• DOI:10.1017/jfm.2017.814
• Corpus ID: 119204612

Advective balance in pipe-formed vortex rings

@article{Shariff2017AdvectiveBI,
  title={Advective balance in pipe-formed vortex rings},
  author={Karim Shariff and Paul S. Krueger},
  journal={Journal of Fluid Mechanics},
  year={2017},
  volume={836},
  pages={773 - 796}
}

• K. Shariff, P. Krueger
• Published 12 December 2017
• Mathematics, Physics
• Journal of Fluid Mechanics

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 $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r\approx F(\unicode[STIX]{x1D713},t)$ , is achieved within the region (called the vortex ring bubble) enclosed by the dividing streamline. Here $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r$ is the ratio of azimuthal vorticity to cylindrical radius, and $\unicode[STIX]{x1D713}$ is the Stokes streamfunction in the frame of the ring. Some, but not all, of the $Re$ dependence in the time evolution of $F(\unicode[STIX]{x1D713},t)$ can be captured by introducing a scaled time $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D708}t$ , where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. When $\unicode[STIX]{x1D708}t/D^{2}\gtrsim 0.02$ , the shape of $F(\unicode[STIX]{x1D713})$ is dominated by the linear-in- $\unicode[STIX]{x1D713}$ component, the coefficient of the quadratic term being an order of magnitude smaller. An important feature is that, as the dividing streamline ( $\unicode[STIX]{x1D713}=0$ ) is approached, $F(\unicode[STIX]{x1D713})$ 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.