Galerkin reduced-order model on the logarithmic lattice in turbulent channel flow

Reduced-order models (ROMs) based on Galerkin projection have been shown to successfully capture the dynamics of turbulence typically at transitional Reynolds numbers, but their applications to high Reynolds numbers remain important challenges. In this study, taking the controllability modes of the eddy-viscosity-enhanced linearised Navier-Stokes operator about turbulent mean flow as the basis functions in the wall-normal direction, Galerkin-based ROMs with logarithmically spaced plane Fourier modes (i.e. the logarithmic lattice) are examined in channel flow up to Reτ ≈ 200. As in previous study (Omurtag & Sirovich, Theoret. Comput. Fluid Dynamics, 115-127, vol 13, 1999), a large number of bases for the mean Fourier mode are required to predict the turbulent statistics accurately. On the contrary, only a moderate number of bases for the fluctuation Fourier modes are required to capture turbulent fluctuations. The inclusion of more bases for the fluctuation modes mainly improves the local dissipation, the role of which is found to be crucial for accurate mean velocity prediction. Finally, the smallest horizontal length scales required on the logarithmic lattice still scale in viscous inner units. In the final presentation, this issue will be discussed further with the results at higher Reynolds numbers up to Reτ ≈ 1000.