Transient perturbation growth of vortex rings

Perturbations to a vortex ring with largest transient energy growth are calculated and applied to drive the vortex ring to evolve. Optimal initial perturbations appear at azimuthal wave numbers β = 0 and β = 9. The former was described as 'axial flow' and is not related to the breakdown of the ring, while the latter is critical to the ring breakdown process as observed in previous direct numerical simulation (DNS) with random initial noise input. In the present work, DNS is conducted to study the nonlinear development of the optimal perturbations, and good agreement with past studies is achieved.