Best Selling Books by Jacques Philippe

In many applications, there is growing interest to use limited pressure observations to estimate the flow behavior. In this dissertation, we represent the flow field by the positions and strengths of a collection of point vortices. We perform the assimilation of the pressure observations with the ensemble Kalman filter (EnKF), which builds a Monte-Carlo approximation of the Kalman gain. As a result of the limited ensemble size, the estimated Kalman gain suffers from ill-conditioning issues, sampling errors, and spurious long-range correlations. We propose two strategies to resolve these problems in the context of flow estimation. First, we show that the flow estimator introduced by Darakananda et al. (Phys. Rev. Fluids 3, 124701 (2018)) is greatly improved by replacing the stochastic version of the EnKF with the ensemble transform Kalman filter (ETKF): a deterministic version of the EnKF that reduces sampling errors. We assess this improved flow estimator on two challenging flow configurations: a flat plate is subjected to strong and overlapping disturbances applied near the leading edge to mimic flow actuation, and a flat plate is placed in the wake of a cylinder. The ETKF significantly improves the estimation of the flow field. Second, predominant methods for regularizing the EnKF suppress correlations at long distances. In incompressible fluid problems, the observations are given by elliptic partial differential equations, e.g. the pressure Poisson equation. Distance localization is not applicable here, as we cannot distinguish the slowly decaying physical correlations from the spurious long-range ones. In elliptic inverse problems, we observe that a low-dimensional projection of the observations is only informative of a low-dimensional subspace of the state space. We introduce the low-rank EnKF (LREnKF): a novel version of the EnKF that leverages this structure. We identify the most informative directions of the state and observation spaces as the leading eigenvectors of Gramian matrices based on the sensitivity of the observation operator. From the rapid spectral decay, we can estimate a lower-dimensional Kalman gain in the low-dimensional subspace spanned by the leading eigenvectors, hence reducing the variance of the estimator. The LREnKF avoids any ad-hoc tuning by adaptively identifying the dimensions of the informative subspace. We show the LREnKF significantly improves the estimate of the stochastic EnKF on two potential flow examples.