Lagrangian heat transport in turbulent three-dimensional convection

TL;DR

This study identifies Lagrangian coherent sets in turbulent convection flows, revealing that these regions contribute significantly less to heat transport, using advanced clustering methods on large-scale numerical simulations.

Contribution

The paper introduces a novel combination of graph Laplacian and SEBA for improved detection of Lagrangian coherent sets in turbulent convection.

Findings

Coherent sets contribute one-third less to heat transport.

Method improves cluster detection compared to traditional approaches.

Lagrangian analysis correlates with Eulerian flow patterns.

Abstract

Spatial regions that do not mix effectively with their surroundings and thus contribute less to the heat transport in fully turbulent three-dimensional Rayleigh-B\'{e}nard flows are identified by Lagrangian trajectories that stay together for a longer time. These trajectories probe Lagrangian coherent sets (CS) which we investigate here in direct numerical simulations in convection cells with square cross section of aspect ratio $\Gamma = 16$, Rayleigh number $Ra = 10^{5}$, and Prandtl numbers $Pr = 0.1, 0.7$ and $7$. The analysis is based on $N=524,288$ Lagrangian tracer particles which are advected in the time-dependent flow. Clusters of trajectories are identified by a graph Laplacian with a diffusion kernel, which quantifies the connectivity of trajectory segments, and a subsequent sparse eigenbasis approximation (SEBA) for cluster detection. The combination of graph Laplacian and SEBA leads to a significantly improved cluster identification that is compared with the large-scale patterns in the Eulerian frame of reference. We show that the detected CS contribute by a third less to the global turbulent heat transport for all investigated $Pr$ compared to the trajectories in the spatial complement. This is realized by monitoring Nusselt numbers along the tracer trajectory ensembles, a dimensionless local measure of heat transfer.