Non-normal dynamics of co-rotating vortex pairs

Transient energy growth of disturbances to co-rotating vortex pairs is investigated in an analysis where vortex core expansion and vortex merging are included. The dynamics of pairs are compared to those of individual vortices in order to highlight the effect of vortex interaction. Three typical vortex pair cases are studied, with the pairs comprised respectively of individually asymptotically stable vortices, inviscidly unstable vortices, and viscously unstable vortices. In each case the asymptotic stability of the pair system reflects that of the individual vortices. For the asymptotically stable case, large linear transient energy growth of optimal perturbations occurs for time horizons corresponding to vortex merging. Linear transient disturbance energy growth exhibited by pairs in this stable case is two to three orders of magnitude larger than that for a corresponding individual vortex. For the inviscidly unstable case, the optimal perturbation takes the form of a superposition of two individual helical unstable modes, while where the individual vortices are viscously unstable, the optimal disturbances within each core have similar spatial distributions to the individually stable case. Direct numerical simulations of stable pairs seeded by optimal initial perturbations have been carried out. For axially invariant cases, the sign of perturbation has an effect, as well as magnitude; the sign dependence relates to whether or not the perturbation adds to or subtracts from the swirl of the base flow. For a two-dimensional perturbation that adds to the swirl of the base flow, seeding with the linear optimal disturbance at a relative energy level 1 × 10 induces the pair to rotate more rapidly and approximately halves the time required for merger. Direct numerical simulation shows that the optimal three-dimensional perturbation can induce the vortex system to break up before merging occurs.