Universal aspects of small-scale motions in turbulence

Two aspects of small-scale turbulence are currently regarded universal, as they have been reported for a wide variety of turbulent flows. Firstly, the vorticity vector has been found to display a preferential alignment with the eigenvector corresponding to the intermediate eigenvalue of the strain rate tensor; and secondly, the joint probability density function (p.d.f.) of the second and third invariant of the velocity gradient tensor, Q and R, has a characteristic teardrop shape. This paper provides an explanation for these universal aspects in terms of a spatial organization of coherent structures, which is based on an evaluation of the average flow pattern in the local coordinate system defined by the eigenvectors of the strain rate tensor. The approach contrasts with previous investigations, which have relied on assumed model flows. The present average flow patterns have been calculated for existing experimental (particle image velocimetry) or numerical (direct numerical simulation) datasets of a turbulent boundary layer (TBL), a turbulent channel flow and for homogeneous isotropic turbulence. All results show a shear-layer structure consisting of aligned vortical motions, separating two larger-scale regions of relatively uniform flow. Because the directions of maximum and minimum strain in a shear layer are in the plane normal to the vorticity vector, this vector aligns with the remaining strain direction, i.e. the intermediate eigenvector of the strain rate tensor. Further, the QR joint p.d.f. for these average flow patterns reveals a shape reminiscent of the teardrop, as seen in many turbulent flows. The above-mentioned organization of the small-scale motions is not only found in the average patterns, but is also frequently observed in the instantaneous velocity fields of the different turbulent flows. It may, therefore, be considered relevant and universal.