We study the mechanisms of centrifugal instability and its eventual self-limitation, as well as regenerative instability on a vortex column with a circulation overshoot (potentially unstable) via direct numerical simulations of the incompressible Navier-Stokes equations. The perturbation vorticity (ω ) dynamics are analysed in cylindrical (r, θ, z) coordinates in the computationally accessible vortex Reynolds number, Re(≡circulation/viscosity), range of 500-12 500, mostly for the axisymmetric mode (azimuthal wavenumber m = 0). Mean strain generates azimuthally oriented vorticity filaments (i.e. filaments with azimuthal vorticity, ω θ ), producing positive Reynolds stress necessary for energy growth. This ω θ in turn tilts negative mean axial vorticity, −Ω z (associated with the overshoot), to amplify the filament, thus causing instability. (The initial energy growth rate (σ r ), peak energy (G max ) and time of peak energy (T p ) are found to vary algebraically with Re.) Limitation of vorticity growth, also energy production, occurs as the filament moves the overshoot outward, hence lessening and shifting |−Ω z |, while also transporting the core +Ω z , to the location of the filament. We discover that a basic change in overshoot decay behaviour from viscous to inviscid occurs at Re ∼ 5000. We also find that the overshoot decay time has an asymptotic limit of 45 turnover times with increasing Re. After the limitation, the filament generates negative Reynolds stress, concomitant energy decay and hence self-limitation of growth; these inviscid effects are enhanced further by viscosity. In addition, the filament transports angular momentum radially inward, which can produce a new circulation overshoot and renewed instability. Energy decays at the Re studied, but, at higher Re, regenerative growth of energy is likely due to the renewed mean shearing. New generation of overshoot and Reynolds stress is examined using a helical (m = 1) perturbation. Regenerative energy growth, possibly resulting in even vortex breakup, can be triggered by this new overshoot at practical Re (∼10 6 for trailing vortices), which are currently beyond the computational capability.
Growth of optimal transient perturbations to an Oseen vortex column into the nonlinear regime is studied via direct numerical simulation (DNS) for Reynolds number, Re (≡ circulation/viscosity), up to 10000. An optimal bending-wave transient mode is obtained from linear analysis and used as the initial condition. (DNS of a vortex column embedded in finer-scale turbulence reveals that optimal modes are preferentially excited during vortex–turbulence interaction.) Tilting of the optimal mode's radial vorticity perturbation into the azimuthal direction and its concomitant stretching by the column's strain field produces positive Reynolds stress, hence kinetic energy growth. Modes experiencing the largest growth are those with initial vorticity localized at a ‘critical radius’ outside the core, such that this perturbation vorticity resonantly induces core waves. Resonant forcing leads to growth of perturbation energy concentrated within the core. Moderate-amplitude (~5%) perturbations cause significant distortion of the core and generate secondary filament-like spiral structures (‘threads’) outside the core. As the mode evolves into the nonlinear regime, radially outward self-advection of thread dipoles accelerates growth arrest by removing the perturbation from the critical radius and disrupting resonant forcing. With increasing Re, the evolving vorticity patterns become more chaotic, more turbulent-like (finer scaled, contorted vorticity), and persist longer. This suggests that at typical Re (~106), nonlinear transient growth may indeed be able to break up, hence induce rapid decay of, column vortices – highly relevant for addressing the aircraft wake hazard crisis and the looming air traffic capacity crisis. In addition, we discover a regenerative transient growth scenario in which threads induce secondary perturbations closer to the vortex column. A parent–offspring regenerative mechanism is postulated and verified by DNS. There is a clear trend towards stronger regenerative growth with increasing Re. These results, showing an important role of transient growth in turbulent vortex decay, are highly relevant to the prediction and control of vortex-dominated flows.
External turbulence-induced axial flow in an incompressible, normal-mode stable Lamb-Oseen (two-dimensional) vortex column is studied via direct numerical simulations of the Navier-Stokes equations. Azimuthally oriented vorticity filaments, formed from external turbulence, advect radially towards or away from the vortex axis (depending on the filament's swirl direction), resulting in a net induced axial flow in the vortex core; axial flow increases with increasing vortex Reynolds number (Re = vortex circulation/viscosity). This contrasts the viscous mechanism for axial flow generation downstream of a lifting body, wherein an axial pressure gradient is produced by viscous diffusion of the swirl (Batchelor, J. Fluid Mech., vol. 20, 1964, pp. 645-658). Analysis of the self-induced motion of an arbitrarily curved external filament shows that any non-axisymmetric filament undergoes radial advection. We then studied the evolution of a vortex column starting with an imposed optimal transient growth perturbation. For a range of Re values, axial flow develops and initially grows as (time) 5/2 before decreasing after two turnover times; for Re = 10 000 -the highest computationally achievable -axial flow at late times becomes sufficiently strong to induce vortex instability. Contrary to a prior claim of a parent-offspring mechanism at the outer edge of the core, vorticity tilting within the core by axial flow is the underlying mechanism producing energy growth. Thus, external perturbations in practical flows (at Re ∼ 10 7 ) produce destabilizing axial flow, possibly leading to the sought-after vortex breakup.
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