Abstract
In the current study, a hierarchy of control-oriented Galerkin models is proposed targeting least-dimensional representations at different operating conditions. These models are employed for passive as well as active actuation. In passive control, a linearised model is shown to reproduce a wake stabilization experiment of Strykowski & Sreenivasan (1990). In particular, the effect and optimal position of control wires are accurately predicted. In active closed-loop control, focus is also placed on experiment. Here, control design requires a model which has on the hand a sufficiently broad dynamic range and is on the other hand low-dimensional enough for online computation. POD Galerkin models have a desirable mathematical structure and dimension for an online capable control design but tend to be over-optimised for the reference conditions. The resulting limited dynamic bandwidth is associated with the underlying expansion modes which change their shape at different operating conditions. To increase that model bandwidth, a novel continuous mode interpolation technique is proposed. The mode interpolation smoothly connects not only different operating conditions, but also stability and POD modes and even flows at different boundary conditions. In addition, the extrapolation of modes outside the design conditions is illustrated. The interpolated modes enable ‘least-order’ Galerkin models keeping the dimension from a single operating condition but resolving several states. These models are well suited for control design. The mode interpolation technique is demonstrated for three benchmark problems, the flow around circular cylinder, a NACA airfoil and an Ahmed body.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Gad-el-Hak, M.: Flow Control: Passive, Active and Reactive Flow Management. Cambridge University Press (2000)
Fiedler, H. & Fernholz, H.H.: On the management and control of turbulent shear flows. Progr. Aeronaut. Sci. 27 (1990) 305–387
Becker, R., Garwon, M., Gutknecht, C., Bärwolff, G., King, R.: Regelung aerodynamischer Strömungen am Beispiel einer rückwärtsgewandten Stufe (transl.: Control of aerodynamics flows examplified for the backward-facing step). Automatisierungstechnik 50 (2002) 79–86
Gerhard, J., Pastoor, M., King, R., Noack, B.R., Dillmann, A., Morzyński, M., Tadmor, G.: Model-based control of vortex shedding using low-dimensional Galerkin models. AIAA-Paper 2003-4262, 33rd AIAA Fluids Conference and Exhibit, Orlando, Florida, USA (2003)
Rowley, C., Juttijudata, V.: Control and estimation of oscillating cavity flows. In 57th Annual Meeting of the Division of Fluid Dynamics of the American Physical Society. (2004) (Bulletin Am. Phys. Soc., Vol. 49, No. 9, AG.005)
Acton, F.: Numerical Methods that Work. Harper & Row, Publishers, New York, Evanston and London (1970)
Noack, B.R., Afanasiev, K., Morzyński, M., Tadmor, G., Thiele, F.: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497 (2003) 335–363
Luchtenburg, D., Tadmor, G., Lehmann, O., Noack, B.R., King, R., Morzyński, M.: Tuned POD Galerkin models for transient feedback regulation of the cylinder wake. AIAA-Paper 2006-1407, 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, USA (2006)
Stuart, J.: On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4 (1958) 1–21
Stuart, J.: Nonlinear stability theory. Ann. Rev. Fluid Mech. 3 (1971) 347–370
Holmes, P., Lumley, J., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1998)
Ladyzhenskaya, O.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, London (1963)
Fletcher, C.: Computational Galerkin Methods. Springer, New York (1984)
Siegel, S., Cohen, K., Seigel, J., McLaughlin, T.: Proper orthogonal decomposition snapshot selection for state estimation of feedback controlled flows. AIAA-Paper 2006-1400, AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, USA (2006)
Noack, B.R., Eckelmann, H.: A low-dimensional Galerkin method for the three-dimensional flow around a circular cylinder. Phys. Fluids 6 (1994) 124–143
Noack, B.R., Eckelmann, H.: A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270 (1994) 297–330
Noack, B.R., Papas, P., Monkewitz, P.: The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523 (2005) 339–365
