On the final states of two-dimensional unbounded flows

A high-accuracy numerical study on the evolution of two-dimensional unbounded flows with the Hermite pseudo-spectral solver is presented. Our simulations clearly show that the simple Oseen vortex always appears in the late stage for every initial condition with non-zero circulation ($\Omega \neq 0$). In general, the theoretical time adopted to describe the Oseen vortex and the simulating time in numerical investigations are not the same, and their difference ($T_{diff}$) is in inverse proportion to the viscosity for the same initial condition. In particular, a perturbed monopole will also eventually relax into an Oseen vortex which shows obvious difference from the original monopole no matter how small the perturbation is. This difference can be well represented by the time gap ($T_{gap}$) between the theoretical time of two monopoles, and the type and amplitude of the perturbation determine the value of $T_{gap}$.