Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-06-11T00:28:47.368Z Has data issue: false hasContentIssue false
  • English
  • Français

Evolution of dissipative fluid flows with imposed helicity conservation

Published online by Cambridge University Press:  06 January 2023

Zhaoyuan Meng Open the ORCID record for Zhaoyuan Meng [Opens in a new window] ,
Weiyu Shen Open the ORCID record for Weiyu Shen [Opens in a new window]  and
Yue Yang Open the ORCID record for Yue Yang [Opens in a new window]
Zhaoyuan Meng
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, PR China
Weiyu Shen
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, PR China
Yue Yang*
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, PR China HEDPS-CAPT, Peking University, Beijing 100871, PR China
*
†Email address for correspondence: yyg@pku.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose the helicity-conserved Navier–Stokes (HCNS) equation by modifying the non-ideal force term in the Navier–Stokes (NS) equation. The corresponding HCNS flow has strict helicity conservation, and retains major NS dynamics with finite dissipation. Using the helical wave decomposition, we show that the pentadic interaction of Fourier helical modes in the HCNS dynamics is more complex than the triadic interaction in the NS dynamics, and enhanced variations for left- and right-handed helicity components cancel each other in the HCNS flow to keep the invariant helicity. A comparative study of HCNS and NS flow evolutions with direct numerical simulation elucidates the influence of the helicity conservation on flow structures and statistics in the vortex reconnection and isotropic turbulence. First, the HCNS flow evolves towards a Beltrami state with a $-4$ scaling law of the energy spectrum at high wavenumbers at long times. Second, large-scale flow structures are almost identical during the viscous reconnection of vortex tubes in the two flows, whereas many more small-scale helical structures are generated via the pentadic mode interaction in the HCNS flow than in the NS flow. Moreover, we demonstrate that parity breaking at small scales can trigger a notable helicity variation in the NS flow. These findings hint that the helicity may not be conserved in the inviscid limit of the NS flow.

JFM classification

Mathematical Foundations: Topological fluid dynamics Vortex Flows: Vortex dynamics Turbulent Flows: Isotropic turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 954 , 10 January 2023 , A36
Check for updates
Copyright
© The Author(s), 2023. 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

Alexakis, A. 2017 Helically decomposed turbulence. J. Fluid Mech. 812, 752–770.CrossRefGoogle Scholar
Alexakis, A. & Biferale, L. 2018 Cascades and transitions in turbulent flows. Phys. Rep.-Rev. Sec. Phys. Lett. 767–769, 1–101.Google Scholar
Alexakis, A. & Biferale, L. 2022 $\lambda$–Navier–Stokes turbulence. Phil. Trans. R. Soc. A-Math. Phys. Engng Sci. 380, 20210243.CrossRefGoogle ScholarPubMed
André, J.C. & Lesieur, M. 1977 Influence of helicity on the evolution of isotropic turbulence at high Reynolds number. J. Fluid Mech. 81, 187–207.CrossRefGoogle Scholar
Arnold, V.I. 1992 Topological methods in hydrodynamics. Annu. Rev. Fluid Mech. 24, 145–166.CrossRefGoogle Scholar
