en>Michael Hardy at 01:27, 13 January 2014

2014-01-13T01:27:08Z

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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-similarity|self-similar]] flow.
An inward, radial flow, tends to concentrate [[vorticity]] in a narrow column around the symmetry axis. On the same time, [[viscosity|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 <math>(r,z,\phi)</math> coordinates. Assuming axial symmetry (no <math>\phi</math>-dependence), the [[vorticity equation]] is solved by the flow field:

:<math>v_r=-\frac12 \alpha r,</math>
:<math>v_z=\alpha z,</math>
:<math>v_\phi=v_\phi(r),</math>

where <math>\alpha>0</math> 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 <math>z</math>-direction, where it becomes

:<math>\frac{D\zeta}{D t}=\zeta\frac{\partial v_z}{\partial z}+\nu\nabla^2\zeta,</math>

where <math>D/Dt</math> denotes the [[convective derivative]] and <math>\nu</math> 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

:<math>\zeta=\zeta_0 \exp(-\frac{\alpha r^2}{4\nu}),</math>

where <math>\zeta_0</math> is a constant.<ref>{{cite book | last=Lautrup|first= Benny | title=Physics of Continuous Matter|edition= 2nd | publisher=CRC Press | year=2011 | isbn=1420077007}}</ref> The vorticity is thus distributed as a [[Gaussian function|Gaussian]] of width

:<math>R=2\sqrt{\frac{\nu}{\alpha}}.</math>

==References==
{{Reflist}}

[[Category:Fluid dynamics| ]]

Approximate tangent space - Revision history

en>Michael Hardy at 01:27, 13 January 2014