Advective balance in pipe-formed vortex rings

Vorticity distributions in axisymmetric vortex rings produced by a piston-pipe apparatus are numerically studied over a range of Reynolds numbers, $\mathrm{Re}$, and stroke-to-diameter ratios, $L/D$. It is found that a state of advective balance, such that $\zeta \equiv \omega_\phi/r \approx F(\psi, t)$, is achieved within the region (called the vortex ring bubble) enclosed by the dividing streamline. Here $\zeta \equiv\omega_\phi/r$ is the ratio of azimuthal vorticity to cylindrical radius, and $\psi$ is the Stokes streamfunction in the frame of the ring. Some but not all of the $\mathrm{Re}$ dependence in the time evolution of $F(\psi, t)$ can be captured by introducing a scaled time $\tau = \nu t$, where $\nu$ is the kinematic viscosity. When $\nu t/D^2 \gtrsim 0.02$, the shape of $F(\psi)$ is dominated by the linear-in-$\psi$ component, the coefficient of the quadratic term being an order of magnitude smaller. An important feature is that as the dividing streamline ($\psi = 0$) is approached, $F(\psi)$ tends to a non-zero intercept which exhibits an extra $\mathrm{Re}$ dependence. This and other features are explained by a simple toy model consisting of the one-dimensional cylindrical diffusion equation. The key ingredient in the model responsible for the extra $\mathrm{Re}$ dependence is a Robin-type boundary condition, similar to Newton's law of cooling, that accounts for the edge layer at the dividing streamline.