Topological Analysis of Ensembles of Hydrodynamic Turbulent Flows -- An Experimental Study

Abstract

1: Topological Data Analysis protocols applied on an ensemble dataset of a Kelvin-Helmholtz instability. (a) The 180 members of the ensemble obtained with variations of timesteps, interpolation schemes, orders, resolutions and Riemann solvers (Tab. 1). (b)The top cluster represents the time separation of t 0 and t 1 for the flows S 1 and S 2 with the Wasserstein distance and the bottom cluster with the L 2 -norm. Red lines show the timestep separation with our clustering method whereas the sphere colors are the ground truth, illustrating the limitation of the L 2 -norm. (c) Persistence curve protocol: Differences between integrals of persistence curves (gray area) of the enstrophy computed with a SLAU2 solver, an order 7 TENO scheme and a resolution of 1024 × 1024 for various configurations (S 1 at t 0 , S 2 and S 3 at t 1 ). These integral differences exhibit the appearance of vortices (critical points) as the time increases. (d) Outlier distance protocol: Wasserstein distance matrix for 5 configurations S 1 (t 0 , HLLC), S 2 (t 1 , Roe), S 3 (t 1 , HLLC), S 4 (t 2 , Roe), S 5 (t 2 , HLLC) computed with an order 7 WENO-Z interpolation scheme at 512 × 512. The sum of each row the configuration maximizing this distance between solvers and timesteps, here S 1 . (e) Unsupervised classification: Wasserstein distance matrix for the previous configurations with an order 7 WENO-Z interpolation scheme at 256 × 256. The clustering based on the Wasserstein distance and colored according to the Kmeans clustering method successfully segments the time steps.