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Cluster-based feedback control of turbulent post-stall separated flows

  • Aditya G. Nair (a1) (a2), Chi-An Yeh (a1) (a3), Eurika Kaiser (a2), Bernd R. Noack (a4) (a5) (a6) (a7), Steven L. Brunton (a2) and Kunihiko Taira (a1) (a3)...


We propose a cluster-based control strategy for feedback control of post-stall separated flows over an airfoil. The present approach partitions the flow trajectories (force measurements) into clusters, which correspond to characteristic coarse-grained phases in a low-dimensional feature space. A feedback control law (using blowing/suction actuation) is then sought for each cluster state through iterative evaluation and downhill simplex search to minimize power consumption in aerodynamic flight. The optimized control laws re-route the flow trajectories to the aerodynamically favourable regions in the feature space in a model-free manner. Utilizing a limited number of sensor measurements for both clustering and optimization, these feedback laws were determined in only $O(10)$ iterations. The objective of the present work is not necessarily to suppress flow separation but to minimize the desired cost function to achieve enhanced aerodynamic performance. The present approach is applied to the control of two- and three-dimensional separated flows over a NACA 0012 airfoil in large-eddy simulations at an angle of attack of $9^{\circ }$ , Reynolds number $Re=23\,000$ and free-stream Mach number $M_{\infty }=0.3$ . The optimized control laws avoid the intermittent occurrence of long-period shedding associated with high-drag clusters, thus lowering the mean drag. The present work aims to address some of the challenges associated with feedback control design for turbulent separated flows at moderate Reynolds number.


