Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-14T17:38:25.217Z Has data issue: false hasContentIssue false
  • English
  • Français

Tracking vortex surfaces frozen in the virtual velocity in non-ideal flows

Published online by Cambridge University Press:  25 January 2019

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

Abstract

We demonstrate that, if a globally smooth virtual circulation-preserving velocity exists, Kelvin’s and Helmholtz’s theorems can be extended to some non-ideal flows which are viscous, baroclinic or with non-conservative body forces. Then we track vortex surfaces frozen in the virtual velocity in the non-ideal flows, based on the evolution of a vortex-surface field (VSF). For a flow with a viscous-like diffusion term normal to the vorticity, we obtain an explicit virtual velocity to accurately track vortex surfaces in time. This modified flow is dissipative but prohibits reconnection of vortex lines. If a globally smooth virtual velocity does not exist, an approximate virtual velocity can still facilitate the tracking of vortex surfaces in non-ideal flows. In a magnetohydrodynamic Taylor–Green flow, we find that the conservation of vorticity flux is significantly improved in the VSF evolution convected by the approximate virtual velocity instead of the physical velocity, and the spurious vortex deformation induced by the Lorentz force is eliminated.

JFM classification

Mathematical Foundations: Topological fluid dynamics Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 863 , 25 March 2019 , pp. 513 - 544
Copyright
© 2019 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

