Direct numerical simulation of turbulence and microphysics in the Pi Chamber

The Pi Chamber is a cloud chamber at Michigan Technological University that utilizes moist turbulent Rayleigh-Bénard flow between two temperature-controlled, saturated plates to create cloud conditions in a controlled laboratory setting. This experimental apparatus has been the source of numerous scientific studies but also offers an advantageous platform with which to test numerical modeling approaches. In this study, the primary goal is to use direct numerical simulation (DNS) with Lagrangian aerosol/droplet microphysics to recreate, as realistically as possible, the conditions inside the Pi Chamber. The biggest discrepancies between the DNS and laboratory setups are the Rayleigh number ($\mathrm{Ra}=7.9\times {10}^6$ in the DNS) and the use of periodic lateral boundary conditions. Nonetheless, numerical experiments are conducted for two published Pi Chamber cases: steady aerosol injection and the resulting statistically steady-state cloud and transient conditions when aerosol injection is shut off. Generally speaking, the DNS is able to capture many of the salient features observed in the Pi Chamber experiments, both qualitatively and quantitatively, including microphysical details and influences on the fluctuating ambient saturation in the chamber. From the DNS, Lagrangian statistics are interrogated which are otherwise inaccessible from the experimental view. In particular, the supersaturation fluctuations seen by droplets are observed to deviate from a Gaussian distribution—a common assumption in stochastic modeling—and the probability distribution of droplet lifetime does not adhere to the expected behavior assuming solid particles settling in a quiescent medium.

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Figure 1

Representative snapshot of Pi Chamber DNS. Colored isosurfaces are the $q_v=12.5\mathrm{g}{\mathrm{kg}}^{-1}$ and $q_v=17.5\mathrm{g}{\mathrm{kg}}^{-1}$ surfaces, and black dots represent the computational droplets.

Figure 2

(a) Volume-averaged relative humidity $\langle \mathrm{RH}\rangle$ and (b) standard deviation ${\sigma }_{\mathrm{RH}}^V$ of volume-averaged relative humidity as a function of time for all aerosol injection rates.

Figure 3

Temporally and horizontally averaged profiles of (a) relative humidity $\overline{\mathrm{RH}}$, (b) its standard deviation ${\sigma }_{\mathrm{RH}}$, (c) water vapor mixing ratio ${\overline{q}}_v$, and (d) temperature $\overline{T}$ as a function of height across the domain. Line colors are the same as in Fig. 2.

Figure 4

PDFs of supersaturation fluctuation $s^{\prime }$, as defined by deviation from horizontal mean. Panels (a)–(f) refer to the increasing injection rates, starting from the unladen case in panel (a). Solid lines: PDF computed from the flow at the domain centerplane at multiple times; dashed lines: Gaussian fit to the centerplane PDF; circles: PDF constructed from the local values of $s^{\prime }$ taken along at least ${10}^4$ droplet trajectories.

Figure 5

(a) Time series of number concentrations of aerosols (dotted), droplets (dashed), and total (solid), as defined by the activation state. (b) Steady-state aerosol number concentration $n_a$ as a function of injection rate. For the highest two aerosol injection rates, $n_a$ is the last recorded value. (c) Steady-state droplet number concentration $n_d$ as a function of injection rate. Hollow symbols are from Chandrakar et al. [7], and colors refer to aerosol injection rates [see legend of Fig. 4].

Figure 6

[(a)–(c)] The droplet mean diameter $\overline{d}$, standard deviation ${\sigma }_d$, and relative dispersion ${\sigma }_d/\overline{d}$ as a function of time for the various injection rates. [(d)–(f)] The steady-state $\overline{d}$, ${\sigma }_d$, and ${\sigma }_d/\overline{d}$ versus $\dot{n}$. Hollow symbols are the experimental measurements from Chandrakar et al. [7].

Figure 7

(a) Total number concentration ${\overline{n}}_T$ (droplets and aerosols), horizontally and temporally averaged, as a function of height; (b) temporal and horizontal mean diameter as a function of height.

Figure 8

Droplet size distributions for all injection rates. The large spike centered at $d<1\mu \mathrm{m}$ represents unactivated aerosols.

Figure 9

(a) PDF of total aerosol residence time ${\tau }_l$, from injection until removal from the domain; (b) PDF of activation times ${\tau }_{l,\mathrm{act}}$, from when a droplet is first activated to when it either deactivates or is removed from the domain; (c) PDF only of times between activation and removal ${\tau }_{l,\mathrm{act}\text{−}\mathrm{sed}}$; (d) PDF of $N_{\mathrm{act}}$, the number of times aerosols have been activated during their lifetime in the domain.

Figure 10

(a) Mean droplet lifetime $\langle {\tau }_l\rangle$ as a function of mean droplet diameter $\langle d\rangle$ for each of the injection rates $\dot{n}$. (b) Same, but for the time from activation until removal. The dashed black line is a power-law fit $\langle {\tau }_l\rangle \propto {\langle d\rangle }^{\lambda }$ for a fitted value of $\lambda$ given in the legends, and the dotted black line is the estimate given by ${\tau }_{\mathrm{Stokes}}$ for each droplet diameter.

Figure 11

In the center is the PDF of total lifetime ${\tau }_l$ for the $\dot{n}=1{\mathrm{cm}}^{-3}{\min }^{-1}$ case from Fig. 9. For each of the three lifetime ranges indicated by the arrows, sample trajectories are given showing the droplet height, local relative humidity, and diameter. The nonmonotonic PDF of ${\tau }_l$ at small lifetimes is clearly due to the periodic circulations which exist in Rayleigh-Bénard turbulence.

Figure 12

Transient response of (a) the volume-mean RH and (b) relative humidity variance ${\sigma }_{\mathrm{RH}}^2$ as a function of time after stopping aerosol injection.

Figure 13

Transient response of (a) number concentrations of droplets, aerosols, and their sum. (b) The transient Damköhler number Da based on the current droplet number density and mean radius (dashed line is $\mathrm{Da}=1$); (c) mean droplet diameter; and (d) standard deviation of droplet diameter ${\sigma }_d$, as a function of time after stopping aerosol injection.

Figure 14

The cloud DSD as a function of time for each of the transient cases. Colors represent the logarithm of the probability and range between $-5$ to $-1$. The color scale is the same in all panels.