On the final states of two-dimensional unbounded flows

TL;DR

This study uses high-accuracy numerical simulations to show that two-dimensional unbounded flows with non-zero circulation tend to evolve into an Oseen vortex, with the relaxation time influenced by viscosity and initial perturbations.

Contribution

It demonstrates that all such flows ultimately relax into an Oseen vortex and quantifies the relationship between relaxation time, viscosity, and initial perturbations.

Findings

Oseen vortex appears in late stages for all initial conditions with non-zero circulation.

The difference between theoretical and simulation times is inversely proportional to viscosity.

Perturbed monopoles also relax into Oseen vortices, with the relaxation time depending on perturbation type and amplitude.

Abstract

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 $T_{gap}$.