  The expected universality of small-scale properties of turbulent flows
implies isotropic properties of the velocity gradient tensor in the very large
Reynolds number limit. Using direct numerical simulations, we determine the
tensors formed by n = 2 and 3 velocity gradients at a single point in turbulent
homogeneous shear flows, and in the log-layer of a turbulent channel flow, and
we characterize the departure of these tensors from the corresponding isotropic
prediction. Specifically, we separate the even components of the tensors,
invariant under reflexion with respect to all axes, from the odd ones, which
identically vanish in the absence of shear. Our results indicate that the
largest deviation from isotropy comes from the odd component of the third
velocity gradient correlation function, especially from the third moment of the
derivative along the normal direction of the streamwise velocity component. At
the Reynolds numbers considered (R_lambda ~ 140), we observe that these second
and third order correlation functions are significantly larger in turbulent
channel flows than in homogeneous shear flow. Overall, our work demonstrates
that a mean shear leads to relatively simple structure of the velocity gradient
tensor. How isotropy is restored in the very large Reynolds limit remains to be
understood.
