  In this article, we provide a methodology to reconstruct high-Reynolds number
turbulent mean-flows from few time-averaged measurements. A turbulent flow over
a backward-facing step at Re = 28275 is considered to illustrate the potential
of the approach. The data-assimilation procedure, based on a variational
approach, consists in correcting a given baseline model by tuning
space-dependent source terms such that the corresponding solution matches
available measurements (obtained here from direct-numerical simulations). The
baseline model chosen here consists in Reynolds-Averaged-Navier-Stokes (RANS)
equations closed with the turbulence Spalart-Allmaras model. We investigate two
possible tuning functions: a source term in the momentum equations, which is
able to compensate for the deficiencies in the modeling of the Reynolds
stresses by the Boussinesq approximation and a source term in the turbulence
equation, which modifies the balance between the eddy-viscosity production and
dissipation. The quality of the mean-flow reconstruction strongly depends on
the baseline model and on the quantity of measurements. In the case of many
measurements, very accurate reconstructions of the mean-flow are obtained with
the model corrected by the source term in the momentum equations, while the
reconstruction is more approximate when tuning the source term in the
turbulence model. In the case of few measurements, this "rigidity" of the
corrected turbulence model is favourably used and allows the best mean-flow
reconstruction. The flexibility / rigidity of a model is further discussed in
the light of a singular-value decomposition of the linear input / output
operator between source term and measurements.
