Adjoint-variational estimation of near-wall turbulence from outer observations*

Turbulence in the vicinity of the wall and the associated surface stresses are challenging to probe experimentally.  A recent study demonstrated that, given fully resolved outer measurements, the wall-attached horizontal layer can synchronize to the true flow state and evolution, as long as the thickness of the wall layer is less than thirty viscous units (M. Wang and T. Zaki, 2022, J. Fluid Mech. 943, A4).   Beyond this critical thickness, synchronization is not possible, and the accuracy of estimating the wall layer has never been examined.  This problem is studied using adjoint-variational data assimilation, where we seek a Navier-Stokes solution that reproduces the outer flow observations and predicts the unknown near-wall turbulence.  The reconstructed wall layer remains almost identical to the true flow when the measurement height is within fifty wall units, and the estimation accuracy deteriorates as the observations are further separated from the wall. We also demonstrate that the accuracy of the optimally estimated near-wall flow is robust when filtered and subsampled outer observations are adopted.