Lamb–Oseen vortex

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

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

where ${\displaystyle \Gamma }$ is the circulation of the vortex core. This lead Navier-Stokes equations to reduce to

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

which when is subjected to the conditions that is regular at ${\displaystyle r=0}$ and becomes unity as ${\displaystyle r\rightarrow \infty }$, leads to[3]

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

where ${\displaystyle \nu }$ is the kinematic viscosity of the fluid. At ${\displaystyle t=0}$, we have a potential vortex with concentrated vorticity at the ${\displaystyle z}$ axis; and this vorticity diffuses away as time passes.

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

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

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

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

where ρ is the constant density[4]

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

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

that leads to the equation

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

Self-similar solution exists for the coordinate ${\displaystyle \eta =r/\phi (t)}$, provided ${\displaystyle \phi \phi '+\gamma \phi ^{2}=a}$, where ${\displaystyle a}$ is a constant, in which case ${\displaystyle g=1-\mathrm {e} ^{-a\eta ^{2}/2\nu }}$. The solution for ${\displaystyle \phi (t)}$ may be written according to Rott (1958)[5] as

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

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