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Closed-loop control of an open cavity flow using reduced-order models

Published online by Cambridge University Press:  30 November 2009

ALEXANDRE BARBAGALLO ,
DENIS SIPP  and
PETER J. SCHMID
ALEXANDRE BARBAGALLO
Affiliation:
ONERA-DAFE, 8 rue des Vertugadins, 92190 Meudon, France Laboratoire d'Hydrodynamique (LadHyX), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
DENIS SIPP
Affiliation:
ONERA-DAFE, 8 rue des Vertugadins, 92190 Meudon, France
PETER J. SCHMID*
Affiliation:
Laboratoire d'Hydrodynamique (LadHyX), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
*
†Email address for correspondence: [email protected]
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Abstract

The control of separated fluid flow by reduced-order models is studied using the two-dimensional incompressible flow over an open square cavity at Reynolds numbers where instabilities are present. Actuation and measurement locations are taken on the upstream and downstream edge of the cavity. A bi-orthogonal projection is introduced to arrive at reduced-order models for the compensated problem. Global modes, proper orthogonal decomposition (POD) modes and balanced modes are used as expansion bases for the model reduction. The open-loop behaviour of the full and the reduced systems is analysed by comparing the respective transfer functions. This analysis shows that global modes are inadequate to sufficiently represent the input–output behaviour whereas POD and balanced modes are capable of properly approximating the exact transfer function. Balanced modes are far more efficient in this process, but POD modes show superior robustness. The performance of the closed-loop system corroborates this finding: while reduced-order models based on POD are able to render the compensated system stable, balanced modes accomplish the same with far fewer degrees of freedom.

