A robust computational framework for variational data assimilation of mean flows with sparse measurements corrupted by strong outliers

A variational framework for the reconstruction of time-averaged mean flows using a sparse set of observations with large magnitudes of noise (referred to as outliers), is presented. The observations constitute a set of point-wise measurements of the flow field with outliers at certain measurement locations and are incorporated into a numerical simulation governed by the two-dimensional, incompressible Reynolds-averaged Navier–Stokes (RANS) equations with an unknown momentum forcing. This forcing, which corresponds to the divergence of the Reynolds stress tensor, is calculated from a direct-adjoint optimization procedure to reduce the deviation between the measured and estimated velocity fields. ℓ2, ℓ1, Huber, and hybrid loss functions are used to represent the discrepancy in the velocity field between the measurements and the predictions. A variety of algorithms are considered to solve the optimization problem with these loss functions and a performance comparison in terms of the quality and physical features of the recovered flow field is presented. The Huber loss function performed best as it remained robust to strong outliers in the measurements with its ℓ1 contribution and also ensured the uniqueness of the optimal solution with its ℓ2 contribution. Huber loss functions restrict the effect of outliers at the local measurement locations, thereby not affecting the quality of the high-dimensional reconstructed flow field. The hybrid loss function, a modified form of the continuous Huber loss function, also recovered the flow field with high accuracy. We demonstrate the performance of the data assimilation framework for the case of two-dimensional laminar flow around a circular cylinder at Re=100. We then extend the analysis to the case of two-dimensional laminar flow over a backward-facing step at Re=500, to further assess the efficacy and robustness of the data assimilation framework.