0
Research Papers: Fundamental Issues and Canonical Flows

# A Unified Energy Feature of Vortex Rings for Identifying the Pinch-Off Mechanism

[+] Author and Article Information
Yang Xiang, Suyang Qin

J.C. Wu Center for Aerodynamics,
School of Aeronautics and Astronautics,
Shanghai Jiao Tong University,
Shanghai 200240, China

Hong Liu

J.C. Wu Center for Aerodynamics,
School of Aeronautics and Astronautics,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: hongliu@sjtu.edu.cn

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 14, 2017; final manuscript received July 20, 2017; published online September 20, 2017. Assoc. Editor: Francine Battaglia.

J. Fluids Eng 140(1), 011203 (Sep 20, 2017) (12 pages) Paper No: FE-17-1154; doi: 10.1115/1.4037506 History: Received March 14, 2017; Revised July 20, 2017

## Abstract

Owing to the limiting effect of energy, vortex rings cannot grow indefinitely and thus pinch off. In this paper, experiments on the vortex rings generated using a piston-cylinder apparatus are conducted so as to investigate the pinch-off mechanisms and identify the limiting effect of energy. Both theoretical and experimental results show that the generated vortex rings share a unified energy feature, regardless of whether they are pinched-off or not. Moreover, the unified energy feature is quantitatively described by a dimensionless energy number γ, defined as $γ=(E/I2Γωmax)$ and exhibiting a critical value γring = 0.14 ± 0.01 for the generated vortex rings. This unified energy feature reflects the limiting effect of energy and specifies the target of vortex ring formation. Furthermore, based on the tendency of γ during vortex ring formation, criteria for determining the two timescales, i.e., pinch-off time and separation time, which correspond to the onset and end of pinch-off process, respectively, are suggested.

<>
Your Session has timed out. Please sign back in to continue.

