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.





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.