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Dynamics of a trefoil knotted vortex

Published online by Cambridge University Press:  27 July 2021

Jie Yao Open the ORCID record for Jie Yao [Opens in a new window] ,
Yue Yang Open the ORCID record for Yue Yang [Opens in a new window]  and
Fazle Hussain Open the ORCID record for Fazle Hussain [Opens in a new window]
Jie Yao
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX79409, USA
Yue Yang
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing100871, PR China
Fazle Hussain*
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX79409, USA
*
†Email address for correspondence: [email protected]
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Abstract

A slender trefoil knotted vortex is studied using direct numerical simulation of the Navier–Stokes equations for vortex Reynolds numbers ($Re \equiv \varGamma /\nu$, circulation/viscosity) up to 12 000. For initially zero twist ($T_{w,0}=0$), neither the writhe $W_r$ nor the global helicity $H$ is conserved. Initially $W_r$ slowly decreases, then suddenly drops during reconnection and becomes almost constant thence; its evolution is almost $Re$ independent. Before reconnection, $H$ also gradually decreases but sharply increases during reconnection. The evolution of $H$ after reconnection strongly depends on $Re$. While steadily decreasing at low $Re$, $H$ significantly increases before eventually decaying at high $Re$. Flow visualization, helicity decomposition and helical wave decomposition reveal that significant amounts of positive and negative twist helicities are simultaneously generated before and during reconnection. Also, the small leading and large trailing rings resulting from asymmetric reconnection have respectively negative and positive twists, which then decay differently due to different initial values, geometries and mutual induction. In particular, at high $Re$, the twist in the small ring, under stretching by the large trailing ring, decays much faster and even switches sign to become positive by the writhe-to-twist conversion – the main reason for the ‘transient growth’ of $H$. Simulations with non-zero initial twists ($T_{w,0}=7.48$ and $22.48$) reveal that the overall dynamics is similar to the $T_{w,0}=0$ case. Hence, the evolution of the trefoil knotted vortex is mainly governed by $W_r$, not $T_w$, although the latter is found to play an essential role in enstrophy growth as well as energy cascade.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex interactions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A19
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Alamri, S.Z., Youd, A.J. & Barenghi, C.F. 2008 Reconnection of superfluid vortex bundles. Phys. Rev. Lett. 101 (21), 215302.CrossRefGoogle ScholarPubMed
Archer, P.J., Thomas, T.G. & Coleman, G.N. 2008 Direct numerical simulation of vortex ring evolution from the laminar to the early turbulent regime. J. Fluid Mech. 598, 201–226.CrossRefGoogle Scholar
Ashurst, W.T. & Meiron, D.I. 1987 Numerical study of vortex reconnection. Phys. Rev. Lett. 58 (16), 1632.CrossRefGoogle ScholarPubMed
Baggaley, A.W. & Barenghi, C.F. 2011 Spectrum of turbulent kelvin-waves cascade in superfluid helium. Phys. Rev. B 83 (13), 134509.CrossRefGoogle Scholar
Batchelor, G.K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Bewley, G.P., Paoletti, M.S., Sreenivasan, K.R. & Lathrop, D.P. 2008 Characterization of reconnecting vortices in superfluid helium. Proc. Natl Acad. Sci. 105 (37), 13707–13710.CrossRefGoogle ScholarPubMed
Biferale, L., Musacchio, S. & Toschi, F. 2012 Inverse energy cascade in three-dimensional isotropic turbulence. Phys. Rev. Lett. 108 (16), 164501.CrossRefGoogle ScholarPubMed
Boratav, O.N., Pelz, R.B. & Zabusky, N.J. 1992 Reconnection in orthogonally interacting vortex tubes: direct numerical simulations and quantifications. Phys. Fluids A 4 (3), 581–605.CrossRefGoogle Scholar