Alfvén, H. 1943 On the existence of electromagnetic-hydromagnetic waves. Ark. Mat. Astron. Fys. 29, 1–7.Google Scholar
Aluie, H.2009 Hydrodynamic and magnetohydrodynamic turbulence: invariants, cascades, and locality. PhD thesis, Johns Hopkins University, Baltimore, MD.Google Scholar
Brachet, M. E., Meiron, D. I., Orszag, S. A., Nickel, B. G., Morf, R. H. & Frisch, U. 1983 Small-scale structure of the Taylor–Green vortex. J. Fluid Mech. 130, 411–452.10.1017/S0022112083001159Google Scholar
Chen, S., Eyink, G. L., Wan, M. & Xiao, Z. 2006 Is the Kelvin theorem valid for high Reynolds number turbulence? Phys. Rev. Lett. 97, 144505.10.1103/PhysRevLett.97.144505Google Scholar
Chern, A.2017 Fluid dynamics with incompressible Schrödinger flow. PhD thesis, California Institute of Technology, Pasadena, CA.Google Scholar
Chern, A., Knöppel, F., Pinkall, U., Schröder, P. & Weißmann, S. 2016 Schrödinger’s smoke. ACM Trans. Graph. 35, 77.10.1145/2897824.2925868Google Scholar
Cheviakov, A. F. & Oberlack, M. 2014 Generalized Ertel’s theorem and infinite hierarchies of conserved quantities for three-dimensional time-dependent Euler and Navier–Stokes equations. J. Fluid Mech. 760, 368–386.10.1017/jfm.2014.611Google Scholar
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765–777.10.1063/1.857730Google Scholar
Chorin, A. J. 1994 Vorticity and Turbulence. Springer.10.1007/978-1-4419-8728-0Google Scholar
Chorin, A. J. & Bernard, P. S. 1973 Discretization of a vortex sheet, with an example of roll-up. J. Comput. Phys. 13, 423–429.10.1016/0021-9991(73)90045-4Google Scholar
Clebsch, A. 1859 Ueber die Integration der hydrodynamischen Gleichungen. J. Reine Angew. Math. 56, 1–10.10.1515/crll.1859.56.1Google Scholar
Constantin, P. 2001 An Eulerian–Lagrangian approach to the Navier–Stokes equations. Commun. Math. Phys. 216, 663–686.10.1007/s002200000349Google Scholar
Constantin, P. & Iyer, G. 2008 A stochastic Lagrangian representation of the three-dimensional incompressible Navier–Stokes equations. Commun. Pure Appl. Maths 61, 330–345.10.1002/cpa.20192Google Scholar
Cottet, G.-H. & Koumoutsakos, P. 2000 Vortex Methods: Theory and Practice. Cambridge University Press.10.1017/CBO9780511526442Google Scholar
Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Elsässer, W. M. 1950 The hydromagnetic equations. Phys. Rev. 79, 183–183.10.1103/PhysRev.79.183Google Scholar
Ertel, H. 1942 Ein neuer hydrodynamischer Wirbelsatz. Meteorol. Z. 59, 271–281.Google Scholar
Eyink, G., Vishniac, E., Lalescu, C., Aluie, H., Kanov, K., Bürger, K., Burns, R., Meneveau, C. & Szalay, A. 2013 Flux-freezing breakdown in high-conductivity magnetohydrodynamic turbulence. Nature 497, 466.10.1038/nature12128Google Scholar
Eyink, G. L. 2009 Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models. J. Math. Phys. 50, 083102.10.1063/1.3193681Google Scholar
Gibbon, J. D., Galanti, B. & Kerr, R. M. 2000 Stretching and compression of vorticity in the 3D Euler equations. In Turbulence Structure and Vortex Dynamics, pp. 23–34. Cambridge University Press.Google Scholar
Gibbon, J. D. & Holm, D. D. 2010 The dynamics of the gradient of potential vorticity. J. Phys. A: Math. Theor. 43, 172001.Google Scholar
Gottlieb, S. & Shu, C.-W. 1998 Total variation diminishing Runge–Kutta schemes. Math. Comput. 67, 73–85.10.1090/S0025-5718-98-00913-2Google Scholar
Greene, J. M. 1993 Reconnection of vorticity lines and magnetic lines. Phys. Fluids B 5, 2355.10.1063/1.860718Google Scholar
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 1–26.10.1017/S0022112004002526Google Scholar
He, P. & Yang, Y. 2016 Construction of initial vortex-surface fields and Clebsch potentials for flows with high-symmetry using first integrals. Phys. Fluids 28, 037101.10.1063/1.4943368Google Scholar
Helmholtz, H. 1858 Uber integrale der hydrodynamischen Gleichungen welche den Wirbel-bewegungen ensprechen. J. Reine Angew. Math. 55, 25–55.10.1515/crll.1858.55.25Google Scholar
Hornig, G. & Schindler, K. 1996 Magnetic topology and the problem of its invariant definition. Phys. Plasmas 3, 781.10.1063/1.871778Google Scholar
Hou, T. Y. & Li, R. 2007 Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226, 379–397.10.1016/j.jcp.2007.04.014Google Scholar
Huang, M., Küpper, T. & Masbaum, N. 1997 Computation of invariant tori by the Fourier methods. SIAM J. Sci. Comput. 18, 918–942.Google Scholar
Hunt, J. C. R., Wray, A. & Moin, P. 1988 Eddies, stream, and convergence zones in turbulent flows. In Center for Turbulence Research Report, Stanford, CA, USA, pp. 193–208.Google Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.10.1017/S0022112095000462Google Scholar
Jiang, G. S. & Shu, C.-W. 1996 Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228.10.1006/jcph.1996.0130Google Scholar
