Nonlinear estimation in turbulent channel flows

We design a nonlinear estimator for channel flows at $Re_{\tau}=180$ and $590$. The nonlinear estimator uses a linear estimator structure based on the linearised Navier-Stokes equations and explicitly calculates the nonlinear forcing from the estimated velocities in physical space. The goal is to use the velocities at one wall-normal height to estimate the velocities at other wall-normal heights. The estimation performance is compared among the nonlinear estimator, the linear estimator and the linear estimator augmented with eddy viscosity. At $Re_{\tau}=180$, the nonlinear estimator and the linear estimator augmented with eddy viscosity outperform the linear estimator in terms of estimating the velocity magnitudes, structures and energy transfer (production and dissipation) across the channel height. The limitations of using measurement data at one wall-normal height are discussed. At $Re_{\tau}=590$, the nonlinear estimator does not work well with only one measurement plane, whereas the linear estimator augmented with eddy viscosity performs well. The performance of the nonlinear estimator at $Re_{\tau}=590$ is significantly enhanced by providing multiple measurement planes.