Control of Fluid Flow pp 189-212 | Cite as

# Vortex-based Control Algorithms

## 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.

## Keywords

Proper Orthogonal Decomposition Vortex Pair Point Vortex Hamiltonian Structure Impulsive Control## Preview

Unable to display preview. Download preview PDF.

## 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.zbMATHCrossRefGoogle 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.zbMATHCrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.zbMATHCrossRefMathSciNetGoogle Scholar - [7]I. Aranson, H. Levine, and L. Tsimring. Controlling spatiotemporal chaos.
*Physical Review Lett*., 72:2561–2564, 1994.CrossRefGoogle Scholar - [8]H. Aref. Integrable, chaotic, and turbulent vortex motion in two-dimensional flows.
*Annual Review of Fluid Mechanics*, 15:345–389, 1983.CrossRefMathSciNetGoogle Scholar - [9]V.I. Arnold.
*Mathematical Methods of Classical Mechanics*. Springer-Verlag, New York, 1978.zbMATHGoogle 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.zbMATHGoogle 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.zbMATHGoogle 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.zbMATHMathSciNetGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.CrossRefMathSciNetGoogle 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.CrossRefGoogle 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.zbMATHCrossRefMathSciNetGoogle Scholar - [21]S.I. Chernyshenko. Stabilization of trapped vortices by alternating blowing suction.
*Physics of Fluids*, 7:802–807, 1995.zbMATHCrossRefGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.zbMATHCrossRefGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.CrossRefGoogle 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.zbMATHCrossRefGoogle 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.zbMATHGoogle Scholar - [28]M. Gad el Hak and D.M. Bushnell. Separation control: review.
*J. of Fluid Engineering*, 113:5–30, 1991.CrossRefGoogle Scholar - [29]L. Friedland. Control of kirchhoff vortices by a resonant strain.
*Physical Review E*, 59:4106–4111, 1999.CrossRefGoogle Scholar - [30]L. Friedland and A.G. Shagalov. Resonant formation and control of 2d symmetric vortex waves.
*Physical Review Lett.*, 85:2941–2944, 2000.CrossRefGoogle Scholar - [31]L. Friedland and A.G. Shagalov. Emergence of nonuniform v-states by synchronization.
*Physics of Fluids*, 14:3074–3086, 2002.CrossRefMathSciNetGoogle Scholar - [32]T. Gerz, F. Holzapfel, and D. Darracq. Commercial aircraft wake vortices.
*Progress in Aerospace Sciences*, 38:181–208, 2002.CrossRefGoogle 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.zbMATHCrossRefGoogle Scholar - [34]E.A. Gillies. Low-dimensional control of the circular cylinder wake.
*Journal of Fluid Mechanics*, 371:157–178, 1998.zbMATHCrossRefMathSciNetGoogle Scholar - [35]E.A. Gillies. Multiple sensor control of vortex shedding.
*AIAA Journal*, 39:748–750, 2001.CrossRefGoogle 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.CrossRefGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.CrossRefMathSciNetGoogle Scholar - [39]D. Greenblatt and I.J. Wygnanski. The control of flow separation by periodic excitation.
*Progress in Aerospace Sciences*, 36:487–545, 2000.CrossRefGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.zbMATHCrossRefMathSciNetGoogle Scholar - [47]X. Huang. Feedback control of vortex shedding from a circular cylinder.
*Experiments in Fluids*, 20:218–224, 1996.CrossRefGoogle Scholar - [48]O. Inoue. Double-frequency forcing on spatially growing mixing layers.
*Journal of Fluid Mechanics*, 234:553–581, 1992.zbMATHCrossRefGoogle Scholar - [49]O. Inoue. Note on multiple-frequency forcing on mixing layers.
*Fluid Dynamics Research*, 16:161–172, 1995.CrossRefGoogle 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.zbMATHCrossRefGoogle 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.zbMATHCrossRefGoogle 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.zbMATHCrossRefMathSciNetGoogle Scholar - [53]J.B. Kadtke, A. Pentek, and G. Pedrizzetti. Controlled capture of a continuous vorticity distribution.
*Physics Letters A*, 204:108–114, 1995.CrossRefGoogle 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.CrossRefGoogle Scholar - [56]P.V. Kokotovic, H.K. Khalil, and J. O’Reilly. Singular perturbation methods in control: analysis and design. Academic, London, 1986.zbMATHGoogle 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.zbMATHCrossRefGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.CrossRefMathSciNetGoogle 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.zbMATHCrossRefGoogle 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.zbMATHGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.CrossRefGoogle Scholar - [68]A.M. Mitchell and J. Delery. Research into vortex breakdown control.
*Progress in Aerospace Sciences*, 37:385–418, 2002.CrossRefGoogle Scholar - [69]V.J. Modi. Moving surface boundary-layer control: a review.
*Journal of Fluids and Structures*, 10:491–500, 1996.CrossRefGoogle 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.zbMATHGoogle 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.zbMATHCrossRefGoogle 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.zbMATHCrossRefGoogle 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.CrossRefMathSciNetGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.CrossRefGoogle Scholar - [80]D. Rockwell. Vortex-body interactions.
*Annual Review of Fluid Mechanics*, 30:199–229, 1998.CrossRefMathSciNetGoogle 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.CrossRefGoogle Scholar - [82]G. N. Silva and R. B. Vinter. Necessary conditions for optimal impulsive control problems.
*SIAM J. Control Optimization*, 35:1829–1846, 1997.zbMATHCrossRefMathSciNetGoogle Scholar - [83]P.R. Spalart. Airplane trailing vortices.
*Annual Review of Fluid Mechanics*, 30:107–138, 1998.CrossRefMathSciNetGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefMathSciNetGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.CrossRefMathSciNetGoogle 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.CrossRefGoogle 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.zbMATHCrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.zbMATHCrossRefMathSciNetGoogle Scholar