Arnold, V.I. 2014 The Asymptotic Hopf Invariant and its Applications, pp. 357–375. Springer Berlin Heidelberg.Google Scholar
Batchelor, G.K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Biferale, L., Musacchio, S. & Toschi, F. 2012 Inverse energy cascade in three-dimensional isotropic turbulence. Phys. Rev. Lett. 108, 164501.CrossRefGoogle ScholarPubMed
Biferale, L., Musacchio, S. & Toschi, F. 2013 Split energy-helicity cascades in three-dimensional homogeneous and isotropic turbulence. J. Fluid Mech. 730, 309–327.CrossRefGoogle Scholar
Biferale, L. & Titi, E.S. 2013 On the global regularity of a helical-decimated version of the 3D Navier–Stokes equations. J. Stat. Phys. 151, 1089–1098.CrossRefGoogle Scholar
Bos, W.J.T. 2021 Three-dimensional turbulence without vortex stretching. J. Fluid Mech. 915, A121.CrossRefGoogle Scholar
Briard, A. & Gomez, T. 2017 Dynamics of helicity in homogeneous skew-isotropic turbulence. J. Fluid Mech. 821, 539–581.CrossRefGoogle Scholar
Brissaud, A., Frisch, U., Leorat, J., Lesieur, M. & Mazure, A. 1973 Helicity cascades in fully developed isotropic turbulence. Phys. Fluids 16, 1366–1367.CrossRefGoogle Scholar
Chen, Q., Chen, S. & Eyink, G.L. 2003 a The joint cascade of energy and helicity in three-dimensional turbulence. Phys. Fluids 15, 361–374.CrossRefGoogle Scholar
Chen, Q., Chen, S., Eyink, G.L. & Holm, D.D. 2003 b Intermittency in the joint cascade of energy and helicity. Phys. Rev. Lett. 90, 214503.CrossRefGoogle ScholarPubMed
Constantin, P. & Majda, A. 1988 The Beltrami spectrum for incompressible fluid flows. Commun. Math. Phys. 115, 435–456.CrossRefGoogle Scholar
Devenport, W.J., Rife, M.C., Liapis, S.I. & Follin, G.J. 1996 The structure and development of a wing-tip vortex. J. Fluid Mech. 312, 67–106.CrossRefGoogle Scholar
Ditlevsen, P.D. & Giuliani, P. 2001 Dissipation in helical turbulence. Phys. Fluids 13, 3508–3509.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Gallavotti, G. 1996 Equivalence of dynamical ensembles and Navier–Stokes equations. Phys. Lett. A 223, 91–95.CrossRefGoogle Scholar
Gallavotti, G. 1997 Dynamical ensembles equivalence in fluid mechanics. Physica D 105, 163–184.CrossRefGoogle Scholar
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.CrossRefGoogle Scholar
Hao, J., Xiong, S. & Yang, Y. 2019 Tracking vortex surfaces frozen in the virtual velocity in non-ideal flows. J. Fluid Mech. 863, 513–544.CrossRefGoogle Scholar
Hussain, F. 1986 Coherent structures and turbulence. J. Fluid Mech. 173, 303–356.CrossRefGoogle Scholar
Ishida, T., Davidson, P.A. & Kaneda, Y. 2006 On the decay of isotropic turbulence. J. Fluid Mech. 564, 455–475.CrossRefGoogle Scholar
Jaccod, A. & Chibbaro, S. 2021 Constrained reversible system for Navier–Stokes turbulence. Phys. Rev. Lett. 127, 194501.CrossRefGoogle ScholarPubMed
Kaneda, Y., Ishihara, T., Itakura, K. & Uno, A. 2003 Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Phys. Fluids 15, 21–24.CrossRefGoogle Scholar
Kerr, R.M. 2018 Trefoil knot timescales for reconnection and helicity. Fluid Dyn. Res. 50, 011422.CrossRefGoogle Scholar
Kida, S. & Takaoka, M. 1987 Bridging in vortex reconnection. Phys. Fluids 30, 2911–2914.CrossRefGoogle Scholar
Kida, S. & Takaoka, M. 1988 Reconnection of vortex tubes. Fluid Dyn. Res. 3, 257–261.CrossRefGoogle Scholar
