Curvature-induced deformations of the vortex rings generated at the exit of a rectangular duct

Rectangular air jets of aspect ratio $2$ are studied at $Re=UD_{h}/\unicode[STIX]{x1D708}=17\,750$ using particle image velocimetry and hot-wire anemometry as they develop naturally or under acoustic forcing. The velocity spectra and the spatial theory of linear stability characterize the fundamental ($f_{n}$) and subharmonic ($f_{n}/2$) modes corresponding to the Kelvin–Helmholtz roll-up and vortex pairing, respectively. The rectangular cross-section of the jet deforms into elliptic/circular shapes downstream due to axis switching. Despite the apparent rotation of the vortex rings or the jet cross-section, the axis-switching phenomenon occurs due to reshaping into rounder geometries. By enhancing the vortex pairing, excitation at $f_{n}/2$ shortens the potential core, increases the jet spread rate and eliminates the overshoot typically observed in the centreline velocity fluctuations. Unlike circular jets, the skewness and kurtosis of the rectangular jets demonstrate elevated anisotropy/intermittency levels before the end of the potential core. The axis-switching location is found to be variable by the acoustic control of the relative expansion/contraction rates of the shear layers in the top (longer edge), side (shorter edge) and diagonal views. The self-induced vortex deformations are demonstrated by the spatio-temporal evolution of the phase-locked three-dimensional ring structures. The curvature-induced velocities are found to reshape the vortex ring by imposing nonlinear azimuthal perturbations occurring at shorter wavelengths with time/space evolution. Eventually, the multiple high-curvature/high-velocity regions merge into a single mode distribution. In the plane of the top view, the self-induced velocity distribution evolves symmetrically while the tilted ring results in the asymmetry of the azimuthal perturbations in the side view as the side closer to the acoustic source rolls up in more upstream locations.

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

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.

On the scaling of propagation of periodically generated vortex rings

The propagation of periodically generated vortex rings (period $T$) is numerically investigated by imposing pulsed jets of velocity $U_{jet}$ and duration $T_{s}$ (no flow between pulses) at the inlet of a cylinder of diameter $D$ exiting into a tank. Because of the step-like nature of pulsed jet waveforms, the average jet velocity during a cycle is $U_{ave}=U_{jet}T_{s}/T$. By using $U_{ave}$ in the definition of the Reynolds number ($Re=U_{ave}D/\unicode[STIX]{x1D708}$, $\unicode[STIX]{x1D708}$: kinematic viscosity of fluid) and non-dimensional period ($T^{\ast }=TU_{ave}/D=T_{s}U_{jet}/D$, i.e. equivalent to formation time), then based on the results, the vortex ring velocity $U_{v}/U_{jet}$ becomes approximately independent of the stroke ratio $T_{s}/T$. The results also show that $U_{v}/U_{jet}$ increases by reducing $Re$ or increasing $T^{\ast }$ (more sensitive to $T^{\ast }$) according to a power law of the form $U_{v}/U_{jet}=0.27T^{\ast 1.31Re^{-0.2}}$. An empirical relation, therefore, for the location of vortex ring core centres ($S$) over time ($t$) is proposed ($S/D=0.27T^{\ast 1+1.31Re^{-0.2}}t/T_{s}$), which collapses (scales) not only our results but also the results of experiments for non-periodic rings. This might be due to the fact that the quasi-steady vortex ring velocity was found to have a maximum of 15 % difference with the initial (isolated) one. Visualizing the rings during the periodic state shows that at low $T^{\ast }\leqslant 2$ and high $Re\geqslant 1400$ here, the stopping vortices become unstable and form hairpin vortices around the leading ones. However, by increasing $T^{\ast }$ or decreasing $Re$ the stopping vortices remain circular. Furthermore, rings with short $T^{\ast }=1$ show vortex pairing after approximately one period in the downstream, but higher $T^{\ast }\geqslant 2$ generates a train of vortices in the quasi-steady state.

Shear/rotation competition during the roll-up of acoustically excited shear layers

Naturally developing and acoustically excited shear layers at the Reynolds numbers $Re_{\unicode[STIX]{x1D703}_{0}}=U\unicode[STIX]{x1D703}_{0}/\unicode[STIX]{x1D708}=85{-}945$ are studied using the hot-wire (HW) anemometry and particle image velocimetry (PIV), with a focus on the shear/rotation competition during the initial Kelvin–Helmholtz (KH) roll-up. Velocity spectra and the spatial linear stability (LST) analysis characterize the fundamental ($f_{n}$) and its subharmonic ($f_{n}/2$) mode interacting due to the vortex pairing. For $276\leqslant Re_{\unicode[STIX]{x1D703}_{0}}\leqslant 780$, the root-mean-square (r.m.s.) of the streamwise turbulence intensity shows a double-peaking phenomenon, i.e. major and minor peaks of the $u_{rms}$ coexist towards the high-speed (HS) and the low-speed (LS) sides, respectively. The single/double-peaked $u_{rms}$ profiles are found to be correlated with the scattered/organized distribution of the shear/rotation, demonstrating a transitioning character with the downstream distance, $Re_{\unicode[STIX]{x1D703}_{0}}$ and the upstream turbulence levels. The rotating vortex cores and the corresponding peripheral shear regions, demonstrate the phase reversal of the velocity fluctuations with respect to the HS and the LS sides. Excitation at $f_{n}$ increases the vortex count by 21 %, advances the location of the first KH roll-up and hence also the minor peak formation location. Due to the enhanced pairing at the $f_{n}/2$ forcing, the vortex count reduces by 23 %. Before merging into the downstream rotation core, the upstream vortex is shifted towards the HS side and the major peak is accordingly augmented. Actuation advances the transition to the nonlinear state, as well as the saturation of the amplification factor. The volumetric topologies of the shear/rotation loops tracked in consecutive phases during the period of the acoustic excitation, separate from the edge and grow in time–space due to the viscous diffusion. The shearing and rotating loops are found to be associated with the thinning (elongation) and expansion (accumulation) of the vorticity, respectively.

Geometry-driven collective ordering of bacterial vortices

Geometry-induced transition of vortex pairing in bacterial collective motion.