Turbulent Friction Drag Reduction:

From Feedback to Predetermined,

and Feedback Again

Koji Fukagata

Department of Mechanical Engineering

Keio University, JAPAN

http://kflab.jp

5th Symposium on Fluid-Structure-Sound Interactions and Control (FSSIC2019)

27-30 August 2019, Chania, Crete island, Greece

Plenary Lecture

2/46 Outline

1. From Feedback to

Predetermined Control

– Streamwise traveling wave

– Uniform blowing

2. Feedback Control Again

– Resolvent analysis

3. Toward the Future

– Machine learning

3/46 Why friction drag is larger in turbulent flows?

• Exchange of momentum by these vortices

4/46

• Opposition control (Choi et al., J. Fluid Mech., 1994)

– Sensors: wall-normal velocity sensors above the wall

– Actuators: blowing/suction continuously distributed on the wall

– To oppose the quasi-streamwise vortices

– About 20% drag reduction in DNS of low-Re channel flow

Then, let us suppress these vortices

→ Opposition control by Choi et al. (1994)

(Iwamoto, private commun.)

z

y

suction blowing

v(x,0,z,t) v(x, y

d,z,t)

5/46 But one cannot place sensors above the wall?

• Different approaches to substitute this idealization …

– Assume control law and optimize the parameters

• Neural network (Lee et al., PoF 1997)

• Genetic algorithm --- used also in wind tunnel experiment (Yoshino, Suzuki, and Kasagi, JFST 2009)

– Suboptimal control theory

• Spanwise wall-shear or wall pressure (Lee et al., JFM 1998)

– They also used streamwise wall-shear, but did not observe drag reduction --- we come back to this later

• Streamwise wall-shear (from Taylor expansion of the near-wall Reynolds shear stress) (Fukagata & Kasagi, IJHFF 2004)

– State estimation techniques

• Kalman filter：Lee et al. (2001), Högberg et al. (2003), Hoepffner et al. (2005), Chevalier et al. (2006)

• Neural network：Milano & Koumoutsakos (2002)

• High-order Taylor expansion：Bewley & Protas (2004)

6/46

Feedback system for turbulence control ・・・ 6% drag reduction

GA-based control (wind tunnel experiment) (Yoshino et al., J. Fluid Sci. Technol. 3, 2008;

Kasagi, Suzuki, and Fukagata, Annu. Rev. Fluid Mech. 41, 2009)

7/46

Big issue toward practical application:

Physical diameter and frequency of QSV

SMA： Shape Memory Alloy

Elec.：Electro-magnetic,

Electro-static, Piezo,

Electro-strictive, etc.

PA：Plasma Actuator SMA

Elec.

PA

Physical diameter and

frequency of QSV

(Kasagi, Suzuki, and Fukagata,

Annu. Rev. Fluid Mech. 41, 2009)

8/46

From feedback control to predetermined

control…

• Feedback (closed-loop) control

– Sensors, actuators, controller

– Potentially effective

– Big hurdle for hardware development (especially if the QSVs are targeted at)

• Predetermined (open-loop) control without sensors

– Perhaps less difficult to make hardware

– But how?

Flow

Controller Sensors Actuators

Flow

Controller Actuators

9/46 To start with predetermined control…

• Relationship between friction drag and turbulence statistics for a

fully-developed incompressible channel flow --- FIK identity (Fukagata, Iwamoto, and Kasagi, Phys. Fluids 14, 2002)

• Even if we do not know anything about vortices, if we can reduce

the near-wall RSS, then we can reduce drag!

– Feedback body force

(Fukagata et al., Proc. SMART-6, 2005)

– Upstream traveling wave-like

blowing/suction (Min et al., J. Fluid Mech., 2006)

Drag lower than laminar flow (sub-laminar drag*)

* Although re-laminarization is the best in terms of energy saving (Bewley, J. Fluid Mech., 2009; Fukagata et al., Physica D, 2009)

1

0

laminar drag turbulent contribution (=weighted integral of RSS)

1224 (1 )( )

Ref

b

C y u v dy

10/46

Primary drag reduction mechanism by traveling

wave-like blowing/suction (Min et al., J. Fluid Mech., 2006; Mamori et al., Phys. Rev. E., 2010)

• Wave of blowing/suction traveling in the upstream

direction

“Negative” Reynolds shear stress

Net flow in the downstream direction

= “Pumping effect” in the direction opposite to the

wave (Hœpffner & Fukagata, JFM 2009)

External pressure gradient

required to keep the flow

rate (constant) is reduced

= “Drag reduction”

(+ Turbulence modification) (Animation: Mamori, MEng Thesis, 2008)

11/46

But, traveling wave-like blowing/suction device

is still difficult to make in practice

• Last sentence in Min et al. (2006)’s paper

However, a moving surface with wavy motion would

produce a similar effect, since wavy walls with small

amplitudes can be approximated by surface blowing

and suction.

