# Smoothed Particle Hydrodynamics Modeling with Advanced Boundary Conditions for Two-Dimensional Dam-Break Floods

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. The Numerical Model

#### 2.1. SPH “Semi-Analytic Approach” of SPHERA for the Boundary Condition Scheme

^{−3}), δ

_{ij}is Kronecker’s delta (unitless), $\underset{\_}{u}\equiv \left(u,v,w\right)$ stands for the velocity vector (m s

^{−1}), and t is the time (s). One needs to calculate the momentum and continuity equations at each position of the fluid particle by using SPH formalism and including the boundary terms in the computation. One considers the discretization of the momentum and continuity equations, as provided by the SPH approximation of the first derivative of a generic function (f), according to a semi-analytic approach (“

_{SA}”; [58]):

_{b}” (neighbouring particles with volume ω) in the kernel support of the computational fluid particle (“

_{0}”). In the truncated portion of the kernel support V

_{h}’ (m

^{3}), the generic function f (pressure, velocity, or density, alternatively) can be defined and computed under the assumption of the semi-analytic approach (“

_{SA}”), as developed by the authors of [42]:

_{SA}” values are added in the functions and derivatives within the V

_{h}’ to consider a null normal gradient of reduced pressure at the frontier interface (while considering uniform density):

_{j}” and using the semi-analytic method for the treatment of boundary conditions (second term on the right-hand side):

_{s}(kg m

^{−3}s

^{−1}) represents a fluid-body interaction term. The notation $\langle \rangle $ indicates the SPH particle integral approximation. The approximation of the momentum equation is:

_{s}(m s

^{−2}) represents the acceleration term due to the fluid-body interactions, υ

_{M}(m

^{2}s

^{−1}) is the artificial viscosity [59], m (kg) is the particle mass, and r (m) is the relative distance between the neighboring and the computational particle.

#### 2.2. The SPHERA Scheme for the Transport of Solid Bodies as a Boundary Treatment Scheme

_{s}” depends on the fluid particle “

_{0}” under current investigation; accordingly, the interaction is represented by the subscript “

_{s,0}”. A unique pressure value for each body particle (free-slip conditions) is derived by applying an SPH interpolation over all the pressure values coming from the fluid-body particle interactions:

#### 2.3. Time Integration Scheme

## 3. Case Study

_{1}= 300 mm in the reservoir, and H

_{2}= 265 mm, H

_{3}= 514 mm, and H

_{4}= 762.5 mm, all downstream of the gate). Given that the phenomenon under analysis was a gravity current flow with a horizontal bed, the Reynolds and the Weber numbers, based on both the distance between the gate and the wall and the predicted velocity of the wave front [60], were 3.8 × 10

^{6}and 1.64 × 10

^{5}, respectively. Under these conditions of heavy turbulence in the boundary layer triggered by the advancing wave front, surface tension effects were negligible. The pressures were measured by five piezo-resistive sensors with an acquisition frequency of 20 kHz. Four were in the central line, and an additional one was placed halfway towards the back wall, to detect the presence of any 3D effects of the frontal impact wave. In detail, sensor 1 was positioned at 3 mm from the bottom of the tank, while sensors 2, 3, and 4 were, respectively, at 15 mm, 30 mm, and 80 mm above the bed; sensor 5 was installed at the same height as sensor 2.

## 4. Results and Discussion

_{1}, H

_{2}, H

_{3}, and H

_{4}compared with those obtained using the first (green line) and the second (blue line) boundary treatment techniques. The numerical reconstruction of the free-surface profile from the gate release to the arrival of the secondary wave is pictured in Figure 3. The numerical and Lobovsky’s experimental [55] results at locations H

_{1}, H

_{2}, H

_{3}, and H

_{4}have been compared with the longitudinal water surface profiles at several time instants from the gate release to the arrival of the secondary wave, obtained by the theoretical parabolic profile of Ritter [60]. The SPH simulation reproduces the theoretical solution well, as shown in Figure 4.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Experimental set-up: (

**a**) water level measuring position on the left (lateral view); (

**b**) pressure sensor locations on the right (frontal view). Dimensions are in millimeters.

**Figure 2.**Two-dimensional dam-break phenomena: field snapshots of the absolute value of velocity at several time instants: (

**a**) t = 0.15 s; (

**b**) t = 0.42 s; (

**c**) t = 0.60 s; (

**d**) t = 0.90 s; (

**e**) t = 1.06 s; and (

**f**) t = 1.30 s.

**Figure 3.**Comparison between the experimental and numerical water level elevations at locations: (

**a**) H

_{1}; (

**b**) H

_{2}; (

**c**) H

_{3}; and (

**d**) H

_{4}over time, until the arrival of the secondary wave.

**Figure 4.**Comparison between the experimental and the numerical water level elevations at locations: (

**a**) H

_{1}; (

**b**) H

_{2}; (

**c**) H

_{3}; and (

**d**) H

_{4}with the Ritter theoretical solution at t = 0.1 s, t = 0.2 s, t = 0.3 s, and t = 0.4 s, until the arrival of the secondary wave.

**Figure 5.**Water surface elevation comparison between the numerical results and the experimental data from the secondary wave arrival: (

**a**) H

_{1}; (

**b**) H

_{2}; (

**c**) H

_{3}; and (

**d**) H

_{4}.

**Figure 6.**Temporal evolution of pressure simulated at: (

**a**) P1; (

**b**) P2; (

**c**) P3; and (

**d**) P4 located on the downstream vertical wall of the channel. Comparison between experimental and numerical data in the literature.

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**MDPI and ACS Style**

Mirauda, D.; Albano, R.; Sole, A.; Adamowski, J.
Smoothed Particle Hydrodynamics Modeling with Advanced Boundary Conditions for Two-Dimensional Dam-Break Floods. *Water* **2020**, *12*, 1142.
https://doi.org/10.3390/w12041142

**AMA Style**

Mirauda D, Albano R, Sole A, Adamowski J.
Smoothed Particle Hydrodynamics Modeling with Advanced Boundary Conditions for Two-Dimensional Dam-Break Floods. *Water*. 2020; 12(4):1142.
https://doi.org/10.3390/w12041142

**Chicago/Turabian Style**

Mirauda, Domenica, Raffaele Albano, Aurelia Sole, and Jan Adamowski.
2020. "Smoothed Particle Hydrodynamics Modeling with Advanced Boundary Conditions for Two-Dimensional Dam-Break Floods" *Water* 12, no. 4: 1142.
https://doi.org/10.3390/w12041142