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Vortex breakdown of compressible subsonic swirling flows in a finite-length straight circular pipe

Published online by Cambridge University Press:  16 September 2015

Zvi Rusak ,
Jung J. Choi ,
Nicholas Bourquard  and
Shixiao Wang
Zvi Rusak*
Affiliation:
Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA
Jung J. Choi
Affiliation:
Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA
Nicholas Bourquard
Affiliation:
Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA
Shixiao Wang
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

A global analysis of steady states of inviscid compressible subsonic swirling flows in a finite-length straight circular pipe is developed. A nonlinear partial differential equation for the solution of the flow stream function is derived in terms of the inlet flow specific total enthalpy, specific entropy and circulation functions. The equation reflects the complicated thermo–physical interactions in the flows. Several types of solutions of the resulting nonlinear ordinary differential equation for the columnar case together with a flow force condition describe the outlet state of the flow in the pipe. These solutions are used to form the bifurcation diagram of steady compressible flows with swirl as the inlet swirl level is increased at a fixed inlet Mach number. The approach is applied to two profiles of inlet flows, solid-body rotation and the Lamb–Oseen vortex, both with a uniform axial velocity and temperature. The computed results provide for each inlet flow profile theoretical predictions of the critical swirl levels for the appearance of vortex breakdown states as a function of the inlet Mach number, suggesting that the results are robust for a variety of inlet swirling flows. The analysis sheds light on the dynamics of compressible flows with swirl and vortex breakdown, and shows the delay in the appearance of breakdown with increase of the inlet axial flow Mach number in the subsonic range of operation. The present theory is limited to axisymmetric dynamics of swirling flows in pipes where the wall boundary layer is thin and attached and does not interact with the flow in the bulk.

JFM classification

Compressible Flows: Compressible Flows Vortex Flows: Vortex breakdown Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 781 , 25 October 2015 , pp. 3 - 27
Copyright
© 2015 Cambridge University Press 

