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

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.

💡 Research Summary

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This paper presents a comprehensive experimental investigation of three‑dimensional velocity‑gradient statistics in Rayleigh–Bénard convection (RBC) carried out in a horizontally extended cell with an aspect ratio of Γ = 25, a configuration that mimics mesoscale convection in the atmosphere, mantle, and oceans. The experiments cover Rayleigh numbers Ra ranging from 3.7 × 10⁵ to 4.8 × 10⁶ and Prandtl numbers Pr between 5 and 7.1, using water as the working fluid. By varying the temperature difference ΔT between the heated bottom plate and the cooled top plate, the authors systematically explore how increasing buoyancy forcing influences small‑scale turbulence.

A key methodological advance is the use of moderately dense Lagrangian particle tracking (LPT) combined with the “Shake the Box” (STB) reconstruction algorithm and the VIC# volume‑reconstruction technique. Neutral‑buoyancy polyamide particles (55 µm diameter) are seeded in the flow, illuminated by a pulsed white‑light LED, and recorded simultaneously by four high‑resolution sCMOS cameras positioned at ≈35° to the vertical. The resulting particle trajectories are processed to obtain a fully three‑dimensional, time‑resolved velocity field whose spatial resolution reaches the Kolmogorov length scale η. This enables direct computation of all nine components of the velocity‑gradient tensor ∂ui/∂xj throughout the measurement volume.

The experimental velocity‑gradient data are benchmarked against direct numerical simulations (DNS) performed at Ra = 1 × 10⁶ and Pr = 6.6. PDFs, low‑order moments, skewness, and flatness of each gradient component show excellent agreement between experiment and DNS, confirming that the LPT‑based reconstruction faithfully captures the fine‑scale structure of turbulent convection. The authors also demonstrate statistical convergence up to the sixth‑order moment, a non‑trivial achievement given the intermittent nature of the flow.

The main physical findings concern the non‑Gaussian statistics of velocity gradients, vorticity components, kinetic‑energy dissipation ε, and local enstrophy Ω, examined at several heights within the lower half of the cell. The probability of extreme (high‑amplitude) gradients increases markedly when moving from the bulk toward the heated bottom plate, and the same trend is observed when Ra is increased at a fixed height. This reflects an enhancement of small‑scale intermittency near the boundary and under stronger buoyancy forcing. The PDFs of the horizontal‑derivative components (∂ui/∂x, ∂ui/∂y) are more symmetric than those involving the vertical derivative (∂ui/∂z), indicating persistent anisotropy close to the plate.

A particularly novel result is the observation of the left‑tail scaling of the normalized dissipation and enstrophy PDFs. In doubly‑logarithmic plots, the low‑amplitude side of the ε/⟨ε⟩ distribution follows a power‑law with exponent 3/2, while the Ω/⟨Ω⟩ distribution follows an exponent 1/2. These exponents were previously reported only in high‑resolution DNS of homogeneous isotropic turbulence (Gotoh & Yang 2022). Their experimental verification demonstrates that the theoretical predictions for the left‑tail behavior, which is associated with relatively frequent, low‑intensity events, hold also in wall‑bounded, buoyancy‑driven turbulence.

The study further discusses how the large‑scale superstructures (alternating up‑ and down‑flows) characteristic of high‑aspect‑ratio convection cells modulate the small‑scale statistics. Near the bottom plate, the large‑scale circulation (LSC) exhibits stronger fluctuations, leading to a higher incidence of intense dissipation and enstrophy events. In contrast, the bulk region displays more homogeneous statistics. The authors also note that increasing Ra reduces anisotropy in the horizontal plane, moving the flow toward isotropy, yet the vertical direction remains distinct due to the presence of the thermal boundary layer.

Finally, the paper situates its contributions within the broader context of experimental turbulence measurement techniques. Traditional single‑point methods such as Laser Doppler Velocimetry (LDV) or hot‑wire anemometry rely on Taylor’s frozen‑turbulence hypothesis and suffer from spatial resolution limits. Two‑dimensional PIV can estimate pseudo‑dissipation but cannot capture all nine gradient components. The presented 3D LPT approach overcomes these limitations, providing non‑intrusive, high‑resolution access to the full velocity‑gradient tensor and derived quantities.

In summary, the authors deliver the first laboratory confirmation that the left‑tail scaling laws (3/2 for dissipation, 1/2 for enstrophy) derived from DNS also manifest in real turbulent convection. They show that small‑scale intermittency intensifies both with increasing Rayleigh number and proximity to the heated plate, and they validate the fidelity of Lagrangian particle‑tracking reconstructions against DNS. These results furnish valuable benchmark data for turbulence modeling, especially for geophysical and engineering applications involving high‑aspect‑ratio buoyancy‑driven flows.