Optimal prediction of wall stress and pressure from outer velocity observations

Probing near-wall turbulence experimentally is challenging, which leads to difficulty in accurately evaluating the wall stresses and pressure fluctuations. When measurements satisfy the conditions for synchronization (Wang & Zaki, J. Fluid Mech 943, 2022), the near-wall flow can be reconstructed with machine accuracy. When synchronization is not possible, we determine the most accurate estimation of the near-wall region using adjoint-variational data assimilation. The estimated flow field satisfies the Navier-Stokes equations, optimally reproduces the available data, and provides full access to the unknown near-wall turbulence and wall shear stresses and pressure. When the thickness of the unobserved layer is up to fifty viscous units, the reconstructed velocity and pressure fields are almost identical to the true state. As the observations are further separated from the wall, only the inner-layer large-scale structures are accurately estimated. These trends are explained using the coherence spectrum between inner and outer flow quantities: longer wavelength structures have larger coherence depth towards the wall. Lastly, the robustness of the estimation accuracy is demonstrated using filtered and sub-sampled outer observations.