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Vortex ring formation process in starting jets with uniform background co- and counter-flow

Published online by Cambridge University Press:  07 August 2023

Jianwei Zhu Open the ORCID record for Jianwei Zhu [Opens in a new window] ,
Guoqing Zhang Open the ORCID record for Guoqing Zhang [Opens in a new window] ,
Lei Gao Open the ORCID record for Lei Gao [Opens in a new window]  and
S.C.M. Yu Open the ORCID record for S.C.M. Yu [Opens in a new window]
Jianwei Zhu
Affiliation:
Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, PR China
Guoqing Zhang*
Affiliation:
Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, PR China
Lei Gao
Affiliation:
School of Aeronautics and Astronautics, Sichuan University, Chengdu 610065, PR China
S.C.M. Yu
Affiliation:
Department of Aeronautical and Aviation Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, PR China
*
†Email address for correspondence: zhanggq@bit.edu.cn
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Abstract

The formation process of the leading vortex ring in starting jets with uniform background co- and counter-flow has been studied numerically for $-0.5\leq R_v\leq 0.5$, where $R_v$ is the ratio of background velocity to jet velocity. For the cases with background counter-flow, the normal formation process of the leading vortex ring would be destroyed when $R_v<-0.4$, i.e. the trailing jet would overtake the leading vortex ring through the centre, a phenomenon reminiscent of vortex leapfrogging. As the velocity ratio $R_v$ increases, the formation number $F_{t^*}$ decreases from $9.6$ at $R_v=-0.4$ to $1.92$ at $R_v=0.5$. An analytical model based on the kinematic criterion has been developed so as to describe the relationship between the formation number $F_{t^*}$ and velocity ratio $R_v$. A linear relationship between the vortex core parameter and stroke ratio of starting jet ($\varepsilon \sim k_1L/D$) for the Norbury vortex ring has been established and used effectively to close the model. For co-flow with $0< R_v\leq 0.5$, the results from this model are consistent with the present numerical simulation and the experiments by Krueger et al. (J. Fluid Mech., vol. 556, 2006, pp. 147–166). For counter-flow, two different equations are proposed for $-0.4\leq R_v\leq -0.2$ and $-0.2< R_v<0$, respectively.

