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We theoretically analyze the effect of density/refractive-index gradients on the measurement precision of Volumetric Particle Tracking Velocimetry (V-PTV) and Background Oriented Schlieren (BOS) experiments by deriving the Cramer-Rao lower bound (CRLB) for the 2D centroid estimation process. A model is derived for the diffraction limited image of a particle or dot viewed through a medium containing density gradients that includes the effect of various experimental parameters such as the particle depth, viewing direction and f-number. Using the model we show that non-linearities in the density gradient field lead to blurring of the particle/dot image. This blurring amplifies the effect of image noise on the centroid estimation process, leading to an increase in the CRLB and a decrease in the measurement precision. The ratio of position uncertainties of a dot in the reference and gradient images is a function of the ratio of the dot diameters and dot intensities. We term this parameter the Amplification Ratio (AR), and we propose a methodology for estimating the position uncertainties in tracking-based BOS measurements. The theoretical predictions of the particle/dot position estimation variance from the CRLB are compared to ray tracing simulations with good agreement. The uncertainty amplification is also demonstrated on experimental BOS images of flow induced by a spark discharge, where we show that regions of high amplification ratio correspond to regions of density gradients. This analysis elucidates the dependence of the position error on density and refractive-index gradient induced distortion parameters, provides a methodology for accounting its effect on uncertainty quantification and provides a framework for optimizing experiment design.
We study statistical properties after a sudden episode of wind for water waves propagating in one direction. A wave with random initial conditions is propagated using a forced-damped higher order Nonlinear Schrodinger equation (NLS). During the wind
We propose an improved density integration methodology for Background Oriented Schlieren (BOS) measurements that overcomes the noise sensitivity of the commonly used Poisson solver. The method employs a weighted least-squares (WLS) optimization of th
We investigate the capability of neural network-based model order reduction, i.e., autoencoder (AE), for fluid flows. As an example model, an AE which comprises of a convolutional neural network and multi-layer perceptrons is considered in this study
Convection over a wavy heated bottom wall in the air flow has been studied in experiments with the Rayleigh number $sim 10^8$. It is shown that the mean temperature gradient in the flow core inside a large-scale circulation is directed upward, that c
We derive the complex Ginzburg-Landau equation for the dynamical self-diffraction of optical waves in a nonlinear cavity. The case of the reflection geometry of wave interaction as well as a medium that possesses the cubic nonlinearity (including a l