0
Review Article

Closed-Loop Turbulence Control: Progress and Challenges

[+] Author and Article Information
Steven L. Brunton

Department of Mechanical Engineering
and eScience Institute,
University of Washington,
Seattle, WA 98195

Bernd R. Noack

Institut PPRIME, CNRS - Université de
Poitiers - ENSMA, UPR 3346,
Département Fluides, Thermique, Combustion,
CEAT,
F-86036 Poitiers Cedex, France
Institut für Strömungsmechanik,
Technische Universität Braunschweig,
D-38108 Braunschweig, Germany

Manuscript received November 12, 2014; final manuscript received July 25, 2015; published online August 26, 2015. Assoc. Editor: Jörg Schumacher.

Appl. Mech. Rev 67(5), 050801 (Aug 26, 2015) (48 pages) Paper No: AMR-14-1091; doi: 10.1115/1.4031175 History: Received November 12, 2014; Revised July 25, 2015

Closed-loop turbulence control is a critical enabler of aerodynamic drag reduction, lift increase, mixing enhancement, and noise reduction. Current and future applications have epic proportion: cars, trucks, trains, airplanes, wind turbines, medical devices, combustion, chemical reactors, just to name a few. Methods to adaptively adjust open-loop parameters are continually improving toward shorter response times. However, control design for in-time response is challenged by strong nonlinearity, high-dimensionality, and time-delays. Recent advances in the field of model identification and system reduction, coupled with advances in control theory (robust, adaptive, and nonlinear) are driving significant progress in adaptive and in-time closed-loop control of fluid turbulence. In this review, we provide an overview of critical theoretical developments, highlighted by compelling experimental success stories. We also point to challenging open problems and propose potentially disruptive technologies of machine learning and compressive sensing.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Fish, F. E. , and Lauder, G. V. , 2006, “ Passive and Active Flow Control by Swimming Fishes and Mammals,” Annu. Rev. Fluid Mech., 38, pp. 193–224. [CrossRef]
Ahlborn, B. K. , 2004, Zoological Physics, Springer-Verlag, Berlin.
Dean, B. , and Bhushan, B. , 2010, “ Shark-Skin Surfaces for Fluid-Drag Reduction in Turbulent Flow: A Review,” Philos. Trans. R. Soc. A, 368(1929), pp. 4775–4806. [CrossRef]
Bechert, D. W. , Bruse, M. , Hage, W. , van der Hoeven, J. G. T. , and Hoppe, G. , 1997, “ Experiments on Drag-Reducing Surfaces and Their Optimization With an Adjustable Geometry,” J. Fluid Mech., 338, pp. 59–87. [CrossRef]
Gilliéron, P. , and Kourta, A. , 2010, “ Aerodynamic Drag Reduction by Vertical Splitter Plates,” Exp. Fluids, 48(1), pp. 1–16. [CrossRef]
Grandemange, M. , Ricot, D. , Vartanian, C. , Ruiz, T. , and Cadot, O. , 2014, “ Characterisation of the Flow Past Real Road Vehicles With Blunt Afterbodies,” Int. J. Aerodyn., 4(1), pp. 24–42. [CrossRef]
Pfeiffer, J. , and King, R. , 2012, “ Multivariable Closed-Loop Flow Control of Drag and Yaw Moment for a 3D Bluff Body,” AIAA Paper No. 2012-2802.
Gad-el Hak, M. , 1989, “ Flow Control,” ASME Appl. Mech. Rev., 42(10), pp. 261–293. [CrossRef]
Gad-el Hak, M. , and Tsai, H. M. , 2006, Transition and Turbulence Control, Vol. 8, World Scientific, Singapore.
Kim, J. , 2011, “ Physics and Control of Wall Turbulence for Drag Reduction,” Philos. Trans. R. Soc. A, 369(1940), pp. 1396–1411. [CrossRef]
Baker, C. , Jones, J. , Lopez-Calleja, F. , and Munday, J. , 2004, “ Measurements of the Cross Wind Forces on Trains,” J. Wind Eng. Ind. Aerodyn., 92(7), pp. 547–563. [CrossRef]
Baker, C. , 2010, “ The Flow Around High Speed Trains,” J. Wind Eng. Ind. Aerodyn., 98(6), pp. 277–298. [CrossRef]
Schetz, J. A. , 2001, “ Aerodynamics of High-Speed Trains,” Annu. Rev. Fluid Mech., 33(1), pp. 371–414. [CrossRef]
Gad-el Hak, M. , 1996, “ Modern Developments in Flow Control,” ASME Appl. Mech. Rev., 49(7), pp. 365–379. [CrossRef]
Barber, T. J. , 1999, private communication.
King, R. , 2007, “ Active Flow Control,” Notes on Numerical Fluid Mechanics and Interdisciplinary Design, Vol. 95, Springer, Berlin.
King, R. , 2010, “ Active Flow Control II,” Notes on Numerical Fluid Mechanics and Interdisciplinary Design, Vol. 108, Springer, Berlin.
Liepmann, H. , and Nosenchuck, D. , 1982, “ Active Control of Laminar-Turbulent Transition,” J. Fluid Mech., 118, pp. 201–204. [CrossRef]
Roussopoulos, K. , 1993, “ Feedback Control of Vortex Shedding at Low Reynolds Numbers,” J. Fluid Mech., 248, pp. 267–296. [CrossRef]
Kim, J. , and Bewley, T. , 2007, “ A Linear Systems Approach to Flow Control,” Annu. Rev. Fluid Mech., 39, pp. 383–417. [CrossRef]
Sipp, D. , Marquet, O. , Meliga, P. , and Barbagallo, A. , 2010, “ Dynamics and Control of Global Instabilities in Open-Flows—A Linearized Approach,” ASME Appl. Mech. Rev., 63(3), p. 030801. [CrossRef]
Lee, C. , Kim, J. , Babcock, D. , and Goodman, R. , 1997, “ Application of Neural Networks to Turbulence Control for Drag Reduction,” Phys. Fluids, 9(6), pp. 1740–1747. [CrossRef]
Medjo, T. T. , Temam, R. , and Ziane, M. , 2008, “ Optimal and Robust Control of Fluid Flows: Some Theoretical and Computational Aspects,” ASME Appl. Mech. Rev., 61(1), p. 010802. [CrossRef]
Bewley, T. R. , 2001, “ Flow Control: New Challenges for a New Renaissance,” Prog. Aerosp. Sci., 37(1), pp. 21–58. [CrossRef]
Greenblatt, D. , and Wygnanski, I. J. , 2000, “ The Control of Flow Separation by Periodic Excitation,” Prog. Aerosp. Sci., 36(7), pp. 487–545. [CrossRef]
Bushnell, D. M. , and McGinley, C. B. , 1989, “ Turbulence Control in Wall Flows,” Annu. Rev. Fluid Mech., 21, pp. 1–20. [CrossRef]
Moin, P. , and Bewley, T. , 1994, “ Feedback Control of Turbulence,” ASME Appl. Mech. Rev., 47(6S), pp. S3–S13. [CrossRef]
Lumley, J. , and Blossey, P. , 1998, “ Control of Turbulence,” Annu. Rev. Fluid Mech., 30, pp. 311–327. [CrossRef]
Gutmark, E. J. , Schadow, K. C. , and Yu, K. H. , 1994, “ Methods for Enhanced Turbulence Mixing in Supersonic Shear Flows,” ASME Appl. Mech. Rev., 47(6S), pp. S188–S192. [CrossRef]
Aamo, O. M. , and Krstić, M. , 2002, Flow Control by Feedback: Stabilization and Mixing, Springer-Verlag, London.
Dimotakis, P. E. , 2005, “ Turbulent Mixing,” Annu. Rev. Fluid Mech., 37, pp. 329–356. [CrossRef]
Mankbadi, R. R. , 1992, “ Dynamics and Control of Coherent Structures in Turbulent Jets,” ASME Appl. Mech. Rev., 45(6), pp. 219–248. [CrossRef]
Dowling, A. P. , and Morgans, A. S. , 2005, “ Feedback Control of Combustion Oscillations,” Annu. Rev. Fluid Mech., 37, pp. 151–182. [CrossRef]
Rowley, C. , and Williams, D. , 2006, “ Dynamics and Control of High-Reynolds Number Flows Over Open Cavities,” Annu. Rev. Fluid Mech., 38, pp. 251–276. [CrossRef]
Choi, H. , Jeon, W.-P. , and Kim, J. , 2008, “ Control of Flow Over a Bluff Body,” Annu. Rev. Fluid Mech., 40, pp. 113–139. [CrossRef]
Cattafesta, L. , 2011, “ Actuators for Active Flow Control,” Annu. Rev. Fluid Mech., 43, pp. 247–272. [CrossRef]
Chen, K. K. , and Rowley, C. W. , 2011, “ H2 Optimal Actuator and Sensor Placement in the Linearised Complex Ginzburg-Landau System,” J. Fluid Mech., 681, pp. 241–260. [CrossRef]
Bradshaw, P. , Ferriss, D. H. , and Johnson, R. , 1964, “ Turbulence in the Noise-Producing Region of a Circular Jet,” J. Fluid Mech., 19(4), pp. 591–624. [CrossRef]
Brown, G. L. , and Roshko, A. , 1974, “ On Density Effects and Large Structure in Turbulent Mixing Layers,” J. Fluid Mech., 64(4), pp. 775–816. [CrossRef]
Kim, J. , 2003, “ Control of Turbulent Boundary Layers,” Phys. Fluids, 15(5), pp. 1093–1105. [CrossRef]
Siauw, W. , Bonnet, J.-P. , Tensi, J. , Cordier, L. , Noack, B. R. , and Cattafesta, L. I. , 2010, “ Transient Dynamics of the Flow Around a NACA0015 Airfoil Using Fluid Vortex Generators,” Int. J. Heat Fluid Flow, 31(3), pp. 450–459. [CrossRef]
Shaqarin, T. , 2014, private communication.
Choi, H. , Moin, P. , and Kim, J. , 1994, “ Active Turbulence Control for Drag Reduction in Wall-Bounded Flows,” J. Fluid Mech., 262, pp. 75–110. [CrossRef]
Gerhard, J. , Pastoor, M. , King, R. , Noack, B. R. , Dillmann, A. , Morzyński, M. , and Tadmor, G. , 2003, “ Model-Based Control of Vortex Shedding Using Low-Dimensional Galerkin Models,” AIAA Paper No. 2003-4262.
Pastoor, M. , Henning, L. , Noack, B. R. , King, R. , and Tadmor, G. , 2008, “ Feedback Shear Layer Control for Bluff Body Drag Reduction,” J. Fluid Mech., 608, pp. 161–196. [CrossRef]
Samimy, M. , Debiasi, M. , Caraballo, E. , Serrani, A. , Yuan, X. , Little, J. , and Myatt, J. , 2007, “ Feedback Control of Subsonic Cavity Flows Using Reduced-Order Models,” J. Fluid Mech., 579, pp. 315–346. [CrossRef]
Vukasinovic, B. , Rusak, Z. , and Glezer, A. , 2010, “ Dissipative, Small-Scale Actuation of a Turbulent Shear Layer,” J. Fluid Mech., 656, pp. 51–81. [CrossRef]
Luchtenburg, D. M. , Günter, B. , Noack, B. R. , King, R. , and Tadmor, G. A. , 2009, “ Generalized Mean-Field Model of the Natural and Actuated Flows Around a High-Lift Configuration,” J. Fluid Mech., 623, pp. 283–316. [CrossRef]
Aider, J.-L. , 2014, private communication.
Gordon, M. , and Soria, J. , 2002, “ PIV Measurements of a Zero-Net-Mass-Flux Jet in Cross Flow,” Exp. Fluids, 33(6), pp. 863–872. [CrossRef]
Cater, J. E. , and Soria, J. , 2002, “ The Evolution of Round Zero-Net-Mass-Flux Jets,” J. Fluid Mech., 472, pp. 167–200. [CrossRef]
Zhang, P. , Wang, J. , and Feng, L. , 2008, “ Review of Zero-Net-Mass-Flux Jet and Its Application in Separation Flow Control,” Sci. China Ser. E, Technol. Sci., 51(9), pp. 1315–1344. [CrossRef]
Cattafesta, L. N. , Garg, S. , and Shukla, D. , 2001, “ Development of Piezoelectric Actuators for Active Flow Control,” AIAA J., 39(8), pp. 1562–1568. [CrossRef]
Gallas, Q. , Holman, R. , Nishida, T. , Carroll, B. , Sheplak, M. , and Cattafesta, L. , 2003, “ Lumped Element Modeling of Piezoelectric-Driven Synthetic Jet Actuators,” AIAA J., 41(2), pp. 240–247. [CrossRef]
Glezer, A. , and Amitay, M. , 2002, “ Synthetic Jets,” Annu. Rev. Fluid Mech., 34, pp. 503–529. [CrossRef]
Smith, B. L. , and Glezer, A. , 1998, “ The Formation and Evolution of Synthetic Jets,” Phys. Fluids, 10(9), pp. 2281–2297. [CrossRef]
Holman, R. , Utturkar, Y. , Mittal, R. , Smith, B. L. , and Cattafesta, L. , 2005, “ Formation Criterion for Synthetic Jets,” AIAA J., 43(10), pp. 2110–2116. [CrossRef]
You, D. , and Moin, P. , 2008, “ Active Control of Flow Separation Over an Airfoil Using Synthetic Jets,” J. Fluids Struct., 24(8), pp. 1349–1357. [CrossRef]
Moreau, E. , 2007, “ Airflow Control by Non-Thermal Plasma Actuators,” J. Phys. D: Appl. Phys., 40(3), p. 605. [CrossRef]
Hanson, R. E. , Lavoie, P. , and Naguib, A. M. , 2010, “ Effect of Plasma Actuator Excitation for Controlling Bypass Transition in Boundary Layers,” AIAA Paper No. 2010-1091.
Hanson, R. E. , Bade, K. M. , Belson, B. A. , Lavoie, P. , Naguib, A. M. , and Rowley, C. W. , 2014, “ Feedback Control of Slowly-Varying Transient Growth by an Array of Plasma Actuators,” Phys. Fluids, 26(2), p. 024102. [CrossRef]
Huang, J. , Corke, T. C. , and Thomas, F. O. , 2006, “ Plasma Actuators for Separation Control of Low-Pressure Turbine Blades,” AIAA J., 44(1), pp. 51–57. [CrossRef]
Roth, J. R. , Sherman, D. M. , and Wilkinson, S. P. , 2000, “ Electrohydrodynamic Flow Control With a Glow-Discharge Surface Plasma,” AIAA J., 38(7), pp. 1166–1172. [CrossRef]
Post, M. L. , and Corke, T. C. , 2004, “ Separation Control on High Angle of Attack Airfoil Using Plasma Actuators,” AIAA J., 42(11), pp. 2177–2184. [CrossRef]
Hanson, R. E. , Lavoie, P. , Naguib, A. M. , and Morrison, J. F. , 2010, “ Transient Growth Instability Cancelation by a Plasma Actuator Array,” Exp. Fluids, 49(6), pp. 1339–1348. [CrossRef]
Ho, C.-M. , and Tai, Y.-C. , 1996, “ Review: MEMS and Its Applications for Flow Control,” ASME J. Fluids Eng., 118(3), pp. 437–447. [CrossRef]
Ho, C.-M. , and Tai, Y.-C. , 1998, “ Micro-Electro-Mechanical Systems (MEMS) and Fluid Flows,” Annu. Rev. Fluid Mech., 30, pp. 579–612.
Naguib, A. , Christophorou, C. , Alnajjar, E. , Nagib, H. , Huang, C. , and Najafi, K. , 1997, “ Arrays of MEMS-Based Actuators for Control of Supersonic Jet Screech,” AIAA Paper No. 1997-1963.
Löfdahl, L. , and Gad-el-Hak, M. , 1999, “ MEMS Applications in Turbulence and Flow Control,” Prog. Aeronaut. Sci., 35(2), pp. 101–203. [CrossRef]
Huang, C. , Christophorou, C. , Najafi, K. , Naguib, A. , and Nagib, H. M. , 2002, “ An Electrostatic Microactuator System for Application in High-Speed Jets,” Microelectromech. Syst., J., 11(3), pp. 222–235. [CrossRef]
Suzuki, H. , Kasagi, N. , and Suzuki, Y. , 2004, “ Active Control of an Axisymmetric Jet With Distributed Electromagnetic Flap Actuators,” Exp. Fluids, 36(3), pp. 498–509. [CrossRef]
Kasagi, N. , Suzuki, Y. , and Fukagata, K. , 2009, “ Microelectromechanical Systems-Based Feedback Control of Turbulence for Skin Friction Reduction,” Annu. Rev. Fluid Mech., 41, pp. 231–251. [CrossRef]
Wu, J. , Wang, L. , and Tadmor, J. , 2007, “ Suppression of the Von Karman Vortex Street Behind a Circular Cylinder by a Traveling Wave Generated by a Flexible Surface,” J. Fluid Mech., 574, pp. 365–391. [CrossRef]
Thiria, B. , Goujon-Durand, S. , and Wesfreid, J. E. , 2006, “ The Wake of a Cylinder Performing Rotary Oscillations,” J. Fluid Mech., 560, pp. 123–147. [CrossRef]
