In this paper we investigate the control of flow problems where the control objective is to reduce the oscillation amplitude while keeping the frequency of oscillation between predefined limits at all times. The governing equations are simplified to obtain the oscillatory mode dynamics, after which the conditions that the control parameters must satisfy in order to achieve the desired objective are derived in detail. The results obtained are illustrated on a physical application example, namely cavity flow control, where it is seen that the controller is successful in achieving the control goal.
Aamo, O.M., Krstic, M., and Bewley, T.R., 2003, "Control of mixing by boundary feedback in 2d channel flow," Automatica39(9), 1597-1606.
2.
Amitay, M., Smith, D.R., Kibens, V., Parekh, D.E., and Glezer, A., 2001, "Aerodynamic flow control over an unconventional airfoil using synthetic jet actuators," AIAA Journal39(3), 361-370.
3.
Banaszuk, A., Ariyur, K.B., Krstic, M., and Jacobson, C.A., 2004, "An adaptive algorithm for control of combustion instability ," Automatica40(11), 1965-1972.
4.
Baramov, L., Tutty, O.R., and Rogers, E., 2004, "h∞ control of non-periodic two-dimensional channel flow," IEEE Transactions on Control Systems Technology12(1), 111-122.
5.
Bewley, T.R., 2001, "Flow control: new challenges for a new Renaissance," Progress in Aerospace Sciences37(1), 21-58.
6.
Camphouse, R.C., 2005, "Boundary feedback control using proper orthogonal decomposition models," Journal of Guidance, Control, and Dynamics28, 931-938.
7.
Caraballo, E., Little, J., Debiasi, M., and Samimy, M., 2007, "Development and implementation of an experimental based reduced-order model for feedback control of subsonic cavity flows," Journal of Fluids Engineering129, 813-824.
8.
Cohen, K., Siegel, S., McLaughlin, T., Gillies, E., and Myatt, J., 2005, "Closed-loop approaches to control of a wake flow modeled by the Ginzburg-Landau equation," Computers and Fluids34(8), 927-949.
9.
Cortelezzi, L., Speyer, J.L., Lee, K.H., and Kim, J., 1998, "Robust reduced-order control of turbulent channel flows via distributed sensors and actuators," in Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, 1998.
10.
Efe, M.O. and Ozbay, H., 2004, "Low dimensional modelling and Dirichlet boundary controller design for Burgers equation," International Journal of Control77(10), 895-906.
11.
Fitzpatrick, K., Feng, Y., Lind, R., Kurdila, A.J., and Mikolaitis, D.W. , 2005, "Flow control in a driven cavity incorporating excitation phase differential," Journal of Guidance, Control, and Dynamics28(1), 63-70.
12.
Gad-el Hak, M., 2000, Flow Control - Passive, Active, and Reactive Flow Management , Cambridge University Press, New York.
13.
Graham, W.R., Peraire, J., and Tang, K.Y., 1999, "Optimal control of vortex shedding using low-order models I - open-loop model development," International Journal for Numerical Methods in Engineering44, 945-972.
14.
Gilarranz, J.L., Traub, L.W., and Rediniotis, O.K. , 2005, "A new class of synthetic jet actuators: Part II: Application to flow separation control," Journal of Fluids Engineering127(2), 377-387.
15.
Hinze, M. and Kunisch, K., 2004, "Second order methods for boundary control of the instationary Navier-Stokes system," Zeitschrift fur Angewandte Mathematik und Mechanik84(3), 171-187.
16.
Hogberg, M., Bewley, T.R., and Henningson, D.S. , 2001, "Linear feedback control and estimation of transition in plane channel flow," Journal of Fluid Mechanics481, 149-175.
17.
Holmes, P., Lumley, J.L., and Berkooz, G., 1996, Turbulence, Coherent Structures, Dynamical Systems, and Symmetry, Cambridge University Press, Cambridge.
18.
Jordan, D.W., and Smith, P., 1999, Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems. Oxford University Press, Oxford.
Kasnakoglu, C. and Serrani, A., 2007, "Attenuation of oscillations in Galerkin systems using center-manifold techniques," European Journal of Control13(5), 529.
21.
Kasnakoglu, C., Serrani, A., and Efe, M.O., 2008, "Control input separation by actuation mode expansion for flow control problems," International Journal of Control81(9), 1475-1492.
22.
Kim, J., 2003, "Control of turbulent boundary layers," Physics of Fluids15, 1093.
