Lagrangian coherent sets in turbulent Rayleigh-Bénard convection

Coherent circulation rolls and their relevance for the turbulent heat transfer in a two-dimensional Rayleigh-Bénard convection model are analyzed. The flow is in a closed cell of aspect ratio four at a Rayleigh number $\mathrm{Ra}={10}^6$ and at a Prandtl number $\mathrm{Pr}=10$. Three different Lagrangian analysis techniques based on graph Laplacians (distance spectral trajectory clustering, time-averaged diffusion maps, and finite-element based dynamic Laplacian discretization) are used to monitor the turbulent fields along trajectories of massless Lagrangian particles in the evolving turbulent convection flow. The three methods are compared to each other and the obtained coherent sets are related to results from an analysis in the Eulerian frame of reference. We show that the results of these methods agree with each other and that Lagrangian and Eulerian coherent sets form basically a disjoint union of the flow domain. Additionally, a windowed time averaging of variable interval length is performed to study the degree of coherence as a function of this additional coarse graining which removes small-scale fluctuations that cause trajectories to disperse quickly. Finally, the coherent set framework is extended to study heat transport.

Figure 1

Two-dimensional RBC benchmark flow with Rayleigh number $\mathrm{Ra}={10}^6$ and Prandtl number $\mathrm{Pr}=10$. Top: instantaneous flow configuration exhibiting a pair of counter-rotating circulation rolls. Temperature contours are shown as colors and the velocity field is indicated by arrows. Bottom: time-averaged flow configuration. Averaging is performed for the total duration of the time integration which is 500 free-fall time units.

Figure 2

Finite-time Lyapunov exponent field computed in forward time over $20t_f$ and $200t_f$ in the top and bottom panels, respectively. Large values are indicated by yellow. Superimposed are particles (white dots) at initial positions ($t=2000t_f$) which belong to two coherent sets as identified by the method described in Sec. 4a (see also Figs. 4 and 7).

Figure 3

Eigenvalues of $2{\varepsilon }^{-1}{Q}^{\mathrm{nw}}$ (circles), $Q^{\mathrm{dm}}$ (crosses), and $(\overline{K},M)$ (diamonds). The leading 10 eigenvalues of the matrices of each approach for the short time interval setting are shown.

Figure 4

Clustering with respect to the second and fifth eigenvectors of $Q^{\mathrm{nw}}$(top) and of $Q^{\mathrm{dm}}$ (center) and the second and third eigenvectors of $(\overline{K},M)$ (bottom). Particles in different coherent sets at time $t=2010t_f$ are colored according to $k$-means clustering.

Figure 5

Particles colored according to eigenvectors of $Q^{\mathrm{nw}}$ (a), (b), $Q^{\mathrm{dm}}$ (c), (d), and $(\overline{K},M)$ (e), (f) in the normalized range $[-1,1]$. Negative (positive) values are indicated by dark (bright) contours. The second and fifth (respectively third) eigenvectors at $t=2010t_f$.

Figure 6

Eigenvalues of $2{\varepsilon }^{-1}{Q}^{\mathrm{nw}}$ (circles), $Q^{\mathrm{dm}}$ (crosses), and $(\overline{K},M)$ (diamonds). Leading 10 eigenvalues of the relevant matrices of each approach for the long time interval setting.

Figure 7

Clustering with respect to the second eigenvector of $Q^{\mathrm{nw}}$ (top), $Q^{\mathrm{dm}}$ (center), and $(\overline{K},M)$ (bottom). Particles in different coherent sets at time $t=2100t_f$ are colored according to $k$-means clustering.

Figure 8

Particles colored according to eigenvectors of $Q^{\mathrm{nw}}$ (a), (b), $Q^{\mathrm{dm}}$ (c), (d), and $(\overline{K},M)$ (e), (f) in the normalized range $[-1,1]$. Negative (positive) values are indicated by dark (bright) contours. The specific eigenvector is indicated at the top of each panel. The first nontrivial eigenvectors at $t=2100t_f$.

Figure 9

Symmetrical difference of the clusters obtained from $Q^{\mathrm{nw}}$ and $Q^{\mathrm{dm}}$ at $t=2010t_f$ (top) and at $t=2100t_f$ (bottom).

Figure 10

Magnitude of the time-averaged advecting velocity field in the plane as a filled contour plot. The averaging time is for 2000 $t_f$. The largest magnitudes are bright contours, the smallest ones are dark contours.

Figure 11

Effect of time averaging in the Lagrangian analysis. Leading 10 eigenvalues of $Q^{\mathrm{nw}}$ of the original trajectories and time-averaged trajectories with window sizes $\mathrm{\Delta }t=5t_f$ and $14t_f$ for a time interval of $200.0t_f$.

Figure 12

Comparison of trajectories for the original and time-averaged Lagrangian analysis. Top: trajectories of six random particles for a time interval of $200.0t_f$. Middle (bottom): six time-averaged trajectories for $\mathrm{\Delta }t=5t_f$ ($\mathrm{\Delta }t=14t_f$).

Figure 13

Short-term heat-coherence analysis in two-dimensional Rayleigh-Bénard flow. Second eigenvector (top) and third eigenvector (bottom) are shown. Negative (positive) values are indicated by dark (bright) colors. Eigenvectors of $Q^{\mathrm{dm}}$ at the initial time.

Figure 14

Same as Fig. 13, but for the final time of the short time interval.

Figure 15

Temperature contour grayscale plot in relation to the heat-coherence evaluation. Initial slice (top) at $t=2000t_f$ and final slice (bottom) at $t=2010t_f$. Dimensionless temperature varies between 0 (bright) and 1(dark).

Figure 16

Long-term heat-coherence analysis in two-dimensional Rayleigh-Bénard flow. Second eigenvector (top) and third eigenvector (top) are shown. Negative (positive) values are indicated by dark (bright) colors.