Expanding the Q-R space to three dimensions

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 -1/3s(ij)s(jk)s(ki) and enstrophy production 1/4w(i)w(j)s(ij) terms. The IQ; R space is a planar projection of the new three-dimensional representation. In the {Q; -sss; wws} space the Lagrangian evolution of the velocity gradient tensor A(ij) is studied via conditional mean trajectories (CMTs) as introduced by Martin et al. (Phys. Fluids, vol. 10, 1998, p. 2012). From an analysis of a numerical data set for isotropic turbulence of Re(lambda) similar to 434, taken from the Johns Hopkins University (JHU) turbulence database, we observe a pronounced cyclic evolution that is almost perpendicular to the Q-R plane. The relatively weak cyclic evolution in the Q-R space is thus only a projection of a much stronger cycle in the {Q; -sss; wws} 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 A(ij), on the other hand, seems the main responsible for intermittently alternating between low and high intensity A(ij) states.