Abstract
Computational simulations of turbulent flows indicate that the regions of low dissipation/enstrophy production feature high degree of local alignment between the velocity and the vorticity, i.e., the flow is locally near-Beltrami. Hence one could envision a geometric scenario in which the persistence of the local near-Beltrami property might be consistent with a (possible) finite-time singularity formation. The goal of this note is to show that this scenario is in fact prohibited if the sine of the angle between the velocity and the vorticity is small enough with respect to the local enstrophy.
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References
Beirao da Veiga, H., Berselli, L.C.: On the regularizing effect of the vorticity direction in incompressible viscous flows. Differ. Integral Equ. 15, 345 (2002)
Beirao da Veiga, H.: Viscous incompressible flows under stress-free boundary conditions. The smoothness effect of near orthogonality or near parallelism between velocity and vorticity. Boll. U.M.I. V, 225 (2012)
Berselli, L.C., Cordoba, D.: On the regularity of the solutions to the 3D Navier–Stokes equations: a remark on the role of the helicity. C. R. Acad. Sci. Paris Ser I 347, 613 (2009)
Biferele, L., Titi, E.: On the global regularity of a helical-decimated version of the 3D Navier–Stokes equations. J. Stat. Phys. 151, 1089 (2013)
Chae, D.: On the regularity conditions of suitable weak solutions of the 3D Navier–Stokes equations. J. Math. Fluid. Mech. 12, 171 (2010)
Choi, Y., Kim, B.-G., Lee, C.: Alignment of velocity and vorticity and the intermittent distribution of helicity in isotropic turbulence. Phys. Rev. E 80, 017301 (2009)
Constantin, P.: Geometric statistics in turbulence. SIAM Rev. 36, 73 (1994)
Constantin, P., Fefferman, C.: Direction of vorticity and the problem of global regularity for the Navier–Stokes equations. Indiana Univ. Math. J. 42, 775 (1993)
Grujić, Z.: Localization and geometric depletion of vortex-stretching in the 3D NSE. Commun. Math. Phys. 290, 861 (2009)
Grujić, Z., Guberović, R.: Localization of analytic regularity criteria on the vorticity and balance between the vorticity magnitude and coherence of the vorticity direction in the 3D NSE. Commun. Math. Phys. 298, 407 (2010)
Holm, D.D., Kerr, R.: Transient vortex events in the initial value problem for turbulence. Phys. Rev. Lett. 88, 244501 (2002)
Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193 (1934)
Mahalov, A., Titi, E.S., Leibovich, S.: Invariant helical subspaces for the Navier–Stokes equations. Arch. Ration. Mech. Anal. 112, 193 (1990)
Pelz, R.B., Yakhot, V., Orszag, S.A.: Velocity-vorticity patterns in turbulent flows. Phys. Rev. Lett. 54, 2505 (1985)
She, Z.-S., Jackson, E., Orszag, S.: Structure and dynamics of homogeneous turbulence: models and simulations. Proc. R. Soc. Lond. A 434, 101 (1991)
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The work of Z.G. is supported in part by the National Science Foundation Grant DMS—1515805.
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Communicated by Paul Newton.
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Farhat, A., Grujić, Z. Local Near-Beltrami Structure and Depletion of the Nonlinearity in the 3D Navier–Stokes Flows. J Nonlinear Sci 29, 803–812 (2019). https://doi.org/10.1007/s00332-018-9504-8
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DOI: https://doi.org/10.1007/s00332-018-9504-8
Keywords
- Navier-Stokes equations
- Regularity of solutions
- Geometric constraints
- Helicity
- Velocity and vorticity directions