# Direct Numerical Simulation of Water Droplets in Turbulent Flow

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Flow over a Spherical Particle

_{rel}the relative velocity between the air and the particle and ${\rho}_{air}$ the density of the air. The flow field can be characterised by the Reynolds number (Re = $\rho vD/\mu $), which directly relates to ${C}_{\mathrm{d}}$. Here, $\rho $, v and $\mu $ are the density of the fluid, the velocity of the particle and the dynamic viscosity of the fluid, respectively. Clift et al. [22] recommended a standard drag curve for a sphere, where ${C}_{\mathrm{d}}$ is plotted as a function of Re. Previous studies provided a detailed discussion including the rich physics phenomenon of the flow field over a sphere at different Reynolds numbers. These include the work of Clift et al. [22], Johnson and Patel [23], Sakamoto and Haniu [24], Leweke et al. [25], Mittal [26], Kim and Durbin [27], Yun et al. [28] and Van Dyke [29].

#### 1.2. Hydrodynamics of Raindrops

#### 1.3. Drag Coefficients and Terminal Velocities of Water Drops

## 2. Numerical Methods

**u**the velocity vector, p the pressure, $\mu $ the dynamic viscosity and

**g**the gravity vector. The surface tension force

**f**is modelled with the balanced-force Continuum Surface Force (CSF) method combined with Height Function (HF) curvature technique proposed by Popinet [40]. The pressure field is computed by solving a Poisson equation implicitly, using a multi-grid algorithm [41].

_{st}**n**of the interface where $\mathbf{n}=-\nabla f/\left(\right)open="|"\; close="|">\nabla f$. The position of the reconstructed plane corresponds to the VOF variable f in each interface cell and is solved through iterations [41].

## 3. Computational Setup

**g**points in the direction of the negative x-axis. We applied the no slip wall condition at the inflow and continuous boundary conditions (Neumann) on the rest of the domain boundaries. In the case of still air, the air flow at 293.15 K enters the computational domain with a uniform velocity. In the case of a water drop falling across turbulent air flow, the random spot method by Kornev and Hassel [43] was applied for the generation of a turbulent inflow. The grid used is fine enough to resolve the smallest turbulent scales. For a 3 mm water drop falling at its terminal velocity, the ratio between the cell size ${L}_{\mathrm{c}}$ and the Kolmogorov length scale ${\lambda}_{\mathrm{k}}={L}_{\mathrm{t}}/{Re}_{\mathrm{t}}^{3/4}$ is around 2.3 with $R{e}_{\mathrm{t}}={\rho}_{\mathrm{air}}\sqrt{\overline{{u}^{\prime 2}}}{L}_{\mathrm{t}}/{\mu}_{\mathrm{air}}$, the turbulence length scale ${L}_{\mathrm{t}}$ = 0.75 mm and a turbulence intensity $Tu$ = 10%.

## 4. Results

#### 4.1. Terminal Velocity

#### 4.2. Drop Shape Deformation and Drop Oscillation

#### 4.3. Flow Field

#### 4.3.1. Instantaneous Flow

_{pb}is thus higher in the case D3L050T10 than that in the still air case. Similar changes in the near wake region were found for the cases with a less intense turbulent inflow, as well as in the cases with the 2 mm water drop. Hence, we think that this might be one potential influential factor that relates to the reduction of the terminal velocity of a water drop.

