# Lamb–Oseen vortex

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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 ${\displaystyle \theta }$–direction in the Lamb–Oseen vortex is:

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

with

• ${\displaystyle r}$ = radius,
• ${\displaystyle r_{c}(t)={\sqrt {4\nu t+r_{c}(0)^{2}}}}$ = core radius of vortex,
• ${\displaystyle \nu }$ = viscosity, and
• ${\displaystyle \Gamma }$ = circulation contained in the vortex.

The radial velocity is equal to zero.

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

${\displaystyle \omega _{z}(r,t)={\frac {\Gamma }{\pi r_{c}(t)^{2}}}\exp \left(-{\frac {r^{2}}{r_{c}^{2}(t)}}\right),}$

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

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

where ${\displaystyle r_{\max }(t)={\sqrt {\alpha }}r_{c}(t)}$ is the radius at which ${\displaystyle v_{\max }}$ 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

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

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; 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.