Approximate tangent space

In fluid dynamics, the Burgers vortex is an exact solution to the Navier–Stokes equations governing viscous flow. The Burgers vortex describes a stationary, self-similar flow. An inward, radial flow, tends to concentrate vorticity in a narrow column around the symmetry axis. On the same time, viscous diffusion tends to spread the vorticity. The stationary Burgers vortex arises when the two effects balance.

The Burgers vortex, apart from serving as an illustration of the vortex stretching mechanism, may describe such flows as tornados, where the vorticity is provided by continuous convection-driven vortex stretching.

Flow field

The flow for the Burgers vortex is described in cylindrical $(r,z,\phi )$ coordinates. Assuming axial symmetry (no $\phi$-dependence), the vorticity equation is solved by the flow field:

$v_{r}=-{\frac {1}{2}}\alpha r,$

$v_{z}=\alpha z,$

$v_{\phi }=v_{\phi }(r),$

where $\alpha >0$ is a constant. The flow satisfies the continuity equation by the two first of the above equations. The vorticity equation only gives a non-trivial component in the $z$-direction, where it becomes

${\frac {D\zeta }{Dt}}=\zeta {\frac {\partial v_{z}}{\partial z}}+\nu \nabla ^{2}\zeta ,$

where $D/Dt$ denotes the convective derivative and $\nu$ the viscosity. Note that the first term on the right-hand side is the vortex stretching term which tends to amplify the vorticity, while he second term, due to viscosity, attenuates (or rather spreads) vorticity. The solution can be found as

$\zeta =\zeta _{0}\exp(-{\frac {\alpha r^{2}}{4\nu }}),$

where $\zeta _{0}$ is a constant. The vorticity is thus distributed as a Gaussian of width

$R=2{\sqrt {\frac {\nu }{\alpha }}}.$

References

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