Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T06:42:42.090Z Has data issue: false hasContentIssue false
  • English
  • Français

Polarized vortex reconnection

Published online by Cambridge University Press:  12 July 2021

Jie Yao Open the ORCID record for Jie Yao [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
Fazle Hussain*
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX79409, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Polarized vortical structures (i.e. with axial flow, thus coiled vortex lines) are generic to turbulent flows – hence the importance of their dynamics, interactions and cascade. Direct numerical simulations of two anti-parallel polarized vortex tubes are performed for vortex Reynolds numbers $Re$ ($\equiv \varGamma /\nu$) up to $9000$ and initial polarization strength $q$ (ratio of peak axial to azimuthal velocities) between $0$ and $4/3$. For both counter- and co-polarized cases, although the reconnection is delayed as $q$ increases – mainly due to weakened self-induction – it is more rapid and more complete for small $q$. Enstrophy growth and energy cascade are suppressed for weak polarization ($q < 1/2$) due to depleted nonlinearity, but are enhanced for strong polarization ($q > 1/2$) due to instability and/or transient growth. When counter-polarized, numerous structures with both positive and negative helicity densities (i.e. $\pm h$) are generated. For large $q$, strong axial flows opposite to the initial flows occur – causing polarization reversals. For the co-polarized cases, although $+h$ predominates, $-h$ structures also form and interact with positive ones – leading to helicity cascade to small scales. As $Re$ increases, small scales are more numerous: for counter-polarized cases, the threads undergo successive reconnections in a cascade – akin to the unpolarized case; for co-polarized cases, the newly formed vortex ring breaks up with numerous hairpin vortices wrapping around it. Increasing $q$ alters the energy spectrum in the inertial range with a scaling varying from $k^{-5/3}$ for the unpolarized case to $k^{-7/3}$ for the strongly polarized case, which seems to be associated with the enhanced vortex spiralling. In addition, for the strongly co-polarized cases, a $k^{-4/3}$ helicity spectrum develops. Furthermore, most of the energy and helicity in the inertial range with scale $L$ transfer to scales between $0.3L$ and $0.4L$. Therefore, polarization can significantly alter the dynamics of vortex reconnection as well as turbulence cascade.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex interactions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 922 , 10 September 2021 , A19
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Biferale, L., Musacchio, S. & Toschi, F. 2012 Inverse energy cascade in three-dimensional isotropic turbulence. Phys. Rev. Lett. 108 (16), 164501.CrossRefGoogle ScholarPubMed
Blanco-Rodríguez, F.J. & Le Dizès, S. 2017 Curvature instability of a curved Batchelor vortex. J. Fluid Mech. 814, 397–415.CrossRefGoogle Scholar
Boffetta, G. & Ecke, R.E. 2012 Two-dimensional turbulence. Annu. Rev. Fluid Mech. 44, 427–451.CrossRefGoogle Scholar
Borue, V. & Orszag, S.A. 1997 Spectra in helical three-dimensional homogeneous isotropic turbulence. Phys. Rev. E 55 (6), 7005–7009.CrossRefGoogle Scholar
Brissaud, A., Frisch, U., Léorat, J., Lesieur, M. & Mazure, A. 1973 Helicity cascades in fully developed isotropic turbulence. Phys. Fluids 16 (8), 1366–1367.CrossRefGoogle Scholar
Buaria, D., Pumir, A. & Bodenschatz, E. 2020 Self-attenuation of extreme events in Navier–Stokes turbulence. Nat. Commun. 11 (1), 1–7.CrossRefGoogle ScholarPubMed
Canuto, C., Hussaini, M.Y., Quarteroni, A. & Thomas, A. Jr. 2012 Spectral Methods in Fluid Dynamics. Springer Science & Business Media.Google Scholar
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. 2010 Vortex ring with swirl: a numerical study. Phys. Fluids 22 (9), 097101.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
Crow, S.C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8 (12), 2172–2179.CrossRefGoogle Scholar
