Sparse data assimilation for under-resolved large-eddy simulations

Abstract: The need for accurate and fast scale-resolving simulations of fluid flows, where turbulent dispersion is a crucial physical feature, is evident. Large-eddy simulations (LES) are computationally more affordable than direct numerical simulations, but their accuracy depends on sub-grid scale models and the quality of the computational mesh. In order to compensate related errors, a data assimilation approach for LES is devised in this work. The presented method is based on variational assimilation of sparse time-averaged velocity reference data. Working with the time-averaged LES momentum equation allows to employ a stationary discrete adjoint method. Therefore, a stationary corrective force in the unsteady LES momentum equation is iteratively updated within the gradient-based optimization framework in conjunction with the adjoint gradient. After data assimilation, corrected anisotropic Reynolds stresses are inferred from the stationary corrective force. Ultimately, this corrective force that acts on the mean velocity is replaced by a term that scales the velocity fluctuations through nudging of the corrected anisotropic Reynolds stresses. Efficacy of the proposed framework is demonstrated for turbulent flow over periodic hills and around a square cylinder. Coarse meshes are leveraged to further enhance the speed of the optimization procedure. Time- and spanwise-averaged velocity reference data from high-fidelity simulations is taken from the literature. Our results demonstrate that adjoint-based assimilation of averaged velocity enables the optimization of the mean flow, vortex shedding frequency (i.e., Strouhal number), and anisotropic Reynolds stresses. This highlights the superiority of scale-resolving simulations such as LES over simulations based on the (unsteady) Reynolds-averaged equations.