• Question: Can it simply be substituted by wall

deformation?

12/46

Blowing/suction Wall deformation

Pumping effect (Hœpffner & Fukagata,

JFM 2009)

Upstream wave reduces

RSS

Flow is induced in the

direction opposite the

wave

Downstream wave

reduces RSS

Flow is induced in the

direction same as the

wave

Receptivity/Stability (Lee et al., PoF 2008;

Moarref & Jovanovic,

JFM 2010)

Flow is stabilized by

downstream wave of wall-

normal velocity on the

wall

???

Drag reduction

1. Upstream wave (Min et al., JFM 2006)

(pumping effect)

2. Downstream wave (Mamori et al., PoF 2014)

(stabilizing effect)

Downstream wave?

(pumping effect

+

stabilizing effect)

Summary of existing knowledge and prediction

13/46

,0 wu

tv d

u , d： Displacements of walls (varicose mode)

a : Velocity amplitude (0.05 … 0.3)

c : Phase speed (－3 … 3)

k : Wavenumber (1 … 4)

• Constant flow rate at

Reb = 2Ubh/n = 5600

DNS with traveling wave-like wall deformation (Nakanishi, Mamori, and Fukagata, Int. J. Heat Fluid Flow 35, 2012)

• Boundary conditions

Deformation velocity

a cos(k(x1－ct))

All nondimensionalized by

Twice bulk-mean velocity 2Ub*

and channel half width d *

14/46

DNS with traveling wave-like wall deformation (Nakanishi, Mamori, and Fukagata, Int. J. Heat Fluid Flow 35, 2012)

Animation: Uchino & Fukagata, CFD Symp, Japan, 2013

15/46 -d

P/d

x

No deformation

a = 0.2, c = 1, k = 2

a = 0.1, c = 1, k = 4

a = 0.2, c = 1, k = 4

Time trace of mean pressure gradient (=drag) (Nakanishi, Mamori, and Fukagata, Int. J. Heat Fluid Flow 35, 2012)

• Upstream wave (c < 0): drag increase, or no change

• Downstream wave (c > 0): drag reduction

a = 0.1, c = －1, k = 2

Laminar drag

t

16/46

Drag reduction effects under different sets of

parameters (Nakanishi, Mamori, and Fukagata, Int. J. Heat Fluid Flow 35, 2012)

no control

no control

100%D

dP dP

dx dxR

dP

dx

0

0

( )100%

p a

P

W W WR

W

W0 : Pumping power for

plane channel flow

Wp : Pumping power

Wa : Actuation power

Label : (RD, RP)

Relaminalization (unstable)

Relaminarization

Drag increase

Drag reduction

Drag reduction: 69%

Net power saving: 65% Deformation period

Deformation

amplitude

Drag reduction rate Power saving rate

17/46

Drag reduction mechanism by downstream

traveling wave-like wall-deformation (Nakanishi, Mamori, and Fukagata, Int. J. Heat Fluid Flow 35, 2012)

• Negative RSS

“Pumping” in the same direction as

the wave

External pressure gradient required to

keep the flow rate (constant) is reduced

= “Drag reduction”

• Stabilization Relaminraization

or

• Destabilization (inflectional instability)

at larger deformation amplitude

Periodic oscillation

+

18/46

Reynolds number effect (Nabae, Kawai, amd Fukagata, Proc. ETMM-12 (2018); IJHFF under review)

Complementary to the question at the conference

• DNS at different Re and predictions at higher Re

under constant pressure gradient (CPG) condition

Mean velocity profile. Prediction of drag reduction rate, RD.

Model A, optimistic estimate;

Model B, conservative estimate.

19/46 Experiment @TUAT

• A rubber sheet driven by a single piezo actuator

• Streamwise traveling wave confirmed by laser

reflection

• About 10% drag reduction by streamwise traveling

wave of wall deformation on one side

Figures are omitted in this distributed version.

