Under Axial Stretching

These MPEG files are animations of the motions of vortex lines in three-dimensional, inviscid, incompressible flow. In each case they show the evolution of an isolated vortex as represented by a number of vortex lines. Each vortex line represents the center of vortex tube with a smooth distribution of vorticity in its core. While the cores of these vortex tubes do not change size or shape with time, a collection of them together can accurately represent the motions of a physical vortex.

The motivation for this work was to study the effects of axial stretching on the instability and transition to turbulence of such vortices. We also show how vortex axisymmetrization, an essentially two-dimensional process, can occur for a period of time before three-dimensional dynamics takes over.

Further discussion can be found in:

Nolan, David S., 2001: The stabilizing effects of axial stretching
on turbulent vortex dynamics. *Phys. Fluids*, **13**,
1724-1738. Download PDF (Acrobat).

The following animations show the evolution of a vortex with a smooth distribution of vorticity in its core, as represented by 18 vortex lines. The vortex is radius one and periodic in the axial direction with length 8. The total circulation of the vortex is equal to 1. We add to the vortex a Gaussian-shaped kink which quickly produces "turbulence." The inital conditions for this and the subsequent simulations are also augmented with a small amount of random noise which helps ensure a rapid transition to fully three-dimensional dynamics.

Stable, Perturbed Vortex without Stretching. ||| Same, with semi-transparent vortex lines.

We then embed the vortex in an unbounded, cylindrical deformation field with a constant rate of strething dw/dz=0.01 everywhere. The development of turbulence is subsantially suppressed.

Stable, Perturbed Vortex under Axial Stretching.

The next two simulations show the evolution of a vortex which contains a patch of opposite-signed vorticity. The positive vortex lines are blue and the negative vortex lines are red. In this case, the vortex is again radius one, but rather than being periodic, it terminates at the bottom against a solid, free-slip lower boundary. The flow is embedded in a cylindrical deformation field with dw/dz=0.02 everywhere. The vortex lines shown are the only those from z=0 to z=8. The vortices are continuously stretched upwards by the deformation field, and elements which pass beyond z=16 are disregarded. The ejection of opposite signed vorticity is best seen from the top view.

Ejection of Opposite-Signed Vorticity. ||| Same, Top View.

In these last two movies, we show the evolution of a stable vortex which interacts with four weaker vortices, alternating in sign, just outside of the vortex at r=2. The boundary conditions are the same as in the ejection case, and dw/dz=0.01. The simulations show how the exterior vortices with same sign are preferentially advected into the larger vortex as opposed the opposite signed vorticity, which remains outside. Again, the top view is best. Observe how half of each of the same-sign vortices are drawn completely into the main vortex.

Axisymmetrization of Wavenumber Two Anomalies. ||| Same, Top View.

This work was supported by the Applied Mathematical Sciences Subprogram of the Office of Energy Research, Department of Energy, under contract DE-AC03-76SF00098. The three-dimensional animations of the vortex lines were created by David Adalsteinsson. I would also like to thank Alexandre Chorin for his advice and support.