Bretherton, F.P. 1970 A note on Hamilton's principle for perfect fluids. J. Fluid Mech. 44 (1), 19–31.CrossRefGoogle Scholar
Brissaud, A., Frisch, U., Leorat, J., Lesieur, M. & Mazure, A. 1973 Helicity cascades in fully developed isotropic turbulence. Phys. Fluids 16 (8), 1366–1367.CrossRefGoogle Scholar
Canuto, C., Hussaini, M.Y., Quarteroni, A. & Zhang, T.J. 2012 Spectral Methods in Fluid Dynamics. Springer Science & Business Media.Google Scholar
Chatelain, P., Kivotides, D. & Leonard, A. 2003 Reconnection of colliding vortex rings. Phys. Rev. Lett. 90 (5), 054501.CrossRefGoogle ScholarPubMed
Chen, Q., Chen, S. & Eyink, G.L. 2003 The joint cascade of energy and helicity in three-dimensional turbulence. Phys. Fluids 15 (2), 361–374.CrossRefGoogle Scholar
Cheng, M., Lou, J. & Lim, T.T. 2016 Evolution of an elliptic vortex ring in a viscous fluid. Phys. Fluids 28 (3), 037104.CrossRefGoogle Scholar
Daryan, H., Hussain, F. & Hickey, J.-P. 2020 Aeroacoustic noise generation due to vortex reconnection. Phys. Rev. Fluids 5, 062702.CrossRefGoogle Scholar
Dyson, F.W. 1893 The potential of an anchor ring. Part II. Phil. Trans. R. Soc. Lond A 184, 1041–1106.Google Scholar
Fonda, E., Meichle, D.P., Ouellette, N.T., Hormoz, S. & Lathrop, D.P. 2014 Direct observation of kelvin waves excited by quantized vortex reconnection. Proc. Natl Acad. Sci. 111, 4707–4710.CrossRefGoogle ScholarPubMed
Fonda, E., Sreenivasan, K.R. & Lathrop, D.P. 2019 Reconnection scaling in quantum fluids. Proc. Natl Acad. Sci. 116 (6), 1924–1928.CrossRefGoogle ScholarPubMed
Galantucci, L., Baggaley, A.W., Parker, N.G. & Barenghi, C.F. 2019 Crossover from interaction to driven regimes in quantum vortex reconnections. Proc. Natl Acad. Sci. 116 (25), 12204–12211.CrossRefGoogle ScholarPubMed
Hide, R. 1989 Superhelicity, helicity and potential vorticity. Geophys. Astrophys. Fluid Dyn. 48 (1–3), 69–79.CrossRefGoogle Scholar
Hussain, A.K.M.F. 1986 Coherent structures and turbulence. J. Fluid Mech. 173, 303–356.CrossRefGoogle Scholar
Hussain, F. & Duraisamy, K. 2011 Mechanics of viscous vortex reconnection. Phys. Fluids 23 (2), 021701.CrossRefGoogle Scholar
Iyer, K.P., Sreenivasan, K.R. & Yeung, P.K. 2017 Reynolds number scaling of velocity increments in isotropic turbulence. Phys. Rev. E 95, 021101.CrossRefGoogle ScholarPubMed
Jaque, R.S. & Fuentes, O.V. 2017 Reconnection of orthogonal cylindrical vortices. Eur. J. Mech. (B/Fluids) 62, 51–56.CrossRefGoogle Scholar
Kerr, R.M. 2013 Swirling, turbulent vortex rings formed from a chain reaction of reconnection events. Phys. Fluids 25 (6), 065101.CrossRefGoogle Scholar
Kerr, R.M. 2018 a Enstrophy and circulation scaling for Navier–Stokes reconnection. J. Fluid Mech. 839, R2.CrossRefGoogle Scholar
Kerr, R.M. 2018 b Topology of interacting coiled vortex rings. J. Fluid Mech. 854, R2.CrossRefGoogle Scholar
Kerr, R.M. 2018 c Trefoil knot timescales for reconnection and helicity. Fluid Dyn. Res. 50 (1), 011422.CrossRefGoogle Scholar
Kessar, M., Plunian, F., Stepanov, R. & Balarac, G. 2015 Non-kolmogorov cascade of helicity-driven turbulence. Phys. Rev. E 92 (3), 031004.CrossRefGoogle ScholarPubMed
Kida, S. & Takaoka, M. 1987 Bridging in vortex reconnection. Phys. Fluids 30 (10), 2911–2914.CrossRefGoogle Scholar
Kida, S. & Takaoka, M. 1988 Reconnection of vortex tubes. Fluid Dyn. Res. 3 (1–4), 257–261.CrossRefGoogle Scholar
Kida, S. & Takaoka, M. 1994 Vortex reconnection. Annu. Rev. Fluid Mech. 26 (1), 169–177.CrossRefGoogle Scholar
Kida, S., Takaoka, M. & Hussain, F. 1991 Collision of two vortex rings. J. Fluid Mech. 230, 583–646.CrossRefGoogle Scholar
Kimura, Y. & Moffatt, H.K. 2017 Scaling properties towards vortex reconnection under Biot–Savart evolution. Fluid Dyn. Res. 50 (1), 011409.CrossRefGoogle Scholar
