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Vortex pairing in jets as a global Floquet instability: modal and transient dynamics

Published online by Cambridge University Press:  16 January 2019

Léopold Shaabani-Ardali ,
Denis Sipp  and
Lutz Lesshafft Open the ORCID record for Lutz Lesshafft [Opens in a new window]
Léopold Shaabani-Ardali*
Affiliation:
LadHyX, École polytechnique–CNRS, 91120 Palaiseau, France DAAA, ONERA, Université Paris-Saclay, 92190 Meudon, France
Denis Sipp
Affiliation:
DAAA, ONERA, Université Paris-Saclay, 92190 Meudon, France
Lutz Lesshafft
Affiliation:
LadHyX, École polytechnique–CNRS, 91120 Palaiseau, France
*
†Email address for correspondence: [email protected]
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Abstract

The spontaneous pairing of rolled-up vortices in a laminar jet is investigated as a global secondary instability of a time-periodic spatially developing vortex street. The growth of subharmonic perturbations, associated with vortex pairing, is analysed both in terms of modal Floquet instability and in terms of transient growth dynamics. The article has the double objective to outline a toolset for the global analysis of time-periodic flows, and to leverage such an analysis for a fresh view on the vortex pairing phenomenon. Axisymmetric direct numerical simulations (DNS) of jets with single-frequency inflow forcing are performed, in order to identify combinations of the Reynolds and Strouhal numbers for which vortex pairing is naturally observed. The same DNS calculations are then repeated with an added time-delay control term, which artificially suppresses pairing, so as to obtain time-periodic unpaired base flows for linear stability analysis. It is demonstrated that the natural occurrence of vortex pairing in nonlinear DNS coincides with a linear subharmonic Floquet instability of the underlying unpaired vortex street. However, DNS results suggest that the onset of pairing involves much stronger temporal growth of subharmonic perturbations than that predicted by modal Floquet analysis, as well as a spatial distribution of these fast-growing perturbation structures that is inconsistent with the unstable Floquet mode. Singular value decomposition of the phase-shift operator (the operator that maps a given perturbation field to its state one flow period later) is performed for an analysis of optimal transient growth in the vortex street. Non-modal mechanisms near the jet inlet are thus found to provide a fast route towards the limit-cycle regime of established vortex pairing, in good agreement with DNS observations. It is concluded that modal Floquet analysis accurately predicts the parameter regime where sustained vortex pairing occurs, but that the bifurcation scenario under typical conditions is dominated by transient growth phenomena.

JFM classification

Wakes/Jets: Jets Wakes/Jets: Vortex streets
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 862 , 10 March 2019 , pp. 951 - 989
Copyright
© 2019 Cambridge University Press 

