Vortex tube

A vortex tube is a bundle of vortex lines bounded by a surface to which vorticity is tangent. Circulation measures its vorticity flux. Stretching a nearly incompressible tube reduces its cross-sectional area and increases its vorticity. The Burgers vortex is a local model in which this concentration balances viscous diffusion.

A vortex filament is the ideal thin-core limit of a vortex tube, concentrating a finite circulation on a curve. In an unbounded incompressible fluid with velocity tending to zero at infinity, its induced velocity field follows the vorticity version of the Biot-Savart law. The ideal field is singular on the filament; finite-core models are needed there. A closed circular filament is a thin 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 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.