Direct numerical simulation of vortex ring evolution from the laminar to the early turbulent regime

Abstract

Direct numerical simulation is used to study the temporal development of single vortex rings at various Reynolds numbers and core thicknesses. Qualitative differences between the evolution of thin-and thick-core rings are observed leading to a correction factor to the classical equation for the ring translational velocity. We compare the obtained linear modal growth rates with previous work, highlighting the role of the wake in triply periodic numerical simulations. The transition from a laminar to a turbulent ring is marked by the rearrangement of the outer core vorticity into a clearly defined secondary structure. The onset of the fully turbulent state is associated with shedding of the structure in a series of hairpin vortices. A Lagrangian particle analysis was performed to determine the ring entrainment and detrainment properties and to investigate the possibility of an axial flow being generated around the circumference of the core region prior to the onset of turbulence.

Optimal transient growth on a vortex ring and its transition via cascade of ringlets

“…At t = 8, shown in figure 6(l), secondary vortical structures are formed around the original vortex core apart from the finger-like structures outside the core. These secondary structures have previously been termed as 'halo' vorticity and were also observed in previous DNS and experimental works (Bergdorf et al 2007;Archer et al 2008;Sullivan et al 2008). The finger-like structures are outcomes of the optimal transient energy growth, and their forms are similar to the well reported lobe structures, which however are located in the vortex core (Shadden, Dabiri & Marsden 2006).…”

“…From the outcome of the corresponding optimal initial perturbations (not shown here), these optimal growths are associated with the core of the ring and dominated by the elliptic instability induced by the elliptic streamlines of the base flow in the core region (Kerswell 2002;Blanco-Rodríguez & Le Dizèz 2016a,b). The wavenumber of the elliptic unstable modes has been found to vary according to the Transient growth on a vortex ring and its transition via cascade of ringlets ring's slenderness ratio (core/ring radius) and Re, and 8-10 azimuthal standing waves have been observed in previous DNSs and experiments (Archer et al 2008;Sullivan et al 2008). Over short times, e.g.…”

“…Mao and F. Hussain (Re ≡ circulation/viscosity) and core thicknesses (James & Madnia 1996;Bergdorf, Koumoutsakos & Leonard 2007;Archer, Thomas & Coleman 2008). These studies detail various instabilities, highlighting the formation of secondary vortical structures, which deposit hairpin vortices into the ring's wake.…”

“…After initialisation, the unperturbed base flow can be obtained by two-dimensional DNS over a time period of τ = 25 and saved every dT = 0.1. A quick relaxation process is adopted, as the Lamb-Oseen vortex profile along a curved axis is not an exact solution of the NS equations (Bergdorf et al 2007;Archer et al 2008;Mao, Sherwin & Blackburn 2012). The velocity boundary conditions ofū = 0.225,v = 0 andw = 0 are prescribed on the far-field boundaries to account for the self-induced axial motion.…”

“…These studies detail various instabilities, highlighting the formation of secondary vortical structures, which deposit hairpin vortices into the ring's wake. In this process, the onset and growth of instability modes with azimuthal wavenumbers 6, 8, 9 and 10 were observed, depending on the ring's Re and core thickness (Bergdorf et al 2007;Archer et al 2008). All these DNS studies applied random noise as the initial perturbation, and focused on identifying high wavenumber modes concentrated around the core of the vortex ring.…”