Onset of chaos in helical vortex breakdown at low Reynolds number

Abstract

The nonlinear dynamics of a swirling wake flow stemming from a Graboswksi-Berger vortex [Grabowski and Berger, J. Fluid Mech. 75, 525 (1976)] in a semi-infinite domain is addressed at low Reynolds numbers for a fixed swirl number S = 1.095, defined as the ratio between the characteristic tangential velocity and the centerline axial velocity. In this system, only pure hydrodynamic instabilities develop and interact through the quadratic nonlinearities of the Navier-Stokes equations. Such interactions lead to the onset of chaos at a Reynolds value of Re = 220. This chaotic state is reached by following a Ruelle-Takens-Newhouse scenario, which is initiated by a Hopf bifurcation (the spiral vortex breakdown) as the Reynolds number increases. At larger Reynolds value, a frequency synchronization regime appears followed by a chaotic state again. This scenario is corroborated by nonlinear time series analyses. Stability analysis around the time-average flow and temporal-azimuthal Fourier decomposition of the nonlinear flow distributions both identify successfully the developing vortices and provide deeper insight into the development of the flow patterns leading to this route to chaos. Three single-helical vortices are involved: the primary spiral associated with the spiral vortex breakdown, a downstream spiral, and a near-wake spiral. As the Reynolds number increases, the frequencies of these vortices become closer, increasing their interactions by nonlinearity to eventually generate a strong chaotic axisymmetric oscillation.

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

“…The red dots are the neutrally stable limit of the flow for the azimuthal wavenumber . These curves accurately fall on the stable–unstable limit of the bifurcation branches, which has already been observed for swirling flows (Meliga & Gallaire 2011; Pasche, Avellan & Gallaire 2018 a ). The blue dots represent the neutrally stable limit of the flow concerning the azimuthal wavenumber , i.e.…”

“…The displacement of the obstacle can be modelled as a stationary m = 1 mode that by nonlinear interactions produces an axisymmetric pulsation of the flow. This mechanism has been reported for the spiral vortex breakdown (Pasche et al 2018a) and hydraulic turbines (Pasche et al 2019) as a plunging wave. The quasiperiodic or chaotic state persists for increasing lateral displacements up to the point where the vortex stabilizes as a horseshoe vortex with a strong and weak secondary vortex circumventing the obstacle.…”

“…This mechanism has been reported for the spiral vortex breakdown (Pasche et al. 2018 a ) and hydraulic turbines (Pasche et al. 2019) as a plunging wave.…”

“…2019) and in the spiral vortex breakdown where several single helical modes generate axisymmetric pulsations (Pasche et al. 2018 a ).

Figure 13.Amplitude Fourier spectrum decomposed in azimuthal wavenumber of the 3-D DNS at for azimuthal velocity and , , and .

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Dynamics and bifurcations of laminar annular swirling and non-swirling jets

“…The fluids community has invested significant theoretical efforts towards understanding and modelling the dynamics of many swirling flows. Though a detailed bifurcation analysis of swirling annular jets is lacking, similar analyses have been performed for several laminar swirling flow configurations, including the Grabowski-Berger vortex model (Meliga, Gallaire & Chomaz 2012;Pasche, Avellan & Gallaire 2018) and swirling circular jets (Meliga & Gallaire 2011;Montagnani 2018;Moise & Mathew 2019;Douglas et al 2021b;Douglas & Lesshafft 2022). Classical linear stability analyses of parallel swirling jets and wakes have also been performed (Loiseleux, Chomaz & Huerre 1998;Loiseleux, Delbende & Huerre 2000).…”