Measuring unsteady drag of the flow around a sphere based on time series displacement measurements using physics-informed neural networks

Surface drag increases energy usage and emissions in numerous transport and engineering applications, with surface treatment being one technique aimed at reducing surface skin friction and hence, surface drag. One approach to investigate the performance of surface treatments is based on the measurement of the terminal velocities of settling spheres with and without surface treatment. This relatively affordable method is limited to the terminal velocity region, as the analysis of the data during the accelerating phase of the sphere requires the differentiation of the experimental data, which results in significant noise and drag measurements with high uncertainty. This paper compares two neural network approaches to determine the temporal derivatives of experimental data. Specifically, the ensemble averaged experimental data and physics information is used to define a model for a settling sphere, which is used to test the performance of neural networks trained on noisy displacement data using an approach based on the governing differential equation and a direct approach, with varying loss term weights, measurement error levels and temporal resolutions. The results show that the neural network trained directly on the experimental displacement data is able to reduce the uncertainty of the temporal derivatives to a level similar to that of the differential equation approach when initial conditions are included in the training. Finally, the two approaches are applied to experimental data to determine the unsteady drag. This paper demonstrates that neural networks can be used to determine the temporal derivatives of experimental data with relatively low uncertainty, particularly where the system physics is at least partially known.