Lamb–Oseen vortex

In fluid dynamics, the Lamb–Oseen vortex models a line vortex that decays due to viscosity. This vortex is named after Horace Lamb and Carl Wilhelm Oseen.^[1]

Vector plot of the Lamb–Oseen vortex.

The mathematical model for the flow velocity in the circumferential $\theta$–direction in the Lamb–Oseen vortex is:

$V_\theta(r,t) = \frac{\Gamma}{2\pi r} \left(1-\exp\left(-\frac{r^2}{r_c^2(t)}\right)\right),$

with

• $r$ = radius,
• $r_c(t)=\sqrt{4\nu t}$ = core radius of vortex,
• $\nu$ = viscosity, and
• $\Gamma$ = circulation contained in the vortex.

The radial velocity is equal to zero.

The associated vorticity distribution^[2] in the vortex-filament-direction (here $\hat{z}$) can be found with the curl:

$\omega_z(r,t) = \frac{\Gamma}{\pi r_c(t)^2} \exp\left(-\frac{r^2}{r_c^2(t)}\right),$

An alternative definition is to use the peak tangential velocity of the vortex rather than the total circulation

$V_\theta\left( r \right) = V_{\theta \max} \left( 1 + \frac{1}{2\alpha} \right) \frac{r_\max}{r} \left[ 1 - \exp \left( - \alpha \frac{r^2}{r_\max^2} \right) \right],$

where $r_{\max} (t)=\sqrt{\alpha} r_c(t)$ is the radius at which $v_\max$ is attained, and the number α = 1.25643, see Devenport et al.^[3]

The pressure field simply ensures the vortex rotates in the circumferential direction, providing the centripetal force

${\partial p \over \partial r} = \rho {v^2 \over r},$

where ρ is the constant density^[4]

References

1. Saffman, P. G.; Ablowitz, Mark J.; J. Hinch, E.; Ockendon, J. R.; Olver, Peter J. (1992). Vortex dynamics. Cambridge: Cambridge University Press. ISBN 0-521-47739-5. p. 253.
2. Wu, J. Z.; Ma, H. Y.; Zhou M. D.; (2006). Vorticity and Vortex Dynamics. Berlin: Springer-Verlag. p. 262. ISBN 3-540-29027-3. p. 262.
3. W.J. Devenport, M.C. Rife, S.I. Liapis and G.J. Follin (1996). "The structure and development of a wing-tip vortex". Journal of Fluid Mechanics 312: 67–106. Bibcode:1996JFM...312...67D. doi:10.1017/S0022112096001929.
4. G.K. Batchelor (1967). An Introduction to Fluid Dynamics. Cambridge University Press.