23.
Kobayashi, T. and Oya, M., 2003, "Nonlinear boundary control of coupled Burgers’ equations," Control and Cybernetics32(2), 245-258.
24.
Krstic, M., 1999, "On global stabilization of Burgers’ equation by boundary control," Systems and Control Letters37(3), 123-141.
25.
Landau, L.D. and Lifshitz, E.M., 1987, Course of Theoretical Physics, vol. 6: Fluid Mechanics. Pergamon Press, Oxford.
26.
Noack, B.R., Afanasiev, K., Morzynski, M., Tadmor, G., and Thiele, F., 2003, "A hierarchy of low-dimensional models for the transient and post-transient cylinder wake," Journal of Fluid Mechanics497, 335-363.
27.
Noack, B.R. and Eckelmann, H., 1994, "A global stability analysis of the steady and periodic cylinder wake," Journal of Fluid Mechanics270, 297-330.
28.
Noack, B.R., Papas, P., and Monketwitz, P.A. , 2005, "The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows ," Journal of Fluid Mechanics523, 339-365.
29.
Noack, B.R., Tadmor, G., and Morzynski, M., 2004, "Low-dimensional models for feedback flow control. Part I: Empirical Galerkin models," in Proceedings of the 2nd AIAA Flow Control Conference, Portland, OR. AIAA Paper 2004-2408.
30.
Ocvirk, G., Munroe, M., Tang, T., Oleschuk, R., Westra, K., and Harrison, D.J., 2000, "Electrokinetic control of fluid flow in native poly (dimethylsiloxane) capillary electrophoresis devices," Electrophoresis21, 107- 115.
31.
Park, H.M. and Lee, M.W., 2000, "Boundary control of the Navier-Stokes equation by empirical reduction of modes," Computer Methods in Applied Mechanics and Engineering188(1-3), 165-186.
32.
Rowley, C.W., Colonius, T., and Murray, R.M., 2004, "Model reduction for compressible flows using POD and Galerkin projection," Physica D189(1-2), 115-129.
33.
Rowley, C.W. and Juttijudata, V., 2005, "Model-based control and estimation of cavity flow oscillations ," in Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain, 2005.
34.
Rowley, C.W. and Marsden, J.E., 2000, "Reconstruction equations and the Karhunen-Loeve expansion for systems with symmetry," Physica D142(1-2), 1-19.
35.
Samimy, M., Debiasi, M., Caraballo, E., Serrani, A., Yuan, X., Little, J., and Myatt, J.H., 2007, "Feedback control of subsonic cavity flows using reduced-order models," Journal of Fluid Mechanics579, 315-346.
36.
Seiler, K., Fan, Z.H., Fluri, K., and Harrison, D.J., 1994, "Electroosmotic pumping and valveless control of fluid flow within a manifold of capillaries on a glass chip," Analytical Chemistry66(20), 3485-3491.
37.
Singh, S.N., Myatt, J.H., Addington, G.A., Banda, S., and Hall, J.K., 2001, "Optimal feedback control of vortex shedding using proper orthogonal decomposition models," ASME Journal of Fluids Engineering123(3), 612-618.
38.
Sirovich, L., 1987, "Turbulence and the dynamics of coherent structures," Quarterly Journal of Applied Mathematics45(3), 561-590.
39.
Smaoui, N., 2005, Boundary and distributed control of the viscous Burgers equation. Journal of Computational and Applied Mathematics182(1), 91-104.
40.
Stuart, J.T., 2006, "On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The basic behaviour in plane Poiseuille flow", Journal of Fluid Mechanics Digital Archive9(3), 353-370.
41.
Tadmor, G., Noack, B.R., Dillmann, A., Gerhard, J., Pastoor, M., King, R., and Morzynski, M.2003, "Control, observation and energy regulation of wake flow instabilities", in Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HI, 2003.
42.
Tadmor, G., Noack, B.R., Morzynski, M., and Siegel, S., 2004, "Low-dimensional models for feedback flow control. Part II: Controller design and dynamic estimation," in Proceedings of the 2nd AIAA Flow Control Conference, Portland, OR, 2004. AIAA Paper 2004-2409.
43.
Wu, J., Lu, X., Denny, A.G., Fan, M., and Wu, J.M., 1998, "Post-stall flow control on an airfoil by local unsteady forcing," Journal of Fluid Mechanics371, 21-58.