Davidson, P.A., Morishita, K. & Kaneda, Y. 2008 On the generation and flux of enstrophy in isotropic turbulence. J. Turbul. 9, N42.CrossRefGoogle Scholar
Doan, N.A.K., Swaminathan, N., Davidson, P.A. & Tanahashi, M. 2018 Scale locality of the energy cascade using real space quantities. Phys. Rev. Fluids 3 (8), 084601.CrossRefGoogle Scholar
Doering, C.R. 2009 The 3D Navier–Stokes problem. Annu. Rev. Fluid Mech. 41, 109–128.CrossRefGoogle Scholar
Domaradzki, J.A. & Rogallo, R.S. 1990 Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence. Phys. Fluids A 2 (3), 413–426.CrossRefGoogle Scholar
Duraisamy, K. & Lele, S.K. 2008 Evolution of isolated turbulent trailing vortices. Phys. Fluids 20 (3), 035102.CrossRefGoogle Scholar
Fonda, E., Sreenivasan, K.R. & Lathrop, D.P. 2019 Reconnection scaling in quantum fluids. Proc. Natl Acad. Sci. USA 116 (6), 1924–1928.CrossRefGoogle ScholarPubMed
Frisch, U. 1995 Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Gargan-Shingles, C., Rudman, M. & Ryan, K. 2016 The linear stability of swirling vortex rings. Phys. Fluids 28 (11), 114106.CrossRefGoogle Scholar
Goto, S., Saito, Y. & Kawahara, G. 2017 Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers. Phys. Rev. Fluids 2 (6), 064603.CrossRefGoogle Scholar
Grauer, R. & Sideris, T.C. 1991 Numerical computation of 3D incompressible ideal fluids with swirl. Phys. Rev. Lett. 67 (25), 3511.CrossRefGoogle ScholarPubMed
Hattori, Y., Blanco-Rodríguez, F.J. & Le Dizès, S. 2019 Numerical stability analysis of a vortex ring with swirl. J. Fluid Mech. 878, 5–36.CrossRefGoogle Scholar
Heaton, C.J. & Peake, N. 2007 Transient growth in vortices with axial flow. J. Fluid Mech. 587, 271–301.CrossRefGoogle Scholar
Hide, R. 1989 Superhelicity, helicity and potential vorticity. Geophys. Astrophys. Fluid Dyn. 48 (1–3), 69–79.CrossRefGoogle Scholar
Hou, T.Y. & Li, R. 2006 Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations. J. Nonlinear Sci. 16 (6), 639–664.CrossRefGoogle Scholar
Hou, T.Y. & Li, R. 2008 Blowup or no blowup? The interplay between theory and numerics. Physica D 237 (14–17), 1937–1944.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
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.CrossRefGoogle Scholar
Kang, D., Yun, D. & Protas, B. 2020 Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows. J. Fluid Mech. 893, A22.CrossRefGoogle Scholar
Kerr, R.M. 1993 Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids A 5 (7), 1725–1746.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 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
Kimura, Y. & Moffatt, H.K. 2018 A tent model of vortex reconnection under Biot–Savart evolution. J. Fluid Mech. 834, R1.CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301–305.Google Scholar
Kraichnan, R.H. 1973 Helical turbulence and absolute equilibrium. J. Fluid Mech. 59 (4), 745–752.CrossRefGoogle Scholar
Lacaze, L., Ryan, K. & Le Dizes, S. 2007 Elliptic instability in a strained Batchelor vortex. J. Fluid Mech. 577, 341–361.CrossRefGoogle Scholar
Leung, T., Swaminathan, N. & Davidson, P.A. 2012 Geometry and interaction of structures in homogeneous isotropic turbulence. J. Fluid Mech. 710, 453–481.CrossRefGoogle Scholar
Lundgren, T.S. 1982 Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25 (12), 2193–2203.CrossRefGoogle Scholar
Mayer, E.W. & Powell, K.G. 1992 Viscous and inviscid instabilities of a trailing vortex. J. Fluid Mech. 245, 91–114.CrossRefGoogle Scholar
McGavin, P. & Pontin, D.I. 2018 Vortex line topology during vortex tube reconnection. Phys. Rev. Fluids 3 (5), 054701.CrossRefGoogle Scholar
McGavin, P. & Pontin, D.I. 2019 Reconnection of vortex tubes with axial flow. Phys. Rev. Fluids 4 (2), 024701.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
Moffatt, H.K. 1988 Generalised vortex rings with and without swirl. Fluid Dyn. Res. 3 (1–4), 22.CrossRefGoogle 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. 2014 Helicity and singular structures in fluid dynamics. Proc. Natl Acad. Sci. USA 111 (10), 3663–3670.CrossRefGoogle ScholarPubMed
