Self-limiting and regenerative dynamics of perturbation growth on a vortex column

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 (Gmax) and time of peak energy (Tp) 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\sim 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( ∼ 106 for trailing vortices), which are currently beyond the computational capability.