# Comparison between the Lagrangian and Eulerian Approach in Simulation of Free Surface Air-Core Vortices

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. The Lagrangian Approach

_{j}and ρ

_{j}are representatives of the mass and density of particle “j” while N corresponds to the total number of the particles in the field of influence of particle “i”.

^{2}s) and the stress tensor ${\overrightarrow{\tau}}_{ij}$ in the SPS method could be defined in accordance with Equation (4). The SPS method is applied aiming to exert the LES turbulence model [44].

#### 2.2. The Eulerian Approach

## 3. The Physical Model

#### 3.1. The Experimental Setup

#### 3.2. Flow Measurement Techniques

#### 3.2.1. Velocity Measurement

#### 3.2.2. Water Surface Measurement

#### 3.3. The Procedure of the Experiments

## 4. The Numerical Model

## 5. Results

#### 5.1. The Air Core Vortex

#### 5.2. The Tangential Velocity Distribution

_{in}) and the radius of the intake. It should be pointed out that in the physical model the velocity field close to the core could not be assessed due to the light scattering in that region. According to Figure 3 it can be observed that the Eulerian method performed more efficiently in presentation of the tangential velocity component. The maximum difference between the results of the numerical model and the experimental data corresponds to 7.4% and 23.3% for the Eulerian and Lagrangian approach respectively.

#### 5.3. The Radial Velocity Distribution

#### 5.4. The Free Surface

_{r}) has been made dimensionless relative to the water elevation over the intake axis (H

_{0}) and the depth of the fluid in the vicinity of the borders (H). Comparison of the numerical simulations with the experimental results demonstrated that although the error with regard to the depth of the air core is high, the trends of both methods are similar to the experimental results with the Lagrangian model following more accurately the free surface curvature.

#### 5.5. Sensitivity Analysis

_{intake}> 2.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**The tangential velocity distribution in the physical model, the Eulerian, and Lagrangian models: (

**a**) test 1; (

**b**) test 2; (

**c**) test 3.

**Figure 4.**The radial velocity distribution in the physical model, the Eulerian, and Lagrangian models: (

**a**) test 1; (

**b**) test 2; (

**c**) test 3.

**Figure 6.**Sensitivity analysis (tangential velocity, test 2): (

**a**) Eulerian model; (

**b**) Lagrangian model.

Tests | Flow Rate Lit/s | Velocity at the Intake m/s | Intake Froude Number | Elevation of the Velocity Measurement Plane from the Bottom of the Tank (cm) |
---|---|---|---|---|

1 | 0.2 | 0.2 | 0.34 | 5 |

2 | 9 | |||

3 | 13 |

Particle/Mesh Size | Eulerian Model | Lagrangian Model | ||
---|---|---|---|---|

Computational Cost | Error % | Computational Cost | Error % | |

2 mm | 40 h | 2.1 | 168 h | 3.7 |

2.5 mm | 19 h | 5.7 | 97 h | 18.1 |

3 mm | 12.5 h | 9.2 | 58 h | 29.6 |

3.5 mm | 6 h | 14.8 | 29 h | - |

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**MDPI and ACS Style**

Azarpira, M.; Zarrati, A.R.; Farrokhzad, P.
Comparison between the Lagrangian and Eulerian Approach in Simulation of Free Surface Air-Core Vortices. *Water* **2021**, *13*, 726.
https://doi.org/10.3390/w13050726

**AMA Style**

Azarpira M, Zarrati AR, Farrokhzad P.
Comparison between the Lagrangian and Eulerian Approach in Simulation of Free Surface Air-Core Vortices. *Water*. 2021; 13(5):726.
https://doi.org/10.3390/w13050726

**Chicago/Turabian Style**

Azarpira, Maryam, Amir Reza Zarrati, and Pouya Farrokhzad.
2021. "Comparison between the Lagrangian and Eulerian Approach in Simulation of Free Surface Air-Core Vortices" *Water* 13, no. 5: 726.
https://doi.org/10.3390/w13050726