Bergmann, M. , Cordier, L. , and Brancher, J.-P. , 2005, “ Optimal Rotary Control of the Cylinder Wake Using Proper Orthogonal Decomposition Reduced Order Model,” Phys. Fluids, 17(9), p. 097101. [CrossRef]
Wiener, N. , 1948, Cybernetics or Control and Communication in the Animal and the Machine, 1st ed., MIT Press, Boston.
Kolmogorov, A. , 1941, “ The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Number,” Dokl. Akad. Nauk. SSSR, 30, pp. 9–13.
Kolmogorov, A. , 1941, “ On Degeneration (Decay) of Isotropic Turbulence,” Dokl. Akad. Nauk SSSR, 31, pp. 538–540.
Landau, L. D. , and Lifshitz, E. M. , 1987, “ Fluid Mechanics,” Course of Theoretical Physics, 2nd ed., Vol. 6, Pergamon Press, Oxford, UK.
Pope, S. , 2000, Turbulent Flows, 1st ed., Cambridge University Press, Cambridge, UK.
Lee, M. , Malaya, N. , and Moser, R. D. , 2013, “ Petascale Direct Numerical Simulation of Turbulent Channel Flow on Up to 786 k Cores,” International Conference on High Performance Computing, Networking, Storage and Analysis (SC'13), Denver, CO, Nov. 17–21, p. 61.
Kaneda, Y. , Ishihara, T. , Yokokawa, M. , Itakura, K. , and Uno, A. , 2003, “ Energy Dissipation Rate and Energy Spectrum in High Resolution Direct Numerical Simulations of Turbulence in a Periodic Box,” Phys. Fluids, 15(2), pp. L21–L24. [CrossRef]
Moore, G. E. , 1965, “ Cramming More Components Onto Integrated Circuits,” Electronics, 38(8), pp. 114–117.
Lumley, J. , 1970, Stochastic Tools in Turbulence, Academic Press, New York.
Holmes, P. , Lumley, J. L. , Berkooz, G. , and Rowley, C. W. , 2012, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd ed., Cambridge University Press, Cambridge, UK.
Sirovich, L. , 1987, “ Turbulence and the Dynamics of Coherent Structures, Part I—Coherent Structures,” Q. Appl. Math., XLV(3), pp. 561–571.
Golub, G. H. , and Reinsch, C. , 1970, “ Singular Value Decomposition and Least Squares Solutions,” Numer. Math., 14(5), pp. 403–420. [CrossRef]
Golub, G. , and Kahan, W. , 1965, “ Calculating the Singular Values and Pseudo-Inverse of a Matrix,” J. Soc. Ind. Appl. Math., Ser. B: Numer. Anal., 2(2), pp. 205–224. [CrossRef]
Trefethen, L. N. , and Bau, D., III. , 1997, Numerical Linear Algebra, Vol. 50, SIAM, Philadelphia.
Antoulas, A. C. , 2005, Approximation of Large-Scale Dynamical Systems, SIAM, Philadelphia.
Pearson, K. , 1901, “ On Lines and Planes of Closest Fit to Systems of Points in Space,” Philos. Mag., 2(7–12), pp. 559–572. [CrossRef]
Hotelling, H. , 1933, “ Analysis of a Complex of Statistical Variables Into Principal Components,” J. Educ. Psychol., 24(6), pp. 417–441. [CrossRef]
Karhunen, K. , 1946, “ Zur Spektraltheorie Stochastischer Prozesse,” Ann. Acad. Sci., Fennicae, Ser. A. I., Math.-Phys., 37, pp. 1–79.
Lorenz, E. , 1956, “ Empirical Orthogonal Functions and Statistical Weather Prediction,” Department of Meteorology, Statistical Forecasting Project, MIT, Cambridge, MA, Scientific Report No. 1.
Andino, M. Y. , Wallace, R. D. , Glauser, M. N. , Camphouse, R. C. , Schmit, R. F. , and Myatt, J. H. , 2011, “ Boundary Feedback Flow Control: Proportional Control With Potential Application to Aero-Optics,” AIAA J., 49(1), pp. 32–40. [CrossRef]
Willcox, K. , and Peraire, J. , 2002, “ Balanced Model Reduction Via the Proper Orthogonal Decomposition,” AIAA J., 40(11), pp. 2323–2330. [CrossRef]
Rowley, C. , 2005, “ Model Reduction for Fluids Using Balanced Proper Orthogonal Decomposition,” Int. J. Bifurcation Chaos, 15(3), pp. 997–1013. [CrossRef]
Schmid, P. J. , and Sesterhenn, J. , 2008, “ Dynamic Mode Decomposition of Numerical and Experimental Data,” 61st Annual Meeting of the APS Division of Fluid Dynamics, San Antonio, TX, Nov. 23–25, American Physical Society, College Park, MD, pp. 208.
Schmid, P. J. , 2010, “ Dynamic Mode Decomposition for Numerical and Experimental Data,” J. Fluid Mech., 656, pp. 5–28. [CrossRef]
Rowley, C. W. , Mezić, I. , Bagheri, S. , Schlatter, P. , and Henningson, D. , 2009, “ Spectral Analysis of Nonlinear Flows,” J. Fluid Mech., 645, pp. 115–127. [CrossRef]
Tu, J. H. , Rowley, C. W. , Luchtenburg, D. M. , Brunton, S. L. , and Kutz, J. N. , 2014, “ On Dynamic Mode Decomposition: Theory and Applications,” J. Comput. Dyn., 1(2), pp. 391–421. [CrossRef]
Tu, J. H. , Rowley, C. W. , Kutz, J. N. , and Shang, J. K. , 2014, “ Spectral Analysis of Fluid Flows Using Sub-Nyquist-Rate PIV Data,” Exp. Fluids, 55(9), p. 1805.
Koopman, B. O. , 1931, “ Hamiltonian Systems and Transformation in Hilbert Space,” Proc. Natl. Acad. Sci., 17(5), pp. 315–318. [CrossRef]
Mezić, I. , and Banaszuk, A. , 2004, “ Comparison of Systems With Complex Behavior,” Phys. D: Nonlinear Phenom., 197(1), pp. 101–133. [CrossRef]
Mezić, I. , 2005, “ Spectral Properties of Dynamical Systems, Model Reduction and Decompositions,” Nonlinear Dyn., 41(1–3), pp. 309–325. [CrossRef]
Budišić, M. , Mohr, R. , and Mezić, I. , 2012, “ Applied Koopmanism,” Chaos: Interdiscip. J. Nonlinear Sci., 22(4), p. 047510. [CrossRef]
Mezić, I. , 2013, “ Analysis of Fluid Flows Via Spectral Properties of the Koopman Operator,” Annu. Rev. Fluid Mech., 45, pp. 357–378. [CrossRef]
Schmid, P. J. , and Hennigson, D. S. , 2001, Stability and Transition in Shear Flows, Springer, New York.
Schmid, P. J. , 2007, “ Nonmodal Stability Theory,” Annu. Rev. Fluid Mech., 39, pp. 129–162. [CrossRef]
Theofilis, V. , 2011, “ Global Linear Instability,” Annu. Rev. Fluid Mech., 43, pp. 319–352. [CrossRef]
Schmid, P. J. , and Brandt, L. , 2014, “ Analysis of Fluid Systems: Stability, Receptivity, Sensitivity,” ASME Appl. Mech. Rev., 66(2), p. 024803. [CrossRef]
Grosch, C. E. , and Salwen, H. , 1978, “ The Continuous Spectrum of the Orr-Sommerfeld Equation Part I—The Spectrum and the Eigenfunctions,” J. Fluid Mech., 87, pp. 33–54. [CrossRef]
Salwen, H. , and Grosch, C. E. , 1981, “ The Continuous Spectrum of the Orr-Sommerfeld Equation. Part 2—Eigenfunction Expansions,” J. Fluid Mech., 104, pp. 445–465. [CrossRef]
Joseph, D. D. , 1976, “ Stability of Fluid Motions I & II,” Springer Tracts in Natural Philosophy, Vols. 26 and 27, Springer, New York.
Boberg, L. , and Brosa, U. , 1988, “ Onset of Turbulence in a Pipe,” Z. Naturforsch., 43a, pp. 697–726.
Trefethen, L. N. , Trefethen, A. E. , Reddy, S. C. , and Driscoll, T. A. , 1993, “ Hydrodynamic Stability Without Eigenvalues,” Science, 261(5121), pp. 578–584. [CrossRef] [PubMed]
Belson, B. A. , Tu, J. H. , and Rowley, C. W. , 2014, “ Algorithm 945: modred—A Parallelized Model Reduction Library,” ACM Trans. Math. Software, 40(4), p. 30. [CrossRef]
von Karman, T. , 1912, “ Über Den Mechanismus des Widerstands, den Ein Bewegter Korper in Einer Flüssigkeit Erfährt,” Göttinger Nachrichten, Math. Phys. Kl., 1912, pp. 547–556.
Föppl, L. , 1913, “ Wirbelbewegung hinter einem Kreiszylinder,” Sitzb. d. k. Bayer. Akad. d. Wiss., 1, pp. 1.
Suh, Y. , 1993, “ Periodic Motion of a Point Vortex in a Corner Subject to a Potential Flow,” J. Phys. Soc. Jpn., 62, pp. 3441–3445. [CrossRef]
Noack, B. R. , Mezić, I. , Tadmor, G. , and Banaszuk, A. , 2004, “ Optimal Mixing in Recirculation Zones,” Phys. Fluids, 16(4), pp. 867–888. [CrossRef]
Lugt, H. , 1996, Introduction to Vortex Theory, Vortex Flow Press, Potomac, MA.
Cottet, G. H. , and Koumoutsakos, P. , 2000, Vortex Methods—Theory and Practice, Cambridge University Press, Cambridge, UK.
Wu, J.-Z. , Ma, H.-Y. , and Zhou, M.-D. , 2006, Vorticity and Vortex Dynamics, 1st ed., Springer, Berlin.
Adrian, R. , and Moin, P. , 1988, “ Stochastic Estimation of Organized Turbulent Structure: Homogeneous Shear Flow,” J. Fluid Mech., 190, pp. 531–559. [CrossRef]
Nicoud, F. , Baggett, J. , Moin, P. , and Cabot, W. , 2001, “ Large Eddy Simulation Wall-Modeling Based on Suboptimal Control Theory and Linear Stochastic Estimation,” Phys. Fluids, 13(10), pp. 2968–2984. [CrossRef]
Bonnet, J.-P. , Cole, D. , Delville, J. , Glauser, M. N. , and Ukeiley, L. S. , 1998, “ Stochastic Estimation and Proper Orthogonal Decomposition—Complementary Techniques for Identfying Structure,” Exp. Fluids, 17(5), pp. 307–314. [CrossRef]
Glauser, M. N. , Higuchi, H. , Ausseur, J. , and Pinier, J. , 2004, “ Feedback Control of Separated Flows,” AIAA Paper No. 2004-2521.
Ausseur, J. M. , Pinier, J. T. , Glauser, M. N. , Higuchi, H. , and Carlson, H. , 2006, “ Experimental Development of a Reduced-Order Model for Flow Separation Control,” AIAA Paper No. 2006-1251.
Tinney, C. , Coiffet, F. , Delville, J. , Hall, A. , Jordan, P. , and Glauser, M. , 2006, “ On Spectral Linear Stochastic Estimation,” Exp. Fluids, 41(5), pp. 763–775. [CrossRef]
Hudy, L. M. , Naguib, A. , and Humphreys, W. M. , 2007, “ Stochastic Estimation of a Separated-Flow Field Using Wall-Pressure-Array Measurements,” Phys. Fluids, 19(2), p. 024103. [CrossRef]
Pinier, J. T. , Ausseur, J. M. , Glauser, M. N. , and Higuchi, H. , 2007, “ Proportional Closed-Loop Feedback Control of Flow Separation,” AIAA J., 45(1), pp. 181–190. [CrossRef]
Farrell, B. F. , and Ioannou, P. J. , 2001, “ State Estimation Using a Reduced-Order Kalman Filter,” J. Atmos. Sci., 58(23), pp. 3666–3680. [CrossRef]
King, R. , and Gilles, E. , 1985, “ Multiple Kalman Filters for Early Detection of Hazardous States,” International Conference Industrial Process Modelling and Control, Hangzhou, China, June 6–9, pp. 130–138.
Tu, J. H. , Griffin, J. , Hart, A. , Rowley, C. W. , III, L. N. C. , and Ukeiley, L. S. , 2013, “ Integration of Non-Time-Resolved PIV and Time-Resolved Velocity Point Sensors for Dynamic Estimation of Velocity Fields,” Exp. Fluids, 54(2), p. 1429.
Welch, G. , and Bishop, G. , 1995, “ An Introduction to the Kalman Filter ,” University of North Carolina, Chapel Hill, NC, Technical Report 95-041.
Busse, F. H. , 1991, “ Numerical Analysis of Secondary and Tertiary States of Fluid Flow and Their Stability Properties,” Appl. Sci. Res., 48(3–4), pp. 341–351. [CrossRef]
Noack, B. R. , and Eckelmann, H. , 1994, “ A Global Stability Analysis of the Steady and Periodic Cylinder Wake,” J. Fluid Mech., 270, pp. 297–330. [CrossRef]
Fletcher, C. A. J. , 1984, Computational Galerkin Methods, 1st ed., Springer, New York.
Holmes, P. , Lumley, J. L. , and Berkooz, G. , 1998, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 1st ed., Cambridge University Press, Cambridge, UK.
Juang, J. N. , and Pappa, R. S. , 1985, “ An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction,” J. Guid., Control, Dyn., 8(5), pp. 620–627. [CrossRef]
Juang, J. N. , 1994, Applied System Identification, Prentice-Hall, Upper Saddle River, NJ.
Ljung, L. , 2001, “ Black-Box Models From Input–Output Measurements,” 18th IEEE Instrumentation and Measurement Technology Conference (IMTC 2001), Budapest, May 21–23, pp. 138–146.
Ljung, L. , 1999, System Identification: Theory for the User, Prentice-Hall, Upper Saddle River, NJ.
Crouch, P. , 1981, “ Dynamical Realizations of Finite Volterra Series,” SIAM J. Control Optim., 19(2), pp. 177–202. [CrossRef]
Boyd, S. , Chua, L. O. , and Desoer, C. A. , 1984, “ Analytical Foundations of Volterra Series,” IMA J. Math. Control Inf., 1(3), pp. 243–282. [CrossRef]
Boyd, S. , and Chua, L. O. , 1985, “ Fading Memory and the Problem of Approximating Nonlinear Operators With Volterra Series,” IEEE Trans. Circuits Syst., 32(11), pp. 1150–1161. [CrossRef]
Lesiak, C. , and Krener, A. J. , 1978, “ The Existence and Uniqueness of Volterra Series for Nonlinear Systems,” IEEE Trans. Autom. Control, 23(6), pp. 1090–1095. [CrossRef]
Brockett, R. W. , 1976, “ Volterra Series and Geometric Control Theory,” Automatica, 12(2), pp. 167–176. [CrossRef]
Krstić, M. , Smyshlyaev, A. , and Vazquez, R. , 2006, “ Boundary Control of PDEs and Applications to Turbulent Flows and Flexible Structures,” IEEE Chinese Control Conference (CCC 2006), Harbin, China, Aug. 7–11, pp. PL–4–PL–16.
Floriani, E. , de Wit, T. D. , and Le Gal, P. , 2000, “ Nonlinear Interactions in a Rotating Disk Flow: From a Volterra Model to the Ginzburg–Landau Equation,” Chaos: Interdiscip. J. Nonlinear Sci., 10(4), pp. 834–847. [CrossRef]
Tromp, J. C. , and Jenkins, J. E. A. , 1990, “ Volterra Kernel Identification Scheme Applied to Aerodynamic Reactions,” AIAA Paper No. 90-2803.
Prazenica, R. J. , Reisenthel, P. H. , Kurdila, A. J. , and Brenner, M. J. , 2007, “ Volterra Kernel Extrapolation for Modeling Nonlinear Aeroelastic Systems at Novel Flight Conditions,” J. Aircr., 44(1), pp. 149–162. [CrossRef]
Balajewicz, M. , and Dowell, E. , 2012, “ Reduced-Order Modeling of Flutter and Limit-Cycle Oscillations Using the Sparse Volterra Series,” J. Aircr., 49(6), pp. 1803–1812. [CrossRef]
Balikhin, M. , Bates, I. , and Walker, S. , 2001, “ Identification of Linear and Nonlinear Processes in Space Plasma Turbulence Data,” Adv. Space Res., 28(5), pp. 787–800. [CrossRef]
Vazquez, R. , and Krstić, M. , 2007, Control of Turbulent and Magnetohydrodynamic Channel Flows: Boundary Stabilization and State Estimation, Springer, New York.
Estrada, T. , Happel, T. , Hidalgo, C. , Ascasibar, E. , and Blanco, E. , 2010, “ Experimental Observation of Coupling Between Turbulence and Sheared Flows During LH Transitions in a Toroidal Plasma,” Europhys. Lett., 92(3), p. 35001. [CrossRef]
Smola, A. J. , and Schölkopf, B. , 2004, “ A Tutorial on Support Vector Regression,” Stat. Comput., 14(3), pp. 199–222. [CrossRef]
