Deflection and trapping of a counter-rotating vortex pair by a flat plate

The interaction of a counter-rotating vortex pair (dipole) with a flat plate in its path is studied numerically. The vortices are initially separated by a distance $D$ (dipole size) and placed far upstream of a plate of length $L$. The plate is centered on the dipole path and inclined relative to it at an incident angle ${\beta }_i$. At first, the plate is held fixed in place. The vortices approach the plate, travel around it, and then leave as a dipole with unchanged velocity but generally a different travel direction, measured by a transmitted angle ${\beta }_t$. For certain plate angles the transmitted angle is highly sensitive to changes in the incident angle. The sensitivity increases as the dipole size decreases relative to the plate length. In fact, for sufficiently small values of $D/L$, singularities appear: near critical values of ${\beta }_i$, the dipole trajectory undergoes a topological discontinuity under changes of ${\beta }_i$ or $D/L$. The discontinuity is characterized by a jump in the winding number of one vortex around the plate, and in the time that the vortices take to leave the plate. The jumps occur repeatedly in a self-similar, fractal fashion, within a region near the critical values of ${\beta }_i$, showing the existence of incident angles that trap the vortices, which never leave the plate. The number of these trapping regions increases as the parameter $D/L$ decreases, and the dependence of the motion on ${\beta }_i$ becomes increasingly complex. The simulations thus show that even in this apparently simple scenario, the inviscid dynamics of a two-point-vortex system interacting with a stationary wall is surprisingly rich. The results are then applied to separate an incoming stream of dipoles by an oscillating plate.

Figure 1

(a)–(c) Trajectories of single point vortices in simple domains. (d) Speed of vortex in panel (c) vs time $t$, normalized by the time $T$ required for one rotation. The vortex moves around the plate edges at $t/T=0.25,0.75$.

Figure 2

(a) Sketch showing a dipole consisting of two counter-rotating point vortices with circulation $\pm \mathrm{\Gamma }$ separated by a distance $D$, and a plate of length $L$ centered on the dipole path at an angle. (b) Definition of incident and transmitted angles ${\beta }_i$ and ${\beta }_t$, and shift coordinates $(x_s,y_s)$.

Figure 3

Conformal map $\zeta =f\left(z\right)$ from outside the plate to outside the cylinder. (a) Point vortices $z_k=x_k+i{y}_k$ and plate in the $z$ plane. (b) Rotation into the $w$ plane, $w=e^{i\beta }z$. (c) Map to the $\zeta$ plane, $\zeta =w+{(w^2+1)}^{1/2}$.

Figure 4

Vortex trajectories for $D/L=\pi /4$ and (a) ${\beta }_i=0^{\circ }$, (b) ${22}^{\circ }$, (c) ${48}^{\circ }$, (d) ${67}^{\circ }$, (e) ${80}^{\circ }$, (f) ${90}^{\circ }$.

Figure 5

(a) Transmitted angle ${\beta }_t$, (b) derivative $d{\beta }_t/d{\beta }_i$, (c) trajectory shift coordinates $x_s$ and $y_s$, and (d) time delay $\tau$, vs ${\beta }_i$, for $D/L=\pi /4$. The solid dots in panel (a) denote the pairs $({\beta }_i,{\beta }_t)$ plotted in Fig. 4.

Figure 6

Vortex trajectories for ${\beta }_i={30}^{\circ }$ and (a) $D/L=\pi /4$, (b) $0.5\pi /4$, (c) $0.45\pi /4$, (d) $0.25\pi /4$.

Figure 7

Closeup of figure 6.

Figure 8

Hamiltonian $H\left(t\right)$ for ${\beta }_i={30}^{\circ }$ and $D/L=0.45\pi /4$. (a) $|H\left(t\right)-H_0|$ vs $t$, where $H_0=H\left(0\right)$, computed with the indicated values of $\mathrm{\Delta }t$. (b) $\underset{t\in [0,60]}{max}|H\left(t\right)-H_0|$ vs $\mathrm{\Delta }t$. The dashed line has slope three.

Figure 9

Transmitted angle ${\beta }_t-{\beta }_i$ vs ${\beta }_i$, for (a) $D/L=\pi /4$, (b) $0.5\pi /4$, (c) $0.45\pi /4$, and (d) $0.25\pi /4$.

Figure 10

Singular behavior for $D/L=0.25\pi /4$, computed with conformal mapping technique (method 2) with $\mathrm{\Delta }t=0.0002$. (a)–(c) Winding number $\alpha$ vs ${\beta }_i$, at three different scales. (d)–(f) Time delay $\tau$ vs ${\beta }_i$, at three different scales.

Figure 11

Winding number and time delay $\tau$ computed with vortex sheet discretization (method 1), using $N=160$, $\mathrm{\Delta }t=0.0002$, for $D/L$ and ${\beta }_i$ as in Figs. 10 and 10.

Figure 12

Convergence of vortex pair position for $D/L=0.25\pi /4$, $\beta =29.04133$, at $t=60$. The point coordinates ${\mathbf{x}}_N$ are those computed with method 1 with $N=60$, 80, 120, 160, 240, 330. The point coordinates $\mathbf{x}$ are those computed with method 2. The values of $\mathrm{\Delta }t$ are as indicated. The dashed line has slope ten.

Figure 13

Winding number $\alpha$ for $D/L=0.167\pi /4$, vs ${\beta }_i$. Panels (b)–(d) show closeups of the region whose width is indicated by arrows in panels (a)–(c).

Figure 14

Diagram showing the position of trapping regions centered at ${\beta }_i^c$, as a function of $D/L$. Panels (b) and (c) are closeups of the boxed regions in panel (a).

Figure 15

The plate oscillates with period ${\tau }_p=40$. Dipoles are inserted at $y=3$ at time intervals ${\tau }_v$ with $2{\tau }_v=3{\tau }_p$. The position of all vortices is shown at the indicated times $t/{\tau }_p$ (solid dots), with all previous trajectories shown as curves.

Figure 16

Trajectories of the first 12 dipoles inserted at $y=3$ at time intervals ${\tau }_v$, where (a) $2{\tau }_v=3{\tau }_p$, (b) $3{\tau }_v=5{\tau }_v$, (c) $4{\tau }_v=7{\tau }_v$. In all cases, the plate oscillation period is ${\tau }_p=40$.

Figure 17

Trajectories of single vortices moving near a plate in otherwise inviscid, irrotational flow with vanishing velocity at $\infty$. The dots denote the initial vortex position at $(0,\epsilon )$, for $\epsilon =0.2$, 0.1, 0.05, 0.025. The vortices rotate clockwise around the plate. Panel (b) shows a closeup of panel (a) near the right plate edge.

Figure 18

Time $T$ taken by a single vortex for one rotation around the plate, scaled by $\epsilon$, vs $\epsilon$, where $\epsilon$ is the initial vortex distance from the plate.

Figure 19

Vortex speed, scaled by $1/\epsilon$, vs (a) normalized time $t$ and (b) a scaled normalized time near the right edge, at which $t/T=0.25$. Results are shown for $\epsilon =0.2$, 0.1, 0.05, 0.025.