#### 4.3.2. Mean Flow Parameters

#### 4.4. Internal Circulation

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${\mathbf{v}}_{\mathbf{t}}$ | terminal velocity | ($\mathrm{m}/\mathrm{s}$) |

DNS | Direct Numerical Simulation | |

FS3D | Free Surface 3D | |

MAC | Marker and Cell | |

VOF | volume of fluid method | |

PLIC | Piecewise Linear Interface Calculation | |

∇ | Nabla-Operator | |

$TKE$ | turbulent kinetic energy | (−) |

D | equivalent droplet diameter | ($\mathrm{m}$) |

${C}_{\mathrm{d}}$ | drag coefficient | (−) |

$\rho $ | density | ($\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$) |

f | volume fraction variable | (−) |

$\theta $ | separation angle | (${}^{\circ}$) |

Re | Reynolds number | |

St | Strouhal number | |

We | Weber number | |

$\mathbf{u}$ | velocity vector | ($\mathrm{m}/\mathrm{s}$) |

t | time | ($\mathrm{s}$) |

p | pressure | ($\mathrm{N}/{\mathrm{m}}^{2}$) |

$\mu $ | dynamic viscosity | ($\mathrm{Pa}$$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$$\mathrm{s}$) |

$\mathbf{g}$ | gravitational acceleration | ($\mathrm{m}/{\mathrm{s}}^{2}$) |

${\mathbf{f}}_{\mathbf{st}}$ | surface tension force | ($\mathrm{N}/{\mathrm{m}}^{3}$) |

l | liquid phase | |

$gas$ | gas phase | |

$\mathbf{n}$ | normal vector at the interface | (−) |

CFL | Courant–Friedrich–Lewy | |

$st$ | surface tension | |

$Tu$ | turbulence intensity | (−) |

${L}_{\mathrm{t}}$ | turbulence length scale | ($\mathrm{m}$) |

${L}_{\mathrm{c}}$ | cell size | ($\mathrm{m}$) |

${\lambda}_{\mathrm{k}}$ | Kolmogorov length scale | ($\mathrm{m}$) |

${u}^{\prime}$ | streamwise velocity fluctuations | ($\mathrm{m}/\mathrm{s}$) |

${v}^{\prime}$ | transverse velocity fluctuations | ($\mathrm{m}/\mathrm{s}$) |

${w}^{\prime}$ | spanwise velocity fluctuations | ($\mathrm{m}/\mathrm{s}$) |

a | semi-major axis length, measured in y-direction | ($\mathrm{m}$) |

${a}^{\prime}$ | semi-major axis length, measured in z-direction | ($\mathrm{m}$) |

b | semi-minor axis length | ($\mathrm{m}$) |

${A}_{\mathrm{dr}}$ | drop surface area | (${\mathrm{m}}^{2}$) |

${A}_{\mathrm{sp}}$ | surface area of a sphere | (${\mathrm{m}}^{2}$) |

$\overline{a}$ | time-averaged length of semi-major axis | ($\mathrm{m}$) |

$-{C}_{\mathrm{pb}}$ | base suction coefficient | (−) |

$\left|\omega \right|$ | vorticity magnitude | ($1/\mathrm{s}$) |

${C}_{\mathrm{p}}$ | pressure coefficient | (−) |

${\overline{\theta}}_{s}$ | time-averaged flow separation angle | (${}^{\circ}$) |

$\overline{}$ | time average | |

L | length of the recirculation region | ($\mathrm{m}$) |

$-{C}_{\mathrm{pmin}}$ | minimum pressure coefficient on the wake axis | (−) |

${L}_{\mathrm{v}}$ | vortex formation length | ($\mathrm{m}$) |

## Appendix A. Studies for Grid Resolution and Domain Size

**Table A1.**Setup for the studies of the grid resolution, the size of the simulation domain and the position of water drop in the main flow direction.

Test Series | Grid-Cells in a Drop Diameter | Domain Width | Wake Length | Drop Position | Case Number |
---|---|---|---|---|---|

Grid Resolution | 13.3 | 7.2D | 10.8D | 3.6D | a1 |

15.6 | a2 | ||||

18.3 | a3 | ||||

26.7 | a4 | ||||

Wake Length | 26.7 | 7.2D | 10.8D | 3.6D | b1 |

15.6D | b2 | ||||

20.4D | b3 | ||||

24.4D | b4 | ||||

Domain Width | 26.7 | 7.2D | 18D | 6.0D | c1 |

12.0D | c2 | ||||

16.8D | c3 | ||||

Drop Centre in the Stream-wise Direction | 26.7 | 12D | 15.6D | 3.6D | d1 |

6.0D | d2 | ||||

8.4D | d3 | ||||

10.8D | d4 |

**Figure A1.**Studies of the grid resolution, the domain width and wake length, as well as the position of water drop in the stream-wise direction.

