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Vortex-based Control Algorithms

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Control of Fluid Flow

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 330))

Abstract

In high Reynolds number flows, interactions between coherent vortical structures are key to understanding dynamics. In this paper we review methods of control that rely on this observation. Control of vortex dynamics is pursued using a variety of reduced-order representations of the dynamics, such as point vortices, vortex blobs and vortex patches. Control algorithms designed based on such representations are applied in many contexts of which we review a few of the most common ones, such as wake vorticity control and recirculation zone vortex control. We also review some of our own work in the area of vortex control concentrated on two-vortex merging interactions. The methods used are those of averaging in the case of limited control authority and the nonlinear control method of flat coordinates.

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References

  1. O.M. Aamo, M. Krstiç, and T.R. Bewley. Control of mixing by boundary feedback in 2d channel flow. Automatica, 39:1597–1606, 2003.

    Article  MATH  Google Scholar 

  2. F. Abergel and R. Temam. On some control problems in fluid mechanics. Theoretical and Computational Problems in Fluid Dynamics, 1:303–325, 1990.

    Article  MATH  Google Scholar 

  3. S. Acharya, T.A. Myrum, and S. Inamdar. Subharmonic excitation of the shearlayer between 2 ribs — vortex interaction and pressure field. AIAA Journal, 29:1390–1399, 1991.

    Article  Google Scholar 

  4. K. Afanasiev and M. Hinze. Adaptive control of a wake flow using Proper Orthogonal Decomposition. Technische Universitat Berlin Preprint, 1999.

    Google Scholar 

  5. C.R. Anderson, Y.C. Chen, and J.S. Gibson. Control and identification of vortex wakes. Journal of Dynamic Systems Measurement and Control-Transactions of the ASME, 122:298–305, 2000.

    Article  Google Scholar 

  6. J.M. Anderson, K. Streitlien, D.S. Barrett, and M.S. Triantafyllou. Oscillating foils of high propulsive efficiency. Journal of Fluid Mechanics, 360:41–72, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  7. I. Aranson, H. Levine, and L. Tsimring. Controlling spatiotemporal chaos. Physical Review Lett., 72:2561–2564, 1994.

    Article  Google Scholar 

  8. H. Aref. Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Annual Review of Fluid Mechanics, 15:345–389, 1983.

    Article  MathSciNet  Google Scholar 

  9. V.I. Arnold. Mathematical Methods of Classical Mechanics. Springer-Verlag, New York, 1978.

    MATH  Google Scholar 

  10. V.I. Arnold. Geometrical Methods in the Theory of Ordinary Differential Equations, volume XI. New York-Heidelberg-Berlin: Springer-Verlag, New York-Heidelberg-Berlin, 1983.

    MATH  Google Scholar 

  11. J. Baillieul. The geometry of controlled mechanical systems. In Mathematical Control Theory, J. Baillieul and J.C. Williams, Eds., pages 322–354, 1999.

    Google Scholar 

  12. B. Bamieh and I. Meziç. A framework for destabilization of dynamical systems and mixing enhancement. Proc. 30th IEEE CDC, page Paper 4980, 2001.

    Google Scholar 

  13. G. K. Batchelor. An introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 1967.

    MATH  Google Scholar 

  14. P. Berggren. Numerical solution of a flow control problem: vorticity reduction by dynamic boundary action. SIAM Journal in Control and Optimization, 19:829–860, 1998.

    MATH  MathSciNet  Google Scholar 

  15. T.R. Bewley, R. Temam, and M. Ziane. A general framework for robust control in fluid mechanics. Physica D, 138:360–392, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  16. N.N. Bogolyubov and Yu. A. Mitropolsky. Asymptotic Methods in the Theory of Nonlinear Oscillations, volume 537. Gordon and Breach Science Publ., New York, 1961.

    Google Scholar 

  17. F. Bullo. Averaging and vibrational control of mechanical systems. SIAM Journal on Control and Optimization, 41:542–562, 1999.

    Article  MathSciNet  Google Scholar 

  18. D.M. Bushnell. Aircraft drag reduction — a review. Proceedings of The institution of Mechanical Engineers Part G — Journal of Aerospace Engineering, 217:1–18, 2003.

