Lagrangian acceleration in Rayleigh-Bénard convection at various aspect ratios

A laboratory investigation using three-dimensional particle tracking velocimetry is performed to uncover the distinctive Lagrangian dynamics of turbulent Rayleigh-Bénard convection at aspect ratios $\mathrm{\Gamma }=1.2$ and 2, where emphasis is placed on the Lagrangian statistics of thermal plumes. A laterally extended investigation volume allowed us to capture the dynamics of the relatively high vertical motion. Approximately $3.5\times {10}^7$ events were analyzed at Rayleigh numbers of $\text{Ra}\approx 1.1\times {10}^{10}$ for $\mathrm{\Gamma }=1.2$ and $\text{Ra}\approx 2.8\times {10}^9$ for $\mathrm{\Gamma }=2$. The probability density functions (PDFs) of the vertical Lagrangian acceleration exhibited heavier tails compared with the lateral counterpart away from the near-wall boundary layer, which evidenced higher Lagrangian intermittency in the vertical direction. The second-order, conditional acceleration PDFs with various vertical-velocity magnitude thresholds indicate that the intense vortical motion promotes stronger vertical acceleration with respect to the lateral. A comparison with previous works reveals the influence of roll dynamics induced by $\mathrm{\Gamma }$ on the distribution of the vertical acceleration. The pair dispersion is explored and showed $t^2$ behavior under the local Kolmogorov timescale (${\tau }_{\eta }$) for both aspect ratios, with some departure at scales sufficiently lower than ${\tau }_{\eta }$.

Figure 1

Basic schematic of the experimental setup illustrating the six-degree-of-freedom traversing system allocating the four-camera participle tracking velocimetry and 80-W laser illumination, as well as heating and cooling systems.

Figure 2

(a) Sample of $\approx 30000$, short-time ($\mathrm{\Delta }t=0.12$ s) trajectories; colors represent vertical velocity. (b) Single trajectory within $\mathrm{\Delta }t\approx 1.5$ s near the center of the cell; colors denote acceleration magnitude.

Figure 3

Mean velocity field of the (a) $x$ component, $u_x$, (b) $x$ component, $u_y$, and (c) $z$ component, $u_z$. (d) Lateral velocity profile along the vertical axis $z/H$ at $x/H=0$. (e) Vertical-velocity component along the lateral axis $x/H$ at $z/H=0$. The solid and dotted lines show PIV measurements by Xia et al. [30] at $\text{Ra}=0.4\times {10}^{10}$ and $\text{Ra}=3.5\times {10}^{10}$.

Figure 4

PDF of the Lagrangian acceleration components. (a) Case with vertical-velocity threshold $\left|\right|u_z{\tau }_{\eta }/\eta \left|\right|<35$ Lagrangian acceleration PDF of low vertical velocity. (b) Entire span of the data. Measurements by Ni et al. [13] in the center of a convective cell and by Liot et al. [14] in the extended vertical span are included for reference. Around $3.5\times {10}^7$ events are included in the analysis.

Figure 5

Second-order moment of the vertical acceleration PDF with various thresholds of the vertical velocity.

Figure 6

Acceleration PDFs of the (a) vertical and (b) lateral components for various aspect ratios. DNS simulation by Schumacher et al. ($\mathrm{\Gamma }=2$ [12] and 4 [19]) are included for reference.

Figure 7

Particle pair dispersion $R^2={\langle {[\mathrm{\Delta }\left(t\right)-\mathrm{\Delta }\left(0\right)]}^2\rangle }_L$ of a sub-Kolmogorov scale timescale in turbulent convection at various aspect ratios. The inset shows the $t^{-2}$-compensated pair dispersion.

Figure 8

PDF of the pair dispersion at various time steps for the (a), (c) vertical and (b), (d) lateral components. (a), (b) Time step in the vicinity of $2\times {10}^{-1}$, and (c), (d) smaller time steps $\lesssim 1\times {10}^{-1}$ (included DNS data from Schumacher [12]).