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On the global nonlinear stability of a near-critical swirling flow in a long finite-length pipe and the path to vortex breakdown

Published online by Cambridge University Press:  28 September 2012

Z. Rusak ,
S. Wang ,
L. Xu  and
S. Taylor
Z. Rusak*
Affiliation:
Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA
S. Wang
Affiliation:
Department of Mathematics, University of Auckland, 38 Princes Street, Auckland, 1142, New Zealand
L. Xu
Affiliation:
Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA
S. Taylor
Affiliation:
Department of Mathematics, University of Auckland, 38 Princes Street, Auckland, 1142, New Zealand
*
†Email address for correspondence: rusakz@rpi.edu
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Abstract

The dynamics of a perturbed incompressible, inviscid, axisymmetric, near-critical swirling flow in a long, finite-length, straight, circular pipe is studied through a weakly nonlinear analysis. The flow is subjected to non-periodic inlet and outlet conditions. The long-wave approach involves a rescaling of the axial distance and time. It results in a separation of the perturbation’s structure into a critical standing wave in the radial direction and an evolving wave in the axial direction, that is described by a nonlinear model problem. The approach is first validated by establishing the bifurcation of non-columnar states from the critical swirl and the linear stability modes of these states. Examples of the flow dynamics at various near-critical swirl levels in response to different initial perturbations demonstrate the important role of the nonlinear steepening terms in perturbation dynamics. The computed dynamics shows quantitative agreement with results from numerical simulations that are based on the axisymmetric Euler equations for various swirl levels and as long as perturbations are small, thereby verifying the accuracy of each computation and capturing the essence of flow dynamics. Results demonstrate the various stages of the flow dynamics, specifically during the transition to vortex breakdown states. They reveal the evolution of faster-than-exponential and shape-changing modes as perturbations grow into the vortex breakdown process. These explosive modes provide the sudden and abrupt nature of the vortex breakdown phenomenon. Further analysis of the model problem shows the important role of the nonlinear evolution of perturbations and its relevance to the transfer of the perturbation’s kinetic energy between the boundaries and flow bulk, the evolution of perturbations in practical concentrated vortex flows, and the design of control methods of vortex flows. A robust feedback control method to stabilize a solid-body rotation flow in a pipe at a wide range of swirl levels above critical is developed. The applicability of this method to stabilizing medium and small core-size vortices is also discussed.

JFM classification

Flow Control: Control theory Vortex Flows: Vortex breakdown Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 712 , 10 December 2012 , pp. 295 - 326
Copyright
©2012 Cambridge University Press

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References

Ash, R. L. & Khorrami, M. R. 1995 Vortex stability. In Fluid Vortices (ed. Green, S. I.), chap. 8, pp. 317–372. Kluwer Academic.CrossRefGoogle Scholar
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 behavior of vortex breakdown in tubes. Comput. Fluids 23, 913–937.CrossRefGoogle 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.CrossRefGoogle Scholar
Bruecker, C. & 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.CrossRefGoogle Scholar
Buntine, J. D. & Saffman, P. G. 1935 Inviscid swirling flows and vortex breakdown. Proc. R. Soc. Lond. A 449, 139–153.Google Scholar
Darmofal, D. L. 1996 Comparisons of experimental and numerical results for axisymmetric vortex breakdown in pipes. Comput. Fluids 25, 353–371.CrossRefGoogle Scholar
Gallaire, F. & Chomaz, J.-M. 2004 The role of boundary conditions in a simple model of incipient vortex breakdown. Phys. Fluids 16 (2), 274.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.CrossRefGoogle Scholar
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamics and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463–476.CrossRefGoogle Scholar
Kribus, A. & Leibovich, S. 1994 Instability of strongly nonlinear-waves in vortex flows. J. Fluid Mech. 269, 247–264.CrossRefGoogle Scholar
Leclaire, B. & Sipp, D. 2010 A sensitivity study of vortex breakdown onset to upstream boundary conditions. J. Fluid Mech. 645, 81–119.CrossRefGoogle Scholar
Leibovich, S. 1984 Vortex stability and breakdown: survey and extension. AIAA J. 22, 1192–1206.CrossRefGoogle Scholar
Leibovich, S. & Kribus, A. 1990 Large amplitude wavetrains and solitary waves in vortices. J. Fluid Mech. 216, 459–504.CrossRefGoogle Scholar
Leibovich, S. & Randall, J. D. 1972 Solitary waves in concentrated vortices. J. Fluid Mech. 52, 625–635.CrossRefGoogle Scholar
Leibovich, S. & Stewartson, K. 1983 A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335–356.CrossRefGoogle Scholar
Lopez, J. M. 1994 On the bifurcation structure of axisymmetric vortex breakdown in a constricted pipe. Phys. Fluids 6, 3683–3693.CrossRefGoogle Scholar
Malkiel, E., Cohen, J., Rusak, Z. & Wang, S. 1996 Axisymmetric vortex breakdown in a pipe: theoretical and experimental studies. In Proceedings of the 36th Israel Annual Conference on Aerospace Sciences, pp. 24–34.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.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. Aeronautical Research Council Technical Report CP508, HMSO.Google Scholar
Randall, J. D. & Leibovich, S. 1973 The critical state: a trapped wave model of vortex breakdown. J. Fluid Mech. 53, 48–493.Google Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. Ser. A 93, 148–154.Google Scholar
Ruith, M. R., Chen, P., Meiburg, E. & Maxworthy, T. 2003 Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation. J. Fluid Mech. 486, 331–378.CrossRefGoogle Scholar
Rusak, Z. 1998 The interaction of near-critical swirling flows in a pipe with inlet vorticity perturbations. Phys. Fluids 10 (7), 1672–1684.CrossRefGoogle Scholar
Rusak, Z. & Judd, K. P. 2001 The stability of non-columnar swirling flows in diverging streamtubes. Phys. Fluids 13 (10), 2835–2844.CrossRefGoogle Scholar
Rusak, Z., Judd, K. P. & Wang, S. 1997 The effect of small pipe divergence on near critical swirling flows. Phys. Fluids 9 (8), 2273–2285.CrossRefGoogle 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
Sarpkaya, T. 1971 On stationary and travelling vortex breakdowns. J. Fluid Mech. 45, 545–559.CrossRefGoogle Scholar
Sarpkaya, T. 1995 Turbulent vortex breakdown. Phys. Fluids 7 (10), 2301–2303.CrossRefGoogle Scholar
Squire, H. B. 1960 Analysis of the vortex breakdown phenomenon. In Miszallaneen der Angewandten Mechanik, pp. 306–312. Akademie.Google Scholar
Synge, L. 1933 The stability of heterogeneous liquids. Trans. R. Soc. Can. 27, 1–18.Google Scholar
Wang, S. 2009 On the nonlinear stability of inviscid axisymmetric swirling flows in a pipe of finite length. Phys. Fluids 21, 084104.CrossRefGoogle Scholar
Wang, S. & Rusak, Z. 1996 On the stability of an axisymmetric rotating flow in a pipe. Phys. Fluids 8 (4), 1007–1016.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Xu, L. 2012 Vortex flow stability, dynamics and feedback stabilization. PhD dissertation, Rensselaer Polytechnic Institute.Google Scholar