Studying turbulence structure near the wall in hydrodynamic flows: An approach based on the Schur decomposition of the velocity gradient tensor

The Schur decomposition of the velocity gradient tensor (VGT) is an alternative to the classical Cauchy-Stokes decomposition into rotation rate and strain rate components. Recently, there have been several strands of work that have employed this decomposition to examine the physics of turbulence dynamics, including approaches that combine the Schur and Cauchy-Stokes formalisms. These are briefly reviewed before the latter approach is set out. This partitions the rotation rate and strain rate tensors into normal/local and non-normal/non-local contributions. We then study the relation between the VGT dynamics and ejection-sweep events in a channel flow boundary-layer. We show that the sweeps in particular exhibit novel behaviour compared with either the other quadrants, or the flow in general, with a much-reduced contribution to the dynamics from the non-normal terms above the viscous sub-layer. In particular, the reduction in the production term that is the interaction between the non-normality and the normal straining reduces in the log-layer as a consequence of an absence of alignment between the non-normal vorticity and the strain rate eigenvectors. There have been early forays into using the Schur transform approach for subgrid-scale modelling in large-eddy simulation (LES) and this would appear to be an exciting way forward.