Data-driven estimation of scalar quantities from planar velocity measurements by deep learning applied to temperature in thermal convection

The measurement of the transport of scalar quantities within flows is oftentimes laborious, difficult or even unfeasible. On the other hand, velocity measurement techniques are very advanced and give high-resolution, high-fidelity experimental data. Hence, we explore the capabilities of a deep learning model to predict the scalar quantity, in our case temperature, from measured velocity data. Our method is purely data-driven and based on the u-net architecture and, therefore, well-suited for planar experimental data. We demonstrate the applicability of the u-net on experimental temperature and velocity data, measured in large aspect ratio Rayleigh–Bénard convection at Pr=7.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Pr} =7.1$$\end{document} and Ra=2×105,4×105,7×105\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Ra} =2\times 10^5,4\times 10^5,7\times 10^5$$\end{document}. We conduct a hyper-parameter optimization and ablation study to ensure appropriate training convergence and test different architectural variations for the u-net. We test two application scenarios that are of interest to experimentalists. One, in which the u-net is trained with data of the same experimental run and one in which the u-net is trained on data of different Ra\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Ra}$$\end{document}. Our analysis shows that the u-net can predict temperature fields similar to the measurement data and preserves typical spatial structure sizes. Moreover, the analysis of the heat transfer associated with the temperature showed good agreement when the u-net is trained with data of the same experimental run. The relative difference between measured and reconstructed local heat transfer of the system characterized by the Nusselt number Nu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Nu}$$\end{document} is between 0.3 and 14.1% depending on Ra\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Ra}$$\end{document}. We conclude that deep learning has the potential to supplement measurements and can partially alleviate the expense of additional measurement of the scalar quantity.