An EnKF-based Flow State Estimator for Aerodynamic Problems

Regardless of the plant model, robust flow estimation based on limited measurements remains a major challenge in successful flow control applications. Aiming to combine the robustness of a high-dimensional representation of the dynamics with the cost efficiency of a low-order approximation of the state covariance matrix, a flow state estimator based on the Ensemble Kalman Filter (EnKF) is applied to two-dimensional flow past a cylinder and an airfoil at high angle of attack and low Reynolds number.  For development purposes, we use the numerical algorithm as both the estimator and as a surrogate for the measurements. In a perfect-model framework, a reduced number of either pressure sensors on the surface of the body or sparsely placed velocity probes in the wake are sufficient to accurately estimate the instantaneous flow state.  Because the dynamics of these flows are restricted to a low-dimensional manifold of the state space, a small ensemble size is sufficient to yield the correct asymptotic behavior. The relative importance of each sensor location is evaluated by analyzing how they influence the estimated flow field, and optimal locations for pressure sensors are determined. 

However, model inaccuracies are ubiquitous in practical applications. Covariance inflation is used to enhance the estimator performance in the presence of unmodeled freestream perturbations. A combination of parametric modeling and augmented state methodology is used to successfully estimate the forces on immersed bodies subjected to deterministic and random gusts. The robustness of high-dimensional representation of the dynamics to the choice of parameters such as the Reynolds number is inherited by the estimator, which was shown to successfully estimate the reference Reynold number on the fly.

Spatial and temporal discretization can constitute a second source of errors which can render numerical solutions a biased representation of reality. Left unaccounted for, biased forecast and observation models can lead to poor estimator performance. In this work, we propose a low-rank representation for the bias whose dynamics are represented by a colored-noise process. System state and bias parameters are simultaneously tracked online with the Ensemble Kalman Filter (EnKF) algorithm. The proposed methodology is demonstrated to achieve a 70% error reduction for the problem of estimating the state of the two-dimensional low-Re flow past a flat plate at a high angle of attack using an ensemble of coarse-mesh simulations and pressure measurements at the surface of the body, compared to a bias-blind estimator. Strategies to determine the bias statistics and to deal with nonlinear observation functions in the context of ensemble methods are discussed.