Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-17T13:21:13.741Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear feedback stabilization of point-vortex equilibria near a Kasper wing

Published online by Cambridge University Press:  18 August 2017

R. Nelson Open the ORCID record for R. Nelson [Opens in a new window] ,
B. Protas Open the ORCID record for B. Protas [Opens in a new window]  and
T. Sakajo
R. Nelson*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
B. Protas
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L85 4K1, Canada
T. Sakajo
Affiliation:
Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper concerns feedback stabilization of point-vortex equilibria above an inclined thin plate and a three-plate configuration known as the Kasper wing in the presence of an oncoming uniform flow. The flow is assumed to be potential and is modelled by the two-dimensional incompressible Euler equations. Actuation has the form of blowing and suction localized on the main plate and is represented in terms of a sink–source singularity, whereas measurement of pressure across the plate serves as system output. We focus on point-vortex equilibria forming a one-parameter family with locus approaching the trailing edge of the main plate and show that these equilibria are either unstable or neutrally stable. Using methods of linear control theory we find that the system dynamics linearized around these equilibria is both controllable and observable for almost all actuator and sensor locations. The design of the feedback control is based on the linear–quadratic–Gaussian (LQG) compensator. Computational results demonstrate the effectiveness of this control and the key finding of this study is that Kasper wing configurations are in general not only more controllable than their single-plate counterparts, but also exhibit larger basins of attraction under LQG feedback control. The feedback control is then applied to systems with additional perturbations added to the flow in the form of random fluctuations of the angle of attack and a vorticity shedding mechanism. Another important observation is that, in the presence of these additional perturbations, the control remains robust, provided the system does not deviate too far from its original state. Furthermore, except in a few isolated cases, introducing a vorticity-shedding mechanism enhanced the effectiveness of the control. Physical interpretation is provided for the results of the controllability and observability analysis as well as the response of the feedback control to different perturbations.

JFM classification

Flow Control: Instability control Nonlinear Dynamical Systems: Low-dimensional models Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 827 , 25 September 2017 , pp. 121 - 154
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baker, H. F. 1897 Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press.CrossRefGoogle Scholar
Bewley, T. R. 2001 Flow control: new challenges for a new Renaissance. Prog. Aerosp. Sci. 37, 21–58.CrossRefGoogle Scholar
Bewley, T. R. & Liu, S. 1998 Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 305–349.CrossRefGoogle Scholar
Bunyakin, A. V., Chernyshenko, S. I. & Stepanov, B. Y. 1998 High-Reynolds-number Batchelor-model asymptotics of a flow past an aerofoil with a vortex trapped in a cavity. J. Fluid Mech. 358, 283–297.CrossRefGoogle Scholar
Cortelezzi, L. 1996 Nonlinear feedback control of the wake past a plate with a suction point on the downstream wall. J. Fluid Mech. 327, 303–324.CrossRefGoogle Scholar
Crowdy, D. 2006 Calculating the lift on a finite stack of cylindrical aerofoils. Proc. R. Soc. Lond. A 462, 1387–1407.Google Scholar
Crowdy, D. 2012 Conformal slit maps in applied mathematics. ANZIAM J. 53, 171–189.Google Scholar
Crowdy, D. & Marshall, J. 2005 Analytical formulae for the Kirchhoff–Routhpath function in multiply connected domains. Proc. R. Soc. Lond. A 461, 2477–2501.Google Scholar
Crowdy, D. & Marshall, J. S. 2007 Computing the Schottky–Klein prime function on the Schottky double of planar domains. Comput. Meth. Funct. Theor. 7, 293–308.CrossRefGoogle Scholar
Donelli, R., Iannelli, P., Chernyshenko, S., Iollo, A. & Zannetti, L. 2009 Flow models for a vortex cell. AIAA 47, 451–467.CrossRefGoogle Scholar
Feng, L.-H., Choi, K.-S. & Wang, J.-J. 2015 Flow control over an airfoil using virtual Gurney flaps. J. Fluid Mech. 767, 595–626.CrossRefGoogle Scholar
Gallizio, F., Iollo, A., Protas, B. & Zannetti, L. 2010 On continuation of inviscid vortex patches. Physica D 239, 190–201.Google Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.CrossRefGoogle Scholar
Huang, M. K. & Chow, C. Y. 1982 Trapping of a free vortex by Joukowski airfoils. AIAA 20, 292–298.CrossRefGoogle Scholar
Kasper, W.1974 Aircraft wing with vortex generation. US Patent 3 831 885.Google Scholar
Kruppa, E.1977 A wind tunnel investigation of the Kapser vortex concept. In AIAA 13th Annual Meeting and Technical Display Incorporating the Forum on the Future of Air Transportation. American Institute of Aeronautics and Astronautics. AIAA Paper 1977-310.Google Scholar
Mathworks, The2014 MATLAB and Control System Toolbox Release 2014b.Google Scholar
Nelson, R. & Sakajo, T. 2014 Trapped vortices in multiply connected domains. Fluid Dyn. Res. 46, 061402.CrossRefGoogle Scholar
Nelson, R. & Sakajo, T.2016 Robustness of point vortex equilibria in the vicinity of a Kasper Wing. In preparation for submission.Google Scholar
Newton, P. K. 2001 The N–Vortex Problem. Analytical Techniques. Springer.CrossRefGoogle Scholar
Newton, P. K. & Chamoun, G. 2007 Construction of point vortex equilibria via Brownian ratchets. Proc. R. Soc. Lond. A 463, 1525–1540.Google Scholar
Protas, B. 2004 Linear feedback stabilization of laminar vortex shedding based on a point vortex model. Phys. Fluids 16, 4473.CrossRefGoogle Scholar
Protas, B. 2007 Center manifold analysis of a point-vortex model of vortex shedding with control. Physica D 228, 179–187.Google Scholar
Protas, B. 2008 Vortex dynamics models in flow control problems. Nonlinearity 21, R203–R250.CrossRefGoogle Scholar
Ramesh, K., Gopalarathnam, A., Granlund, K., Ol, M. V. & Edwards, J. R. 2014 Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding. J. Fluid Mech. 751, 500–538.CrossRefGoogle Scholar
Rival, D. E., Kriegseis, J., Schaub, P., Widmann, A. & Tropea, C. 2014 Characteristic length scales for vortex detachment on plunging profiles with varying leading-edge geometry. Exp. Fluids 55 (1), 1660.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Saffman, P. G. & Sheffield, J. S. 1977 Flow over a wing with an attached free vortex. Stud. Appl. Maths 57, 107–117.CrossRefGoogle Scholar
Smith, D. R., Amitay, M., Kibens, V., Parekh, D. E. & Glezer, A.1998 Modification of lifting body aerodynamics using synthetic jet actuators. AIAA Paper 98-0209.CrossRefGoogle Scholar
Stengel, R. F. 1994 Optimal Control and Estimation. Dover.Google Scholar
Storms, B. L. & Jang, C. S. 1994 Lift enhancement of an airfoil using a Gurney flap and vortex generators. J. Aircraft 31, 542–547.CrossRefGoogle Scholar
Strykowski, P. J. & Sreenivasan, K. R. 1990 On the formation and suppression of vortex shedding at low Reynolds numbers. J. Fluid Mech. 218, 71–107.CrossRefGoogle Scholar
Xia, X. & Mohseni, K.2012 Lift evaluation of a 2d flapping flat plate. Preprint, arXiv:1205.6853.Google Scholar
Zannetti, L. & Iollo, A. 2003 Passive control of the vortex wake past a flat plate at incidence. Theor. Comput. Fluid Dyn. 16, 211–230.CrossRefGoogle Scholar