Kivotides, D., Vassilicos, J.C., Samuels, D.C. & Barenghi, C.F. 2001 Kelvin waves cascade in superfluid turbulence. Phys. Rev. Lett. 86 (14), 3080.CrossRefGoogle ScholarPubMed
Kleckner, D. & Irvine, W.T. 2013 Creation and dynamics of knotted vortices. Nat. Phys. 9 (4), 253–258.CrossRefGoogle Scholar
Koplik, J. & Levine, H. 1993 Vortex reconnection in superfluid helium. Phys. Rev. Lett. 71 (9), 1375.CrossRefGoogle ScholarPubMed
Laing, C.E., Ricca, R.L. & Sumners, D.W. 2015 Conservation of writhe helicity under anti-parallel reconnection. Sci. Rep. 5, 9224.CrossRefGoogle ScholarPubMed
Lee, M. & Moser, R.D. 2015 Direct numerical simulation of turbulent channel flow up to Re 5200. J. Fluid Mech. 774, 395–415.CrossRefGoogle Scholar
Lesieur, M. 2012 Turbulence in Fluids. Springer Science & Business Media.Google Scholar
Levy, Y., Degani, D. & Seginer, A. 1990 Graphical visualization of vortical flows by means of helicity. AIAA J. 28 (8), 1347–1352.CrossRefGoogle Scholar
Maggioni, F., Alamri, S., Barenghi, C.F. & Ricca, R.L. 2010 Velocity, energy, and helicity of vortex knots and unknots. Phys. Rev. E 82 (2), 026309.CrossRefGoogle ScholarPubMed
Mariani, R. & Kontis, K. 2010 Experimental studies on coaxial vortex loops. Phys. Fluids 22 (12), 126102.CrossRefGoogle Scholar
McKeown, R., Monico, R.O., Pumir, A., Brenner, M.P. & Rubinstein, S.M. 2018 Cascade leading to the emergence of small structures in vortex ring collisions. Phys. Rev. Fluids 3 (12), 124702.CrossRefGoogle Scholar
Melander, M.V. & Hussain, F. 1988 Cut-and-connect of two antiparallel vortex tubes. In Studying Turbulence Using Numerical Simulation Databases, vol. 2, pp. 257–286. Center for Turbulence Research.Google Scholar
Melander, M.V. & Hussain, F. 1993 Polarized vorticity dynamics on a vortex column. Phys. Fluids A 5 (8), 1992–2003.CrossRefGoogle Scholar
Melander, M.V. & Hussain, F. 1994 Core dynamics on a vortex column. Fluid Dyn. Res. 13 (1), 1–37.CrossRefGoogle Scholar
Melander, M.V. & Zabusky, N.J. 1988 Interaction and “apparent” reconnection of 3D vortex tubes via direct numerical simulations. Fluid Dyn. Res. 3 (1–4), 247–250.Google Scholar
Moffatt, H.K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35 (1), 117–129.CrossRefGoogle Scholar
Moffatt, H.K. & Ricca, R.L. 1992 Helicity and the Călugăreanu invariant. Proc. R. Soc. Lond. A 439, 411–429.Google Scholar
Oberti, C. & Ricca, R.L. 2019 Influence of winding number on vortex knots dynamics. Sci. Rep. 9 (1), 17284.CrossRefGoogle ScholarPubMed
Orlandi, P., Pirozzoli, S. & Carnevale, G.F. 2012 Vortex events in euler and Navier–Stokes simulations with smooth initial conditions. J. Fluid Mech. 690, 288–320.CrossRefGoogle Scholar
Pohl, W.F. 1968 The self-linking number of a closed space curve. J. Math. Mech. 17, 975–985.Google Scholar
Pradeep, D.S. & Hussain, F. 2004 Effects of boundary condition in numerical simulations of vortex dynamics. J. Fluid Mech. 516, 115–124.CrossRefGoogle Scholar
Pradeep, D.S. & Hussain, F. 2010 Vortex dynamics of turbulence–coherent structure interaction. Theor. Comput. Fluid Dyn. 24 (1–4), 265–282.CrossRefGoogle Scholar
Priest, E. & Forbes, T. 2000 Magnetic Reconnection: MHD Theory and Applications. Cambridge University Press.CrossRefGoogle Scholar
Proment, D. & Krstulovic, G. 2020 A matching theory to characterize sound emission during vortex reconnection in quantum fluids. Phys. Rev. Fluids 5, 104701.Google Scholar
Pumir, A. & Kerr, R.M. 1987 Numerical simulation of interacting vortex tubes. Phys. Rev. Lett. 58 (16), 1636.CrossRefGoogle ScholarPubMed
van Rees, W.M., Hussain, F. & Koumoutsakos, P. 2012 Vortex tube reconnection at $Re= 104$. Phys. Fluids 24 (7), 075105.CrossRefGoogle Scholar
Sadlo, F., Hausner, P., Hofmann, L., Jung, P., Karch, G., Peikert, R., Pilz, L., Roth, M. & Sdeo, K. 2019 VCG ParaView plugins.Google Scholar