    Article  Google Scholar 

  19. X. Carton, G. Maze, and B. Legras. A two-dimensional vortex merger in an external strain field. J. of Turbulence, 3, art. no.045, 2002.

    Google Scholar 

  20. C. Cerretelli and C.H.K. Williamson. The physical mechanism for vortex merging. Journal of Fluid Mechanics, 475:41–77, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  21. S.I. Chernyshenko. Stabilization of trapped vortices by alternating blowing suction. Physics of Fluids, 7:802–807, 1995.

    Article  MATH  Google Scholar 

  22. S.S. Collis, K. Ghayour, M. Heinkenschloss, M. Ulbrich, and S. Ulbrich. Optimal control of unsteady compressible viscous flows. International Journal for Numerical Methods in Fluids, 40:1401–1429, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  23. L. Cortelezzi. Nonlinear feedback control of the wake past a plate with a suction point on the downstream wall. Journal of Fluid Mechanics, 327:303–324, 1996.

    Article  MATH  Google Scholar 

  24. L. Cortelezzi, Y.C. Chen, and H.L. Chang. Nonlinear feedback control of the wake past a plate: From a low-order model to a higher-order model. Physics of Fluids, 9:2009–2022, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  25. L. Cortelezzi, A. Leonard, and J.C. Doyle. An example of active circulation control of the unsteady separated flow past a semiinfinite plate. Journal of Fluid Mechanics, 260:127–154, 1994.

    Article  Google Scholar 

  26. D. D’Alessandro, M. Dahleh, and I. Meziç. Control of mixing in fluid flow: A maximun entropy approach. IEEE Transactions on Automatic Control, 44:1852–1863, 1999.

    Article  MATH  Google Scholar 

  27. Ming de Zhou and I. Wygnanski. The response of a mixing layer formed between parallel streams to a concomitant excitation at two frequencies. Journal of Fluid Mechanics, 441:139–168, 2001.

    MATH  Google Scholar 

  28. M. Gad el Hak and D.M. Bushnell. Separation control: review. J. of Fluid Engineering, 113:5–30, 1991.

    Article  Google Scholar 

  29. L. Friedland. Control of kirchhoff vortices by a resonant strain. Physical Review E, 59:4106–4111, 1999.

    Article  Google Scholar 

  30. L. Friedland and A.G. Shagalov. Resonant formation and control of 2d symmetric vortex waves. Physical Review Lett., 85:2941–2944, 2000.

    Article  Google Scholar 

  31. L. Friedland and A.G. Shagalov. Emergence of nonuniform v-states by synchronization. Physics of Fluids, 14:3074–3086, 2002.

    Article  MathSciNet  Google Scholar 

  32. T. Gerz, F. Holzapfel, and D. Darracq. Commercial aircraft wake vortices. Progress in Aerospace Sciences, 38:181–208, 2002.

    Article  Google Scholar 

  33. O. Ghattas and J. Bark. Optimal control of 2-d and 3-d incompressible navierstokes flows. Journal of Computational Physics, 136:231–244, 1997.

    Article  MATH  Google Scholar 

  34. E.A. Gillies. Low-dimensional control of the circular cylinder wake. Journal of Fluid Mechanics, 371:157–178, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  35. E.A. Gillies. Multiple sensor control of vortex shedding. AIAA Journal, 39:748–750, 2001.

    Article  Google Scholar 

  36. R. Gopalkrishnan, M.S. Triantafyllou, G.S. Triantafyllou, and D. Barrett. Active vorticity control in a shear-flow using a flapping foil. Journal of Fluid Mechanics, 274:1–21, 1994.

    Article  Google Scholar 

  37. W.R. Graham, J. Peraire, and K.Y. Tang. Optimal control of vortex shedding using low-order models. part i — open-loop model development. International Journal for Numerical Methods in Engineering, 44:945–972, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  38. W.R. Graham, J. Peraire, and K.Y. Tang. Optimal control of vortex shedding using low-order models. part ii — model-based control. International Journal for Numerical Methods in Engineering, 44:973–990, 1999.

    Article  MathSciNet  Google Scholar 

  39. D. Greenblatt and I.J. Wygnanski. The control of flow separation by periodic excitation. Progress in Aerospace Sciences, 36:487–545, 2000.

