# A 3D Fully Non-Hydrostatic Model for Free-Surface Flows with Complex Immersed Boundaries

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Governing Equations

#### 2.2. Numerical Solution by FDM

#### 2.3. Numerical Solution with FDM and VOS-Based Immersed Boundary Method

#### 2.4. Volume Fraction Transport Equation

#### 2.5. Solution Procedure

- Propose the initial and the boundary conditions for a given time-step and use the VOF reconstruction method via Equation (23) to compute a cell-center estimate of the volume fraction and free-surface position.
- Solve the ${\mathrm{cx}}^{\mathrm{n}}$, ${\mathrm{cy}}^{\mathrm{n}},$ and ${\mathrm{cz}}^{\mathrm{n}}$ using Equations (10)–(12) via FDM.
- Solve the right-hand side D of pressure Poisson equation (Equation 16) using iterative method to obtain the pressure, P
^{n+1/2}based on point-successive over relaxation iteration. - Solve the velocity, u, v, and w using Equations (7)–(9) via a split operating step FDM.
- Using the ${V}_{\mathrm{GC}}$ via Equation (20) to calculate the forcing function, ${\mathrm{F}}_{\mathrm{x}}^{\mathrm{n}+1}$, ${\mathrm{F}}_{\mathrm{y}}^{\mathrm{n}+1}$, and ${\mathrm{F}}_{\mathrm{z}}^{\mathrm{n}+1}$ in Equations (7)–(9) include the effect of immersed body.
- In the fractional time-step, the values of velocities, forcing functions, and pressure variables calculated from the previous time-step were used to resolve the steps 1 to 5 to obtain new values. The above numerical procedure was reiterated until the prescribed time was reached.

## 3. Results and Discussion

#### 3.1. Dam-Break Flow Passing a Cylinder

#### 3.2. Partial Dam-Break Flow

#### 3.3. Dam-Break Surge Passing Four Aligned Square or Circular Cylinders

_{x}= 0.02 in the x direction in the area downstream of the dam. The same boundary conditions are used as described in Section 3.2. Figure 15 shows the time history of the impact force on each square and circular cylinder. As expected, the square and circular cylinders at the far side, marked as 1, reveal the smaller force loadings on the bodies, which monochromatically increases with the time between 0 to 10 s. The other three cylinders marked as 2 to 3 demonstrate analogous variations of the impact forces with time, with the appearance of the force peaks as depicted in Figure 15b–d.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Schematic diagram of 3D dam-break flows over a (

**a**) vertical square cylinder, (

**b**) vertical rectangular cylinder with a different aspect ratio ($\mathsf{\alpha}=\mathrm{W}/\mathrm{L}$), (

**c**) vertical circular cylinder. The volume of water is 0.4 $\times $ 0.61 $\times $ 0.3 m

^{3}.

**Figure 4.**Time history of the horizontal velocity upstream the square obstacle at the location (0.754, 0, 0.026).

**Figure 6.**Variations of the impact forces with respect to time for the different aspect ratios of the vertical rectangular cylinder.

**Figure 7.**Simulation of the 3D dam-break flows over a vertical rectangular obstacle ($\mathsf{\alpha}=0.25$).

**Figure 8.**Simulation of the 3D dam-break flows over a vertical rectangular obstacle ($\mathsf{\alpha}=1.5$).

**Figure 11.**Layout of the two-dimensional dam break simulation in the benchmark study of Fennema and Chaudhry [42].

**Figure 12.**Surface elevation downstream the partial dam break simulated using the present model and compared with the predicted result of LABSWE [42].

**Figure 13.**Temporal evolution of free surface elevations and velocity distribution. The flow separations downstream of the broken dam are predicted.

**Figure 14.**Geometry of a dam break surge with four aligned square and circular cylinders downstream.

**Figure 15.**Time variation of the impact force on the individual square and cylinder where the slope bed is equal to 0.02 at the downstream for four cylinders (

**a**) cylinder 1; (

**b**) cylinder 2; (

**c**) cylinder 3; (

**d**) cylinder 4.

**Figure 16.**Evolutions of the free surface elevations and velocity distribution with four square cylinders where the S

_{x}= 0.02 at the downstream.

**Figure 17.**Evolution of free surface elevations and velocity distribution with four circular cylinders where the S

_{x}= 0.02 at the downstream.

**Table 1.**Comparisons between the CPU times (s) with different time-steps and grid-sizes for the case of the dam-break flow passing a square cylinder.

Δt (s) | Grid Size (m) | ||
---|---|---|---|

0.02 | 0.01 | 0.005 | |

0.001 | 48 | 156 | 508 |

0.0005 | 75 | 226 | 711 |

0.00025 | 105 | 306 | 938 |

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**MDPI and ACS Style**

Lo, D.-C.; Tsai, Y.-S. A 3D Fully Non-Hydrostatic Model for Free-Surface Flows with Complex Immersed Boundaries. *Water* **2022**, *14*, 3803.
https://doi.org/10.3390/w14233803

**AMA Style**

Lo D-C, Tsai Y-S. A 3D Fully Non-Hydrostatic Model for Free-Surface Flows with Complex Immersed Boundaries. *Water*. 2022; 14(23):3803.
https://doi.org/10.3390/w14233803

**Chicago/Turabian Style**

Lo, Der-Chang, and Yuan-Shiang Tsai. 2022. "A 3D Fully Non-Hydrostatic Model for Free-Surface Flows with Complex Immersed Boundaries" *Water* 14, no. 23: 3803.
https://doi.org/10.3390/w14233803