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Feedback control of three-dimensional optimal disturbances using reduced-order models

Published online by Cambridge University Press:  29 March 2011

ONOFRIO SEMERARO ,
SHERVIN BAGHERI ,
LUCA BRANDT  and
DAN S. HENNINGSON
ONOFRIO SEMERARO*
Affiliation:
Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
SHERVIN BAGHERI
Affiliation:
Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
LUCA BRANDT
Affiliation:
Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
DAN S. HENNINGSON
Affiliation:
Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
*
†Email address for correspondence: onofrio@mech.kth.se
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Abstract

The attenuation of three-dimensional wavepackets of streaks and Tollmien–Schlichting (TS) waves in a transitional boundary layer using feedback control is investigated numerically. Arrays of localized sensors and actuators (about 10–20) with compact spatial support are distributed near the rigid wall equidistantly along the spanwise direction and connected to a low-dimensional (r = 60) linear quadratic Gaussian controller. The control objective is to minimize the disturbance energy in a domain spanned by a number of proper orthogonal decomposition modes. The feedback controller is based on a reduced-order model of the linearized Navier–Stokes equations including the inputs and outputs, computed using a snapshot-based balanced truncation method. To account for the different temporal and spatial behaviour of the two main instabilities of boundary-layer flows, we design two controllers. We demonstrate that the two controllers reduce the energy growth of both TS wavepackets and streak packets substantially and efficiently, using relatively few sensors and actuators. The robustness of the controller is investigated by varying the number of actuators and sensors, the Reynolds number and the pressure gradient. This work constitutes the first experimentally feasible simulation-based control design using localized sensing and acting devices in conjunction with linear control theory in a three-dimensional setting.