For further information, please refer to:

I. Suzuki, T. Shimura, A. Mitsuishi, K. Iwamoto, and A. Murata, “Experimental study

on drag reduction effect with traveling wave control using PIV measurement,”

Proceedings of the ASME-JSME-KSME 2019, 8th Joint Fluids Engineering

Conference (AJKFluids2019), July 28-August 1, 2019, San Francisco, CA, USA, Paper

No. AJKFluids2019-4855 (2019).

20/46

Drag reduction by uniform blowing (UB) (Kametani & Fukagata, J. Fluid Mech. 681, 2011)

• With uniform blowing: Turbulence

is enhanced, but drag is reduced!

Decomposition of contributions

using the FIK identity

Contribution

of mean

wall-normal flux

Spatial

development

Turbulent

contribution

Total

White: Vortex cores

Color: Wall shear stress

21/46

FIK identity for spatially developing TBL (Kametani & Fukagata, J. Fluid Mech. 681, 2011)

Complementary to the question at the conference

22/46

Wind-tunnel experiment of uniform blowing (Eto, Kondo, Fukagata, and Tokugawa, AIAA J. 57, 2019)

K. Eto, working with the low turbulence wind-

tunnel at JAXA (Clark-Y model, Rec ~ 106,

Blowing velocity = 0.14% of free-stream velocity)

23/46

Wind-tunnel experiment of uniform blowing (Eto, Kondo, Fukagata, and Tokugawa, AIAA J. 57, 2019)

• cf of the uncontrolled flow:

– Modified Clauser Chart Method (Dixit & Ramesh, Exp. Fluids, 2009)

– Law of the wall with pressure gradient (Nickels, J. Fluid Mech., 2004)

• cf with blowing: Transformation by the modified Stevenson’s law

(Vigdorovich, Phys. Fluids, 2016), then fitted to Nickels’ law of the wall

24/46 Outline

1. From Feedback to

Predetermined Control

– Streamwise traveling wave

– Uniform blowing

2. Feedback Control Again

– Resolvent analysis

3. Toward the Future

– Machine learning

25/46

• Governing equations

– Continuity equation

– Navier-Stokes equation (NSE)

• NSE in Fourier domain : wavenumber-propagating speed combination

• Express as a matrix form

Resolvent analysis (McKeon & Sharma JFM 2010)

Streamwise wavenumber 𝑘𝑥 ，

Spanwise wavenumber 𝑘𝑧，

Propagation speed 𝑐 = 𝜔/𝑘𝑥．

(𝜔 : temporal frequency)

Linear operator

Input :

Output :

𝒌 = (𝑘𝑥, 𝑘𝑧, 𝑐)

Resolvent operator

All variables are normalized

by friction velocity 𝑢𝜏 and

channel half-height ℎ.

Friction Reynolds number

𝑅𝑒𝜏 = 𝑢𝜏ℎ/𝜈

(𝜈 : viscocity)

26/46

• Singular value decomposition (SVD)

– Rank-1 mode is highly amplified (e.g.

McKeon et al. PoF2013)

– Singular value comes in pairs due to

the wall-normal symmetry (Moarref et al.

JFM2013)

• 𝜎𝒌,1 = 𝜎𝒌,2 ≫ 𝜎𝒌,3 = 𝜎𝒌,4 ≫ ⋯ > 0

– In the following, we use 𝑢𝒌 = 𝑢𝒌,1 ,

𝜎𝒌 = 𝜎𝒌,1 , 𝑓𝒌 = 𝑓𝒌,1

Resolvent analysis (McKeon & Sharma JFM 2010)

(・)∗ : complex conjugate

Rank-1 velocity structure resembling

NW cycle in the spanwise-wall normal

plane

𝑦+

𝑢

27/46

NW cycle

VLSMs

Case SP (𝜶+ = 𝟎. 𝟓※)

Effect of Lee et al. (1998)’s suboptimal control

in spectral space (Nakashima, Fukagata, and Luhar, J. Fluid Mech. 828, 2017)

• Normalized change of squared singular value

relative to the uncontrolled case at 𝑹𝒆𝝉 = 𝟐𝟎𝟎𝟎

Red: attenuation

Blue: amplification ※𝛼+ is set to match the amplitude of the control-input used in the previous DNS (Lee et al. JFM1998)

Note that Case ST does not achieve drag reduction in DNS.