Schölkopf, B. , and Smola, A. J. , 2002, Learning With Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press, Cambridge, MA.
Suykens, J. A. , and Vandewalle, J. , 1999, “ Least Squares Support Vector Machine Classifiers,” Neural Process. Lett., 9(3), pp. 293–300. [CrossRef]
Doyle, J. C. , 1978, “ Guaranteed Margins for LQG Regulators,” IEEE Trans. Autom. Control, 23(4), pp. 756–757. [CrossRef]
Doyle, J. C. , and Stein, G. , 1981, “ Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis,” IEEE Trans. Autom. Control, 26(1), pp. 4–16. [CrossRef]
Glover, K. , and Doyle, J. C. , 1988, “ State-Space Formulae for All Stabilizing Controllers That Satisfy an H -Norm Bound and Relations to Risk Sensitivity,” Syst. Control Lett., 11(3), pp. 167–172. [CrossRef]
Doyle, J. C. , Glover, K. , Khargonekar, P. P. , and Francis, B. A. , 1989, “ State-Space Solutions to Standard H2 and H Control Problems,” IEEE Trans. Autom. Control, 34(8), pp. 831–847. [CrossRef]
Schlinker, R. , Simonich, J. , Shannon, D. , Reba, R. , Colonius, T. , Gudmundsson, K. , and Ladeinde, F. , 2009, “ Supersonic Jet Noise From Round and Chevron Nozzles: Experimental Studies,” AIAA Paper No. 2009-3257.
Skogestad, S. , and Postlethwaite, I. , 1996, Multivariable Feedback Control, Wiley, Chichester, UK.
Dullerud, G. E. , and Paganini, F. , 2000, “ A Course in Robust Control Theory: A Convex Approach,” Texts in Applied Mathematics, Springer, Berlin.
Scott Collis, S. , Joslin, R. D. , Seifert, A. , and Theofilis, V. , 2004, “ Issues in Active Flow Control: Theory, Control, Simulation, and Experiment,” Prog. Aerosp. Sci., 40(4), pp. 237–289. [CrossRef]
Rowley, C. W. , and Batten, B. A. , 2008, “ Dynamic and Closed-Loop Control,” Fundamentals and Applications of Modern Flow Control (Progress in Astronautics and Aeronautics, Vol. 231), American Institute of Aeronautics and Astronautics, Reston, VA, pp. 115–148.
Bagheri, S. , Hoepffner, J. , Schmid, P. J. , and Henningson, D. S. , 2009, “ Input–Output Analysis and Control Design Applied to a Linear Model of Spatially Developing Flows,” ASME Appl. Mech. Rev., 62(2), p. 020803. [CrossRef]
Fabbiane, N. , Semeraro, O. , Bagheri, S. , and Henningson, D. S. , 2014, “ Adaptive and Model-Based Control Theory Applied to Convectively Unstable Flows,” ASME Appl. Mech. Rev., 66(6), p. 060801.
Devasia, S. , Chen, D. , and Paden, B. , 1996, “ Nonlinear Inversion-Based Output Tracking,” IEEE Trans. Autom. Control, 41(7), pp. 930–942. [CrossRef]
Krstić, M. , and Banaszuk, A. , 2006, “ Multivariable Adaptive Control of Instabilities Arising in Jet Engines,” Control Eng. Pract., 14(7), pp. 833–842. [CrossRef]
Bewley, T. R. , Temam, R. , and Ziane, M. , 2000, “ A General Framework for Robust Control in Fluid Mechanics,” Phys. D, 138(3–4), pp. 360–392. [CrossRef]
King, R. , Active Flow and Combustion Control, Vol. 127, Springer International Publishing, Cham, Switzerland.
Kerstens, W. , Pfeiffer, J. , Williams, D. , King, R. , and Colonius, T. , 2011, “ Closed-Loop Control of Lift for Longitudinal Gust Suppression at Low Reynolds Numbers,” AIAA J., 49(8), pp. 1721–1728. [CrossRef]
Devasia, S. , 2002, “ Should Model-Based Inverse Inputs be Used as Feedforward Under Plant Uncertainty?” IEEE Trans., Autom. Control, 47(11), pp. 1865–1871. [CrossRef]
Chen, K. K. , and Rowley, C. W. , 2013, “ Normalized Coprime Robust Stability and Performance Guarantees for Reduced-Order Controllers,” IEEE Trans. Autom. Control, 58(4), pp. 1068–1073. [CrossRef]
Businger, P. A. , and Golub, G. H. , 1969, “ Algorithm 358: Singular Value Decomposition of a Complex Matrix [F1, 4, 5],” Commun. ACM, 12(10), pp. 564–565. [CrossRef]
Ho, B. L. , and Kalman, R. E. , 1965, “ Effective Construction of Linear State-Variable Models From Input/Output Data,” 3rd Annual Allerton Conference on Circuit and System Theory, Monticello, IL, Oct. 20–22, pp. 449–459.
Moore, B. C. , 1981, “ Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction,” IEEE Trans. Autom. Control, 26(1), pp. 17–32. [CrossRef]
Berkooz, G. , Holmes, P. , and Lumley, J. , 1993, “ The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows,” Annu. Rev. Fluid Mech., 25, pp. 539–575. [CrossRef]
Ilak, M. , and Rowley, C. W. , 2008, “ Modeling of Transitional Channel Flow Using Balanced Proper Orthogonal Decomposition,” Phys. Fluids, 20(3), p. 034103. [CrossRef]
Lall, S. , Marsden, J. E. , and Glavaški, S. , 1999, “ Empirical Model Reduction of Controlled Nonlinear Systems,” International Federation of Automatic Control (IFAC) World Congress, Beijing, July 5–9, pp. 473–478.
Lall, S. , Marsden, J. E. , and Glavaški, S. , 2002, “ A Subspace Approach to Balanced Truncation for Model Reduction of Nonlinear Control Systems,” Int. J. Rob. Nonlinear Control, 12(6), pp. 519–535. [CrossRef]
Laub, A. J. , Heath, M. T. , Paige, C. , and Ward, R. , 1987, “ Computation of System Balancing Transformations and Other Applications of Simultaneous Diagonalization Algorithms,” IEEE Trans. Autom. Control, 32(2), pp. 115–122. [CrossRef]
Sirovich, L. , 1987, “ Turbulence and the Dynamics of Coherent Structures, Part III—Dynamics and Scaling,” Q. Appl. Math., XLV, pp. 583–590.
Sirovich, L. , 1987, “ Turbulence and the Dynamics of Coherent Structures, Part II—Symmetries and Transformations,” Q. Appl. Math., XLV, pp. 573–582.
Ma, Z. , Ahuja, S. , and Rowley, C. W. , 2011, “ Reduced Order Models for Control of Fluids Using the Eigensystem Realization Algorithm,” Theor. Comput. Fluid Dyn., 25(1), pp. 233–247. [CrossRef]
Luchtenburg, D. M. , and Rowley, C. W. , 2011, “ Model Reduction Using Snapshot-Based Realizations,” Bull. Am. Phys. Soc., 56, p. BAPS.2011.DFD.H19.4.
Tu, J. H. , and Rowley, C. W. , 2012, “ An Improved Algorithm for Balanced POD Through an Analytic Treatment of Impulse Response Tails,” J. Comput. Phys., 231(16), pp. 5317–5333. [CrossRef]
Juang, J. N. , Phan, M. , Horta, L. G. , and Longman, R. W. , 1991, “ Identification of Observer/Kalman Filter Markov Parameters: Theory and Experiments,” NASA Langley Research Center, Hampton, VA, NASA Technical Memorandum No. 104069.
Phan, M. , Juang, J. N. , and Longman, R. W. , 1992, “ Identification of Linear-Multivariable Systems by Identification of Observers With Assigned Real Eigenvalues,” J. Astronaut. Sci., 40(2), pp. 261–279.
Phan, M. , Horta, L. G. , Juang, J. N. , and Longman, R. W. , 1993, “ Linear System Identification Via an Asymptotically Stable Observer,” J. Optim. Theory Appl., 79(1), pp. 59–86. [CrossRef]
Proctor, J. L. , Brunton, S. L. , and Kutz, J. N. , 2014, “ Dynamic Mode Decomposition With Control: Using State and Input Snapshots to Discover Dynamics,” arXiv:1409.6358.
Barkley, D. , and Tuckerman, L. S. , 1999, “ Stability Analysis of Perturbed Plane Couette Flow,” Phys. Fluids, 11(5), pp. 1187–1195. [CrossRef]
Bayly, B. J. , Orszag, S. A. , and Herbert, T. , 1988, “ Instability Mechanisms in Shear-Flow Transition,” Annu. Rev. Fluid Mech., 20(1), pp. 359–391. [CrossRef]
Orszag, S. A. , and Patera, A. T. , 1983, “ Secondary Instability of Wall-Bounded Shear Flows,” J. Fluid Mech., 128, pp. 347–385. [CrossRef]
Ruelle, D. , and Takens, F. , 1971, “ On the Nature of Turbulence,” Commun. Math. Phys., 20(3), pp. 167–192. [CrossRef]
Aamo, O. M. , Krstić, M. , and Bewley, T. R. , 2003, “ Control of Mixing by Boundary Feedback in 2D Channel Flow,” Automatica, 39(9), pp. 1597–1606. [CrossRef]
Bagheri, S. , and Henningson, D. S. , 2011, “ Transition Delay Using Control Theory,” Philos. Trans. R. Soc. A, 369(1940), pp. 1365–1381. [CrossRef]
Abergel, F. , and Temam, R. , 1990, “ On Some Control Problems in Fluid Mechanics,” Theor. Comput. Fluid Dyn., 1(6), pp. 303–325. [CrossRef]
Jameson, A. , 2003, “ Aerodynamic Shape Optimization Using the Adjoint Method” (VKI Lecture Series on Aerodynamic Drag Prediction and Reduction), von Karman Institute of Fluid Dynamics, Rhode-St-Genese, Belgium.
Reuther, J. J. , Jameson, A. , Alonso, J. J. , Rimlinger, M. J. , and Saunders, D. , 1999, “ Constrained Multipoint Aerodynamic Shape Optimization Using an Adjoint Formulation and Parallel Computers, Part 1,” J. Aircr., 36(1), pp. 51–60. [CrossRef]
Jameson, A. , Martinelli, L. , and Pierce, N. , 1998, “ Optimum Aerodynamic Design Using the Navier–Stokes Equations,” Theor. Comput. Fluid Dyn., 10(1–4), pp. 213–237. [CrossRef]
Reuther, J. , Jameson, A. , Farmer, J. , Martinelli, L. , and Saunders, D. , 1996, “ Aerodynamic Shape Optimization of Complex Aircraft Configurations Via an Adjoint Formulation,” Research Institute for Advanced Computer Science, NASA Ames Research Center, Mountain View, CA, Report No. NASA-CR-203275.
Choi, H. , Temam, R. , Moin, P. , and Kim, J. , 1993, “ Feedback Control for Unsteady Flow and Its Application to the Stochastic Burgers Equation,” J. Fluid Mech., 253, pp. 509–543. [CrossRef]
Bewley, T. , and Moin, P. , 1994, “ Optimal Control of Turbulent Channel Flows,” Act. Control Vib. Noise, ASME DE-Vol. 75, pp. 221–227.
Lee, C. , Kim, J. , and Choi, H. , 1998, “ Suboptimal Control of Turbulent Channel Flow for Drag Reduction,” J. Fluid Mech., 358, pp. 245–258. [CrossRef]
Bewley, T. R. , Moin, P. , and Temam, R. , 2001, “ DNS-Based Predictive Control of Turbulence: An Optimal Benchmark for Feedback Algorithms,” J. Fluid Mech., 447, pp. 179–225. [CrossRef]
Collis, S. S. , Chang, Y. , Kellogg, S. , and Prabhu, R. , 2000, “ Large Eddy Simulation and Turbulence Control,” AIAA Paper No. 2000-2564.
Bewley, T. , and Liu, S. , 1998, “ Optimal and Robust Control and Estimation of Linear Paths to Transition,” J. Fluid Mech., 365, pp. 305–349. [CrossRef]
Baramov, L. , Tutty, O. R. , and Rogers, E. , 2000, “ Robust Control of Plane Poiseuille Flow,” AIAA Paper No. 2000-2684.
Högberg, M. , Bewley, T. R. , and Henningson, D. S. , 2003, “ Linear Feedback Control and Estimation of Transition in Plane Channel Flow,” J. Fluid Mech., 481, pp. 149–175. [CrossRef]
Högberg, M. , and Henningson, D. S. , 2002, “ Linear Optimal Control Applied to Instabilities in Spatially Developing Boundary Layers,” J. Fluid Mech., 470, pp. 151–179. [CrossRef]
Chevalier, M. , Hœpffner, J. , Åkervik, E. , and Henningson, D. , 2007, “ Linear Feedback Control and Estimation Applied to Instabilities in Spatially Developing Boundary Layers,” J. Fluid Mech., 588, pp. 163–187. [CrossRef]
Åkervik, E. , Hœpffner, J. , Ehrenstein, U. , and Henningson, D. S. , 2007, “ Optimal Growth, Model Reduction and Control in Separated Boundary-Layer Flow Using Global Eigenmodes,” J. Fluid Mech., 579, pp. 305–314. [CrossRef]
Ahuja, S. , Rowley, C. W. , Kevrekidis, I. G. , Wei, M. , Colonius, T. , and Tadmor, G. , 2007, “ Low-Dimensional Models for Control of Leading-Edge Vortices: Equilibria and Linearized Models,” AIAA Paper No. 2007-709.
Colonius, T. , and Taira, K. , 2008, “ A Fast Immersed Boundary Method Using a Nullspace Approach and Multi-Domain Far-Field Boundary Conditions,” Comput. Methods Appl. Mech. Eng., 197(25–28), pp. 2131–2146. [CrossRef]
Taira, K. , and Colonius, T. , 2007, “ The Immersed Boundary Method: A Projection Approach,” J. Comput. Phys., 225(2), pp. 2118–2137. [CrossRef]
Bagheri, S. , Brandt, L. , and Henningson, D. , 2009, “ Input–Output Analysis, Model Reduction and Control of the Flat-Plate Boundary Layer,” J. Fluid Mech., 620, pp. 263–298. [CrossRef]
Semeraro, O. , Bagheri, S. , Brandt, L. , and Henningson, D. S. , 2011, “ Feedback Control of Three-Dimensional Optimal Disturbances Using Reduced-Order Models,” J. Fluid Mech., 677, pp. 63–102. [CrossRef]
Illingworth, S. J. , Morgans, A. S. , and Rowley, C. W. , 2010, “ Feedback Control of Flow Resonances Using Balanced Reduced-Order Models,” J. Sound Vib., 330(8), pp. 1567–1581. [CrossRef]
Illingworth, S. J. , Morgans, A. S. , and Rowley, C. W. , 2012, “ Feedback Control of Cavity Flow Oscillations Using Simple Linear Models,” J. Fluid Mech., 709, pp. 223–248. [CrossRef]
Semeraro, O. , Bagheri, S. , Brandt, L. , and Henningson, D. S. , 2013, “ Transition Delay in a Boundary Layer Flow Using Active Control,” J. Fluid Mech., 731, pp. 288–311. [CrossRef]
Moarref, R. , and Jovanović, M. R. , 2012, “ Model-Based Design of Transverse Wall Oscillations for Turbulent Drag Reduction,” J. Fluid Mech., 707, pp. 205–240. [CrossRef]
Cortelezzi, L. , Lee, K. , Kim, J. , and Speyer, J. , 1998, “ Skin-Friction Drag Reduction Via Robust Reduced-Order Linear Feedback Control,” Int. J. Comput. Fluid Dyn., 11(1–2), pp. 79–92. [CrossRef]
Cortelezzi, L. , and Speyer, J. , 1998, “ Robust Reduced-Order Controller of Laminar Boundary Layer Transitions,” Phys. Rev. E, 58(2), pp. 1906–1910. [CrossRef]
Lee, K. H. , Cortelezzi, L. , Kim, J. , and Speyer, J. , 2001, “ Application of Reduced-Order Controller to Turbulent Flows for Drag Reduction,” Phys. Fluids, 13(5), pp. 1321–1330. [CrossRef]
Kasagi, N. , Hasegawa, Y. , and Fukagata, K. , 2009, “ Toward Cost-Effective Control of Wall Turbulence for Skin Friction Drag Reduction,” Advances in Turbulence XII, Springer, Berlin, pp. 189–200.
Fukagata, K. , Kobayashi, M. , and Kasagi, N. , 2010, “ On the Friction Drag Reduction Effect by a Control of Large-Scale Turbulent Structures,” J. Fluid Sci. Technol., 5(3), pp. 574–584. [CrossRef]
Mamori, H. , Fukagata, K. , and Hoepffner, J. , 2010, “ Phase Relationship in Laminar Channel Flow Controlled by Traveling-Wave-Like Blowing or Suction,” Phys. Rev. E, 81(4), p. 046304. [CrossRef]
