Mean- and unsteady-flow reconstruction with one or two time-resolved measurements

In this article, we propose a methodology to reconstruct, in a single step, the mean- and unsteady properties of a flow from very few time-resolved measurements. The procedure is based on the {\it a priori} alignement of Fourier- and Resolvent-modes over energetic frequencies, which is a common feature in shear-dominated transitional flows. Hence, the Reynolds-stresses, which determine the mean-flow, may be approximated from a series of Resolvent modes, which discretize the fluctuation field in the frequency space and whose amplitudes can be tuned thanks to few measurements. In practice, we solve a nonlinear optimization problem (with only few parameters) based on a model coupling strongly the equations governing the mean-flow and the Resolvent modes. The input data for the assimilation procedure may be very sparse, typically one or two pointwise measurements. This technique is applied to two distinct physical configurations, one "oscillator" flow with periodic fluctuations (squared-section cylinder) and one "noise amplifier" flow with a broadband frequency spectrum (backward-facing step).