Burgers vortex

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:

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

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

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

where ${\displaystyle \alpha >0}$ (strain rate) and ${\displaystyle \Gamma >0}$ (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[2]

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

The equation is integrated with the condition ${\displaystyle g(\infty )=1}$ 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

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

Intuitively the flow can be understood by looking at the three terms in the vorticity equation for ${\displaystyle \omega _{z}}$. The axial velocity ${\displaystyle v_{z}}$ intensifies the vorticity of the vortex core at the axis by vortex stretching. The intensified vorticity tries to diffuse outwards radially, but prevented by radial vorticity convection due to ${\displaystyle v_{r}}$. The three-way balance establishes a steady solution.

In 1959, Roger D. Sullivan extended the Burgers vortex solution by considering the solution of the form[3]

${\displaystyle v_{r}=-\alpha r+{\frac {1}{r}}f(\eta ),}$

${\displaystyle v_{z}=2\alpha z\left[1-{\frac {f'(\eta )}{2\nu }}\right],}$

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

where ${\displaystyle \eta =\alpha r^{2}/(2\nu )}$. The functions ${\displaystyle f(\eta )}$ and ${\displaystyle g(\eta )}$ are given by

${\displaystyle f(\eta )=6\nu (1-e^{-\eta }),}$

${\displaystyle g(\eta )={\frac {\nu }{\alpha }}\int _{0}^{\eta }\exp \left[-t+3\int _{0}^{t}{\frac {1-e^{-s}}{s}}\mathrm {d} s\right]\mathrm {d} t.}$

For Burgers vortex ${\displaystyle v_{r}<0}$, ${\displaystyle v_{\theta }>0}$ and ${\displaystyle v_{z}/z>0}$ are always positive, Sullivans result shows that ${\displaystyle v_{r}>0}$ for ${\displaystyle \eta \leq 2.821}$ and ${\displaystyle v_{z}/z<0}$ for ${\displaystyle \eta \leq 1.099}$. Thus Sullivan vortex resembles Burgers vortex for ${\displaystyle \eta >2.821}$, but develops a two-cell structure near the axis due to the sign change of ${\displaystyle v_{z}/z}$.

• Kerr–Dold vortex