Skip to main content Accessibility help
×
Home

Development and Application of a Reduced Order Model for the Control of Self-Sustained Instabilities in Cavity Flows

  • Kaushik Kumar Nagarajan (a1), Laurent Cordier (a2) and Christophe Airiau (a3)

Abstract

Flow around a cavity is characterized by a self-sustained mechanism in which the shear layer impinges on the downstream edge of the cavity resulting in a feedback mechanism. Direct Numerical Simulations of the flow at low Reynolds number has been carried out to get pressure and velocity fluctuations, for the case of un-actuated and multi frequency actuation. A Reduced Order Model for the isentropic compressible equations based on the method of Proper Orthogonal Decomposition has been constructed. The model has been extended to include the effect of control. The Reduced Order dynamical system shows a divergence in time integration. A method of calibration based on the minimization of a linear functional of error, to the sensitivity of the modes, is proposed. The calibrated low order model is used to design a feedback control of cavity flows based on an observer design. For the experimental implementation of the controller, a state estimate based on the observed pressure measurements is obtained through a linear stochastic estimation. Finally the obtained control is introduced into the Direct Numerical Simulation to obtain a decrease in spectra of the cavity acoustic mode.

Copyright

Corresponding author

References

Hide All
[1]Rowley, C. W., Colonius, T., Basu, A. J., On self-sustained oscillations in two-dimensional compressible flow over rectangular cavities, J. Fluid Mech. 455 (2002) 315346.
[2]Gloerfelt, X., Bailly, C., Juveé, D., Direct computation of the noise radiated by a subsonic cavity flow and application of integral methods, J. Sound Vib. 266 (1) (2003) 119146.
[3]Bres, G. A., Colonius, T., Three-dimensional instabilities in compressible flow over open cavities, J. Fluid Mech. 599 (2008) 309339.
[4]Larcheveque, L., Sagaut, P., Labbe, O., Large-eddy simulation of a subsonic cavity flow including asymmetric three-dimensional effects, J. Fluid Mech. 577 (2007) 105126.
[5]Rowley, C. W., Williams, D. R., Dynamics and control of high-reynolds number flow over cavities, Annual Review of Fluid Mechanics 38 (2006) 251276.
[6]Bergmann, M., Cordier, L., Brancher, J.-P., Optimal rotary control of the cylinder wake using POD Reduced Order Model, Phys. Fluids 17 (9) (2005) 097101:121.
[7]Luchtenburg, D. M., Guenther, B., Noack, B. R., King, R., Tadmor, G., A generalized mean-field model of the natural and high-frequency actuated flow around a high-lift configuration, J. Fluid Mech. 623 (2009) 283316.
[8]Rowley, C. W., Colonius, T., Murray, R. M., Model reduction for compressible flows using POD and Galerkin projection, Physica D. Nonlinear Phenomena 189 (1-2) (2004) 115129.
[9]Gloerfelt, X., Compressible Proper Orthogonal Decomposition/Galerkin reduced order model of self sustained oscillations in a cavity, Phys. Fluids 20 (2008) 115105.
[10]Weller, J., Lombardi, E., Iollo, A., Robust model identification of actuated vortex wakes, Physica D: Nonlinear Phenomena 238 (2009) 416427.
[11]Kasnakoglu, C., Reduced order modeling, nonlinear analysis and control methods for flow control problems, Ph.D. thesis, Ohio State University (2007).
[12]Samimy, M., Debiasi, M., Caraballo, E., Serrani, A., Yuan, X., Little, J., Myatt, J., Feedback Control of Subsonic Cavity Flows Using Reduced-order Models, J. Fluid Mech. 579 (2007) 315346.
[13]Galletti, B., Bruneau, C.-H., Zannetti, L., Iollo, A., Low-order modelling of laminar flow regimes past a confined square cylinder, J. Fluid Mech. 503 (2004) 161170.
[14]Couplet, M., Basdevant, C., Sagaut, P., Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling J. Comp. Phys. 207 (2005) 192220.
[15]Perret, L., Collin, E., Delville, J., Polynomial identification of POD based low-order dynamical system, Journal of Turbulence 7 (2006) 115.
[16]Kalb, V. L., Deane, A. E., An intrinsic stabilization scheme for proper orthogonal decomposition based low-dimensional models, Phys. Fluids 19 (2007) 054106.
