  Data assimilation (DA) plays a crucial role in extracting valuable
information from flow measurements in fluid dynamics problems. Often only
time-averaged data is available, which poses challenges for DA in the context
of unsteady flow problems. Recent works have shown promising results in
optimizing Reynolds-averaged Navier-Stokes (RANS) simulations of stationary
flows using sparse data through variational data assimilation, enabling the
reconstruction of mean flow profiles.
  In this study we perform three-dimensional variational data assimilation of
sparse time-averaged data into an unsteady RANS (URANS) simulation by means of
a stationary divergence-free forcing term in the URANS equations. Efficiency
and speed of our method are enhanced by employing coarse URANS simulations and
leveraging the stationary discrete adjoint method for the time-averaged URANS
equations. The data assimilation codes were developed in-house using OpenFOAM
for the URANS simulations as well as for the solution of the adjoint problem,
and Python for the gradient-based optimization.
  Our results demonstrate that data assimilation of sparse time-averaged
velocity measurements not only enables accurate mean flow reconstruction, but
also improves the flow dynamics, specifically the vortex shedding frequency. To
validate the efficacy of our approach, we applied it to turbulent flows around
cylinders of various shapes at Reynolds numbers ranging from 3000 to 22000. Our
findings indicate that data points near the cylinder play a crucial role in
improving the vortex shedding frequency, while additional data points further
downstream are necessary to also reconstruct the time-averaged velocity field
in the wake region.
