# Direct Numerical Simulation of a Warm Cloud Top Model Interface: Impact of the Transient Mixing on Different Droplet Population

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Methodology and Simulation Setup

#### 2.1. Evolutive Equations for the Fluid Flow

#### 2.2. Lagrangian Equations for the Cloud Droplets

#### 2.3. Numerical Method of the DNS

#### 2.4. Simulation Setup

**R25**,

**R18**and

**R6**; whereas, the carrier airflow initial conditions remained the same for all these simulations. A sketch of the simulation domain is shown in Figure 1a. Gravitational force (as presented in Figure 1a) acts on both the fluid flow (in form of buoyancy forces B in Equation (2)) and on the momentum of the cloud droplets in Equation (8). In this cloud top simulation setup, gravity therefore acts in downwards direction, causing heavier droplets to settle down towards the bottom boundary of the cloudy volume.

#### 2.4.1. Initial Setup for the Fluid Flow

#### 2.4.2. Initial Setup for the Cloud Droplets

## 3. Results of Transient Evolution

#### 3.1. Time Evolution of the Fluid Flow

#### 3.2. Time Evolution of the Cloud Droplet Population

**a**), (

**c**), (

**e**) of Figure 10) after 3 initial eddy ($t/{\tau}_{0}=3$) turnover time is presented. These droplets were transported to the subsaturated clear air region due to detrainment from the near interface region of the cloudy part of the domain. Due to subsaturation, only the droplets from simulations with initial 25 and 18 $\mathsf{\mu}$m droplet populations are observed to survive the entire simulation. The impact of gravitational settlement is observed to be very pronounced for the larger droplet population, leading to a short residence time in the subsaturated area (two out of the three droplets were back to the cloudy supersaturated region of the domain almost immediately, see Figure 10a). Whereas, the other remaining droplet was trapped in some eddy to follow lateral movement inside the clear air region. In Figure 10b, these droplets are observed not to follow the fluid velocity exactly but rather show negative ${v}_{3}$ indicating stronger influences of gravitational forces on these droplets. The sample droplets from the simulation with initial 18 $\mathsf{\mu}$m droplet population shows comparatively less influence under gravitational forces and remains entrapped in the eddies inside the clear air region of the domain (Figure 10c), which produces a continuous size reduction due to local subsaturation (Figure 10d). Whereas, local subsaturation played most important impact on the samples of the droplets from the simulation with initial 6 $\mathsf{\mu}$m droplet population. After being detrained to the subsaturated clear air region, these droplets cloud not return back to the saturated cloudy part of the domain due to decay in TKE inside the domain (Figure 10e) and eventually evaporated completely in the middle of the simulation duration (Figure 10f).

## 4. Discussions and Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Simulation setup: (

**a**) scheme of the three dimensional simulation domain and the boundary conditions; (

**b**) comparison between the TKE spectrum $E\left(k\right)$ of present simulation with the infield measurements; (

**c**) initial distribution of the rms of velocity fluctuations ${u}^{\prime}$ and TKE dissipation rate $\langle \epsilon \rangle $; and (

**d**) simulated initial profile of temperature $\langle T\rangle $ and water vapour density $\langle {\rho}_{v}\rangle $.

**Figure 2.**Transient evolution of the flow inside the clear air and cloudy region: (

**a**) decay of TKE E and dissipation rate $\epsilon $; (

**b**) Taylor micro-scale Reynolds number $R{e}_{\lambda}$; (

**c**) longitudinal and transversal integral length scales L; and (

**d**) average cloud droplet Stokes number $St$ and settling parameter ${S}_{v}$ of the different droplet populations. Plots (

**a**–

**c**) are from simulation

**R25**. Differences are small among the three simulations.

**Figure 3.**Transient evolution of flow statistics: (

**a**) TKE $\langle E\rangle $; (

**b**) kurtosis of vertical velocity fluctuations $K\left({u}_{3}^{\prime}\right)$; (

**c**) mean water vapour density $\langle {\rho}_{v}\rangle $; (

**d**) mean temperature $\langle T\rangle $; (

**e**) variance of water vapour fluctuations $\langle {\rho}_{v}^{\prime 2}\rangle $; (

**f**) variance of temperature fluctuations $\langle {T}^{\prime 2}\rangle $; (

**g**) skewness of water vapour fluctuations $S\left({\rho}_{v}^{\prime}\right)$; and (

**h**) skewness of temperature fluctuations $S\left({T}^{\prime}\right)$ evolution. All plots present horizontal plane averaged quantities as in Figure 1c,d. The presented data are from simulation

**R25**. Differences are small among the three simulations.

**Figure 4.**Transient evolution of supersaturation: (

**a**) mean of supersaturation $\langle S\rangle $; (

**b**) variance $\langle {S}^{{}^{\prime}2}\rangle $; (

**c**) skewness $S\left({\mathit{S}}^{\prime}\right)$; and (

**d**) kurtosis $K\left({\mathit{S}}^{\prime}\right)$ of supersaturation fluctuations. All plots present horizontal plane averaged quantities.

