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, Re, and stroke-to-diameter ratios, L/D. It is found that a state of advective balance, such that zeta equivalent to omega(phi) /r approximate to F(Psi, t), is achieved within the region (called the vortex ring bubble) enclosed by the dividing streamline. Here zeta equivalent to 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 Re dependence in the time evolution of F(Psi, t)can be captured by introducing a scaled time iota =upsilon t, where upsilon is the kinematic viscosity. When upsilon t/D-2 greater than or similar to 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 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 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.