Estimation of turbulent channel flow at  based on the wall measurement using a simple sequential approach

The unsteady flow estimation problem of wall-bounded turbulence, numerically benchmarked by Chevalier et al. ( J. Fluid Mech. , vol. 552, 2006, pp. 167–187), is re-tackled with simple approaches. A turbulent channel flow at $Re_{\unicode[STIX]{x1D70F}}=100$ with periodic boundary conditions is reconstructed with linear stochastic estimation only based on the wall measurement, i.e. the wall shear stress in the streamwise and spanwise directions as well as the wall pressure over the entire wavenumber space. The results reveal that instantaneous information on the wall governs the success of the estimation in the vicinity of the wall ( $y^{+}\lesssim 20$ ). The degrees of agreement are equivalent to those reported by Chevalier et al. using the extended Kalman filter as well as the ensemble Kalman filter performed in this study. This suggests that the instantaneous information on the wall dictates the reconstruction rather than the prediction step in these state observers solving the dynamical system. Subsequently, we feed the velocity components given by the linear stochastic estimation via the body-force term into the Navier–Stokes system: such an observer slightly improves the estimation in the log layer, indicating a small benefit of involving a dynamical system but over-suppression of turbulent motions beyond the viscous sublayer due to their low correlation with the wall measurement. Errors in the estimation grow in the buffer layer and prevent further reconstruction toward the centreline even if we relax the feedback forcing and let the flow evolve nonlinearly through the observer. We also discuss the flow components truly reconstructible from the wall measurement, which has limited degrees of freedom and poor correlation across wavenumbers.