Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-19T16:53:21.695Z Has data issue: false hasContentIssue false
  • English
  • Français

A sensitivity study of vortex breakdown onset to upstream boundary conditions

Published online by Cambridge University Press:  29 January 2010

BENJAMIN LECLAIRE  and
DENIS SIPP
BENJAMIN LECLAIRE*
Affiliation:
Office National d'Etudes et de Recherche Aérospatiales (ONERA), Department of Fundamental and Experimental Aerodynamics, 8 rue des Vertugadins, 92190 Meudon, France
DENIS SIPP
Affiliation:
Office National d'Etudes et de Recherche Aérospatiales (ONERA), Department of Fundamental and Experimental Aerodynamics, 8 rue des Vertugadins, 92190 Meudon, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper theoretically investigates the influence of the upstream boundary conditions on the bifurcation structure leading to vortex breakdown. The axisymmetric flow of an inviscid fluid in a pipe of constant cross-section and finite axial length is considered. Solutions bifurcating from the columnar solution at criticality are analysed via a weakly nonlinear expansion and computed in the fully nonlinear regime using numerical continuation, until a centreline recirculation is found at the pipe outlet. Bifurcation diagrams are determined for a parametric family of inflows describing a wide range of axial and azimuthal profiles, the third inlet condition being chosen either as a fixed azimuthal vorticity or as a vanishing radial velocity. Including the traditional picture given by Wang & Rusak (J. Fluid Mech., vol. 340, 1997a, p. 177), six different diagrams are found to be possible. In particular, a scenario of smooth transition to breakdown may exist as the swirl is increased, with no loss of stability and no hysteresis, breakdown appearing for swirl levels larger than the critical swirl in a pipe. This transition involves a new type of flow akin to a pre-breakdown flow. Our results, furthermore, suggest that rigidly rotating Poiseuille flow could correspond to the limit for which breakdown is impossible because it is predicted at infinitely large swirl numbers. We finally find that flows with a large rotational core are particularly sensitive to an accurate modelling of the upstream boundary conditions, weakly confined vortices being much more robust.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 645 , 25 February 2010 , pp. 81 - 119
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Althaus, W., Bruecker, Ch. & Weimer, M. 1995 Breakdown of slender vortices. In Fluid Vortices: Fluid Mechanics and Its Applications (ed. Green, S. I.), pp. 373–426. Kluwer.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593–629.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
Billant, P., Chomaz, J.-M. & Huerre, P. 1998 Experimental study of vortex breakdown in swirling jets. J. Fluid Mech. 376, 183–219.CrossRefGoogle Scholar
Brown, G. L. & Lopez, J. M. 1990 Axisymmetric vortex breakdown. II. 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
Delery, J. 1994 Aspects of vortex breakdown. Prog. Aerospace Sci. 30, 1–59.CrossRefGoogle Scholar
Faler, J. H. & Leibovich, S. 1978 An experimental map of the internal structure of a vortex breakdown. J. Fluid Mech. 86, 313–335.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), 274–286.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
Goldshtik, M. & Hussain, F. 1997 The nature of inviscid vortex breakdown. Phys. Fluids 9, 263.CrossRefGoogle Scholar
Govaerts, W. 2000 Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM.CrossRefGoogle Scholar
Grabowski, W. J. & Berger, S. A. 1976 Solutions of the Navier–Stokes equations for vortex breakdown. J. Fluid Mech. 75, 525–544.CrossRefGoogle Scholar
