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On the active feedback control of a swirling flow in a finite-length pipe

Published online by Cambridge University Press:  25 November 2013

Shixiao Wang ,
Zvi Rusak ,
Steve Taylor  and
Rui Gong
Shixiao Wang*
Affiliation:
Department of Mathematics, University of Auckland, 38 Princes Street, Auckland, 1142, New Zealand
Zvi Rusak
Affiliation:
Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA
Steve Taylor
Affiliation:
Department of Mathematics, University of Auckland, 38 Princes Street, Auckland, 1142, New Zealand
Rui Gong
Affiliation:
Department of Mathematics, University of Auckland, 38 Princes Street, Auckland, 1142, New Zealand
*
†Email address for correspondence: [email protected]
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Abstract

The physical properties of a recently proposed feedback-stabilization method of a vortex flow in a finite-length straight pipe are studied for the case of a solid-body rotation flow. In the natural case, when the swirl ratio is beyond a certain critical level, linearly unstable modes appear in sequence as the swirl level is increased. Based on an asymptotic long-wave (long-pipe) approach, the global feedback control method is shown to enforce the decay in time of the perturbation’s kinetic energy and thereby quench all of the instability modes for a swirl range above the critical swirl level. The effectiveness of an extended version of this feedback flow control approach is further analysed through a detailed mode analysis of the full linear control problem for a solid-body rotation flow in a finite-length pipe that is not necessarily long. We first rigourously prove the asymptotic decay in time of all modes with real growth rates. We then compute the growth rate and shape of all modes according to the full linearized control problem for swirl levels up to 50 % above the critical level. We demonstrate that the flow is stabilized in the whole swirl range and can be even further stabilized for higher swirl levels. However, the control effectiveness is sensitive to the choice of the feedback control gain. A potentially best range of the gain is identified. An inadequate level of gain, either insufficient or excessive, could lead to a marginal control or failure of the control method at high swirl levels. The robustness of the proposed control law to stabilize both initial waves and continuous inlet flow perturbations and the elimination of the vortex breakdown process are demonstrated through numerical computations.

JFM classification

Flow Control: Control theory Vortex Flows: Vortex breakdown Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 737 , 25 December 2013 , pp. 280 - 307
Copyright
©2013 Cambridge University Press 

