Some integral relationships and their applications to flow control

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1 IPAM Workshop Mathematics of Turbulence Workshop IV: Turbulence in Engineering Applications November 17-21, 2014 Some integral relationships and their applications to flow control Koji Fukagata Department of Mechanical Engineering Keio University, Japan

2 1. Introduction

3 Two major approaches for active control of turbulence 3/50 Feedback control Sensors, actuators, controller Potentially effective Big hurdle for hardware development (especially if the QSVs are targeted at) Actuators Flow Controller Sensors Predetermined control without sensors Less difficult to make hardware? Suitable input is less clear (e.g. for friction drag reduction) Actuators Flow

4 Feedback control (wind tunnel experiment) (Yoshino et al., J. Fluid Sci. Technol., 2009; also introduced in Kasagi et al., Annu. Rev. Fluid Mech., 2009) 4/50 Feedback system for turbulence control 6% drag reduction

5 Big issue toward practical application: Physical length- and time-scales of QSV 5/50 (Kasagi, Suzuki, and Fukagata, Annu. Rev. Fluid Mech., 2009)

6 2. Integral relationship between the skin friction drag and the turbulent statistics

7 Fundamental question in fluids engineering 7/50 Question: Take a straight pipe, for instance Pressure gradient (or friction drag) given Flow rate? Flow rate given Pressure gradient (or friction drag)? Answer: Laminar flow: Analytical solution (Hagen-Poiseuille) C Turbulent flow f 16 Re b where C Rely on experiments, simulations, and (semi-)empirical formula What is the relationship between turbulent statistics and drag? How does it become when a control is applied? *The same arguments hold for heat transfer problems f w (1/ 2) U 2 b, Re b U b (2 R)

8 Integral relationship between turbulent statistics and drag (1) (Fukagata, Iwamoto, and Kasagi, Phys. Fluids, 2002) 8/50 Fully-developed channel flow (the simplest case) Starting point: Reynolds-Averaged Navier-Stokes eq. dp d 1 du 0 ( uv ) dx dy Re b dy Nondimensionalized by twice bulkmean velocity 2U b * and channel halfwidth * Asterisk: dimensional quantity f f f : f : f Mean Fluctuation Re b 2U * * b *

9 Integral relationship between turbulent statistics and drag (2) (Fukagata, Iwamoto, and Kasagi, Phys. Fluids, 2002) 9/50 Triple integration of Reynolds-Averaged N-S eq. 1 st integration Stress balance 1 du C f ( u v ) (1 y) Re dy 8 b Relationships below are used at each step C f dp * w 8 * *2 Ub / 2 dx Reb dy y0 8 du 2 nd integration Mean velocity profile C f u( y) Re b 8 y y2 2 y 0 (uv )dy u y0 0 (dy dy) 3 rd integration Bulk-mean velocity 1 2 Re b 1 y C f 24 (uv )dy dy 0 0 U b 1 2 (since the velocity is nondimensionalized by 2U b* )

10 Integral relationship between turbulent statistics and drag (3) (Fukagata, Iwamoto, and Kasagi, Phys. Fluids, 2002) 10/50 Convert double integral to single integral By integration by parts 1 y ( u v )dy 1 y dy 1 (uv )dy dy y y (uv )dy (1 y)(uv )dy 1 0 y(uv )dy Or, by iterated integral (Yoshizawa, priv. commun., 2008) Y Relationship for a fully-developed channel flow 1 12 C f 12 2(1 y)( u v ) dy Re b ( u v ) dy (1 Y)( uv) *Essentially the same relationship has been derived also by Bewley and Aamo (JFM 2004) Y 1 Y = y y uv

11 Contributions of different effects 11/50 More general form (for channel flow) 2 12 C f 12 (1 y)( u v ) dy (III) (IV) (V) Re I b 0 II I. Laminar drag II. Turbulent contribution = Weighted integral of the Reynolds shear stress III. Contribution of spatio-temproral development IV. Contribution of body force, additional stress (e.g., polymer) V. Contribution from wall boundary (e.g., uniform blowing/suction)