Grimshaw, R. 1990 Resonant flow of a rotating fluid past an obstacle: the general case. Stud. Appl. Math. 83, 249–269.CrossRefGoogle Scholar
Grimshaw, R. & Yi, Z. 1993 Resonant generation of finite-amplitude waves by the uniform flow of a uniformly rotating fluid past an obstacle. Mathematika 40, 30–50.CrossRefGoogle Scholar
Hafez, M., Kuruvila, G. & Salas, M. D. 1986 Numerical study of vortex breakdown. Appl. Numer. Math. 2 (3–5), 291–302.CrossRefGoogle Scholar
Hanazaki, H. 1996 On the wave excitation and the formation of recirculation eddies in an axisymmetric flow of uniformly rotating fluids. J. Fluid Mech. 322, 165–200.CrossRefGoogle Scholar
Keller, H. B. 1977 Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory, pp. 359–384. Academic Press.Google Scholar
Keller, J. J., Egli, W. & Exley, J. 1985 Force- and loss-free transitions between flow states. J. Appl. Math. Phys. 36, 854–889.Google Scholar
Kopecky, R. M. & Torrance, K. E. 1973 Initiation and structure of axisymmetric eddies in a rotating stream. Comput. Fluids 1, 289–300.CrossRefGoogle Scholar
Kuznetsov, Y. A. 2004 Elements of Applied Bifurcation Theory. Springer.CrossRefGoogle Scholar
Leclaire, B., Jacquin, L. & Sipp, D. 2007 a The generating conditions of a high-Reynolds-number swirling jet. In Fifth Turbulence and Shear Flow Phenomena Sympos., Munich, Germany, August 27–29.Google Scholar
Leclaire, B., Sipp, D. & Jacquin, L. 2007 b Near-critical swirling flow in a contracting duct: the case of plug axial flow with solid-body rotation. Phys. Fluids 19 (9), 091701.CrossRefGoogle Scholar
Leibovich, S. 1984 Vortex stability and breakdown: survey and extension. AIAA J. 22 (9), 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
Liang, H. & Maxworthy, T. 2005 An experimental investigation of swirling jets. J. Fluid Mech. 525, 115–159.CrossRefGoogle Scholar
Lopez, J. M. 1994 On the bifurcation structure of axisymmetric vortex breakdown in a constricted pipe. Phys. Fluids 6 (11), 3683–3693.CrossRefGoogle Scholar
Lucca-Negro, O. & O'Doherty, T. 2001 Vortex breakdown: a review. Progr. Energy Combust. Sci. 27 (4), 431–481.CrossRefGoogle 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
Mishra, A., Tadmor, G. & Rusak, Z. 2008 On the dynamics of axisymmetric vortex breakdown in a pipe. In Fifth AIAA Theoretical Fluid Mechanics Conf., Seattle, WA, June 23–26.Google 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. ARC Tech. Rep. CP-508.Google Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148–154.Google 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.CrossRefGoogle Scholar
Rusak, Z. & Judd, K. P. 2001 The stability of noncolumnar 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. & Meder, C. C. 2004 Near-critical swirling flow in a slightly contracting pipe. AIAA J. 42 (11), 2284–2293.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
Salas, M. D. & Kuruvila, G. 1989 Vortex breakdown simulation: a circumspect study of the steady, laminar, axisymmetric model. Comput. Fluids 17 (1), 247–262.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
Snyder, D. O. & Spall, R. E. 2000 Numerical simulation of bubble-type vortex breakdown within a tube-and-vane apparatus. Phys. Fluids 12 (3), 603–608.CrossRefGoogle Scholar
Squire, H. B. 1960 Analysis of the vortex breakdown phenomenon, part I. Imperial College, Aero. Dept. Rep. 102.Google Scholar
Wang, S. & Rusak, Z. 1996 a On the stability of an axisymmetric rotating flow in a pipe. Phys. Fluids 8 (4), 1007–1016.CrossRefGoogle Scholar
Wang, S. & Rusak, Z. 1996 b On the stability of non-columnar swirling flows. Phys. Fluids 8 (4), 1017–1023.CrossRefGoogle Scholar
Wang, S. & Rusak, Z. 1997 a 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. 1997 b The effect of slight viscosity on a near-critical swirling flow in a pipe. Phys. Fluids 9 (7), 1914–1927.CrossRefGoogle Scholar