A majority of numerical experiments of the Navier-Stokes equations, lacking physical boundaries, have been conducted under periodic boundary conditions so far. In this paper, in order to access the effect of periodicity imposed upon the flow properties, we take up specifically two-dimensional incompressible flows and carry out numerical simulations on the whole plane to compare with those under periodic boundaries, with or without adjusting the Reynolds number. We solve the Navier-Stokes equations on a square domain using a finite-difference scheme to simulate flows on ${\mathbb{R}}^2$. After checking the time evolution of the Oseen vortex with the exact solution, we simulate merging of like-signed vortices to compare it with that under periodic boundaries. Generally speaking, we find that flows decay in norms faster on ${\mathbb{T}}^2$ than on ${\mathbb{R}}^2$, even when the Reynolds number is adjusted. We also simulate merging of three localized vortices that generates finer spatial structure in order to study the decay law of the total enstrophy and spatial patterns in vorticity. In this case, norms on ${\mathbb{T}}^2$ decay in a manner very close to how those on ${\mathbb{R}}^2$ do, but still marginally faster than those on ${\mathbb{R}}^2$. We also study the power law of the energy spectrum on ${\mathbb{R}}^2$, comparing it with the predictions of, for example, Gilbert's spiral model including $E\left(k\right)\sim k^{-11/3}$, which sits at the Sulem-Frisch borderline.

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• Turbulence
• Vortex dynamics
• Direct numerical simulations
• Navier-Stokes equation