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Advective balance in pipe-formed vortex rings

Published online by Cambridge University Press:  12 December 2017

Karim Shariff Open the ORCID record for Karim Shariff [Opens in a new window]  and
Paul S. Krueger Open the ORCID record for Paul S. Krueger [Opens in a new window]
Karim Shariff*
Affiliation:
NASA Ames Research Center, Moffett Field, CA 94035, USA
Paul S. Krueger*
Affiliation:
Department of Mechanical Engineering, Southern Methodist University, P.O. Box 750337, Dallas, TX 75275-0337, USA
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

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.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , pp. 773 - 796
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Copyright
© 2017 Cambridge University Press 

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