  A non-intrusive data assimilation methodology is developed to improve the
statistical predictions of large-eddy simulations (LES). The
ensemble-variational (EnVar) approach aims to minimize a cost function that is
defined as the discrepancy between LES predictions and reference statistics
from experiments or, in the present demonstration, independent direct numerical
simulations (DNS). This methodology is applied to adjust the Smagorinsky
subgrid model and obtain data assimilated LES (DA-LES) which accurately
estimate the statistics of turbulent channel flow. To separately control the
mean and fluctuations of the modeled subgrid tensor, and ultimately the first-
and second-order flow statistics, two types of model corrections are
considered. The first one optimizes the wall-normal profile of the Smagorinsky
coefficient, while the second one introduces an adjustable steady forcing in
the momentum equations to independently act on the mean flow. Using these two
elements, the data assimilation procedure can satisfactorily modify the subgrid
model and accurately recover reference flow statistics. The retrieved subgrid
model significantly outperforms more elaborate baseline models such as dynamic
and mixed models, in a posteriori testing. The robustness of the present data
assimilation methodology is assessed by changing the Reynolds number and
considering grid resolutions that are away from usual recommendations. Taking
advantage of the stochastic formulation of EnVar, the developed framework also
provides the uncertainty of the retrieved model.
