Enhancing modal synchronisation in wall-bounded turbulence via sequential variational data assimilation

This study explores partial synchronisation in turbulent channel flows using sequential variational data assimilation with sparse observations, emphasising the roles of model and observation uncertainties. Unlike previous work that focused on synchronisation using direct numerical simulation, this study considers synchronisation under imperfect models and noisy data. In the first part, a synchronisation map is constructed, revealing invariance with respect to variations in the predictive model, Reynolds number and mesh resolution. Full synchronisation emerges above a critical level of equivalent observation density. At lower observation densities, modal synchronisation is observed, where the energies of dominant modes evolve independently of initial conditions. As data become sparser, the system transitions to a non-synchronisation regime, with assimilated flows exhibiting minimal correlation with the observations. The second part of this study uses the master flow interpolated from down-sampled sparse observations. The delay-coordinate strategy is introduced to enhance the modal synchronisation. Results indicate that the optimal $\sigma$ lies near the threshold between modal synchronisation and non-synchronisation. This demonstrates that the modal synchronisation serves as a critical prerequisite for leveraging historical information in data assimilation to improve the accuracy of turbulence reconstruction. These findings extend the scope of synchronisation theory and provide valuable guidance for advancing data assimilation methodologies.