## Appendix B. Validations for the Drop Deformation

**Figure A2.**Comparison of the axis ratios, the axis ratio amplitudes and the oscillation frequencies of the simulated 3 mm and 2 mm water drops in stagnant air: (

**a**) Axis ratio. (

**b**) Axis ratio amplitude. (

**c**) Oscillation frequency. The figure is adapted from [30].

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**Figure 2.**Terminal velocities and the corresponding Reynolds numbers for water drops of various diameters in stagnant air; data taken from Gunn and Kinzer [11].

**Figure 4.**The black curve illustrates the time evolution of the simulated fall velocity of a 3 mm water drop in stagnant air. The dashed line displays the corresponding terminal velocity according to Gunn and Kinzer [11] (${\mathbf{v}}_{\mathbf{t}}$ = 8.06 m/s).

**Figure 5.**Simulated fall velocities of the 3 mm and 2 mm water drops over time: (

**a**) D = 3 mm. (

**b**) D = 2 mm. In the 3 mm drop case D3T0, the curve is obtained from the validation case. For all other cases, the initial mean velocity of the incoming air flow was set to 7.8 m/s and 6.2 m/s for the 3 mm and 2 mm drops, respectively.

**Figure 6.**Temporal variations in the axis lengths and surface area of the simulated 3 mm water drop in stagnant air.

**Figure 7.**An example for the shape change of the simulated 3 mm water drop, falling in stagnant air.

**Figure 8.**Comparison of the instantaneous (stream-wise) velocity field of a 3 mm water drop falling in still air (

**left**) and that falling in turbulent air flow (

**right**).

**Figure 9.**Instantaneous streamlines coloured with the normalised stream-wise velocity (

**left**), the vorticity magnitude (

**middle**) and the normalised pressure (

**right**) for the case D3T0, at about 2.3 s.

**Figure 10.**Instantaneous streamlines coloured with the normalised stream-wise velocity (

**left**), the vorticity magnitude (

**middle**) and the normalised pressure (

**right**) for the case D3L050T10, at about 1.6 s.

**Figure 11.**An example of time-averaged flow over a droplet, showing the position for the base pressure coefficient −${\overline{C}}_{\mathrm{pb}}$ and wake recirculation bubble.

**Figure 12.**Comparison of the time-averaged stream-wise velocity along the wake symmetry axis for the simulated sphere with Re = 1000 (dashed line), the 3 mm water drop in still air with Re = 916 (dashed line) and the previous studies. The figure is adapted from [46].

**Figure 13.**Comparisons of the Strouhal numbers for the simulated 3 mm and 2 mm water drops in stagnant air, the simulated sphere at Re = 1000 and the spheres from the previous studies. The figure is reproduced from [28] and modified with our simulation results, with the permission of AIP Publishing.

**Figure 14.**Simulated base suction pressure coefficient −${\overline{C}}_{\mathrm{pb}}$ with respect to the length of the recirculation region $\overline{L}/D$, as well as the minimum axis pressure coefficient −${\overline{C}}_{\mathrm{pmin}}$ with respect to the wake formation length ${\overline{L}}_{\mathrm{v}}/D$ (exact values and the corresponding cases are shown in Table 5).

**Figure 15.**Drag coefficients of the simulated water droplets with respect to the lengths of their wake recirculation areas (exact values and the corresponding cases are shown in Table 5).

**Figure 16.**Strouhal numbers of the simulated water drops with respect to their drag coefficients (exact values and the corresponding cases are shown in Table 5).

