Lamb–Oseen vortex

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

$v_{r}=0,\quad v_{\theta }={\frac {\Gamma }{2\pi r}}g(r,t),\quad v_{z}=0.$

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

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

which, subject to the conditions that it is regular at $r=0$ and becomes unity as $r\rightarrow \infty$, 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

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

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

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

where ρ is the constant density

Generalized Oseen vortex

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

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

that leads to the equation

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

Self-similar solution exists for the coordinate $\eta =r/\varphi (t)$, provided $\varphi \varphi '+\gamma \varphi ^{2}=a$, where $a$ is a constant, in which case $g=1-\mathrm {e} ^{-a\eta ^{2}/2\nu }$. The solution for $\varphi (t)$ may be written according to Rott (1958) as

$\varphi ^{2}=2a\exp \left(-2\int _{0}^{t}\gamma (s)\,\mathrm {d} s\right)\int _{c}^{t}\exp \left(2\int _{0}^{u}\gamma (s)\,\mathrm {d} s\right)\,\mathrm {d} u,$

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 $c=-\infty$, $\varphi ^{2}=a/k$, a Burgers vortex is a obtained. For arbitrary $c$, the solution becomes $\varphi ^{2}=a(1+\beta \mathrm {e} ^{-2kt})/k$, where $\beta$ is an arbitrary constant. As $t\rightarrow \infty$, Burgers vortex is recovered.

See also

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