Assimilation of experimental measurements of wall turbulence in direct numerical simulations.

We adopt adjoint-variational data assimilation (4DVar) to reconstruct all the scales of a turbulent boundary layer, from three-dimensional tomographic particle tracking velocimetry (TPTV) measurements. The utility of 4DVar for this task was previously demonstrated using synthetic measurements from independent direct numerical simulation (DNS) [T.A. Zaki, Annu. Rev. Fluid Mech., vol. 57, (2025), pp. 311–33], and is herein applied to experimental measurements. The friction Reynolds number of the experiment is Re_τ > 3000, and the first wall-normal measurement location is outside the viscous layer. The available velocity data in the outer flow are spatio-temporally under-resolved, but are sufficient to ensure successful data assimilation. The measurement time series is divided into windows of duration T⁺, determined based on the Lyapunov time scale, and the assimilation is performed in parallel for 300 windows. Each assimilation task involves iteratively estimating the initial condition for the window, which minimizes the discrepancy between the Navier–Stokes solution and the available measurements. We show that the assimilation recovers the missing near-wall turbulent structures, wall shear stress, and wall-pressure fluctuations. We also discuss the implications of data assimilation on the cost of simulating high-Reynolds number flows, which becomes computationally affordable because the experimental measurements capture the influence of the very large scales. As such, the assimilation can be performed in a smaller volume than standalone DNS, and focus on the generation of the unresolved scales in the available measurements.