Instabilities of counter-rotating vortices

Here we simulate and visualize two vortex-tube instabilities (+ bonus). The volumetric visualization set-up is similar to that presented on the page of the vortex-rings collision.

1. Crow instability

The Crow instability (after the work of Crow (1970)¹) concerns two counter-rotating vortex tubes. The Crow mechanism describes a so-called long-wavelength perturbation (i.e. the wavelength of the perturbation is much larger than the vortex-core size, see movie) in the two vortex tubes. This disturbance may be subject to growth, upto a point where the different tubes connect to each other. It can be observed in the two trailing vortices behind an airplane (see figure), where the dynamics are made visible by the condensation trails.

Here we simulate a single periodic disturbance of two vortex tubes in a periodic box. You can clearly see the recombination process.

Elliptic instability

The long-wavelength Crow instability is contrasted with the short-wavelength Elliptic instability that can occur for two counter-rotating vortex pairs. The streamlines of two counter-rotating vortices can be oval in shape (elliptical), and unstable to short wavelength perturbations. These disturbances can then result in vortex-tube tilting, leading to secondary filaments. In this simulation we use the Lamb-Chaplygin dipole model of initialize a dipolar vortex with elliptic stream lines. It is the same dipole model that was used in the 2D vortex near a sharp edge case.

Vorticity field and stream-lines of the Lamb-Chaplygin dipole

The movie visualizes the elliptic instability, again using a volumetric rendering of the $\lambda_2$ criterion.

In order to find (or exceed) the limits of conveying visual information, we also use volumetric rendering to visualize the $\lambda_2$ field, but the color coding is not according to the magnitude of the $\lambda_2$ field itself (only the opacity is), but the vorticity-vector components $(\omega_x, \omega_y, \omega_z)$ are mapped to the $(r,g,b)$ channels (red, green, blue). Initially, the vorticity is directed in the $x$ direction, so the vortex tubes are colored red.

For experimental en more numerical visualizations, the interested reader in referred to the excellent work of McKeown (2020)². The online supplementary material also contains some very nice movies (tip!).

Bonus: Untying a vortex trefoil

References

Crow, S. C. (1970). Stability theory for a pair of trailing vortices. AIAA journal, 8(12), 2172-2179.↩︎
McKeown, R., Ostilla-Mónico, R., Pumir, A., Brenner, M. P., & Rubinstein, S. M. (2020). Turbulence generation through an iterative cascade of the elliptical instability. Science advances, 6(9), eaaz2717.↩︎