## References

Shariff, K. , and Leonard, A. , 1992, “ Vortex Rings,” Annu. Rev. Fluid Mech., 24(1), pp. 235–279.
Saffman, P. G. , 1981, “ Dynamics of Vorticity,” J. Fluid Mech., 106, pp. 49–58.
Gharib, M. , Rambod, E. , and Shariff, K. , 1998, “ A Universal Time Scale for Vortex Ring Formation,” J. Fluid Mech., 360, pp. 121–140.
Dabiri, J. O. , 2009, “ Optimal Vortex Formation as a Unifying Principle in Biological Propulsion,” Annu. Rev. Fluid Mech., 41(1), pp. 17–33.
Ruiz, L. A. , Whittlesey, R. W. , and Dabiri, J. O. , 2011, “ Vortex-Enhanced Propulsion,” J. Fluid Mech., 668, pp. 5–32.
Whittlesery, R. W. , and Dabiri, J. O. , 2013, “ Optimal Vortex Formation in a Self-Propelled Vehicle,” J. Fluid Mech., 737, pp. 78–104.
Gao, L. , and Yu, S. C. M. , 2010, “ A Model for the Pinch-Off Process of the Leading Vortex Ring in a Starting Jet,” J. Fluid Mech., 656, pp. 205–222.
Benjamin, T. , 1976, “ The Alliance of Practical and Analytical Insights Into the Nonlinear Problems of Fluid Mechanics,” Applications of Methods of Functional Analysis to Problems in Mechanics, Vol. 503, P Germain and B. Nayroles, eds., Springer, Berlin, pp. 8–28.
Kelvin, L. , 1880, “ Vortex Statics,” Philos. Mag., 10(60), pp. 97–109.
Wang, X. X. , and Wu, Z. N. , 2010, “ Stroke-Averaged Lift Forces Due to Vortex Rings and Their Mutual Interactions for a Flapping Flight Model,” J. Fluid Mech., 654, pp. 453–472.
Rayner, J. M. V. , 1979, “ A Vortex Theroy of Animal Flight—Part 1: The Vortex Wake of a Hovering Animal,” J. Fluid Mech., 91(4), pp. 697–730.
Dickinson, M. H. , 1996, “ Unsteady Mechanism of Force Generation in Aquatic and Aerial Locomotion,” Am. Zool., 36(6), pp. 537–554.
Maxworthy, T. , 1977, “ Some Experimental Studies of Vortex Rings,” J. Fluid Mech., 81(03), pp. 465–495.
Didden, N. , 1997, “ On the Formation of Vortex Rings: Rolling-up and Production of Circulation,” Z. Angew. Math. Phys., 30(1), pp. 101–116.
Lim, T. T. , 1997, “ A Note on the Leapfrogging Between Two Coaxial Vortex Rings at Low Reynolds Numbers,” Phys. Fluids., 9(1), pp. 239–241.
Sullivan, I. S. , Niemela, J. J. , Hershberger, R. E. , Bolster, D. , and Donnelly, R. J. , 2008, “ Dynamics of Thin Vortex Rings,” J. Fluid Mech., 609, pp. 319–347.
Chen, B. , Wang, Z. W. , Li, G. J. , and Wang, Y. C. , 2015, “ Experimental Investigation of the Evolution and Head-on Collision of Elliptic Vortex Rings,” ASME J. Fluid Eng., 138(3), p. 031203.
Weigand, A. , and Gharib, M. , 1995, “ Turbulent Vortex Ring/Free Surface Interaction,” ASME J. Fluid Eng., 117(3), pp. 374–381.
Shaffman, P. G. , 1970, “ The Velocity of Viscous Vortex Rings,” Stud. Appl. Math., 49(4), pp. 371–380.
Weigand, A. , and Gharib, M. , 1997, “ On the Evolution of Laminar Vortex Rings,” Exp. Fluids., 22(6), pp. 447–3457.
Fraenkel, L. E. , 1972, “ Example of Steady Vortex Rings of Small Cross-Section in an Ideal Fluid,” J. Fluid Mech., 51(1), pp. 119–135.
Krueger, P. S. , 2005, “ An Over-Pressure Correction to the Slug Model for Vortex Ring Circulation,” J. Fluid Mech., 545, pp. 427–443.
Rosenfeld, M. , Katija, K. , and Dabiri, J. O. , 2009, “ Circulation Generation and Vortex Ring Formation by Conic Nozzles,” ASME J. Fluid Eng., 131(9), p. 091204.
Bourne, K. , Wahono, S. , and Ooi, A. , 2017, “ Numerical Investigation of Vortex Ring Ground Plane Interactions,” ASME J. Fluid Eng., 139(7), p. 071105.
Dabiri, J. O. , and Gharib, M. , 2005, “ The Role of Optimal Vortex Formation in Biological Fluid Transport,” Proc. R. Soc. B., 272(1572), pp. 1557–1560.
Dabiri, J. O. , Colin, S. P. , Costello, J. H. , and Gharib, M. , 2005, “ Flow Patterns Generated by Oblate Medusan Jellyfish: Field Measurements and Laboratory Analyses,” J. Exp. Biol., 208(7), pp. 1257–1265. [PubMed]
Costello, J. H. , Colin, S. P. , and Dabiri, J. O. , 2008, “ Medusan Morphospace: Phylogenetic Constraints, Biomechanical Solutions, and Ecological Consequences,” Invertebr. Biol., 127(3), pp. 265–290.
Shusser, M. , Rosenfeld, M. , Dabiri, J. O. , and Gharib, M. , 2006, “ Effect of Time-Dependent Piston Velocity Program on Vortex Ring Formation in a Piston/Cylinder Arrangement,” Phys. Fluids, 18, p. 033601.
Krueger, P. S. , Dabiri, J. O. , and Gharib, M. , 2006, “ The Formation Number of Vortex Rings Formed in Uniform Background Co-Flow,” J. Fluid Mech., 556, pp. 147–166.
Dabiri, J. O. , and Gharib, M. , 2005, “ Starting Flow Through Nozzles With Temporally Variable Exit Diameter,” J. Fluid Mech., 538, pp. 111–136.
Rosenfeld, M. , Rambod, E. , and Gharib, M. , 1998, “ Circulation and Formation Number of Laminar Vortex Rings,” J. Fluid Mech., 376, pp. 297–318.
Shusser, M. , and Gharib, M. , 2000, “ Energy and Velocity of a Forming Vortex Ring,” Phys. Fluids., 12(3), p. 618.
Gao, L. , and Yu, S. C. M. , 2012, “ Development of the Trailing Shear Layer in a Starting Jet During Pinch-Off,” J. Fluid Mech., 700, pp. 382–405.
O'Farrell, C. , and Dabiri, J. O. , 2010, “ A Lagrangian Approach to Identifying Vortex Pinch-Off,” Chaos, 20(1), p. 017513. [PubMed]
Mohseni, K. , 1998, “ A Model for Universal Time Scale of Vortex Ring Formation,” Phys. Fluids., 10(10), pp. 2436–2438.
O'Farrel, C. , and Dabiri, J. O. , 2014, “ Pinch-Off of Non-Axisymmetric Vortex Rings,” J. Fluid Mech., 740, pp. 61–96.
Fabris, D. , and Liepmann, D. , 1997, “ Vortex Ring Structure at Late Stages of Formation,” Phys. Fluids, 9(9), pp. 2801–2803.
Norbury, J. , 1973, “ A Family of Steady Vortex Rings,” J. Fluid Mech., 57(3), pp. 417–431.
Haller, G. , 2005, “ An Objective Definition of a Vortex,” J. Fluid Mech., 525, pp. 1–26.
Krieg, M. , and Mohseni, K. , 2013, “ On Approximating the Translational Velocity of Vortex Rings,” ASME J. Fluid Eng., 135(12), p. 124501.
Akhmetov, D. G. , 2008, “ Loss of Energy During the Motion of a Vortex Ring,” J. Appl. Mech. Tech. Phys., 49(1), pp. 18–22.
Dabiri, J. O. , 2006, “ Note on the Induced Lagrangian Drift and Added-Mass of a Vortex,” J. Fluid Mech., 547, pp. 105–113.
Dabiri, J. O. , and Gharib, M. , 2004, “ Fluid Entrainment by Isolated Vortex Rings,” J. Fluid Mech., 511, pp. 311–331.
Dora, C. L. , Murugan, T. , De, S. , and Das, D. , 2014, “ Role of Slipstream Instability in Formation of Counter-Rotating Vortex Rings Ahead of a Compressible Vortex Ring,” J. Fluid Mech., 753, pp. 29–48.

