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

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 $\mathrm{\Gamma }=16$ 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 et al. [Phys. Rev. E 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 pseudotrajectories, which is applied in this context for the first time. A small coherent subset of the pseudotrajectories 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.

Figure 1

Lagrangian tracer particle distributions in the turbulent convection cell at different times. Tracers with a vertical position of $0\le z<H/3$ are colored in red, $H/3\le z<2H/3$ in cyan, and $2H/3\le z\le H$ in blue. Times in free-fall time units are $t=1.3T_f$ (a), 2.6 $T_f$ (b), 6.5 $T_f$ (c), 10.4 $T_f$ (d), 30.1 $T_f$ (e), and 49.8 $T_f$ (f).

Figure 2

Characteristic Lagrangian times and scales of turbulent superstructures. (a), (b) Typical Lagrangian particle tracks showing the vertical particle coordinate $z\left(t\right)$ for individual tracers no. 2 (a) and no. 2045 (b). The horizontal lines at $z=0.2$ and $0.8H$ in both panels indicate the heights that were used to determine the Lagrangian turnover times. (c) Probability density function (PDF) of Lagrangian turnover time ${\tau }_{\mathrm{to}}^L$. The dashed vertical line indicates the mean Lagrangian turnover time ${\overline{\tau }}_{\mathrm{to}}^L=(19.1\pm 12.8)T_f$. The root mean square (rms) value from this statistics is $23.0T_f$. (d) PDF of the characteristic Lagrangian scale ${\lambda }_U^L$. The vertical line indicates again the mean with ${\overline{\lambda }}_U^L=(2.7\pm 0.9)H$. This stands for the diameter of a pair of rolls. The rms value is $3.2H$.

Figure 3

Lagrangian tracer pair dispersion versus time. Ballistic, Richardson-like intermediate and diffusive scaling are indicated in addition to all relevant times and outer scales in the problem. The total dispersion is shown together with its two contributions, the vertical and horizontal pair dispersions. Times are given in units of the free-fall time $T_f=H/U_f$, pair dispersion in units of $H^2$. Again $\tau \approx {\tau }^L$.

Figure 4

Time evolution of the Lagrangian superstructures. (a)–(c) Time-averaged temperature field in the midplane. Averaging times are $N_t\mathrm{\Delta }t=2.6$, 10.4, and 30.1 free-fall time units, respectively. (c) The characteristic scale ${\lambda }_{\mathrm{\Theta }}$ of the turbulent superstructures is indicated by a double arrow. (d)–(f) Lagrangian trajectory clusters obtained from the leading eigenvectors of the graph Laplacian. Particles that belong to the same spectral cluster at time $N_t\mathrm{\Delta }t$ are colored equally. The background contours are the ridges of the maximum finite-time Lyapunov exponent (12). Ridges and clusters are indicated with respect to the initial Lagrangian particle position. (g)–(i) Corresponding eigenvalue spectra of the graph Laplacian. The spectral gap between eigenvalues no. 8–9, 12–13, and 14–15 is used to detect $k=8$, 12, and 14 trajectory clusters by the $k$-means clustering algorithm, respectively. The cutoff parameter is $\epsilon =0.75$.

Figure 5

Perspective view of the whole Lagrangian tracer ensemble at times $t=0T_f$ (a), 10.4 $T_f$ (b), 30.1 $T_f$ (c), and 69.4 $T_f$ (d). The particles are colored with respect to their trajectory cluster membership as computed for the case $N_t\mathrm{\Delta }t=30.1T_f$ [cf. Fig. 4].

Figure 6

Time evolution of four individual trajectories (in blue) and their corresponding pseudotrajectories (in red) plotted for the time interval $t/T_f\in$ [0, 80]. The asterisks at the pseudotrajectories indicate the positions $\overline{\mathit{X}}\left(n\widehat{\tau }\right)$ for $n$ integer. Two trajectories are displayed in a three-dimensional perspective (a) and in two projections (c), (e). The same sequence follows for another set of two trajectories in panel (b) and panels (d) and (f), respectively.

Figure 7

Time evolution of the Lagrangian pseudotrajectories for total integration times $T_{\mathrm{end}}=40T_f\approx 2{\tau }^L/3$ (a), (f), $60T_f\approx {\tau }^L$ (b), (g), $80T_f\approx 4{\tau }^L/3$ (c), (h), $100T_f\approx 5{\tau }^L/3$ (d), (i), and $120T_f\approx 2{\tau }^L$ (e), (j). The panels in the upper row show the midpoints of the trajectories that belong to the clusters as given in Table 1. The panels in the bottom row show noisy trajectories.