Cascades and reconnection in interacting vortex filaments

The reconnection of two interacting vortex tubes is a fundamental process in fluid mechanics, which, at very high Reynolds numbers, is associated with the formation of intense velocity gradients. Reconnection is usually studied using two antiparallel tubes which are destabilized via the long-wavelength Crow instability, leading to a very symmetric configuration and to a strong flattening of the cores into thin sheets. Here, we consider the interaction of two initially straight tubes at an angle of $\beta$ and show that by relaxing some symmetries of the problem, a rich phenomenology appears. When the angle between the two tubes is close to $\beta ={90}^{\circ }$, their interaction leads to pairing of small portions of antiparallel tubes, followed by the formation of thin and localized vortex sheets as a precursor of reconnection. The subsequent breakdown of these sheets involves a twisting of the paired sheets, followed by the appearance of a localized cloud of small-scale vortex structures. By decreasing $\beta$, we show that reconnection involves increasingly larger portions of tubes, whose cores are subsequently destabilizing, leaving behind more small-scale vortices. At the smallest values of the angle $\beta$ studied, the two vortices break down through a mechanism, which leads to a cascadelike process of energy conveyance across length scales similar to what was found for previous studies of antiparallel vortex tubes ($\beta =0$) which imposed no symmetries. While, in all cases, the interaction of two vortices depends on the initial condition, the rapid formation of fine-scale vortex structures appears to be a robust feature, possibly universal at very high Reynolds numbers.

Figure 1

Schematic of DNS initial configuration. The thick red and blue lines represent the initial positions of the two vortex filaments, with the arrow indicating the circulation direction. (a) Three-dimensional (3D) schematic. (b) Side view of the $yz$ plane, which shows the definition of the spacing between the filaments, $d$. (c) Top view of the $xz$ plane, which shows the definition of $\beta$ and $b$. The vorticity distribution is invariant under the symmetry resulting from two mirror symmetries with respect to the planes $P_1$ and $P_2$: $(x,y,z)\to (x,-y,-z), ({\omega }_x,{\omega }_y,{\omega }_z)\to ({\omega }_x,-{\omega }_y,-{\omega }_z)$, and $(u_x,u_y,u_z)\to (u_x,-u_y,-u_z)$. The distance between the two tubes in the $y$ direction is $d=0.9$; the circulation is chosen here to be $\mathrm{\Gamma }=1$, and the core radius $\sigma =0.4/\sqrt{2}\approx 0.28$.

Figure 2

Vortex topology changes during a reconnection between initially perpendicular vortex tubes ($b=1$ and ${\mathrm{Re}}_{\mathrm{\Gamma }}=4000$, run 3 in Table 1). (a),(b) In a subdomain of size $3.68\times 3.68\times 3.68$ around the center of the computational box, the isovorticity contours corresponding to ${\omega }_{th}=2.5$ at $t=15.2$, when the tubes begin to interact, and at $t=26.4$, after the interaction. These dynamics reveal a change of topology of the vortex lines and the generation of small-scale vortices during the reconnection. (c),(d) The same vorticity field, bandpass filtered between $k_{<}=\sqrt{2}{k}_f$ and $k_{>}=2\sqrt{2}{k}_f$, with $k_f=2.3$, which removes the small-scale features of the flow and showcases only the change of topology of the vortex tubes. The value of the isosurface is 1 in (c) and (d). Full videos of the process are available in the Supplemental Material [40].

Figure 3

Vortex topology changes during a reconnection between tubes at an acute angle ($\beta \approx 28.1^{\circ }, b=4$ and ${\mathrm{Re}}_{\mathrm{\Gamma }}=4000$, run 11 in Table 1). Only a subdomain of the flow, of size $\sim 5.2\times 5.2\times 8.5$, is shown around the central region where the vortices interact. (a),(b) Isovorticity contours corresponding to ${\omega }_{th}=2.35$ at $t=16$, when the tubes are paired, and at $t=25.5$, after the interaction, respectively. These panels show the generation of fine-scale vortices, similar to the collision of two antiparallel vortices [24]. (c),(d) The bandpassed vorticity field, between the wave numbers $k_{<}=\sqrt{2}{k}_f$ and $k_{>}=2\sqrt{2}{k}_f$ (where $k_f=2.3$), at the same times as (a) and (b). The large-scale portions of the vortex tubes at the middle of the interaction zone break down by $t=25.5$, and the tubes reconnect at the edges of the interaction zone. The value of the isosurface ${\omega }_{th}$ in (c) and (d) is 0.85. Full videos of the process are available in the Supplemental Material [40].