**Figure 17.**Reynolds stress component $\overline{{u}^{\prime}{v}^{\prime}}$/${u}_{\infty}^{2}$ for flow past the simulated sphere at Re = 1000.

**Figure 18.**Comparison of the Reynolds stresses and turbulent kinetic energy of the 3 mm water drops falling in stagnant and turbulent surroundings: (

**a**) $\overline{{u}^{\prime}{u}^{\prime}}$/${u}_{\infty}^{2}$. (

**b**) $\overline{{v}^{\prime}{v}^{\prime}}$/${u}_{\infty}^{2}$. (

**c**) $\overline{{u}^{\prime}{v}^{\prime}}$/${u}_{\infty}^{2}$. (d) Turbulent kinetic energy. From left to right: cases D3T0, D3L050T10, D3L025T10, D3L050T05, D3L025T05.

**Figure 19.**Comparison of the Reynolds stresses and turbulent kinetic energy of the 2 mm water drops falling in stagnant and turbulent surroundings: (

**a**) $\overline{{u}^{\prime}{u}^{\prime}}$/${u}_{\infty}^{2}$. (

**b**) $\overline{{v}^{\prime}{v}^{\prime}}$/${u}_{\infty}^{2}$. (

**c**) $\overline{{u}^{\prime}{v}^{\prime}}$/${u}_{\infty}^{2}$. (d) Turbulent kinetic energy. From left to right: cases D2T0, D2L050T10, D2L025T10, D2L050T05, D2L025T05.

**Figure 20.**Streamlines of the 3 mm (Re = 1614) and 2 mm (Re = 916) water drops falling in stagnant air at their terminal velocities: (

**a**) Time-averaged streamlines of the 3 mm (left) and 2 mm (right) water drop. (

**b**) Streamlines of the 3 mm water drop at two different time steps. (

**c**) Streamlines of the 2 mm water drop at two different time steps.

Air Density $\mathit{\rho}$ _{gas} (kg/m${}^{3}$) | Water Density $\mathit{\rho}$ _{l} (kg/m${}^{3}$) | Air Viscosity $\mathit{\mu}$ _{gas} (Pa$\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}$s) | Water Viscosity $\mathit{\mu}$ _{l} (Pa$\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}$s) | Surface Tension $\mathit{\sigma}$ (N/m) |
---|---|---|---|---|

1.2045 | 998.2 | 18.2 × 10 ${}^{-6}$ | 1 × 10 ${}^{-3}$ | 72.75 × 10 ${}^{-3}$ |

**Table 2.**Terminal velocities, corresponding Reynolds numbers and drag coefficients for 3 mm and 2 mm water drops falling in stagnant air, according to Gunn and Kinzer [11].

Equivalent Drop Diameter (mm) | Terminal Velocity (m/s) | Reynolds Number | Drag Coefficient |
---|---|---|---|

3.0 | 8.06 | 1613 | 0.503 |

2.0 | 6.49 | 866 | 0.517 |

**Table 3.**Comparison of the terminal velocities and drag coefficients for the 3 mm and 2 mm water drops in turbulent and stagnant air flows.

Test Cases | Terminal Velocity (m/s) | Drag Coefficient | Velocity Decrease | Reynolds Number | Weber Number |
---|---|---|---|---|---|

D3T0 | 8.13 | 0.492 | - | 1614 | 3.28 |

D3L050T10 | 7.55 | 0.570 | 7.1% | - | - |

D3L025T10 | 7.73 | 0.544 | 4.9% | - | - |

D3L050T05 | 7.69 | 0.550 | 5.4% | - | - |

D3L025T05 | 7.83 | 0.530 | 3.7% | - | - |

D2T0 | 6.92 | 0.453 | - | 916 | 1.59 |

D2L050T10 | 6.64 | 0.492 | 4.0% | - | - |

D2L025T10 | 6.77 | 0.473 | 2.2% | - | - |

D2L050T05 | 6.73 | 0.479 | 2.7% | - | - |

D2L025T05 | 6.85 | 0.462 | 1.0% | - | - |

**Table 4.**Comparison of the time-averaged drop axis ratios, horizontal axis lengths, axis ratio amplitudes and oscillation frequencies of the simulated falling water drops in both stagnant and turbulent surroundings.

