Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T21:46:12.001Z Has data issue: false hasContentIssue false
  • English
  • Français

Vortex impingement onto an axisymmetric obstacle – subcritical bifurcation to vortex breakdown

Published online by Cambridge University Press:  15 January 2021

S. Pasche Open the ORCID record for S. Pasche [Opens in a new window] ,
F. Avellan  and
F. Gallaire
S. Pasche*
Affiliation:
Linné FLOW centre, Department of Mechanics, KTH, SE-100 44Stockholm, Sweden
F. Avellan
Affiliation:
Laboratory for Hydraulic Machines, Ecole Polytechnique Fédérale de Lausanne, CH-1007 Lausanne, Switzerland
F. Gallaire
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
*
†Email address for correspondence: simon.pasche@alumni.epfl.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

A swirling wake flow submitted to an adverse pressure gradient is studied by bifurcation analysis, modal analysis and direct numerical simulations. In contrast to experiments in diverging tubes, the adverse pressure gradient is imposed by the presence of a downstream axisymmetric obstacle centred on the vortex axis. Different adverse pressure gradients are investigated by modifying the obstacle radius, which results in the deceleration of the vortex axial velocity. Hence, vortex breakdown occurs for a sufficiently large pressure gradient. We observe a spiral vortex breakdown type without any recirculation bubble, which contrasts with classical spiral vortex breakdown developing in the bubble wake. A weakly nonlinear analysis is performed to characterize this self-sustained instability. The resulting Landau equation reveals the sub-critical character of this Hopf bifurcation, highlighting a sub-critical vortex breakdown. In addition, the stabilization mechanism of this spiral vortex breakdown caused by an off-centre displacement of the downstream axisymmetric obstacle is investigated by direct numerical simulations. Nonlinear dynamics, such as a quasi-periodic state, is observed as a consequence of nonlinear interactions between the spiral vortex breakdown and the misalignment of the obstacle before the stabilization occurs.

JFM classification

Vortex Flows: Vortex breakdown
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 910 , 10 March 2021 , A36
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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