    Article  Google Scholar 

  40. M. Gunzburger, L. Hou, and T. Svobodny T. Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and neumann controls. Mathematics of Computation, 57:123–151, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  41. M. Gunzburger, L. Hou, and T. Svobodny T. Boundary velocity control of incompressible flow with an application to viscous drag reduction. SIAM Journal of Control and Optimization, 30:167–181, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  42. M. Gunzburger and S. Manservisi. The velocity tracking problem for Navier-Stokes flows with bounded distributed controls. SIAM Journal of Control and Optimization, 37:1913–1945, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  43. C.M. Ho and L.S. Huang. Subharmonics and vortex merging in mixing layers. Journal of Fluid Mechanics, 38:43–69, 1982.

    Google Scholar 

  44. C. Homescu, I.M. Navon, and Z. Li. Suppression of vortex shedding for flow around a circular cylinder using optimal control. International Journal For Numerical Methods in Fluids, 119:443–473, 2002.

    Google Scholar 

  45. L. Hou, S.S. Ravindran, and Y. Yan. Numerical solutions of optimal distributed control problems for incompressible flows. International Journal of Computational Fluid Dynamics, 8:99–114, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  46. L. Hou and Y. Yan. Dynamics and approximations of a velocity tracking problem for the Navier-Stokes flows with piecewise distributed controls. SIAM Journal on Control and Optimization, 35:1847–1885, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  47. X. Huang. Feedback control of vortex shedding from a circular cylinder. Experiments in Fluids, 20:218–224, 1996.

    Article  Google Scholar 

  48. O. Inoue. Double-frequency forcing on spatially growing mixing layers. Journal of Fluid Mechanics, 234:553–581, 1992.

    Article  MATH  Google Scholar 

  49. O. Inoue. Note on multiple-frequency forcing on mixing layers. Fluid Dynamics Research, 16:161–172, 1995.

    Article  Google Scholar 

  50. A. Iollo and L. Zannetti. Optimal control of a vortex trapped by an airfoil with a cavity. Flow, Turbulence & Combustion, 65:417–30, 2000.

    Article  MATH  Google Scholar 

  51. A. Iollo and L. Zannetti. Trapped vortex optimal control by suction and blowing at the wall. European Journal of Mechanics B-Fluids, 20:7–24, 2001.

    Article  MATH  Google Scholar 

  52. K. Ito and S.S. Ravindran. A reduced-order method for simulation and control of fluid flows. Journal of Computational Physics, 143:403–425, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  53. J.B. Kadtke, A. Pentek, and G. Pedrizzetti. Controlled capture of a continuous vorticity distribution. Physics Letters A, 204:108–114, 1995.

    Article  Google Scholar 

  54. D. E. Kirk. Optimal Control Theory: An Introduction. Prentice Hall, New York, 1970.

    Google Scholar 

  55. K. Koenig and A. Roshko. An experimental study of geometrical effects on the drag and flow field of two bluff bodies separated by a gap. Journal of Fluid Mechanics, 156:167, 1985.

    Article  Google Scholar 

  56. P.V. Kokotovic, H.K. Khalil, and J. O’Reilly. Singular perturbation methods in control: analysis and design. Academic, London, 1986.

    MATH  Google Scholar 

  57. K. Kwon and H. Choi. Control of laminar vortex shedding behind a circular cylinder using splitter plates. Physics of Fluids, 8:479–486, 1996.

    Article  MATH  Google Scholar 

  58. N. E. Leonard and P. S. Krishnaprasad. Motion control of drift-free leftinvariant systems on Lie groups. IEEE Transactions on Automatic Control, 40:1539–1554, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  59. F. Li and N. Aubry. Feedback control of a flow past a cylinder via transverse motion. Physics of Fluids, 15:2163–2176, 2003.

    Article  MathSciNet  Google Scholar 

  60. Z.J. Li, I.M. Navon, M.Y. Hussaini, and F.A. Le Dimet. Optimal control of cylinder wakes via suction and blowing. Computers & Fluids, 32:149–171, 2003.