Test Cases | Axis Ratio | Axis Length 2$\overline{\mathit{a}}$ (mm) | Axis Ratio Amplitude | Oscillation Frequency |
---|---|---|---|---|

D3T0 | 0.881 | 3.119 | 0.086 | 59 |

D3L050T10 | 0.887 | 3.114 | 0.144 | 66 |

D3L025T10 | 0.886 | 3.115 | 0.091 | 62 |

D3L050T05 | 0.887 | 3.114 | 0.088 | 60 |

D3L025T05 | 0.884 | 3.117 | 0.107 | 61 |

D2T0 | 0.942 | 2.040 | 0.064 | 116 |

D2L050T10 | 0.942 | 2.038 | 0.098 | 120 |

D2L025T10 | 0.942 | 2.038 | 0.067 | 116 |

D2L050T05 | 0.942 | 2.040 | 0.083 | 116 |

D2L025T05 | 0.942 | 2.039 | 0.075 | 117 |

**Table 5.**Comparison of statistical flow parameters for the simulated water drops falling in stagnant air and turbulent surroundings, e.g., the flow separation angle, base suction pressure, length of the wake recirculation, minimum pressure coefficient, vortex formation length and Strouhal numbers.

${\overline{\mathit{\theta}}}_{\mathit{s}}$${(}^{\circ})$ | −${\overline{\mathit{C}}}_{\mathbf{pb}}$ | $\overline{\mathit{L}}/\mathit{D}$ | −${\overline{\mathit{C}}}_{\mathbf{pmin}}$ | ${\overline{\mathit{L}}}_{\mathit{v}}/\mathit{D}$ | $\mathit{S}\mathit{t}$ | |
---|---|---|---|---|---|---|

D3T0 | 88.2 | 0.243 | 2.82 | 0.313 | 1.787 | 0.263 |

D3L050T10 | 93.6 | 0.341 | 1.23 | 0.448 | 0.400 | 0.532 |

D3L025T10 | 92.3 | 0.320 | 1.56 | 0.424 | 0.625 | 0.430 |

D3L050T05 | 95.1 | 0.338 | 1.47 | 0.438 | 0.588 | 0.451 |

D3L025T05 | 88.7 | 0.306 | 1.78 | 0.407 | 0.813 | 0.413 |

D2T0 | 90.6 | 0.236 | 2.50 | 0.303 | 1.450 | 0.167 |

D2L050T10 | 96.0 | 0.297 | 1.36 | 0.388 | 0.475 | 0.393 |

D2L025T10 | 92.2 | 0.270 | 1.79 | 0.354 | 0.775 | 0.314 |

D2L050T05 | 93.7 | 0.282 | 1.71 | 0.371 | 0.738 | 0.313 |

D2L025T05 | 90.8 | 0.258 | 2.09 | 0.334 | 1.000 | 0.257 |

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**MDPI and ACS Style**

Ren, W.; Reutzsch, J.; Weigand, B.
Direct Numerical Simulation of Water Droplets in Turbulent Flow. *Fluids* **2020**, *5*, 158.
https://doi.org/10.3390/fluids5030158

**AMA Style**

Ren W, Reutzsch J, Weigand B.
Direct Numerical Simulation of Water Droplets in Turbulent Flow. *Fluids*. 2020; 5(3):158.
https://doi.org/10.3390/fluids5030158

**Chicago/Turabian Style**

Ren, Weibo, Jonathan Reutzsch, and Bernhard Weigand.
2020. "Direct Numerical Simulation of Water Droplets in Turbulent Flow" *Fluids* 5, no. 3: 158.
https://doi.org/10.3390/fluids5030158