Kametani, Y. , and Fukagata, K. , 2011, “ Direct Numerical Simulation of Spatially Developing Turbulent Boundary Layers With Uniform Blowing or Suction,” J. Fluid Mech., 681, pp. 154–172. [CrossRef]
Nakanishi, R. , Mamori, H. , and Fukagata, K. , 2012, “ Relaminarization of Turbulent Channel Flow Using Traveling Wave-Like Wall Deformation,” Int. J. Heat Fluid Flow, 35, pp. 152–159. [CrossRef]
Kasagi, N. , Hasegawa, Y. , Fukagata, K. , and Iwamoto, K. , 2012, “ Control of Turbulent Transport: Less Friction and More Heat Transfer,” ASME J. Heat Transfer, 134(3), p. 031009. [CrossRef]
Rathnasingham, R. , and Breuer, K. S. , 1997, “ System Identification and Control of a Turbulent Boundary Layer,” Phys. Fluids, 9(7), pp. 1867–1869. [CrossRef]
Rathnasingham, R. , and Breuer, K. S. , 2003, “ Active Control of Turbulent Boundary Layers,” J. Fluid Mech., 495, pp. 209–233. [CrossRef]
Rowley, C. W. , 2002, “ Modeling, Simulation, and Control of Cavity Flow Oscillations,” Ph.D. thesis, California Institute of Technology, Pasadena, CA.
Cattafesta, L. , Shukla, D. , Garg, S. , and Ross, J. , 1999, “ Development of an Adaptive Weapons-Bay Suppression System,” AIAA Paper No. 1999-1901.
Cattafesta, L. , Williams, D. , Rowley, C. , and Alvi, F. , 2003, “ Review of Active Control of Flow-Induced Cavity Resonance,” AIAA Paper No. 2003-3567.
Cattafesta, L. N., III , Song, Q. , Williams, D. R. , Rowley, C. W. , and Alvi, F. S. , 2008, “ Active Control of Flow-Induced Cavity Oscillations,” Prog. Aerosp. Sci., 44(7), pp. 479–502. [CrossRef]
Rowley, C. W. , Colonius, T. , and Basu, A. J. , 2002, “ On Self-Sustained Oscillations in Two-Dimensional Compressible Flow Over Rectangular Cavities,” J. Fluid Mech., 455, pp. 315–346. [CrossRef]
Rowley, C. W. , Colonius, T. , and Murray, R. M. , 2000, “ POD Based Models of Self-Sustained Oscillations in the Flow Past an Open Cavity,” AIAA Paper No. 2000-1969.
Rowley, C. , Colonius, T. , and Murray, R. , 2004, “ Model Reduction for Compressible Flows Using POD and Galerkin Projection,” Physica D, 189(1–2), pp. 115–129. [CrossRef]
Rowley, C. W. , Williams, D. R. , Colonius, T. , Murray, R. M. , MacMartin, D. G. , and Fabris, D. , 2002, “ Model-Based Control of Cavity Oscillations. Part II: System Identification and Analysis,” AIAA Paper No. 2002-0972.
Samimy, M. , Debiasi, M. , Caraballo, E. , Malone, J. , Little, J. , Özbay, H. , Efe, M. , Yan, X. , Yuan, X. , DeBonis, J. , Myatt, J. , and Camphouse, R. , 2004, “ Exploring Strategies for Closed-Loop Cavity Flow Control,” AIAA Paper No. 2004-0576.
Rowley, C. W. , Williams, D. R. , Colonius, T. , Murray, R. M. , and Macmynowski, D. G. , 2006, “ Linear Models for Control of Cavity Flow Oscillations,” J. Fluid Mech., 547, pp. 317–330. [CrossRef]
Samimy, M. , Debiasi, M. , Caraballo, E. , Serrani, A. , Yuan, X. , and Little, J. , 2007, “ Reduced-Order Model-Based Feedback Control of Subsonic Cavity Flows—An Experimental Approach,” Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), Vol. 25, Springer, Berlin, pp. 211–230.
Efe, M. , Debiasi, M. , Yan, P. , Özbay, H. , and Samimy, M. , 2005, “ Control of Subsonic Cavity Flows by Neural Networks—Analytical Models and Experimental Validation,” AIAA Paper No. 2005-294.
Belson, B. A. , Semeraro, O. , Rowley, C. W. , and Henningson, D. S. , 2013, “ Feedback Control of Instabilities in the Two-Dimensional Blasius Boundary Layer: The Role of Sensors and Actuators,” Phys. Fluids, 25(5), p. 054106. [CrossRef]
Hervé, A. , Sipp, D. , Schmid, P. J. , and Samuelides, M. , 2012, “ A Physics-Based Approach to Flow Control Using System Identification,” J. Fluid Mech., 702, pp. 26–58. [CrossRef]
Semeraro, O. , Pralits, J. O. , Rowley, C. W. , and Henningson, D. S. , 2013, “ Riccati-Less Approach for Optimal Control and Estimation: An Application to Two-Dimensional Boundary Layers,” J. Fluid Mech., 731, pp. 394–417. [CrossRef]
Weller, J. , Camarri, S. , and Iollo, A. , 2009, “ Feedback Control by Low-Order Modelling of the Laminar Flow Past a Bluff Body,” J. Fluid Mech., 634, pp. 405–418. [CrossRef]
Stuart, J. , 1958, “ On the Non-Linear Mechanics of Hydrodynamic Stability,” J. Fluid Mech., 4(1), pp. 1–21. [CrossRef]
Stuart, J. , 1971, “ Nonlinear Stability Theory,” Annu. Rev. Fluid Mech., 3, pp. 347–370. [CrossRef]
Schumm, M. , Berger, E. , and Monkewitz, P. , 1994, “ Self-Excited Oscillations in the Wake of Two-Dimensional Bluff Bodies and Their Control,” J. Fluid Mech., 271, pp. 17–53. [CrossRef]
Dusek, J. , Le Gal, P. , and Fraunié, P. , 1994, “ A Numerical and Theoretical Study of the First Hopf Bifurcation in a Cylinder Wake,” J. Fluid Mech., 264, pp. 59–80. [CrossRef]
Bourgeois, J. A. , Martinuzzi, R. J. , and Noack, B. R. , 2013, “ Generalised Phase Average With Applications to Sensor-Based Flow Estimation of the Wall-Mounted Square Cylinder Wake,” J. Fluid Mech., 736, pp. 316–350. [CrossRef]
Luchtenburg, M., Tadmor, G. , Lehmann, O. , Noack, B. R. , King, R. , and Morzyński, M. , 2006, “ Tuned POD Galerkin Models for Transient Feedback Regulation of the Cylinder Wake,” 44th AIAA Aerospace Sciences Meeting, Reno, NV, Jan. 9–12, AIAA Paper 2006-1407.
Tadmor, G. , Lehmann, O. , Noack, B. R. , Cordier, L. , Delville, J. , Bonnet, J.-P. , and Morzyński, M. , 2011, “ Reduced Order Models for Closed-Loop Wake Control,” Philos. Trans. R. Soc. A, 369(1940), pp. 1513–1523. [CrossRef]
King, R. , Seibold, M. , Lehmann, O. , Noack, B. R. , Morzyński, M. , and Tadmor, G. , 2005, “ Nonlinear Flow Control Based on a Low Dimensional Model of Fluid Flow,” Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems (Lecture Notes in Control and Information Sciences, Vol. 322), T. Meurer , K. Graichen , and E. Gilles , eds., Springer, Berlin, pp. 369–386.
Bergmann, M. , and Cordier, L. , 2008, “ Optimal Control of the Cylinder Wake in the Laminar Regime by Trust-Region Methods and POD Reduced Order Models,” J. Comput. Phys., 227(16), pp. 7813–7840. [CrossRef]
Parezanovic, V. , Laurentie, J.-C. , Duriez, T. , Fourment, C. , Delville, J. , Bonnet, J.-P. , Cordier, L. , Noack, B. R. , Segond, M. , Abel, M. , Shaqarin, T. , and Brunton, S. L. , 2015, “ Mixing Layer Manipulation Experiment—From Periodic Forcing to Machine Learning Closed-Loop Control,” J. Flow Turbul. Combust., 94(1), pp. 155–173. [CrossRef]
Aleksic, K. , Luchtenburg, D. M. , King, R. , Noack, B. R. , and Pfeiffer, J. , 2010, “ Robust Nonlinear Control Versus Linear Model Predictive Control of a Bluff Body Wake,” AIAA Paper No. 2010-4833.
Duriez, T. , Parezanovic, V. , Laurentie, J.-C. , Fourment, C. , Delville, J. , Bonnet, J.-P. , Cordier, L. , Noack, B. R. , Segond, M. , Abel, M. , Gautier, N. , Aider, J.-L. , Raibaudo, C. , Cuvier, C. , Stanislas, M. , and Brunton, S. L. , 2014, “ Closed-Loop Control of Experimental Shear Flows Using Machine Learning,” AIAA Paper No. 2014-2219.
Luchtenburg, D. M. , Schlegel, M. , Noack, B. R. , Aleksić, K. , King, R. , Tadmor, G. , and Günther, B. , 2010, “ Turbulence Control Based on Reduced-Order Models and Nonlinear Control Design,” Active Flow Control II (Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Vol. 108), R. King , ed., Springer-Verlag, Berlin, pp. 341–356.
Farazmand, M. M. , Kevlahan, N. K.-R. , and Protas, B. , 2011, “ Controlling the Dual Cascade of Two-Dimensional Turbulence,” J. Fluid Mech., 668, pp. 202–222. [CrossRef]
Schlegel, M. , Noack, B. R. , Comte, P. , Kolomenskiy, D. , Schneider, K. , Farge, M. , Scouten, J. , Luchtenburg, D. M. , and Tadmor, G. , 2009, “ Reduced-Order Modelling of Turbulent Jets for Noise Control,” Numerical Simulation of Turbulent Flows and Noise Generation: Results of the DFG/CNRS Research Groups FOR 507 and FOR 508 (Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM)), Springer-Verlag, Berlin, pp. 3–27.
John, C. , Noack, B. R. , Schlegel, M. , Tröltzsch, F. , and Wachsmuth, D. , 2010, “ Optimal Boundary Control Problems Related to High-Lift Configurations,” Active Flow Control II (Notes on Numerical Fluid Mechanics and Multidisciplinary Design), R. King , ed., Springer-Verlag, Berlin.
Cordier, L. , Noack, B. R. , Daviller, G. , Delvile, J. , Lehnasch, G. , Tissot, G. , Balajewicz, M. , and Niven, R. , 2013, “ Control-Oriented Model Identification Strategy,” Exp. Fluids, 54, p. 1580. [CrossRef]
Noack, B. R. , Morzyński, M. , and Tadmor, G. E. , 2011, Reduced-Order Modelling for Flow Control (CISM Courses and Lectures, Vol. 528), Springer-Verlag, Berlin.
Morzyński, M. , Stankiewicz, W. , Noack, B. R. , Thiele, F. , and Tadmor, G. , 2006, “ Generalized Mean-Field Model for Flow Control Using Continuous Mode Interpolation,” AIAA Paper No. 2006-3488.
Sapsis, T. P. , and Majda, A. , 2013, “ Statistically Accurate Low-Order Models for Uncertainty Quantification in Turbulent Dynamical Systems,” Proc. Natl. Acad. Sci., 110(34), pp. 13705–13710. [CrossRef]
Mitchell, T. M. , 1997, Machine Learning, McGraw-Hill, Maidenhead, UK.
Duda, R. O. , Hart, P. E. , and Stork, D. G. , 2000, Pattern Classification, Wiley-Interscience, New York. [PubMed] [PubMed]
Bishop, C. M. , 2006, Pattern Recognition and Machine Learning, Vol. 1, Springer, New York.
Murphy, K. P. , 2012, Machine Learning: A Probabilistic Perspective, MIT Press, Cambridge, MA.
Fleming, P. J. , and Purshouse, R. C. , 2002, “ Evolutionary Algorithms in Control Systems Engineering: A Survey,” Control Eng. Pract., 10(11), pp. 1223–1241. [CrossRef]
Krstić, M. , and Wang, H. , 2000, “ Stability of Extremum Seeking Feedback for General Nonlinear Dynamic Systems,” Automatica, 36(4), pp. 595–601. [CrossRef]
Ariyur, K. B. , and Krstić, M. , 2003, Real-Time Optimization by Extremum-Seeking Control, Wiley, Hoboken, NJ.
Beaudoin, J. , Cadot, O. , Aider, J. , and Wesfreid, J. E. , 2006, “ Bluff-Body Drag Reduction by Extremum-Seeking Control,” J. Fluids Struct., 22(6), pp. 973–978. [CrossRef]
Beaudoin, J.-F. , Cadot, O. , Aider, J.-L. , and Wesfreid, J.-E. , 2006, “ Drag Reduction of a Bluff Body Using Adaptive Control Methods,” Phys. Fluids, 18(8), p. 085107. [CrossRef]
Becker, R. , King, R. , Petz, R. , and Nitsche, W. , 2007, “ Adaptive Closed-Loop Control on a High-Lift Configuration Using Extremum Seeking,” AIAA J., 45(6), pp. 1382–1392. [CrossRef]
Banaszuk, A. , Zhang, Y. , and Jacobson, C. A. , 2000, “ Adaptive Control of Combustion Instability Using Extremum-Seeking,” American Control Conference (ACC), Chicago, June 28–30, Vol. 1, pp. 416–422.
Banaszuk, A. , Ariyur, K. B. , Krstić, M. , and Jacobson, C. A. , 2004, “ An Adaptive Algorithm for Control of Combustion Instability,” Automatica, 40(11), pp. 1965–1972. [CrossRef]
Banaszuk, A. , Narayanan, S. , and Zhang, Y. , 2003, “ Adaptive Control of Flow Separation in a Planar Diffuser,” AIAA Paper No. 2003-617.
Maury, R. , Keonig, M. , Cattafesta, L. , Jordan, P. , and Delville, J. , 2012, “ Extremum-Seeking Control of Jet Noise,” Aeroacoustics, 11(3–4), pp. 459–474. [CrossRef]
Gelbert, G. , Moeck, J. P. , Paschereit, C. O. , and King, R. , 2012, “ Advanced Algorithms for Gradient Estimation in One- and Two-Parameter Extremum Seeking Controllers,” J. Process Control, 22(4), pp. 700–709. [CrossRef]
Wiederhold, O. , King, R. , Noack, B. R. , Neuhaus, L. , Neise, W. , Enghard, L. , and Swoboda, M. , 2009, “ Extensions of Extremum-Seeking Control to Improve the Aerodynamic Performance of Axial Turbomachines,” AIAA Paper No. 092407.
Krieger, J. P. , and Krstić, M. , 2011, “ Extremum Seeking Based on Atmospheric Turbulence for Aircraft Endurance,” J. Guid. Control Dyn., 34(6), pp. 1876–1885. [CrossRef]
Killingsworth, N. J. , and Krstić, M. , 2006, “ PID Tuning Using Extremum Seeking: Online, Model-Free Performance Optimization,” IEEE Control Syst. Mag., 26(1), pp. 70–79. [CrossRef]
Krstić, M. , Krupadanam, A. , and Jacobson, C. , 1999, “ Self-Tuning Control of a Nonlinear Model of Combustion Instabilities,” IEEE Trans. Control Syst. Technol., 7(4), pp. 424–436. [CrossRef]
Koumoutsakos, P. , 1997, “ Active Control of Turbulent Channel Flow,” Center for Turbulence Research, Stanford University, Stanford, CA, Annual Research Briefs, C, pp. 289–297.
Pamiès, M. , Garnier, E. , Merlen, A. , and Sagaut, P. , 2007, “ Response of a Spatially Developing Turbulent Boundary Layer to Active Control Strategies in the Framework of Opposition Control,” Phys. Fluids, 19(10), p. 108102. [CrossRef]
Iwamoto, K. , Fukagata, K. , Kasagi, N. , and Suzuki, Y. , 2005, “ Friction Drag Reduction Achievable by Near-Wall Turbulence Manipulation at High Reynolds Numbers,” Phys. Fluids, 17(1), p. 011702. [CrossRef]
Chung, Y. M. , and Talha, T. , 2011, “ Effectiveness of Active Flow Control for Turbulent Skin Friction Drag Reduction,” Phys. Fluids, 23(2), p. 025102. [CrossRef]
Rebbeck, H. , and Choi, K.-S. , 2001, “ Opposition Control of Near-Wall Turbulence With a Piston-Type Actuator,” Phys. Fluids, 13(8), pp. 2142–2145. [CrossRef]
Endo, T. , Kasagi, N. , and Suzuki, Y. , 2000, “ Feedback Control of Wall Turbulence With Wall Deformation,” Int. J. Heat Fluid Flow, 21(5), pp. 568–575. [CrossRef]
Fukagata, K. , and Kasagi, N. , 2002, “ Active Control for Drag Reduction in Turbulent Pipe Flow,” Engineering Turbulence Modelling and Experiments 5, W. Rodi and N. Fueyo , eds., Elsevier Science, Oxford, UK, pp. 607–616.
