Lagrangian heat transport in turbulent three-dimensional convection

Spatial regions that do not mix effectively with their surroundings and, thus, contribute less to the heat transport in fully turbulent three-dimensional Rayleigh-Bénard flows are identified by Lagrangian trajectories that stay together for a longer time. These trajectories probe Lagrangian coherent sets (CSs) which we investigate here in direct numerical simulations in convection cells with a square cross section of aspect ratio $\mathrm{\Gamma }=16$, Rayleigh number $\mathrm{Ra}={10}^5$, and Prandtl numbers $\mathrm{Pr}=0.1,0.7$, and 7. The analysis is based on $N=524288$ 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 CSs contribute by a third less to the global turbulent heat transport for all investigated Prandtl numbers 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.

Figure 1

Visualization of Lagrangian coherent sets. (a) The 40 369 Lagrangian tracers which correspond to Lagrangian CSs are shown for $\mathrm{Ra}={10}^5,\mathrm{Pr}=7$, and $\mathrm{\Gamma }=16$ (run 3). The highlighted particles comprise the cluster cores during the chosen time interval $[t_0,t_0+\mathrm{\Delta }t]$ with $\mathrm{\Delta }t=17$, whereas the shown tracer positions correspond to the time $t_0+\mathrm{\Delta }t/2$. Contours of the temperature field, which is averaged over the same time interval, are displayed at three faces. The bottom plane stands for ${\langle T(z=0.05)\rangle }_{\mathrm{\Delta }t}$. (b) A representative subset of 1600 Lagrangian trajectories is shown. They form the different clusters in (a).

Figure 2

A typical eigenspectrum (from top downwards) resulting for the graph Laplacian matrix $\mathit{L}$ in run 2. In order to highlight eigengaps between two subsequent eigenvalues ${\lambda }^{\left(n\right)}$, indicator ${\chi }_n$ is used, see Eq. (8).

Figure 3

Results of the SEBA. (a) Scatter plot of maximum likelihood $m_i$ of tracer ${\mathit{X}}_i(t_0+\mathrm{\Delta }t/2)$ with $i=1,...,N$ to belong to one of the $N_{\mathrm{CS}}=82$ clusters taken from the large value of ${\chi }_{82}$ in Fig. 2. Different clusters result for subsequent thresholding with $\zeta \approx 0.7$ in (b) and $\zeta =0.94$ in (c). Black lines indicate the isotherms ${\langle T(z=0.5)\rangle }_{\mathrm{\Delta }t}=0.5$.

Figure 4

Prandtl number dependence of the PDFs of tracer particles captured in the Lagrangian CSs and the RPs. (a) PDF of the $z$ coordinate of the tracers in CSs and RPs. (b) PDF of the temperature $T$ along the trajectory in CSs and (c) in RPs. The PDFs are computed for ten disjoint time intervals of the evolution and arithmetically averaged subsequently. The threshold value is $\zeta =0.94$.

Figure 5

Quantitative analysis and visualization of the contribution of Lagrangian CSs to heat transfer. (a) PDF of ${\langle {\mathrm{Nu}}_{\mathrm{CS}}^L\rangle }_{\mathrm{\Delta }t}$, (b) PDF of ${\langle {\mathrm{Nu}}_{\mathrm{RP}}^L\rangle }_{\mathrm{\Delta }t}$. The inset in (a) is a one-to-one comparison of the central region of the PDFs for Pr = 0.7. All colors equal that of Fig. 4. (c)–(e) Data in vertical cuts with a slice thickness of 0.2 through the convection layers are shown for (c) Pr =0.1, (d) 0.7, and (e) 7. The corresponding temperature field, that is averaged over $\mathrm{\Delta }t$, is shown as background. Tracers that belong to Lagrangian CSs in this slice are indicated by black dots at time $t_0+\mathrm{\Delta }t/2$. The gray vectors indicate velocity projections on the plane for tracers that belong to RPs. The threshold value is again $\zeta =0.94$.