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

:$v_r=0, \quad v_\theta=\fracg(r,t), \quad v_z=0.$

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

:$\frac = \nu\left(\frac - \frac \frac\right)$

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

:$g(r,t) = 1-\mathrm^,$

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

:$\omega_z(r,t) = \frac \mathrm^.$

The pressure field simply ensures the vortex rotates in the circumferential direction, providing the centripetal force
:$= \rho ,$
where ''ρ'' is the constant density

Generalized Oseen vortex

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

:$v_r=-\gamma(t) r, \quad v_\theta= \fracg(r,t), \quad v_z = 2\gamma(t) z$

that leads to the equation

:$\frac -\gamma r\frac = \nu \left(\frac - \frac \frac\right).$

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 $g=1-\mathrm^$. The solution for $\varphi(t)$ may be written according to Rott (1958)Rott, N. (1958). On the viscous core of a line vortex. Zeitschrift für angewandte Mathematik und Physik ZAMP, 9(5-6), 543–553. as

:$\varphi^2= 2a\exp\left(-2\int_0^t\gamma(s)\,\mathrm s\right)\int_c^t\exp\left(2\int_0^u \gamma(s)\,\mathrm s\right)\,\mathrmu,$

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 $\varphi^2=a(1+\beta \mathrm^)/k$, 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.

References

{{DEFAULTSORT:Lamb-Oseen vortex
Vortices
Equations of fluid dynamics