JFM classification

Flow Control: Flow Control Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 641 , 25 December 2009 , pp. 1 - 50
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Ahuja, S. & Rowley, C. W. 2008 Low-dimensional models for feedback stabilization of unstable steady states. AIAA Paper 2008-553.CrossRefGoogle Scholar
Åkervik, E., Hœffner, J., Ehrenstein, U. & Henningson, D. S. 2007 Optimal growth, model reduction and control in a separated boundary-layer flow using global modes. J. Fluid Mech. 579, 305–314.CrossRefGoogle Scholar
Amestoy, P. R., Duff, I. S., Koster, J. & L'Excellent, J.-Y. 2001 A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23 (1), 15–41.CrossRefGoogle Scholar
Antoulas, A. C. 2005 Approximation of Large-Scale Dynamical Systems. SIAM.CrossRefGoogle Scholar
Antoulas, A., Sorensen, D. & Gugercin, S. 2001 A survey of model reduction methods for large-scale systems. Contemp. Math. 280, 193–219.CrossRefGoogle Scholar
Bagheri, S., Brandt, L. & Henningson, D. S. 2009 a Input–output analysis, model reduction and control of the flat-plate boundary layer. J. Fluid Mech. 620, 263–298.Google Scholar
Bagheri, S., Hœpffner, J., Schmid, P. J. & Henningson, D. S. 2009 b Input–output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev. 62 (2), 020803-1-27.Google Scholar
Bai, Z. 2002 Krylov subspace techniques for reduced-order modelling of large-scale dynamical systems. Appl. Numer. Math. 43, 9–44.Google Scholar
Barbagallo, A., Sipp, D., Jacquin, L. & Schmid, P. J. 2008 Control of an incompressible cavity flow using a reduced model based on global modes. In Fifth AIAA Theoretical Fluid Mechanics Conference, Seattle. AIAA paper 2008-3904.Google Scholar
Barkley, D., Gomes, M. G. & Henderson, R. D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167–190.Google Scholar
Bergman, M., Cordier, L. & Brancher, J.-P. 2006 Optimal rotary control of the cylinder wake using POD reduced-order model. Phys. Fluids 17, 305–314.Google Scholar
Berkooz, G., Holmes, P. & Lumley, J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539–575.CrossRefGoogle Scholar
Bewley, T. R. 2001 Flow control: new challenges for a new renaissance. Prog. Aerosp. Sci. 37, 21–58.CrossRefGoogle Scholar
Blackburn, H. M., Barkley, D. & Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271–304.CrossRefGoogle Scholar
Bres, G. A. & Colonius, T. 2008 Three-dimensional instabilities in compressible flow over open cavities. J. Fluid Mech. 599, 309–339.CrossRefGoogle Scholar
Buffoni, M., Camarri, S. & Iollo, A. 2006 Low-dimensional modelling of a confined three-dimensional wake flow. J. Fluid Mech. 569, 141–150.Google Scholar
Burl, J. B. 1999 Linear Optimal Control. ℋ2 and ℋ∞ Methods. Addison-Wesley.Google Scholar
Cattafesta, L. N., Song, Q., Williams, D. R., Rowley, C. W. & Alvi, F. 2008 Active control of flow-induced cavity oscillations. Prog. Aerosp. Sci. 44, 459–502.CrossRefGoogle Scholar
Cattafesta, L. N., Williams, D. R., Rowley, C. W. & Alvi, F. S. 2003 Review of active flow control of flow-induced cavity resonance. AIAA Paper 2003-3567 33rd AIAA Fluid Dyn. Conf., Orlando, FL, June 2003.Google Scholar
Chevalier, M., Hœpffner, J., Åkervik, E. & Henningson, D. S. 2007 Linear feedback control and estimation applied to instabilities in spatially developing boundary layers. J. Fluid Mech. 588, 163–187.CrossRefGoogle Scholar
Crouch, J. D., Garbaruk, A. & Magidov, D. 2007 Predicting the onset of flow unsteadiness based on global instability. J. Comput. Phys. 224, 924–940.CrossRefGoogle Scholar
Delville, J., Cordier, L. & Bonnet, J. P. 1998 Large-scale structure identification and control in turbulent shear flows. In Flow Control: Fundamentals and Practice (eds. Gad-el-Hak, M., Pollard, A., Bonnet, J.-P.) pp. 199–273. Springer.CrossRefGoogle Scholar
Ding, Y. & Kawahara, M. 1999 Three-dimensional linear stability analysis of incompressible viscous flows using finite element method. Intl J. Numer. Meth. Fluids 31, 451–479.Google Scholar
Ehrenstein, U. & Gallaire, F. 2005 On two-dimensional temporal modes in spatially evolving open flows. J. Fluid Mech. 536, 209–218.Google Scholar
Ehrenstein, U. & Gallaire, F. 2008 Optimal perturbations and low-frequency oscillations in a separated boundary-layer flow. In Fifth AIAA Theoretical Fluid Mechanics Conference, Seattle. AIAA paper 2008-4232.Google Scholar
Faure, T. M., Adrianos, P., Lusseyran, F. & Pastur, L. 2007 Visualizations of the flow inside an open cavity at medium-range Reynolds numbers. Exp. Fluids 42, 169–184.CrossRefGoogle Scholar
Freund, R. 2003 Model reduction methods based on Krylov subspaces. Acta Numer. 12, 267–319.Google Scholar
Galletti, B., Bottaro, A., Bruneau, C. H. & Iollo, A. 2007 Accurate model reduction of transient and forced wakes. Eur. J. Mech. B 26, 354–366.CrossRefGoogle Scholar
Henningson, D. S. & Åkervik, E. 2008 The use of global modes to understand transition and perform flow control. Phys. Fluids 20, 031302.CrossRefGoogle Scholar
Hœpffner, J., Chevalier, M., Bewley, T. R. & Henningson, D. S. 2005 State estimation in wall-bounded flow systems. Part 1. Laminar flows. J. Fluid Mech. 534, 263–294.CrossRefGoogle Scholar
Högberg, M., Bewley, T. R. & Henningson, D. S. 2003 Linear feedback control and estimation of transition in plane channel flow. J. Fluid Mech. 481, 149–175.Google Scholar
