# Transporting Particles with Vortex Rings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulation

## 3. Analytical Approximations

#### 3.1. Velocity Field

#### 3.2. Temporal Evolution of a Vortex Ring

#### 3.3. Particle Concentration

#### 3.4. Particle Loss Mechanism

## 4. Experimental Setup and Measurements

## 5. Results and Discussions

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Akhmetov, D.G. Vortex Rings; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Meleshko, V.V.; Gourjii, A.A.; Krasnopolskaya, T.S. Vortex rings: History and state of the art. J. Math. Sci.
**2012**, 187, 772–808. [Google Scholar] [CrossRef] - Fraenkel, L.E. Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech.
**1972**, 51, 119–135. [Google Scholar] [CrossRef] - Auerbach, D. Stirring properties of vortex rings. Phys. Fluids A
**1991**, 3, 1351–1355. [Google Scholar] [CrossRef] - Tarasov, V.F.; Yakushev, V.I. Transport in a turbulent vortex ring. J. Appl. Mech. Tech. Phys.
**1975**, 15, 106–110. [Google Scholar] [CrossRef] - Nikulin, V.V. Mass Exchange between the Atmosphere of Turbulent Vortex Ring and the Surrounding Medium. Fluid Dyn.
**2021**, 56, 473–480. [Google Scholar] [CrossRef] - Rakhsha, M.; Kees, C.E.; Negrut, D. Lagrangian vs. Eulerian: An Analysis of Two Solution Methods for Free-Surface Flows and Fluid Solid Interaction Problems. Fluids
**2021**, 6, 460. [Google Scholar] [CrossRef] - Moraes, P.G.d.; Oliveira, M.A.d.; Bimbato, A.M.; Pereira, L.A.A. A Lagrangian Description of Buoyancy Effects on Aircraft Wake Vortices from Wing Tips near a Heated Ground Plane. Energies
**2022**, 15, 6995. [Google Scholar] [CrossRef] - Griffiths, D.A.; Ananthan, S.; Leishman, J.G. Predictions of Rotor Performance in Ground Effect Using a Free-Vortex Wake Model. J. Am. Helicopter Soc.
**2005**, 50, 302–314. [Google Scholar] [CrossRef] - Akhmetov, D.G.; Lugovtsov, B.A.; Maletin, V.A. Vortex Powder Method for Extinguishing a Fire on Spouting Gas—Oil Wells. In Prevention of Hazardous Fires and Explosions; Springer: Dordrecht, The Netherlands, 1999; pp. 319–328. [Google Scholar] [CrossRef]
- An, D.; Warning, A.; Yancey, K.G.; Chang, C.T.; Kern, V.R.; Datta, A.K.; Steen, P.H.; Luo, D.; Ma, M. Mass production of shaped particles through vortex ring freezing. Nat. Commun.
**2016**, 7, 12401. [Google Scholar] [CrossRef] - Akagi, F.; Haraga, I.; ichi Inage, S.; Akiyoshi, K. Effect of sneezing on the flow around a face shield. Phys. Fluids
**2020**, 32, 127105. [Google Scholar] [CrossRef] - Sullivan, I.S.; Niemela, J.J.; Hershberger, R.E.; Bolster, D.; Donnelly, R.J. Dynamics of thin vortex rings. J. Fluid Mech.
**2008**, 609, 319–347. [Google Scholar] [CrossRef] - Kida, S.; Takaoka, M.; Hussain, F. Collision of two vortex rings. J. Fluid Mech.
**1991**, 230, 583–646. [Google Scholar] [CrossRef] - Cheng, M.; Lou, J.; Lim, T.T. Collision and reconnection of viscous elliptic vortex rings. Phys. Fluids
**2019**, 31, 067107. [Google Scholar] [CrossRef] - Mouallem, J.; Daryan, H.; Wawryk, J.; Pan, Z.x.f.x.a.; Hickey, J.P. Targeted particle delivery via vortex ring reconnection. Phys. Fluids
**2021**, 33, 103305. [Google Scholar] [CrossRef] - Domon, K.; Ishihara, O.; Watanabe, S. Mass Transport by a Vortex Ring. J. Phys. Soc. Jpn.
**2000**, 69, 120–123. [Google Scholar] [CrossRef] - Uchiyama, T.; Yano, C.; Degawa, T. Generation and Transport of Solid Particle Clusters Using a Vortex Ring Launched into Water. Int. J. Chem. Eng. Appl.
**2017**, 8, 253–260. [Google Scholar] [CrossRef] - James, S.; Madnia, C.K. Direct numerical simulation of a laminar vortex ring. Phys. Fluids
**1996**, 8, 2400–2414. [Google Scholar] [CrossRef] - Danaila, I.; Hélie, J. Numerical simulation of the postformation evolution of a laminar vortex ring. Phys. Fluids
**2008**, 20, 073602. [Google Scholar] [CrossRef] - Weigand, A.; Gharib, M. On the decay of a turbulent vortex ring. Phys. Fluids
**1994**, 6, 3806–3808. [Google Scholar] [CrossRef] - Rosenfeld, M.; Rambod, E.; Gharib, M. Circulation and formation number of laminar vortex rings. J. Fluid Mech.
**1998**, 376, 297–318. [Google Scholar] [CrossRef] - Sullivan, J.P.; Widnall, S.E.; Ezekiel, S. Study of Vortex Rings Using a Laser Doppler Velocimeter. AIAA J.
**2012**, 10, 11. [Google Scholar] [CrossRef] - Dziedzic, M.; Leutheusser, H.J. An experimental study of viscous vortex rings. Exp. Fluids
**1996**, 21, 315–324. [Google Scholar] [CrossRef] - Fung, J.C.H. Residence time of inertial particles in a vortex. J. Geophys. Res. Ocean.
**2000**, 105, 14261–14272. [Google Scholar] [CrossRef] - Saffman, P.G. The Velocity of Viscous Vortex Rings. Stud. Appl. Math.
**1970**, 49, 371–380. [Google Scholar] [CrossRef] - Tinaikar, A.; Advaith, S.; Basu, S. Understanding evolution of vortex rings in viscous fluids. J. Fluid Mech.
**2018**, 836, 873–909. [Google Scholar] [CrossRef] - Guide, C.M.U. Mixer Module User’s Guide; COMSOL: Stockholm, Sweden, 2021. [Google Scholar]
- Visuri, O.; Wierink, G.A.; Alopaeus, V. Investigation of drag models in CFD modeling and comparison to experiments of liquid–solid fluidized systems. Int. J. Miner. Process.
**2012**, 104–105, 58–70. [Google Scholar] [CrossRef] - Silva, R.; Cotas, C.; Garcia, F.A.P.; Faia, P.M.; Rasteiro, M.G. Particle Distribution Studies in Highly Concentrated Solid-liquid Flows in Pipe Using the Mixture Model. Procedia Eng.
**2015**, 102, 1016–1025. [Google Scholar] [CrossRef] - Scase, M.M.; Dalziel, S.B. An experimental study of the bulk properties of vortex rings translating through a stratified fluid. Eur. J. Mech. B Fluids
**2006**, 25, 302–320. [Google Scholar] [CrossRef] - Tracker Video Analysis and Modeling Tool for Physics Education. 2023. Available online: https://physlets.org/tracker/ (accessed on 23 May 2023).