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References

Althaus, W., Brücker, Ch. & Weimer, M. 1995 Breakdown of slender vortices. In Fluid Vortices, pp. 373–426. Kluwer Academic.CrossRefGoogle Scholar
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14 (4), 593–629.CrossRefGoogle Scholar
Beran, P. S. 1994 The time-asymptotic behavior of vortex breakdown in tubes. Comput. Fluids 23 (7), 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, 419–527.CrossRefGoogle Scholar
Brown, G. L. & Lopez, J. M. 1990 Axisymmetric vortex breakdown. Part 2. Physical mechanisms. J. Fluid Mech. 221, 553–576.CrossRefGoogle Scholar
Buntine, J. D. & Saffman, P. G. 1995 Inviscid swirling flows and vortex breakdown. Proc. R. Soc. Lond. A 449, 139–153.Google Scholar
Escudier, M. 1988 Vortex breakdown: observations and explanations. Prog. Aeronaut. Sci. 25, 189–229.Google Scholar
Gallaire, F. O., Ruith, M., Meiburg, E., Chomaz, J. M. & Huerre, P. 2006 Spiral vortex breakdown as a global mode. J. Fluid Mech. 549, 71–80.Google Scholar
Hall, M. G. 1972 Vortex breakdown. Annu. Rev. Fluid Mech. 4, 195–217.CrossRefGoogle Scholar
Herrada, M. A., Prez-Sabroid, M. & Barrero, A. 2003 Vortex breakdown in compressible flows in pipes. Phys. Fluids 15, 2208–2215.CrossRefGoogle Scholar
Keller, J. J., Egli, W. & Exley, W. 1985 Force- and loss-free transitions between flow states. Z. Angew. Math. Phys. 36, 854–889.CrossRefGoogle Scholar
Kuruvila, G. & Salas, M.1990 Three-dimensional simulation of vortex breakdown, NASA TM 102664.CrossRefGoogle Scholar
Leibovich, S. 1984 Vortex stability & breakdown: survey and extension. AIAA J. 22, 1192–1206.Google Scholar
Leibovich, S. & Kribus, A. 1990 Large-amplitude wavetrains and solitary waves in vortices. J. Fluid Mech. 216, 459–504.Google Scholar
Liang, H. & Maxworthy, T. 2005 An experimental investigation of swirling jets. J. Fluid Mech. 525, 115–159.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
Luginsland, T. & Kleiser, L. 2015 Mach number influence on vortex breakdown in compressible, subsonic swirling nozzle-jet flows. In Direct and Large-Eddy Simulation IX, ERCOFTAC Series, vol. 20, pp. 311–317. Springer.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
Melville, R.1996 The role of compressibility in free vortex breakdown. AIAA Paper No. 96-2075.CrossRefGoogle Scholar
Novak, F. & Sarpkaya, T. 2000 Turbulent vortex breakdown at high Reynolds numbers. AIAA J. 38, 825–834.Google Scholar
Randall, J. D. & Leibovich, S. 1973 The critical state: a trapped wave model of vortex breakdown. J. Fluid Mech. 58, 481–493.CrossRefGoogle Scholar
Renac, F., Sipp, D. & Jacquin, L. 2007 Criticality of compressible rotating flows. Phys. Fluids 19, 018101.CrossRefGoogle Scholar
Rusak, Z. 1998 The interaction of near-critical swirling flows in a pipe with inlet azimuthal vorticity perturbations. Phys. Fluids 10 (7), 1672–1684.Google Scholar
Rusak, Z., Choi, J. J. & Lee, J.-H. 2007 Bifurcation and stability of near-critical compressible swirling flows. Phys. Fluids 19 (11), 114107.Google Scholar
Rusak, Z. & Judd, K. P. 2001 The stability of non-columnar swirling flows in diverging streamtubes. Phys. Fluids 13 (10), 2835–2844.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.CrossRefGoogle Scholar
Rusak, Z. & Lee, J.-H. 2002 The effect of compressibility on the critical swirl of vortex flows in a pipe. J. Fluid Mech. 461, 301–319.Google Scholar
Rusak, Z. & Lee, J.-H. 2004 On the stability of a compressible axisymmetric rotating flow in a pipe. J. Fluid Mech. 501, 25–42.Google Scholar
Rusak, Z. & Meder, C. C. 2004 Near-critical swirling flow in a slightly contracting pipe. AIAA J. 42 (11), 2284–2293.Google Scholar
Rusak, Z. & Wang, S. 2014 Wall-separation and vortex-breakdown zones in a solid-body rotation flow in a rotating finite-length straight circular pipe. J. Fluid Mech. 759, 321–359.Google Scholar
Rusak, Z., Wang, S., Xu, L. & Taylor, S. 2012 On the global nonlinear stability of a near-critical swirling flow in a long finite-length pipe and the path to vortex breakdown. J. Fluid Mech. 712, 295–326.CrossRefGoogle Scholar
Rusak, Z., Whiting, C. & Wang, S. 1998 Axisymmetric breakdown of a Q-vortex in a pipe. AIAA J. 36 (10), 1848–1853.Google Scholar
Sarpkaya, T. 1995 Turbulent vortex breakdown. Phys. Fluids 7 (10), 2301–2303.CrossRefGoogle Scholar
Squire, H. B. 1956 Rotating fluids. In Surveys in Mechanics (ed. Batchelor, G. K. & Davies, R. M.), pp. 139–161. Cambridge University Press.Google Scholar
Thompson, P. A. 1982 Compressible Fluid Dynamics. McGraw-Hill.Google Scholar
Visbal, M. & Gordnier, R. 1995 Compressibility effects on vortex breakdown onset above a 75-degree sweep delta wing. J. Aircraft 32 (5), 936–942.Google Scholar
Wang, S. & Rusak, Z. 1997a 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. 1997b The effect of slight viscosity on a near-critical swirling flow in a pipe. Phys. Fluids 9 (7), 1914–1927.Google Scholar