Three-dimensional velocity gradient statistics in a mesoscale convection laboratory experiment

TL;DR

The study addresses how small-scale velocity-gradient statistics and intermittency behave in Rayleigh–Bénard convection at a mesoscale. It combines spatio-temporally resolved 3D velocity fields obtained via Shake-The-Box Lagrangian particle tracking and VIC# reconstruction in a $\Gamma=25$ cell, with direct numerical simulations using a spectral-element method for validation at $Ra = 1\times10^6$, $Pr = 6.6$. The authors report agreement between experiment and DNS across all nine velocity-gradient components, and reveal non-Gaussian PDFs with increasing tails as $Ra$ rises and near the plates, including left-tail slopes of $3/2$ for dissipation and $1/2$ for enstrophy. These results confirm enhanced small-scale intermittency in RBC and demonstrate the viability of full 3D gradient statistics in mesoscale convection, informing turbulence modeling and extreme-event analyses in geophysical and engineering contexts. The work also points toward higher-$Ra$ experiments and temperature-field coupling to further illuminate the velocity–temperature interplay in turbulent convection.

Abstract

We present three-dimensional velocity gradient statistics from Rayleigh--Bénard convection experiments in a horizontally extended cell of aspect ratio 25, a paradigm for mesoscale convection. The Rayleigh number $Ra$ ranges from $3.7 \times 10^5$ to $4.8 \times 10^6$, and the Prandtl number $Pr$ from 5 to 7.1. Spatio-temporally resolved volumetric data are reconstructed from moderately dense Lagrangian particle tracking measurements. All nine components of the velocity gradient tensor from the experiments show good agreement with those from direct numerical simulations, both conducted at $Ra = 1 \times 10^6$ and $Pr = 6.6$. The focus of our analysis is on non-Gaussian velocity gradient statistics. Specifically, we examine the probability density functions (PDFs) of components of the velocity gradient tensor, vorticity components, kinetic energy dissipation, and local enstrophy at different heights in the bottom half of the cell. The probability of high-amplitude derivatives increases from the bulk to the bottom plate. A similar trend is observed with increasing $Ra$ at fixed height. Both indicate enhanced small-scale intermittency of the velocity field. Furthermore, doubly-logarithmic plots of the PDFs of normalized energy dissipation and local enstrophy at all heights show that the left tails follow slopes of $3/2$ and $1/2$, respectively, in agreement with numerical results. In general, the left tails of the dissipation and local enstrophy distributions show higher probability values with increasing proximity towards the plate, compared to those in the bulk.

Figures (11)

• Figure 1: (a) Schematic of the experimental setup of the Rayleigh-Bénard flow and (b) snapshot showing particles in the measurement volume in the bottom half of the cell at $Ra=4.8\times10^6$. Particles are coloured according to their vertical velocities, with red and yellow indicating positive values, while cyan and blue indicate negative ones. Velocity varies between -6.5 mm s$^{-1}$ and 6 mm s$^{-1}$. The bottom plate is located at $Z=-11$mm. Clearly visible is the turbulent superstructure pattern of the up- and downflows.
• Figure 2: Vortical structures by Q-criterion at a randomly chosen time instance for $Ra=4.8\times10^6$ and $Pr=5$ within the measurement volume. The iso-surfaces of Q within the range of $0.3$ and $0.5$ (cyan-green-yellow-red) are shown.
• Figure 3: Comparison of probability density functions of the nine velocity gradient tensor components and three vorticity components at the mid-plane for $Ra=1.0\times10^6$ and $Pr=6.6$. The quantities shown are $\partial u_x/\partial x$ in (a), $\partial u_y/\partial x$ in (b), $\partial u_z/\partial x$ in (c), $\partial u_x/\partial y$ in (d), $\partial u_y/\partial y$ in (e), $\partial u_z/\partial y$ in (f), $\partial u_x/\partial z$ in (g), $\partial u_y/\partial z$ in (h), $\partial u_z/\partial z$ in (i), $\omega_x$ in (j), $\omega_y$ in (k) and $\omega_z$ in (l). All quantities are normalised by their respective root-mean-square values. Inset figures show the statistical convergences of higher order velocity derivative statistics for the experimental results shown in the main figures. We plot $\chi^n p(\chi)$ versus $\chi$. Here, $n=2$ and 4 for red and blue solid lines, respectively. Note that the y-axis is in logarithmic units.
• Figure 4: Probability density functions of vorticity components at the mid-plane for four different $Ra$. The quantities shown are $\omega_x$ in (a), $\omega_y$ in (b) and $\omega_z$ in (c).
• Figure 5: Probability density functions of velocity gradients for four different planes in $z$-direction at the highest $Ra=4.8\times10^6$. The quantities shown are $\partial u_x/\partial x$ in (a), $\partial u_y/\partial x$ in (b), $\partial u_x/\partial z$ in (c) and , $\partial u_z/\partial z$ in (d). $z=0.48$ (black line), $z=0.21$ (red line), $z=0.14$ (green line), $z=0.10$ (blue line), Gaussian reference (dashed, gray line).