Vortex ring

A vortex ring is a vortex tube bent into a closed loop, often approximately a circle. Its vorticity runs around the loop, while the induced velocity field passes through the centre and returns outside. For a thin ring of radius $r$ and circulation $\Gamma$, the Biot-Savart law for incompressible vorticity gives an axial velocity $u_z(z)=\Gamma r^2/[2(r^2+z^2{)}^{3/2}]$, measured relative to the ring centre, with its sign set by the circulation orientation. A vortex ring is not a loop of fluid whose velocity everywhere merely follows the circular vortex line.

A separated vortex ring is a ring-shaped recirculating flow structure that remains detached from the body producing it while occupying a relatively stable downstream position in the body's frame. It differs from a ring attached to the body and from a ring that is shed and then travels away. The stable pattern does not require the same parcel of fluid to remain permanently in one place.

The term was used for the flow behind a dandelion seed's pappus. Flow visualization showed a detached recirculation ring downstream of the porous bristles. Air passing through the structure, rather than simply being excluded by a solid disk, was important to the observed pattern.^[ref]

To test the role of structure, the investigators compared microfabricated disks with different porosities. A disk with a pappus-like porosity reproduced the characteristic flow behavior, supporting a mechanism involving permeability and geometry rather than a special biological material alone.^[ref]

For an invented geometric example, a screen with total projected area $10\mathrm{m}{\mathrm{m}}^2$ and solid area $1\mathrm{m}{\mathrm{m}}^2$ has open-area fraction $0.9$. That fraction alone does not fully specify its aerodynamic behavior: the distribution and shape of openings, fluid properties and flow speed also matter. This is not the measured porosity of the paper's seeds.

The example shows why an apparently empty part of a structure can control its drag. A vortex ring is a property of the surrounding velocity field, so visualizing that field can reveal mechanisms invisible in a static outline of the object.

A long array of equal co-oriented vortex rings, spaced by $l\ll r$, approximates an azimuthal vortex sheet on a cylinder of radius $r$. The circulation per unit axial length is $\Gamma /l$. The conditions $\nabla \cdot \mathbf{u}=0$ and $\nabla \times \mathbf{u}=\mathbf{\omega }$ are mathematically analogous to the magnetostatic equations for a solenoid. Away from its ends, an array long compared with $r$ produces nearly uniform axial flow $\Gamma /l$ inside and weak flow outside. A finite array has return flow outside connecting its two ends. The uniform-core approximation fails close to individual vortex cores and near the ends, and the field requires a stated circulation orientation and background flow.