# Smoothed Particle Hydrodynamics Simulations of Water Flow in a 90° Pipe Bend

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations and Numerical Framework

#### 2.1. Exact Equations

#### 2.2. LES Filtering

#### 2.3. The SPH Method

#### 2.4. Boundary Conditions

#### 2.5. Time Marching Scheme

**and**

## 3. Pipe Bend Model Description

#### Validation and Particle Independence Test

## 4. Results

#### 4.1. Laminar Flow

#### 4.2. Turbulent Flow

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Schematic diagram of the bend geometry and dimensions. The vertical lines on the outer arc of the pipe mark the stations where the numerical results are compared with the experimental data.

**Figure 3.**Numerical horizontal (

**left**) and vertical (

**right**) streamwise velocity profiles for $N=$ 1,080,924, 2,104,126 and 4,042,185 particles as compared with the experimental data at ${S}_{2}=0.58D$ upstream of the bend inlet for Re = 43,000. Asymptotic convergence of the numerical solution to the experimental measurements is obtained for $N=$ 4,042,185 SPH particles.

**Figure 4.**Comparison of the numerical normalized horizontal (

**left**) and vertical (

**right**) streamwise velocity profiles with the experimental measurements at ${S}_{2}=0.58D$ upstream of the bend inlet for Re $=500$ and $N=$ 4,042,185, 8,064,442 and 16,056,513 particles.

**Figure 5.**Comparison of the numerical normalized horizontal (

**left**) and vertical (

**right**) streamwise velocity profiles with the experimental measurements at ${S}_{2}=0.58D$ upstream of the bend inlet for Re $=1093$ and $N=$ 4,042,185, 8,064,442 and 16,056,513 particles.

**Figure 6.**Cross-sectional streamwise velocity maps at planes $\theta ={30}^{\xb0}$, ${60}^{\xb0}$ and ${75}^{\xb0}$ through the bend and ${S}_{3}=1D$ downstream of the bend outlet for Re $=500$ (

**left**) and Re$=1093$ (

**right**). The numbers on the color bars indicate the values of the streamwise velocity normalized to ${v}_{\mathrm{B}}$.

**Figure 7.**Comparison of the numerical normalized horizontal (

**left**) and vertical (

**right**) streamwise velocity profiles with the experimental measurements at ${S}_{2}=0.58D$ upstream of the bend inlet for Re = 43,000 and $N=$ 4,042,185, 8,064,442 and 16,056,513 particles.

**Figure 8.**Comparison of the numerical normalized horizontal (

**left**) and vertical (

**right**) turbulence intensity profiles with the experimental measurements at ${S}_{2}=0.58D$ upstream of the bend inlet for Re = 43,000 and $N=$ 4,042,185, 8,064,442 and 16,056,513 particles.

**Figure 9.**Cross-sectional streamwise velocity maps at planes $\theta ={30}^{\xb0}$, ${60}^{\xb0}$ and ${75}^{\xb0}$ in the bend and at stations ${S}_{3}=1D$ and ${S}_{5}=6D$ downstream of the bend outlet for Re = 43,000. The numbers on the color bars indicate the values of the streamwise velocity normalized to ${v}_{\mathrm{B}}$.

**Figure 10.**Variation of the wall static pressure along the inner and outer arc of the bend at Re = 43,000. The numerically calculated values (filled dots) are compared with Enayet et al.’s [1] experimental measurements (open circles). The data are taken at station ${S}_{1}$ upstream of the bend inlet, at angles $\theta ={15}^{\xb0}$, ${30}^{\xb0}$, ${60}^{\xb0}$ and ${75}^{\xb0}$ in the bend and at stations ${S}_{3}$ and ${S}_{4}$ downstream of the bend outlet for azimuthal angles $\varphi ={0}^{\xb0}$, ${45}^{\xb0}$, ${90}^{\xb0}$, ${135}^{\xb0}$ and ${180}^{\xb0}$ on the circular wall of the pipe.

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**MDPI and ACS Style**

Sigalotti, L.D.G.; Alvarado-Rodríguez, C.E.; Klapp, J.; Cela, J.M.
Smoothed Particle Hydrodynamics Simulations of Water Flow in a 90° Pipe Bend. *Water* **2021**, *13*, 1081.
https://doi.org/10.3390/w13081081

**AMA Style**

Sigalotti LDG, Alvarado-Rodríguez CE, Klapp J, Cela JM.
Smoothed Particle Hydrodynamics Simulations of Water Flow in a 90° Pipe Bend. *Water*. 2021; 13(8):1081.
https://doi.org/10.3390/w13081081

**Chicago/Turabian Style**

Sigalotti, Leonardo Di G., Carlos E. Alvarado-Rodríguez, Jaime Klapp, and José M. Cela.
2021. "Smoothed Particle Hydrodynamics Simulations of Water Flow in a 90° Pipe Bend" *Water* 13, no. 8: 1081.
https://doi.org/10.3390/w13081081