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References

Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593–629.CrossRefGoogle Scholar
Beran, P. S. 1994 The time-asymptotic behaviour of vortex breakdown in tubes. Comput. Fluids 23, 913–937.Google Scholar
Beran, P. S. & Culick, F. E. C. 1992 The role of non-uniqueness in the development of vortex breakdown in tubes. J. Fluid Mech. 242, 491–527.Google Scholar
Brücker, Ch. & Althaus, W. 1995 Study of vortex breakdown by particle tracking velocimetry (PTV), Part 3: time-dependent structure and development of breakdown modes. Exp. Fluids 18, 174–186.Google Scholar
Delery, J. M. 1994 Aspects of vortex breakdown. Prog. Aerosp. 30 (1), 1–59.Google Scholar
Faler, J. H. & Leibovich, S. 1977 Disrupted states of vortex flow and vortex breakdown. J. Phys. Fluids 20, 1385–1400.CrossRefGoogle Scholar
Gallaire, F., Chomaz, J.-M. & Huerre, P. 2004 Closed-loop control of vortex breakdown: a model study. J. Fluid Mech. 511, 67–93.Google Scholar
Garg, A. K. & Leibovich, S. 1979 Spectral characteristics of vortex breakdown flow fields. Phys. Fluids 22, 2053.Google Scholar
Hall, M. G. 1972 Vortex breakdown. Annu. Rev. Fluid Mech. 4, 195–217.Google Scholar
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Knole, M. & Sattelmayer, T. 2009 Interaction of heat release and vortex breakdown during flame flashback driven by combustion induced vortex breakdown. Exp. Fluids 47 (45), 627–635.Google Scholar
Leibovich, S. 1978 The structure of vortex breakdown. Annu. Rev. Fluid Mech. 10, 221–246.Google Scholar
Leibovich, S. 1984 Vortex stability and breakdown: survey and extension. AIAA J. 22, 1192–1206.Google Scholar
Leibovich, S. & Randall, J. D. 1972 Solitary waves in concentrated vortices. J. Fluid Mech. 52, 625–635.Google Scholar
Long, R. R. 1953 Steady motion around a symmetrical obstacle moving along the axis of a rotating liquid. J. Met. 10, 197–203.2.0.CO;2>CrossRefGoogle Scholar
Lopez, J. M. 1994 On the bifurcation structure of axisymmetric vortex breakdown in a constricted pipe. Phys. Fluids 6, 3683–3693.Google Scholar
Mattner, T. W., Joubert, P. N. & Chong, M. S. 2002 Vortical flow. Part 1. Flow through a constant diameter pipe. J. Fluid Mech. 463, 259–291.Google Scholar
Meliga, P. & Gallaire, F. 2011 Control of axisymmetric vortex breakdown in a constricted pipe: nonlinear steady states and weakly nonlinear asymptotic expansions. Phys. Fluids 23, 084102.CrossRefGoogle Scholar
Mitchell, A. M. & Delery, J. 2001 Research into vortex breakdown control. Prog. Aerosp. Sci. 37 (4), 385–418.Google Scholar
Muruganandam, T. M., Nair, S, Scarborough, D., Neumeier, Y, Jagoda, J., Lieuwen, T., Seitzman, J. & Zinn, B. 2005 Active control of lean blowout for turbine engine combustors. J. Propul. Power 21 (5), 807–814.CrossRefGoogle Scholar
O’Neil, P. J., Roos, F. W., Barnett, R. M. & Hawk, J. D. 1989, Investigation of flow characteristics of developed vortex, McDonnell Aircraft Company, Report NADC-89114-60, St Louis, MO.Google Scholar
Paschereit, C. O. & Gutmark, E. 2002 Proportional control of combustion instabilities in a simulated gas-turbine combustor. J. Propul. Power 18 (6), 1298–1304.CrossRefGoogle Scholar
Peckham, D. H. & Atkinson, S. A. 1957, Preliminary results of low speed wind tunnel tests on a gothic wing of aspect ratio 1.0, Technical Report CP508, Aeronautical Research Council London.Google Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. Ser. A 93, 148–154.Google Scholar
Rusak, Z. & Lamb, D. 1999 Prediction of vortex breakdown in leading-edge vortices above slender delta wings. J. Aircraft 36 (4), 659–667.Google Scholar
Rusak, Z., Wang, S. & Whiting, C. H. 1998 The evolution of a perturbed vortex in a pipe to axisymmetric vortex breakdown. J. Fluid Mech. 366, 211–237.CrossRefGoogle Scholar
Rusak, Z., Wang, S., Xu, L. & Taylor, S. 2012 On the global nonlinear stability of near-critical swirling flow in a long finite-length pipe and the path to vortex breakdown. J. Fluid Mech. 712, 295–326.Google Scholar
Rusak, Z., Whiting, C. H. & Wang, S. 1998 Axisymmetric breakdown of a Q-vortex in a pipe. AIAA J. 36 (10), 1848–1853.Google Scholar
Sarpkaya, T. 1971 On stationary and travelling vortex breakdowns. J. Fluid Mech. 45, 545–559.Google Scholar
Sarpkaya, T. 1995 Turbulent vortex breakdown. Phys. Fluids 7 (10), 2301–2303.CrossRefGoogle Scholar
Snyder, D. & Spall, R. 2000 Numerical simulation of bubble-type vortex breakdown within a tube-and-vane apparatus. Phys. Fluids 12 (3), 603–608.Google Scholar
Squire, H. B. 1960 Analysis of the vortex breakdown phenomenon. In Miszallaneen der Angewandten Mechanik, pp. 306–312. Akademie.Google Scholar
Wang, S. 2008 A novel method for analysing the global stability of inviscid columnar swirling flow in a finite pipe. Phys. Fluids 20, 074101.Google Scholar
Wang, S. 2009 On the nonlinear stability of inviscid axisymmetric swirling flows in a pipe of finite length. Phys. Fluids 21, 084104.Google Scholar
Wang, S. & Rusak, Z. 1996 On the stability of an axisymmetric rotating flow in a pipe. Phys. Fluids 8 (4), 1007–1016.Google Scholar
Wang, S. & Rusak, Z. 1997 The dynamics of a swirling flow in a pipe and transition to axisymmetric vortex breakdown. J. Fluid Mech. 340, 177–223.Google Scholar
Wang, S. & Rusak, Z. 2011 Energy transfer mechanism of the instability of an axisymmetric swirling flow in a finite-length pipe. J. Fluid Mech. 679, 505–543.Google Scholar
Xu, L. 2012, Vortex flow stability, dynamics and feedback stabilization. Ph.D. dissertation, Rensselaer Polytechnic Institute.Google Scholar