Uncertainty Quantification in density estimation from Background Oriented Schlieren (BOS) measurements

TL;DR

This paper introduces a novel uncertainty quantification method for density measurements obtained from Background Oriented Schlieren (BOS), utilizing cross-correlation algorithms and a Poisson solver to provide local, instantaneous uncertainty bounds.

Contribution

The paper develops a new methodology for propagating displacement uncertainties through density estimation in BOS using a Poisson solver, validated with synthetic and experimental data.

Findings

Uncertainty bounds align well with true uncertainties in synthetic tests.

Sharp density changes increase local uncertainty in experimental flow.

Uncertainty propagates monotonically away from boundary conditions in the Poisson solver.

Abstract

We present an uncertainty quantification methodology for density estimation from Background Oriented Schlieren (BOS) measurements, in order to provide local, instantaneous, a-posteriori uncertainty bounds on each density measurement in the field of view. Displacement uncertainty quantification algorithms from cross-correlation based Particle Image Velocimetry (PIV) are used to estimate the uncertainty in the dot pattern displacements obtained from cross-correlation for BOS and assess their feasibility. In order to propagate the displacement uncertainty through the density integration procedure, we also develop a novel methodology via the Poisson solver using sparse linear operators. Testing the method using synthetic images of a Gaussian density field showed agreement between the propagated density uncertainties and the true uncertainty. Subsequently the methodology is experimentally demonstrated for supersonic flow over a wedge, showing that regions with sharp changes in density lead to an increase in density uncertainty throughout the field of view, even in regions without these sharp changes. The uncertainty propagation is influenced by the density integration scheme, and for the Poisson solver the density uncertainty increases monotonically on moving away from the regions where the Dirichlet boundary conditions are specified.