**Figure 2.**Example of the initial velocity field used to form the rings in simulations. The white rectangle indicates the area where the colloidal particles were initially placed. The color bar indicates the absolute value of the velocity at each point.

**Figure 3.**Time evolution of a vortex ring: (

**lower panel**) experimental pictures, and (

**upper panel**) simulation.

**Figure 5.**Modulus of the vorticity $\left|\mathrm{curl}\right(\mathit{v}\left)\right|$, dilute concentration n, and a fit comprising two Gaussian curves in the vortex cross-section orthogonal to the propagation direction.

**Figure 6.**Correlation between the vorticity and dilute concentration at a fixed time moment and the best linear fit. Different points in the plot correspond to different spatial points.

**Figure 7.**Coefficient $\mathcal{N}\left(t\right)$ calculated with Equation (9) and extracted from Comsol simulations as a ratio $n/\left|\mathrm{curl}\right(\mathit{v}\left)\right|$.

**Figure 8.**Experimental setup: (1) video camera; (2) source of a dye or colloidal particles; (3) automatic drive for control of vortex ring formation; (4) propagating vortex ring; (5) outlet of the injection tube; (6) UV flashlight.

**Figure 10.**Dependence of the vertical coordinate of a vortex ring (

**a**) and its velocity (

**b**) on time. Experimental data are shown with dotted lines, with errors indicated by shaded areas; theoretical data are shown with solid lines; numerical simulation results are shown with dashed lines. Different colors correspond to different initial Reynolds numbers.

**Figure 11.**Time-dependent coefficients of the theoretical function describing the vortex ring shape. Panel (

**a**) shows the distance ${x}_{1}$ from the center of the ring to one of its cores, so ${x}_{1}$ makes sense of the main radius of the ring R, it appears in (10). Panels (

**b**–

**d**) show the coefficients of ${f}_{u}=a\phantom{\rule{0.166667em}{0ex}}\mathrm{exp}(-{r}^{c}/b)$, a, b and c, respectively. Also, the moment of time ${t}_{0}$ is marked in all the figures, it makes sense of the end of the transition period of the ring. Since in modeling the ring is formed according to the primary approximate formula, at the beginning of its evolution it naturally changes shape to its characteristic appearance. Therefore, for approximately the first second, the shape of the ring is extremely non-standard, therefore, the coefficients change very sharply at the beginning, but after the time ${t}_{0}$, they acquire a monotone dependence.

**Figure 12.**Time dependence of the relative number of particles ${N}_{rel}\left(t\right)=N/{N}_{0}$ carried by vortex rings. The experimental and modeling results are shown with light blue and violet colors, respectively. Theoretical calculation results are shown with a dashed line. The shaded areas’ width illustrates the error.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gulinyan, V.; Kuzikov, F.; Podgornyi, R.; Shirkin, D.; Zakharov, I.; Sadrieva, Z.; Korobkov, M.; Muzychenko, Y.; Kudlis, A.
Transporting Particles with Vortex Rings. *Fluids* **2023**, *8*, 315.
https://doi.org/10.3390/fluids8120315

**AMA Style**

Gulinyan V, Kuzikov F, Podgornyi R, Shirkin D, Zakharov I, Sadrieva Z, Korobkov M, Muzychenko Y, Kudlis A.
Transporting Particles with Vortex Rings. *Fluids*. 2023; 8(12):315.
https://doi.org/10.3390/fluids8120315

**Chicago/Turabian Style**

Gulinyan, Van, Fedor Kuzikov, Roman Podgornyi, Daniil Shirkin, Ivan Zakharov, Zarina Sadrieva, Maxim Korobkov, Yana Muzychenko, and Andrey Kudlis.
2023. "Transporting Particles with Vortex Rings" *Fluids* 8, no. 12: 315.
https://doi.org/10.3390/fluids8120315