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![Optical rays geometry when a second pattern at a different height under the container is used. A second displacement QQ′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$QQ^{'}$$\end{document} can be computed. Knowing the difference of height hp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_p$$\end{document} between the patterns, the angle ia\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i_a$$\end{document} can be computed from δr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta r$$\end{document}, the difference of both displacements PP′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$PP^{'}$$\end{document} and QQ′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$QQ^{'}$$\end{document}. A relationship can be established between ia\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i_a$$\end{document} and the incidence angle i by applying the Snell–Descartes law at each interface. This relationship does not depend on the liquid height](publication/362364366/figure/fig4/AS:11431281081037121@1661479008328/Optical-rays-geometry-when-a-second-pattern-at-a-different-height-under-the-container-is.png)
Optical rays geometry when a second pattern at a different height under the container is used. A second displacement QQ′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$QQ^{'}$$\end{document} can be computed. Knowing the difference of height hp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_p$$\end{document} between the patterns, the angle ia\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i_a$$\end{document} can be computed from δr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta r$$\end{document}, the difference of both displacements PP′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$PP^{'}$$\end{document} and QQ′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$QQ^{'}$$\end{document}. A relationship can be established between ia\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i_a$$\end{document} and the incidence angle i by applying the Snell–Descartes law at each interface. This relationship does not depend on the liquid height
Source publication
The Schlieren method intends to reveal the elevation of a refractive fluid–fluid interface. The method is based on a comparison of images of a single pattern placed at the bottom of the container. Accurate measurements can be obtained with a simple and low-cost optical setup. However, it is restricted to weak interface deformations, weak slopes and...
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In this manuscript, we study links between nonlinear equations of motion and equations attained by taking geometric differential description of Hasimoto map of pseudo null curves and some physical consequences. First, we find the equations of Hasimoto transformation occurred by adapting Hasimoto map for pseudo null curves. Then, we investigate the...
Citations
... Digital image correlation (DIC) techniques, specifically cross-correlation, have been used extensively for the tracking of features in FS-SS experiments. Other alternatives, such as Fourier demodulation (Diaz et al. 2022;Jain 2020;Jain et al. 2021b;Metzmacher et al. 2022;Shimazaki et al. 2022;Wildeman 2018), have also been performed on FS-SS experiments. However, the displacement fields produce unreliable data for steep surface gradients (Jain 2020). ...
The free surface at an air–water interface can provide information regarding bathymetric complexities, as well as the subsurface flow. We present a comparison of the performance of two recent advances in light-based optical techniques for free surface measurements, total internal reflection-deflectometry and moon-glade background-oriented schlieren, with the more established method of free surface synthetic schlieren. We make use of an optical flow algorithm over the more traditional digital image correlation, in order to obtain higher spatial resolution data across the imaged free surface domain. The optical flow algorithm presents additional benefits, such as computational efficiency and robustness in capturing large displacements and straining of tracked features. The three optical techniques are assembled in synchronization to image two free surface conditions: (1) a free surface being impinged upon by an underlying turbulent, free-shear flow and (2) a random and irregular wave field induced by a free jet. Using the high-resolution measurements, we provide insight on the emergence of multiple free surface dynamics for a turbulent free surface. We present a comprehensive discussion on the benefits and drawbacks of each technique, including suggestions on the suitability of each technique for several experimental constraints.
Graphical abstract
... Digital Image Correlation (DIC) techniques, specifically cross-correlation, have been used extensively for the tracking of features in FS-SS experiments. Other alternatives, such as Fourier Demodulation (Diaz et al., 2022;Jain, 2020;Metzmacher et al., 2022;Shimazaki et al., 2022;Wildeman, 2018) have also been performed on FS-SS experiments. However, the displacement fields produce unreliable data for steep surface gradients (Jain, 2020). ...
The free surface at an air-water interface can provide information regarding bathymetric complexities, as well as the subsurface flow. We present a comparison of the performance of two recent advances in light-based optical techniques for free surface measurements, Total Internal Reflection-Deflectometry and Moon-Glade Background Oriented Schlieren, with the more established method of Free-Surface Synthetic Schlieren. We make use of an optical flow algorithm over the more traditional Digital Image Correlation, in order to obtain higher spatial resolution data across the imaged free surface domain. The Optical flow algorithm presents additional benefits, such as computational efficiency and robustness in capturing large displacements and straining of tracked features. The three optical techniques are assembled in synchronization to image two free surface conditions: (1) a free surface being impinged upon by an underlying turbulent, free-shear flow and (2) a random and irregular wave field induced by a free jet. Using the high-resolution measurements, we provide insight on the emergence of multiple free surface dynamics for a turbulent free surface and dissect the local hydrodynamics of the free surface for several wave forcings. We present a comprehensive discussion on the benefits and drawbacks of each technique, including suggestions on the suitability of each technique for several experimental constraints.
