Vorticity and angular momentum

Question

In Taylor-Couette flow, the interior fluid becomes fully turbulent if the relative angular velocity of the cylinders is high enough. The turbulent fluid has a vorticity distribution, and each of the (time-averaged) vorticity vectors must point along the rotation axis. If one increases the relative velocity even more, the strength of the turbulence (and its vorticity) also increases, as does the total angular momentum of the fluid. What is the quantitative relation between the two?

Answer 1

The total angular momentum of a 2D flow is given by the integral $$ L_z=\rho \int_D d^2x\; \hat{n} \cdot (\vec r \times\vec v), $$ where $\hat n$ is the unit vector normal to the plane, and $\rho$ is the mass density of the fluid.

Rewriting $\vec r = \nabla(r^2/2)$, and integrating by parts gives $$L_z = \int_D d^2x\; \hat{n} \cdot\left(\nabla {r^2 \over 2}\right) \times \vec v = \int_D d^2x\; \hat{n} \cdot\nabla \times \left({r^2 \over 2} \vec v\right)-\int_D d^2x\; \hat{n} \cdot{r^2 \over 2} \nabla \times \vec v. $$ The first term is a surface integral of a curl, which can be rewritten as a boundary term via Stoke's theorem. The second term is an integral over the vorticity $\varpi = \hat n \cdot \nabla \times \vec v$, weighted by the square of the radius:

$$L_z = \oint_{\partial D} d\vec l \cdot {\rho r^2 \vec v \over 2} -\int_D d^2x\; {\rho r^2 \varpi\over 2}. $$

If the flow on the boundary vanishes, then the total angular momentum $L_z$ is given in terms of the vorticity $\varpi$ as

$$L_z = -{\rho \over 2}\int_D d^2x\; r^2 \varpi. $$

Answer 2

Around each vortex of vorticity $n$ the rotor of velocity is: $$ \oint v \ dl = 2 \pi \kappa $$ Where $\kappa$ is some constant ( in case of superfluids $\frac{n \hbar}{m}, n \in \mathbb{Z}$). At the same time, using $v = \Omega \times r $, $\Omega$ - angular velocity, and $\text{rot} v = 2 \Omega$. Let us consider some circle of unit area, on one hand taking the integral around this circle: $$ \oint v \ dl = 2 \pi \kappa \nu $$ Where we $\nu$ denotes the average vorticity of the fluid. And at the same time $$ \oint v \ dl = 2 \Omega $$ Where we have applied the Stokes formula. So finally one gets: $$ \Omega = \pi \kappa \nu $$