JFM classification

Wakes/Jets: Jets Vortex Flows: Vortex shedding
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 968 , 10 August 2023 , A26
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Ai, J.J., Yu, S.C.M., Law, A.W.K. & Chua, L.P. 2005 Vortex dynamics in starting square water jets. Phys. Fluids 17 (1), 014106.CrossRefGoogle Scholar
Anderson, E.J. & Demont, M.E. 2000 The mechanics of locomotion in the squid Loligo pealei: locomotory function and unsteady hydrodynamics of the jet and intramantle pressure. J. Expl Biol. 203 (18), 2851–2863.CrossRefGoogle ScholarPubMed
Anderson, E.J. & Grosenbaugh, M.A. 2005 Jet flow in steadily swimming adult squid. J. Expl Biol. 208 (6), 1125–1146.CrossRefGoogle ScholarPubMed
Bi, X. & Zhu, Q. 2019 Fluid-structure investigation of a squid-inspired swimmer. Phys. Fluids 31 (10), 101901.CrossRefGoogle Scholar
Cheng, J.Y. & Demont, M.E. 1996 Jet-propelled swimming in scallops: swimming mechanics and ontogenic scaling. Can. J. Zool. 74 (9), 1734–1748.CrossRefGoogle Scholar
Dabiri, J.O. 2009 Optimal vortex formation as a unifying principle in biological propulsion. Annu. Rev. Fluid Mech. 41, 17–33.CrossRefGoogle Scholar
Dabiri, J.O., Colin, S.P., Costello, J.H. & Gharib, M. 2005 Flow patterns generated by oblate medusan jellyfish: field measurements and laboratory analyses. J. Expl Biol. 208 (7), 1257–1265.CrossRefGoogle ScholarPubMed
Dabiri, J.O. & Gharib, M. 2003 Sensitivity analysis of kinematic approximations in dynamic medusan swimming models. J. Expl Biol. 206 (20), 3675–3680.CrossRefGoogle ScholarPubMed
Dabiri, J.O. & Gharib, M. 2004 a Delay of vortex ring pinchoff by an imposed bulk counterflow. Phys. Fluids 16 (4), L28–L30.CrossRefGoogle Scholar
Dabiri, J.O. & Gharib, M. 2004 b Fluid entrainment by isolated vortex rings. J. Fluid Mech. 511, 311–331.CrossRefGoogle Scholar
Dabiri, J.O. & Gharib, M. 2005 Starting flow through nozzles with temporally variable exit diameter. J. Fluid Mech. 538, 111–136.CrossRefGoogle Scholar
Didden, N. 1979 On the formation of vortex rings: rolling-up and production of circulation. Z. Angew. Math. Phys. 30 (1), 101–116.CrossRefGoogle Scholar
Fraenkel, L.E. 1972 Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech. 51 (1), 119–135.CrossRefGoogle Scholar
Gao, L. 2011 The pinch-off process of the leading vortex ring in a starting jet. PhD thesis, School of Mechanical and Aerospace Engineering, Nanyang Technological University Singapore.Google Scholar
Gao, L., Wang, X., Yu, S.C.M. & Chyu, M.K. 2020 Development of the impulse and thrust for laminar starting jets with finite discharged volume. J. Fluid Mech. 902, A27.CrossRefGoogle Scholar
Gao, L. & 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, 205–222.CrossRefGoogle Scholar
Gao, L. & Yu, S.C.M. 2012 Development of the trailing shear layer in a starting jet during pinch-off. J. Fluid Mech. 700, 382–405.CrossRefGoogle Scholar
Gao, L., Yu, S.C.M., Ai, J.J. & Law, A.W.K. 2008 Circulation and energy of the leading vortex ring in a gravity-driven starting jet. Phys. Fluids 20 (9), 093604.CrossRefGoogle Scholar
Gharib, M., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121–140.CrossRefGoogle Scholar
Glezer, A. 1988 The formation of vortex rings. Phys. Fluids 31 (12), 3532–3542.CrossRefGoogle Scholar
de Guyon, G. & Mulleners, K. 2022 Estimating the non-dimensional energy of vortex rings by modelling their roll-up. J. Fluid Mech. 940, R2.CrossRefGoogle Scholar
Hill, M.J.M. 1894 VI. On a spherical vortex. Phil. Trans. R. Soc. Lond. A 185, 213–245.Google Scholar
Jiang, H., Costello, J.H. & Colin, S.P. 2021 Fluid dynamics and efficiency of colonial swimming via multijet propulsion at intermediate Reynolds numbers. Phys. Rev. Fluids 6 (1), 013103.CrossRefGoogle Scholar
Jiang, H. & Grosenbaugh, M.A. 2006 Numerical simulation of vortex ring formation in the presence of background flow with implications for squid propulsion. Theor. Comput. Fluid Dyn. 20 (2), 103–123.CrossRefGoogle Scholar
Krieg, M. & Mohseni, K. 2013 Modelling circulation, impulse and kinetic energy of starting jets with non-zero radial velocity. J. Fluid Mech. 719, 488–526.CrossRefGoogle Scholar
Krieg, M. & Mohseni, K. 2021 A new kinematic criterion for vortex ring pinch-off. Phys. Fluids 33 (3), 037120.CrossRefGoogle Scholar