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References

Arbey, H. & Ffowcs Williams, J. E. 1984 Active cancellation of pure tones in an excited jet. J. Fluid Mech. 149, 445–454.Google Scholar
Arratia, C., Caulfield, C. P. & Chomaz, J.-M. 2013 Transient perturbation growth in time-dependent mixing layers. J. Fluid Mech. 717, 90–133.Google Scholar
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57 (9), 1435–1458.Google Scholar
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.Google Scholar
Becker, H. A. & Massaro, T. A. 1968 Vortex evolution in a round jet. J. Fluid Mech. 31 (03), 435–448.Google Scholar
Blackburn, H. M., Sherwin, S. J. & Barkley, D. 2008 Convective instability and transient growth in steady and pulsatile stenotic flows. J. Fluid Mech. 607, 267–277.Google Scholar
Bogey, C. & Bailly, C. 2010 Influence of nozzle-exit boundary-layer conditions on the flow and acoustic fields of initially laminar jets. J. Fluid Mech. 663, 507–538.Google Scholar
Borisov, A. V., Kilin, A. A. & Mamaev, I. S. 2013 The dynamics of vortex rings: leapfrogging, choreographies and the stability problem. Regular Chaotic Dyn. 18 (1–2), 33–62.Google Scholar
Borisov, A. V., Kilin, A. A., Mamaev, I. S. & Tenenev, V. A. 2014 The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid. Fluid Dyn. Res. 46 (3), 031415.Google Scholar
Bradley, T. A. & Ng, T. T. 1989 Phase-locking in a jet forced with two frequencies. Exp. Fluids 7 (1), 38–48.Google Scholar
Brancher, P. & Chomaz, J.-M. 1997 Absolute and convective secondary instabilities in spatially periodic shear flows. Phys. Rev. Lett. 78 (4), 658–661.Google Scholar
Bridges, J. E. & Hussain, A. K. M. F. 1987 Roles of initial condition and vortex pairing in jet noise. J. Sound Vib. 117 (2), 289–311.Google Scholar
Broze, G. & Hussain, F. 1994 Nonlinear dynamics of forced transitional jets: periodic and chaotic attractors. J. Fluid Mech. 263, 93–132.Google Scholar
Broze, G. & Hussain, F. 1996 Transitions to chaos in a forced jet: intermittency, tangent bifurcations and hysteresis. J. Fluid Mech. 311, 37–71.Google Scholar
Cheng, M. & Chang, H.-C. 1992 Subharmonic instabilities of finite-amplitude monochromatic waves. Phys. Fluids A 4 (3), 505–523.Google Scholar
Cheng, M., Lou, J. & Lim, T. T. 2015 Leapfrogging of multiple coaxial viscous vortex rings. Phys. Fluids 27 (3), 031702.Google Scholar
Delbende, I., Piton, B. & Rossi, M. 2015 Merging of two helical vortices. Eur. J. Mech. (B/Fluids) 49, 363–372.Google Scholar
Felli, M., Camussi, R. & Di Felice, F. 2011 Mechanisms of evolution of the propeller wake in the transition and far fields. J. Fluid Mech. 682, 5–53.Google Scholar
Floquet, G. 1883 Sur les équations différentielles linéaires à coefficients périodiques. Ann. Sci. École Norm. Sup. 12, 47–88.Google Scholar
Garnaud, X., Lesshafft, L., Schmid, P. J. & Huerre, P. 2013 Modal and transient dynamics of jet flows. Phys. Fluids 25 (4), 044103.Google Scholar
Hajj, M. R., Miksad, R. W. & Powers, E. J. 1992 Subharmonic growth by parametric resonance. J. Fluid Mech. 236, 385–413.Google Scholar
Hajj, M. R., Miksad, R. W. & Powers, E. J. 1993 Fundamental–subharmonic interaction: effect of phase relation. J. Fluid Mech. 256, 403–426.Google Scholar
Hall, K. C., Thomas, J. P. & Clark, W. S. 2002 Computation of unsteady nonlinear flows in cascades using a harmonic balance technique. AIAA J. 40 (5), 879–886.Google Scholar
Hecht, F. 2012 New development in FreeFem + +. J. Numer. Math. 20 (3–4), 251–265.Google Scholar
Ho, C.-M. & Huang, L.-S. 1982 Subharmonics and vortex merging in mixing layers. J. Fluid Mech. 119, 443–473.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22 (1), 473–537.Google Scholar
Husain, H. S. & Hussain, F. 1989 Subharmonic resonance in a shear layer. In Advances in Turbulence 2, pp. 96–101. Springer.Google Scholar
Husain, H. S. & Hussain, F. 1995 Experiments on subharmonic resonance in a shear layer. J. Fluid Mech. 304, 343–372.Google Scholar
Hussain, A. K. M. F. & Zaman, K. B. M. Q. 1980 Vortex pairing in a circular jet under controlled excitation. Part 2. Coherent structure dynamics. J. Fluid Mech. 101 (03), 493–544.Google Scholar
Inoue, O. 2002 Sound generation by the leapfrogging between two coaxial vortex rings. Phys. Fluids 14 (9), 3361–3364.Google Scholar
Jallas, D., Marquet, O. & Fabre, D. 2017 Linear and nonlinear perturbation analysis of the symmetry breaking in time-periodic propulsive wakes. Phys. Rev. E 95 (6), 063111.Google Scholar
Johnson, H. G., Brion, V. & Jacquin, L. 2016 Crow instability: nonlinear response to the linear optimal perturbation. J. Fluid Mech. 795, 652–670.Google Scholar