JFM classification

Boundary Layers: Boundary layer stability Flow Control: Control theory Flow Control: Flow Control
Type
Papers
Information
Journal of Fluid Mechanics , Volume 677 , 25 June 2011 , pp. 63 - 102
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Ahuja, S. 2009 Reduction methods for feedback stabilization of fluid flows. PhD thesis, Princeton University, New Jersey.Google Scholar
Ahuja, S. & Rowley, C. W. 2010 Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators. J. Fluid Mech. 645, 447–478.CrossRefGoogle Scholar
Åkervik, E., Ehrenstein, U., Gallaire, F. & Henningson, D. S. 2008 Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. B/Fluids 27, 501–513.CrossRefGoogle Scholar
Anderson, B. & Moore, J. 1990 Optimal Control: Linear Quadratic Methods. Prentice Hall.Google Scholar
Bagheri, S., Åkervik, E., Brandt, L. & Henningson, D. S. 2009 a Matrix-free methods for the stability and control of boundary layers. AIAA J. 47, 1057–1068.CrossRefGoogle Scholar
Bagheri, S., Brandt, L. & Henningson, D. S. 2009 b Input–output analysis, model reduction and control design of the flat-plate boundary layer. J. Fluid Mech. 620, 263–298.CrossRefGoogle Scholar
Bagheri, S., Hœpffner, J., Schmid, P. J. & Henningson, D. S. 2009 c Input–output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev. 62 (2), 1–27.CrossRefGoogle Scholar
Bamieh, B., Paganini, F. & Dahleh, M. 2002 Distributed control of spatially invariant systems. IEEE Trans. Autom. Control 47 (7), 1091–1107.CrossRefGoogle Scholar
Barbagallo, A., Sipp, D. & Schmid, P. J. 2009 Closed-loop control of an open cavity flow using reduced-order models. J. Fluid Mech. 641, 1–50.CrossRefGoogle Scholar
Bewley, T. R. 2001 Flow control: new challenges for a new renaissance. Prog. Aerosp. Sci. 37, 21–58.CrossRefGoogle Scholar
Bewley, T. R. & Liu, S. 1998 Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 305–349.CrossRefGoogle Scholar
Bewley, T. R., Moin, P. & Temam, R. 2001 DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J. Fluid Mech. 447, 179–225.CrossRefGoogle Scholar
Butler, K. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Chevalier, M., Hœpffner, J., Åkervik, E. & Henningson, D. S. 2007 a Linear feedback control and estimation applied to instabilities in spatially developing boundary layers. J. Fluid Mech. 588, 163–187.CrossRefGoogle Scholar
Chevalier, M., Hœpffner, J., Bewley, T. R. & Henningson, D. S. 2006 State estimation in wall-bounded flow systems. Part 2. Turbulent flows. J. Fluid Mech. 552, 167–187.CrossRefGoogle Scholar
Chevalier, M., Schlatter, P., Lundbladh, A. & Henningson, D. S. 2007 b A pseudo-spectral solver for incompressible boundary layer flows. Trita-Mek 7. KTH Mechanics, Stockholm, Sweden.Google Scholar
Choi, H., Moin, P. & Kim, J. 1993 Direct numerical simulation of turbulent flow over riblets. J. Fluid Mech. 255, 503–539.CrossRefGoogle Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: Non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Cortelezzi, L., Speyer, J. L., Lee, K. H. & Kim, J. 1998 Robust reduced-order control of turbulent channel flows via distributed sensors and actuators. In IEEE 37th Conf. on Decision and Control, pp. 1906–1911.Google Scholar
Curtain, R. & Zwart, H. 1995 An Introduction to Infinite-Dimensional Linear Systems Theory. Springer.CrossRefGoogle Scholar
Doyle, J. C. 1978 Guaranteed margins for LQG regulators. IEEE Trans. Autom. Control 23, 756–757.CrossRefGoogle Scholar
Doyle, J. C., Glover, K., Khargonekar, P. P. & Francis, B. A. 1989 State-space solutions to standard H2 and H∞ control problems. IEEE Trans. Autom. Control 34, 831–847.CrossRefGoogle Scholar
Dullerud, E. G. & Paganini, F. 1999 A Course in Robust Control Theory. A Convex Approach. Springer Verlag.Google Scholar
Glover, K. 1984 All optimal Hankel-norm approximations of linear multivariable systems and the l ∞-error bounds. Intl J. Control 39, 1115–1193.CrossRefGoogle Scholar
Green, M. & Limebeer, J. N. 1995 Linear Robust Control. Prentice Hall.Google Scholar
Grundmann, S. & Tropea, C. 2008 Active cancellation of artificially introduced Tollmien–Schlichting waves using plasma actuators. Exp. Fluids 44 (5), 795–806.CrossRefGoogle Scholar
Hammond, E. P., Bewley, T. R. & Moin, P. 1998 Observed mechanisms for turbulence attenuation and enhancement in opposition-controlled wall-bounded flows. Phys. Fluids 10 (9), 2421–2423.CrossRefGoogle Scholar
Ho, C. M. & Tai, Y. 1998 Micro-electro-mechanical systems MEMS and fluid flows. Annu. Rev. Fluid Mech. 30, 579–612.CrossRefGoogle Scholar
Högberg, M. & Bewley, T. R. 2000 Spatially localized convolution kernels for feedback control of transitional flows. In IEEE 39th Conf. on Decision and Control, pp. 3278–3283.Google Scholar
Högberg, M., Bewley, T. R. & Henningson, D. S. 2003 a Linear feedback control and estimation of transition in plane channel flow. J. Fluid Mech. 481, 149–175.CrossRefGoogle Scholar
Högberg, M., Bewley, T. R. & Henningson, D. S. 2003 b Relaminarization of Re τ = 100 turbulence using gain scheduling and linear state-feedback control flow. Phys. Fluids 15, 3572–3575.CrossRefGoogle Scholar
