### Abstract

Analytical formulae, predicted by recently developed vortex ring models, in the limit of small Reynolds numbers (R e), are compared with numerical solutions of the underlying equation for vorticity and experimental data. Particular attention is focused on the recently developed generalised vortex ring model in which the time evolution of the thickness of the vortex ring core L is approximated as a t^{b}, where a and b are constants (1 / 4 ≤ b ≤ 1 / 2). This model incorporates both the laminar model for b = 1 / 2 and the fully turbulent model for b = 1 / 4. A new solution for the normalised vorticity distribution is found in the form ω_{0} + R e ω_{1}, where ω_{0} is the value of normalised vorticity predicted by the classical Phillips solution. This solution shows the correct trends in the redistribution of vorticity due to the Reynolds number effect, and it predicts the increase in the volume of fluid carried inside the vortex ring. It is emphasised that although the structures of vortex rings predicted by analytical formulae, based on the linear approximation, and numerical calculations for arbitrary R e are visibly different for realistic Reynolds numbers, the values of integral characteristics, such as vortex ring translational velocity and energy, predicted by both approaches, turn out to be remarkably close. The values of velocities in the region of maximal vorticity, predicted by the generalised vortex ring model, are compared with the results of experimental studies of vortex ring-like structures in gasoline engine-like conditions with a high-pressure (100 bar) injector. The data analysis is focused on the direct measurements of droplet axial velocities in the regions of maximal vorticity. Most of the values of these velocities lie between the theoretically predicted values corresponding to the later stage of vortex ring development between b = 1 / 4 (fully developed turbulence) and 1 / 2 (laminar case).

Original language | English |
---|---|

Pages (from-to) | 208-216 |

Number of pages | 9 |

Journal | European Journal of Mechanics, B/Fluids |

Volume | 29 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

*European Journal of Mechanics, B/Fluids*,

*29*(3), 208-216. https://doi.org/10.1016/j.euromechflu.2010.01.002

**Dynamics of vortex rings and spray-induced vortex ring-like structures.** / Kaplanski, F.; Sazhin, S. S.; Begg, S.; Fukumoto, Yasuhide; Heikal, M.

Research output: Contribution to journal › Article

*European Journal of Mechanics, B/Fluids*, vol. 29, no. 3, pp. 208-216. https://doi.org/10.1016/j.euromechflu.2010.01.002

}

TY - JOUR

T1 - Dynamics of vortex rings and spray-induced vortex ring-like structures

AU - Kaplanski, F.

AU - Sazhin, S. S.

AU - Begg, S.

AU - Fukumoto, Yasuhide

AU - Heikal, M.

PY - 2010/5/1

Y1 - 2010/5/1

N2 - Analytical formulae, predicted by recently developed vortex ring models, in the limit of small Reynolds numbers (R e), are compared with numerical solutions of the underlying equation for vorticity and experimental data. Particular attention is focused on the recently developed generalised vortex ring model in which the time evolution of the thickness of the vortex ring core L is approximated as a tb, where a and b are constants (1 / 4 ≤ b ≤ 1 / 2). This model incorporates both the laminar model for b = 1 / 2 and the fully turbulent model for b = 1 / 4. A new solution for the normalised vorticity distribution is found in the form ω0 + R e ω1, where ω0 is the value of normalised vorticity predicted by the classical Phillips solution. This solution shows the correct trends in the redistribution of vorticity due to the Reynolds number effect, and it predicts the increase in the volume of fluid carried inside the vortex ring. It is emphasised that although the structures of vortex rings predicted by analytical formulae, based on the linear approximation, and numerical calculations for arbitrary R e are visibly different for realistic Reynolds numbers, the values of integral characteristics, such as vortex ring translational velocity and energy, predicted by both approaches, turn out to be remarkably close. The values of velocities in the region of maximal vorticity, predicted by the generalised vortex ring model, are compared with the results of experimental studies of vortex ring-like structures in gasoline engine-like conditions with a high-pressure (100 bar) injector. The data analysis is focused on the direct measurements of droplet axial velocities in the regions of maximal vorticity. Most of the values of these velocities lie between the theoretically predicted values corresponding to the later stage of vortex ring development between b = 1 / 4 (fully developed turbulence) and 1 / 2 (laminar case).

AB - Analytical formulae, predicted by recently developed vortex ring models, in the limit of small Reynolds numbers (R e), are compared with numerical solutions of the underlying equation for vorticity and experimental data. Particular attention is focused on the recently developed generalised vortex ring model in which the time evolution of the thickness of the vortex ring core L is approximated as a tb, where a and b are constants (1 / 4 ≤ b ≤ 1 / 2). This model incorporates both the laminar model for b = 1 / 2 and the fully turbulent model for b = 1 / 4. A new solution for the normalised vorticity distribution is found in the form ω0 + R e ω1, where ω0 is the value of normalised vorticity predicted by the classical Phillips solution. This solution shows the correct trends in the redistribution of vorticity due to the Reynolds number effect, and it predicts the increase in the volume of fluid carried inside the vortex ring. It is emphasised that although the structures of vortex rings predicted by analytical formulae, based on the linear approximation, and numerical calculations for arbitrary R e are visibly different for realistic Reynolds numbers, the values of integral characteristics, such as vortex ring translational velocity and energy, predicted by both approaches, turn out to be remarkably close. The values of velocities in the region of maximal vorticity, predicted by the generalised vortex ring model, are compared with the results of experimental studies of vortex ring-like structures in gasoline engine-like conditions with a high-pressure (100 bar) injector. The data analysis is focused on the direct measurements of droplet axial velocities in the regions of maximal vorticity. Most of the values of these velocities lie between the theoretically predicted values corresponding to the later stage of vortex ring development between b = 1 / 4 (fully developed turbulence) and 1 / 2 (laminar case).

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U2 - 10.1016/j.euromechflu.2010.01.002

DO - 10.1016/j.euromechflu.2010.01.002

M3 - Article

VL - 29

SP - 208

EP - 216

JO - European Journal of Mechanics, B/Fluids

JF - European Journal of Mechanics, B/Fluids

SN - 0997-7546

IS - 3

ER -