Lamb–Oseen vortex

Oseen looked for a solution for the Navier–Stokes equations in cylindrical coordinates $(r,\backslash theta,z)$ with velocity components $(v\_r,v\_\backslash theta,v\_z)$ of the form

:$v\_r=0,\; \backslash quad\; v\_\backslash theta=\backslash frac\{\backslash Gamma\}\{2\backslash pi\; r\}g(r,t),\; \backslash quad\; v\_z=0.$

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

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

which, subject to the conditions that it is regular at $r=0$ and becomes unity as $r\backslash rightarrow\backslash infty$, leads toDrazin, P. G., & Riley, N. (2006). The Navier–Stokes equations: a classification of flows and exact solutions (No. 334). Cambridge University Press.

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

where $\backslash nu$ 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

:$\backslash omega\_z(r,t)\; =\; \backslash frac\{\backslash Gamma\}\{4\backslash pi\; \backslash nu\; t\}\; \backslash mathrm\{e\}^\{-r^2/4\backslash nu\; t\}.$

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

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

where ρ is the constant density{{cite book | author=G.K. Batchelor |author-link=George Batchelor | year=1967 | title= An Introduction to Fluid Dynamics |publisher=Cambridge University Press }}

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

:$v\_r=-\backslash gamma(t)\; r,\; \backslash quad\; v\_\backslash theta=\; \backslash frac\{\backslash Gamma\}\{2\backslash pi\; r\}g(r,t),\; \backslash quad\; v\_z\; =\; 2\backslash gamma(t)\; z$

that leads to the equation

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

Self-similar solution exists for the coordinate $\backslash eta=r/\backslash varphi(t)$, provided $\backslash varphi\backslash varphi\text{'}\; +\backslash gamma\; \backslash varphi^2=a$, where $a$ is a constant, in which case $g=1-\backslash mathrm\{e\}^\{-a\backslash eta^2/2\backslash nu\}$. The solution for $\backslash varphi(t)$ 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

:$\backslash varphi^2=\; 2a\backslash exp\backslash left(-2\backslash int\_0^t\backslash gamma(s)\backslash ,\backslash mathrm\{d\}\; s\backslash right)\backslash int\_c^t\backslash exp\backslash left(2\backslash int\_0^u\; \backslash gamma(s)\backslash ,\backslash mathrm\{d\}\; s\backslash right)\backslash ,\backslash mathrm\{d\}u,$

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

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

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Category:Vortices

Category:Equations of fluid dynamics