Kelvin wave and soliton propagation in classical viscous vortex filaments

Abstract

Vortex filaments are highly rotating localized structures of fluids that admits several types of excitation. Here, we study them by using numerical simulations of the three-dimensional incompressible Navier-Stokes equations. We first address the propagation of Kelvin waves, helicoidal excitations propagating along the filament, and measure their dispersion relation which turns out to be in good agreement with the original Lord Kelvin predictions. Then, inspired by the connection between vortex line dynamics and an integrable system, we show numerically the existence of solitons propagating along vortex filaments and study the collision of two of such structures. Finally, we show numerically the experimental feasibility of studying vortex solitons in the lab, by proposing an experiment for their generation.

Figures (6)

• Figure 1: (Left) Sketch explaining the local induced approximation (LIA) model \ref{['eq:LIA']}.(Center) Kelvin waves propagating in a vortex filament $Re_{\rm v}=5\times 10^{3}$, $L_z/a_0=256$. (Right) A Hasimoto soliton with $A=0.45$, $\lambda=0.65$, $Re_{\rm v}=2.5\times 10^{3}$, $L_z/a_0=512$. Visualizations display three-dimensional rendering of the enstrophy field $|{\bf \omega (x)}|^2$.
• Figure 2: (Left) Kelvin waves propagating in a vortex filament $Re_{\rm v}=1.7\times 10^{3}$, $L_z/a_0=512$. Kelvins wave have a global amplitude of $C_0=0.07$ (see text). The white dashed line is the KW asymptotic prediction \ref{['eq:KWdispRel']} and the green dashed line is the full dispersion relation given in the SI. The inset displays the evolution of the core size, where $\tau_6=2\pi/\omega_{k_6}$, with $k_6=6\times2\pi/L_z$. The dispersion relation has been calculated in the grey area.
• Figure 3: (Top) Hasimoto soliton propagating in a vortex filament with $A=0.45$, $\lambda=0.65$, $Re_{\rm v}=2.5\times 10^{3}$, $L_z/a_0=512$ (same of Fig.\ref{['fig:visKW_soliton']}). (Bottom) Soliton profile centered at $z=0$. The left inset displays the real and imaginary parts of the soliton amplitude at its maximum ($z=0$), showing a coherent oscillation. The right inset shows the width of the soliton as a function of time.
• Figure 4: Collision of two Hasimoto solitons propagating in a vortex filament with $A=0.45$, $\lambda=0.65$, $Re_{\rm v}=2.5\times 10^{3}$, $L_z/a_0=512$. (Top) Three-dimensional rendering of the enstrophy field $|{\bf \omega (x)}|^2$ showing the two colliding vortex solitons with a close up into the emission of a vortex ring. (Bottom)$z-t$ diagram of the solitons showing their collision and the two surviving solitons.
• Figure 5: Generation of a soliton by the collision of a vortex ring of radius $R=0.6$ (visible at the top left corner), both the ring and the filament have a $Re_{\rm v}=2.5\times 10^{3}$, $L_z/a_0=512$.(Top) Three-dimensional rendering of the enstrophy field $|{\bf \omega (x)}|^2$. (Bottom left)$z-t$ diagram of the soliton amplitude. Collision takes place around $t=8$. (Bottom right) Soliton profile centered at $z=0$ for different times.