Scheeler, M.W., Kleckner, D., Proment, D., Kindlmann, G.L. & Irvine, W.T. 2014 Helicity conservation by flow across scales in reconnecting vortex links and knots. Proc. Natl Acad. Sci. 111 (43), 15350–15355.CrossRefGoogle ScholarPubMed
Scheeler, M.W., van Rees, W.M., Kedia, H., Kleckner, D. & Irvine, W.T. 2017 Complete measurement of helicity and its dynamics in vortex tubes. Science 357 (6350), 487–491.CrossRefGoogle ScholarPubMed
Schmid, P.J., Henningson, D.S. & Jankowski, D.F. 2002 Stability and transition in shear flows. Applied Mathematical Sciences, vol. 142. Appl. Mech. Rev. 55 (3), B57–B59.Google Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.Google Scholar
Shelley, M.J., Meiron, D.I. & Orszag, S.A. 1993 Dynamical aspects of vortex reconnection of perturbed anti-parallel vortex tubes. J. Fluid Mech. 246, 613–652.CrossRefGoogle Scholar
Siggia, E.D. 1985 Collapse and amplification of a vortex filament. Phys. Fluids 28 (3), 794–805.CrossRefGoogle Scholar
Sujudi, D. & Haimes, R. 1995 Identification of swirling flow in 3-d vector fields. In 12th Computational Fluid Dynamics Conference, p. 1715. The American Institute of Aeronautics and Astronautics.CrossRefGoogle Scholar
Tian, S., Gao, Y., Dong, X. & Liu, C. 2018 Definitions of vortex vector and vortex. J. Fluid Mech. 849, 312–339.CrossRefGoogle Scholar
Vazquez, M. & Sumners, D.W. 2004 Tangle analysis of gin site-specific recombination. Math. Proc. Camb. Phil. Soc. 136, 565–582.CrossRefGoogle Scholar
Villois, A., Proment, D. & Krstulovic, G. 2017 Universal and nonuniversal aspects of vortex reconnections in superfluids. Phys. Rev. Fluids 2 (4), 044701.CrossRefGoogle Scholar
Virk, D., Hussain, F. & Kerr, R.M. 1995 Compressible vortex reconnection. J. Fluid Mech. 304, 47–86.CrossRefGoogle Scholar
Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids A 4 (2), 350–363.CrossRefGoogle Scholar
Waleffe, F. 1993 Inertial transfers in the helical decomposition. Phys. Fluids A 5 (3), 677–685.CrossRefGoogle Scholar
Wu, J.Z., Ma, H.Y. & Zhou, M.D. 2007 Vorticity and Vortex Dynamics. Springer Science & Business Media.Google Scholar
Xiong, S. & Yang, Y. 2019 Construction of knotted vortex tubes with the writhe-dependent helicity. Phys. Fluids 31 (4), 047101.CrossRefGoogle Scholar
Xiong, S. & Yang, Y. 2020 Effects of twist on the evolution of knotted magnetic flux tubes. J. Fluid Mech. 895, A28.Google Scholar
Yao, J. & Hussain, F. 2018 Toward vortex identification based on local pressure-minimum criterion in compressible and variable density flows. J. Fluid Mech. 850, 5–17.Google Scholar
Yao, J. & Hussain, F. 2020 a On singularity formation via viscous vortex reconnection. J. Fluid Mech. 888, R2.CrossRefGoogle Scholar
Yao, J. & Hussain, F. 2020 b A physical model of turbulence cascade via vortex reconnection sequence and avalanche. J. Fluid Mech. 883, A51.CrossRefGoogle Scholar
Yao, J. & Hussain, F. 2020 c Separation scaling for viscous vortex reconnection. J. Fluid Mech. 900, R4.CrossRefGoogle Scholar
Yao, J. & Hussain, F. 2021 Polarized vortex reconnection. J. Fluid Mech. 922, A19.Google Scholar
Yu, Z., Chapelier, J.-B. & Scalo, C. 2018 Coherent-vorticity preserving large-eddy simulation of trefoil knotted vortices. In 2018 AIAA Aerospace Sciences Meeting, p. 2090. The American Institute of Aeronautics and Astronautics.Google Scholar
Zhao, X. & Scalo, C. 2019 Direct numerical simulation of trefoil knotted vortices. APS pp. Q12–003.Google Scholar
Zhao, X., Yu, Z., Chapelier, J.-B. & Scalo, C. 2021 Direct numerical and large-eddy simulation of trefoil knotted vortices. J. Fluid Mech. 910, A31.CrossRefGoogle Scholar
Zhao, Y., Yang, Y. & Chen, S. 2016 Vortex reconnection in the late transition in channel flow. J. Fluid Mech. 802, R4.CrossRefGoogle Scholar
Zuccher, S., Caliari, M., Baggaley, A.W. & Barenghi, C.F. 2012 Quantum vortex reconnections. Phys. Fluids 24 (12), 125108.CrossRefGoogle Scholar