Variational assimilation of sparse time-averaged data for efficient adjoint-based optimization of unsteady RANS simulations

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.