Lamb–Oseen vortex

Generalized Oseen vortex

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

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

that leads to the equation

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

Self-similar solution exists for the coordinate ${\displaystyle \eta =r/\phi (t)}$, provided ${\displaystyle \phi {\phi }^{\prime }+\gamma {\phi }^2=a}$, 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) 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 ${\displaystyle c=-\mathrm{\infty }}$, ${\displaystyle {\phi }^2=a/k}$, 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 ${\displaystyle t\to \mathrm{\infty }}$, Burgers vortex is recovered.