Monte Carlo-based a posteriori uncertainty quantification for background-oriented schlieren measurements

Background-oriented schlieren (BOS) technique is a density-based optical measurement technique. BOS measurement is similar to particle image velocimetry (PIV) ones in terms of experiments design and the computation of the displacements. However, in the BOS technique, the reconstruction of the refractive index field involves further mathematical calculations, which depend on the flow geometry, such as Poisson solver, Abel inversion, algebraic reconstruction technique, and filtered back-projection. This lengthy combination of experimental measurements, cross-correlation evaluation, and mathematical computation complicates the uncertainty quantification of the reconstructed field. In this study, we present a detailed approach for an a posteriori estimation of uncertainty when using BOS measurements to reconstruct the refractive index/density field. The proposed framework is based on the Monte Carlo simulation (MCS) method and can consider all kinds of sources of error, ranging from experimental measurements to those arising from image processing. The key features of this methodology are its capacity to handle different mathematical reconstruction procedures and the ease with which it can integrate additional sources of error. We demonstrate this method first by using synthetic images and a Poisson solver with mixed boundary conditions in a 2D domain. The accuracy of the proposed approach is assessed by comparing analytical and MCS results. Then, the modular nature of the proposed framework is experimentally demonstrated using a combination of Abel inversion and inverse gradient techniques to reconstruct a 3D axisymmetric density field around a transonic projectile in free-flight. The results are compared with computational fluid dynamics (CFD) and show high levels of agreement with only limited discrepancies, which are attributed to the space-filtering effect within cross-correlation resulting from shock waves.