Background Oriented Schlieren (BOS) imaging is an effective technique for visualizing and quantifying fluid flow. BOS is cost-effective and flexible, unlike other methods like Particle Image Velocimetry (PIV) and Laser-Induced Fluorescence (LIF). It relies on the distortion of objects in a density-varying medium due to light refraction, with digital image correlation or optical flow algorithms used for analysis. Despite advancements, quantifying complete fluid velocity and pressure fields from BOS images remains challenging. Existing algorithms, mostly based on cross-correlation, are optimized for PIV and provide sparse velocity vectors. Direct pressure estimation requires additional methods. The reconstruction of three-dimensional velocity fields from Tomographic BOS (Tomo-BOS) is an open area in experimental fluid mechanics.

Researchers from the Division of Applied Mathematics, Brown University, LaVision GmbH, Anna-Vandenhoeck-Ring, Germany, and  LaVision Inc., Michigan Ave., Ypsilanti, USA, have developed a method employing Physics-Informed Neural Networks (PINNs) to deduce complete 3D velocity and pressure fields from 3D temperature snapshots obtained through Tomo-BOS imaging. PINNs integrate fluid flow physics and visualization data seamlessly, enabling inference with limited experimental data. The method is validated using synthetic data and applied successfully to Tomo-BOS data, accurately inferring velocity and pressure fields over an espresso cup. 

The study discusses using Schlieren features in sequential images and the sensitivity of physical properties in PINN for estimating 2-D pressure fields. The researchers conduct a Tomo-BOSPINN experiment with downsampling data to investigate the sensitivity of physical properties in the estimation process. The training data is sampled with a time interval of 0.1 s, and the relative L2-norm temperature error is calculated for unseen data using the trained parameters. The researchers compare the inferred velocity field with the displacement determined from Schlieren-tracking and agree. The proposed Tomo-BOSPINN method can accurately guess the full temperature and velocity fields.

The PINN algorithm, functioning as a  data assimilation technique, predicts velocity and pressure fields by analyzing visualization data across a spatio-temporal domain. Unlike conventional data assimilation methods, the efficiency of which relies heavily on accurately choosing initial guesses for velocity and pressure conditions, the PINN algorithm doesn’t require such information. In PINN, the trainable variables are the parameters of the neural network, not the conventional control variables. This distinction eliminates the need to specify initial and boundary conditions for velocity or pressure, simplifying the implementation of the algorithm.

The study presents the results of the Tomo-BOSPINN experiment, which utilizes Schlieren features in sequential images to estimate 2-D pressure fields. The researchers report the residuals of the momentum equations in the x, y, and z directions, with an average residual in the order of 10^-4 m s^-2. Velocity profiles along a horizontal line at various time instances are compared between Tomo-BOSPINN and planar PIV results. The researchers acknowledge the support from the PhILMS grant under the grant number DE-SC0019453.

In conclusion, the researchers have developed a machine-learning algorithm based on PINNs for estimating velocity and pressure fields from temperature data in Tomo-BOS experiments. PINNs integrate governing equations and temperature data without requiring CFD solvers, allowing simultaneous inference of velocity and pressure without initial or boundary conditions. The method is evaluated through a 2D buoyancy-driven flow simulation, demonstrating accurate performance with sparse and noisy data. A Tomo-BOS experiment on flow over an espresso cup successfully infers 3D velocity and pressure fields from reconstructed temperature data, showing the versatility of PINNs with either planar or tomographic BOS data. The flexibility of the proposed method suggests its potential for various fluid mechanics problems, marking a promising direction in experimental fluid mechanics.

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