Expanding the Q-R space to three dimensions

Beat Lthi, Markus Holzner, Arkady Tsinober

Research output: Contribution to journalArticlepeer-review


The two-dimensional space spanned by the velocity gradient invariants Q and R is expanded to three dimensions by the decomposition of R into its strain production 1/3sijsjkski and enstrophy production 1/4ωiωjsij terms. The {Q; R} space is a planar projection of the new three-dimensional representation. In the {Q; sss; s} space the Lagrangian evolution of the velocity gradient tensor Aij is studied via conditional mean trajectories (CMTs) as introduced by Martn et al. (Phys. Fluids, vol. 10, 1998, p. 2012). From an analysis of a numerical data set for isotropic turbulence of Reλ ̃ 434, taken from the Johns Hopkins University (JHU) turbulence database, we observe a pronounced cyclic evolution that is almost perpendicular to the QR plane. The relatively weak cyclic evolution in the QR space is thus only a projection of a much stronger cycle in the {Q; sss;ωω s} space. Further, we find that the restricted Euler (RE) dynamics are primarily counteracted by the deviatoric non-local part of the pressure Hessian and not by the viscous term. The contribution of the Laplacian of Aij, on the other hand, seems the main responsible for intermittently alternating between low and high intensity Aij states.

Original languageEnglish
Pages (from-to)497-507
Number of pages11
JournalJournal of Fluid Mechanics
StatePublished - Dec 2009


  • Dynamics
  • Isotropic
  • Theory


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