  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 through a stationary corrective forcing
term in the LES momentum equation working with the time-averaged LES momentum
equation allows to employ a stationary discrete adjoint method. Therefore, the
stationary forcing in the unsteady LES momentum equation is iteratively updated
within the gradient-based optimization framework in conjunction with the
adjoint gradient. The implementation is based on a three-dimensional LES solver
in OpenFOAM, which is extended to allow the computation of adjoint quantities
and, ultimately, the adjoint gradient.
  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 turbulent kinetic energy. This highlights the superiority
of scale-resolving simulations such as LES over simulations based on the
(unsteady) Reynolds-averaged equations.
