A weakly nonlinear mechanism for mode selection in swirling jets

Abstract

Global linear and nonlinear bifurcation analysis is used to revisit the spiral vortex breakdown of nominally axisymmetric swirling jets. For the parameters considered herein, stability analyses single out two unstable linear modes of azimuthal wavenumber $m= \ensuremath{-} 1$ and $m= \ensuremath{-} 2$, bifurcating from the axisymmetric breakdown solution. These modes are interpreted in terms of spiral perturbations wrapped around and behind the axisymmetric bubble, rotating in time in the same direction as the swirling flow but winding in space in the opposite direction. Issues are addressed regarding the role of these modes with respect to the existence, mode selection and internal structure of vortex breakdown, as assessed from the three-dimensional direct numerical simulations of Ruith et al. (J. Fluid Mech., vol. 486, 2003, pp. 331–378). The normal form describing the leading-order nonlinear interaction between modes is computed and analysed. It admits two stable solutions corresponding to pure single and double helices. At large swirl, the axisymmetric solution bifurcates to the double helix which remains the only stable solution. At low and moderate swirl, it bifurcates first to the single helix, and subsequently to the double helix through a series of subcritical bifurcations yielding hysteresis over a finite range of Reynolds numbers, the estimated bifurcation threshold being in good agreement with that observed in the direct numerical simulations. Evidence is provided that this selection is not to be ascribed to classical mean flow corrections induced by the existence of the unstable modes, but to a non-trivial competition between harmonics. Because the frequencies of the leading modes approach a strong $2$:$1$ resonance, an alternative normal form allowing interactions between the $m= \ensuremath{-} 2$ mode and the first harmonics of the $m= \ensuremath{-} 1$ mode is computed and analysed. It admits two stable solutions, the double helix already identified in the non-resonant case, and a single helix differing from that observed in the non-resonant case only by the presence of a slaved, phase-locked harmonic deformation. On behalf of the finite departure from the $2$:$1$ resonance, the amplitude of the slaved harmonic is however low, and the effect of the resonance on the bifurcation structure is merely limited to a reduction of the hysteresis range.

Coherent structures in a swirl injector at Re = 4800 by nonlinear simulations and linear global modes

“…The swirl number for spiral vortex breakdown is similar in different swirling flows: S = 0.915 for a vortex breakdown bubble (Ruith et al 2003;Meliga et al 2012a) and S = 0.88 for a swirling jet (Oberleithner et al 2011). In the position of the wavemaker in this flow, the swirl number based on the average velocities is Sw = 1.13.…”

“…Furthermore, the strength of the next highest spectral peak (the double-helical mode, POD mode 3-4) is an order of magnitude lower than that of the precessing vortex core. Hence, any nonlinear interactions with other oscillators or coherent structures (such as for the swirling flow in Meliga et al (2012a)) are also likely to be weak and can be ignored.…”

“…The vortex breakdown bubble of the Grabowski vortex (Grabowski & Berger 1976) has been studied by DNS (Ruith et al 2003), by weakly nonlinear analysis (Meliga et al 2012a) and by global temporal stability and sensitivity analyses (Gallaire et al 2006;Qadri et al 2013). The base flow for the Grabowski vortex is axisymmetric and swirling, with a uniform inflow profile for the axial velocity, and a potential vortex for the swirl velocity.…”

Obstacle-induced spiral vortex breakdown

“…They have shown that the single helix observed at low swirl can be viewed as the manifestation of a so-called nonlinear steep global mode (see Huerre et al 2000 for a review), triggered by a transition from convective to absolute local instability in the lee of the axisymmetric bubble. Global linear analysis relaxing this quasiparallel assumption has recently confirmed the interpretation of helical vortex breakdown as a global instability (Meliga et al 2012;Qadri et al 2013).…”

A self-consistent formulation for the sensitivity analysis of finite-amplitude vortex shedding in the cylinder wake

“…For instance, Barkley (2006) shows that the frequency prediction issuing from a linear global stability analysis fails by a large amount, even at Reynolds numbers as low as Re = 80. In the same fashion, the Stuart-Landau amplitude equation describing the onset of limit-cycle oscillations, albeit derived rigorously from the NSE using a multiple-scale expansion (Stuart 1960, see also Sipp & Lebedev 2007;Meliga et al 2009aMeliga et al , 2012a for an application to spatially developing flows) fails to provide correct amplitude and frequency corrections at Reynolds numbers above the bifurcation threshold by only 10%. This is because these approaches are perturbative in nature, and build all oscillating fields as successive-order corrections to the eigenmode feeding on the neutrally stable base flow, whose spatial structure differs considerably from that of the saturated nonlinear oscillation (Dušek et al 1994;Noack et al 2003).…”