(・)𝑐 : controlled

(・)0 : no-control

How about Case ST,

which was reported

drag increment

in the previous DNS

by Lee et al. (1998)?

𝜙𝑘 = 𝛼𝑖𝑘𝑧𝐾

𝜕𝑤𝑘

𝜕𝑦𝑤

A

NW cycle

VLSMs

Case ST (𝜶+ = 𝟏𝟎※)

𝜙𝑘 = −𝛼𝑖𝑘𝑥𝐾

𝜕𝑢𝑘𝜕𝑦

𝑤

28/46

Numerical result using in-house DNS code※

(Nakashima, Fukagata, and Luhar, J. Fluid Mech. 828, 2017)

• Instantaneous contour of the control input 𝝓𝒌+ at the lower wall

(Just after the control begins)

Case SP

𝝓𝒌 = 𝛼𝑖𝑘𝑧𝐾

𝝏𝒘𝒌

𝝏𝒚𝑤

Case ST

𝝓𝒌 = −𝛼𝑖𝑘𝑥𝐾

𝝏𝒖𝒌

𝝏𝒚𝑤

Case ST leads to

the energetic

spanwise structures

※Fukagata, Kasagi, and Koumoutsakos, Phys. Fluids, 2006

Time-averaged wavenumber

spectra for the control input

In Case ST,

noticeable input concentrate

around the length scale

𝜆𝑥+, 𝜆𝑧

+ ≈ (3 × 102, 5 × 102).

Case SP targets at QSV

(Quasi-Streamwise Vortices)

The reason why

Case ST does not

work in the previous

DNS (Lee et al. JFM1998)

Red: blowing

Blue: suction

29/46

Proposal of ‘new’ control law (Kawagoe, Nakashima, Luhar, and Fukagata, J. Fluid Mech. 866, 2019)

• Modification of Case ST based on resolvent analysis

– Result from DNS is consistent with the prediction of resolvent analysis

– About 10% drag reduction is obtained by Case STz

Case ST : 𝜙 = −𝛼𝑖𝑘𝑥

𝐾

𝜕𝑢

𝜕𝑦

𝑤 𝐂𝐚𝐬𝐞 𝐒𝐓𝐳 : 𝝓 = −𝜶

𝒊𝒌𝒛

𝑲

𝝏𝒖

𝝏𝒚

𝒘

A

NW

cycle

Case ST

DNS

Dotted : Uncontrolled

Black : Control for all region (CA)

Blue : Control for large scale (CL)

Red : Control for small scale (CS)

10%

Drag

decrease

Drag

Increase

NW

cycle

Case STz

30/46 Summary on turbulence control

• Extensive investigation has made since DNS has

matured in late 1980’s

– Feedback control

• Succeeded at least theoretically and numerically

• Big hurdle in hardware development

– Predetermined control

• Has a great potential

• Higher possibility of practical application

• Remaining issues

– Further development of control methods more suitable

for practical implementation

– Hardware development

31/46 Outline

1. From Feedback to

Predetermined Control

– Streamwise traveling wave

– Uniform blowing

2. Feedback Control Again

– Resolvent analysis

3. Toward the Future

– Machine learning

32/46 Turbulence Big Data

• Large-scale computer simulations and high-resolution

measurement

Accumulation of Turbulence Big Data

– Feature extraction by using linear theories

• Proper Orthogonal Decomposition (POD)

• Dynamic Mode Decomposition (DMD)

• Resolvent analysis

– Linear theories cannot extract nonlinear dynamics

– Feature extraction methods which can capture the nonlinearity are

desired

Our goal

33/46 To start with…

• Is it possible to study large-sized spatio-temporal data

of turbulent field using ML?

– Spatio-temporal reconstruction of cross-sectional

velocity field in turbulent channel flow

(Fukami, Nabae, Kawai, and Fukagata, Phys. Rev. Fluids 4, 064603, 2019)

– Super-resolution of two-dimensional turbulence

(Fukami, Fukagata, and Taira, J. Fluid Mech. 870, 106-120, 2019)

• Is it possible to derive the governing equations for the

extracted low-dimensional modes?

