# Lattice Boltzmann Solver for Multiphase Flows: Application to High Weber and Reynolds Numbers

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## Abstract

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## 1. Introduction

## 2. Theoretical Background

#### 2.1. Target Macrosopic System

#### 2.2. LB Formulation for Conservative Phase-Field Equation

#### 2.3. LB Model for Flow Field

## 3. Numerical Applications

#### 3.1. Static Droplet: Surface-Tension Measurement

#### 3.2. Rayleigh–Taylor Instability

#### 3.3. Turbulent 3D Rayleigh–Taylor Instability

#### 3.4. Droplet Splashing on Thin Liquid Film

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Hermite Polynomials and Coefficients

## References

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**Figure 1.**Changes in pressure difference around droplet for different surface tensions and droplet radii. Red, blue, and black symbols illustrate results from present study with $\sigma ={10}^{-1},{10}^{-3}$, and ${10}^{-6}$, respectively.

**Figure 2.**(

**Left**) Evolution of interface for Rayleigh–Taylor instability for (

**top row**) Re = 256 and (

**bottom row**) Re = 2048 at different times: (from

**left**to

**right**) $t/T=$1, 2, 3, 4, and 5. (

**Right**) Position of penetrating spike over time: (black) Re = 256 and (red) Re = 2048. (plain lines) Results and (symbols) data from [19].

**Figure 3.**(

**Left**) Interface for Rayleigh–Taylor instability at $t/T=$ 5 and Re = 256 for three different resolutions (

**left**to

**right**) ${L}_{x}$ = 150, 300, and 600. (

**Right**) Position of penetrating spike over time: (black) ${L}_{x}$ = 600, (red) ${L}_{x}$ = 300, and (blue) ${L}_{x}$ = 150.

**Figure 4.**(

**Left**) Interface for Rayleigh–Taylor instability at $t/T$ = 5 and Re = 2048 for three different resolutions (

**left**to

**right**) ${L}_{x}$ = 150, 300, and 600. (

**Right**) Position of penetrating spike over time: (black) ${L}_{x}$ = 600, (red) ${L}_{x}$ = 300, and (blue) ${L}_{x}$ = 150.

**Figure 5.**(

**Left**) Evolution of interface for 3D Rayleigh–Taylor instability for Re = 1000 at different times: (from

**left**to

**right**) $t/T$ = 1.9, 3.9, 5.8, 7.8, and 9.7. (

**Right**) Position of penetrating spike over time: (plain lines) Results and (symbols) data from [41].

**Figure 7.**Impact of circular droplet on liquid sheet at different We and Re numbers with ${\rho}_{h}/{\rho}_{l}=1000$ and ${\nu}_{l}/{\nu}_{h}=15$. (black) Re = 200 and We = 220, (red) Re = 1000 and We = 220, and (blue) Re = 1000 and We = 2200.

**Figure 8.**Evolution of spreading radius ${r}_{K}$ as function of time for droplet impact on liquid film case. Circular symbols designate 2D simulations: (black) Re = 200 and We = 220, (red) Re = 1000 and We = 220, and (blue) Re = 1000 and We = 2200. Rectangular symbols belong to 3D simulation with Re = 1000 and We = 8000. Dashed line is $\frac{{r}_{K}}{D}=1.1\sqrt{t/T}$.

**Figure 9.**Impact of spherical droplet on thin liquid sheet at We = 8000 and Re = 1000 at different times with ${\rho}_{h}/{\rho}_{l}=1000$ and ${\nu}_{l}/{\nu}_{h}=15$.

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**MDPI and ACS Style**

Hosseini, S.A.; Safari, H.; Thevenin, D.
Lattice Boltzmann Solver for Multiphase Flows: Application to High Weber and Reynolds Numbers. *Entropy* **2021**, *23*, 166.
https://doi.org/10.3390/e23020166

**AMA Style**

Hosseini SA, Safari H, Thevenin D.
Lattice Boltzmann Solver for Multiphase Flows: Application to High Weber and Reynolds Numbers. *Entropy*. 2021; 23(2):166.
https://doi.org/10.3390/e23020166

**Chicago/Turabian Style**

Hosseini, Seyed Ali, Hesameddin Safari, and Dominique Thevenin.
2021. "Lattice Boltzmann Solver for Multiphase Flows: Application to High Weber and Reynolds Numbers" *Entropy* 23, no. 2: 166.
https://doi.org/10.3390/e23020166