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.Oseen, C. W. (1912). Uber die Wirbelbewegung in einer reibenden Flussigkeit. Ark. Mat. Astro. Fys., 7, 14–26. p. 253.
Mathematical description 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
where is the kinematic viscosity of the fluid. At , we have a potential vortex (free or irrotational 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 is the constant density. Below are two methods by which Oseen's vortex can be derived.
Generalized Oseen vortex 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)Rott, N. (1958). On the viscous core of a line vortex. Zeitschrift für angewandte Mathematik und Physik ZAMP, 9(5-6), 543–553. 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 Rankine vortex and Kaufmann (Scully) vortex – Common simplified approximations for a viscous vortex.
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