Burgers vortex

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 ${\displaystyle (r,\theta ,z)}$ coordinates. Assuming axial symmetry (no ${\displaystyle \theta }$-dependence), the flow field associated with the axisymmetric stagnation point flow is considered:

${\displaystyle v_r=-\alpha r,}$

${\displaystyle v_z=2\alpha z,}$

${\displaystyle v_{\theta }=\frac{\mathrm{\Gamma }}{2\pi r}g(r),}$

where ${\displaystyle \alpha >0}$ (strain rate) and ${\displaystyle \mathrm{\Gamma }>0}$ (circulation) are constants. The flow satisfies the continuity equation by the two first of the above equations. The azimuthal momentum equation of the Navier–Stokes equations then reduces to

${\displaystyle r\frac{d^{2}g}{d{r}^2}+\left(\frac{\alpha r^2}{\nu }-1\right)\frac{dg}{dr}=0}$

where ${\displaystyle \nu }$ is the kinematic viscosity of the fluid. The equation is integrated with the condition ${\displaystyle g(\mathrm{\infty })=1}$ so that at infinity the solution behaves like a potential vortex, but at finite location, the flow is rotational. The choice ${\displaystyle g(0)=0}$ ensures ${\displaystyle v_{\theta }=0}$ at the axis. The solution is

${\displaystyle g=1-\exp \left(-\frac{\alpha r^2}{2\nu }\right).}$

The vorticity equation only gives a non-trivial component in the ${\displaystyle z}$-direction, given by

${\displaystyle {\omega }_z=\frac{\alpha \mathrm{\Gamma }}{2\pi \nu }\exp \left(-\frac{\alpha r^2}{2\nu }\right).}$

Intuitively the flow can be understood by looking at the three terms in the vorticity equation for ${\displaystyle {\omega }_z}$,

${\displaystyle -\alpha r\frac{d{\omega }_z}{dr}=2\alpha {\omega }_z+\frac{\nu }{r}\frac{d}{dr}\left(r\frac{d{\omega }_z}{dr}\right).}$

The first term on the right-hand side of the above equation corresponds to vortex stretching which intensifies the vorticity of the vortex core due to the axial-velocity component ${\displaystyle v_z=2\alpha z}$. The intensified vorticity tries to diffuse outwards radially due to the second term on the right-hand side, but is prevented by radial vorticity convection due to ${\displaystyle v_r=-\alpha r}$ that emerges on the left-hand side of the above equation. The three-way balance establishes a steady solution. The Burgers vortex is a stable solution of the Navier–Stokes equations.

One of the important property of the Burgers vortex that was shown by Jan Burgers is that the total viscous dissipation rate ${\displaystyle \mathrm{\Phi }}$ per unit axial length is independent of the viscosity, indicating that dissipation by the Burgers vortex is non-zero even in the limit ${\displaystyle \nu \to 0}$. For this reason, it serves as a suitable candidate in modelling and understanding stretched-vortex tubes observed in turbulent flows. The total dissipation rate per unit axial length is, in incompressible flows, simply equal to the total enstrophy per unit length, which is given by

${\displaystyle \mathrm{\Phi }=\nu {\int }_0^{2\pi }{\int }_0^{\mathrm{\infty }}{\omega }_z^{2}rdrd\theta =\frac{\alpha {\mathrm{\Gamma }}^2}{4\pi }.}$

Unsteady evolution to Burgers's vortex

An exact solution of the time dependent Navier Stokes equations for arbitrary function ${\displaystyle \alpha =\alpha (t)}$ is available. In particular, when ${\displaystyle \alpha }$ is constant, the vorticity field ${\displaystyle \omega (r,t)}$ with an arbitrary initial distribution ${\displaystyle \mathrm{\Omega }(r)=\omega (r,0)}$ is given by

${\displaystyle \omega (r,t)=\frac{\alpha }{2\pi \nu (1-e^{-2\alpha t})}\iint \mathrm{\Omega }\left(\sqrt{{\xi }_1^2+{\eta }_1^2}\right)\exp \left[\frac{-(x-{\xi }_1{e}^{-\alpha t}{)}^2-(y-{\eta }_1{e}^{-\alpha t}{)}^2}{(2\nu /\alpha )(1-e^{-2\alpha t})}\right]d{\xi }_{1}d{\eta }_1.}$

As ${\displaystyle t\to \mathrm{\infty }}$, the asymptotic behaviour is given by

${\displaystyle \omega (r,t)=\frac{\alpha }{2\pi \nu }\exp \left(-\frac{\alpha r^2}{2\nu }\right)\left[\mathrm{\Gamma }+e^{-2\alpha t}{\int }_0^{\mathrm{\infty }}\mathrm{\Omega }(s)\left(\frac{\alpha r^2}{2\nu }-1\right)\left(\frac{\alpha s^2}{2\nu }-1\right)2\pi sds+O(e^{-4\alpha t})\right],\mathrm{\Gamma }={\int }_0^{\mathrm{\infty }}\mathrm{\Omega }(s)2\pi sds}$

Thus, provided ${\displaystyle \mathrm{\Gamma }\ne 0}$, an arbitrary vorticity distribution approaches the Burgers' vortex. If ${\displaystyle \mathrm{\Gamma }=0}$, say in the case where the initial condition is composed of two equal and opposite vortices, then the first term is zero and the second term implies that vorticity decays to zero as ${\displaystyle t\to \mathrm{\infty }.}$