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 $r\to \mathrm{\infty }$, leads to

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

where $\nu$ is the kinematic viscosity of the fluid. At $t=0$, we have a potential 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 ρ is the constant density

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 $\eta =r/\phi (t)$, provided $\phi {\phi }^{\prime }+\gamma {\phi }^2=a$, where $a$ is a constant, in which case $g=1-{\mathrm{e}}^{-a{\eta }^2/2\nu }$. The solution for $\phi (t)$ may be written according to Rott (1958) as

${\phi }^2=2a\exp (-2\underset{0}{\overset{t}{\int }}\gamma (s)\mathrm{d}s)\underset{c}{\overset{t}{\int }}\exp (2\underset{0}{\overset{u}{\int }}\gamma (s)\mathrm{d}s)\mathrm{d}u,$

where $c$ is an arbitrary constant. For $\gamma =0$, the classical Lamb–Oseen vortex is recovered. The case $\gamma =k$ corresponds to the axisymmetric stagnation point flow, where $k$ is a constant. When $c=-\mathrm{\infty }$, ${\phi }^2=a/k$, a Burgers vortex is a obtained. For arbitrary $c$, the solution becomes ${\phi }^2=a(1+\beta {\mathrm{e}}^{-2kt})/k$, where $\beta$ is an arbitrary constant. As $t\to \mathrm{\infty }$, Burgers vortex is recovered.

See also

• The Rankine vortex and Kaufmann (Scully) vortex are common simplified approximations for a viscous vortex.