Lamb–Oseen vortex

Template:Short description 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.

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

Oseen looked for a solution for the Navier–Stokes equations in cylindrical coordinates $(r,\theta ,z)$ with velocity components $(v_r,v_{\theta },v_z)$ of the form

$v_r=0,v_{\theta }=\frac{\mathrm{\Gamma }}{2\pi r}g(r,t),v_z=0.$

where $\mathrm{\Gamma }$ is the circulation of the vortex core. Navier–Stokes equations lead to

$\frac{\partial g}{\partial t}=\nu (\frac{{\partial }^{2}g}{\partial r^2}-\frac{1}{r}\frac{\partial g}{\partial r})$

which, subject to the conditions that it is regular at $r=0$ and becomes unity as Template:Tmath, leads to

$g(r,t)=1-{\mathrm{e}}^{-r^2/4\nu t},$

where $\nu$ is the kinematic viscosity of the fluid. At Template:Tmath, we have a potential vortex (free or irrotational vortex) with concentrated vorticity at the $z$-axis; and this vorticity diffuses away as time passes.

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

${\omega }_z(r,t)=\frac{\mathrm{\Gamma }}{4\pi \nu t}{\mathrm{e}}^{-r^2/4\nu t}.$

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

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

where $\rho$ 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

$v_r=-\gamma (t)r,v_{\theta }=\frac{\mathrm{\Gamma }}{2\pi r}g(r,t),v_z=2\gamma (t)z$

that leads to the equation

$\frac{\partial g}{\partial t}-\gamma r\frac{\partial g}{\partial r}=\nu (\frac{{\partial }^{2}g}{\partial r^2}-\frac{1}{r}\frac{\partial g}{\partial r}).$

Self-similar solution exists for the coordinate Template:Tmath, provided Template:Tmath, where $a$ is a constant, in which case Template:Tmath. The solution for $\phi (t)$ may be written according to Rott (1958) as

${\phi }^2=2a\exp (-2{\int }_0^t\gamma (s)\mathrm{d}s){\int }_c^t\exp (2{\int }_0^u\gamma (s)\mathrm{d}s)\mathrm{d}u,$

where $c$ is an arbitrary constant. For Template:Tmath, the classical Lamb–Oseen vortex is recovered. The case $\gamma =k$ corresponds to the axisymmetric stagnation point flow, where $k$ is a constant. When Template:Tmath, Template:Tmath, a Burgers vortex is a obtained. For arbitrary Template:Tmath, the solution becomes Template:Tmath, where $\beta$ is an arbitrary constant. As Template:Tmath, Burgers vortex is recovered.

See also

• Rankine vortex and Kaufmann (Scully) vortex – Common simplified approximations for a viscous vortex.