On the concentration distribution in turbulent thermals

TL;DR

This work investigates the internal concentration distribution in turbulent thermals through coordinated laboratory experiments and direct numerical simulations at large Reynolds numbers. By analyzing moments and probability density functions, the authors demonstrate that the concentration field becomes self-similar in time away from the source and that the distribution of c/⟨c⟩ follows an exponential form for intermediate concentrations, largely independent of Re and Sc across the tested ranges. Dimensional analysis predicts the key scalings for bulk properties and moments, predicting ⟨c⟩ ∼ t^{-3/2} and a time-invariant normalized PDF, which the data corroborate except near undiluted cores. Diffusion and Schmidt number have limited impact on the bulk PDF, though cores and fine-scale structures show diffusivity-dependent differences, highlighting a dynamic balance between homogenization and continual entrainment that governs mixing in buoyancy-driven turbulent plumes.

Abstract

Turbulent thermals emerge in a wide variety of geophysical and industrial flows, such as atmospheric cumulus convection and pollutant dispersal in oceans and lakes. When a buoyant fluid mass rises, or sinks, heat and mass transfers occur by the engulfment of the fresh surrounding fluid inside the thermal - a process that spans over multiple scales from macroscopic entrainment of ambient fluid to microscopic diffusive processes. Turbulent thermals are typically investigated through their integral properties (radius, depth, entrainment rate). However, mixing processes depend on the internal distribution of concentration or temperature inside a thermal, which remains poorly constrained. Here, we use laboratory fluid dynamics experiments and direct numerical simulations to investigate the mixing of a passive scalar in turbulent thermals with large Reynolds numbers. We track the evolution of the concentration field, computing its moments and the probability density function. The concentration distribution exhibits self-similarity over time, except at high concentrations, possibly because of the presence of undiluted cores. These distributions are well approximated by an exponential probability density function. Although diffusion has a strong effect on the spatial structure of the concentration field, we observe no significant effect of diffusivity on the concentration distributions in the investigated range of Peclet numbers.

Figures (11)

• Figure 1: Schematic of the experimental setup using laser-induced fluorescence on a turbulent thermal flow.
• Figure 2: (a,b) Evolution of a turbulent thermal for experiments with $Re = 3452$ for an experiment ($Pe=8.64 \times 10^6$) and a simulation $Pe=3452$, respectively. Note that the concentration scale changes between each time step for better visualization. On each image, the time is normalized by $t_g$. (c,d) Close-up snapshots of the last time step for the experiment (a) and simulation (b).
• Figure 3: (a) Snapshot of the concentration field at $t/t_g = 51.25$ for one of the three experiments at $Re = 4175$. Colored lines denote the mask applied to the snapshot to produce the PDF of the concentration in (b, c). (b) and (c) PDF of the concentration as a function of the concentration in log-log or semi-log, respectively. The dashed black line represents the PDF of the whole snapshot. The gray dotted line shows the PDF of the thermal using a loose threshold. The inside of the thermal comprises all pixels inside the cyan and red masks. Brown and cyan lines in (b,c) (and mask in (a)) denote the different parts of the thermal: wake and cores, respectively.
• Figure 4: (a) Snapshot of the concentration field at $t/t_g = 31.6$ for one of the simulations at $Re = 4175$. Colored lines denote the mask applied to the snapshot to produce the PDF of the concentration in (b, c). (b) and (c) PDF of the concentration as a function of the concentration in log-log or semi-log, respectively. The dashed black line represents the PDF of the whole snapshot. The gray dotted line shows the PDF of the thermal using a loose threshold. The inside of the thermal comprises all pixels inside the cyan and red masks. Brown and cyan lines in (b,c) (and mask in (a)) denote the different parts of the thermal: wake and cores, respectively.
• Figure 5: (a) Mean of concentration $c/c_0$ as a function of time $t/t_g$ for the average of three experiments at each $Re$ (solid lines) and the average of two simulations at each $Re$ (dashed lines). (b) Maximum concentration $c/c_0$ as a function of time $t/t_g$. In (a) and (b), the gray and black dotted lines show the power laws $t^{-3/2}$ and $t^{-0.84}$ as in zlla2013. (c) The ratio between the maximum concentration and its mean as a function of time $t/t_g$. The red dashed line shows the ratio of the two previous power laws. Results of zlla2013 (Z2013) span only between 40 and 100 $t/t_g$. Note that all power laws intend to show the slope, i.e., they have an arbitrary pre-factor.