[17]Cordier, L., El Majd, B. Abou, Favier, J., Calibration of POD Reduced-Order models using Tikhonov regularization, Int. J. Numer. Meth. Fluids 63 (2) (2009) 269296.
[18]Gugercin, S., Antoulas, A. C., A survey of model reduction by balanced truncation and some new results, International Journal of Control 77 (8) (2004) 748766.
[19]Moore, B., Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Transactions on Automatic Control 26 (1981) 1732.
[20]Rowley, C. W., Model reduction for fluids using balanced proper orthogonal decomposition, International Journal of Bifurcation and Chaos 15 (3) (2005) 9971013.
[21]Barbagallo, A., Sipp, D., Schmid, P., Closed-loop control of an open cavity flow using reduced order models, J. Fluid Mech. 641 (2009) 150.
[22]Sirovich, L., Turbulence and the dynamics of coherent structures, Quarterlyof Applied Mathematics XLV (3) (1987) 561590.
[23]Holmes, P., Lumley, J. L., Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge U. K., , 1996.
[24]Cordier, L., Bergmann, M., Proper Orthogonal Decomposition: an overview., in: Lecture series 2002-04 on the post-processing of experimental and numerical data., Von Karman Institut for Fluid Dynamics., 2002.
[25]Caraballo, E., Kasnakoglu, C., Serrani, A., Samimy, M., Control input separation methods for reduced-order model-based feedback flow control, AIAA Journal 46 (9) (2008) 23062322.
[26]Rempfer, D., On low-dimensional Galerkin models for fluid flow, Theor. Comput. Fluid Dyn. 14 (2000) 7588.
[27]Noack, B. R., Papas, P., Monkewitz, P. A., The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows, J. Fluid Mech. 523 (2005) 339365.
[28]Iollo, A., Lanteri, S., Desideri, J. A., Stability properties of POD-Galerkin approximations for the compressible Navier-Stokes equations, Tech. Rep. 3589, INRIA (1998).
[29]Delprat, N., Rossiter’s formula: A simple spectral model for a complex amplitude modulation process?, Phys. Fluids 18 (2006) 071703.
[30]Jordan, P., Tinney, C., Stalnov, O., Schlegel, M., Noack, B. R., Identifying noisy and quiet modes in a jet, in: AIAA Paper 20073702, 2007.
[31]Sirisup, S., Karniadakis, G. E., A spectral viscosity method for correcting the long-term behavior of POD model J. Comp. Phys. 194 (2004) 92116.
[32]Graham, W. R., Peraire, J., Tang, K. T., Optimal Control of Vortex Shedding Using Low Order Models. Part 1. Open-Loop Model Development, Int. J. for Numer. Meth. in Engrg. 44 (7) (1999) 945972.
[33]Graham, W. R., Peraire, J., Tang, K. T., Optimal Control of Vortex Shedding Using Low Order Models. Part 2: Model-based control, Int. J. for Numer. Meth. in Engrg. 44 (7) (1999) 973990.
[34]Fahl, M., Trust-region methods for flow control based on Reduced Order Modeling, Ph.D. thesis, Trier University (2000).
[35]Bergmann, M., Cordier, L., Optimal control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models J. Comp. Phys. 227 (2008) 78137840.
[36]Adrian, R. J., Moin, P., Stochastic estimation of organized turbulent structure: homogeneous shear flow, J. Fluid Mech. 190 (1988) 531 559.
[37]Bonnet, J.-P., Cole, D., Delville, J., Glauser, M., Ukeiley, L., Stochastic estimation and proper orthogonal decomposition: Complementary techniques for identifying structures, Exp. Fluids 17 (1994) 307314.
[38]Buffoni, M., Camarri, S., Iollo, E., Lombardi, E., V, S. M., A non-linear observer for unsteady three-dimensional flows J. Comp. Phys. 227 (4) (2008) 26262643.
[39]Bewley, T. R., Liu, S., Optimal and robust control and estimation of linear paths to transition, J. Fluid Mech. 365 (1998) 305349.
[40]Bagheri, S., Hoepffner, J., Schmid, P. J., Henningson, D. S., Input-output analysis and control design applied to a linear model of spatially developing flows, App. Mech. Rev. 62 (2009) 127.
[41]Nagarajan, K. K., Analysis and control of self-sustained instabilities in a cavity using reduced order modelling, Ph.D. thesis, Institut National Polytechnique de Toulouse (2010).

Keywords

Related content

Powered by UNSILO

Development and Application of a Reduced Order Model for the Control of Self-Sustained Instabilities in Cavity Flows

  • Kaushik Kumar Nagarajan (a1), Laurent Cordier (a2) and Christophe Airiau (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.