King, R., Seibold, M., Lehmann, O., Noack, B.R., Morzyński, M., Tadmor, G.: Nonlinear flow control based on a low dimensional model of fluid flow. In Meurer, T., Graichen, K., Gilles, E., eds.: Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems, LNCIS 322. Springer (2005) 369–386
Noack, B.R., Tadmor, G., Morzyński, M.: Low-dimensional models for feedback flow control. Part I: Empirical Galerkin models. AIAA-Paper 2004-2408 (invited contribution), 2nd AIAA Flow Control Conference, Portland, Oregon, USA (2004)
Tadmor, G., Noack, B.R., Morzyński, M., Siegel, S.: Low-dimensional models for feedback flow control. Part II: Controller design and dynamic estimation. AIAA-Paper 2004-2409 (invited contribution), 2nd AIAA Flow Control Conference, Portland, Oregon, USA (2004)
Busse, F.: Numerical analysis of secondary and tertiary states of fluid flow and their stability properties. Appl. Sci. Res. 48 (1991) 341–351
Boberg, L., Brosa, U.: Onset of turbulence in a pipe. Z. Naturforsch. 43a (1988) 697–726
Rummler, B.: Zur Lösung der instationären inkompressiblen Navier-Stokesschen Gleichungen in speziellen Gebieten (transl.: On the solution of the incompressible Navier-Stokes equations in some domains). Habilitationsschrift, Fakultät für Mathematik, Ottovon-Guericke-Universität Magdeburg (2000)
Bewley, T., Liu, S.: Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365 (1998) 305–349
Dušek, J., Le Gal, P., Fraunie, P.: A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake. J. Fluid Mech. 264 (1994) 59–80
Zielinska, B., Wesfreid, J.: On the spatial structure of global modes in wake flow. Phys. Fluids 7 (1995) 1418–1424
Noack, B.R., Tadmor, G., Morzyński, M.: Actuation models and dissipative control in empirical Galerkin models of fluid flows. In The 2004 American Control Conference, Boston, MA, USA (2004) Paper FrP15.6 0001-0006
Rediniotis, O., Ko, J., Kurdila, A.: Reduced order nonlinear Navier-Stokes models for synthetic jets. J. Fluids Enrng. 124 (2002) 433–443
Deane, A., Kevrekidis, I., Karniadakis, G., Orszag, S.: Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinders. Phys. Fluids A 3 (1991) 2337–2354
Khibnik, A., Narayanan, S., Jacobson, C., Lust, K.: Analysis of low dimensional dynamics of flow separation. In Continuation Methods in Fluid Dynamics. Notes on Numerical Fluid Mechanics, vol. 74, Vieweg (2000) 167–178 Proceedings of ERCOFTAC and Euromech Colloquium 383, Aussois, France (1998)
Ma, X., Karniadakis, G.: A low-dimensional model for simulating three-dimensional cylinder flow. J. Fluid Mech. 458 (2002) 181–190
Jørgensen, B., Sørensen, J., Brøns, M.: Low-dimensional modeling of a driven cavity flow with two free parameters. Theoret. Comput. Fluid Dynamics 16 (2003) 299–317
Bergmann, M., Cordier, L., Brancher, J.P.: Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced order model. Phys. Fluids 17 (2005) 097101–097121
Morzyński, M., Afanasiev, K., Thiele, F.: Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis. Comput. Meth. Appl. Mech. Enrgrg. 169 (1999) 161–176
Zebib, A.: Stability of viscous flow past a circular cylinder. J. Engr. Math. 21 (1987) 155–165
Unal, M., Rockwell, D.: On the vortex formation from a cylinder; Part2. Control by a splitter-plate interference. J. Fluid Mech. 190 (1987) 513–529
Mittal, S.: Effect of a “slip” splitter plate on vortex shedding from a cylinder. Phys. Fluids 9 (2003) 817–820
Strykowski, P., Sreenivasan, K.: On the formation and suppression of vortex’ shedding’ at low Reynolds numbers. J. Fluid Mech. 218 (1990) 71–107
Lehmann, O., Luchtenburg, M., Noack, B.R., King, R., Morzyński, M., Tadmor, G.: Wake stabilization using POD Galerkin models with interpolated modes. In 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005, Seville, Spain (2005) Invited Paper 1608
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Morzyński, M., Stankiewicz, W., Noack, B.R., King, R., Thiele, F., Tadmor, G. (2007). Continuous Mode Interpolation for Control-Oriented Models of Fluid Flow. In: King, R. (eds) Active Flow Control. Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), vol 95. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71439-2_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-71439-2_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-71438-5
Online ISBN: 978-3-540-71439-2
eBook Packages: EngineeringEngineering (R0)