# SPH Modeling of Water-Related Natural Hazards

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## Abstract

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## 1. Introduction

## 2. Two-Phase Coupled Dynamics

#### 2.1. Scouring and Sediment Transport

_{s}, and a magnification factor, η, representing the numerical parameters to be tuned. When the apparent viscosity is lower than the maximum viscosity, the sediment is treated as a non-Newtonian fluid of Bingham type and solid particles are set in motion with constant viscosity μ

_{s}(green curve in Figure 1). The strategy of introducing an upper viscosity limit for the sediment was also followed in [51] in the WCSPH simulation of complex problems in the field of marine engineering; below this maximum limit, the work in [51] adopted a variable apparent viscosity calculated through the M–C theory for the soil phase. The SH critical condition does not require the introduction of a numerical threshold for the viscosity of the solid phase. However, in [7], both the M–C and SH approaches require tuning of the mechanical parameters of the bottom sediment such as the angle of internal friction, φ, and sediment viscosity, μ

_{s}, that became numerical parameters to fit the experimental eroded profile. This may be not be practical when calibration data are not available for the investigated problem.

_{s}of the solid (granular) phase and is characteristic of bed-load transport and fast landslides (see also Section 2.2). In the frictional regime, the mixture (or apparent) viscosity, μ, is calculated as a weighted sum of the pure fluid viscosity μ

_{f}, and the frictional viscosity μ

_{fr}, the latter being evaluated on the basis of the mean effective stress σ’

_{m}, angle of internal friction φ, and the second invariant I

_{2}of the rate of the deformation tensor of the sediment. The frictional viscosity increases as the shear rate tends to zero, in accordance with the pseudo-plastic rheological behavior (dashed blue curve in Figure 1). To avoid the unbounded growth of apparent viscosity of the mixture, a threshold (or maximum) viscosity μ

_{max}is introduced with a physical meaning. Threshold viscosity acts when approaching the zero shear rate; mixture particles with an apparent viscosity higher than the threshold viscosity are considered in the elastic–plastic regime of soil deformation where the kinetic energy of solid particles is relatively small and the frictional regime of the packing limit in the KTGF does not apply. For these reasons, the threshold viscosity is assigned to those particles that are excluded from the SPH computation (continuous red curve in Figure 1; below μ

_{max}, the red curve coincides with the dashed blue curve of the pseudoplastic model). The excluded particles represent a fixed boundary with suitable values of the relevant physical properties and are included in the neighbor list of the nearby moving particle. The value of the threshold viscosity does not require tuning or calibration, but it should be selected for the specific problem; the value assigned to the threshold viscosity is the higher value that does not influence the numerical results significantly. Further increase of the value assigned to the threshold viscosity only affects the computational time because it determines an increase in the maximum value that can be assumed by the apparent mixture viscosity during the computation. The relationship between the mixture viscosity and the time step value that assures the stability of the adopted explicit time-stepping scheme is given by Equation 2.33 in [8]. When the numerical stability of the time integration scheme is dominated by the viscous criterion, the threshold viscosity reduces the computational time.

_{HBP}of the sediment:

_{c}in the HBP model was evaluated in two different calibration procedures depending on the erosion criterion that holds for modeling the yielding mechanism of sediment. The qualitative representation of Equation (1) with n < 1 is denoted by the blue dashed curve in Figure 1.

_{bcr}, replacing the yield stress parameter τ

_{c}, in the HBP model.

_{y}, replacing the yield stress parameter, τ

_{c}, in the HBP model to be defined to determine the apparent viscosity for the sediment.

_{v}, computed for an interface particle as the ratio of the sediment particle volume to the total particle volume within the interaction domain. The onset of suspended transport is determined by c

_{v}falling below the threshold value of 0.3 and the suspended sediment viscosity is computed through [54] the experimental colloidal equation, which is more simple to implement with respect to the piece-wise function adopted in [51]. The density of suspended particles is computed by solving the mass balance equation. Even if, in some cases, the percentage gap between the experimental and numerically predicted maximum scouring depth is significant, it can be seen that the scour process is affected by several random factors and therefore reliable predictions of scouring effects are quite difficult to obtain, even with experimental modeling. The sediment dynamics models based on a synthetic rheological law (e.g., [6]) assume the same rheological behavior for the bed-load transport (frictional regime of KTGF), suspension for dense granular flows (kinetic-collisional regime of KTGF),a and suspension for diluted granular flows (kinetic regime of KTGF). This feature provides advantages and drawbacks with respect to KTGF-based sediment dynamics models (e.g., [8]). The model of [6] can reproduce several sediment transport regimes (not only bed-load transport), but is not coherent with KTGF, some parameters require tuning procedures, and non-transported granular material (e.g., landslides) is not treated.

