Uncertainty quantification in density estimation from background-oriented Schlieren measurements

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.

Improvement in systematic error in background-oriented schlieren results by using dynamic backgrounds

“…The applicability of optical flow algorithms, albeit sparse optical flow algorithms, was studied and confirmed by Atcheson et al (2009). For the numerical calculations, the distortion of the reference images is computer-generated, a technique established by Schreier et al (2000), Atcheson et al (2009) and more recently Rajendran et al (2019). This allows for a study of the effectiveness and efficiency of different background changing schemes insulated from any random environmental effects.…”

Two regime cooling in flow induced by a spark discharge

“…Experimental uncertainties in the BOS measurements were propagated through the complete density reconstruction according to [47]. Uncertainties in the displacement field were calculated using Image Matching [41], then propagated through the optical layout followed by the Poisson solver to obtain uncertainty in the projected density field.…”

Uncertainty amplification due to density/refractive index gradients in background-oriented schlieren experiments

“…Then, the position uncertainty for a given dot in the gradient image can be calculated by multiplying the position uncertainty for the corresponding dot in the reference image and the amplification ratio. The uncertainty in the density field can then be obtained by propagating the displacement uncertainties through the BOS measurement chain [20], [21].…”