Fukagata, K. , and Kasagi, N. , 2003, “ Drag Reduction in Turbulent Pipe Flow With Feedback Control Applied Partially to Wall,” Int. J. Heat Fluid Flow, 24(4), pp. 480–490. [CrossRef]
Fukagata, K. , and Kasagi, N. , 2004, “ Suboptimal Control for Drag Reduction Via Suppression of Near-Wall Reynolds Shear Stress,” Int. J. Heat Fluid Flow, 25(3), pp. 341–350. [CrossRef]
Farrell, B. F. , and Ioannou, P. J. , 1996, “ Turbulence Suppression by Active Control,” Phys. Fluids, 8(5), pp. 1257–1268. [CrossRef]
Luhar, M. , Sharma, A. S. , and McKeon, B. J. , 2014, “ Opposition Control Within the Resolvent Analysis Framework,” J. Fluid Mech., 749, pp. 597–626. [CrossRef]
Cheng, B. , and Titterington, D. M. , 1994, “ Neural Networks: A Review From a Statistical Perspective,” Statistical Science, 9(1), pp. 2–30. [CrossRef]
Haykin, S. , 2004, Neural Networks: A Comprehensive Foundation, Prentice Hall, Upper Saddle River, NJ.
Müller, S. , Milano, M. , and Koumoutsakos, P. , 1999, “ Application of Machine Learning Algorithms to Flow Modeling and Optimization,” Center for Turbulence Research Annual Research Briefs, Stanford University, Stanford, CA, pp. 169–178.
Milano, M. , and Koumoutsakos, P. , 2002, “ Neural Network Modeling for Near Wall Turbulent Flow,” J. Comput. Phys., 182(1), pp. 1–26. [CrossRef]
Oja, E. , 1992, “ Principal Components, Minor Components, and Linear Neural Networks,” Neural Networks, 5(6), pp. 927–935. [CrossRef]
Oja, E. , 1997, “ The Nonlinear PCA Learning Rule in Independent Component Analysis,” Neurocomputing, 17(1), pp. 25–45. [CrossRef]
Karhunen, J. , and Joutsensalo, J. , 1994, “ Representation and Separation of Signals Using Nonlinear PCA Type Learning,” Neural Networks, 7(1), pp. 113–127. [CrossRef]
Nair, A. G. , and Taira, K. , 2015, “ Network-Theoretic Approach to Sparsified Discrete Vortex Dynamics,” J. Fluid Mech., 768, pp. 549–571. [CrossRef]
Ciresan, D. , Meier, U. , and Schmidhuber, J. , 2012, “ Multi-Column Deep Neural Networks for Image Classification,” IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Providence, RI, June 16–21, pp. 3642–3649.
Dean, J. , Corrado, G. , Monga, R. , Chen, K. , Devin, M. , Mao, M. , Senior, A. , Tucker, P. , Yang, K. , Le, Q. V. , and Ng, A. Y. , 2012, “ Large Scale Distributed Deep Networks,” Advances in Neural Information Processing Systems 25, F. Pereira , C. J. C. Burges , L. Bottou , and K. Q. Weinberger , eds., Curran Associates, Inc., Red Hook, NY. pp. 1223–1231.
Hinton, G. , Deng, L. , Yu, D. , Dahl, G. E. , Mohamed, A.-R. , Jaitly, N. , Senior, A. , Vanhoucke, V. , Nguyen, P. , Sainath, T. N. , and Kingsbury, B. , 2012, “ Deep Neural Networks for Acoustic Modeling in Speech Recognition: The Shared Views of Four Research Groups,” Signal Process. Mag., 29(6), pp. 82–97. [CrossRef]
Holland, J. H. , 1975, Adaptation in Natural and Artificial Systems: An Introductory Analysis With Applications to Biology, Control, and Artificial Intelligence, University of Michigan Press, Ann Arbor, MI.
Davis, L. , 1991, Handbook of Genetic Algorithms, Vol. 115, Van Nostrand Reinhold, New York.
Goldberg, D. E. , 2006, Genetic Algorithms, Pearson Education India, Delhi, India.
Koza, J. R. , 1992, Genetic Programming: On the Programming of Computers by Means of Natural Selection, Vol. 1, MIT Press, Cambridge, MA.
Koza, J. R. , Bennet, F. H., III , and Stiffelman, O. , 1999, “ Genetic Programming as a Darwinian Invention Machine,” Genetic Programming, Springer, Berlin, pp. 93–108.
Koumoutsakos, P. , Freund, J. , and Parekh, D. , 2001, “ Evolution Strategies for Automatic Optimization of Jet Mixing,” AIAA J., 39(5), pp. 967–969. [CrossRef]
Buche, D. , Stoll, P. , Dornberger, R. , and Koumoutsakos, P. , 2002, “ Multiobjective Evolutionary Algorithm for the Optimization of Noisy Combustion Processes,” IEEE Trans. Systems, Man, and Cybernet., Part C, 32(4), pp. 460–473. [CrossRef]
Poncet, P. , Cottet, G.-H. , and Koumoutsakos, P. , 2005, “ Control of Three-Dimensional Wakes Using Evolution Strategies,” C. R. Mec., 333(1), pp. 65–77. [CrossRef]
Fukagata, K. , Kern, S. , Chatelain, P. , Koumoutsakos, P. , and Kasagi, N. , 2008, “ Evolutionary Optimization of an Anisotropic Compliant Surface for Turbulent Friction Drag Reduction,” J. Turbul., 9(35), pp. 1–17.
Gazzola, M. , Vasilyev, O. V. , and Koumoutsakos, P. , 2011, “ Shape Optimization for Drag Reduction in Linked Bodies Using Evolution Strategies,” Comput. Struct., 89(11), pp. 1224–1231. [CrossRef]
Hansen, N. , Niederberger, A. S. , Guzzella, L. , and Koumoutsakos, P. , 2009, “ A Method for Handling Uncertainty in Evolutionary Optimization With an Application to Feedback Control of Combustion,” IEEE Trans. Evol. Comput., 13(1), pp. 180–197. [CrossRef]
Noack, B. R. , Duriez, T. , Cordier, L. , Segond, M. , Abel, M. , Brunton, S. L. , Morzyński, M. , Laurentie, J.-C. , Parezanovic, V. , and Bonnet, J.-P. , 2013, “ Closed-Loop Turbulence Control With Machine Learning Methods,” Bull. Am. Phys. Soc., 58(18), p. 418.
Parezanović, V. , Duriez, T. , Cordier, L. , Noack, B. R. , Delville, J. , Bonnet, J.-P. , Segond, M. , Abel, M. , and Brunton, S. L. , 2014, “ Closed-Loop Control of an Experimental Mixing Layer Using Machine Learning Control,” preprint arXiv:1408.3259.
Gautier, N. , Aider, J.-L. , Duriez, T. , Noack, B. R. , Segond, M. , and Abel, M. , 2015, “ Closed-Loop Separation Control Using Machine Learning,” J. Fluid Mech., 770, pp. 442–457. [CrossRef]
Duriez, T. , Parezanović, V. , Cordier, L. , Noack, B. R. , Delville, J. , Bonnet, J.-P. , Segond, M. , and Abel, M. , 2014, “ Closed-Loop Turbulence Control Using Machine Learning,” preprint arXiv:1404.4589.
Gautier, N. , 2014, “ Flow Control Using Optical Sensors,” Ph.D. thesis, Ecole Doctorale: Sciences Mécaniques, Acoustique, Électronique & Robotique (UPMC), ESPCI, Laboratoire PMMH, Paris.
Gunzburger, M. D. , 2003, Perspectives in Flow Control and Optimization, Vol. 5, SIAM, Philadelphia.
Williams, D. , and MacMynowski, D. , “ Brief History of Flow Control,” Fundamentals and Applications of Modern Flow Control, Vol. 231, R. Joslin and D. Miller , eds., American Institute of Aeronautics and Astronautics, Reston, VA, pp. 1–20.
Schlichting, H. , 1979, Boundary-Layer Theory, 7th ed., McGraw-Hill, New York.
Fiedler, H. , and Fernholz, H.-H. , 1990, “ On the Management and Control of Turbulent Shear Flows,” Prog. Aeronaut. Sci., 27(4), pp. 305–387. [CrossRef]
McComb, D. , 1991, The Physics of Fluid Turbulence, 1st ed., Clarendon Press, Oxford, UK.
Frisch, U. , 1995, Turbulence, 1st ed., Cambridge University Press, Cambridge, UK.
Taylor, H. , 1947, “ The Elimination of Diffuser Separation by Vortex Generators,” United Aircraft Corporation, East Hartford, CT, Technical Report No. R.4012-3.
Lorenz, E. N. , 1963, “ Deterministic Nonperiodic Flow,” J. Atmos. Sci., 20(2), pp. 130–141. [CrossRef]
Ott, E. , Grebogi, C. , and Yorke, J. A. , 1990, “ Controlling Chaos,” Phys. Rev. Lett., 64(23), p. 2837. [CrossRef]
Schöll, E. , and Schuster, H. G. , 2007, Handbook of Chaos Control, Wiley-VCH, Weinheim, Germany.
Aubry, N. , Holmes, P. , Lumley, J. L. , and Stone, E. , 1988, “ The Dynamics of Coherent Structures in the Wall Region of a Turbulent Boundary Layer,” J. Fluid Mech., 192, pp. 115–173. [CrossRef]
Glauser, M. N. , Leib, S. J. , and George, W. K. , 1987, Coherent Structures in the Axisymmetric Turbulent Jet Mixing Layer, Springer, Berlin.
George, W. K. , 1988, “ Insight Into the Dynamics of Coherent Structures From a Proper Orthogonal Decomposition,” Symposium on Near Wall Turbulence, Dubrovnik, Yugoslavia, May 16–20.
Glauser, M. N. , and George, W. K. , 1992, “ Application of Multipoint Measurements for Flow Characterization,” Exp. Therm. Fluid Sci., 5(5), pp. 617–632. [CrossRef]
Guyot, D. , Paschereit, C. O. , and Raghu, S. , 2009, “ Active Combustion Control Using a Fluidic Oscillator for Asymmetric Fuel Flow Modulation,” Int. J. Flow Control, 1(2), pp. 155–166. [CrossRef]
Bobusch, B. C. , Woszidlo, R. , Bergada, J. , Nayeri, C. N. , and Paschereit, C. O. , 2013, “ Experimental Study of the Internal Flow Structures Inside a Fluidic Oscillator,” Exp. Fluids, 54(6), p. 1559. [CrossRef]
Vallikivi, M. , Hultmark, M. , Bailey, S. , and Smits, A. , 2011, “ Turbulence Measurements in Pipe Flow Using a Nano-Scale Thermal Anemometry Probe,” Exp. Fluids, 51(6), pp. 1521–1527. [CrossRef]
Bailey, S. C. , Kunkel, G. J. , Hultmark, M. , Vallikivi, M. , Hill, J. P. , Meyer, K. A. , Tsay, C. , Arnold, C. B. , and Smits, A. J. , 2010, “ Turbulence Measurements Using a Nanoscale Thermal Anemometry Probe,” J. Fluid Mech., 663, pp. 160–179. [CrossRef]
Hultmark, M. , Vallikivi, M. , Bailey, S. , and Smits, A. , 2012, “ Turbulent Pipe Flow at Extreme Reynolds Numbers,” Phys. Rev. Lett., 108(9), p. 094501. [CrossRef] [PubMed]
Daniel, T. L. , 1988, “ Forward Flapping Flight From Flexible Fins,” Can. J. Zool., 66(3), pp. 630–638. [CrossRef]
Anderson, J. M. , Streitlien, K. , Barrett, D. S. , and Triantafyllou, M. S. , 1998, “ Oscillating Foils of High Propulsive Efficiency,” J. Fluid Mech., 360, pp. 41–72. [CrossRef]
Triantafyllou, M. S. , and Triantafyllou, G. S. , 1995, “ An Efficient Swimming Machine,” Sci. Am., 272(3), pp. 64–71. [CrossRef]
Allen, J. J. , and Smits, A. J. , “ Energy Harvesting Eel,” J. Fluids Struct., 15(3–4), pp. 629–640.
Combes, S. A. , and Daniel, T. L. , 2001, “ Shape, Flapping and Flexion: Wing and Fin Design for Forward Flight,” J. Exp. Biol., 204(12), pp. 2073–2085. [PubMed]
Clark, R. P. , and Smits, A. J. , 2006, “ Thrust Production and Wake Structure of a Batoid-Inspired Oscillating Fin,” J. Fluid Mech., 562, pp. 415–429. [CrossRef] [PubMed]
Buchholz, J. H. , and Smits, A. J. , 2008, “ The Wake Structure and Thrust Performance of a Rigid Low-Aspect-Ratio Pitching Panel,” J. Fluid Mech., 603(May), pp. 331–365. [CrossRef] [PubMed]
Song, A. , Tian, X. , Israeli, E. , Galvao, R. , Bishop, K. , Swartz, S. , and Breuer, K. , 2008, “ Aeromechanics of Membrane Wings With Implications for Animal Flight,” AIAA J., 46(8), pp. 2096–2106. [CrossRef]
Taira, K. , and Colonius, T. , 2008, “ Effect of Tip Vortices in Low-Reynolds-Number Poststall Flow Control,” AIAA J., 47(3), pp. 749–756 . [CrossRef]
Taira, K. , and Colonius, T. , 2009, “ Three-Dimensional Flows Around Low-Aspect-Ratio Flat-Plate Wings at Low Reynolds Numbers,” J. Fluid Mech., 623, pp. 187–207. [CrossRef]
Whittlesey, R. W. , Liska, S. C. , and Dabiri, J. O. , 2010, “ Fish Schooling as a Basis for Vertical-Axis Wind Turbine Farm Design,” Bioinspiration Biomimetics, 5(3), p. 035005. [CrossRef] [PubMed]
Faruque, I. , and Humbert, J. S. , 2010, “ Dipteran Insect Flight Dynamics. Part 1: Longitudinal Motion About Hover,” J. Theor. Biol., 264(2), pp. 538–552. [CrossRef] [PubMed]
Faruque, I. , and Humbert, J. S. , 2010, “ Dipteran Insect Flight Dynamics. Part 2: Lateral–Directional Motion About Hover,” J. Theor. Biol., 265(3), pp. 306–313. [CrossRef] [PubMed]
Humbert, J. S. , and Hyslop, A. M. , 2010, “ Bioinspired Visuomotor Convergence,” IEEE Trans. Rob., 26(1), pp. 121–130. [CrossRef]
Shelley, M. J. , and Zhang, J. , 2011, “ Flapping and Bending Bodies Interacting With Fluid Flows,” Annu. Rev. Fluid Mech., 43, pp. 449–465. [CrossRef]
Leftwich, M. C. , Tytell, E. D. , Cohen, A. H. , and Smits, A. J. , 2012, “ Wake Structures Behind a Swimming Robotic Lamprey With a Passively Flexible Tail,” J. Exp. Biol., 215(3), pp. 416–425. [CrossRef] [PubMed]
Dewey, P. A. , Carriou, A. , and Smits, A. J. , 2012, “ On the Relationship Between Efficiency and Wake Structure of a Batoid-Inspired Oscillating Fin,” J. Fluid Mech., 691, pp. 245–266. [CrossRef]
Nawroth, J. C. , Lee, H. , Feinberg, A. W. , Ripplinger, C. M. , McCain, M. L. , Grosberg, A. , Dabiri, J. O. , and Parker, K. K. , 2012, “ A Tissue-Engineered Jellyfish With Biomimetic Propulsion,” Nat. Biotechnol., 30, pp. 792–797. [CrossRef] [PubMed]
Roth, E. , Sponberg, S. , and Cowan, N. , 2014, “ A Comparative Approach to Closed-Loop Computation,” Curr. Opin. Neurobiol., 25, pp. 54–62. [CrossRef] [PubMed]
Cowan, N. J. , Ankarali, M. M. , Dyhr, J. P. , Madhav, M. S. , Roth, E. , Sefati, S. , Sponberg, S. , Stamper, S. A. , Fortune, E. S. , and Daniel, T. L. , 2014, “ Feedback Control as a Framework for Understanding Tradeoffs in Biology,” Integr. Comp. Biol., 54(2), pp. 223–237.
Dickinson, M. H. , and Gotz, K. G. , 1996, “ The Wake Dynamics and Flight Forces of the Fruit Fly Drosophila melanogaster ,” J. Exp. Biol., 199(9), pp. 2085–2104. [PubMed]
Sane, S. P. , and Dickinson, M. H. , 2001, “ The Control of Flight Force by a Flapping Wing: Lift and Drag Production,” J. Exp. Biol., 204(15), pp. 2607–2626. [PubMed]
Frye, M. A. , and Dickinson, M. H. , 2001, “ Fly Flight: A Model for the Neural Control of Complex Behavior,” Neuron, 32(3), pp. 385–388. [CrossRef] [PubMed]
Ghose, K. , Horiuchi, T. K. , Krishnaprasad, P. S. , and Moss, C. F. , 2006, “ Echolocating Bats Use a Nearly Time-Optimal Strategy to Intercept Prey,” PLoS Biol., 4(5), p. e108. [CrossRef] [PubMed]
Hedenström, A. , Johansson, L. , Wolf, M. , Von Busse, R. , Winter, Y. , and Spedding, G. , 2007, “ Bat Flight Generates Complex Aerodynamic Tracks,” Science, 316(5826), pp. 894–897. [CrossRef] [PubMed]
