Coherent structures in statistically stationary homogeneous shear turbulence

Abstract

The three-dimensional vortex clusters, and the structures based on the quadrant classification of the intense tangential Reynolds stress (Qs), are studied in direct numerical simulations of statistically stationary homogeneous shear turbulence (HST) at Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}\approx 50{-}250$, with emphasis on comparisons with turbulent channels (CHs). The Qs and vortex clusters in HST are found to be versions of the corresponding detached (in the sense of del Álamo et al. (J. Fluid Mech., vol. 561 (2006), pp. 329–358)) structures in CHs, although statistically symmetrised with respect to the substitution of sweeps by ejections and vice versa. In turn, these are more symmetric versions of the corresponding attached Qs and clusters. In both flows, only co-gradient sweeps and ejections larger than the local Corrsin scale are found to couple with the shear. They are oriented anisotropically, and are responsible for carrying most of the total Reynolds stress. Most large eddies in CHs are attached to the wall, but it is shown that this is probably a geometric consequence of their size, rather than the reason for their dynamical significance. Most small Q structures associated with different quadrants are far from each other in comparison to their size, but those that are close to each other tend to form quasi-streamwise trains of groups of a sweep and an ejection paired side by side in the spanwise direction, with a vortex cluster in between, generalising to three dimensions the corresponding arrangement of attached eddies in CHs. These pairs are organised around an inclined large-scale conditional vortex ‘roller’, and it is shown that the composite structure tends to be located at the interface between high- and low-velocity streaks, as well as in strong ‘co-gradient’ shear layers that separate streaks of either sign in which velocity is more uniform. It is further found that the conditional rollers are terminated by ‘hooks’ reminiscent of hairpins, both upright and inverted. The inverted hook weakens as the structures approach the wall, while the upright one changes little. At the same time, the inclination of the roller with respect to the mean velocity decreases from $45^{\circ }$ in HST to quasi-streamwise for wall-attached eddies. Many of these observations are generalised to intense Reynolds stresses formed with different pairs of velocity components, and it is shown that most properties of the small structures can be traced to their definitions, rather than to their dynamics. It is concluded that the larger Reynolds-stress structures are associated with shear turbulence, rather than with the presence of a wall, while the smaller ones are generic to turbulence in general, whether sheared or not.