Three-dimensional variational data assimilation of separated flows using time-averaged experimental data

We present a novel framework for assimilating planar PIV experimental data using a variational approach to enhance the predictions of the Spalart-Allmaras RANS turbulence model. Our method applies three-dimensional constraints to the assimilation of mean velocity data, incorporating a corrective forcing term in the momentum equations. The advantages of this approach are highlighted through a direct comparison with traditional two-dimensional assimilation using the same experimental dataset. We demonstrate its efficacy by assimilating the deep stall flow over a NACA0012 airfoil at a $15^\circ$ angle of attack and a chord-based Reynolds number of $Re_c \approx 7.5 \times 10^4$. We find that in two-dimensional assimilation, the corrective forcing term compensates not only for physical modeling errors but also for the lack of divergence in the experimental data. This conflation makes it difficult to isolate the effects of measurement inconsistencies from deficiencies in the turbulence model. In contrast, three-dimensional assimilation allows the corrective forcing term to primarily address experimental setup errors while enabling the turbulence model to more accurately capture the flow physics. We establish the superiority of three-dimensional assimilation by demonstrating improved agreement in reconstructed quantities, including pressure, lift force, and Reynolds shear stress.