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 coordinates. Assuming axial symmetry (no -dependence), the flow field associated with the axisymmetric stagnation point flow is considered:

where (strain rate) and (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]

The equation is integrated with the condition so that at infinity the solution behaves like a potential vortex, but at finite location, the flow is rotational. The choice ensures at the axis. The solution is

The vorticity equation only gives a non-trivial component in the -direction, given by

Intuitively the flow can be understood by looking at the three terms in the vorticity equation for . The axial velocity 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 . 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]

where . The functions and are given by

For Burgers vortex , and are always positive, Sullivans result shows that for and for . Thus Sullivan vortex resembles Burgers vortex for , but develops a two-cell structure near the axis due to the sign change of .

See also

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.
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