The topology of the footprints of wall-turbulence

When studying the topology of turbulent flows, the three invariants of the velocity gradient tensor are often used. For incompressible flow the first invariant $P$ is zero and the topology of the flow structures can be investigated in terms of the second and third invariants, $Q$ and $R$ respectively. For example, isosurfaces of $Q$ above a certain threshold are often used in an attempt to identify vortical structures in the flow. In wall-turbulence, however, these invariants are zero on a no slip wall. Therefore, analysis tools relying on these invariants cannot be used to topologically study the footprint of turbulence on the wall. In this paper, it is proposed that the ``flow'' field on a wall can be described by a no slip Taylor-series expansion. This provides a new tensor relating skin friction to streamwise and spanwise coordinate. Like the velocity gradient tensor, it is possible to define invariants $\mathcal{P}$, $\mathcal{Q}$ and $\mathcal{R}$ of the so-called ``no slip'' tensor. It will also be shown that it may be possible to investigate the topology of the flow field on a no slip wall in terms of these invariants.