Kida, S. & Takaoka, M. 1994 Vortex reconnection. Annu. Rev. Fluid Mech. 26, 169–177.CrossRefGoogle Scholar
Kleckner, D. & Irvine, W.T.M. 2013 Creation and dynamics of knotted vortices. Nat. Phys. 9, 253–258.CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 On degeneration of isotropic turbulence in an incompressible viscous liquid. Dokl. Akad. Nauk SSSR 31, 538–540.Google Scholar
Kraichnan, R.H. 1970 Diffusion by a random velocity field. Phys. Fluids 13, 22–31.CrossRefGoogle Scholar
Kurgansky, M.V. 2017 Helicity in dynamic atmospheric processes. Izv. Acad. Nauk SSSR Atmos. Ocean. Phys. 53, 127–141.CrossRefGoogle Scholar
Laing, C.E., Ricca, R.L. & Sumners de, W.L. 2015 Conservation of writhe helicity under anti-parallel reconnection. Sci. Rep. 5, 9224.CrossRefGoogle ScholarPubMed
Moffatt, H.K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129.CrossRefGoogle Scholar
Moffatt, H.K. 2014 Helicity and singular structures in fluid dynamics. Proc. Natl Acad. Sci. USA 111, 3663–3670.CrossRefGoogle ScholarPubMed
Moffatt, H.K. 2017 Helicity – invariant even in a viscous fluid. Science 357, 448–449.CrossRefGoogle Scholar
Moffatt, H.K. 2021 Some topological aspects of fluid dynamics. J. Fluid Mech. 914, P1.CrossRefGoogle Scholar
Moffatt, H.K. & Ricca, R.L. 1992 Helicity and the Călugăreanu invariant. Proc. R. Soc. A-Math. Phys. Engng Sci. 439, 411–429.Google Scholar
Moffatt, H.K. & Tsinober, A. 1992 Helicity in laminar and turbulent flow. Annu. Rev. Fluid Mech. 24, 281–312.CrossRefGoogle Scholar
Moreau, J.J. 1961 Constantes d'un îlot tourbillonnaire en fluide parfait barotrope. C. R. Acad. Sci. Paris 252, 2810–2812.Google Scholar
Pelz, R.B., Shtilman, L. & Tsinober, A. 1986 The helical nature of unforced turbulent flows. Phys. Fluids 29, 3506.CrossRefGoogle Scholar
Plunian, F., Teimurazov, A., Stepanov, R. & Verma, M.K. 2020 Inverse cascade of energy in helical turbulence. J. Fluid Mech. 895, A13.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Ricca, R.L., Samuels, D.C. & Barenghi, C.F. 1999 Evolution of vortex knots. J. Fluid Mech. 391, 29–44.CrossRefGoogle Scholar
Ruan, S., Xiong, S., You, J. & Yang, Y. 2022 Generation of streamwise helical vortex loops via successive reconnections in early pipe transition. Phys. Fluids 34, 054112.CrossRefGoogle Scholar
Sahoo, G., Alexakis, A. & Biferale, L. 2017 Discontinuous transition from direct to inverse cascade in three-dimensional turbulence. Phys. Rev. Lett. 118, 164501.CrossRefGoogle ScholarPubMed
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. USA 111, 15350–15355.CrossRefGoogle ScholarPubMed
She, Z.-S. & Jackson, E. 1993 Constrained Euler system for Navier–Stokes turbulence. Phys. Rev. Lett. 70, 1255–1258.CrossRefGoogle ScholarPubMed
Shen, W., Yao, J., Hussain, F. & Yang, Y. 2022 Topological transition and helicity conversion of vortex knots and links. J. Fluid Mech. 943, A41.CrossRefGoogle Scholar
Slomka, J. & Dunkel, J. 2017 Spontaneous mirror-symmetry breaking induces inverse energy cascade in 3D active fluids. Proc. Natl Acad. Sci. USA 114, 2119–2124.CrossRefGoogle ScholarPubMed
Sreenivasan, K.R. 1984 On the scaling of the turbulence energy dissipation rate. Phys. Fluids 27, 1048–1051.CrossRefGoogle Scholar
Sreenivasan, K.R. 1998 An update on the dissipation rate in homogeneous turbulence. Phys. Fluids 10, 528–529.CrossRefGoogle Scholar