12 Relationships in other geometries (Fukagata, Iwamoto, and Kasagi, Phys. Fluids, 2002) 12/50 Pipe flow C f r z Reb r uu rdr Nondimensionalized by 2U * b and pipe radius R * Re b 2U * R * b * ZPG boundary layer 1 4(1 d ) C f 4 (1 y)( u v ) dy Re 1 2 uu 2 (1 y) 0 Contribution of spatial development 0 x uv y d y Contribution of mean wall-normal flux Nondimensionalized by freestream velocity U * and 99% boundary layer thickness 99% * Re 2U * * 99% * d : Nondimensionalized displacement thickness

13 Wall unit of uncontrolled flow Example 1: Opposition-controlled pipe flow (Fukagata, Iwamoto, and Kasagi, Phys. Fluids, 2002) 13/50 Decomposition of contributions I (Laminar drag) II (Turb. Contrib.) Sum (C f ) No control Controlled C f Re 2r u / r u / z rdr b 1 0 I II Weighted Reynolds shear stress Amount of drag reduction

Example 2: Spatially-developing ZPG turbulent boundary layer

development Turbulent contribution Total White: Vortex cores

14 Example 2: Spatially-developing ZPG turbulent boundary layer with uniform blowing/suction (Kametani & Fukagata, J. Fluid Mech., 2011) 14/50 Decomposition of contributions Spatial development Turbulent contribution Total White: Vortex cores Color: Wall shear stress With uniform blowing: Turbulence is enhanced, but drag is reduced! Contribution of mean wall-normal flux

15 Example 3: Drag reduction by surfactant addition (Yu, Li, and Kawaguchi, Int. J. Heat Fluid Flow, 2004) 15/50 Integral relationship for Giesekus fluid (channel flow) C xy C f 24 u v (1 y) dy (1 y) dy Re 0 0 b We Contribution to C f % Viscous contribution Turbulence contribution Viscoelastic contribution 46% 34% Newtonian 14% 1 2 Surfactant 20% A similar analysis can be made also for polymer addition (White & Mungal, Annu. Rev. Fluid Mech., 2008)

16 Some other extensions 16/50 Arbitrary-shaped straight duct (const. pres. grad.) (Subragaglia & Sugiyama, Physcia D, 2007) Volume average Stokes flow Corresponding to (1 y) weight Compressible flow (Gomez, Flutet, and Sagaut, Phys. Rev. E, 2009) Contribution from variable viscosity Contribution from viscosity fluctuations (small)

17 Some other extensions (cont d) 17/50 Relationship between wall heat flux and turbulent statistics (Kasagi, Hasegawa, Fukagata, and Iwamoto, J. Heat Transfer, 2012) Constant temperature difference condition Stanton number Prandtl number Constant heat flux condition Dimensionless temperature Partial flow rate Partial flow rate deviation from laminar flow Uniform heat generation condition (omitted here)

18 3. Application to drag reduction control

19 To start with predetermined control 19/50 For a fully-developed incompressible channel flow (review) (Fukagata et al., Phys. Fluids, 2002) 12 C f 24 (1 y)( u v ) dy Re b laminar drag Even if we do not know anything about vortices, if we can reduce the 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 turbulent contribution (=weighted integral of RSS)

20 y x U(y) Traveling wave-like blowing/suction (Min, Kang, Speyer, and Kim, J. Fluid Mech., 2006) c wavenumber Drag normalized in a different way c Laminar 2 1 drag "Turbulent" contribution, D 20/ D 2 Re ( y)( u v ) dy Downstream wave Drag increase Upstream wave Drag decrease Negative Reynolds shear stress! wavespeed

Negative Reynolds shear stress(rss) --- linear analysis 21/50 Away from the wall:u and v are orthogonal (same as inviscid) Near the wall: phase shift in u due to viscosity (Min

21 Negative Reynolds shear stress(rss) --- linear analysis 21/50 Away from the wall:u and v are orthogonal (same as inviscid) Near the wall: phase shift in u due to viscosity (Min et al., J. Fluid Mech., 2006; Mamori et al., Phys. Rev. E, 2010) v u u v Viscous phase shift Negative RSS slightly exceeds! (Mamori, Fukagata, and Hœpffner, Phys. Rev. E, 2010)

Primary drag reduction mechanism by traveling wave-like blowing/suction 22/50 Wave of blowing/suction traveling in the upstream direction Negative Reynolds shear stress Net flow in the downstream

22 Primary drag reduction mechanism by traveling wave-like blowing/suction 22/50 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) External pressure gradient required to keep the flow rate (constant) is reduced = Drag reduction (+ Turbulence modification) (Animation: Mamori, MEng Thesis, 2008)

But, traveling wave-like blowing/suction device is difficult to make in practice 23/50 Last sentence in Min et al (2006) s paper However, a moving surface with wavy motion

23 But, traveling wave-like blowing/suction device is difficult to make in practice 23/50 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?

