On late stages of transition in round jets

Abstract

A round jet is one of the simplest canonical axisymmetric free shear flows that occur both in engineering applications and natural phenomena. A laminar-turbulent transition process is inherent to its development involving multiple stages of instabilities – a topic of intense research due to its importance in mixing and noise generation. This study focuses on the last two stages of this transition process, .i.e. instability of vortex rings and its breakdown to turbulence. Vortex rings are subjected to two types of short wavelength azimuthal insta- bilities: elliptic and curvature, with the former most commonly seen during the transition of round jets. A global linear stability analysis was performed to examine elliptic instability, and the results were compared with asymptotic theories and Direct Numerical Simulations (DNS). The stability analysis explored two types of base flows: Gaussian and equilibrated – a skewed Gaussian, which differ in their core vorticity distribution. The growth rates of the stationary modes are found to be very sensitive to the vorticity distribution, in a way not accounted for in the asymptotic theories. This is demonstrated by the equilibrated ring results where a 9% difference in inviscid growth rates is observed for rings with the same slenderness ratio but evolved with different Reynolds numbers, leading to slight variations in their vorticity distribution. However, inviscid growth rates are very close to asymptotic theory predictions if corrections for the ring radius evolution and equilibrated distribution are considered, while for Gaussian rings, these are smaller by 19 − 33%. This difference in Gaussian rings is attributed to the absence of a local contribution in the vorticity distribution that comes from the deformation of the ring core due to the strain field of induced velocity, which gets naturally included for equilibrated rings. The inclusion of viscosity reduces the growth rates due to damping, which agree very well with those extracted from the DNS for equilibrated rings, but there are differences for the Gaussian rings. Additionally, for viscous cases non-zero frequency rotating modes were isolated for both the base flow types which appeared in multiple branches, physically corresponding to both curvature and elliptic insta- bilities. The large-scale structures from the nonlinear evolution of elliptic instability modes were next explored using DNS and linear stability analysis. The evolution of the most unstable mode leads to the formation of an inner core, wrapped around by halo vorticity, which even- tually breaks down due to nonlinear interactions, yielding a turbulent vortex ring with the ejection of multiple hairpin vortices in the wake of the ring. The evolution of an isolated hairpin vortex was studied with other vortex structures during the transition modelled as a background flow: a uniformly convecting stream and a uniform shear. In all the cases inves- tigated in this work, the entire hairpin vortex moved upward due to its self-induced velocity, and due to large curvature, its tip rose above its plane, resulting in the two legs approaching each other upstream of the tip. Further evolution leads to the viscous vortex reconnection process at the point of closest approach, which splits the hairpin into a vortex ring and a second hairpin. In this study, with the addition of a background flow the reconnection plane gets convected downstream, while three stages of the reconnection process: inviscid advec- tion, bridging and threading, observed in other configurations, are identified. Reconnection occurs early and at increasingly smaller timescales with an increase in Reynolds number, while the addition of a positive and negative shear accelerates and decelerates the onset of the reconnection process, unlike a uniformly convecting stream. The present work identifies the vortex reconnection process as an important mechanism for the formation of small-scales during the last stages of breakdown of a laminar vortex ring into turbulence.