This paper treats the question of control of two-dimensional incompressible, unsteady wake flow behind a circular cylinder at Reynolds number Re=100. Two finite-dimensional lower order models based on proper orthogonal decomposition (POD) are considered for the control system design. Control action is achieved via cylinder rotation. Linear optimal control theory is used for obtaining stabilizing feedback control systems. An expression for the region of stability of the system is derived. Simulation results for 18-mode POD models obtained using the control function and penalty methods are presented. These results show that in the closed-loop system mode amplitudes asymptotically converge to the chosen equilibrium state for each flow model for large perturbations in the initial states.
Issue Section:
Technical Papers
1.
Hou
, L. S.
, and Ravindran
, S. S.
, 1997
, “A Penalized Neuman Control Approach for Solving an Optimal Dirichlet Control Problem for Navier-Stokes Equations
,” SIAM J. Control Optim.
, 36
, pp. 1795
–1814
.2.
Ito
, K.
, and Ravindran
, S. S.
, 1998
, “A Reduced Order Method for Simulation and Control of Fluid Flows
,” J. Comput. Phys.
, 143
, pp. 403
–425
.3.
Burns, J. A., and Ou, Y. R., 1994, “Feedback Control of the Driven Cavity Problem Using LQR Design,” Proc. 33rd IEEE Conf. on Decision and Control, Florida, pp. 289-284.
4.
Joslin
, R. D.
, Gunzburger
, M. D.
, Nicolaides
, R.
, Erlebacher
, G.
, and Hussaini
, M. Y.
, 1997
, “A Self-Contained, Automated Methodology for Optimal Flow Control Validated for Transition Delay
,” AIAA J.
, 35
, pp. 816
–824
.5.
Sritharan, S. S., 1992, “An Optimal Control Problem in Exterior Hydrodynamics,” Proceedings of the Royal Society of Edinburgh, Vol. 121A, pp. 5–32.
6.
Seifert
, A.
, Darabi
, A.
, and Wyganski
, I.
, 1996
, “Delay of Airfoil Stall by Periodic Excitation
,” J. Aircr.
, 33
, No. 4
No. 4
.7.
Modi
, V. J.
, Mokhtarian
, F.
, Fernando
, M.
, and Yokomizo
, T.
, 1991
, “Moving Surface Boundary-Layer Control as Applied to Two-Dimensional Airfoils
,” J. Aircr.
, 28
, pp. 104
–112
.8.
Roussopoulos
, K.
, 1993
, “Feedback Control of Vortex Shedding at Low Reynold’s Numbers
,” J. Fluid Mech.
, 248
, pp. 267
–296
.9.
Maddlon, D. V., Collier, F. S., Montoya, F. S., and Land, C. K., 1989, “Transition Flight Experiments on a Swept Wing with Suction,” AIAA Paper 89-1893.
10.
Smith, B. L., and Glezer, A., 1997, “Vectoring and Small Scale Motions Effected in Free Shear Flows Using Synthetic Jet Actuators,” AIAA Paper 97-0213.
11.
Rediniotis, O. K., Ko, J., Yue, X., and Kurdila, A. J., 1999, “Synthetic Jets, their Reduced Order Modeling and Applications to Flow Control,” AIAA Paper 99-1000.
12.
Karhunen, K., 1946, “Zur Spectral Theorie Stochasticher Prozesse,” Annales Academiae Scientiarum Fennicae, Mthematica-Physica, 37.
13.
Loeve, M., 1945, “Functiona Aleatoire de Second Ordre,” Comptes Rendus Academie des Sciences, Paris.
14.
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
.15.
Berkooz
, G.
, Holmes
, P.
, and Lumley
, J. L.
, 1993
, “The Proper Orthogonal Decomposition in the Analysis of Turbulent Channel Flow
,” Annu. Rev. Fluid Mech.
, 25
, No. 5
, pp. 539
–575
.16.
Rajaee
, M.
, Karlson
, S. K.
, and Sirovich
, L.
, 1994
, “Low Dimensional Description of Free Shear Flow Coherent Structures and their Dynamic Behavior
,” J. Fluid Mech.
, 258
, pp. 1401
–1402
.17.
Rempfer
, D.
, 1996
, “Investigations of Boundary Layer Transition via Galerkin Projections on Empirical Eigen Functions
,” Phys. Fluids
, 8
, No. 1
, pp. 175
–188
.18.
Ravindran, S. S., 1999, “Proper Orthogonal Decomposition in Optimal Control of Fluids,” NASA/TM-1999-209113.
19.
Graham
, W. R.
, Peraire
, J.
, and Tang
, K. Y.
, 1999
, “Optimal Control of Vortex Shedding Using Low Order Models, Part I: Open-loop Model Development
,” Int. J. Numer. Methods Eng.
, 44
, No. 7
, pp. 945
–972
.20.
Graham
, W. R.
, Peraire
, J.
, and Tang
, K. Y.
, 1999
, “Optimal Control of Vortex Shedding Using Low Order Models, Part II: Model Based Control
,” Int. J. Numer. Methods Eng.
, 44
, No. 7
, pp. 9973
–990
.21.
Donea
, J.
, Giuliani
, S.
, Laval
, H.
, and Quartapelle
, L.
, 1982
, “Finite Element Solution of the Unsteady Navier-Stokes Equations by a Fractional Step Method
,” Comput. Methods Appl. Mech. Eng.
, 30. pp.
53
–73
.22.
Zienkiewicz
, O. C.
, Szmelter
, J.
, and Peraire
, J.
, 1990
, “Compressible and Incompressible Flow; An Algorithm for all Seasons
,” Comput. Methods Appl. Mech. Eng.
, 78
, pp. 105
–121
.23.
Chen, C-T., 1999, Linear Systems Theory and Design, Oxford University Press, New York, pp. 151-152.
24.
Athans, M., and Falb, P., 1966, Optimal Control, McGraw-Hill, New York.
25.
Vidyasagar, M., 1993, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cliffs, NJ.
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