Riskin, D. K. , Willis, D. J. , Iriarte-Díaz, J. , Hedrick, T. L. , Kostandov, M. , Chen, J. , Laidlaw, D. H. , Breuer, K. S. , and Swartz, S. M. , 2008, “ Quantifying the Complexity of Bat Wing Kinematics,” J. Theor. Biol., 254(3), pp. 604–615. [CrossRef] [PubMed]
Hubel, T. Y. , Hristov, N. I. , Swartz, S. M. , and Breuer, K. S. , 2009, “ Time-Resolved Wake Structure and Kinematics of Bat Flight,” Exp. Fluids, 46(5), pp. 933–943. [CrossRef]
Fish, F. E. , and Hui, C. A. , 1991, “ Dolphin Swimming—A Review,” Mamm. Rev., 21(4), pp. 181–195. [CrossRef]
Fish, F. E. , 1996, “ Transitions From Drag-Based to Lift-Based Propulsion in Mammalian Swimming,” Am. Zool., 36(6), pp. 628–641.
Dickinson, M. H. , Lehmann, F. O. , and Sane, S. P. , 1999, “ Wing Rotation and the Aerodynamic Basis of Insect Flight,” Science, 284(5422), pp. 1954–1960. [CrossRef] [PubMed]
Birch, J. , and Dickinson, M. , 2001, “ Spanwise Flow and the Attachment of the Leading-Edge Vortex on Insect Wings,” Nature, 412(6848), pp. 729–733. [CrossRef] [PubMed]
Sane, S. P. , 2003, “ The Aerodynamics of Insect Flight,” J. Exp. Biol., 206(23), pp. 4191–4208. [CrossRef] [PubMed]
Liao, J. C. , Beal, D. N. , Lauder, G. V. , and Triantafyllou, M. S. , 2003, “ Fish Exploiting Vortices Decrease Muscle Activity,” Science, 302(5650), pp. 1566–1569. [CrossRef] [PubMed]
Tytell, E. D. , and Lauder, G. V. , 2004, “ The Hydrodynamics of Eel Swimming. I. Wake Structure,” J. Exp. Biol., 207(11), pp. 1825–1841. [CrossRef] [PubMed]
Lauder, G. V. , and Tytell, E. D. , 2005, “ Hydrodynamics of Undulatory Propulsion,” Fish Physiol., 23, pp. 425–468.
Videler, J. J. , Samhuis, E. J. , and Povel, G. D. E. , 2004, “ Leading-Edge Vortex Lifts Swifts,” Science, 306(5703), pp. 1960–1962. [CrossRef] [PubMed]
Wang, Z. J. , 2005, “ Dissecting Insect Flight,” Annu. Rev. Fluid Mech., 37, pp. 183–210. [CrossRef]
Dabiri, J. O. , 2009, “ Optimal Vortex Formation as a Unifying Principle in Biological Propulsion,” Annu. Rev. Fluid Mech., 41, pp. 17–33. [CrossRef]
Wu, T. Y. , 2011, “ Fish Swimming and Bird/Insect Flight,” Annu. Rev. Fluid Mech., 43, pp. 25–58. [CrossRef]
Collett, T. S. , and Land, M. F. , 1975, “ Visual Control of Flight Behaviour in the Hoverfly Syritta pipiens L.,” J. Comp. Physiol. A, 99(1), pp. 1–66. [CrossRef]
Fayyazuddin, A. , and Dickinson, M. H. , 1996, “ Haltere Afferents Provide Direct, Electronic Input to a Steering Motor Neuron in the Blowfly, Calliphora,” J. Neurosci., 16(16), pp. 5225–5232. [PubMed]
Fox, J. L. , and Daniel, T. L. , 2008, “ A Neural Basis for Gyroscopic Force Measurement in the Halteres of Holorusia,” J. Comp. Physiol. A, 194(10), pp. 887–897. [CrossRef]
Sane, S. P. , Dieudonne, A. , Willis, M. A. , and Daniel, T. L. , 2007, “ Antennal Mechanosensors Mediate Flight Control in Moths,” Science, 315(5813), pp. 863–866. [CrossRef] [PubMed]
Brown, R. E. , and Fedde, M. R. , 1993, “ Airflow Sensors in the Avian Wing,” J. Exp. Biol., 179(1), pp. 13–30.
Sterbing-D'Angelo, S. J. , and Moss, C. F. , 2014, “ Air Flow Sensing in Bats,” Flow Sensing in Air and Water, Springer, Berlin, pp. 197–213.
Sterbing-D'Angelo, S. , Chadha, M. , Chiu, C. , Falk, B. , Xian, W. , Barcelo, J. , Zook, J. M. , and Moss, C. F. , 2011, “ Bat Wing Sensors Support Flight Control,” Proc. Natl. Acad. Sci., 108(27), pp. 11291–11296. [CrossRef]
Dickinson, B. , 2010, “ Hair Receptor Sensitivity to Changes in Laminar Boundary Layer Shape,” Bioinspiration Biomimetics, 5(1), p. 016002. [CrossRef]
Massey, T. , Kapur, R. , Dabiri, F. , Vu, L. N. , and Sarrafzadeh, M. , 2007, “ Localization Using Low-Resolution Optical Sensors,” IEEE International Conference on Mobile Adhoc and Sensor Systems (MASS 2007), Pisa, Italy, Oct. 8–11.
Giannetti, F. , and Luchini, P. , 2007, “ Structural Sensitivity of the First Instability of the Cylinder Wake,” J. Fluid Mech., 581, pp. 167–197. [CrossRef]
Hof, B. , de Lozar, A. , Avila, M. , Tu, X. , and Schneider, T. M. , 2010, “ Eliminating Turbulence in Spatially Intermittent Flows,” Science, 327(5972), pp. 1491–1494. [CrossRef] [PubMed]
McKeon, B. J. , 2010, “ Controlling Turbulence,” Science, 327(5927), pp. 1462–1463. [CrossRef] [PubMed]
Avila, K. , Moxey, D. , de Lozar, A. , Avila, M. , Barkley, D. , and Hof, B. , 2011, “ The Onset of Turbulence in Pipe Flow,” Science, 333(6039), pp. 192–196. [CrossRef] [PubMed]
Brunton, B. W. , Brunton, S. L. , Proctor, J. L. , and Kutz, J. N. , 2013, “ Optimal Sensor Placement and Enhanced Sparsity for Classification,” preprint arXiv:1310.4217.
Proctor, J. L. , Brunton, S. L. , Brunton, B. W. , and Kutz, J. N. , 2014, “ Exploiting Sparsity and Equation-Free Architectures in Complex Systems (Invited Review),” Eur. Phys. J. Spec. Top., 223(13), pp. 2665–2684. [CrossRef]
Hey, A. J. , Tansley, S. , and Tolle, K. M. , 2009, The Fourth Paradigm: Data-Intensive Scientific Discovery, Microsoft Research, Redmond, WA.
Allaire, D. , Biros, G. , Chambers, J. , Ghattas, O. , Kordonowy, D. , and Willcox, K. , 2012, “ Dynamic Data Driven Methods for Self-Aware Aerospace Vehicles,” Proc. Comput. Sci., 9, pp. 1206–1210. [CrossRef]
Kutz, J. N. , 2013, Data-Driven Modeling & Scientific Computation: Methods for Complex Systems & Big Data, Oxford University Press, Oxford, UK.
Candès, E. J. , 2006, “ Compressive Sampling,” International Congress of Mathematics, Madrid, Aug. 22–30, Vol. 3, pp. 1433–1452.
Donoho, D. L. , 2006, “ Compressed Sensing,” IEEE Trans. Inf. Theory, 52(4), pp. 1289–1306. [CrossRef]
Baraniuk, R. G. , 2007, “ Compressive Sensing,” IEEE Signal Process. Mag., 24(4), pp. 118–120. [CrossRef]
Tropp, J. A. , and Gilbert, A. C. , 2007, “ Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit,” IEEE Trans. Inf. Theory, 53(12), pp. 4655–4666. [CrossRef]
Candès, E. J. , and Wakin, M. B. , 2008, “ An Introduction to Compressive Sampling,” IEEE Signal Process. Mag., 25(2), pp. 21–30. [CrossRef]
Willert, C. E. , and Gharib, M. , 1991, “ Digital Particle Image Velocimetry,” Exp. Fluids, 10(4), pp. 181–193. [CrossRef]
Nyquist, H. , 1928, “ Certain Topics in Telegraph Transmission Theory,” Trans. AIEE, 47(2), pp. 617–644.
Shannon, C. E. , 1948, “ A Mathematical Theory of Communication,” Bell Syst. Tech. J., 27(3), pp. 379–423. [CrossRef]
Petra, S. , and Schnörr, C. , 2009, “ TomoPIV Meets Compressed Sensing,” Pure Math. Appl., 20(1–2), pp. 49–76.
Becker, F. , Wieneke, B. , Petra, S. , Schröder, A. , and Schnörr, C. , 2012, “ Variational Adaptive Correlation Method for Flow Estimation,” IEEE Trans. Image Process., 21(6), pp. 3053–3065. [CrossRef] [PubMed]
Bai, Z. , Wimalajeewa, T. , Berger, Z. , Wang, G. , Glauser, M. , and Varshney, P. K. , 2013, “ Physics Based Compressive Sensing Approach Applied to Airfoil Data Collection and Analysis,” AIAA Paper No. 2013-0772.
Bai, Z. , Wimalajeewa, T. , Berger, Z. , Wang, G. , Glauser, M. , and Varshney, P. K. , 2014, “ Low-Dimensional Approach for Reconstruction of Airfoil Data Via Compressive Sensing,” AIAA J., 53(4), pp. 920–933. [CrossRef]
Candès, E. J. , Romberg, J. , and Tao, T. , 2006, “ Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information,” IEEE Trans. Inf. Theory, 52(2), pp. 489–509. [CrossRef]
Candès, E. J. , Romberg, J. , and Tao, T. , 2006, “ Stable Signal Recovery From Incomplete and Inaccurate Measurements,” Commun. Pure Appl. Math., 59(8), pp. 1207–1223. [CrossRef]
Candès, E. J. , and Tao, T. , 2006, “ Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?” IEEE Trans. Inf. Theory, 52(12), pp. 5406–5425. [CrossRef]
Boyd, S. , and Vandenberghe, L. , 2009, Convex Optimization, Cambridge University Press, Cambridge, UK.
Mathelin, L. , and Gallivan, K. A. , 2012, “ A Compressed Sensing Approach for Partial Differential Equations With Random Input Data,” Commun. Comput. Phys., 12(4), pp. 1–36.
Schaeffer, H. , Caflisch, R. , Hauck, C. D. , and Osher, S. , 2013, “ Sparse Dynamics for Partial Differential Equations,” Proc. Natl. Acad. Sci. U.S.A., 110(17), pp. 6634–6639. [CrossRef] [PubMed]
Mackey, A. , Schaeffer, H. , and Osher, S. , “ On the Compressive Spectral Method,” Multiscale Model. & Simul., 12(4), pp. 1800–1827.
Tran, G. , Schaeffer, H. , Feldman, W. M. , and Osher, S. J. , 2014, “ An L1 Penalty Method for General Obstacle Problems,” preprint arXiv:1404.1370.
Shi, J. V. , Yin, W. , Sankaranarayanan, A. C. , and Baraniuk, R. G. , “ Video Compressive Sensing for Dynamic MRI,” BMC Neurosci., 13(Suppl 1), p. 183. [CrossRef]
Jovanović, M. R. , Schmid, P. J. , and Nichols, J. W. , 2014, “ Sparsity-Promoting Dynamic Mode Decomposition,” Phys. Fluids, 26(2), p. 024103. [CrossRef]
Brunton, S. L. , Proctor, J. L. , and Kutz, J. N. , “ Compressive Sampling and Dynamic Mode Decomposition,” arXiv:1312.5186.
Gueniat, F. , Mathelin, L. , and Pastur, L. , 2015, “ A Dynamic Mode Decomposition Approach for Large and Arbitrarily Sampled Systems,” Phys. Fluids, 27(2), p. 025113. [CrossRef]
Bright, I. , Lin, G. , and Kutz, J. N. , 2013, “ Compressive Sensing and Machine Learning Strategies for Characterizing the Flow Around a Cylinder With Limited Pressure Measurements,” Phys. Fluids, 25(12), p. 127102. [CrossRef]
Brunton, S. L. , Tu, J. H. , Bright, I. , and Kutz, J. N. , 2014, “ Compressive Sensing and Low-Rank Libraries for Classification of Bifurcation Regimes in Nonlinear Dynamical Systems,” SIAM J. Appl. Dyn. Syst., 13(4), pp. 1716–1732. [CrossRef]
Tayler, A. B. , Holland, D. J. , Sederman, A. J. , and Gladden, L. F. , 2012, “ Exploring the Origins of Turbulence in Multiphase Flow Using Compressed Sensing MRI,” Phys. Rev. Lett., 108(26), p. 264505. [CrossRef] [PubMed]
Branicki, M. , and Majda, A. J. , 2014, “ Quantifying Bayesian Filter Performance for Turbulent Dynamical Systems Through Information Theory,” Commun. Math. Sci., 12(5), pp. 901–978. [CrossRef]
Bourguignon, J.-L. , Tropp, J. , Sharma, A. , and McKeon, B. , 2014, “ Compact Representation of Wall-Bounded Turbulence Using Compressive Sampling,” Phys. Fluids, 26(1), p. 015109. [CrossRef]
Fu, X. , Brunton, S. L. , and Kutz, J. N. , 2014, “ Classification of Birefringence in Mode-Locked Fiber Lasers Using Machine Learning and Sparse Representation,” Opt. Express, 22(7), pp. 8585–8597. [CrossRef] [PubMed]
Brunton, S. L. , Fu, X. , and Kutz, J. N. , 2014, “ Self-Tuning Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron., 20(5), p. 1101408. [CrossRef]
Wright, J. , Yang, A. , Ganesh, A. , Sastry, S. , and Ma, Y. , 2009, “ Robust Face Recognition Via Sparse Representation,” IEEE Trans. Pattern Anal. Mach. Intell. (PAMI), 31(2), pp. 210–227. [CrossRef]
Kaiser, E. , Noack, B. R. , Cordier, L. , Spohn, A. , Segond, M. , Abel, M. , Daviller, G. , and Niven, R. K. , 2014, “ Cluster-Based Reduced-Order Modelling of a Mixing Layer,” J. Fluid Mech., 754, pp. 365–414. [CrossRef]
Burkardt, J. , Gunzburger, M. , and Lee, H.-C. , 2004, “ Centroidal Voronoi Tessellation-Based Reduced-Order Modeling of Complex Systems,” SIAM J. Sci. Comput., 28(2), pp. 459–484.
Schneider, T. M. , Eckhardt, B. , and Vollmer, J. , 2007, “ Statistical Analysis of Coherent Structures in Transitional Pipe Flow,” Phys. Rev. E, 75(6), pp. 66–313. [CrossRef]
Gear, C. W. , Kevrekidis, I. G. , and Theodoropoulos, C. , “ ‘Coarse’ Integration/Bifurcation Analysis Via Microscopic Simulators: Micro-Galerkin Methods,” Comput. Chem. Eng., 26(7–8), pp. 941–963.
Gorban, A. , Kazantzis, N. K. , Kevrekidis, I. G. , Öttinger, H. , and Theodoropoulos, C. , eds., 2006, Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, Springer-Verlag, Berlin.
Kevrekidis, I. G. , Gear, C. W. , Hyman, J. M. , Kevrekidis, P. G. , Runborg, O. , and Theodoropoulos, C. , 2003, “ Equation-Free, Coarse-Grained Multiscale Computation: Enabling Microscopic Simulators to Perform System-Level Analysis,” Commun. Math. Sci., 1(4), pp. 715–762. [CrossRef]
Sirisup, S. , Karniadakis, G. E. , Xiu, D. , and Kevrekidis, I. G. , 2005, “ Equation-Free/Galerkin-Free POD-Assisted Computation of Incompressible Flows,” J. Comput. Phys., 207(2), pp. 568–587. [CrossRef]
Xiu, D. , and Karniadakis, G. E. , 2002, “ The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations,” SIAM J. Sci. Comput., 24(2), pp. 619–644. [CrossRef]
Xiu, D. , and Karniadakis, G. E. , 2003, “ Modeling Uncertainty in Flow Simulations Via Generalized Polynomial Chaos,” J. Comput. Phys., 187(1), pp. 137–167. [CrossRef]
Xiu, D. , 2010, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, NJ.
Grosek, J. , and Kutz, J. N. , 2014, “ Dynamic Mode Decomposition for Real-Time Background/Foreground Separation in Video,” preprint arXiv:1404.7592.
Hemati, M. S. , Williams, M. O. , and Rowley, C. W. , 2014, “ Dynamic Mode Decomposition for Large and Streaming Datasets,” preprint arXiv:1406.7187.
Dawson, S. , Hemati, M. , Williams, M. , and Rowley, C. , 2014, “ Characterizing and Correcting for the Effect of Sensor Noise in the Dynamic Mode Decomposition,” Bull. Am. Phys. Soc., 59(20), p. 428.
Aref, H. , 1984, “ Stirring by Chaotic Advection,” J. Fluid Mech., 143, pp. 1–21. [CrossRef]
Wiener, N. , 1938, “ The Homogeneous Chaos,” Am. J. Math., 60(4), pp. 897–936. [CrossRef]
Wan, X. , and Karniadakis, G. E. , 2005, “ An Adaptive Multi-Element Generalized Polynomial Chaos Method for Stochastic Differential Equations,” J. Comput. Phys., 209(2), pp. 617–642. [CrossRef]