24 Blowing/suction vs wall deformation (without external pressure gradient) 24/50 Wall deformation Blowing/suction Black point: Fluid particle (marker) Color: Pressure (Hœpffner & Fukagata, J. Fluid Mech., 2009)

25 Blowing/suction vs wall deformation (without external pressure gradient) 25/50 Wall deformation Blowing/suction Direction of wave Net Flow Net Flow (Hœpffner & Fukagata, J. Fluid Mech., 2009)

26 Summary of existing knowledge and prediction 26/50 Pumping effect (Hœpffner & Fukagata, JFM 2009) Blowing/suction Direction opposite to the wave Wall deformation Direction same as the wave Stability (Lee et al., PoF 2008; Moarref & Jovanovic, JFM 2010) Flow is stabilized by downstream wave of wallnormal velocity on the wall Drag reduction 1. Upstream wave (Min et al., JFM 2006) --- unstable? 2. Downstream wave (Mamori et al., PoF 2014) --- stable Downstream wave? --- stable? Question: Is the drag really reduced by a downstream traveling wave-like wall deformation?

27 DNS with traveling wave-like wall deformation (Nakanishi, Mamori, and Fukagata, Int. J. Heat Fluid Flow, 2012) 27/50 Boundary conditions u w 0, v d t Deformation velocity a cos(k(x 1 -ct)) u, d : Displacements of walls (varicose mode) a : Velocity amplitude ( ) c : Phase speed (-3 3) k : Wavenumber (1 4) Constant flow rate at Re b = 2U b h/ = 5600 Initial field: Fully developed turbulent flow in plane channel All nondimensionalized by Twice bulk-mean velocity 2U b * and channel half width *

28 DNS in a bit detail (Nakanishi et al., Int. J. Heat Fluid Flow, 2012) 28/50 Coordinate transform (Kang & Choi, Phys. Fluids, 2000) x x, 1 1 x3 3 x (1 ), 2 x2 u d (streamwise) (wall-normal) (spanwise) ( u, d : displacements of upper and lower walls) x Continuity eq. ui x i S Navier-Stokes eq. u t i ( uiu x j j ) p x i 1 Re 2 ui x x j j P x i i1 S i DNS Code: Based on the FDM code for a plane channel flow (Fukagata et al., Phys. Fluids, 2006)

29 DNS with traveling wave-like wall deformation (Nakanishi et al., Int. J. Heat Fluid Flow, 2012) Animation: Uchino & Fukagata, CFD Symp, Japan, /50 Relaminarization About 70% drag reduction, 65% reduction in net power

Downstream wave (c > 0): drag reduction a = 0.

30 -dp/dx t Time trace of mean pressure gradient (=drag) (Nakanishi et al., Int. J. Heat Fluid Flow, 2012) 30/50 Upstream wave (c < 0): drag increase, or no change Downstream wave (c > 0): drag reduction a = 0.1, c = -1, k = 2 No deformation a = 0.2, c = 1, k = 4 a = 0.1, c = 1, k = 4 Laminar drag a = 0.2, c = 1, k = 2

dp dp dx no control dx 100% dp dx no control Relaminarization R P W0 ( Wp Wa) 100% W 0 Deformation amplitude Drag

31 Drag reduction effects under different sets of parameters (Nakanishi et al., Int. J. Heat Fluid Flow, 2012) 31/50 Drag reduction rate Power saving rate Label : (R D, R P ) Relaminalization (unstable) R D dp dp dx no control dx 100% dp dx no control Relaminarization R P W0 ( Wp Wa) 100% W 0 Deformation amplitude Drag increase Deformation period Drag reduction W 0 : Pumping power for plane channel flow W p : Pumping power W a : Actuation power Drag reduction: 69% Power saving: 65%

Heat Fluid Flow, 2012) a 6, c 30, 560 32/50 No

32 Flow field (a = 0.2, c = 1, k = 2) (Nakanishi et al., Int. J. Heat Fluid Flow, 2012) a 6, c 30, /50 No control : Instantaneous phase average : Plane channel : Laminar RSS Streamwise velocity Relaminarized case

