Fig 1.
We assume the existence of a slow manifold on which each fluid lives.
We extract synthetic data as vectors in from computational simulations. These data are possibly noisy, see the zoomed detail. An example sloshing phenomenon is represented as the red line in phase space. Given the inherent high-dimensionality of the manifold, nonlinear dimensionality reduction techniques will also be applied. This will project the data to a lower dimensionality embedding space in
, with d ≪ D. We develop structure-preserving integration schemes to integrate the evolution of the system in this low-order manifold. These are then mapped back to the physical space in
so as to obtain meaningful results.
Fig 2.
Standard classification of the fluid families considered herein.
In the case of Newtonian fluids, their properties are constant over time and show a linear response. Their flow index n is then set to 1, and their yield stress to τ0 = 0. In contrast, shear thinning fluids start flowing when the stimulus is greater than the yield stress τ0 > 0. For these fluids, n > 1. In this work, fluids whose behavior can be assimilated to shear thinning have been considered. Fluids that incorporate some kind of plastic behavior need for a special treatment and have not been considered yet.
Table 1.
Characteristics of the fluids considered in this work.
Fig 3.
Example of frame binarization.
Original frame (top) versus binarized one (bottom). The picture is transformed firstly to gray scale for gentle binarization. Noise is also filtered to detect a smooth surface, which is highlighted in red in the top figure.
Fig 4.
Free surface detection and tracking in video sequence.
The points selected to belong to the free surface are highlighted in red over the original frame for verification.
Fig 5.
This picture represents the evolution of the eigenvalues for the first 10 k-PCA modes.
We distinguish three modes that stand out with regard to the others. This fact justifies the reduction to the embedded manifold. As a result, we aim to provide with a more manageable and efficient system for classification.
Fig 6.
By employing k-PCA we reach a manifold of 3 dimensions where the different fluids, represented by one color each, remain clustered.
Table 2.
Results obtained from classification after random forest training with k-fold cross validation.
Fig 7.
Representation of the quantifiable comparison of the real liquid and the replica.
Free surface is defined as a function of its height at different points. These heights are compared in a same snapshot to evaluate the reconstruction error.
Fig 8.
Snapshots employed for comparison between the real liquid and the digital twin.
The free surface reconstruction has been evaluated to compute the error.
Table 3.
Numerical results of the experimental validation.
Snapshot number refer to the ones shown in Fig 8.
Fig 9.
The picture shows the functioning of the stereo system.
The camera moves freely, and its movement is related to the origin position through the extrinsic parameters. In computer vision, at least two images are required for the 3D reconstruction of a point. The camera performs continuous triangulations so as to export the depth of each pixel relating the 2D matches detected among right and left lenses.