Assessment of three-dimensional density measurements from tomographic background-oriented schlieren (BOS)

Abstract

Tomographic background-oriented schlieren (BOS) is used to measure three-dimensional instantaneous density fields in turbulent flows. Density gradients are related to the refractive index through the Gladstone-Dale relation. Based on path-integrated apparent displacements in a background pattern from path variations in light rays travelling through the flow, a tomographic reconstruction of the three-dimensional refractive index gradients can be obtained. A Poisson equation of the reconstructed gradients is solved to obtain the refractive index field. We examine four sources of measurement error: (i) defocus blurring; (ii) spatial averaging in the finite difference scheme for solving the Poisson equation; (iii) limited-view tomographic reconstruction; and (iv) displacement field noise. Synthetic BOS displacements are generated from the instantaneous density fields of a heated jet at a Reynolds number of 10 000 and jet exit centreline-to-ambient density ratio of 0.83 computed via DNS. Synthetic background displacements are produced by raytracing through the refractive index field obtained using the Gladstone-Dale relation. The virtual BOS setup consists of 15 cameras placed circumferentially around the jet axis, based on a proposed experimental setup. We show that defocus blurring has the greatest impact on the measurement accuracy, using the RMS and peak errors between the measured and true DNS fields as metrics; its impact is greatest in the transitional region and decreases steadily downstream. Sources (ii) and (iv) have marginal impact (for noise 15% of mean displacements). Three reconstruction methods are tested: (i) filtered back-projection (FBP); (ii) an algebraic reconstruction technique (ART); and (iii) sequential FBP-ART. ART and FBP-ART are similarly accurate, producing under half the errors of FBP, which suffers from significant reconstruction artefacts. The reconstruction error is modest compared to the impact of defocus blurring.