Figure 4

Close-up view of sheet formation near the reconnection event of initially perpendicular vortex tubes (run 3, $\beta ={90}^{\circ }, b=1, {\mathrm{Re}}_{\mathrm{\Gamma }}=4000$). Isovorticity contours of the flow are shown in a subdomain of size $2.45\times 2.45\times 2.45$ around the central region where vorticity is the most amplified, at (a) $t=17$ (${\omega }_{th}=3.6$), (b) $t=19$ (${\omega }_{th}=3.8$), and (c) $t=20$ (${\omega }_{th}=4.0$). The figure reveals the formation of a flattened region of amplified vorticity where the two tubes locally contact.

Figure 5

Structure of the sheets formed during reconnection (run 3, $\beta ={90}^{\circ }, b=1, {\mathrm{Re}}_{\mathrm{\Gamma }}=4000$). (a) A more detailed view of the region where the vortices interact is shown by zooming in (size of the domain shown: $1.95\times 1.95\times 1.95$), and rotating the domain at $t=20, {\omega }_{th}=4.0$ [compare with Fig. 4]. (b) Cross-sectional plot of the $z$ component of vorticity in the middle plane $P_1$ at $z=0$. (c)–(e) Cross-sectional plot of the vorticity $\left|\mathit{\omega }\right|$ in the horizontal planes (c) $x=x_1$, (d) $x=x_2$, and (e) $x=x_3$, with the locations being indicated in (b). The change in the orientation of the isosurfaces, clearly visible, reflects a change in the direction of the vorticity vectors along the $x$ direction, as indicated by the arrows. The isocontour lines are separated by 2.5 in (b) and by 2 in (c)–(e).

Figure 6

Twisting of the vortex sheets and reconnection. Vorticity magnitude isosurface for initially perpendicular vortex tubes in run 3 ($\beta ={90}^{\circ }, {\mathrm{Re}}_{\mathrm{\Gamma }}=4000$) at three consecutive times, following the vortex sheet formation. The size of the subdomain shown is the same as in Fig. 5 ($1.95\times 1.95\times 1.95$). (a) After the vortex cores flatten into sheets, they become twisted as they wrap around each other. (b) The vortex sheets become folded and reorient along the $z=0$ ($P_1$) plane, forming an array of transverse vortex filaments. (c) The two strong vortex sheets annihilate, leaving a tangle of small-scale vortex filaments. The last time shown, $t=23.2$, approximately corresponds to the peak dissipation rate; see Fig. 7. The isovorticity contours of the flow are shown at $t=20$ (${\omega }_{th}=4.75$), $t=21.6$ (${\omega }_{th}=5.25$), $t=22.4$ (${\omega }_{th}=5.60$), and $t=23.2$ (${\omega }_{th}=6.0$). The plane $P_1$ ($z=0$) separates the box shown in the middle.

Figure 7

Global dynamics for initially perpendicular vortex tubes at various Reynolds numbers. The evolution of (a) the mean kinetic energy rate $\langle {\mathit{u}}^2\rangle /2$, (b) the mean dissipation rate $\nu \langle {\mathit{\omega }}^2\rangle$, and (c) the mean sixth moment of the vorticity, ${\langle {\mathit{\omega }}^6\rangle }^{1/3}$, for runs 1–5, at fixed $\beta ={90}^{\circ }$ and increasing values of ${\mathrm{Re}}_{\mathrm{\Gamma }}$. The vertical dashed line in (b) and (c) corresponds to $t=23.2$, which is the last time shown in Fig. 6, and is also very close to the peak dissipation rate. The dashed (full) lines correspond to runs at low (high) resolutions.

