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Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks

Published online by Cambridge University Press:  25 March 2021

Shengze Cai
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Zhicheng Wang
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Frederik Fuest
Affiliation:
LaVision GmbH, Anna-Vandenhoeck-Ring 19, D-37081 Goettingen, Germany
Young Jin Jeon
Affiliation:
LaVision GmbH, Anna-Vandenhoeck-Ring 19, D-37081 Goettingen, Germany
Callum Gray
Affiliation:
LaVision Inc., 211 W. Michigan Ave., Ypsilanti, MI 48197, USA
George Em Karniadakis
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Corresponding

Abstract

Tomographic background oriented Schlieren (Tomo-BOS) imaging measures density or temperature fields in three dimensions using multiple camera BOS projections, and is particularly useful for instantaneous flow visualizations of complex fluid dynamics problems. We propose a new method based on physics-informed neural networks (PINNs) to infer the full continuous three-dimensional (3-D) velocity and pressure fields from snapshots of 3-D temperature fields obtained by Tomo-BOS imaging. The PINNs seamlessly integrate the underlying physics of the observed fluid flow and the visualization data, hence enabling the inference of latent quantities using limited experimental data. In this hidden fluid mechanics paradigm, we train the neural network by minimizing a loss function composed of a data mismatch term and residual terms associated with the coupled Navier–Stokes and heat transfer equations. We first quantify the accuracy of the proposed method based on a two-dimensional synthetic data set for buoyancy-driven flow, and subsequently apply it to the Tomo-BOS data set, where we are able to infer the instantaneous velocity and pressure fields of the flow over an espresso cup based only on the temperature field provided by the Tomo-BOS imaging. Moreover, we conduct an independent PIV experiment to validate the PINN inference for the unsteady velocity field at a centre plane. To explain the observed flow physics, we also perform systematic PINN simulations at different Reynolds and Richardson numbers and quantify the variations in velocity and pressure fields. The results in this paper indicate that the proposed deep learning technique can become a promising direction in experimental fluid mechanics.

Type
JFM Papers
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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Cai et al. supplementary movie 1

Movie of Schlieren images: the temperature-induced Schlieren images from one camera observing the flow over an espresso cup.

Download Cai et al. supplementary movie 1(Video)
Video 8 MB

Cai et al. supplementary movie 2

Movie of PINN results for Tomo-BOS: the 2D temperature, pressure and velocity vectors at $Z=-21$ mm inferred by PINN method.

Download Cai et al. supplementary movie 2(Video)
Video 91 MB

Cai et al. supplementary movie 3

Movie of PIV results: the 2D velocity fields over an espresso cup from the planar PIV experiment.

Download Cai et al. supplementary movie 3(Video)
Video 71 MB
Supplementary material: PDF

Cai et al. supplementary material

Supplementary data

Download Cai et al. supplementary material(PDF)
PDF 6 MB
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Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks
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