From wall observations to turbulence: The difficulty of flow reconstruction

TL;DR

This paper investigates the challenges of reconstructing turbulent flow states from wall measurements using data assimilation, revealing limitations in sensitivity and convergence related to flow structures and observation timing.

Contribution

It introduces a Hessian-based analysis to quantify the difficulty of flow reconstruction from wall data and explores how sensitivity varies with flow regions and measurement timing.

Findings

Eigenmodes decay beyond the buffer layer, indicating weak sensitivity in the bulk flow.

Sensitivity to outer large-scale motions increases when measurement time exceeds 20 wall units.

Adjoint field energy concentrates in the buffer layer, affecting optimization convergence.

Abstract

Estimation of the initial state of turbulent channel flow from spatially and temporally resolved wall data is performed using adjoint-variational data assimilation. The accuracy of the predicted flow deteriorates with distance from the wall, most precipitously across the buffer layer beyond which only large-scale structures are reconstructed. To quantify the difficulty of the state estimation, the Hessian of the associated cost function is evaluated at the true solution. The forward-adjoint duality is exploited to efficiently compute the Hessian matrix from the ensemble-averaged cross-correlation of the adjoint fields due to impulses at the sensing locations and times. Characteristics of the Hessian are examined when observations correspond to the streamwise or spanwise wall shear stress or wall pressure. Most of the eigenmodes decay beyond the buffer layer, thus demonstrating weak sensitivity of wall observations to the turbulence in the bulk. However, when the measurement time $t_m^+ \gtrsim 20$, some streamwise-elongated Hessian eigenfunctions remain finite in the outer flow, which corresponds to the sensitivity of wall observations to outer large-scale motions. Furthermore, we report statistics of the adjoint-field kinetic-energy budget at long times, which are distinctly different from those of the forward model. One notable difference is the high concentration of energy within the buffer layer and its narrower support. A large and exponentially amplifying adjoint field in that region leads to large gradients of the cost function, smaller step size in the gradient-based optimization, and difficulty to achieve convergence especially elsewhere in the domain where the gradients are comparatively small.