– Visualization and interpretation of nonlinear low-

dimensional modes (for a simple problem) (Murata, Fukami, and Fukagata, under review, arXiv:1906.04029 (2019))

– Derivation of ODE using SINDy (Brunton et al., PNAS, 2016) (Murata, Fukami, and Fukagata, Proc. AJK-FLUIDS, 2019) --- omitted today

34/46

Spatio-temporal reconstruction of cross-

sectional velocity field in turbulent channel flow (Fukami, Nabae, Kawai, and Fukagata, Phys. Rev. Fluids 4, 064603, 2019)

• Temporal development of cross-sectional velocity field （u, v, w,

p） in turbulent channel flow at Ret = 180

Cross-sectional velocity field, u’

（DNS）

NN DNS10 1.26t t

z /d

y /d

35/46

Spatio-temporal reconstruction (Fukami, Nabae, Kawai, and Fukagata, Phys. Rev. Fluids 4, 064603, 2019)

• Network structure to learn the relationship between two

discrete times

Surrogate model for time-discretized Navier-Stokes eq.

Network structure. Blue parts are Convolutional Neural Network (CNN) autoencoder.

BackProp

Input

Output

DNS

nq

1

ML

nq

36/46

Spatio-temporal reconstruction (Fukami, Nabae, Kawai, and Fukagata, Phys. Rev. Fluids 4, 064603, 2019)

DNS (answer) Case 1 (2 MLP layers, latent size 3072)

Case 2 (2 MLP layers, latent size 192) Case 3 (3 MLP layers, latent size 3072)

37/46

Spatio-temporal reconstruction (Fukami, Nabae, Kawai, and Fukagata, Phys. Rev. Fluids 4, 064603, 2019)

38/46

Spatio-temporal reconstruction (Fukami, Nabae, Kawai, and Fukagata, Phys. Rev. Fluids 4, 064603, 2019)

• Temporal spectra

• Temporal two-point

correlations

• Integral time scale

No “spurious

oscillation” which

was present in

periodic DNS

177y

13.2y

39/46

Spatio-temporal reconstruction (Fukami, Nabae, Kawai, and Fukagata, Phys. Rev. Fluids 4, 064603, 2019)

• Used as “Turbulent inflow generator”

– Excellent statistics in the

downstream region

– Lower computational cost than

“driver” periodic DNS

• 1/580 under our environment

(CPU 1 core + GPU Tesla K40)

40/46

Super-resolution of two-dimensional turbulence (Fukami, Fukagata, and Taira, J. Fluid Mech. 870, 2019)

• What is Super-resolution?

41/46

Super-resolution of two-dimensional turbulence (Fukami, Fukagata, and Taira, J. Fluid Mech. 870, 2019)

42/46

Super-resolution of two-dimensional turbulence (Fukami, Fukagata, and Taira, J. Fluid Mech. 870, 2019)

43/46

POD vs linear CNN vs nonlinear CNN (Murata, Fukami, and Fukagata, under review, arXiv:1906.04029 (2019))

• Velocity field reconstructed using the first 2 modes

44/46

Visualization of non-linear low-dim. modes (Murata, Fukami, and Fukagata, under review, arXiv:1906.04029 (2019))

• CNN autoencoder structure for visualization (MD-CNN)

45/46

Visualization of non-linear low-dim modes (Murata, Fukami, and Fukagata, under review, arXiv:1906.04029 (2019))

POD applied to

each of the 2

nonlinear modes

extracted using

CNN (i.e., 2

decomposed

fields)

POD of DNS field

(for comparison)

46/46 Summary of “Toward the future”

• Is it possible to study large-sized spatio-temporal data of

turbulent field using ML?

– Spatio-temporal reconstruction of cross-sectional velocity field in

turbulent channel flow

(Fukami, Nabae, Kawai, and Fukagata, Phys. Rev. Fluids 4, 064603, 2019)

– Super-resolution of two-dimensional turbulence

(Fukami, Fukagata, and Taira, J. Fluid Mech. 870, 106-120, 2019)

Seem to work reasonably well. What about in 3D? (ongoing)

• Is it possible to derive the governing equations for the

extracted low-dimensional modes?

– Visualization and interpretation of nonlinear low-dimensional

modes (for a simple problem)

(Murata, Fukami, and Fukagata, under review, arXiv:1906.04029, 2019)

– Derivation of ODE using SINDy (Brunton et al., PNAS, 2016) (Murata, Fukami, and Fukagata, Proc. AJKFluids2019)

Seem to work reasonably well, too. In turbulent flow? (ongoing)

Toward nonlinear ROM for turbulence control design …