What is observable from wall data in turbulent channel flow?

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 estimated flow fields satisfy the Navier–Stokes equations and minimize a cost function defined as the difference between model predictions and the available observations. 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 explain this trend, we examine the domain of dependence of the observations and the Hessian matrix of the state-estimation cost function, both of which are efficiently evaluated using adjoint fields initiated from impulses at wall sensors. Eigenanalysis of the Hessian is performed, and the eigenvalues are related to the capacity to reconstruct flow structures represented by the eigenvectors. 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, and correspond to the sensitivity of wall observations to outer large-scale motions. At much longer observation times, the adjoint field becomes chaotic as it amplifies exponentially, which is indicative of extreme gradients of the cost function and an ill-conditioned Hessian matrix, and both exacerbate the difficulty of estimating turbulence from wall observations.