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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][2]

Vector plot of the Lamb–Oseen vortex velocity field.
Evolution of a Lamb–Oseen vortex in air in real time. Free-floating test particles reveal the velocity and vorticity pattern. (scale: image is 20 cm wide)

Mathematical description

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Oseen looked for a solution for the Navier–Stokes equations in cylindrical coordinates with velocity components of the form

where is the circulation of the vortex core. Navier-Stokes equations lead to

which, subject to the conditions that it is regular at and becomes unity as , leads to[3]

where is the kinematic viscosity of the fluid. At , we have a potential vortex with concentrated vorticity at the axis; and this vorticity diffuses away as time passes.

The only non-zero vorticity component is in the direction, given by

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

where ρ is the constant density[4]

Generalized Oseen vortex

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The generalized Oseen vortex may be obtained by looking for solutions of the form

that leads to the equation

Self-similar solution exists for the coordinate , provided , where is a constant, in which case . The solution for may be written according to Rott (1958)[5] as

where is an arbitrary constant. For , the classical Lamb–Oseen vortex is recovered. The case corresponds to the axisymmetric stagnation point flow, where is a constant. When , , a Burgers vortex is a obtained. For arbitrary , the solution becomes , where is an arbitrary constant. As , Burgers vortex is recovered.

See also

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References

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  1. ^ Oseen, C. W. (1912). Uber die Wirbelbewegung in einer reibenden Flussigkeit. Ark. Mat. Astro. Fys., 7, 14–26.
  2. ^ 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.
  3. ^ Drazin, P. G., & Riley, N. (2006). The Navier–Stokes equations: a classification of flows and exact solutions (No. 334). Cambridge University Press.
  4. ^ G.K. Batchelor (1967). An Introduction to Fluid Dynamics. Cambridge University Press.
  5. ^ Rott, N. (1958). On the viscous core of a line vortex. Zeitschrift für angewandte Mathematik und Physik ZAMP, 9(5-6), 543–553.