Reconnection of vortex tubes with axial flow

This paper addresses the interaction of initially antiparallel vortex tubes containing an axial flow that induces a twisting of the vortex lines around the tube axes, using numerical simulations. Vortex tube configurations with both the same and opposite senses of twist—corresponding to the same and opposite signs of kinetic helicity density—are considered. It is found that the topology of the reconnection process is very different between the two cases. For tubes with the same sense of twist, the reconnection is fully three-dimensional (3D): vortex lines reconnect at a finite angle, and 3D vortex null points may be created. Following reconnection the vortex line topology in both bridge and thread structures exhibits a high degree of complexity. For oppositely twisted tubes the reconnection is locally two-dimensional, occurring along vorticity null lines, that in contrast to the untwisted case are not perpendicular to the tube axes. This leads to a break in the symmetry between the two vortex bridges generated during reconnection. For all cases studied, increasing the twist in the vortex tubes leads to a later, faster, and more complete reconnection process.

Figure 1

(a) Vorticity isosurface and (b) vorticity field lines for the initial condition with nonzero net helicity.

Figure 2

(a) Measured change in twist of each vortex tube at early times (solid) compared to the change in twist predicted by evaluating $\nu {\left\{\int {(\mathbf{\nabla }\times \mathbf{\omega })}_{\parallel }dl\right\}}_{\max }$ (dashed). (b) Twist of field lines around the tube axis plotted on the $x=-3$ boundary at $t^{*}=0.05$ for $q=0.485$. Note that the twist is defined as the number of turns that a vortex line makes around the tube axis between $x=-3$ and $x=3$ (equivalently the angle of rotation around the tube axis in this interval, divided by $2\pi$).

Figure 3

$30\%$ vorticity isosurfaces (left) and vortex lines plotted from contours at 30% of the maximum vorticity on the $x=\pm 3$ boundaries (right), at (a) $t^{*}=0.47$, (b) $t^{*}=0.94$, (c) $t^{*}=1.41$, and (d) $t^{*}=2.34$. From the simulation with net helicity for the tube pair and $q=0.970$ ($v_0=2$).

Figure 4

Contour plots of ${(\mathbf{\nabla }\times \mathbf{\omega })}_{\parallel }$, for $q=0.970$. In all cases the contours are shown in planes that pass through the location of the spatial maximum over the domain of ${(\mathbf{\nabla }\times \mathbf{\omega })}_{\parallel }$ at the corresponding time. (a) $t^{*}=0, y=-1$, (b) $t^{*}=0.89, y=0$, (c) $t^{*}=0.89, z=-6.65$, and (d) $t^{*}=2.39, z=-3.8$.

Figure 5

Selected vortex lines during the interaction of vortex tubes with finite net helicity, with $q=0.970$, at $t^{*}=0.70$. Red and white: thread vortex lines in the vortex sheet that will soon reconnect. Cyan: reconnected bridges. Shading on the end planes and in the volume shows $\left|\mathbf{\omega }\right|$.

Figure 6

3D null points (red squares) and the surrounding unit-vector vorticity field $\widehat{\mathbf{\omega }}$ in (a) the $y=0$ plane and (b) the $x=0$ plane at $t^{*}=0.70$. Background shading shows the out-of-plane component of $\mathbf{\omega }$. From the simulation with net helicity for the tube pair and $q=0.485$.

Figure 7

(a) Vorticity flux in the symmetry plane. (b) Rate of change of vorticity flux in the symmetry plane (note that here and in the following figures this is a rate of change with respect to $t^{*}$). (c) Vorticity flux of the flux rings around the $z$ axis.

Figure 8

Selected vortex lines plotted from the $30\%$ maximum vorticity contour at $x=-3$ at (a) $t^{*}=0.66$, (b) $t^{*}=0.75$, (c) $t^{*}=1.64$, and (d) $t^{*}=2.11$ with $q=0.485$. Change in color indicates crossing a plane $x=3n,n\in \mathbb{Z}$.

Figure 9

$30\%$ vorticity isosurfaces (left) and vortex lines plotted from contours at 30% of the maximum vorticity on the $x=\pm 3$ boundaries (right) at (a) $t^{*}=0.47$, (b) $t^{*}=0.94$, (c) $t^{*}=1.41$, and (d) $t^{*}=2.34$. From the simulation with zero net helicity for the tube pair and $q=0.970$.

Figure 10

Selected vortex lines during the interaction of vortex tubes with zero net helicity, with $q=0.970$, at $t^{*}=0.70$. Red and white: thread vortex lines in the vortex sheet. Cyan: reconnected bridges. Shading on the end planes and in the volume shows $\left|\mathbf{\omega }\right|$. The yellow line indicates the approximate path of the null line.

Figure 11

${\omega }_y$ contours in the dividing plane ($y=0$) for the simulation with $q=0.485$ at (a) $t^{*}=0.47$, (b) $t^{*}=0.70$, (c) $t^{*}=1.17$, and (d) $t^{*}=1.88$.

Figure 12

(a) Maximum values of $\pm {\omega }_y$ in the dividing plane as a function of time. Line styles as in Fig. 7. (b) Max(${\omega }_y$), diamonds, and $-\max (-{\omega }_y)$, crosses, over time in the dividing plane as a function of $q$.

Figure 13

Contours of ${(\mathbf{\nabla }\times \mathbf{\omega })}_{\parallel }$ for the simulation with zero net helicity and $q=0.970$, at (a) $t^{*}=0.94$ and $z=-6.35$ and (b) $t^{*}=1.88$ and $z=-4.75$. In each case the contours are shown in planes that pass through the location of the spatial maximum over the domain of ${(\mathbf{\nabla }\times \mathbf{\omega })}_{\parallel }$ at the corresponding time.

Figure 14

(a) Vorticity flux through the dividing plane over time and (b) rate of change of this flux (derivative with respect to $t^{*}$), with line styles as in Fig. 7. (c) Maximum rate of change of the vorticity flux as a function of $q$.

Figure 15

Selected vortex lines plotted from the $30\%$ maximum vorticity contour at $x=-3$ at (a) $t^{*}=0.75$ and (b) $t^{*}=1.88$. The change in color indicates crossing a plane $x=3n,n\in \mathbb{Z}$.

Figure 16

Comparison of the reconnection rate over time for the zero helicity runs (solid) and nonzero helicity runs (dashed). Colors as in Fig. 2.

Figure 17

Comparison of volume integrated quantities for the zero helicity runs (solid) and nonzero helicity runs (dashed). (a) Normalized enstrophy, (b) normalized kinetic energy, (c) net absolute helicity $|\mathbf{\omega }·\mathbf{v}|$. Colors as in Fig. 2.

Figure 18

(a) Selected vortex lines during the interaction of antiparallel vortex tubes (red, threads; cyan, reconnected bridges). Shading on the end planes and in the volume shows $\left|\mathbf{\omega }\right|$. Taken from the simulation described in Ref. [19]. Vorticity field lines (b) before and (c) after internal 3D reconnection. The shaded sphere in panel (b) represents the nonideal region, within which $(\mathbf{\nabla }\times \mathbf{\omega })·\mathbf{\omega }\ne 0$.