# Fourier Image Analysis of Multiphase Interfaces to Quantify Primary Atomization

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Simulation Method

#### 2.1.1. Incompressible Generalized Finite Difference Method

#### 2.1.2. Multiphase Flow

#### 2.2. Post-Processing

#### 2.2.1. Image Generation

#### 2.2.2. Image Processing

## 3. Results and Discussion

#### 3.1. Verification

#### 3.2. Quantification of Nozzle Atomization for Comparison

#### 3.2.1. Two-Dimensional Spectral Analysis

#### 3.2.2. 1D Spectral Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**A comparison between the (

**a**) instantaneous color gradient magnitude at the particle level and (

**b**) the image generated from the particle data.

**Figure 3.**Comparison between feature shapes in the 2D frequency space and its projection onto the 1D effective frequency space.

**Figure 4.**The real component of the complex exponential weighting of the FFT at ${f}_{eff}=5$ for (

**a**) $\tilde{x}=2$ and $\tilde{y}=3$, (

**b**) $\tilde{x}=3$ and $\tilde{y}=2$, and (

**c**) $\tilde{x}=5$ and $\tilde{y}=0$.

**Figure 7.**A 1D projection of the FFT data for the single circle image with (

**a**) ${r}_{0}=0.1$ and (

**b**) ${r}_{0}=0.25$. The dashed blue line indicates the HFLS.

**Figure 8.**Generated images for the (

**a**) four and (

**b**) nine randomly placed circles with a radius of 0.1.

**Figure 10.**A 1D projection of the FFT data of the (

**a**) four and (

**b**) nine randomly placed circles. The dashed blue line indicates the HFLS.

**Figure 11.**Image processing output for the multi-radius case showing (

**a**) the generated image, (

**b**) the FFT data, and (

**c**) the FFT data projected onto the effective frequency space.

**Figure 12.**Image processing output for the elliptical interface case showing (

**a**) the generated image, (

**b**) the FFT data, and (

**c**) the FFT data projected onto the effective frequency space.

**Figure 15.**Time-averaged results for the cylindrical outlet using (

**a**) density and (

**b**) color gradient magnitude. The density and color gradient magnitude image intensity maps to a range of 0 to 220 $\mathrm{kg}/{\mathrm{m}}^{3}$ and 0 to 0.09 ${\mathrm{m}}^{2}/\mathrm{kg}$, respectively.

**Figure 16.**Time-averaged results for the elliptical outlet using (

**a**) density and (

**b**) color gradient magnitude. The density and color gradient magnitude image intensity maps to a range of 0 to 220 $\mathrm{kg}/{\mathrm{m}}^{3}$ and 0 to 0.09 ${\mathrm{m}}^{2}/\mathrm{kg}$, respectively.

**Figure 17.**FFT of density for the cylindrical case over (

**a**) the total frequency domain and (

**b**) a low-frequency domain.

**Figure 18.**FFT of density for the elliptical case over (

**a**) the total frequency domain and (

**b**) a low-frequency domain.

**Figure 19.**FFT of color gradient magnitude for the cylindrical case over (

**a**) the total frequency domain and (

**b**) a low-frequency domain.

**Figure 20.**FFT of color gradient magnitude for the elliptical case over (

**a**) the total frequency domain and (

**b**) a low-frequency domain.

**Figure 21.**Projected FFT color gradient magnitude as a function of (

**a**) effective frequency and (

**b**) aspect ratio for the cylindrical outlet profile.

**Figure 22.**Projected FFT color gradient magnitude as a function of (

**a**) effective frequency and (

**b**) aspect ratio for the elliptical outlet profile.

**Figure 23.**Percentage increase in the binned color gradient magnitude FFT results over the effective frequency spectrum.

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**MDPI and ACS Style**

Joubert, J.C.; Wilke, D.N.; Pizette, P.
Fourier Image Analysis of Multiphase Interfaces to Quantify Primary Atomization. *Math. Comput. Appl.* **2023**, *28*, 55.
https://doi.org/10.3390/mca28020055

**AMA Style**

Joubert JC, Wilke DN, Pizette P.
Fourier Image Analysis of Multiphase Interfaces to Quantify Primary Atomization. *Mathematical and Computational Applications*. 2023; 28(2):55.
https://doi.org/10.3390/mca28020055

**Chicago/Turabian Style**

Joubert, Johannes C., Daniel N. Wilke, and Patrick Pizette.
2023. "Fourier Image Analysis of Multiphase Interfaces to Quantify Primary Atomization" *Mathematical and Computational Applications* 28, no. 2: 55.
https://doi.org/10.3390/mca28020055