Bayesian inference applied to spatio-temporal reconstruction of flows around a NACA0012 airfoil

In this paper, we shall investigate sequential data assimilation techniques to improve the stability of reduced-order models for fluid flows. The reduced-order model used relies on a Galerkin projection of Navier–Stokes equations on proper orthogonal decomposition (POD) basis vectors estimated from snapshots of the flow fields obtained with time-resolved particle image velocimetry (TR-PIV) measurements. The coefficients of the dynamical system are given through a least-squares regression technique applied to the experimental data and lead to a low-order model which is known to diverge, or damp, rapidly in time if left uncontrolled. In this context, a sequential data assimilation method based on a Bayesian approach is proposed. In this formalism, reduced-order models (ROMs) are modeled with discrete time from the hidden Markov processes. Given the whole trajectories of the POD temporal modes, the state of ROM coefficients initially provided by noisy PIV measurements are re-estimated from a Kalman filtering of the sequential data. Results are obtained for the flow around a NACA0012 airfoil at Reynolds numbers of 1000 and 2000 and angles of attack of 10∘,15∘,20∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{\circ },15^{\circ },20^{\circ }$$\end{document} and 30∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$30^{\circ }$$\end{document}.