Gerritsma, M. , van der Steen, J.-B. , Vos, P. E. J. , and Karniadakis, G. E. , 2010, “ Time-Dependent Generalized Polynomial Chaos,” J. Comput. Phys., 229(22), pp. 8333–8363. [CrossRef]
Luchtenburg, D. M. , Brunton, S. L. , and Rowley, C. W. , 2014, “ Long-Time Uncertainty Propagation Using Generalized Polynomial Chaos and Flow Map Composition,” J. Comput. Phys., 274, pp. 783–802. [CrossRef]
Le Maître, O. P. , and Knio, O. M. , 2010, Spectral Methods for Uncertainty Quantification, Springer, Dordrecht.
Sapsis, T. P. , and Lermusiaux, P. F. , 2009, “ Dynamically Orthogonal Field Equations for Continuous Stochastic Dynamical Systems,” Physica D, 238(23–24), pp. 2347–2360. [CrossRef]
Sapsis, T. P. , and Lermusiaux, P. F. , 2012, “ Dynamical Criteria for the Evolution of the Stochastic Dimensionality in Flows With Uncertainty,” Physica D, 241(1), pp. 60–76. [CrossRef]
Haller, G. , 2001, “ Distinguished Material Surfaces and Coherent Structures in Three-Dimensional Fluid Flows,” Physica D, 149(4), pp. 248–277. [CrossRef]
Haller, G. , 2002, “ Lagrangian Coherent Structures From Approximate Velocity Data,” Phys. Fluids, 14(6), pp. 1851–1861. [CrossRef]
Shadden, S. C. , Lekien, F. , and Marsden, J. E. , 2005, “ Definition and Properties of Lagrangian Coherent Structures From Finite-Time Lyapunov Exponents in Two-Dimensional Aperiodic Flows,” Physica D, 212(3–4), pp. 271–304. [CrossRef]
Green, M. A. , Rowley, C. W. , and Haller, G. , 2007, “ Detection of Lagrangian Coherent Structures in 3D Turbulence,” J. Fluid Mech., 572, pp. 111–120. [CrossRef]
Mathur, M. , Haller, G. , Peacock, T. , Ruppert-Felsot, J. E. , and Swinney, H. L. , 2007, “ Uncovering the Lagrangian Skeleton of Turbulence,” Phys. Rev. Lett., 98(14), p. 144502. [CrossRef] [PubMed]
Brunton, S. L. , and Rowley, C. W. , 2010, “ Fast Computation of FTLE Fields for Unsteady Flows: A Comparison of Methods,” Chaos, 20(1), p. 017503. [CrossRef] [PubMed]
Farazmand, M. , and Haller, G. , 2012, “ Computing Lagrangian Coherent Structures From Their Variational Theory,” Chaos, 22(1), p. 013128. [CrossRef] [PubMed]
Kafiabad, H. A. , Chan, P. W. , and Haller, G. , 2012, “ Lagrangian Detection of Aerial Turbulence for Landing Aircraft,” J. Appl. Meteorol. Climatol., 30(12), pp. 2808–2819.
Shadden, S. C. , Astorino, M. , and Gerbeau, J. F. , 2010, “ Computational Analysis of an Aortic Valve Jet With Lagrangian Coherent Structures,” Chaos, 20(1), p. 017512. [CrossRef] [PubMed]
Wilson, M. M. , Peng, J. , Dabiri, J. O. , and Eldredge, J. D. , 2009, “ Lagrangian Coherent Structures in Low Reynolds Number Swimming,” J. Phys.: Condens. Matter, 21(20), p. 204105. [CrossRef] [PubMed]
Green, M. A. , Rowley, C. W. , and Smits, A. J. , 2011, “ The Unsteady Three-Dimensional Wake Produced by a Trapezoidal Pitching Panel,” J. Fluid Mech., 685, pp. 117–145. [CrossRef]
Peng, J. , and Dabiri, J. O. , 2008, “ The ‘Upstream Wake’ of Swimming and Flying Animals and Its Correlation With Propulsive Efficiency,” J. Exp. Biol., 211(16), pp. 2669–2677. [CrossRef] [PubMed]
Bollt, E. M. , Luttman, A. , Kramer, S. , and Basnayake, R. , 2012, “ Measurable Dynamics Analysis of Transport in the Gulf of Mexico During the Oil Spill,” Int. J. Bifurcation Chaos, 22(3), p. 1230012. [CrossRef]
Lekien, F. , Coulliette, C. , Mariano, A. J. , Ryan, E. H. , Shay, L. K. , Haller, G. , and Marsden, J. E. , 2005, “ Pollution Release Tied to Invariant Manifolds: A Case Study for the Coast of Florida,” Physica D, 210(1–2), pp. 1–20. [CrossRef]
Mezić, I. , Loire, S. , Fonoberov, V. A. , and Hogan, P. , 2010, “ A New Mixing Diagnostic and Gulf Oil Spill Movement,” Science, 330(6003), pp. 486–489. [CrossRef] [PubMed]
Padberg, K. , Hauff, T. , Jenko, F. , and Junge, O. , 2007, “ Lagrangian Structures and Transport in Turbulent Magnetized Plasmas,” New J. Phys., 9(11), p. 400. [CrossRef]
Froyland, G. , and Padberg, K. , 2009, “ Almost-Invariant Sets and Invariant Manifolds—Connecting Probabilistic and Geometric Descriptions of Coherent Structures in Flows,” Physica D, 238(16), pp. 1507–1523. [CrossRef]
Froyland, G. , Santitissadeekorn, N. , and Monahan, A. , 2010, “ Transport in Time-Dependent Dynamical Systems: Finite-Time Coherent Sets,” Chaos, 20(4), p. 043116. [CrossRef] [PubMed]
Tallapragada, P. , and Ross, S. D. , 2013, “ A Set Oriented Definition of Finite-Time Lyapunov Exponents and Coherent Sets,” Commun. Nonlinear Sci. Numer. Simul., 18(5), pp. 1106–1126. [CrossRef]
Dellnitz, M. , Froyland, G. , and Junge, O. , 2001, “ The Algorithms Behind Gaio—Set Oriented Numerical Methods for Dynamical Systems,” Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, B. Fieldler , ed., Springer, Dordrecht, pp. 145–174.
Dellnitz, M. , and Junge, O. , 2002, “ Set Oriented Numerical Methods for Dynamical Systems,” Handbook of Dynamical Systems , Vol. 2, B. Fiedler , ed., Elsevier, Amsterdam, pp. 221–264.
Carlberg, K. , Bou-Mosleh, C. , and Farhat, C. , 2011, “ Efficient Non-Linear Model Reduction Via a Least-Squares Petrov–Galerkin Projection and Compressive Tensor Approximations,” Int. J. Numer. Methods Eng., 86(2), pp. 155–181. [CrossRef]
Avellaneda, M. , and Majda, A. J. , 1990, “ Mathematical Models With Exact Renormalization for Turbulent Transport,” Commun. Math. Phys., 131(2), pp. 381–429. [CrossRef]
Amsallem, D. , Zahr, M. J. , and Farhat, C. , 2012, “ Nonlinear Model Order Reduction Based on Local Reduced-Order Bases,” Int. J. Numer. Methods Eng., 92(10), pp. 891–916. [CrossRef]
Carlberg, K. , Farhat, C. , Cortial, J. , and Amsallem, D. , 2013, “ The GNAT Method for Nonlinear Model Reduction: Effective Implementation and Application to Computational Fluid Dynamics and Turbulent Flows,” J. Comput. Phys., 242, pp. 623–647. [CrossRef]
Everson, R. , and Sirovich, L. , 1995, “ Karhunen–Loeve Procedure for Gappy Data,” JOSA A, 12(8), pp. 1657–1664. [CrossRef]
Willcox, K. , 2006, “ Unsteady Flow Sensing and Estimation Via the Gappy Proper Orthogonal Decomposition,” Comput. Fluids, 35(2), pp. 208–226. [CrossRef]
Barrault, M. , Maday, Y. , Nguyen, N. C. , and Patera, A. T. , 2004, “ An ‘Empirical Interpolation’ Method: Application to Efficient Reduced-Basis Discretization of Partial Differential Equations,” C. R. Math., 339(9), pp. 667–672. [CrossRef]
Chaturantabut, S. , and Sorensen, D. C. , 2010, “ Nonlinear Model Reduction Via Discrete Empirical Interpolation,” SIAM J. Sci. Comput., 32(5), pp. 2737–2764. [CrossRef]
Chaturantabut, S. , and Sorensen, D. C. , 2012, “ A State Space Error Estimate for POD-DEIM Nonlinear Model Reduction,” SIAM J. Numer. Anal., 50(1), pp. 46–63. [CrossRef]
Peherstorfer, B. , Butnaru, D. , Willcox, K. , and Bungartz, H.-J. , 2014, “ Localized Discrete Empirical Interpolation Method,” SIAM J. Sci. Comput., 36(1), pp. A168–A192. [CrossRef]
Majda, A. J. , and Kramer, P. R. , 1999, “ Simplified Models for Turbulent Diffusion: Theory, Numerical Modelling, and Physical Phenomena,” Phys. Rep., 314(4), pp. 237–574. [CrossRef]
Majda, A. J. , Harlim, J. , and Gershgorin, B. , 2010, “ Mathematical Strategies for Filtering Turbulent Dynamical Systems,” Discrete Contin. Dyn. Syst., 27(2), pp. 441–486. [CrossRef]
Majda, A. J. , and Harlim, J. , 2012, Filtering Complex Turbulent Systems, Cambridge University Press, Cambridge, UK.
Maynard Gayme, D. , 2010, “ A Robust Control Approach to Understanding Nonlinear Mechanisms in Shear Flow Turbulence,” Ph.D. thesis, California Institute of Technology, Pasadena, CA.
Marusic, I. , and Hutchins, N. , 2005, “ Experimental Study of Wall Turbulence: Implications for Control,” Transition and Turbulence Control, World Scientific, Singapore, pp. 207–246.
Smits, A. J. , McKeon, B. J. , and Marusic, I. , 2011, “ High-Reynolds Number Wall Turbulence,” Annu. Rev. Fluid Mech., 43(1), pp. 353–375. [CrossRef]
Cacuci, D. G. , Navon, I. M. , and Ionescu-Bujor, M. , 2013, Computational Methods for Data Evaluation and Assimilation, Chapman & Hall, Oxford, UK.
Cordier, L. , Abou El Majd, B. , and Favier, J. , 2010, “ Calibration of POD Reduced-Order Models Using Tikhonov Regularization,” Int. J. Numer. Methods Fluids, 63(2), pp. 269–296.
Kapur, J. N. , and Kevasan, H. K. , 1992, Entropy Optimization Principles With Applications, 1st ed., Academic Press, Boston.
Noack, B. R. , and Niven, R. K. , 2012, “ Maximum-Entropy Closure for a Galerkin System of Incompressible Shear Flow,” J. Fluid Mech., 700, pp. 187–213. [CrossRef]
Noack, B. R. , and Niven, R. K. , 2013, “ A Hierarchy of Maximum-Entropy Closures for Galerkin Systems of Incompressible Flows,” Comput. Math. Appl., 65(10), pp. 1558–1574. [CrossRef]
Andresen, B. , 1983, “ Finite-Time Thermodynamics,” Physics Laboratory II, 1st ed., University of Copenhagen, Copenhagen, Denmark.
Noack, B. R. , Schlegel, M. , Ahlborn, B. , Mutschke, G. , Morzyński, M. , Comte, P. , and Tadmor, G. , 2008, “ A Finite-Time Thermodynamics of Unsteady Fluid Flows,” J. Non-Equilibr. Thermodyn., 33(2), pp. 103–148. [CrossRef]
Noack, B. R. , Schlegel, M. , Morzyński, M. , and Tadmor, G. , 2010, “ System Reduction Strategy for Galerkin Models of Fluid Flows,” Int. J. Numer. Methods Fluids, 63(2), pp. 231–248.
Taira, K. , 2015, private communication.
Watts, D. J. , and Strogatz, S. H. , 1998, “ Collective Dynamics of ‘Small-World' Networks,” Nature, 393(6684), pp. 440–442. [CrossRef] [PubMed]
Barabási, A.-L. , and Albert, R. , 1999, “ Emergence of Scaling in Random Networks,” Science, 286(5439), pp. 509–512. [CrossRef] [PubMed]
Barabási, A.-L. , 2009, “ Scale-Free Networks: A Decade and Beyond,” Science, 325(5939), pp. 412–413. [CrossRef] [PubMed]
Del Genio, C. I. , Gross, T. , and Bassler, K. E. , 2011, “ All Scale-Free Networks are Sparse,” Phys. Rev. Lett., 107(17), p. 178701. [CrossRef] [PubMed]
Barzel, B. , and Barabási, A.-L. , 2013, “ Universality in Network Dynamics,” Nat. Phys., 9(10), pp. 673–681. [CrossRef]
Newman, M. E. , 2003, “ The Structure and Function of Complex Networks,” SIAM Rev., 45(2), pp. 167–256. [CrossRef]
Leonard, N. E. , and Fiorelli, E. , 2001, “ Virtual Leaders, Artificial Potentials and Coordinated Control of Groups,” 40th IEEE Conference on Decision and Control, Orlando, FL, Dec. 4–7, Vol. 3, pp. 2968–2973.
Olfati-Saber, R. , 2006, “ Flocking for Multi-Agent Dynamic Systems: Algorithms and Theory,” IEEE Trans. Autom. Control, 51(3), pp. 401–420. [CrossRef]
Balch, T. , and Arkin, R. C. , 1998, “ Behavior-Based Formation Control for Multirobot Teams,” IEEE Trans. Rob. Autom., 14(6), pp. 926–939. [CrossRef]
Cortes, J. , Martinez, S. , Karatas, T. , and Bullo, F. , 2002, “ Coverage Control for Mobile Sensing Networks,” IEEE International Conference on Robotics and Automation (ICRA '02), Washington, DC, May 11–15, Vol. 2, pp. 1327–1332.
Leonard, N. E. , Paley, D. A. , Lekien, F. , Sepulchre, R. , Fratantoni, D. M. , and Davis, R. E. , 2007, “ Collective Motion, Sensor Networks, and Ocean Sampling,” Proc. IEEE, 95(1), pp. 48–74. [CrossRef]
Milo, R. , Shen-Orr, S. , Itzkovitz, S. , Kashtan, N. , Chklovskii, D. , and Alon, U. , 2002, “ Network Motifs: Simple Building Blocks of Complex Networks,” Science, 298(5594), pp. 824–827. [CrossRef] [PubMed]
Luscombe, N. M. , Babu, M. M. , Yu, H. , Snyder, M. , Teichmann, S. A. , and Gerstein, M. , 2004, “ Genomic Analysis of Regulatory Network Dynamics Reveals Large Topological Changes,” Nature, 431(7006), pp. 308–312. [CrossRef] [PubMed]
Low, S. H. , Paganini, F. , and Doyle, J. C. , 2002, “ Internet Congestion Control,” Control Syst., 22(1), pp. 28–43. [CrossRef]
Doyle, J. C. , Alderson, D. L. , Li, L. , Low, S. , Roughan, M. , Shalunov, S. , Tanaka, R. , and Willinger, W. , 2005, “ The “Robust Yet Fragile” Nature of the Internet,” Proc. Natl. Acad. Sci. U.S.A., 102(41), pp. 14497–14502. [CrossRef] [PubMed]
Rahmani, A. , Ji, M. , Mesbahi, M. , and Egerstedt, M. , 2009, “ Controllability of Multi-Agent Systems From a Graph-Theoretic Perspective,” SIAM J. Control Optim., 48(1), pp. 162–186. [CrossRef]
Liu, Y.-Y. , Slotine, J.-J. , and Barabasi, A.-L. , 2011, “ Controllability of Complex Networks,” Nature, 473(7346), pp. 167–173. [CrossRef] [PubMed]
Lin, F. , Fardad, M. , and Jovanović, M. R. , 2014, “ Algorithms for Leader Selection in Stochastically Forced Consensus Networks,” IEEE Trans. Automat. Control, 59(7), pp. 1789–1802. [CrossRef]
Cowan, N. J. , Chastain, E. J. , Vilhena, D. A. , Freudenberg, J. S. , and Bergstrom, C. T. , 2012, “ Nodal Dynamics, Not Degree Distributions, Determine the Structural Controllability of Complex Networks,” PloS One, 7(6), p. e38398. [CrossRef] [PubMed]
Brockett, R. , 2012, “ Notes on the Control of the Liouville Equation,” Control of Partial Differential Equations (Lecture Notes in Mathematics, Vol. 2048), F. Alabau-Boussouira , R. Brockett , O. Glass , J. Le Rousseau , and E. Zuazua , eds., Springer, Berlin, pp. 101–129.
Hopf, E. , 1951, “ Statistical Hydromechanics and Functional Analysis,” J. Ration. Mech. Anal., 1, pp. 87–123.
Bagheri, S. , 2013, “ Koopman-Mode Decomposition of the Cylinder Wake,” J. Fluid Mech., 726, pp. 596–623. [CrossRef]
Bagheri, S. , 2014, “ Effects of Weak Noise on Oscillating Flows: Linking Quality Factor, Floquet Modes and Koopman Spectrum,” Phys. Fluids, 26(9), p. 094104. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Turbulence control roadmap. For details, see text and the respective sections.