Alekseenko, S.V, Kuibin, P.A., Okulov, V.L. & Shtork, S.I. 1999 Helical vortices in swirl flow. J. Fluid Mech. 382, 195–243.CrossRefGoogle Scholar
Althaus, W., Krause, E., Hofhaus, J. & Weimer, M. 1994 Vortex breakdown: transition between bubble- and spiral-type breakdown. Meccanica 29 (4), 373–382.CrossRefGoogle Scholar
Benjamin, T.B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14 (4), 593–629.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
Citro, V., Tchoufag, J., Fabre, D., Giannetti, F. & Luchini, P. 2016 Linear stability and weakly nonlinear analysis of the flow past rotating spheres. J. Fluid Mech. 807, 62–86.CrossRefGoogle Scholar
Delbende, I., Chomaz, J.-C. & Huerre, P. 1998 Absolute/convective instabilities in the batchelor vortex: a numerical study of the linear impulse response. J. Fluid Mech. 355, 229–254.CrossRefGoogle Scholar
Delery, J.M. 1994 Aspects of vortex breakdown. Prog. Aerosp. Sci. 30 (1), 1–59.Google Scholar
Dusek, J., Gal, P.L. & Fraunié, P. 1994 A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake. J. Fluid Mech. 264, 59–80.CrossRefGoogle Scholar
Escudier, M.P. 1984 Observations of the flow produced in a cylindrical container by a rotating endwall. Exp. Fluids 2 (4), 189–196.CrossRefGoogle Scholar
Fernandez-Feria, R. 1996 Viscous and inviscid instabilities of non-parallel self-similar axisymmetric vortex cores. J. Fluid Mech. 323, 339–365.CrossRefGoogle Scholar
Fischer, P.F., Lottes, J.W. & Kerkemeier, S.G. 2008 nek5000. Available at: http://nek5000.mcs.anl.gov.Google Scholar
Fujimura, K. 1991 Methods of centre manifold and multiple scales in the theory of weakly nonlinear stability for fluid motions. Proc. R. Soc. Lond. A 434, 719–733.Google Scholar
Gallaire, F. & Chomaz, J.M. 2003 Instability mechanisms in swirling flows. Phys. Fluids 15 (9), 2622–2639.Google Scholar
Gallaire, F., Ruith, M., Meiburg, E., Chomaz, J.-M. & Huerre, P. 2006 Spiral vortex breakdown as a global mode. J. Fluid Mech. 549, 71–80.CrossRefGoogle Scholar
Hall, M.G. 1972 Vortex breakdown. Annu. Rev. Fluid Mech. 4, 195–218.CrossRefGoogle Scholar
Heaton, C.J., Nichols, J.W. & Schmid, P.J. 2009 Global linear stability of the non-parallel batchelor vortex. J. Fluid Mech. 629, 139–160.CrossRefGoogle Scholar
Hecht, F. 2012 New development in freefem$++$. J. Numer. Maths 20 (3–4), 251–265.Google Scholar
Herrada, M.A. & Fernandez-Feria, R. 2006 On the development of three-dimensional vortex breakdown in cylindrical regions. Phys. Fluids 18 (8), 084105.Google Scholar
Jochmann, P., Sinigersky, A., Hehle, M., Schäfer, O., Koch, R. & Bauer, H.-J. 2006 Numerical simulation of a precessing vortex breakdown. Intl J. Heat Fluid Flow 27 (2), 192–203.CrossRefGoogle Scholar
Jones, M.C., Hourigan, K. & Thompson, M.C. 2015 A study of the geometry and parameter dependence of vortex breakdown. Phys. Fluids 27 (4), 044102.CrossRefGoogle Scholar
Krause, E. 1985 A contribution to the problem of vortex breakdown. Comput. Fluids 13 (3), 375–381.CrossRefGoogle Scholar
Lambourne, N.C. & Bryer, D.W. 1962 The bursting of leading-edge vortices – some observations and discussion of the phenomenon. Aero. Res. Counc. R&M 3292, 1–35.Google Scholar
Leibovich, S. 1978 The structure of the vortex breakdown. Annu. Rev. Fluid Mech. 10, 221–246.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 (11), 3683–3693.CrossRefGoogle Scholar
Lopez, J.M. 2006 Rotating and modulated rotating waves in transitions of an enclosed swirling flow. J. Fluid Mech. 553, 323–346.CrossRefGoogle Scholar
Lucca-Negro, O. & O'Doherty, T. 2001 Vortex breakdown: a review. Prog. Energy Combust. Sci. 27, 431–481.CrossRefGoogle Scholar
Mager, A. 1972 Dissipation and breakdown of a wing-tip vortex. J. Fluid Mech. 55 (4), 609–628.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
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 (8), 084102.CrossRefGoogle Scholar
Meliga, P., Gallaire, F. & Chomaz, J.-M. 2012 A weakly nonlinear mechanism for mode selection in swirling jets. J. Fluid Mech. 699, 216–262.Google Scholar
Moise, P. & Mathew, J. 2019 Bubble and conical forms of vortex breakdown in swirling jets. J. Fluid Mech. 873, 322–357.Google Scholar
Naumov, I.V. & Podolskaya, I.Yu. 2017 Topology of vortex breakdown in closed polygonal containers. J. Fluid Mech. 820, 263–283.CrossRefGoogle Scholar
Nishi, M., Matsunaga, S., Kubota, T. & Senoo, A. 1984 Surging characteristics of conical and elbow-type draft tubes. In Proceedings of the 12th IAHR Symposium on Hydraulic Machinery and System, Stirling, Scotland.Google Scholar
