Resolvent-based estimation of a turbulent wake

We present a resolvent-based framework for estimating turbulent velocity fluctuations in the wake of a spanwise-periodic NACA0012 airfoil at Mach 0.3, Reynolds number 23 000, and an angle of attack of $6^{\circ }$ . Building on the methodology of Jung et al. (2025, J. Fluid Mech. 1016, A41), we extend the approach to the more complex regime of a turbulent wake, which involves three primary challenges: (i) globally unstable modes in the linearised Navier–Stokes operator, (ii) multi-scale turbulent structures and (iii) high-dimensional datasets. To address these challenges, we employ a data-driven approach that constructs causal resolvent-based estimation kernels from cross-spectral densities obtained via large-eddy simulations. These kernels are derived using the Wiener–Hopf method, which optimally enforces causality, thereby enhancing real-time estimation accuracy. The framework captures the spectral signatures of coherent structures and, through the empirically determined cross-spectral densities, implicitly accounts for the coloured statistics of the nonlinear forcing acting on the linear system. To handle the computational demands of the high-dimensional estimation problem, we utilise parallel algorithms developed within the same framework. We further investigate sensor placement by analysing single-sensor estimation error and coherence with target flow quantities. Results demonstrate accurate causal estimation of streamwise velocity for the spanwise-averaged, spanwise-Fourier-transformed and mid-span flow using limited shear-stress measurements on the surface of the airfoil. This study underscores the potential of the resolvent-based framework for efficient estimation in compressible, turbulent environments.