A statistical description of flow regions with negative streamwise velocity is provided based on simulations of turbulent plane channels in the Reynolds number range $547\leqslant Re_{\unicode[STIX]{x1D70F}}\leqslant 2003$. It is found that regions of backflow are attached and their density per surface area – in wall units – is an increasing function of $Re_{\unicode[STIX]{x1D70F}}$. Their size distribution along the three coordinates reveals that, even though in the mean they appear to be circular in the wall-parallel plane, they tend to become more elongated in the spanwise direction after reaching a certain height. Time-tracking of backflow regions in a $Re_{\unicode[STIX]{x1D70F}}=934$ simulation showed they convect downstream at the mean velocity corresponding to $y^{+}\approx 12$, they seldom interact with other backflow events, their statistical signature extends in the streamwise direction for at least $300$ wall units, and they result from a complex interaction between regions of high and low spanwise vorticity far beyond the viscous sublayer. This could explain why some statistical aspects of these near-wall events do not scale in viscous units; they are dependent on the $Re_{\unicode[STIX]{x1D70F}}$-dependent dynamics further away from the wall.

In this paper we examine the invariants $p$ and $q$ of the reduced $2\times 2$ velocity gradient tensor (VGT) formed from a two-dimensional (2D) slice of an incompressible three-dimensional (3D) flow. Using data from both 2D particle image velocimetry (PIV) measurements and 3D direct numerical simulations of various turbulent flows, we show that the joint probability density functions (p.d.f.s) of $p$ and $q$ exhibit a common characteristic asymmetric shape consistent with $\langle pq\rangle \lt 0$ . An explanation for this inequality is proposed. Assuming local homogeneity we derive $\langle p\rangle = 0$ and $\langle q\rangle = 0$ . With the addition of local isotropy the sign of $\langle pq\rangle $ is proved to be the same as that of the skewness of $\partial {u}_{1} / \partial {x}_{1} $ , hence negative. This suggests that the observed asymmetry in the joint p.d.f.s of $p{{\ndash}}q$ stems from the universal predominance of vortex stretching at the smallest scales. Some advantages of this joint p.d.f. compared with that of $Q{{\ndash}}R$ obtained from the full $3\times 3$ VGT are discussed. Analysing the eigenvalues of the reduced strain-rate matrix associated with the reduced VGT, we prove that in some cases the 2D data can unambiguously discriminate between the bi-axial (sheet-forming) and axial (tube-forming) strain-rate configurations of the full $3\times 3$ strain-rate tensor.