Oberleithner, K., Sieber, M., Nayeri, C.N., Petz, C., Hege, H.-C., Noack, R.R. & Wygnanski, I. 2011 Three-dimensional coherent strucutres in a swirling jet undergoing vortex breakdown: stability analysis and empirical mode construction. J. Fluid Mech. 679, 383–414.CrossRefGoogle Scholar
Oberleithner, K., Stöhr, M., Im, S.H., Arndt, C.M. & Steinberg, A.M. 2015 Formation and flame-induced suppression of the precessing vortex core in a swirl combustor: experiments and linear stability analysis. Combust. Flame 162 (8), 3100–3114.CrossRefGoogle Scholar
Okulov, V.L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319–342.Google Scholar
Pagan, D. & Benay, R. 1987 Vortex breakdown induced by an adverse pressure gradient – experimental and numerical approaches. In 5th Applied Aerodynamics Conference, AIAA meeting Monterey, CA. Available at: https://doi.org/10.2514/6.1987-2478.CrossRefGoogle Scholar
Pasche, S., Avellan, F. & Gallaire, F. 2017 Part load vortex rope as a global unstable mode. Trans. ASME: J. Fluids Engng 139 (5), 051102.Google Scholar
Pasche, S., Avellan, F. & Gallaire, F. 2018 a Onset of chaos in helical vortex breakdown at low Reynolds number. Phys. Rev. Fluids 3, 064701.CrossRefGoogle Scholar
Pasche, S., Gallaire, F. & Avellan, F. 2018 b Predictive control of spiral vortex breakdown. J. Fluid Mech. 842, 58–86.CrossRefGoogle Scholar
Pasche, S., Gallaire, F. & Avellan, F. 2019 Origin of the synchronous pressure fluctuations in the draft tube of Francis turbines operating at part load conditions. J. Fluids Struct. 86, 13–33.CrossRefGoogle Scholar
Pasche, S., Gallaire, F., Dreyer, M. & Farhat, M. 2014 Obstacle-induced spiral vortex breakdown. Exp. Fluids 55, 1784.Google Scholar
Paschereit, C.O., Flohr, P. & Gutmark, E.J. 2002 Combustion control by vortex breakdown stabilization. Trans. ASME: J. Turbomach. 128, 679–688.Google Scholar
Qadri, U.A., Mistry, D. & Juniper, M.P. 2013 Structural sensitivity of spiral vortex breakdown. J. Fluid Mech. 720, 558–581.CrossRefGoogle Scholar
Rockwell, D. 1998 Vortex-body interactions. Annu. Rev. Fluid Mech. 30 (1), 199–229.CrossRefGoogle 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., 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., Whiting, C.H. & Wang, S. 1998 Axisymmetric breakdown of a $q$-vortex in a pipe. AIAA J. 36 (10), 1848–1853.CrossRefGoogle Scholar
Rusak, Z., Zhang, Y., Lee, H. & Wang, S. 2017 Swirling flow states in finite-length diverging or contracting circular pipes. J. Fluid Mech. 819, 678–712.CrossRefGoogle Scholar
Salinger, A.G., Burroughs, E.A, Pawlowski, R.P., Phipps, E.T. & Romero, L.A. 2005 Bifurcation tracking algorithms and software for large scale applications. Intl J. Bifurcation Chaos 15 (03), 1015– 1032.CrossRefGoogle Scholar
Sarpkaya, T. 1971 On stationary and travelling vortex breakdowns. J. Fluid Mech. 45 (3), 545–559.CrossRefGoogle Scholar
Sarpkaya, T. 1974 Effect of the adverse pressure gradient on vortex breakdown. AIAA J. 12 (5), 602–607.Google Scholar
Serre, E. & Bontoux, P. 2002 Vortex breakdown in a three-dimensional swirling flow. J. Fluid Mech. 459, 347–370.CrossRefGoogle Scholar
Shtern, V. & Hussain, F. 1999 Collapse, symmetry breaking, and hysteresis in swirling flows. Annu. Rev. Fluid Mech. 31 (1), 537–566.CrossRefGoogle Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.CrossRefGoogle Scholar
Sorensen, J.N., Naumov, I.V. & Okulov, V.L. 2011 Multiple helical modes of vortex breakdown. J. Fluid Mech. 683, 430–441.CrossRefGoogle Scholar
Sotiropoulos, F., Ventikos, Y. & Lackey, T.C. 2001 Chaotic advection in three-dimensional stationary vortex-breakdown bubbles: šil'nikov's chaos and the devil's staircase. J. Fluid Mech. 444, 257–297.Google Scholar
Spall, R.E., Gatski, T.B. & Ash, R.L. 1990 The structure and dynamics of bubble-type vortex breakdown. Proc. R. Soc. Lond. A 429, 613–637.Google Scholar
Spohn, A., Moiry, M. & Hopfinger, E.J. 1993 Observations of vortex breakdown in an open cylindrical container with a rotating bottom. Exp. Fluids 14, 70.CrossRefGoogle Scholar
Susan-Resiga, R., Vu, T.C., Muntean, S., Ciocan, G.D. & Nennemann, B. 2006 Jet control of the draft tube vortex rope in Francis turbines at partial discharge. In Proceedings of the 23th IAHR Symposium on Hydraulic Machinery and Systems, Yokohama, Japan.Google Scholar
Tammisola, O. & Juniper, M.P. 2016 Coherent structures in a swirl injector at ${R}e = 4800$ by nonlinear simulations and linear global modes. J. Fluid Mech. 792, 620–657.CrossRefGoogle Scholar
Viola, F. 2016 Resonance in swirling wakes and sloshing waves non-normal and sublinear effects. Thesis, EPFL.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.CrossRefGoogle Scholar