    Article  MATH  Google Scholar 

  61. P. Lochak, C. Meunier, and H.S. Dumas. Multiphase averaging for classical systems: with applications to adiabatic theorems. Springer-Verlag, New York, 1988.

    MATH  Google Scholar 

  62. N. Mahir and D. Rockwell. Vortex formation from a forced system of two cylinders. part i: tandem arrangement. Journal of Fluids and Structures, 10:473–489, 1996.

    Article  Google Scholar 

  63. N. Mahir and D. Rockwell. Vortex formation from a forced system of two cylinders. part ii: side-by-side arrangement. Journal of Fluids and Structures, 10:491–500, 1996.

    Article  Google Scholar 

  64. M.V. Melander, N.J. Zabusky, and J.C. McWilliams. Symmetric vortex merger in two dimensions: causes and conditions. Journal of Fluid Mechanics, 195:303–340, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  65. I. Mezić. Controllability, integrability and ergodicity. In Proceedings of the Mohammed Dahleh Symposium, in Lecture Notes in Control and Information Sciences, volume 289, pages 213–229. Springer-Verlag, 2003.

    Google Scholar 

  66. I. Mezić. Controllability of hamiltonian systems with drift: Action-angle variables and ergodic partition. In Proceedings of Conference on Decision and Control, Maui, 2003.

    Google Scholar 

  67. A.M. Mitchell, D. Barberis, P. Molton, and J. Delery. Control of leading-edge vortex breakdown by trailing edge injection. Journal of Aircraft, 39:221–226, 2002.

    Article  Google Scholar 

  68. A.M. Mitchell and J. Delery. Research into vortex breakdown control. Progress in Aerospace Sciences, 37:385–418, 2002.

    Article  Google Scholar 

  69. V.J. Modi. Moving surface boundary-layer control: a review. Journal of Fluids and Structures, 10:491–500, 1996.

    Article  Google Scholar 

  70. P. K. Newton. Vortex Dynamics. Springer-Verlag, New York, 2002.

    Google Scholar 

  71. H. Nijmeijer and A.J. van der Schaft. Nonlinear dynamical control systems. Springer-Verlag, New York, 1990.

    MATH  Google Scholar 

  72. B.R. Noack and I. Mezićand A. Banaszuk. Controling vortex motion and chaotic advection. In Proceedings of the 39th IEEE Conference on Decision and Control, pages 1716–1723, 2000.

    Google Scholar 

  73. B.R. Noack, I. Mezić, G. Tadmor, and A. Banaszuk. Optimal mixing in recirculation zones. Physics of Fluids, pages 867–888, 2004.

    Google Scholar 

  74. S. Ozono. Flow control of vortex shedding by a short splitter plate asymmetrically arranged downstream of a cylinder. Physics of Fluids, 11:2928–2934, 1999.

    Article  MATH  Google Scholar 

  75. D.S. Park, D.M. Ladd, and E.W. Hendricks. Feedback-control of von karman vortex shedding behind a circular-cylinder at low reynolds-numbers. Physics of Fluids, 6:2390–2405, 1994.

    Article  MATH  Google Scholar 

  76. A. Pentek, J.B. Kadtke, and G. Pedrizzetti. Dynamical control for capturing vortices near bluff bodies. Physical Review E, 58:1883–1898, 1998.

    Article  MathSciNet  Google Scholar 

  77. A. Pentek, T. Tel, and Z. Toroczkai. Stabilizing chaotic vortex trajectories: an example of high-dimensional control. Physics Letters A, 224:85–92, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  78. B. Protas. Linear feedback stabilization of laminar vortex shedding based on a point vortex model. Physics of Fluids, pages 4473–4488, 2004.

    Google Scholar 

  79. B. Protas and A. Styczek. Optimal rotary control of the cylinder wake in the laminar regime. Physics of Fluids, 14:2073–2087, 2002.

    Article  Google Scholar 

  80. D. Rockwell. Vortex-body interactions. Annual Review of Fluid Mechanics, 30:199–229, 1998.

    Article  MathSciNet  Google Scholar 

  81. H. Sakamoto and H. Haniu. Optimum suppression of fluid forces acting on a circular-cylinder. Journal of Fluids Engineering-Transactions of The ASME, 116:221–227, 1994.