Sulem, P.L., She, Z., Scholl, H. & Frisch, U. 1989 Generation of large-scale structures in three-dimensional flow lacking parity-invariance. J. Fluid Mech. 205, 341–358.CrossRefGoogle Scholar
Takaoka, M. 1996 Helicity generation and vorticity dynamics in helically symmetric flow. J. Fluid Mech. 319, 125–149.CrossRefGoogle Scholar
Taylor, G.I. & Green, A.E. 1937 Mechanism of the production of small eddies from large ones. Proc. R. Soc. Lond. A-Math. Phys. Engng Sci. 158, 499–521.Google Scholar
Tong, W., Yang, Y. & Wang, S. 2020 Characterizing three-dimensional features of vortex surfaces in the flow past a finite plate. Phys. Fluids 32, 011903.Google Scholar
Truesdell, C. 2018 The Kinematics of Vorticity. Courier Dover Publications.Google Scholar
Tsinober, A. & Levich, E. 1983 On the helical nature of three-dimensional coherent structures in turbulent flows. Phys. Lett. A 99A, 321–324.CrossRefGoogle Scholar
Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids 4, 350–363.CrossRefGoogle Scholar
Woltjer, L. 1958 A theorem on force-free magnetic fields. Proc. Natl Acad. Sci. USA 44, 489–491.CrossRefGoogle ScholarPubMed
Wood, D.H., Mehta, R.D. & Koh, S.G. 1992 Structure of a swirling turbulent mixing layer. Exp. Therm. Fluid Sci. 5, 196–202.CrossRefGoogle Scholar
Wu, J., Ma, H. & Zhou, M. 2015 Vortical Flows. Springer.CrossRefGoogle Scholar
Xiong, S. & Yang, Y. 2019 a Construction of knotted vortex tubes with the writhe-dependent helicity. Phys. Fluids 31, 047101.CrossRefGoogle Scholar
Xiong, S. & Yang, Y. 2019 b Identifying the tangle of vortex tubes in homogeneous isotropic turbulence. J. Fluid Mech. 874, 952–978.CrossRefGoogle Scholar
Xiong, S. & Yang, Y. 2020 Effects of twist on the evolution of knotted magnetic flux tubes. J. Fluid Mech. 895, A28.CrossRefGoogle Scholar
Yan, Z., Li, X., Yu, C., Wang, J. & Chen, S. 2020 Dual channels of helicity cascade in turbulent flows. J. Fluid Mech. 894, R2.CrossRefGoogle Scholar
Yang, Y.-T., Su, W.-D. & Wu, J.-Z. 2010 Helical-wave decomposition and applications to channel turbulence with streamwise rotation. J. Fluid Mech. 662, 91–122.CrossRefGoogle Scholar
Yang, Y.-T. & Wu, J.-Z. 2011 Channel turbulence with spanwise rotation studied using helical wave decomposition. J. Fluid Mech. 692, 137–152.CrossRefGoogle Scholar
Yao, J. & Hussain, F. 2021 Polarized vortex reconnection. J. Fluid Mech. 922, A19.CrossRefGoogle Scholar
Yao, J. & Hussain, F. 2022 Vortex reconnection and turbulence cascade. Annu. Rev. Fluid Mech. 54, 317–347.CrossRefGoogle Scholar
Yao, J., Shen, W., Yang, Y. & Hussain, F. 2022 Helicity dynamics in viscous vortex links. J. Fluid Mech. 944, A41.CrossRefGoogle Scholar
Yao, J., Yang, Y. & Hussain, F. 2021 Dynamics of a trefoil knotted vortex. J. Fluid Mech. 923, A19.CrossRefGoogle Scholar
Zhao, X. & Scalo, C. 2021 Helicity dynamics in reconnection events of topologically complex vortex flows. J. Fluid Mech. 920, A30.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
Zhao, X., Yu, Z., Chapelier, J. & Scalo, C. 2021 Direct numerical and large-eddy simulation of trefoil knotted vortices. J. Fluid Mech. 910, A31.CrossRefGoogle Scholar
Zhu, J., Yang, W. & Zhu, G. 2014 Purely helical absolute equilibria and chirality of (magneto) fluid turbulence. J. Fluid Mech. 739, 479–501.CrossRefGoogle Scholar