# 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

## References

- Biscarini, C.; Francesco, S.D.; Manciola, P. CFD modelling approach for dam break flow studies. Hydrol. Earth Syst. Sci.
**2010**, 14, 705–718. [Google Scholar] [CrossRef] [Green Version] - Akoz, M.S.; Kirkgoz, M.S.; Oner, A.A. Experimental and numerical modeling of a sluice gate flow. J. Hydr. Res.
**2009**, 47, 161–176. [Google Scholar] [CrossRef] - Istrati, D.; Hasanpour, A. Numerical investigation of dam break-induced extreme flooding of bridge superstructures. In Proceedings of the 3rd International Conference on Natural Hazards & Infrastructure, Athens, Greece, 5–7 July 2022. [Google Scholar]
- Tsai, Y.S.; Lo, D.C. A ghost-cell immersed boundary method for wave-structure interaction using a two-phase flow model. Water
**2020**, 12, 3346. [Google Scholar] [CrossRef] - Xiang, T.; Istrati, D.; Yim, S.C.; Buckle, I.G. Tsunami loads on a representative coastal bridge deck: Experimental study and validation of design equations. J. Water Port Coast. Ocean Eng.
**2020**, 146, 04020022. [Google Scholar] [CrossRef] - Hasanpour, A.; Istrati, D. Extreme storm wave impact on elevated coastal buildings. In Proceedings of the 3rd International Conference on Natural Hazards & Infrastructure, Athens, Greece, 5–7 July 2022. [Google Scholar]
- Azadbakht, M. Tsunami and hurricane wave loads on bridge superstructures. Ph.D. Thesis, Oregon State University, Corvallis, OR, USA, 2013. [Google Scholar]
- Xiang, T.; Istrati, D. Assessment of extreme wave impact on coastal decks with different geometries via the arbitrary Lagrangian-Eulerian method. J. Mar. Sci. Eng.
**2021**, 9, 1342. [Google Scholar] [CrossRef] - Istrati, D.; Buckle, I.G. Tsunami Loads on Straight and Skewed Bridges—Part 2: Numerical Investigation and Design Recommendations; Oregon Department of Transportation: Salem, OR, USA, 2021. [Google Scholar]
- Hasanpour, A.; Istrati, D. Reducing extreme flooding loads on essential facilities via elevated structures. In Proceedings of the ASCE Lifelines Conference, Virtual, 31 January–11 February 2022. [Google Scholar]
- Elliot, R.C.; Chaudhry, M.H. A wave propagation model for two-dimensional dam-break flows. J. Hydr. Res.
**1992**, 30, 467–483. [Google Scholar] [CrossRef] - Molls, T.; Chaudhry, M.H. Depth-averaged open-channel flow model. J. Hydr. Engrg.
**1995**, 121, 453–464. [Google Scholar] [CrossRef] - Sanders, B.F. Non-reflecting boundary flux function for finite volume shallow-water models. Adv. Water Resour.
**2001**, 25, 195–202. [Google Scholar] [CrossRef] - Rao, V.S.; Latha, G. A slope modification method for shallow water equations. Int. J. Numer. Meth. Fluids
**1992**, 14, 189–196. [Google Scholar] [CrossRef] - Fraccarollo, L.; Toro, E.F. Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems. J. Hydr. Res.
**1995**, 33, 843–864. [Google Scholar] [CrossRef] - Wang, J.S.; He, Y.S.; Ni, H.G. Two-dimensional free surface flow in branch channels by a finite-volume TVD scheme. Adv. Water Resour.
**2003**, 26, 623–633. [Google Scholar] [CrossRef] - Caleffi, V.; Valiani, A.; Bernini, A. High-order balanced CWENO scheme for movable bed shallow water equations. Adv. Water Resour.
**2007**, 30, 730–741. [Google Scholar] [CrossRef] - Nujic, M. Efficient implementation of non-oscillatory schemes for the computation of free-surface flows. J. Hydr. Res.
**1995**, 3, 101–111. [Google Scholar] [CrossRef] - Savic, L.J.; Holly, F.M., Jr. Dambreak flood waves computed by modified Godunov method. J. Hydr. Res.
**1993**, 1, 187–204. [Google Scholar] [CrossRef] - Glaister, P. Approximate Rieman solutions of shallow water equations. J. Hydr. Res.
**1988**, 26, 293–306. [Google Scholar] [CrossRef] - Harten, A. High resolution schemes for hyperbolic conservation laws. J. Comput. Phys.
**1983**, 49, 357–393. [Google Scholar] [CrossRef] [Green Version] - Yang, J.Y.; Hsu, C.A.; Chang, S.H. Computations of free surface flows, part 2: 2D unsteady bore diffraction. J. Hydr. Res.
