Advective balance in pipe-formed vortex rings

TL;DR

This study numerically investigates vorticity distributions in axisymmetric vortex rings, revealing a state of advective balance within the vortex ring bubble and how it depends on Reynolds number and viscosity effects.

Contribution

It introduces the concept of advective balance in vortex rings and analyzes its dependence on Reynolds number and viscosity through a simple diffusion-based toy model.

Findings

Advective balance is achieved within the vortex ring bubble.

The shape of the vorticity distribution is dominated by linear components at higher viscosity.

Reynolds number influences the vorticity distribution, especially near the dividing streamline.

Abstract

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.