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Amitay, M. & Glezer, A. 2002 Role of actuation frequency in controlled flow reattachment over a stalled airfoil. AIAA J. 40 (2), 209216.
Amitay, M. & Glezer, A. 2006 Aerodynamic flow control using synthetic jet actuators. In Control of Fluid Flow, pp. 4573. Springer.
Anderson, J. D. 1999 Aircraft Performance and Design. McGraw-Hill.
Ariyur, K. B. & Krstic, M. 2003 Real-time Optimization by Extremum-seeking Control. Wiley.
Bagheri, S., Brandt, L. & Henningson, D. S. 2009 Input–output analysis, model reduction and control of the flat-plate boundary layer. J. Fluid Mech. 620, 263298.
Bänsch, E., Benner, P., Saak, J. & Weichelt, H. K. 2015 Riccati-based boundary feedback stabilization of incompressible Navier–Stokes flow. SIAM J. Sci. Comput. 37 (2), A832A858.
Barbagallo, A., Sipp, D. & Schmid, P. J. 2009 Closed-loop control of an open cavity flow using reduced-order models. J. Fluid Mech. 641, 150.
Beaudoin, J.-F., Cadot, O., Aider, J.-L. & Wesfreid, J. E. 2006 Bluff-body drag reduction by extremum-seeking control. J. Fluids Struct. 22 (6–7), 973978.
Benton, S. I. & Visbal, M. R. 2018 High-frequency forcing to mitigate unsteady separation from a bursting separation bubble. Phys. Rev. Fluids 3 (1), 013907.
Bewley, T. R. 2001 Flow control: new challenges for a new renaissance. Prog. Aerosp. Sci. 37 (1), 2158.
Brès, G. A., Ham, F. E., Nichols, J. W. & Lele, S. K. 2017 Unstructured large-eddy simulations of supersonic jets. AIAA J. 55 (4), 11641184.
Brunton, S. L. & Noack, B. R. 2015 Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67, 050801–48.
Brunton, S. L., Proctor, J. L. & Kutz, J. N. 2016 Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA, 201517384.
Carini, M., Pralits, J. O. & Luchini, P. 2015 Feedback control of vortex shedding using a full-order optimal compensator. J. Fluids Struct. 53, 1525.
Chiang, M. M.-T. & Mirkin, B. 2010 Intelligent choice of the number of clusters in k-means clustering: an experimental study with different cluster spreads. J. Classification 27 (1), 340.
Colonius, T. & Williams, D. R. 2011 Control of vortex shedding on two-and three-dimensional aerofoils. Phil. Trans. R. Soc. Lond. A 369 (1940), 15251539.
Debien, A., von Krbek, K. A. F. F., Mazellier, N., Duriez, T., Cordier, L., Noack, B. R., Abel, M. W. & Kourta, A. 2016 Closed-loop separation control over a sharp edge ramp using genetic programming. Exp. Fluids 57 (3), 40.
Duriez, T., Brunton, S. L. & Noack, B. R. 2016 Machine Learning Control-Taming Nonlinear Dynamics and Turbulence. Springer.
Freund, J. B. 1997 Proposed inflow/outflow boundary condition for direct computation of aerodynamic sound. AIAA J. 35 (4), 740742.
G-Michael, T., Gunzburger, M. & Peterson, J. 2018 Clustering approaches to feature change detection. In Detection and Sensing of Mines, Explosive Objects, and Obscured Targets XXIII, vol. 10628, p. 106281G. International Society for Optics and Photonics.
Gautier, N., Aider, J.-L., Duriez, T., Noack, B. R., Segond, M. & Abel, M. 2015 Closed-loop separation control using machine learning. J. Fluid Mech. 770, 442457.
Goutte, C., Toft, P., Rostrup, E., Nielsen, F. Å. & Hansen, L. K. 1999 On clustering fMRI time series. NeuroImage 9 (3), 298310.
Greenblatt, D. & Wygnanski, I. J. 2000 The control of flow separation by periodic excitation. Prog. Aerosp. Sci. 36, 487545.
Hervé, A., Sipp, D., Schmid, P. J. & Samuelides, M. 2012 A physics-based approach to flow control using system identification. J. Fluid Mech. 702, 2658.
Huang, S.-C. & Kim, J. 2008 Control and system identification of a separated flow. Phys. Fluids 20 (10), 101509.
Hunt, J. C. R., Wray, A. A. & Moin, P. 1998 Eddies, streams, and convergence zones in turbulent flows. In Center for Turbulence Research Report CTR-S88, pp. 193208.
Illingworth, S. J., Morgans, A. S. & Rowley, C. W. 2012 Feedback control of cavity flow oscillations using simple linear models. J. Fluid Mech. 709, 223248.
Kaiser, E., Li, R. & Noack, B. R. 2017a On the control landscape topology. In The Proceedings of the 20th World Congress of the International Federation of Automatic Control (IFAC), pp. 15. IFAC.
Kaiser, E., Morzyński, M., Daviller, G., Kutz, J. N., Brunton, B. W. & Brunton, S. L. 2018 Sparsity enabled cluster reduced-order models for control. J. Comput. Phys. 352, 388409.
Kaiser, E., Noack, B. R., Cordier, L., Spohn, A., Segond, M., Abel, M., Daviller, G., Östh, J., Krajnović, S. & Niven, R. K. 2014 Cluster-based reduced-order modelling of a mixing layer. J. Fluid Mech. 754, 365414.
Kaiser, E., Noack, B. R., Spohn, A., Cattafesta, L. N. & Morzyński, M. 2017b Cluster-based control of a separating flow over a smoothly contoured ramp. Theor. Comput. Fluid Dyn. 31 (5–6), 579593.
