Turbulence: The view from the wall

For many practical applications, from naive observation to active control strategies, it is desirable to know the internal state of wall-bounded turbulence. However, practical measurements are limited to the wall, where everything vanishes except for the pressure and the wall-parallel vorticities. We focus on the reconstruction of the second invariant of the velocity gradient tensor, $\Pi$, away from the wall from these three measurements, using its linear relation with the inertial component of the pressure, $p$; $\Nabla^2p= 2\Pi$. Statistically optimal approximations are computed for the velocity fields, $\mathbf{u}$, recovering similar results to those of $\Pi$. Their relative error close to the wall ($y^+\approx 10$) is less than $10\%$ of the turbulent kinetic energy. While both the reconstructions of $\Pi$ and $\mathbf{u}$ capture most of the flow features at $y^+ < 30$, they degrade rapidly, holding reasonable errors only for scales comparable or larger than the distance to the wall, or ``attached''.