State estimation in wall-bounded flow systems. Part 2. Turbulent flows

• Published in: Journal of fluid mechanics Vol. 552; no. 1; pp. 167 - 187
• Main Authors: CHEVALIER, MATTIAS, HŒPFFNER, JÉRÔME, BEWLEY, THOMAS R., HENNINGSON, DAN S.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 10.04.2006
• Subjects: Channel flow; Exact sciences and technology; Filters; Flow profiles; Flow system; Fluid dynamics; Fluid mechanics; Fundamental areas of phenomenology (including applications); Kalman filters; Physics; Turbulence control; Turbulent flow; Turbulent flows, convection, and heat transfer; Digital filtering; Turbulent flow; Pipe flow; Feedback; Digital simulation; Control systems; Modelling; Kalman filters; Stochastic method; State estimation; Turbulence structure; Turbulent laminar transition
• DOI: 10.1017/S0022112005008578
• ISSN: 0022-1120

This work extends the estimator developed in Part 1 of this study to the problem of estimating a turbulent channel flow at $Re_{\tau}\,{=}\,100$ based on a history of noisy measurements on the wall. The key advancement enabling this work is the development and implementation of an efficient technique to extract, from direct numerical simulations, the relevant statistics of an appropriately defined ‘external forcing’ term on the Navier–Stokes equation linearized about the mean turbulent flow profile. This forcing term is designed to account for the unmodelled (nonlinear) terms during the computation of the (linear) Kalman filter feedback gains in Fourier space. Upon inverse transform of the resulting feedback gains computed on an array of wavenumber pairs to physical space, we obtain, as in Part 1, effective and well-resolved feedback convolution kernels for the estimation problem. It is demonstrated that, by applying the feedback so determined, satisfactory correlation between the actual and estimated flow is obtained in the near-wall region. As anticipated, extended Kalman filters (with the nonlinearity of the actual system reintroduced into the estimator model after the feedback gains are determined) outperform standard (linear) Kalman filters on the full system.