Ilak, M. & Rowley, C. W. 2006 Reduced-order modelling of channel flow using travelling POD and balanced POD. In Third AIAA Flow Control Conference, San Francisco, AIAA paper 2006–3194.Google Scholar
Ilak, M. & Rowley, C. W. 2008 Modelling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids 20, 034103.CrossRefGoogle Scholar
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45.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.Google Scholar
Kim, J. 2003 Control of turbulent boundary layers. Phys. Fluids 15, 1093–1105.CrossRefGoogle Scholar
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383–417.Google Scholar
Lall, S., Marsden, J. E. & Glavaski, S. 2002 A subspace approach to balanced truncation for model reduction of nonlinear control systems. Intl J. Robust Nonlinear Control 12, 519–535.CrossRefGoogle Scholar
Lauga, E. & Bewley, T. R. 2003 The decay of stabilizability with Reynolds number in a linear model of spatially developing flows. Proc. R. Soc. Lond. A 459, 2077–2095.CrossRefGoogle Scholar
Lauga, E. & Bewley, T. R. 2004 Performance of a linear robust control strategy on a nonlinear model of spatially developing flows. J. Fluid Mech. 512, 343–374.CrossRefGoogle Scholar
Lee, K. H., Cortelezzi, L., Kim, J. & Speyer, J. 2001 Application of reduced-order controller to turbulent flows for drag reduction. Phys. Fluids 13, 1321–1330.CrossRefGoogle Scholar
Lehoucq, R. B. & Scott, J. A. 1997 Implicitly restarted Arnoldi methods and subspace iteration. SIAM J. Matrix Anal. Appl. 23, 551–562.Google Scholar
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi methods. SIAM Publishing.Google Scholar
Lumley, J. L. 1970 Stochastic Tools in Turbulence. Academic Press.Google Scholar
Ma, Z., Ahuja, S. & Rowley, C. W. Reduced order models for control of fluids using the eigensystem realization algorithm. Theor. Comput. Fluid Dyn. (submitted)Google Scholar
Marquet, O., Sipp, D., Chomaz, J.-M. & Jacquin, L. 2008 Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech. 605, 429–443.Google Scholar
Moin, P. & Bewley, T. R. 1994 Feedback control of turbulence. Appl. Mech. Rev. 47 (6), S3–S13.Google Scholar
Moore, B. 1981 Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26, 17–32.CrossRefGoogle Scholar
Noack, B., Afanasiev, K., Morzynski, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.CrossRefGoogle Scholar
Noack, B. R. & Eckelmann, H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330.CrossRefGoogle Scholar
Or, A. C. & Speyer, J. L. 2008 Model reduction of input–output dynamical systems by proper orthogonal decomposition. J. Guid. Control Dyn. 31–2, 322–328.Google Scholar
Podvin, B., Fraigneau, Y., Lusseyran, F. & Gougat, P. 2006 A reconstruction method for the flow past an open cavity. ASME J. Fluids Engng 128, 531–540.Google Scholar
Rowley, C. W. 2005 Model reduction for fluids using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15, 997–1013.Google Scholar
Rowley, C. W., Ahuja, S., Taira, K. & Colonius, T. 2008 Closed-loop control of leading edge vorticity on 3d wings: simulations and low-dimensional models. In Thirty-eighth Fluid Dynamics Conference and Exhibit, Seattle. AIAA paper 2008-3981.Google Scholar
Rowley, C. W. & Williams, D. R. 2006 Dynamics and control of high-Reynolds-number flow over open cavities. Annu. Rev. Fluid Mech. 38, 251–276.CrossRefGoogle Scholar
Samimy, M., Debiasi, M., Caraballo, E., Serrani, A., Yuan, X., Little, J. & Myatt, J. 2007 Feedback control of subsonic cavity flows using reduced-order models. J. Fluid Mech. 579, 315–346.Google Scholar
Scherpen, J. M. 1993 Balancing for nonlinear systems. Syst. Control Lett. 21, 143–153.CrossRefGoogle Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.Google Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. Q. Appl. Math. 45, 561–590.CrossRefGoogle Scholar
Tadmor, G., Noack, B., Morzynski, M. & Siegel, S. 2004 Low-dimensional models for feedback flow control. Part II. Control design and dynamical estimation. AIAA Paper 2004-2409.CrossRefGoogle Scholar
Theofilis, V. 2000 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39, 249–315.CrossRefGoogle Scholar
Willcox, K. & Peraire, J. 2002 Balanced model reduction via proper orthogonal decomposition. AIAA J. 40, 2323–2330.CrossRefGoogle Scholar
Zebib, A. 1987 Stability of a viscous flow past a circular cylindar. J. Engng Math. 21, 155–165.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
Zhou, K., Salomon, G. & Wu, E. 2002 Robust and Optimal Control. Prentice Hall.Google Scholar
Zielinska, J. A. & Wesfreid, J. E. 1995 On the spatial structure of global modes in wake flow. Phys. Fluids 7 (6), 1418–1424.Google Scholar

Barbagallo et al. supplementary movie

Movie 1. Linearized Direct Numerical Simulation of the cavity at Re=7500 without control. The initial condition is the most unstable mode. The perturbations, displayed by contour of streamwise velocity, are exponentially amplified.

Download Barbagallo et al. supplementary movie(Video)
Video 8.1 MB

Barbagallo et al. supplementary movie

Movie 2. Linearized Direct Numerical Simulation of the cavity at Re=7500 using the "best-control strategy". The initial condition is the most unstable mode. The perturbations, displayed by contour of streamwise velocity, are amplified for T<2. For T>2, the estimator provides a sufficiently accurate estimated state for the control to be effective.

Download Barbagallo et al. supplementary movie(Video)
Video 9.8 MB