Nonshielded multipolar vortices at high Reynolds number

Vortex multipoles—consisting of a core of vorticity closely surrounded by several smaller vorticity concentrations of opposite sign—are obtained from the evolution of vorticity in two-dimensional simulations. Using a meshless vortex method, we obtained triangular and square vortices, surrounded by three and four satellites, respectively. These structures have only been observed before to emerge from zero-circulation initial conditions. We also observed a pentagon vortex. Here, we obtain compound vortices of nonzero total circulation, and suggest a gamut of multipolar asymptotic solutions to the Navier-Stokes equations.

Figure 1

Evolution of the tripolar vortex: 14 equally spaced contour levels of $\omega$; $\mathrm{Re}={10}^4$, $\delta =0.25$.

Figure 2

Evolution of a triangular vortex: 14 equally spaced contour levels of $\omega$; $\mathrm{Re}={10}^4$, $\delta =0.55$.

Figure 3

Emergence of a triangular vortex from the saturated instability of a shielded monopole; 14 equally spaced contours of vorticity (black is negative, white is positive).

Figure 4

Grayscale levels of perturbation vorticity; case $\mathrm{Re}={10}^4$, $\delta =0.55$ (cf. Fig. 2). Black is negative, white is positive.

Figure 5

Shadowed plot of ${\log }_{10}\mid \omega \mid$, emphasizing the boundary of the satellite vortices; case with $\delta =0.3$ and $\mathrm{Re}={10}^4$.

Figure 6

Line contours of perturbation vorticity for a case with $\delta =0.7$ and $\mathrm{Re}={10}^4$, where the flow was perturbed at $t=1600$ by randomly moving all vortex particles.

Figure 7

Plots of $\omega$ for higher multipoles. Top two rows: square vortex; $\mathrm{Re}={10}^4$, $\delta =0.7$, $m=4$. Bottom two rows: pentagon vortex; $\mathrm{Re}={10}^4$, $\delta =1.2$, $m=5$.