Lagrangian velocity and acceleration measurements in plume-rich regions of turbulent Rayleigh-Bénard convection

We report an experimental study of Lagrangian velocity and acceleration in turbulent Rayleigh-Bénard convection using particle tracking velocimetry, with the Rayleigh number Ra spanning from $5.4\times {10}^8$ to $1.3\times {10}^{10}$. The measurements were made in two representative regions of a cylindrical convection cell with aspect ratio unity, where an abundant amount of thermal plumes are passing through constantly. The results are compared with those obtained in the cell's central region, where plumes are passing through much less frequently. It is found that the probability density functions (pdf's) of the three velocity components almost collapse with each other and follow Gaussian distribution for all the regions in the high Ra range, but they behave differently and deviate from the Gaussian function for lower Ra numbers. For the acceleration, the pdf's in all the regions exhibit a stretched exponential form, but for lower Ra cases the amount of stretching is much more pronounced in the plume-abundant regions as compared to that in the cell center. This difference is more evident in terms of acceleration variance: for $\mathrm{Ra}\lesssim 4.3\times {10}^9$, the acceleration variances in the plume-abundant regions are larger than those in the central region and show a different Ra-dependent power law. As Ra number increases, the acceleration variances obtained in different regions gradually merge into a single master curve that follows the Heisenberg-Yaglom prediction for homogeneous and isotropic turbulence. A consistent transitional behavior is also observed in the kinetic energy dissipation rate. Through a detailed examination of the possible balance relations between acceleration and other small-scale properties, our results show that the acceleration in the plume-abundant regions is balanced with a combination of thermal and kinetic energy dissipation rates, which suggests that the turbulent flow in these regions is governed by a mixed dynamics with contributions from both thermal plumes and turbulent background fluctuations. This picture is supported by a modified Heisenberg-Yaglom relation and also by the scaling behaviors of the Eulerian structure functions in the inertial range. The observed transitional behaviors can be understood as a result of the evolution in the circulation path of the large-scale flow, which changes from an ellipse to a more squarish shape with increasing Ra.

Figure 1

(a) A typical shadowgraph image of turbulent RB convection measured at $\mathrm{Ra}\simeq 1\times {10}^{10}$ and $\mathrm{Pr}=4.3$. The squared boxes indicate the two measurement regions with abundant plumes passing through. (b) and (c): Sketches of the experimental setups for particle tracking measurements in the bottom and sidewall regions, respectively. The light green indicates the laser beam and the dark green indicates the measurement volume.

Figure 2

The probability density functions (pdf's) of the three normalized velocity components measured in different regions and at three representative Ra numbers, where $k=x,y$, and $z$. From top to bottom: Center region [(a), (b), (c)], sidewall region [(d), (e), (f)], and bottom region [(g), (h), (i)]. Left panel: $\mathrm{Ra}\simeq 6\times {10}^8$; middle panel: $\mathrm{Ra}\simeq 1.3\times {10}^9$; right panel: $\mathrm{Ra}\simeq 1\times {10}^{10}$. For all the pdf's, red circle, blue triangle, and black square represent the $x,y$, and $z$ components, respectively.

Figure 3

The Reynolds number ${\mathrm{Re}}_{\mathrm{rms}}$ measured in different regions as a function of the Ra number. The data are calculated from the rms value of the Lagrangian velocity and compensated by a ${\mathrm{Pr}}^{-2/3}$ dependence according to the model proposed by Grossmann and Lohse [27, 28, 29, 30].

Figure 4

The probability density functions (pdf's) of the three normalized acceleration components measured in different regions and at three representative Ra numbers, where $k=x,y$, and $z$. From top to bottom: Center region [(a), (b), (c)], sidewall region [(d), (e), (f)], and bottom region [(g), (h), (i)]. Left panel: $\mathrm{Ra}\simeq 6\times {10}^8$; middle panel: $\mathrm{Ra}\simeq 1.3\times {10}^9$; right panel: $\mathrm{Ra}\simeq 1\times {10}^{10}$. For all the pdf's, red circle, blue triangle, and black square represent the $x,y$, and $z$ components, respectively.

Figure 5

The vertical acceleration variance as a function of Ra measured in different regions. The open stars are taken from another experimental study [58] for comparison, which were measured in the plume-abundant region (pink star) and the center region (magenta star) of a similar convection cell, respectively. Black solid line: $\langle a_z^2\rangle \sim {\mathrm{Ra}}^{1.54\pm 0.08}$ for the data measured in the plume-abundant regions with $\mathrm{Ra}\lesssim 4.3\times {10}^9$. Dark-yellow solid line: $\langle a_z^2\rangle \sim {\mathrm{Ra}}^{2.55\pm 0.11}$ for the data measured in the center region. The blue dotted line and red dash-dotted line are the results calculated from ${\langle a^2\rangle }_{\mathrm{buoy}}=\langle {\left(\alpha g\delta T\right)}^2\rangle$ for the bottom and sidewall regions, respectively. See text for the explanation.

Figure 6

The Ra dependence of the kinetic energy dissipation rates measured in different regions. The data are compensated by ${\mathrm{Pr}}^{1.15}$ according to [77]. Black solid line: $\langle {\epsilon }_u\rangle \sim {\mathrm{Ra}}^{1.34\pm 0.07}$ for the data measured in the plume-abundant regions with $\mathrm{Ra}\lesssim 4.3\times {10}^9$. Dark-yellow dashed line: $\langle {\epsilon }_u\rangle \sim {\mathrm{Ra}}^{1.55\pm 0.02}$ for the data measured in the cell center. Inset: Examples of the longitudinal (solid inverted triangle) and transverse (open upright triangle) second-order Eulerian structure functions compensated by the dissipative range scaling $r^2$ and $2r^2$, respectively. They are measured in the plume-abundant regions for $\mathrm{Ra}\simeq 1.9\times {10}^9$ (lower data set) and $\mathrm{Ra}\simeq 5.6\times {10}^9$ (upper data set). The blue horizontal lines indicate the plateau heights that are used to calculate the kinetic energy dissipation rates.

Figure 7

The compensated second-order longitudinal Eulerian structure functions measured in the plume-abundant region (black square) and the center region (red cross). Left panel: Compensated by $r^{2/3}$; right panel: compensated by $r^{4/5}$. Top panel: $\mathrm{Ra}\simeq 1.9\times {10}^9$; bottom panel: $\mathrm{Ra}\simeq 7.0\times {10}^9$. The blue horizontal lines are used to guide the eye.