Holmes, P., Lumley, J. & Berkooz, G. 1996 Turbulence Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.CrossRefGoogle Scholar
Ilak, M., Bagheri, S., Brandt, L., Rowley, C. W. & Henningson, D. 2010 Model reduction of the nonlinear complex Ginzburg–Landau equation. SIAM J. Appl. Dyn. Syst. 9 (4), 1284–1302.CrossRefGoogle Scholar
Ilak, M. & Rowley, C. W. 2008 Modeling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids 20, 034103.CrossRefGoogle Scholar
Joshi, S. S., Speyer, J. L. & Kim, J. 1997 A systems theory approach to the feedback stabilization of infinitesimal and finite-amplitude disturbances in plane Poiseuille flow. J. Fluid Mech. 332, 157–184.CrossRefGoogle Scholar
Kalman, R. E. 1960 A new approach to linear filtering and prediction problems. Trans. ASME D J. Basic Engng 82, 24–45.CrossRefGoogle Scholar
Laub, A. 1991 Invariant subspace methods for the numerical solution of Riccati equations. In The Riccati Equation (ed. Bittaini, S., Laub, A. J. & Willems, J. C.), pp. 163–196. Springer.CrossRefGoogle Scholar
Laub, A., Heath, M., Paige, C. & Ward, R. 1987 Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms. IEEE Trans. Autom. Control 32 (2), 115–122.CrossRefGoogle Scholar
Lewis, F. L. & Syrmos, L. V. 1995 Optimal Control. Wiley.Google Scholar
Lundell, F. 2003 Pulse-width modulated blowing/suction as a flow control actuator. Exp. Fluids 35, 502–504.CrossRefGoogle Scholar
Lundell, F. 2007 Reactive control of transition induced by free-stream turbulence: an experimental demonstration. J. Fluid Mech. 585, 41–71.CrossRefGoogle Scholar
Ma, Z., Ahuja, S. & Rowley, C. W. 2009 Reduced order models for control of fluids using the eigensystem realization algorithm. Theor. Comput. Fluid Dyn. (in press).CrossRefGoogle Scholar
Milling, W. 1981 Tollmien–Schlichting wave cancellation. Phys. Fluids 24 (5), 979–981.CrossRefGoogle Scholar
Monokrousos, A., Åkervik, E., Brandt, L. & Henningson, D. S. 2010 a Global optimal disturbances in the Blasius boundary-layer flow using time-steppers. J. Fluid Mech. 650, 181–214.CrossRefGoogle Scholar
Monokrousos, A., Brandt, L., Schlatter, P. & Henningson, D. S. 2008 DNS and LES of estimation and control of transition in boundary layers subject to free-stream turbulence. Intl J. Heat Fluid Flow 29 (3), 841–855.CrossRefGoogle Scholar
Monokrousos, A., Lundell, F. & Brandt, L. 2010 b Feedback control of boundary layer bypass transition: comparison of a numerical study with experiments. J. AIAA 48 (8), 1848–1851.CrossRefGoogle Scholar
Moore, B. 1981 Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26 (1), 17–32.CrossRefGoogle Scholar
Nordström, J., Nordin, N. & Henningson, D. S. 1999 The fringe region technique and the Fourier method used in the direct numerical simulation of spatially evolving viscous flows. SIAM J. Sci. Comput. 20 (4), 1365–1393.CrossRefGoogle Scholar
Pang, J. & Choi, K.-S. 2004 Turbulent drag reduction by Lorentz force oscillation. Phys. Fluids 16 (5), L35–L38.CrossRefGoogle Scholar
Pernebo, L. & Silverman, L. 1982 Model reduction via balanced state space representations. IEEE Trans. Autom. Control 27, 382–387.CrossRefGoogle Scholar
Quadrio, M. & Ricco, P. 2004 Critical assessment of turbulent drag reduction through spanwise wall oscillations. J. Fluid Mech. 521, 251–271.CrossRefGoogle Scholar
Rempfer, D. & Fasel, H. 1994 Evolution of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid. Mech. 260, 351–375.CrossRefGoogle Scholar
Rowley, C. W. 2005 Model reduction for fluids using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15 (3), 997–1013.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2000 Boundary-Layer Theory. Springer.CrossRefGoogle Scholar
Semeraro, O., Bagheri, S., Brandt, L. & Henningson, D. S. 2010 Linear control of 3D disturbances on a flat-plate. In Seventh IUTAM Symp. on Laminar–Turbulent Transition (ed. Schlatter, P. & Henningson, D. S.), vol. 18. Springer.Google Scholar
Skogestad, S. & Postlethwaite, I. 2005 Multivariable Feedback Control, Analysis to Design, 2nd edn. Wiley.Google Scholar
Smith, B. L. & Glezer, A. 1998 The formation and evolution of synthetic jets. Phys. Fluids 10 (9), 2281–2297.CrossRefGoogle Scholar
Sturzebecher, D. & Nitsche, W. 2003 Active cancellation of Tollmien–Schlichting waves instabilities on a wing using multi-channel sensor actuator systems. Intl J. Heat Fluid Flow 24, 572–583.CrossRefGoogle Scholar
Tempelmann, D., Hanifi, A. & Henningson, D. S. 2010 Optimal disturbances and receptivity in three-dimensional boundary layers. In 5th European Conf. on Computational Fluid Dynamics, Lisbon, Portugal.Google Scholar
White, E. & Saric, W. 2000 Application of variable leading-edge roughness for transition control on swept wings. In 38th AIAA Aerospace Sciences Meeting and Exhibit. AIAA Paper 2000-283. Reno, NV.Google Scholar
Zhou, K., Doyle, J. C. & Glover, K. 2002 Robust and Optimal Control. Prentice Hall.Google Scholar
Zhou, K., Salomon, G. & Wu, E. 1999 Balanced realization and model reduction for unstable systems. Intl J. Robust Nonlinear Control 9, 183–198.3.0.CO;2-E>CrossRefGoogle Scholar