Numerical comparison of two-dimensional Navier-Stokes flows on the whole plane and the periodic domain

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.

Figure 1

Comparison of time evolution of IC 1: the Oseen vortex, numerical results (curves), and exact solutions (symbols). Time runs from top to bottom: $t=0$, 10, and 50.

Figure 2

Plot of normalized enstrophy of IC 1: the Oseen vortex, $Q\left(t\right)/Q\left(0\right)$ on ${\mathbb{R}}^2$ (solid curve), $q\left(t\right)/q\left(0\right)$ with $\tilde{\nu }$ on ${\mathbb{T}}^2$ (dotted curve), and $q\left(t\right)/q\left(0\right)$ with $\nu$ on ${\mathbb{T}}^2$ (dashed curve). The inset shows a semilogarithmic plot of the last quantity. Also plotted is the analytical result $Q\left(t\right)/Q\left(0\right)=\frac{1}{\nu t+1}$ (pluses).

Figure 3

Evolution of normalized enstrophy of IC 2 for two merging vortices: $Q\left(t\right)/Q\left(0\right)$ on ${\mathbb{R}}^2$ (solid curve), $q\left(t\right)/q\left(0\right)$ with $\tilde{\nu }$ on ${\mathbb{T}}^2$ (dashed curve), and $q\left(t\right)/q\left(0\right)$ with $\nu$ on ${\mathbb{T}}^2$ (dotted curve). The inset shows a semilogarithmic plot of the last quantity.

Figure 4

Time evolution of vorticity contours on ${\mathbb{R}}^2$ for IC 2. Six levels of contours are drawn between the maximum and minimum of $\omega (x,y)$ on ${[-L,L]}^2$ with $L=20$.

Figure 5

Time evolution of vorticity contours on ${\mathbb{T}}^2$ for IC 2, with the Reynolds number adjusted using scaled $\tilde{\nu }$. Six levels of contours are drawn between the maximum and minimum of $\tilde{\omega }(x,y)$ on ${[-\pi ,\pi ]}^2$.

Figure 6

Time evolution of vorticity contours on ${\mathbb{T}}^2$ for IC 2 using unscaled $\nu$ without adjusting the Reynolds number. Six levels of contours are drawn between the maximum and minimum of $\tilde{\omega }(x,y)$ on ${[-\pi ,\pi ]}^2$.

Figure 7

Time evolution of normalized enstrophy for IC 3: $Q\left(t\right)/Q\left(0\right)$ on ${\mathbb{R}}^2$ with $\nu =2.5\times {10}^{-4}$ (solid curve) and $q\left(t\right)/q\left(0\right)$ on ${\mathbb{T}}^2$ with $\tilde{\nu }=\nu {(\pi /L)}^2$ (dotted curve) and with $\nu =$ (dashed curve). The inset shows a close-up view of the former two.

Figure 8

Semilogarithmic plot of normalized palinstrophy for IC 3: $P\left(t\right)/P\left(0\right)$ on ${\mathbb{R}}^2$ with $\nu =2.5\times {10}^{-4}$ (solid curve) and $p\left(t\right)/p\left(0\right)$ on ${\mathbb{T}}^2$ with $\tilde{\nu }=\nu {(\pi /L)}^2$ (dotted curve) and with $\nu$ (dashed curve). The inset shows a close-up linear plot of the former two.

Figure 9

Evolution of vorticity contours for IC 3 with $\nu =2.5\times {10}^{-4}$, on ${[-L,L]}^2\subset {\mathbb{R}}^2$ ($L=40$), using thresholds $\omega =3,\pm 3/2^n$ ($n=1,...,6$).

Figure 10

Enstrophy spectrum $Q\left(k\right)$ for IC 3 on ${\mathbb{R}}^2$ at different times. The straight lines are the guide for the BKL $-1$ (dashed curves), SF $-5/3$ (dotted curves), and Saffman $-2$ (dash-dotted curves) slopes.

Figure 11

Enstrophy spectrum $Q\left(k\right)$ for IC 3 averaged over $t=20,...,50$, on ${\mathbb{R}}^2$ for $\nu =2.5\times {10}^{-4}$ (solid curve), $\nu =5\times {10}^{-4}$ (dashed curve), and guide slopes $-1, -5/3$, and $-2$ from above. The exponent looks close to $-1$.

Figure 12

Enstrophy spectrum $Q\left(k\right)$ for IC 3 averaged over $t=50,...,80$, on ${\mathbb{R}}^2$ for $\nu =2.5\times {10}^{-4}$ (solid curve) $\nu =5\times {10}^{-4}$ (dashed curve), and guide slopes $-1, -5/3$, and $-2$ from above. The exponent looks close to $-5/3$.

Figure 13

Evolution of vorticity contours for IC 3, with scaled $\tilde{\nu }$ on ${[-\pi ,\pi ]}^2\subset {\mathbb{T}}^2$, using thresholds $\tilde{\omega }=3,\pm 3/2^n$ ($n=1,...,6$).

Figure 14

Evolution of vorticity contours for IC 3 on ${[-\pi ,\pi ]}^2\subset {\mathbb{T}}^2$, using $\nu$ without adjusting the Reynolds number. The thresholds are $\tilde{\omega }=3,\pm 3/2^n$ ($n=1,...,6$).

Figure 15

Enstrophy spectrum $Q\left(k\right)$ for IC 3 on ${\mathbb{T}}^2$ at different times, using $\nu$ without adjusting the Reynolds number.