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 …
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