Grahic Jump Location
Fig. 2

Applications of closed-loop turbulence control: (a) homogeneous grid turbulence (Reproduced with permission from T. Corke and H. Nagib.); (b) turbulent jet from Bradshaw et al. [38]; (c) Karman vortex street behind a mountain, photo by Bob Cahalan, NASA GSFC; (d) coherent structures in a mixing layer from Brown and Roshko [39]; (e) thunderstorm; (f) automobile in a wind tunnel, photo by Robert G. Bulmahn; (g) high-speed train; (h) cargo ship; (i) passenger jet; (j) Blue Angles fighter jets; (k) automobile engine; (l) turbo jet engine; (m) aircraft engines; (n) wind turbines; (o) heat exchanger flow; (p) rotating mixer; (q) air conditioner; (r) chocolate mixing; and (s) total artificial heart. Images (e) and (g)–(n) are from the website.1 Images (c), (f), and (q)–(s) are from the website.2 Images (o) and (p) were made using the COMSOL Multiphysics® software and are provided courtesy of COMSOL.

Grahic Jump Location
Fig. 5

Schematic illustrating popular choices at the various levels of kinematic and dynamic descriptions of the turbulent system P and choices for designing the controller K.

Grahic Jump Location
Fig. 4

Model hierarchy for control design based on Wiener[76].

Grahic Jump Location
Fig. 3

Heuristics of turbulence control. Here, s are the sensor signals and b are the actuation signals.

Grahic Jump Location
Fig. 6

Open-loop control topology

Grahic Jump Location
Fig. 7

Closed-loop control topology

Grahic Jump Location
Fig. 11

Two degrees-of-freedom control with reference tracking and disturbance rejection

Grahic Jump Location
Fig. 10

General framework for feedback control. The input to the controller is the system measurements s, and the controller outputs an actuation signal b. The exogenous inputs w may refer to a reference wr, disturbances wd, or sensor noise wn. The cost function J may measure the cost associated with inaccuracy of reference tracking, expense of control, etc.

Grahic Jump Location
Fig. 8

Linear-quadratic Gaussian controller. The Kalman filter Kf is a dynamical system that takes sensor measurements s and the actuation signal b to estimate the full-state â. The LQR gain Kr is a matrix that multiplies the full-state to produce an actuation signal b=−Krâ that is optimal with respect to the quadratic cost function in Eq. (18).

Grahic Jump Location
Fig. 9

Feedback control with disturbances and noise

Grahic Jump Location
Fig. 13

Phase portrait of oscillatory linear dynamics (49). The dashed trajectory corresponds to the unactuated dynamics while the solid trajectory corresponds to actuated dynamics with Eq. (50). The chosen parameters are σu= 0.1, σc= −0.1, ωu= 1, g = 1 implying k = 0.4.

Grahic Jump Location
Fig. 12

Schematic of the closed-loop controller for transition delay of a flat-plate boundary layer (Reproduced with permission from Semeraro et al. [225]. Copyright 2013 by Cambridge University Press). Here, ψ corresponds to sensors s and φ corresponds to actuators b.

Grahic Jump Location
Fig. 14

Phase portrait of weakly nonlinear dynamics. The dashed trajectory corresponds to the unactuated dynamics while the solid trajectory corresponds to actuated dynamics. The chosen parameters of Eq. (52) are σu= 0.1, ωu= 1, αu= 1, βu= 1, γu= 0, and the forced decay rate σc= −0.1. The globally stable limit cycle lies on the parabolic inertial manifold.

Grahic Jump Location
Fig. 15

Flow visualization of the experimental wake behind a D-shaped body without (a) and with symmetric low-frequency actuation (b) (Reproduced with permission from Mark Pastoor.) The D-shaped body is shown with five pressure sensors on the rear face, and the arrows at the corners indicate the employed zero net mass flux actuators. (a) Natural wake with vortex shedding and (b) actuated partially stabilized wake.

Grahic Jump Location
Fig. 16

Phase portrait of moderately nonlinear dynamics (54). The dashed trajectory corresponds to the unactuated dynamics while the solid trajectory corresponds to actuated dynamics with Eq. (50). The chosen parameters are enumerated in Table1.

Grahic Jump Location
Fig. 17

Phase portrait of the shift-mode amplitudes a5 and a6, i.e., the slow dynamics in Eq. (54). Same transient solutions as in Fig. 16.

Grahic Jump Location
Fig. 18

Venn diagram for the classification of nonlinearities. Prototypic examples are for (A), the subcritical flow over backward-facing step with noise excitation [251]; for (B), the supercritical onset of vortex shedding [256]; for (C), the suppression of Kelvin–Helmoltz vortices by high-frequency forcing [47]; and for (D), the decay of 2D turbulence [267].

Grahic Jump Location
Fig. 19

Input/output characteristics of different dynamics. Left: actuation command; right: sensor signal without forcing (dark), and sensor signal under periodic forcing (light). From top to bottom: a stable fixed point with periodic excitation (linear dynamics); a stable limit cycle with locking periodic forcing (weakly nonlinear dynamics); a stable limit cycle with high-frequency forcing (moderately nonlinear dynamics); and broadband turbulence under periodic forcing (strongly nonlinear dynamics).

Grahic Jump Location
Fig. 20

Overview of model-free control methods discussed in Sec. 6. Model-free control involves the choice of control law structure as well as the optimization of controller parameters.

Grahic Jump Location
Fig. 21

Schematic illustrating the components of an ESC. A sinusoidal perturbation is added to the best guess of the input b, passing through the plant, and resulting in a sinusoidal output perturbation. The high-pass filter removes the DC gain and results in a zero-mean output perturbation, which is then multiplied (demodulated) by the same input perturbation. This demodulated signal is finally integrated into the best guess b̂ for the optimizing input b.

Grahic Jump Location
Fig. 22

Schematic illustrating ESC on for a static objective function J(b). The output perturbation (light) is in phase when the input is left of the peak value (i.e., b < b*) and out of phase when the input is to the right of the peak (i.e., b > b*). Thus, integrating the product of input and output sinusoids moves b̂ toward b*.

Grahic Jump Location
Fig. 23

Acoustic pressure reduction in combustor experiment with modified ESC algorithm. The main peak is reduced by about a factor of 60 when control is applied (Reproduced with permission from Gelbert et al. [288]. Copyright 2012 by Elsevier).

Grahic Jump Location
Fig. 24

Illustration of the benefits of opposition control (bottom) in contrast to the unforced system (top). Contours of streamwise vorticity are plotted in a cross-flow plane. Negative contours are indicated with dashed lines (Reproduced with permission from Lee et al. [22]. Copyright 1997 by AIP Publishing LLC).

Grahic Jump Location
Fig. 25

Illustration of a possible binary representation of parameters used in GAs. This example has two parameters, each represented with a 3-bit binary number.

Grahic Jump Location
Fig. 26

Illustration of function tree representation used in GP

Grahic Jump Location
Fig. 27

Genetic operations are used to advance generations of individuals in GAs. Operations are elitism (E), replication (R), crossover (C), and mutation (M). For each individual of generation k + 1, after the elitism step, a genetic operation is chosen randomly according to a predetermined probability distribution. The individuals participating in this operation are selected from generation k with probability related to their fitness (e.g., inversely proportional to the cost function).

Grahic Jump Location
Fig. 28

Genetic operations are used to advance generations of functions in GP

Grahic Jump Location
Fig. 29

Schematic of closed-loop feedback control using GP for optimization. Various controllers in a population compete to minimize a cost function J, and the best performing individual controllers may advance to the next generation according to the optimization procedure on the right (Reproduced with permission from Fig. 4 of Duriez et al. [265]. Copyright 2014 by T. Duriez, V. Parezanovic, J.-C. Laurentie, C. Fourment, J. Delville, J.P. Bonnet, L. Cordier, B.R. Noack, M. Segond, M. Abel, N. Gautier, J.L. Aider, C. Raibaudo, C. Cuvier, M. Stanislas, S.L. Brunton).

Grahic Jump Location
Fig. 30

“Pseudovisualizations of the TUCOROM experimental mixing layer demonstrator for three cases: (I) unforced baseline (width W = 100%), (II) the best open-loop benchmark (width W = 155%), and (III) MLC closed-loop control (width W = 167%). The velocity fluctuations recorded by 24 hot-wires probes are shown as contour-plot over the time t (abscissa) and the sensor position y (ordinate). The black stripes above the controlled cases indicate when the actuator is active (taking into account the convective time). The average actuation frequency achieved by the MLC control is comparable to the open-loop benchmark.” The relative mixing cost function of the natural flow, open-loop forcing, and machine-learning control is shown in (b), and the mixing layer is shown in (a) (Reproduced with permission from Parezanovic et al. [263]. Copyright 2015 by Springer). A lower cost function J indicates improved mixing (Reproduced with permission from Duriez et al. [265]. Copyright 2014 by T. Duriez, V. Parezanovic, J.-C. Laurentie, C. Fourment, J. Delville, J.P. Bonnet, L. Cordier, B.R. Noack, M. Segond, M. Abel, N. Gautier, J.L. Aider, C. Raibaudo, C. Cuvier, M. Stanislas, S.L. Brunton).

Grahic Jump Location
Fig. 31

Flow chart illustrating the hierarchy of active control approaches. This diagram is conservative in giving preference to the most established techniques. If the task is optimization or minimization of measurement time, machine-learning control may be an earlier branch. Top panel depicts a turbulent jet from Bradshaw et al. [38].

Grahic Jump Location
Fig. 32

Schematic illustrating a roadmap for future development. We envision the synthesis of classical control theory with data-driven methods for the development of hybrid controllers. Both top-down and bottom-up approaches will contribute to a better understanding of nonlinearities, which will in turn contribute to the development of more effective controllers.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In