## Figures

Fig. 1

Schematic of the experimental setup with a piston-cylinder apparatus

Fig. 2

Velocity programs of piston in six cases

Fig. 3

A dyed vortex ring

Fig. 4

Streamline pattern

Fig. 5

The configuration of vortex ring with different methods. Left: vorticity contour; middle: Q-criterion contour; right: streamline pattern.

Fig. 6

Velocity distribution along the vertical axis

Fig. 7

Normalized circulation of the total jet and the leading vortex ring as a function of formation time for all cases

Fig. 8

Temporal formation of a vortex ring with the Q-criterion and streamline pattern in case C1. Left: flow field based on the Q-criterion, Right: flow field based on the streamline pattern: (a) T* = 1.0, (b) T* = 4.0, (c) T* = 4.8, (d) T* = 6.0, (e) T* = 7.0, and (f) T* = 7.0.

Fig. 10

Formation of the vortex region based on the Q-criterion and its flow visualization in case C2: (a) T* = 3.0 and (b) T* = 6.0

Fig. 11

Evolution of the vortex region based on the Q-criterion and its flow visualization in case C4: (a) T* = 5.8, (b) T* = 6.5, and (c) T* = 8.0

Fig. 9

Circulation of the vortex core of the leading vortex ring during the formation process

Fig. 13

Variation of the dimensionless energy number γ of the total jet against formation time T*

Fig. 14

Variation of the dimensionless energy number γ of the leading vortex ring against formation time T*

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections