0.060425, 0.2922, 0.3847),

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

Computational conditions.

Reb

L1 × L2 × L3

N1 × N2 × N3

∆ξ1+

∆ξ2+

∆ξ3+

Driver5600

4π × 2 × 2π

256 × 96 × 256

8.8

0.9-6.0

4.4

Main

5600

4π × 2 × 2π

256 × 96 × 256

8.8

0.9-6.0

4.4

Driver12600

2π × 2 × π

256 × 192 × 256

8.8

0.9-6.0

4.4

Main

12600

2π × 2 × π

256 × 192 × 256

8.8

0.9-6.0

4.4

Table 2. Each drag coefficient in the uncontrolled and controlled cases and the reduction rate of the drag coefficient whenh+ ≈ 20 and Reb = 5600.

Driver region
Periodicboudary condition

Periodic
bondary condition

Uncontrolled

ContolledRD

CD f ,u [×10−3 ]

8.08

6.07

24.9%

[×10−3 ]

7.99

5.95

25.5%

CDp,l [×10−3 ]

1.20

1.17

2.5%

[×10−3 ]

17.27

13.19

23.6%

Main region
Inflowboundary condition

Streamwise direction

Outflowboundary condition

CD f ,lFigure 1.

The schematic of the computational domains.

CDso that the bump height is h+ ≈ 20, where ()+ denotes
the wall unit based on the friction velocity of the uncontrolled flow. For the case of h+ ≈ 40, (a, a′ , c, c′ ) =(0.1010, 0.120850, 0.2922, 0.3847). The bump is located at
the center of the channel, i.e., b = 2π .

is imposed in both domains. Note that in order to make thetotal mass flux from the wall zero, the second term on the righthand-side in Eq. (6) is added. In this study, the detection plane+≈ 15. The DNS code is based on that of Fukagata
is set to ξ2,det al. (2006) and the code for the driver domain is based onthat of a spatially developing turbulent boundary layer flow
extended by Kametani & Fukagata (2011). The computational
conditions are shown in Table 1.

Governing equations
The governing equations are the incompressible continuity and Navier-Stokes equations in the boundary-fitted coordinates, i.e.,

∂ ui
= −S,
∂ ξi

∂ ui u j1 ∂ 2 ui
dP
∂ ui
∂p
−
+
−
δi1 + Si ,
=−
∂t
∂ξj
∂ ξi Reb ∂ ξ j ξ j d ξ1

Drag coefficients(4)

The total drag coefficient, CD can be decomposed into thefriction drag, CD f and the pressure drag, CDp , i.e.,(5)CD = CD f ,u +CD f ,l +CDp,l ,where S and Si in Eqs. (4) and (5) denote the additional terms
due to the coordinate transformation. All variables are madedimensionless by twice the bulk-mean velocity 2Ub∗ , the channel half-width δ ∗ , and the fluid density ρ ∗ . The bulk Reynolds
number, Reb = 2Ub∗ δ ∗ /ν ∗ is 5600 or 12600, which corresponds to the friction Reynolds number of Reτ = 180 and 360,
respectively, in the uncontrolled flow. Here, the superscript ∗denotes dimensional quantities. All simulations are conductedunder the constant flow rate condition.

where (·)u and (·)l denote quantities on the upper and lower
walls, respectively. Similar to Mori et al. (2017), the friction
and pressure drags can be computed as follows:

CD f =

11(1/2)ρ ∗Ub∗2 L1 L3

Z L1 Z L3

e 1 · τ ∗ · n dx3 dx1 ,

CDp =

1
1
(1/2)ρ ∗Ub∗2 L1 L3

Z L1 Z L3

e 1 · (−p∗w I ) · n dx3 dx1 ,

0

0

0

0

Numerical conditions
The computational domain consists of two regions, i.e., adriver region without a bump and a main region with a bump as
shown in Figure 1. In the driver region, the periodic boundary
condition is applied in the streamwise (ξ1 ) and spanwise (ξ3 )directions, and the no-slip condition is applied on both walls.
In the main region, the inflow and outflow boundary conditions are applied in the streamwise direction and the periodic
boundary condition is applied in the spanwise direction. On
the lower wall (ξ2 = 0), the following velocity, i.e.,ξ3v(ξ1 , 0, ξ3 ) = −v(ξ1 , ξ2,d , ξ3 ) + v(ξ1 , ξ2,d , ξ3 ) ,

(7)

(8)

(9)

where τ ∗ , pw , e 1 and n represent the viscous stress tensor, thepressure on the wall, the streamwise unit vector and the wallnormal unit vector, respectively.

RESULTS AND DISCUSSION
Effects of the opposition control
For the bump height of h+ ≈ 20 at Reb = 5600, the totaldrag reduction rate obtained is 23.6%. Each component of
the drag coefficients in the uncontrolled and controlled cases
and the reduction rate by the control are shown in Table 2.

(6)2

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

(a)

(a)

(b)
(b)

Figure 3. Streamwise velocity in the region around and rear
the bump in the case of h+ ≈ 20 at Reb = 5600: (a) uncontrolled case; (b) controlled case.

Figure 2. Visualization of vortical structures in the uncontrolled and controlled cases (threshhold, Q+ = −0.07; color,
the wall-normal coordinate in the real space, y+ ): (a) uncontrolled case; (b) controlled case.

Focusing on the contributions from the friction drag and thepressure drag, the friction drag decreases by 25.5% and thepressure drag slightly decreases by 2.5%. The friction dragreduction rate in the present study is almost the same as thatin the case without a bump, i.e., 25% (Hammond et al., 1998)
on both walls. Thus, existence of a bump hardly decreases thefriction drag reduction effect.
Figure 2 shows the visualization of vortical structures
identified by the second invariant of velocity gradient tensor,
Q (Hunt et al., 1988). In the upstream region of the bump(x+ < 1000), vortical structures are significantly suppressed by
the control. This trend is basically similar to that in the casewithout a bump (Hammond et al., 1998). Conversely, vorticalstructures around and rear the bump (1200 < x+ < 1400) areenhanced and turbulence is activated by the control.
Figure 3 shows a comparison of streamwise velocity
fields around and rear the bump. The white line in Figure 3
represents u = 0. In the region behind the bump, streamwise
velocity is negative, indicating that the flow is separated fromthe lower wall. Compared to the uncontrolled case, the reattachment position in the controlled case shifts further downstream as shown in Figure 3. It is clearly observed that theflow separation is enhanced downstream by the control.
Figure 4 shows the friction drag and the pressure on thelower wall in the streamwise direction. As shown in Figure 4(a), the friction drag is reduced by the control in the wide region except for the region just behind the bump center where
the flow is separated. On the other hand, the friction drag behind the bump (1200 < x+ < 1400) is not suppressed by thecontrol. Therefore, it can be suggested that the effect of thefriction drag reduction by the opposition control is hardly obtained in the region where the flow separation occurs. Also,
the pressure on the lower wall is not almost modified by thecontrol in the whole region of the channel.
As shown in Figure 5, similar to the case without a bump
(Hammond et al., 1998), streak structures lined up in the spanwise direction at ξ2+ ≈ 8 are suppressed in the regions where

Figure 4. The friction drag coefficient and the pressure on thelower wall in the uncontrolled and controlled cases of h+ ≈ 20
at Reb = 5600 (colored area shows the region where the bump
exists): (a) the friction drag coefficient; (b) pressure on thelower wall.

the friction drag reduction is achieved (x+ < 1000, x+ >1400). Since the suppression of the streak structures wasobserved also in the case without a bump (Hammond et al.,1998), it is implied that the mechanism of the friction drag reduction in this study is basically similar to the case without abump. On the other hand, the streak structures just behind thebump (1200 < x+ < 1400) are weak even in the uncontrolled
case, so that the control effect just behind the bump is little.
The Reynolds shear stress (RSS) at four different locations, i.e., upstream (x+ ≈ 1050), the top (x+ ≈ 1150), downstream (x+ ≈ 1250) of the bump and rear the bump (x+ ≈2000) is shown in Figure 6. Except for the downstream regionof the bump (x+ ≈ 1250), RSS shifts away from the wall, resulting in suppression of RSS near the wall. This trend is similar to the case without a bump. On the other hand, the RSS
in the region near the wall is hardly suppressed at x+ ≈ 1250,
indicating that the flow separation vanishes the control effectbehind the bump.

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

(a)

(b)

Figure 5. Streamwise velocity fluctuation at ξ2+ ≈ 8 in theuncontrolled and controlled cases of h+ ≈ 20 at Reb = 5600:(a) uncontrolled case; (b) controlled case. Solid white line,
the bump; dashed white line, the top of the bump.

(c)

When the opposition control is applied to a turbulent
channel flow with a bump, the friction drag reduction effectis confirmed in the wide region except for the downstream ofthe bump where the flow is separated. On the other hand, it
is also confirmed that the opposition control has a small effecton the pressure drag, and the flow separation is enhanced. On
average, in the case of h+ ≈ 20, the total drag is reduced because the friction drag, which accounts for a large amount of
the total drag, is reduced by the control.

Dependence of the control effect on the bumpheightIn order to investigate the dependence of the drag reduction effect on the bump height, the opposition control isapplied in the case where the bump height is doubled, i.e.,h+ ≈ 40. Each drag coefficient in the uncontrolled and controlled cases and the reduction rate of the drag coefficient whenh+ ≈ 40 and Reb = 5600 are shown in Table 3. Similar to thecase of h+ ≈ 20, the friction drag on both walls is decreased,
and the pressure drag is slightly decreased. However, the friction drag reduction rates on both walls decrease and the pressure drag increases as the height of the bump increases. Thus,
compared to the case of h+ ≈ 20, the total drag reduction rate
becomes small, i.e., RD = 15.0%.
The friction drag coefficient and the pressure on the lower
wall are shown in Figure 7. The friction drag is reduced in the
upstream region of the bump (x+ < 1000). This can be observed also in the case of h+ ≈ 20. On the other hand, unlike
the case of h+ ≈ 20, the friction drag is reduced also in thedownstream region of the bump (x+ ≈ 1500). Streamwise andwall-normal velocities at x+ ≈ 1500 where the friction drag is
reduced are shown in Figure 8. As shown in Figure 8 (a), thenegative streamwise velocity near the lower wall is enhancedby the control, and it is clear that the flow separation is en-(d)Figure 6. Comparison of Reynolds shear stress in the caseof h+ ≈ 20 at Reb = 5600: (a) upstream (x+ ≈ 1050); (b) top(x+ ≈ 1150); (c) downstream (x+ ≈ 1250); (d) the region rear
the bump (x+ ≈ 2000).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 3. Each drag coefficient in the uncontrolled and controlled cases and the reduction rate of the drag coefficient whenh+ ≈ 40 and Reb = 5600.

Uncontrolled

ContolledRD[×10−3 ]

8.22

6.44

21.6%

CD f ,l [×10−3 ]

7.71

6.21

19.4%

[×10−3 ]

6.47

6.39

1.2%

22.40

19.04

15.0%

CD f ,uCDp,lCD [×10−3 ]

(a)

(b)

(b)

Figure 8. Streamwise and wall-normal velocities at x+ ≈1500 in the uncontrolled and controlled cases where h+ ≈ 40
and Reb = 5600: (a) streamwise velocity; (b) wall-normal velocity.
Figure 7. The friction drag coefficient and the pressure on thelower wall in the uncontrolled and controlled cases of h+ ≈ 40
at Reb = 5600 (colored area shows the region where the bump
exists): (a) the friction drag coefficient; (b) pressure on thelower wall.

Table 4. Each drag coefficient in the uncontrolled and controlled cases and the reduction rate of the drag coefficient whenh+ ≈ 20 and Reb = 12600.

hanced by the control. As shown in Figure 8 (b), negativewall-normal velocity is suppressed by the control and it is also
due to enhancement of the flow separation. Therefore, it issuggested that the enhancement of the flow separation leads tothe friction drag reduction. It is also noticeable that the friction
drag rear the reattachment position (x+ > 1600) is not reduced(Figure 4 (a)) unlike the case of h+ ≈ 20. Similar to the caseof h+ ≈ 20, the control effect for the pressure drag is little.
In sum, although the control reduces the total drag also inthe case of h+ ≈ 40, the total drag reduction rate is smaller thanthat in the case of h+ ≈ 20. In addition, the flow separation isenhanced by the control in the downstream region of the bump,and it leads to friction drag reduction.

Uncontrolled

ContolledRD

CD f ,u [×10−3 ]

6.55

5.11

21.9%

CD f ,l [×10−3 ]

6.60

4.94

25.1%

CDp,l [×10−3 ]

0.92

0.93

−1.2%

CD [×10−3 ]

14.07

10.98

21.9%

cording to Chang et al. (2002), the friction drag reduction rateby the opposition control decreases as the Reynolds number
increases in the case without a bump, and the same trend isobserved on the upper wall in the case with the bump. On theother hand, the friction drag reduction rate on the lower wall issimilar to that in the case at Reb = 5600. In other words, thedependence of the control effect on the Reynolds number isdifferent whether a bump exists or not. Focusing on the pressure drag, like the case at Reb = 5600, the pressure drag isnearly unchanged.
Figure 9 shows the friction drag and the pressure on thelower wall in the streamwise direction. Similar to the case at
Reb = 5600 as shown in Figure 4, reduction of the friction
drag can be observed in a wide range except for the vicinity of the bump. In the downstream region of the bump, thefriction drag is reduced behind x+ ≈ 1300, while the friction
drag reduction is observed behind x+ ≈ 1400 in the case at