    Article  Google Scholar 

  82. G. N. Silva and R. B. Vinter. Necessary conditions for optimal impulsive control problems. SIAM J. Control Optimization, 35:1829–1846, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  83. P.R. Spalart. Airplane trailing vortices. Annual Review of Fluid Mechanics, 30:107–138, 1998.

    Article  MathSciNet  Google Scholar 

  84. S. Tang and N. Aubry. Suppression of vortex shedding inspired by a lowdimensional model. Journal of Fluids And Structures, 14:443–468, 2000.

    Article  Google Scholar 

  85. J.S. Tao, X.Y. Huang XY, and W.K. Chan. Flow visualization study on feedback control of vortex shedding from a circular cylinder. Journal of Fluids and Structures, 10:965–970, 1996.

    Article  Google Scholar 

  86. M.S. Triantafyllou, A.H. Techet, Q. Zhu, D.N. Beal, F.S. Hover, and D.K.P. Yue. Vorticity control in fish-like propulsion and maneuvering. Integrative And Comparative Biology, 42:1026–1031, 2002.

    Article  Google Scholar 

  87. M.S. Triantafyllou, G.S. Triantafyllou, and D.K.P. Yue. Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech., 32:33–53, 2000.

    Article  MathSciNet  Google Scholar 

  88. U. G. Vaidya and I. Mezić. Controllability of a class of area-preserving twist maps. UCSB Preprint, 2002.

    Google Scholar 

  89. D.L. Vainchtein and I. Mezić. Control of a vortex pair using a weak external flow. Journal of Turbulence, 3, 2002.

    Google Scholar 

  90. D.L. Vainchtein and I. Mezić. Optimal control of a co-rotating vortex pair: Averaging and impulsive control. Physica D, 192:63–82, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  91. Y. Wang, G. Haller, A. Banaszuk, and G. Tadmor. Closed-loop lagrangian separation control in a bluff body shear flow model. Physics of Fluids, 15:2251–2266, 2003.

    Article  MathSciNet  Google Scholar 

  92. H.M. Warui and N. Fujisawa. Feedback control of vortex shedding from a circular cylinder by cross-flow cylinder oscillations. Experiments in Fluids, 21:49–56, 1996.

    Article  Google Scholar 

  93. M.J. Wolfgang, J.M. Anderson, M.A. Grosenbaugh, D.K.P. Yue, and M.S. Triantafyllou. Near-body flow dynamics in swimming fish. The Journal of Experimental Biology, 202:2303–2327, 1999.

    Google Scholar 

  94. T. Yang. Impulsive control. IEEE Transactions on Automatic Control, 44:1081–1083, 1999.

    Article  MATH  Google Scholar 

  95. I. Yasuda and G.R. Flierl. Two-dimensional asymmetric vortex merger: merger dynamics and critical merger distance. Dynamics of Atmospheres and Oceans, 26:159–181, 1997.

    Article  Google Scholar 

  96. J.H. Yu and C.F. Driscoll. Diocotron wave echoes in a pure electron plasma. IEEE Transactions on Plasma Science, 30:24–25, 2002.

    Article  Google Scholar 

  97. L. Zannetti and A. Iollo. Passive control of the vortex wake past a flat plate at incidence. Theoretical And Computational Fluid Dynamics, 16:211–230, 2003.

    Article  Google Scholar 

  98. Q. Zhu, M.J. Wolfgang, D.K.P. Yue, and M.S. Triantafyllou. Three-dimensional flow structures and vorticity control in fish-like swimming. J. of Fluid Mechanics, 468:1–28, 2002.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of California Santa Barbara, CA, 93106, USA

    Dmitri Vainchtein & Igor Meziç

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  1. Dmitri Vainchtein
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  2. Igor Meziç
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  1. ETH Zürich, CH-8092 Zürich, Switzerland

    Petros Koumoutsakos (Professor of Computational Science) (Professor of Computational Science)

  2. Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, CA, 93106, USA

    Igor Mezic

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Vainchtein, D., Meziç, I. (2006). Vortex-based Control Algorithms. In: Koumoutsakos, P., Mezic, I. (eds) Control of Fluid Flow. Lecture Notes in Control and Information Sciences, vol 330. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-36085-8_9

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