Moffatt, H.K. & Kimura, Y. 2019 a Towards a finite-time singularity of the Navier–Stokes equations. Part 1. Derivation and analysis of dynamical system. J. Fluid Mech. 861, 930–967.CrossRefGoogle Scholar
Moffatt, H.K. & Kimura, Y. 2019 b Towards a finite-time singularity of the Navier–Stokes equations. Part 2. Vortex reconnection and singularity evasion. J. Fluid Mech. 870, R1.CrossRefGoogle Scholar
Moffatt, H.K. & Ricca, R.L. 1995 Helicity and the Călugăreanu invariant. In Knots and Applications, pp. 251–269. World Scientific.CrossRefGoogle Scholar
Moore, D.W. & Saffman, P.G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346 (1646), 413–425.Google Scholar
Motoori, Y. & Goto, S. 2019 Generation mechanism of a hierarchy of vortices in a turbulent boundary layer. J. Fluid Mech. 865, 1085–1109.CrossRefGoogle Scholar
Naitoh, T., Okura, N., Gotoh, T. & Kato, Y. 2014 On the evolution of vortex rings with swirl. Phys. Fluids 26 (6), 067101.CrossRefGoogle Scholar
Ostilla-Mónico, R., McKeown, R., Brenner, M.P., Rubinstein, S.M. & Pumir, A. 2021 Cascades and reconnection in interacting vortex filaments. arXiv:2102.11133.CrossRefGoogle 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
Pullin, D.I. & Lundgren, T.S. 2001 Axial motion and scalar transport in stretched spiral vortices. Phys. Fluids 13 (9), 2553–2563.CrossRefGoogle Scholar
van Rees, W.M., Hussain, F. & Koumoutsakos, P. 2012 Vortex tube reconnection at $Re= 10^{4}$. Phys. Fluids 24 (7), 075105.CrossRefGoogle Scholar
Roy, C., Schaeffer, N., Le Dizès, S. & Thompson, M. 2008 Stability of a pair of co-rotating vortices with axial flow. Phys. Fluids 20 (9), 094101.CrossRefGoogle Scholar
Scheeler, M.W., Kleckner, D., Proment, D., Kindlmann, G.L. & Irvine, W.T.M. 2014 Helicity conservation by flow across scales in reconnecting vortex links and knots. Proc. Natl Acad. Sci. USA 111 (43), 15350–15355.CrossRefGoogle ScholarPubMed
Scheeler, M.W., van Rees, W.M., Kedia, H., Kleckner, D. & Irvine, W.T.M. 2017 Complete measurement of helicity and its dynamics in vortex tubes. Science 357 (6350), 487–491.CrossRefGoogle ScholarPubMed
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
Stewartson, K. & Brown, S.N. 1985 Near-neutral centre-modes as inviscid perturbations to a trailing line vortex. J. Fluid Mech. 156, 387–399.CrossRefGoogle Scholar
Taylor, G.I. & Green, A.E. 1937 Mechanism of the production of small eddies from large ones. Proc. R. Soc. Lond. A 158 (895), 499–521.Google 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
Virk, D., Melander, M.V. & Hussain, F. 1994 Dynamics of a polarized vortex ring. J. Fluid Mech. 260, 23–55.CrossRefGoogle Scholar
Wu, J.-Z., Ma, H.-Y. & Zhou, M.-D. 2007 Vorticity and Vortex Dynamics. Springer Science & Business Media.Google Scholar
Yan, Z., Li, X. & Yu, C. 2020 a Scale locality of helicity cascade in physical space. Phys. Fluids 32 (6), 061705.CrossRefGoogle Scholar
Yan, Z., Li, X., Yu, C., Wang, J. & Chen, S. 2020 b Dual channels of helicity cascade in turbulent flows. J. Fluid Mech. 894, R2.CrossRefGoogle 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., Yang, Y. & Hussain, F. 2021 Dynamics of a trefoil knotted vortex. J. Fluid Mech. (submitted).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

Yao and Hussain supplementary movie 1

Isosurface of vorticity magnitude shaded with contours of axial vorticity for counter-polarized cases at Re=5000.

Download Yao and Hussain supplementary movie 1(Video)
Video 5.3 MB

Yao and Hussain supplementary movie 2

Isosurface of vorticity magnitude shaded with contours of axial vorticity for co-polarized cases at Re=5000.

Download Yao and Hussain supplementary movie 2(Video)
Video 11.3 MB

Yao and Hussain supplementary movie 3

Isosurface of vorticity magnitude shaded with contours of axial vorticity for counter-polarized q=1 cases at Re=9000.

Download Yao and Hussain supplementary movie 3(Video)
Video 2.4 MB

Yao and Hussain supplementary movie 4

Isosurface of vorticity magnitude shaded with contours of axial vorticity for co-polarized q=1 cases at Re=9000.
Download Yao and Hussain supplementary movie 4(Video)
Video 3.4 MB