Probing turbulent superstructures in Rayleigh-Bénard convection by Lagrangian trajectory clusters

Abstract

We analyze large-scale patterns in three-dimensional turbulent convection in\na horizontally extended square convection cell by Lagrangian particle\ntrajectories calculated in direct numerical simulations. A simulation run at a\nPrandtl number Pr $=0.7$, a Rayleigh number Ra $=10^5$, and an aspect ratio\n$\\Gamma=16$ is therefore considered. These large-scale structures, which are\ndenoted as turbulent superstructures of convection, are detected by the\nspectrum of the graph Laplacian matrix. Our investigation, which follows\nHadjighasem {\\it et al.}, Phys. Rev. E {\\bf 93}, 063107 (2016), builds a\nweighted and undirected graph from the trajectory points of Lagrangian\nparticles. Weights at the edges of the graph are determined by a mean dynamical\ndistance between different particle trajectories. It is demonstrated that the\nresulting trajectory clusters, which are obtained by a subsequent $k$-means\nclustering, coincide with the superstructures in the Eulerian frame of\nreference. Furthermore, the characteristic times $\\tau^L$ and lengths\n$\\lambda_U^L$ of the superstructures in the Lagrangian frame of reference agree\nvery well with their Eulerian counterparts, $\\tau$ and $\\lambda_U$,\nrespectively. This trajectory-based clustering is found to work for times\n$t\\lesssim \\tau\\approx\\tau^L$. Longer time periods $t\\gtrsim \\tau^L$ require a\nchange of the analysis method to a density-based trajectory clustering by means\nof time-averaged Lagrangian pseudo-trajectories, which is applied in this\ncontext for the first time. A small coherent subset of the pseudo-trajectories\nis obtained in this way consisting of those Lagrangian particles that are\ntrapped for long times in the core of the superstructure circulation rolls and\nare thus not subject to ongoing turbulent dispersion.\n