Abstract
At high Reynolds number, the interaction between two vortex tubes leads to intense velocity gradients, which are at the heart of fluid turbulence. This vorticity amplification comes about through two different instability mechanisms of the initial vortex tubes, assumed anti-parallel and with a mirror plane of symmetry. At moderate Reynolds number, the tubes destabilize via a Crow instability, with the nonlinear development leading to strong flattening of the cores into thin sheets. These sheets then break down into filaments which can repeat the process. At higher Reynolds number, the instability proceeds via the elliptical instability, producing vortex tubes that are perpendicular to the original tube directions. In this work, we demonstrate that these same transition between Crow and Elliptical instability occurs at moderate Reynolds number when we vary the initial angle $\beta$ between two straight vortex tubes. We demonstrate that when the angle between the two tubes is close to $\pi/2$, the interaction between tubes leads to the formation of thin vortex sheets. 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. At smaller values of the angle $\beta$ between the two tubes, the breakdown mechanism changes to an elliptic cascade-like mechanism. Whereas the interaction of two vortices depends on the initial condition, the rapid formation of fine-scales vortex structures appears to be a robust feature, possibly universal at very high Reynolds numbers.
Highlights
Many experiments have demonstrated that the interaction of two vortex tubes coming close together eventually leads to a change of topology of the vortex lines through a process known as vortex reconnection [1,2,3]
Vortex reconnection is a fundamental process in fluid mechanics, and it has been postulated to play a significant role in fluid phenomena such as the turbulent energy cascade [4], noise generation [5], and the transfer of helicity across topologically distinct vortices [6]
When the tubes are initially almost perpendicular to each other (β ≈ 90◦), the interacting vortex tubes locally contact where they overlap, flattening into a pair of intense, slender vortex sheets. This resembles the classically studied reconnection found in the case of a pair of straight vortex tubes with a perturbation symmetric with respect to the plane separating the two vortices [11,15,16,17,18,19,20,42,43], and which involves the flattening of the vortex tubes into intense, slender vortex sheets on either side of the symmetry plane
Summary
Many experiments have demonstrated that the interaction of two vortex tubes coming close together eventually leads to a change of topology of the vortex lines through a process known as vortex reconnection [1,2,3]. While it is clear that reconnections are still present in hydrodynamical fluids with Re 1 [23], their appearance could be restricted to a rather small subset of initial conditions that either enforce many symmetries on the vortices or stabilize the core through spin [20], as simulations have shown the long-range interaction between vortices tends to align them in an antiparallel manner [9,35] This would mean that reconnections (understood as topological changes) could become rarer in classical fluids than in idealized models, even if they remain important in the former case from a theoretical and mathematical perspective [36]. For β 53.1◦ (b 2), the mechanism that prevails in the interaction between the two tubes is the formation of transverse vortex tubes, as observed, formally, when β → 0 (b → ∞) due to the presence of the elliptical instability [24]
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