Krueger, P.S. 2005 An over-pressure correction to the slug model for vortex ring circulation. J. Fluid Mech. 545, 427–443.CrossRefGoogle Scholar
Krueger, P.S., Dabiri, J.O. & Gharib, M. 2003 Vortex ring pinchoff in the presence of simultaneously initiated uniform background co-flow. Phys. Fluids 15 (7), L49–L52.CrossRefGoogle Scholar
Krueger, P.S., Dabiri, J.O. & Gharib, M. 2006 The formation number of vortex rings formed in uniform background co-flow. J. Fluid Mech. 556, 147–166.CrossRefGoogle Scholar
Krueger, P.S. & Gharib, M. 2003 The significance of vortex ring formation to the impulse and thrust of a starting jet. Phys. Fluids 15 (5), 1271–1281.CrossRefGoogle Scholar
Krueger, P.S., Moslemi, A.A., Nichols, J.T., Bartol, I.K. & Stewart, W.J. 2008 Vortex rings in bio-inspired and biological jet propulsion. In Advances in Science and Technology, vol. 58, pp. 237–246. Trans Tech.CrossRefGoogle Scholar
Lim, T.T. & Nickels, T.B. 1995 Vortex rings. In Fluid Vortices (ed. S.I. Green), pp. 95–153. Springer.CrossRefGoogle Scholar
Linden, P.F. 2011 The efficiency of pulsed-jet propulsion. J. Fluid Mech. 668, 1–4.CrossRefGoogle Scholar
Linden, P.F. & Turner, J.S. 2001 The formation of ‘optimal’ vortex rings, and the efficiency of propulsion devices. J. Fluid Mech. 427, 61–72.CrossRefGoogle Scholar
Linden, P.F. & Turner, J.S. 2004 ‘Optimal’ vortex rings and aquatic propulsion mechanisms. Proc. R. Soc. Lond. B 271 (1539), 647–653.CrossRefGoogle ScholarPubMed
Luo, Y., Xiao, Q., Zhu, Q. & Pan, G. 2021 Jet propulsion of a squid-inspired swimmer in the presence of background flow. Phys. Fluids 33 (3), 031909.CrossRefGoogle Scholar
Ma, X., Gong, X. & Jiang, N. 2022 Experimental study of vortex formation in pulsating jet flow by time-resolved particle image velocimetry. Phys. Fluids 34 (3), 035105.CrossRefGoogle Scholar
Mohseni, K. 2006 Pulsatile vortex generators for low-speed maneuvering of small underwater vehicles. Ocean Engng 33 (16), 2209–2223.CrossRefGoogle Scholar
Mohseni, K. & Gharib, M. 1998 A model for universal time scale of vortex ring formation. Phys. Fluids 10 (10), 2436–2438.CrossRefGoogle Scholar
Moslemi, A.A. & Krueger, P.S. 2010 Propulsive efficiency of a biomorphic pulsed-jet underwater vehicle. Bioinspir. Biomim. 5 (3), 036003.CrossRefGoogle ScholarPubMed
Moslemi, A.A. & Krueger, P.S. 2011 The effect of Reynolds number on the propulsive efficiency of a biomorphic pulsed-jet underwater vehicle. Bioinspir. Biomim. 6 (2), 026001.CrossRefGoogle ScholarPubMed
Nichols, J.T. & Krueger, P.S. 2012 Effect of vehicle configuration on the performance of a submersible pulsed-jet vehicle at intermediate Reynolds number. Bioinspir. Biomim. 7 (3), 036010.CrossRefGoogle ScholarPubMed
Norbury, J. 1973 A family of steady vortex rings. J. Fluid Mech. 57 (3), 417–431.CrossRefGoogle Scholar
Rosenfeld, M., Rambod, E. & Gharib, M. 1998 Circulation and formation number of laminar vortex rings. J. Fluid Mech. 376, 297–318.CrossRefGoogle Scholar
Ruiz, L.A., Whittlesey, R.W. & Dabiri, J.O. 2011 Vortex-enhanced propulsion. J. Fluid Mech. 668, 5–32.CrossRefGoogle Scholar
Saffman, P.G. 1975 On the formation of vortex rings. Stud. Appl. Maths 54 (3), 261–268.CrossRefGoogle Scholar
Shariff, K. & Leonard, A. 1992 Vortex rings. Annu. Rev. Fluid Mech. 24 (1), 235–279.CrossRefGoogle Scholar
Shusser, M. & Gharib, M. 2000 Energy and velocity of a forming vortex ring. Phys. Fluids 12 (3), 618–621.CrossRefGoogle Scholar
Shusser, M., Gharib, M. & Mohseni, K. 1999 A new model for inviscid vortex ring formation. In 30th Fluid Dynamics Conference, p. 3805. American Institute of Aeronautics and Astronautics.CrossRefGoogle Scholar
Whittlesey, R.W. & Dabiri, J.O. 2013 Optimal vortex formation in a self-propelled vehicle. J. Fluid Mech. 737, 78–104.CrossRefGoogle Scholar
Zhang, X., Wang, J. & Wan, D. 2020 CFD investigations of evolution and propulsion of low speed vortex ring. Ocean Engng 195, 106687.CrossRefGoogle Scholar
Zhao, W., Frankel, S.H. & Mongeau, L.G. 2000 Effects of trailing jet instability on vortex ring formation. Phys. Fluids 12 (3), 589–596.CrossRefGoogle Scholar
Zhu, J., Zhang, G., Gao, L. & Yu, S.C.M. 2022 The formation process of annular starting jets. J. Fluid Mech. 949, A47.CrossRefGoogle Scholar
Zhu, J., Zhang, G., Gao, L. & Yu, S.C.M. 2023 The circulation growth of non-impulsive starting jet. Phys. Fluids 35 (5), 057102.Google Scholar