33 Drag reduction mechanism by downstream traveling wave-like wall-deformation (Nakanishi et al., Int. J. Heat Fluid Flow, 2012) 33/50 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

34 4. Integral relationships on dissipation (or power balance)

35 Power balance in internal flows (Fukagata, Sugiyama, and Kasagi, Physica D, 2009) 35/50 Periodic duct with arbitrary shape Derivation of power balance Starting point: streamwise momentum equation Zero-net-flux blowing/suction Zero-net body force Multiplication by streamwise velocity Integration in volume + vector/tensor operations

36 Integral relationship on dissipation (Fukagata, Sugiyama, and Kasagi, Physica D, 2009) 36/50 W p : Pumping power W a : Actuation power (I): Dissipation from the velocity profile of the Stokes flow at the same flow rate (II): Dissipation due to the mean deviation from the Stokes profile (non-negative) (III): Dissipation due to the fluctuating velocities (non-negative) (IV): Additional term for variable-curvature ducts (zero for constant-curvature ducts)

37 Lower bound of net power (Fukagata, Sugiyama, and Kasagi, Physica D, 2009) 37/50 For straight channel, straight pipe, channel/pipe with constant curvature Pumping power Actuation power Stokes dissipation Deviation from Stokes Fluctuations Lowest net power achievable in a controlled duct flow is the pumping power of the Stokes flow at the same flow rate (not always the same as laminar flow!) For a fully-developed plane channel flow, this is exactly the same as Bewley (JFM 2009) s argument because laminar = Stokes in that case

38 Example 1: Flow in a straight channel with Min et al. (2006) s traveling-wave-like blowing/suction 38/50 The line of constant total power Light gray: Region not allowed by Bewley s conjecture (Bewley, Prog. Aerosp. Sci., 2001) Dark gray: Region not allowed by the correct theorem (Fukagata et al., Physica D, 2009; Bewley, J. Fluid Mech., 2009) Star: Turbulent (DNS, Min et al. (2006) s condition) Circle: Laminar (2D-DNS) (Hœpffner & Fukagata, J. Fluid Mech., 2009)

39 Example 2: Sublaminar drag in uncontrolled flow in a pipe with a constant curvature (Noorani & Schlatter, submitted for publication) 39/50 > e Stokes 0.005

40 For variable curvature ducts: Possibility of Lower-than-Stokes net power? 40/50 Possible situation of S (IV) 2 p ds 0 S Flow S p > 0 0 H L S p > 0 0 Constant suction/blowing Gives a hint to clarify the theoretical limitation of bluff body control, which seems also unclear (Choi et al., Annu. Rev. Fluid Mech., 2008)?

41 Integral relationship on dissipation in an external flow (Naito & Fukagata, Phys. Rev. E, 2014) 41/50 For a flow around a circular cylinder where Deviation from freestream velocity Dissipation Pressure drag Friction drag Additional drag Ideal actuation power

Suboptimal control minimizing dissipation (Naito & Fukagata, Phys. Rev.

only (neglecting the time derivative) neglect Control input (blowing/suction) to

42 Suboptimal control minimizing dissipation (Naito & Fukagata, Phys. Rev. E, 2014) 42/50 Cost function J : Dissipation expressed by the surface quantities only (neglecting the time derivative) neglect Control input (blowing/suction) to minimize the cost function J : Obtained using the suboptimal procedure (Lee et al., J. Fluid Mech., 1998; Min & Choi, J. Fluid Mech., 1999) DNS of a flow around a cylinder: Validated in our previous work (Naito & Fukagata, Phys. Fluids, 2012) Thanks, John!

43 Suboptimal control minimizing dissipation (cont d) (Naito & Fukagata, Phys. Rev. E, 2014) 43/50 Resultant mean blowing/suction profile, as compared to J 1 control (= minimizing the pressure drag: Min & Choi, JFM 1999) Weaker suction in the front half Radial velocity on the cylinder surface Narrower blowing in the rear half Angle from the front stagnation point

44 Suboptimal control minimizing dissipation (cont d) (Naito & Fukagata, Phys. Rev. E, 2014) 44/50 No control J 1 control Present control A bit more parallel due to narrower blowing

45 5. Some other useful relationships for turbulence control

46 Prediction of drag reduction rate in channel flow by ideal damping of near-wall fluctuations (Iwamoto, Fukagata, Kasagi, and Suzuki, Phys. Fluids, 2005) 46/50 Constant flow rate; ideal damping in 0 < y < y d Implicit formula among Re 0, R D, and y d / y 1 1 ln Re ( / ) (1 / 3 )(1 )Re F yd yd RD 0 ( y / )(1 y / 2 )(1 y / )(1 R )Re d d d D 3/2 1/2 1 3/2 1/2 (1 yd / ) (1 RD ) ln (1 yd / ) (1 RD) Re 0 F 0 0 Damping near-wall region (y + < 60) p z, DNS at Re 650, y d + 60

47 Prediction of drag reduction rate of flow in a channel with superhydrophobic surfaces (Fukagata, Kasagi, and Koumouotsakos, Phys. Fluids, 2006) 47/50 Implicit formula among l x, l z, Re 0, and R D where 1 1 R ln Re (1 ) ln 1 Re or an improved model F( l ) F z 0 D 0 F0 RD lx RD 0 2 ( F0 F ) ( F F ) l 0 1 R F 1 R l z D 0 z F( lz ) F F F exp l / a (Original) (Busse & Sandham, Phys. Fluids, 2012) b 0 D z

48 6. Summary

49 Summary 49/50 Integral relationships Formal solutions to the fundamental questions in fluids engineering Connection between drag and turbulent statistics Connection among pumping power, actuation power, and dissipation Convenient tools for turbulence control studies Analysis of control effect Quantification of major contributor(s) Proposal of new control methods Method to reduce RSS, such as traveling waves Good affinity with (sub-)optimal control theory Future directions Connection to dynamics

50 Acknowledgments 50/50 Financial support Institute for Pure and Applied Mathematics (IPAM), for invitation to this workshop Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan Society for the Promotion of Science (JSPS), Japan Aerospace Exploration Agency (JAXA), Railway Technical Research Institute (RTRI) of Japan Co-workers (current affiliation) Univ. Tokyo: Drs. Nobuhide Kasagi (JST), Yuji Suzuki, Kaoru Iwamoto (TUAT), Yosuke Hasegawa, Kazuyasu Sugiyama (Osaka Univ.) ETH: Drs. Petros Koumoutsakos, Stefan Kern, Philippe Chatelain (Louvain Catholique Univ.) KTH: Drs. Henrik Alfredsson, Antonio Segalini, Philipp Schlatter, Ramis Örlü Keio Univ.: Drs. Shinnosuke Obi, Jerôme Hœpffner (UPMC), Yoshitsugu Naka (Tokyo Tech), Hiroya Mamori (TUAT), Simon J. Illingworth (Univ. Melbourne), Hiroshi Naito (JAXA), Yukinori Kametani (KTH), Ms. Rio Nakanishi (Chubu Electric Power), Messrs. Kosuke Higashi (Kobelco), Nobuhito Tomiyama (Daikin), Sho Watanabe (Komatsu), Taichi Igarashi (Kobelco), Keisuke Uchino, Ryosuke Kidogawa, and all the former and current students Research community Drs. John Kim (UCLA), Tom Bewley (UCSD), Haecheon Choi (Seoul National Univ.), Godfrey Mungal (Stanford/SCU), Pierre Sagaut (UPMC), Yasuo Kawaguchi (Tokyo Univ. Sci.), Bo Yu (China Univ. Petroleum), Shu Takagi (Univ. Tokyo), Yuichi Murai (Hokkaido Univ.), Michael Leschziner (Imperial College), Bettina Frohnapfel (KIT), Maurizio Quadrio (Politec Milano), Mihailo Jovanovic (Minnesota Univ.), Ivan Marusic (Univ. Melbourne), Pierre Ricco (Univ. Sheffield), Azad Noorani (KTH), and actually many other people! And, of course, the workshop organizers and all the participants today --- Thank you very much!

The Phase Relationship in Laminar Channel Flow Controlled by Traveling Wave-Like Blowing/Suction. Abstract

The Phase Relationship in Laminar Channel Flow Controlled by Traveling Wave-Like Blowing/Suction Hiroya Mamori, Koji Fukagata, and Jerôme Hœpffner 2 Department of Mechanical Engineering, Keio University,