  The Reynolds-averaged Navier-Stokes (RANS) equations provide a
computationally efficient method for solving fluid flow problems in engineering
applications. However, the use of closure models to represent turbulence
effects can reduce their accuracy. To address this issue, recent research has
explored data-driven techniques such as data assimilation and machine learning.
  An efficient variational data assimilation (DA) approach is presented to
enhance steady-state eddy viscosity based RANS simulations. To account for
model deficiencies, a corrective force term is introduced in the momentum
equation. In the case of only velocity reference data, this term can be
represented by a potential field and is divergence-free. The DA implementation
relies on the discrete adjoint method and approximations for efficient gradient
evaluation.
  The implementation is based on a two-dimensional coupled RANS solver in
OpenFOAM, which is extended to allow the computation of the adjoint velocity
and pressure as well as the adjoint gradient. A gradient-based optimizer is
used to minimize the difference between the simulation results and the
reference data. To evaluate this approach, it is compared with alternative data
assimilation methods for canonical stationary two-dimensional turbulent flow
problems. For the data assimilation, sparsely distributed reference data from
averaged high-fidelity simulation results are used.
  The results suggest that the proposed method achieves the optimization goal
more efficiently compared to applying data assimilation for obtaining the eddy
viscosity, or a field modifying the eddy viscosity, directly. The method works
well for different reference data configurations and runs efficiently by
leveraging coarse meshes.