Figure 8

Global dynamics for vortex tubes at varying initial orientation angles. The evolution of (a) the mean kinetic energy rate $\langle {\mathit{u}}^2\rangle /2$, (b) the mean dissipation rate $\nu \langle {\mathit{\omega }}^2\rangle$, and (c) the mean sixth moment of the vorticity, ${\langle {\mathit{\omega }}^6\rangle }^{1/3}$, for runs 3 ($\beta ={90}^{\circ }, b=1$), 6 ($\beta \approx 77.3^{\circ }, b=5/4$), and 7 ($\beta \approx 67.4^{\circ }, b=3/2$), at fixed ${\mathrm{Re}}_{\mathrm{\Gamma }}=4000$. The local pairing of the vortex tubes leads, in all of these cases, to the formation of vortex sheets.

Figure 9

Interaction and breakdown of nearly antiparallel vortex tubes. The evolution of the vorticity magnitude isosurface for run 11 ($\beta \approx 28.1^{\circ }, b=4, {\mathrm{Re}}_{\mathrm{\Gamma }}=4000$). Only a cubic subdomain, ${[-2.6,2.6]}^3$, surrounding the regions where the vortices interact is shown. The isosurfaces of the vorticity field are shown at $t=15.6$ (left), $t=20.4$ (center), and $t=22.8$ (right); the upper row shows the top view and the bottom row shows the front view. The thresholds are ${\omega }_{th}=2.5$ ($t=15.6$), 3.2 ($t=20.4$), and 3.6 ($t=22.8$). The latest time shown corresponds to the peak dissipation rate.

Figure 10

Vortex core structure at two positions, $z\approx 0.12$ (upper row) and $z\approx 0.23$ (lower row). The times shown are intermediate between the two first times shown in Fig. 9, and correspond to the phase where small-scale structures begin to form. The upper core wraps around the lower core, the more so as the plane is further away from the center ($z$ increases). In addition, (f) begins to show the formation of regions where vorticity gets more intense, as was found in the case for two antiparallel filaments [24].

Figure 11

Global dynamics for vortex tubes initially oriented at shallow angles. The evolution of (a) the mean kinetic energy rate $\langle {\mathit{u}}^2\rangle /2$, (b) the mean dissipation rate $\nu \langle {\mathit{\omega }}^2\rangle$, and (c) the mean sixth moment of the vorticity, ${\langle {\mathit{\omega }}^6\rangle }^{1/3}$, for runs 8 ($\beta \approx 53.1^{\circ }, b=2$), 9 ($\beta \approx 43.6^{\circ }, b=5/2$), 10 ($\beta \approx 36.8^{\circ }, b=3$), and 11 ($\beta \approx 28.1^{\circ }, b=4$), at fixed ${\mathrm{Re}}_{\mathrm{\Gamma }}=4000$.

Figure 12

Global dynamics for initially antiparallel vortex tubes with various noise levels. The evolution of (a) the mean kinetic energy rate $\langle {\mathit{u}}^2\rangle /2$, (b) the mean dissipation rate $\nu \langle {\mathit{\omega }}^2\rangle$, and (c) the mean sixth moment of the vorticity, ${\langle {\mathit{\omega }}^6\rangle }^{1/3}$, for run 12 with varying noise amplitudes where $\beta =0^{\circ }$ and ${\mathrm{Re}}_{\mathrm{\Gamma }}=4000$. The results shown here correspond to a resolution of ${192}^2\times 768$. Results at higher resolution, up to ${320}^2\times 1280$, do not differ appreciably from those shown here. A movie showing the evolution of the solution for the run with the highest level of noise can be found in the Supplemental Material [40].

Figure 13

Spectra of the solutions close to the peak of dissipation. The inset shows the energy dissipation, with the bars indicating the time at which the solution is shown. (a) Run 5 (${\mathrm{Re}}_{\mathrm{\Gamma }}=5400, \beta ={90}^{\circ }$), (b) run 3 (${\mathrm{Re}}_{\mathrm{\Gamma }}=4000, \beta ={90}^{\circ }$), (c) run 11 (${\mathrm{Re}}_{\mathrm{\Gamma }}=4000, \beta =28.1^{\circ }$), and (d) run 12A (${\mathrm{Re}}_{\mathrm{\Gamma }}=4000, \beta =0^{\circ }$). For reference, the Kolmogorov $k^{-5/3}$ spectrum is indicated on all panels.