Data-driven reduced modelling of turbulent Rayleigh-Benard convection using DMD-enhanced Fluctuation-Dissipation Theorem

A data-driven, model-free framework is introduced for calculating Reduced-Order Models (ROMs) capable of accurately predicting time-mean responses to external forcings, or forcings needed for specified responses, e.g., for control, in fully turbulent flows. The framework is based on using the Fluctuation-Dissipation Theorem (FDT) in the space of a limited number of modes obtained from Dynamic Mode Decomposition (DMD). Using the DMD modes as the basis functions, rather than the commonly used Proper Orthogonal Decomposition (POD) modes, resolves a previously identified problem in applying FDT to high-dimensional, non-normal turbulent flows. Employing this DMD-enhanced FDT method (FDT$_\mathrm{DMD}$), a 1D linear ROM with horizontally averaged temperature as state vector, is calculated for a 3D Rayleigh-B\'enard convection system at the Rayleigh number of $10^6$ using data obtained from Direct Numerical Simulation (DNS). The calculated ROM performs well in various tests for this turbulent flow, suggesting FDT$_\mathrm{DMD}$ as a promising method for developing ROMs for high-dimensional, turbulent systems.