# Immersed Boundary Method Application as a Way to Deal with the Three-Dimensional Sudden Contraction

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Investigations

#### 2.1. Mathematical Model

**x**, ${u}_{i}$ is the i-th component of the velocity vector

**u**, p is the pressure, $\rho $ and $\nu $ are the density and the kinematic viscosity of the fluid and ${f}_{i}$ is the i-th component of the force vector

**f**which represent any external force acting on the fluid.

#### 2.2. Numerical Methods

#### Computational Aspects

#### 2.3. Immersed Boundary Method

**u**in the Eulerian domain. Minimizing J with respect to $\mathit{a}\left(\mathit{x}\right)$ leads to the following set of equations:

## 3. Problem Description

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IB | Immersed boundary method |

LDA | Laser-Doppler Anemometry |

MLS | Moving Least Squares Method |

PIV | Particle image velocimetry |

## Appendix A

**Figure A1.**Grid independency at $R{e}_{D}=365$ (

**a**) $z/D=-0.288$, (

**b**) $z/D=-0.104$, (

**c**) $z/D=-0.079$ and (

**d**) $z/D=+0.784$.

Mesh | $\mathit{z}/\mathit{D}=-0.288$ | $\mathit{z}/\mathit{D}=-0.104$ | $\mathit{z}/\mathit{D}=-0.079$ | $\mathit{z}/\mathit{D}=0.784$ | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{u}/\mathit{U}$ | Error (%) | $\mathit{u}/\mathit{U}$ | Error (%) | $\mathit{u}/\mathit{U}$ | Error (%) | $\mathit{u}/\mathit{U}$ | Error (%) | |

$60\times 60\times 168$ | $2.3318$ | $3.2474$ | $3.3511$ | $6.0494$ | $3.5898$ | $6.5112$ | $5.7598$ | $7.6615$ |

$70\times 70\times 196$ | $2.3134$ | $2.4358$ | $3.3024$ | $4.5076$ | $3.5332$ | $4.8333$ | $5.6672$ | $5.9315$ |

$80\times 80\times 224$ | $2.2595$ | $0.0465$ | $3.1744$ | $0.4573$ | $3.3884$ | $0.5364$ | $5.3928$ | $0.8019$ |

$90\times 90\times 252$ | $2.2584$ | $0.0000$ | $3.1600$ | $0.0000$ | $3.3703$ | $0.0000$ | $5.3450$ | $0.0000$ |

Richardson extr. | $2.2545$ | - | $3.1056$ | - | $3.2023$ | - | $5.1650$ | - |

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**Figure 1.**Schematic representation of problem definition for a sudden contraction. Adapted from [26].

**Figure 5.**Pressure (

**a**) distribution along the centreline at various $R{e}_{D}$, (

**b**) pressure loss, (

**c**) pressure fields at $R{e}_{D}=365$, (

**d**) pressure field at $R{e}_{D}=993$ and (

**e**) vena contracta evidence at $R{e}_{D}=993$ (zoom in on the pressure field).

**Figure 6.**Streamline for $R{e}_{D}=365$ (

**a**) upstream region for velocity magnitude [26], (

**b**) upstream regions for velocity magnitude (present work), (

**c**) both regions for axial velocity and (

**d**) both regions for radial velocity.

**Figure 7.**Flow visualization in the 3D domain at $R{e}_{D}=365$. (

**a**) Streamline for different positions and (

**b**) velocity vector for different positions.

**Figure 8.**Comparison of numerical and experimental non-dimensional axial velocity profiles for $R{e}_{D}=365$ (

**a**) $z/D=-0.288$, (

**b**) $z/D=-0.236$, (

**c**) $z/D=-0.079$, (

**d**) $z/D=-0.026$, (

**e**) $z/D=+0.049$ and (

**f**) $z/D=+0.784$.

**Table 1.**Percentage error in maximum velocity at $R{e}_{D}=365$ and $\beta =1.97$ for some positions ($z/D$).

$\mathit{z}/\mathit{D}$ | $\mathit{u}/\mathit{U}$ | ||
---|---|---|---|

Present | Sanchez [26] | Error (%) | |

$-0.288$ | $2.26$ | $2.28$ | $0.88$ |

$-0.236$ | $2.41$ | $2.40$ | $0.26$ |

$-0.079$ | $3.37$ | $3.31$ | $1.70$ |

$-0.026$ | $3.85$ | $3.81$ | $1.08$ |

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**MDPI and ACS Style**

Borges, J.E.; Lourenço, M.; Padilla, E.L.M.; Micallef, C.
Immersed Boundary Method Application as a Way to Deal with the Three-Dimensional Sudden Contraction. *Computation* **2018**, *6*, 50.
https://doi.org/10.3390/computation6030050

**AMA Style**

Borges JE, Lourenço M, Padilla ELM, Micallef C.
Immersed Boundary Method Application as a Way to Deal with the Three-Dimensional Sudden Contraction. *Computation*. 2018; 6(3):50.
https://doi.org/10.3390/computation6030050

**Chicago/Turabian Style**

Borges, Jonatas E., Marcos Lourenço, Elie L. M. Padilla, and Christopher Micallef.
2018. "Immersed Boundary Method Application as a Way to Deal with the Three-Dimensional Sudden Contraction" *Computation* 6, no. 3: 50.
https://doi.org/10.3390/computation6030050