In applications involving the model-based control of transitional wall-bounded flow systems, it is often desired to estimate the interior flow state based on a history of noisy measurements from an array of flush-mounted skin-friction and pressure sensors on the wall. This paper considers this estimation problem, using a Kalman filter based on the linearized Navier-Stokes equations and appropriate stochastic models for the relevant statistics of the initial conditions, sensor noise and external disturbances acting on the system. We show that a physically relevant parameterization of these statistics is key to obtaining well-resolved feedback kernels with appropriate spatial extent for all three types of flow measurement available on the wall. The effectiveness of the resulting Kalman and extended Kalman filters that implement this feedback is verified for both infinitesimal and finite-amplitude disturbances in direct numerical simulations of a perturbed laminar channel flow. The consideration of time-varying feedback kernels is shown to be particularly advantageous in accelerating the convergencc of the estimator from unknown initial conditions. A companion paper (Part 2) considers the extension of such estimators to the case of fully developed turbulence.

This work extends the estimator developed in Part I 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.

The linear perturbations evolving on streamwise boundary layer streaks which yield maximum energy growth are computed. The steady and spanwise-periodic streaks arising from the nonlinear saturation of optimally growing streamwise vortices are considered as base flow. It is shown that significant transient growth may occur for both sinuous antisymmetric perturbations and for varicose symmetric modes. The energy growth is observed at amplitudes significantly below the threshold beyond which the streaks become linearly unstable and is largest for sinuous perturbations, to which the base flow considered first become unstable. The optimal initial condition consists of velocity perturbations localized in the regions of highest shear of the streak base flow, tilted upstream from the wall. The optimal response is still localized in the areas of largest shear but it is tilted in the flow direction. The most amplified perturbations closely resemble the unstable eigenfunctions obtained for streaks of higher amplitudes. The results suggest the possibility of a transition scenario characterized by the non-modal growth of primary perturbations, the streaks, followed by the secondary transient growth of higher frequency perturbations. The implication for turbulent flow is also discussed.

A method is proposed to estimate the covariance of disturbances to a stable linear system when its state covariance is known and a dynamic model is available. This is an issue of fundamental interest for estimation and control of fluid mechanical systems whose dynamics is described by the linearized Navier-Stokes equations. The problem is formulated in terms of a matrix norm minimisation with linear matrix inequality constraint, and solved numerically by means of alternating convex projection. The method is tested on covariance estimation in a low Reynolds number channel flow.

A new method, enabling the computation of steady solutions of the Navier-Stokes equations in globally unstable configurations, is presented. We show that it is possible to reach a steady state by damping the unstable (temporal) frequencies. This is achieved by adding a dissipative relaxation term proportional to the high-frequency content of the velocity fluctuations. Results are presented for cavity-driven boundary-layer separation and a separation bubble induced by an external pressure gradient.