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In this paper we conduct an analysis of the geometrical and vortical statistics in the small scales of helical and nonhelical turbulences generated with direct numerical simulations. Using a filtering approach, the helicity flux from large scales to small scales is represented by the subgrid-scale (SGS) helicity dissipation. The SGS helicity dissipation is proportional to the product between the SGS stress tensor and the symmetric part of the filtered vorticity gradient, a tensor we refer to as the vorticity strain rate. We document the statistics of the vorticity strain rate, the vorticity gradient, and the dual vector corresponding to the antisymmetric part of the vorticity gradient. These results provide new insights into the local structures of the vorticity field. We also study the relations between these quantities and vorticity, SGS helicity dissipation, SGS stress tensor, and other quantities. We observe the following in both helical and nonhelical turbulences: (1) there is a high probability to find the dual vector aligned with the intermediate eigenvector of the vorticity strain rate tensor; (2) vorticity tends to make an angle of 45 with both the most contractive and the most extensive eigendirections of the vorticity strain rate tensor; (3) the vorticity strain rate shows a preferred alignment configuration with the SGS stress tensor; (4) in regions with strong straining of the vortex lines, there is a negative correlation between the third order invariant of the vorticity gradient tensor and SGS helicity dissipation fluctuations. The correlation is qualitatively explained in terms of the self-induced motions of local vortex structures, which tend to wind up the vortex lines and generate SGS helicity dissipation. In helical turbulence, we observe that the joint probability density function of the second and third tensor invariants of the vorticity gradient displays skewed distributions, with the direction of skewness depending on the sign of helicity input. We also observe that the intermediate eigenvalue of the vorticity strain rate tensor is more probable to take negative values. These interesting observations, reported for the first time, call for further studies into their dynamical origins and implications. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3336012]
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