Kibens, V. 1980 Discrete noise spectrum generated by acoustically excited jet. AIAA J. 18 (4), 434–441.Google Scholar
Lust, K. & Roose, D. 1998 An adaptive Newton–Picard algorithm with subspace iteration for computing periodic solutions. SIAM J. Sci. Comput. 19 (4), 1188–1209.Google Scholar
Mankbadi, R. R. 1985 On the interaction between fundamental and subharmonic instability waves in a turbulent round jet. J. Fluid Mech. 160, 385–419.Google Scholar
Meunier, P., Ehrenstein, U., Leweke, T. & Rossi, M. 2002 A merging criterion for two-dimensional co-rotating vortices. Phys. Fluids 14 (8), 2757–2766.Google Scholar
Meynart, R. 1983 Speckle velocimetry study of vortex pairing in a low-Re unexcited jet. Phys. Fluids 26 (8), 2074–2079.Google Scholar
Michalke, A. 1971 Instabilität eines kompressiblen runden Freistrahls unter Berücksichtigung des Einflusses der Strahlgrenzschichtdicke. Z. Flugwiss. 19, 319–328; English translation: NASA Tech. Memo. 75190 (1977).Google Scholar
Monkewitz, P. A. 1988 Subharmonic resonance, pairing and shredding in the mixing layer. J. Fluid Mech. 188, 223–252.Google Scholar
Narayanan, S. & Hussain, F.1997 Chaos control in open flows – experiments in a circular jet. AIAA Paper 97-1822.Google Scholar
Nek 5000 Version 1.0 rc1/svn r1094, Argonne National Laboratory, Illinois. Available:https://nek5000.mcs.anl.gov.Google Scholar
Nichols, J. W. & Lele, S. K. 2011 Global modes and transient response of a cold supersonic jet. J. Fluid Mech. 669, 225–241.Google Scholar
Paschereit, C. O., Wygnanski, I. & Fiedler, H. E. 1995 Experimental investigation of subharmonic resonance in an axisymmetric jet. J. Fluid Mech. 283, 365–407.Google Scholar
Petersen, R. A. 1978 Influence of wave dispersion on vortex pairing in a jet. J. Fluid Mech. 89 (03), 469–495.Google Scholar
Raman, G. & Rice, E. J. 1991 Axisymmetric jet forced by fundamental and subharmonic tones. AIAA J. 29 (7), 1114–1122.Google Scholar
Roose, D., Lust, K., Champneys, A. & Spence, A. 1995 A Newton–Picard shooting method for computing periodic solutions of large-scale dynamical systems. Chaos, Solitons Fractals 5 (10), 1913–1925.Google Scholar
Saad, Y. 2011 Numerical Methods for Large Eigenvalue Problems: Revised Edition. SIAM.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Sánchez, J. & Net, M. 2010 On the multiple shooting continuation of periodic orbits by Newton–Krylov methods. Intl J. Bifurcation Chaos 20 (01), 43–61.Google Scholar
Sánchez, J., Net, M., Garcıa-Archilla, B. & Simó, C. 2004 Newton–Krylov continuation of periodic orbits for Navier–Stokes flows. J. Comput. Phys. 201 (1), 13–33.Google Scholar
Schram, C.2003 Aeroacoustics of subsonic jets: prediction of the sound produced by vortex pairing based on particle image velocimetry. PhD thesis, Technische Universiteit Eindhoven.Google Scholar
Schram, C., Taubitz, S., Anthoine, J. & Hirschberg, A. 2005 Theoretical/empirical prediction and measurement of the sound produced by vortex pairing in a low Mach number jet. J. Sound Vib. 281 (1), 171–187.Google Scholar
Selçuk, C., Delbende, I. & Rossi, M. 2017a Helical vortices: linear stability analysis and nonlinear dynamics. Fluid Dyn. Res. 50 (1), 011411.Google Scholar
Selçuk, C., Delbende, I. & Rossi, M. 2017b Helical vortices: quasiequilibrium states and their time evolution. Phys. Rev. Fluids 2 (8), 084701.Google Scholar
Shaabani-Ardali, L., Sipp, D. & Lesshafft, L. 2017 Time-delayed feedback technique for suppressing instabilities in time-periodic flow. Phys. Rev. Fluids 2 (11), 113904.Google Scholar
Shariff, K. & Leonard, A. 1992 Vortex rings. Annu. Rev. Fluid Mech. 24 (1), 235–279.Google Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.Google Scholar
Thomas, J. P., Dowell, E. H. & Hall, K. C. 2002 Nonlinear inviscid aerodynamic effects on transonic divergence, flutter, and limit-cycle oscillations. AIAA J. 40 (4), 638–646.Google Scholar
Tophøj, L. & Aref, H. 2013 Instability of vortex pair leapfrogging. Phys. Fluids 25 (1), 014107.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.Google Scholar
Vermeer, L. J., Sørensen, J. N. & Crespo, A. 2003 Wind turbine wake aerodynamics. Prog. Aerosp. Sci. 39 (6–7), 467–510.Google Scholar
Winant, C. D. & Browand, F. K. 1974 Vortex pairing: the mechanism of turbulent mixing-layer growth at moderate Reynolds number. J. Fluid Mech. 63 (2), 237–255.Google Scholar
Zaman, K. B. M. Q. & Hussain, A. K. M. F. 1980 Vortex pairing in a circular jet under controlled excitation. Part 1. General jet response. J. Fluid Mech. 101 (03), 449–491.Google Scholar