**Figure 5.**Transient evolution of one dimensional horizontal spectra: (

**a**,

**b**) transversal spectra of ${u}_{3}^{\prime}$; (

**c**,

**d**) spectra of water vapour fluctuations ${\rho}_{v}^{\prime}$; and (

**e**,

**f**) temperature fluctuations ${T}^{\prime}$ respectively at cloud core and at the interface.

**Figure 6.**Visualization: Enstrophy E field across a vertical plane (plane $({x}_{3},{x}_{1})$) is presented in superposition with the supersaturation S field in contour lines and cloud droplets around that plane (thickness of droplets containing slice is 0.0025 m) after 6 initial eddy ($t/{\tau}_{0}=6$) turnover time. Colourbar represents magnitude of the enstrophy in the flow field, red and yellow contour lines represent saturated ($S=0$) and subsaturated ($S=-0.2$) conditions respectively. The sizes of the droplets are proportional to $r/{r}_{in}$ for each population. The panels show the simulations with the droplet populations of (

**a**) 25 $\mathsf{\mu}$m; (

**b**) 18 $\mathsf{\mu}$m; and (

**c**) 6 $\mathsf{\mu}$m initial radius.

**Figure 7.**Probability density functions (PDFs): PDFs of (

**a**–

**c**) cloud droplet radius r; (

**d**–

**f**) droplet surface area growth rate $\mathrm{d}{r}^{2}/\mathrm{d}t$; and (

**g**–

**i**) vertical component of droplet velocity ${v}_{3}$ for 25 $\mathsf{\mu}$m, 18 $\mathsf{\mu}$m, and 6 $\mathsf{\mu}$m initial droplet size populations respectively.

**Figure 8.**Transient evolution in normalized number density of droplets: (

**a**) number density of total number of cloud droplets ${N}_{d}\left(t\right)$; (

**b**) number density of droplets went through condensational growth ${N}_{cond}\left(t\right)$; (

**c**) number density of droplets undergone evaporative size reduction ${N}_{evap}\left(t\right)$; and (

**d**) number density of droplets with one collision ${N}_{coll}\left(t\right)$ as observed for 25 $\mathsf{\mu}$m, 18 $\mathsf{\mu}$m and 6 $\mathsf{\mu}$m initial droplet size populations. No collisional growth is observed for 6 $\mathsf{\mu}$m initial droplet populations.

**Figure 9.**Correlation between fluid and droplet: (

**a**–

**c**) correlation between vertical component of fluid velocity fluctuations ${u}_{3}^{\prime}$ and droplet velocity fluctuations ${v}_{3}^{\prime}$; and (

**d**–

**f**) between supersaturation fluctuations ${S}^{\prime}$ and fluctuations in liquid water content $lw{c}^{\prime}$. Vertical panels from the left to right present correlation diagrams for 25 $\mathsf{\mu}$m, 18 $\mathsf{\mu}$m and 6 $\mathsf{\mu}$m initial droplet size populations respectively.

**Figure 10.**Lagrangian trajectories: A sample of few individual cloud droplet trajectories. (

**a**,

**c**,

**e**) visualization of the time evolution of droplet positions; and (

**b**,

**d**,

**f**) Lagrangian history of the vertical component of droplet velocity ${v}_{3}$, fluid velocity at that droplet position ${u}_{3}$ and their normalized droplet radius $r/{r}_{in}$ up to the end of simulation duration ($t/{\tau}_{0}=18$) or to the end of the droplet life till being completely evaporated (panels

**e**and

**f**) are presented. Colourbar for the left panels represents droplet radius growth rate $\mathrm{d}r/\mathrm{d}t$, thus indicating the droplet positions where condensation or evaporation occur. Size of the droplets are proportional to normalized droplet radius $r/{r}_{in}$.

