Probing turbulent superstructures in Rayleigh-B\'{e}nard convection by Lagrangian trajectory clusters

TL;DR

This paper uses Lagrangian trajectory clustering to identify and analyze large-scale turbulent superstructures in Rayleigh-Bénard convection, demonstrating the method's effectiveness and consistency with Eulerian measures.

Contribution

It introduces a trajectory clustering approach based on graph Laplacian spectra to detect turbulent superstructures, extending analysis to longer times with density-based clustering.

Findings

Trajectory clusters match Eulerian superstructures.

Lagrangian and Eulerian characteristic times and lengths agree.

Longer times require density-based clustering for coherent structures.

Abstract

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