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383417.
Kojima, R., Nonomura, T., Oyama, A. & Fujii, K. 2013 Large eddy simulation of low-Reynolds-number flow over thick and thin NACA airfoils. J. Aircraft 50 (1), 187196.
Kontovasilis, K. P. & Mitrou, N. M. 1995 Markov-modulated traffic with nearly complete decomposability characteristics and associated fluid queueing models. Adv. Appl. Probability 27 (4), 11441185.
Lee, D. & Wiswall, M. 2007 A parallel implementation of the simplex function minimization routine. Comput. Economics 30 (2), 17187.
Leicht, E. A. & Newman, M. E. J. 2008 Community structure in directed networks. Phys. Rev. Lett. 100 (11), 118703.
Lloyd, S. 1982 Least squares quantization in PCM. IEEE Trans. Inf. Theory 28 (2), 129137.
Loiseau, J.-C., Noack, B. R. & Brunton, S. L. 2018 Sparse reduced-order modelling: sensor-based dynamics to full-state estimation. J. Fluid Mech. 844, 459490.
Lomax, R. G. & Hahs-Vaughn, D. L. 2013 Statistical Concepts: A Second Course. Routledge.
Luchtenburg, D. M., Günther, B., Noack, B. R., King, R. & Tadmor, G. 2009 A generalized mean-field model of the natural and high-frequency actuated flow around a high-lift configuration. J. Fluid Mech. 623, 283316.
Luersen, M. A., Le Riche, R. & Guyon, F. 2004 A constrained, globalized, and bounded Nelder–Mead method for engineering optimization. Struct. Multidiscip. Optim. 27 (1–2), 4354.
Mao, X., Blackburn, H. M. & Sherwin, S. J. 2015 Nonlinear optimal suppression of vortex shedding from a circular cylinder. J. Fluid Mech. 775, 241265.
McKay, M. D., Beckman, R. J. & Conover, W. J. 2000 A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 42 (1), 5561.
Munday, P. M. & Taira, K. 2018 Effects of wall-normal and angular momentum injections in airfoil separation control. AIAA J. 56 (5), 18301842.
Nair, A. G., Brunton, S. L. & Taira, K. 2018 Networked-oscillator-based modeling and control of unsteady wake flows. Phys. Rev. E 97 (6), 063107.
Nelder, J. A. & Mead, R. 1965 A simplex method for function minimization. Computer J. 7 (4), 308313.
Newman, M. 2010 Networks: An Introduction. Oxford University Press.
Noack, B., Tadmor, G. & Morzynski, M. 2004 Low-dimensional models for feedback flow control. Part I: Empirical galerkin models. In 2nd AIAA Flow Control Conference, p. 2408.
Noack, B. R. 2019 Closed-loop turbulence control-from human to machine learning (and retour). In Proceedings of the 4th Symposium on Fluid Structure-Sound Interactions and Control (FSSIC) (ed. Zhou, Y., Kimura, M., Peng, G., Lucey, A. D. & Huang, L.), pp. 2332. Springer.
Noack, B. R., Afanasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335363.
Noack, B. R., Morzynski, M. & Tadmor, G. 2011 Reduced-order Modelling for Flow Control. Springer.
Norris, J. R. 1998 Markov Chains. Cambridge University Press.
Pinier, J. T., Ausseur, J. M., Glauser, M. N. & Higuchi, H. 2007 Proportional closed-loop feedback control of flow separation. AIAA J. 45 (1), 181190.
Protas, B. 2004 Linear feedback stabilization of laminar vortex shedding based on a point vortex model. Phys. Fluids 16 (12), 44734488.
Rabault, J., Kuchta, M., Jensen, A., Réglade, U. & Cerardi, N. 2019 Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control. J. Fluid Mech. 865, 281302.
Rokach, L. & Maimon, O. 2005 Clustering methods. In Data Mining and Knowledge Discovery Handbook, pp. 321352. Springer.
Semeraro, O., Bagheri, S., Brandt, L. & Henningson, D. S. 2011 Feedback control of three-dimensional optimal disturbances using reduced-order models. J. Fluid Mech. 677, 63102.
Taira, K. & Nakao, H. 2018 Phase-response analysis of synchronization for periodic flows. J. Fluid Mech. 846, R2.
Tibshirani, R., Walther, G. & Hastie, T. 2001 Estimating the number of clusters in a data set via the gap statistic. J. R. Statist. Soc. B 63 (2), 411423.
Vreman, B. 2004 An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications. Phys. Fluids 16 (10), 36703681.
Wand, M. P. & Jones, M. C. 1994 Kernel Smoothing. CRC Press.
Yeh, C.-A., Munday, P. & Taira, K.2017 Use of local periodic heating for separation control on a NACA 0012 airfoil. AIAA Paper 2017-1451.
Yeh, C.-A. & Taira, K. 2019 Resolvent-analysis-based design of airfoil separation control. J. Fluid Mech. 867, 572610.
Young, G. & Householder, A. S. 1938 Discussion of a set of points in terms of their mutual distances. Psychometrika 3 (1), 1922.
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Cluster-based feedback control of turbulent post-stall separated flows

  • Aditya G. Nair (a1) (a2), Chi-An Yeh (a1) (a3), Eurika Kaiser (a2), Bernd R. Noack (a4) (a5) (a6) (a7), Steven L. Brunton (a2) and Kunihiko Taira (a1) (a3)...


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