# Comparison of Flow Behavior in Saccular Aneurysm Models Using Proper Orthogonal Decomposition

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Aneurysm Models and Fluid

#### 2.2. Velocity Field Measurements

#### 2.3. Pump System

#### 2.4. Test Conditions

#### 2.5. POD

## 3. Results

#### 3.1. Average Flow Field

#### 3.2. POD Modes

#### 3.3. POD Energies

#### 3.4. POD Time-Varying Coefficients

#### 3.5. POD Low-Order Reconstruction

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$CFD$ | Computational Fluid Dynamics; |

${D}_{p}$ | Pipe diameter (m); |

$PIV$ | Particle Image Velocimetry; |

$POD$ | Proper Orthogonal Decomposition; |

$\mathbf{R}$ | Spatial velocity correlation matrix; |

$R{e}_{p}$ | Peak Reynolds number; |

t | Time (s); |

T | Time period (s); |

$\overrightarrow{U}(x,y)$ | Velocity vector; |

$\overrightarrow{u}$ | Velocity component in x-direction (m/s); |

$\overrightarrow{v}$ | Velocity component in y-direction (m/s); |

${V}_{max}$ | Maximum centerline velocity in the pipe (m/s); |

$x,y$ | Cartesian coordinates; |

$\alpha $ | Womersley number; |

${\nu}_{b}$ | Blood kinematic viscosity (m^{2}/s); |

${\nu}_{f}$ | Kinematic viscosity (m^{2}/s); |

${\rho}_{b}$ | Blood density (kg/m^{3}); |

${\rho}_{f}$ | Fluid density (kg/m^{3}); |

$\omega $ | Angular frequency (rad/s); |

$\overrightarrow{{\mathrm{\Psi}}^{i}}$ | ith POD mode; |

${\psi}_{uu}^{i}$ | Streamwise component of ith POD mode; |

${\psi}_{vv}^{i}$ | Transverse component of ith POD mode; |

${\mathrm{\Omega}}_{xy}$ | Domain of interest; |

${\lambda}^{i}$ | Energy captured by ith POD mode; |

⊗ | Tensor product; |

$\langle ...\rangle $ | Ensemble averaging. |

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**Figure 1.**Critical features of the idealized, saccular, aneurysm model for (

**a**) $BF=1.0$ and (

**b**) $BF=1.6$.

**Figure 3.**Average velocity field inside the aneurysm sac for $\alpha =2$ for (

**a**) $Re=50$ and $BF=1.0$, (

**b**) $Re=50$ and $BF=1.6$, (

**c**) $Re=270$ and $BF=1.0$, and (

**d**) $Re=270$ and $BF=1.6$.

**Figure 4.**POD modes presented as streamlines for $\alpha =2$, $R{e}_{p}=270$, and $BF=1.0$. (

**a**) ${\psi}^{1}$, (

**b**) ${\psi}^{2}$, (

**c**) ${\psi}^{3}$, (

**d**) ${\psi}^{4}$, and (

**e**) ${\psi}^{5}$.

**Figure 5.**POD modes presented as streamlines for $\alpha =2$, $R{e}_{p}=270$, and $BF=1.6$. (

**a**) ${\psi}^{1}$, (

**b**) ${\psi}^{2}$, (

**c**) ${\psi}^{3}$, (

**d**) ${\psi}^{4}$, and (

**e**) ${\psi}^{5}$.

**Figure 6.**POD modes presented as streamlines for $\alpha =5$, $R{e}_{p}=270$, and $BF=1.0$. (

**a**) ${\psi}^{1}$, (

**b**) ${\psi}^{2}$, (

**c**) ${\psi}^{3}$, (

**d**) ${\psi}^{4}$, and (

**e**) ${\psi}^{5}$.

**Figure 7.**POD modes presented as streamlines for $\alpha =5$, $R{e}_{p}=270$, and $BF=1.6$. (

**a**) ${\psi}^{1}$, (

**b**) ${\psi}^{2}$, (

**c**) ${\psi}^{3}$, (

**d**) ${\psi}^{4}$, and (

**e**) ${\psi}^{5}$.

**Figure 9.**Time-varying coefficients ${a}_{1}\left(t\right)$–${a}_{5}\left(t\right)$ for $\alpha =5$ and $R{e}_{p}=270$. Grey square markers represent experimental data while dashed lines represent curve fit data. (

**a**–

**e**) $BF=1.0$ and (

**f**–

**j**) $BF=1.6$.

**Figure 10.**POD low-order reconstruction for $R{e}_{p}=270$, $\alpha =2$, and $BF=1.0$ for selected time phases. The velocity reconstruction used three POD modes.

**Figure 11.**POD low-order reconstruction for $R{e}_{p}=270$, $\alpha =2$, and $BF=1.6$ for selected time phases. The velocity reconstruction used ten POD modes.

**Figure 12.**POD low-order reconstruction for $R{e}_{p}=50$, $\alpha =5$, and $BF=1.0$ for selected time phases. The velocity reconstruction used five POD modes.

**Figure 13.**POD low-order reconstruction for $R{e}_{p}=50$, $\alpha =5$, and $BF=1.6$ for selected time phases. The velocity reconstruction used five POD modes.

**Figure 14.**POD low-order reconstruction for $R{e}_{p}=270$, $\alpha =5$, and $BF=1.0$ for selected time phases. The velocity reconstruction used five POD modes.

**Figure 15.**POD low-order reconstruction for $R{e}_{p}=270$, $\alpha =5$, and $BF=1.6$ for selected time phases. The velocity reconstruction used ten POD modes.

BF | Re | $\mathit{\alpha}$ | PIV Images | PIV Frame Rate (Hz) | Pump Frequency (Hz) |
---|---|---|---|---|---|

1.0 | 50 | 2 | 500 | 1.17 | 0.4 |

1.0 | 270 | 2 | 500 | 1.17 | 0.4 |

1.0 | 50 | 5 | 500 | 1.17 | 2.4 |

1.0 | 270 | 5 | 500 | 1.17 | 2.4 |

1.6 | 50 | 2 | 500 | 1.17 | 0.4 |

1.6 | 270 | 2 | 500 | 1.17 | 0.4 |

1.6 | 50 | 5 | 500 | 1.17 | 2.4 |

1.6 | 270 | 5 | 500 | 1.17 | 2.4 |

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**MDPI and ACS Style**

Yu, P.; Durgesh, V. Comparison of Flow Behavior in Saccular Aneurysm Models Using Proper Orthogonal Decomposition. *Fluids* **2022**, *7*, 123.
https://doi.org/10.3390/fluids7040123

**AMA Style**

Yu P, Durgesh V. Comparison of Flow Behavior in Saccular Aneurysm Models Using Proper Orthogonal Decomposition. *Fluids*. 2022; 7(4):123.
https://doi.org/10.3390/fluids7040123

**Chicago/Turabian Style**

Yu, Paulo, and Vibhav Durgesh. 2022. "Comparison of Flow Behavior in Saccular Aneurysm Models Using Proper Orthogonal Decomposition" *Fluids* 7, no. 4: 123.
https://doi.org/10.3390/fluids7040123