# Three-Dimensional Hydrostatic Curved Channel Flow Simulations Using Non-Staggered Triangular Grids

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## Abstract

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## 1. Introduction

## 2. Governing Equations

## 3. Hydrostatic Calculations on Unstructured Prism Grids

## 4. Numerical Discretization

#### 4.1. Integral Form of Equations

#### 4.2. Integral Approximation of the Momentum Equation

#### 4.3. Interpolation of Face-Normal Velocities

## 5. Divergence Noise Analysis

#### 5.1. Divergence Noise Control and Implementation

#### 5.2. An Example of the Divergence Noise Control

## 6. Numerical Results

#### 6.1. Velocity Profiles

#### 6.2. Water Surface Elevations

#### 6.3. Secondary Flow Effects

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Variables defined in the cell-centers for non-staggered grids in each control volume; (

**b**) Prism grids projected into two dimensions.

**Figure 3.**The numerical solution and divergence error of the velocity field calculated by different triangular grids without and with filtering.

**Figure 5.**Curved channel flow with secondary flow circulations. ${R}_{c}$ represents the radius of curvature to the channel centerline, W is the channel width, and d is the channel depth.

**Figure 8.**Comparison of vertical velocity fields $w(r,\theta ,z=-0.5)$ between the non-filtered and filtered hydrostatic case.

**Figure 9.**(

**a**) Transverse water superelevation across different channel sections; (

**b**) Surface elevation distribution along the curved channel at $Fr=0.1$.

**Figure 10.**Three-dimensional streamlines in the curved channel at $Fr=0.4$. Red and blue lines represent primary flow and black lines represent secondary flows.

**Figure 11.**Comparison of streamlines at ${30}^{\circ}$, ${60}^{\circ}$, ${90}^{\circ}$, ${120}^{\circ}$, ${150}^{\circ}$, and ${180}^{\circ}$ cross-sections in filtered hydrostatic calculations between $Fr=0.1$ and $Fr=0.2$.

**Figure 12.**Comparison of streamlines at ${30}^{\circ}$, ${60}^{\circ}$, ${90}^{\circ}$, ${120}^{\circ}$, ${150}^{\circ}$, and ${180}^{\circ}$ cross-sections in filtered hydrostatic calculations between $Fr=0.3$, $Fr=0.4$, and $Fr=0.6$.

**Table 1.**The summary of the use of different models in the simulation of curved channel flows from the previous studies.

Reference | Method | Dimension | Curvature | Surface |
---|---|---|---|---|

Ippen [12] | Experiment | 3D | $22.{5}^{\circ},{45}^{\circ}$ bends | Free surface |

de Vriend [14] | Analytical | 3D | ${180}^{\circ}$ bend | Rigid surface |

Steffler et al. [8] | Analytical Experiment | 2D | ${180}^{\circ}$ bend | Free surface |

Odgaard [9], Odgaard [10] | Numerical | 2D | ${60}^{\circ}$ bend | Rigid surface |

Jin and Steffler [11] | Numerical | 2D | ${90}^{\circ}$ bend | Rigid surface |

Hodskinson and Ferguson [15] | Numerical Experiment | 3D | ${60}^{\circ}$, ${90}^{\circ}$ bends | Rigid surface |

Rameshwaran and Naden [16] | Numerical (RANS) | 3D | ${120}^{\circ}$ bend | Free surface Rigid surface |

Zeng et al. [18] | Numerical (RANS) Experiment | 3D | ${120}^{\circ}$ bend | Free surface |

Kashyap et al. [17] | Numerical (RANS) | 3D | ${135}^{\circ}$ bend | Rigid surface |

Wolfram and Fringer [2] | Numerical (DNS) | 3D | ${180}^{\circ}$ bend | Rigid surface |

Sin [19] | Numerical (RANS) | 3D | ${60}^{\circ},{90}^{\circ}$, ${120}^{\circ},{150}^{\circ}$ bends | Free surface |

Xu et al. [20] | Numerical (RANS) Experiment | 3D | ${180}^{\circ}$ bend | Rigid surface |

Present study | Numerical (DNS) | 3D | ${180}^{\circ}$ bend | Free surface |

**Table 2.**The diagram of divergence, $\mathrm{div}\left(\mathbf{u}\right)$, sequence in hydrostatic calculations.

Hydrostatic Calculations |
---|

(0) ${U}_{f}^{n}={\mathbf{u}}_{f}^{n}\xb7\mathbf{n}$; |

(1) Calculate ${\mathbf{u}}^{n+1}$ and ${\mathrm{v}}^{n+1}$ based on ${U}_{f}^{n}$; Calculate ${U}_{f}^{n+1}={\mathbf{u}}_{f}^{n+1}\xb7\mathbf{n}$; |

(2) Calculate $\mathrm{div}\left({\mathbf{u}}_{H}^{n+1}\right)$ based on ${U}_{f}^{n+1}$; |

(3) ${w}_{k+1}^{n+1}$ based on ${w}_{k}^{n+1}$ and $\mathrm{div}\left({\mathbf{u}}_{H}^{n+1}\right);$ ${\eta}^{n+1}$ based on $\mathrm{div}\left({\mathbf{u}}_{H}^{n+1}\right)$ |

(4) ${p}_{h}^{n+1}=\rho g({\eta}^{n+1}-z)$ and go to step 1 |

**Table 3.**Water superelevations versus Froude numbers ($F{r}_{1}$ and $\Delta {h}_{1}$ correspond to the case: $Fr=0.1$).

$\mathit{Fr}$ | ${\mathit{\eta}}_{\mathit{inner}}$ | ${\mathit{\eta}}_{\mathit{outer}}$ | $\mathbf{\Delta}\mathit{h}$ | ${(\mathit{Fr}/{\mathit{Fr}}_{1})}^{2}$ | $\mathbf{\Delta}\mathit{h}/\mathbf{\Delta}{\mathit{h}}_{1}$ |
---|---|---|---|---|---|

0.1 | 0.0254 | 0.03175 | 0.00635 | - | - |

0.2 | 0.1290 | 0.1575 | 0.0285 | 4 | 4.4 |

0.3 | 0.4133 | 0.4800 | 0.0667 | 9 | 10.5 |

0.4 | 0.9210 | 1.0297 | 0.1087 | 16 | 17.1 |

0.6 | 2.1498 | 2.3752 | 0.2254 | 36 | 35.5 |

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**MDPI and ACS Style**

Zhang, W.; Uh Zapata, M.; Pham Van Bang, D.; Nguyen, K.D.
Three-Dimensional Hydrostatic Curved Channel Flow Simulations Using Non-Staggered Triangular Grids. *Water* **2022**, *14*, 174.
https://doi.org/10.3390/w14020174

**AMA Style**

Zhang W, Uh Zapata M, Pham Van Bang D, Nguyen KD.
Three-Dimensional Hydrostatic Curved Channel Flow Simulations Using Non-Staggered Triangular Grids. *Water*. 2022; 14(2):174.
https://doi.org/10.3390/w14020174

**Chicago/Turabian Style**

Zhang, Wei, Miguel Uh Zapata, Damien Pham Van Bang, and Kim Dan Nguyen.
2022. "Three-Dimensional Hydrostatic Curved Channel Flow Simulations Using Non-Staggered Triangular Grids" *Water* 14, no. 2: 174.
https://doi.org/10.3390/w14020174