Dependence of the control effect on the
Reynolds number
In order to investigate the Reynolds number effect, theopposition control is applied in a flow at a higher Reynolds
number, i.e., Reb = 12600. Here, the bump height is set toh+ ≈ 20. Each drag coefficient in the uncontrolled and controlled cases and the reduction rate of the drag coefficient whenh+ ≈ 20 and Reb = 12600 are shown in Table 4. The total dragreduction rate is slightly smaller than the case at Reb = 5600
as shown in Table 2. The friction drag reduction rate on theupper wall decreases as the Reynolds number increases. Ac-

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a)around the bump depends on the Reynolds number.

CONCLUSIONS

-4

5

×10

Direct numerical simulation (DNS) of feedback control
in a turbulent channel flow with a bump is performed in thepresent study. As the simplest feedback control, the opposition control is applied. We can achieve the total drag reduction by the control, and the drag reduction rate is found tobe nearly unchanged from that in the case without a bump.
It is also found that the control effect on the pressure drag ishardly observed. Furthermore, in order to investigate the dependence of the drag reduction effect on the bump height, DNS
is performed in the case where the bump height is doubled. Although the total drag reduction can be achieved also in the caseof h+ ≈ 40, it is smaller than that in the case of h+ ≈ 20 due
to stronger flow separation and larger pressure drag. In addition, in order to investigate the Reynolds number effect of thecontrol, we apply the control to a flow at a higher Reynolds
number, i.e., Reb = 12600. The total drag reduction rate at
Reb = 12600 is smaller than the case at Reb = 5600, and thedifference of local drag reduction rate can be observed.
In the future, we will clarify a general relationship between the local pressure gradient and the local drag reduction
rate to obtain more universal knowledge on the flow control
effect in practical boundary layer flows and to suggest a more
effective control method.

0
600

800

1000

(b)

Figure 9. The friction drag coefficient and the pressure on thelower wall in the uncontrolled and controlled cases of h+ ≈ 20
at Reb = 12600 (colored area shows the region where the bump
exists): (a) the friction drag coefficient; (b) pressure on thelower wall.

ACKNOWLEGMENTS
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.

REFERENCES

Figure 10. The reduction rate of the local friction drag coefficient at Reb = 5600 and 12600 (colored area shows the region
where the bump exists). Note that the reduction rate is plottedin only the region where the local friction drag coefficient in
the uncontrolled case is positive.

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Choi, H., Moin, P. & Kim, J. 1994 Active turbulence control
for drag reduction in wall-bounded flows. J. Fluid Mech.
262, 75–110.
Fukagata, K., Kasagi, N. & Koumoutsakos, P. 2006 A theoretical prediction of friction drag reduction in turbulent flowby superhydrophobic surfaces. Phys. Fluids 18, 051703.
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mechanisms for turbulence attenuation and enhancement in
opposition-controlled wall-bounded flows. Phys. Fluids 10,
2421–2423.
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and convergence zones in turbulent flows. In Proceedings of
the Summer Program, pp. 193–208. Center for Turbulence
Research.
Kametani, Y. & Fukagata, K. 2011 Direct numerical simulation of spatially developing turbulent boundary layers withuniform blowing or suction. J. Fluid Mech. 681, 154–172.
Kang, S. & Choi, H. 2000 Active wall motions for skin-friction
drag reduction. Phys. Fluids 12, 3301–3304.
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Reb = 5600. It indicates that the separated flow reattatches
in more upstream position at a higher Reynolds number andit leads to the larger friction drag reduction rate on the lower
wall at Reb = 12600. In addition, it can be also observed that
there is almost no change in the pressure drag by the control,
similarly to the case at Reb = 5600.
Furthermore, in order to investigate the Reynolds number
dependence of the friction drag reduction effect by the control in more detail, the local friction drag reduction rates in thecases at Reb = 5600 and 12600 are shown in Figure 10. In just
upstream region of the bump, the friction drag reduction rate
at Reb = 5600 is slightly larger than that at Reb = 12600. This
trend is basically similar to the case without a bump. In thedownstream region of the bump, the local reduction rate of thefriction drag shifts slightly upstream as Reynolds number increases due to the shift of the reattatchment position. Near thereattachment position, the drag reduction rate is significantly
large at both Reynolds numbers.
The total drag reduction rate decreases as the Reynolds
number increases, and the local friction drag reduction effect6