Burgers vortex

In fluid dynamics, the Burgers vortex is an exact solution to the Navier–Stokes equations governing viscous flow, named after Jan Burgers[1]. 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. At 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 ${\displaystyle (r,\theta ,z)}$ 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

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

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.

Sullivan vortex

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 ${\displaystyle v_{z}/z}$.

See also

• Kerr–Dold vortex

References

1. Burgers, J. M. (1948). A mathematical model illustrating the theory of turbulence. In Advances in applied mechanics (Vol. 1, pp. 171-199). Elsevier.
2. Drazin, P. G., & Riley, N. (2006). The Navier-Stokes equations: a classification of flows and exact solutions (No. 334). Cambridge University Press.
3. Roger D. Sullivan. (1959). A two-cell vortex solution of the Navier-Stokes equations. Journal of the Aerospace Sciences, 26(11), 767-768.