Kerr, R. M. 1993 Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids A 5, 1725–1746.10.1063/1.858849Google Scholar
Kida, S. & Takaoka, M. 1991 Breakdown of frozen motion of vorticity field and vorticity reconnection. J. Phys. Soc. Japan 60, 2184–2196.10.1143/JPSJ.60.2184Google Scholar
Kida, S. & Takaoka, M. 1994 Vortex reconnection. Annu. Rev. Fluid Mech. 26, 169–189.10.1146/annurev.fl.26.010194.001125Google Scholar
Lee, E., Brachet, M. E., Pouquet, A., Rosenberg, P. D. & Mininni, D. 2009 Paradigmatic flow for small-scale magnetohydrodynamics: properties of the ideal case and the collision of current sheets. Phys. Rev. E 78, 066401.Google Scholar
Lindsay, K. & Krasny, R. 2001 A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow. J. Comput. Phys. 172, 879–907.10.1006/jcph.2001.6862Google Scholar
Madelung, E. 1927 Quantentheorie in hydrodynamischer form. Z. Phys. 40, 322–326.Google Scholar
Majda, A. J. & Bertozzi, A. L. 2001 Vorticity and Incompressible Flow. Cambridge University Press.10.1017/CBO9780511613203Google Scholar
Moffatt, H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129.10.1017/S0022112069000991Google Scholar
Moffatt, H. K. 1978 Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Moffatt, H. K. 2014 Note on the triad interactions of homogeneous turbulence. J. Fluid Mech. 741, R3.10.1017/jfm.2013.637Google Scholar
Moreau, J. J. 1961 Constantes d’un îlot tourbillonnaire en fluide parfait barotrope. C. R. Acad. Sci. 252, 2810.Google Scholar
Mullin, T. 2011 Experimental studies of transition to turbulence in a pipe. Annu. Rev. Fluid Mech. 43, 1–24.10.1146/annurev-fluid-122109-160652Google Scholar
Newcomb, W. A. 1958 Motion of magnetic lines of force. Ann. Phys. 3, 347–385.10.1016/0003-4916(58)90024-1Google Scholar
Ohkitani, K. & Constantin, P. 2003 Numerical study of the Eulerian–Lagrangian formulation of the Navier–Stokes equations. Phys. Fluids 15, 3251–3254.10.1063/1.1608009Google Scholar
Peng, N. & Yang, Y. 2018 Effects of the Mach number on the evolution of vortex-surface fields in compressible Taylor–Green flows. Phys. Rev. Fluids 3, 013401.10.1103/PhysRevFluids.3.013401Google Scholar
Priest, E. & Forbes, T. 2000 Magnetic Reconnection: MHD Theory and Applications. Cambridge University Press.10.1017/CBO9780511525087Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Taylor, G. I. & Green, A. E. 1937 Mechanism of the production of small eddies from large ones. Proc. R. Soc. Lond. A 158, 499–521.Google Scholar
Thompson, W. (Lord Kelvin) 1869 On vortex motion. Trans. R. Soc. Edinburgh 25, 217–260.10.1017/S0080456800028179Google Scholar
Truesdell, C. 1954 The Kinematics of Vorticity. Indiana University Press.Google Scholar
Wu, J. Z. 2018 Vortex definition and ‘vortex criteria’. Sci. China-Phys. Mech. Astron. 61, 024731.10.1007/s11433-017-9120-8Google Scholar
Wu, J. Z., Ma, H. Y. & Zhou, M. D. 2006 Vorticity and Vortex Dynamics. Springer.10.1007/978-3-540-29028-5Google Scholar
Xiong, S. & Yang, Y. 2017 The boundary-constraint method for constructing vortex-surface fields. J. Comput. Phys. 339, 31–45.10.1016/j.jcp.2017.03.013Google Scholar
Yang, Y. & Pullin, D. I. 2010 On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions. J. Fluid Mech. 661, 446–481.10.1017/S0022112010003125Google Scholar
Yang, Y. & Pullin, D. I. 2011 Evolution of vortex-surface fields in viscous Taylor–Green and Kida–Pelz flows. J. Fluid Mech. 685, 146–164.10.1017/jfm.2011.287Google Scholar
Zhao, Y., Xiong, S., Yang, Y. & Chen, S. 2018 Sinuous distortion of vortex surfaces in the lateral growth of turbulent spots. Phys. Rev. Fluids 3, 074701.10.1103/PhysRevFluids.3.074701Google Scholar
Zhao, Y., Yang, Y. & Chen, S. 2016 Vortex reconnection in the late transition in channel flow. J. Fluid Mech. 802, R4.10.1017/jfm.2016.492Google Scholar
Zhou, H., You, J., Xiong, S., Yang, Y., Thévenin, D. & Chen, S. 2018 Interactions between the premixed flame front and the three-dimensional Taylor–Green vortex. Proc. Combust. Inst.; doi:10.1016/j.proci.2018.08.015.Google Scholar
Zhu, J.-Z.2017 On topological fluid mechanics of non-ideal systems and virtual frozen-in dynamics. arXiv:1702.04447.Google Scholar
Zhu, J.-Z. 2018a Local invariants in non-ideal flows of neutral fluids and two-fluid plasmas. Phys. Fluids 30, 037104.10.1063/1.5020863Google Scholar
Zhu, J.-Z. 2018b Vorticity and helicity decompositions and dynamics with real Schur form of the velocity gradient. Phys. Fluids 30, 031703.10.1063/1.5022684Google Scholar
Zhu, J.-Z., Yang, W. & Zhu, G.-Y. 2014 Purely helical absolute equilibria and chirality of (magneto)fluid turbulence. J. Fluid Mech. 739, 479–501.10.1017/jfm.2013.561Google Scholar
Supplementary material: File

Hao et al. supplementary material

Hao et al. supplementary material 1

Download Hao et al. supplementary material(File)
File 51 KB