**1993**, 31, 403–412. [Google Scholar] [CrossRef] - Yee, H.C. Construction of explicit and implicit symmetric TVD schemes and their applications. J. Comput. Phys.
**1987**, 68, 151–179. [Google Scholar] [CrossRef] [Green Version] - Yang, H.Q.; Przekwas, A.J. A comparative study of advanced shock-capturing schemes applied to Burger’s equation. J. Comput. Phys.
**1992**, 102, 139–159. [Google Scholar] [CrossRef] - Jeng, Y.N.; Payne, U.J. An adaptive TVD limiter. J. Comput. Phys.
**1995**, 118, 229–241. [Google Scholar] [CrossRef] - Wang, J.S.; Ni, H.G.; He, Y.S. Finite-difference TVD scheme for computation of dam-break problems. J. Hydr. Engrg. ASCE
**2000**, 126, 253–262. [Google Scholar] [CrossRef] - Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S.R. Uniformly high-order accurate essentially non-oscillatory schemes, III. J. Comput. Phys.
**1987**, 71, 231–275. [Google Scholar] [CrossRef] - Liu, X.-D.; Osher, S.; Chan, T. Weighted essentially non-oscillatory schemes. J. Comput. Phys.
**1994**, 115, 200–212. [Google Scholar] [CrossRef] [Green Version] - Jiang, G.-S.; Shu, C.-W. Efficient implementation of weighted ENO schemes. J. Comput. Phys.
**1996**, 126, 202–228. [Google Scholar] [CrossRef] [Green Version] - Qiu, J.; Shu, C.-W. On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes. J. Comput. Phys.
**2002**, 183, 187–209. [Google Scholar] [CrossRef] [Green Version] - Wei, Z.; Dalrymple, R.A. Numerical study on mitigating tsunami force on bridges by an SPH model. J. Ocean Eng. Mar. Energy
**2016**, 2, 365–380. [Google Scholar] [CrossRef] [Green Version] - Sarfaraz, M.; Pak, A. SPH numerical simulation of tsunami wave forces impinged on bridge superstructures. Coastal Eng.
**2017**, 121, 145–157. [Google Scholar] [CrossRef] - Hasanpour, A.; Istrati, D.; Buckle, I. Coupled SPH–FEM modeling of tsunami-borne large debris flow and impact on coastal structures. J. Mar. Sci. Eng.
**2021**, 9, 1068. [Google Scholar] [CrossRef] - Hasanpour, A.; Istrati, D.; Buckle, I.G. Multi-physics modeling of tsunami debris impact on bridge decks. In Proceedings of the 3rd International Conference on Natural Hazards & Infrastructure, Athens, Greece, 5–7 July 2022. [Google Scholar]
- Chen, X. A fully hydrodynamic model for three-dimensional, free-surface flows. Int. J. Numer. Meth. Fluids
**2003**, 42, 929–952. [Google Scholar] [CrossRef] - Hirt, C.W.; Nichols, B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys.
**1981**, 39, 201–225. [Google Scholar] [CrossRef] - Lo, D.C. A novel volume-of-solid-based immersed-boundary method for viscous flow with a moving rigid boundary. Numer. Heat Transf. B Fundam.
**2015**, 68, 115–140. [Google Scholar] [CrossRef] - Chorin, A.J. A numerical method for solving incompressible viscous flow problem. J. Comput. Phys.
**1967**, 2, 12–26. [Google Scholar] [CrossRef] - Weymouth, G.D.; Yue, D.K.-P. Conservative Volume-of-Fluid method for free-surface simulations on Cartesian-grids. J. Comput. Phys.
**2010**, 229, 2853–2865. [Google Scholar] [CrossRef] - Raad, P.E.; Bidoae, R. The three-dimensional Eulerian–Lagrangian marker and micro cell method for the simulation of free surface flows. J. Comput. Phys.
**2005**, 203, 668–699. [Google Scholar] [CrossRef] - Xie, Z.; Stoesser, T.; Xia, J. Simulation of three-dimensional free-surface dam-break flows over a cuboid, cylinder, and sphere. J. Hydraul. Eng.
**2021**, 147, 06021009. [Google Scholar] [CrossRef] - Fennema, R.J.; Chaudhry, M.H. Explicit methods for two-dimensional unsteady free-surface flow. J. Hydr. Engrg.
**1990**, 116, 1013–1034. [Google Scholar] [CrossRef] - Frandsen, J.B. Free-Surface Lattice Boltzmann Modeling in Single Phase Flows. In Advanced Numerical Models for Simulating Tsunami Waves and Runup; Liu, P., Yeh, H., Synolakis, C., Eds.; World Scientific: Singapore, 2008; Volume 10. [Google Scholar]

**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