Quantity | Symbol | Value | Unit |
---|---|---|---|

Reference temperature | ${T}_{0}$ | $283.16$ | $\mathrm{K}$ |

Reference atmospheric pressure | ${p}_{0}$ | $92.4$ | $\mathrm{kPa}$ |

Reference air density | ${\rho}_{0}$ | $1.13$ | $\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ |

Reference kinematic viscosity | $\nu $ | $1.56\times {10}^{-5}$ | ${\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

Gravitational acceleration | g | $9.8$ | $\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-2}$ |

Thermal conductivity of the air | ${\lambda}_{T}$ | $2.5\times {10}^{-2}$ | $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

Thermal diffusivity of air | $\kappa $ | $2.2\times {10}^{-5}$ | ${\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

Diffusivity of water vapour | ${\kappa}_{v}$ | $2.54\times {10}^{-5}$ | ${\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

Specific heat of air at constant pressure | ${c}_{p}$ | 1005 | $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}$ |

Latent heat for condensation of water vapour | L | $2.5\times {10}^{6}$ | $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}$ |

Gas constant for water vapour | ${R}_{v}$ | $461.5$ | $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}$ |

Gas constant for air | ${R}_{a}$ | $286.84$ | $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}$ |

Density of liquid water | ${\rho}_{L}$ | 1000 | $\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ |

Reference saturated water vapour density at ${T}_{0}$ | ${\rho}_{vs}\left({T}_{0}\right)$ | $9.4\times {10}^{-3}$ | $\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ |

Constant in Equation (9) | C | $9.22\times {10}^{-11}$ | ${\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

Simulation grid step | $\Delta x$ | $0.001$ | m |

Simulation domain discretization | ${N}_{1}\times {N}_{2}\times {N}_{3}$ | $256\times 256\times 512$ | |

Simulation domain size | ${L}_{{x}_{1}}\times {L}_{{x}_{2}}\times {L}_{{x}_{3}}$ | $0.256\times 0.256\times 0.512$ | ${\mathrm{m}}^{3}$ |

Simulation IDs | |||
---|---|---|---|

Quantity | R25 | R18 | R6 |

Initial Droplet Radius ${r}_{in}\phantom{\rule{0.277778em}{0ex}}\left[\mathsf{\mu}\mathrm{m}\right]$ | 25 | 18 | 6 |

Total number of initial droplets | 286,240 | 286,240 | 286,240 |

Initial droplet number density ${N}_{d}\left(0\right)\phantom{\rule{0.277778em}{0ex}}\left[{\mathrm{cm}}^{-3}\right]$ | 17 | 17 | 17 |

Initial liquid water content $lwc\phantom{\rule{0.277778em}{0ex}}\left[\mathrm{gm}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}\right]$ | $1.12$ | $0.42$ | $0.02$ |

Initial Stokes number $St$ | $1.59$ | $0.82$ | $0.09$ |

Initial rms of velocity fluctuations ${u}^{\prime}$ in cloud $\left[\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}\right]$ | $0.268$ | $0.268$ | $0.268$ |

Initial energy ratio ${E}_{cloud}/{E}_{air}$ | 20 | 20 | 20 |

Temperature difference between cloudy and clear air $\Delta T$ [K] | 4 | 4 | 4 |

Initial integral scale L of cloud and air [m] | $0.0235$ | $0.0235$ | $0.0235$ |

Initial Taylor micro-scale Reynolds no. $R{e}_{\lambda}$ of cloud | 90 | 90 | 90 |

Initial Taylor micro-scale Reynolds no. $R{e}_{\lambda}$ of air | 20 | 20 | 20 |

Simulation time-step $\Delta t$ [s] | $1.224\times {10}^{-4}$ | $1.224\times {10}^{-4}$ | $1.377\times {10}^{-5}$ |

Total simulation duration [s] | $2.1$ | $2.1$ | $2.1$ |

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**MDPI and ACS Style**

Bhowmick, T.; Iovieno, M.
Direct Numerical Simulation of a Warm Cloud Top Model Interface: Impact of the Transient Mixing on Different Droplet Population. *Fluids* **2019**, *4*, 144.
https://doi.org/10.3390/fluids4030144

**AMA Style**

Bhowmick T, Iovieno M.
Direct Numerical Simulation of a Warm Cloud Top Model Interface: Impact of the Transient Mixing on Different Droplet Population. *Fluids*. 2019; 4(3):144.
https://doi.org/10.3390/fluids4030144

**Chicago/Turabian Style**

Bhowmick, Taraprasad, and Michele Iovieno.
2019. "Direct Numerical Simulation of a Warm Cloud Top Model Interface: Impact of the Transient Mixing on Different Droplet Population" *Fluids* 4, no. 3: 144.
https://doi.org/10.3390/fluids4030144