^{3}has been suggested for the reasonable initial density of the TWPs based on studies of bottom sediment movement. Figure 2 shows that the proposed ISPH model can simulate the real-time processes of the 2D overflow induced scouring. The detailed comparisons between numerical and experimental data can be found in [2].

#### 2.2. Fast Landslides and Dense Granular Flows Interacting with Water

_{2}of the rate of deformation tensor; K, μ

_{0}, and μ

_{∞}are three constant parameters that can be conveniently related to common Bingham rheological parameters, namely the yield stress τ

_{B}and viscosity μ

_{B}. From a physical point of view, μ

_{0}and μ

_{∞}denote the viscosity at very low and very high shear rate, respectively. In order to avoid numerical divergence caused by the unbounded growth of effective viscosity as the shear rate approaches zero, μ

_{eff}is limited to a suitably high threshold value, which is set to μ

_{0}= 103μ

_{B}to assure convergence. The test cases simulated in work in [63] considered the flow on inclined surfaces and analyzed the role of the Froude number [65] during the propagation phase, which may be helpful in designing the control works.

_{0}, referred to as limiting viscosity. The effect of limiting viscosity arises in the frictional regime at low deformation rates near the transition zone to the elastic–plastic regime: in this shear rate interval, a constant value μ

_{0}(lower than μ

_{max}) is assigned to the mixture viscosity (see red dot-dashed curve in Figure 1), thus reducing the computational time in the case where the viscous criterion dominates the numerical stability of the time integration scheme. There are other alternative approaches to keep control of computational time in the simulation of high-viscosity flows. In the work of [73], a semi-implicit integration scheme was proposed to overcome the severe time-stepping restrictions caused by the WCSPH explicit integration scheme when simulating highly viscous fluids, as in the case of lava flow with thermal-dependent rheology. According to this approach, only the viscous part of the momentum equation is solved implicitly, thus saving computational time and obtaining an improved quality of the results with respect to the fully explicit scheme.

_{0}, reaching a reasonable compromise with consumed computational time. This test case is provided as tutorial number 35 in the documentation of SPHERA v.9.0.0 that is freely available in [76].

## 3. Flooding in Complex Topography with the Transport of Sediments

## 4. High Performance Computing Solutions for Complex Hydraulic Engineering Problems

## 5. Conclusions and Future Perspectives

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Rheological models for non-cohesive sediment erosion with bed load transport and for dense granular flows. I

_{2}: second invariant of the rate of deformation tensor; μ

_{app}: apparent viscosity.

**Figure 2.**Snapshots of incompressible smoothed particle hydrodynamics (ISPH) calculated equilibrium scouring pit with different overflow depth: (

**a**) η = 3.3 cm; (

**b**) η = 4.7 cm, and (

**c**) η = 6.0 cm.

**Figure 3.**ISPH computed sediment bed scouring process: (

**a**) snapshots of ISPH calculated vorticity in the scouring pit; (

**b**) Snapshots of the ISPH calculated scouring pit.

**Figure 4.**Frames of the 2D smoothed particle hydrodynamics (SPH) simulation of the rainfall induced shallow landslide occurred on April 2009 at Recoaro, Oltrepò Pavese (Northern Italy). (

**a**) Velocity field; (

**b**) Density field.

**Figure 6.**Demonstrative test case for the meshless SPH method applied to a 3D erosional dam-break flood on complex topography [8]. (

**a**) 3D view. (

**b**) Top view. Digital elevation model (grey, with a black vertex triangulation), liquid SPH particles (blue), mixture SPH particles (brown).

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**MDPI and ACS Style**

Manenti, S.; Wang, D.; Domínguez, J.M.; Li, S.; Amicarelli, A.; Albano, R.
SPH Modeling of Water-Related Natural Hazards. *Water* **2019**, *11*, 1875.
https://doi.org/10.3390/w11091875

**AMA Style**

Manenti S, Wang D, Domínguez JM, Li S, Amicarelli A, Albano R.
SPH Modeling of Water-Related Natural Hazards. *Water*. 2019; 11(9):1875.
https://doi.org/10.3390/w11091875

**Chicago/Turabian Style**

Manenti, Sauro, Dong Wang, José M. Domínguez, Shaowu Li, Andrea Amicarelli, and Raffaele Albano.
2019. "SPH Modeling of Water-Related Natural Hazards" *Water* 11, no. 9: 1875.
https://doi.org/10.3390/w11091875