This section illustrates some engineering applications of the SPH method in the simulation of two-phase flows involved in hydraulic engineering problems of practical interest in the field of water-related natural hazards.
Among the variety of problems involving the numerical simulation of multi-phase flows, two topics of concern in the field of water-related natural hazards have been examined in the following: (i) scour with non-cohesive sediment transport induced by rapidly varied free-surface flow and erosion around complex structures; and (ii) fast dynamics of dense granular flows and landslides. Some relevant works on these topics are discussed, pointing out the peculiar aspects of numerical modeling.
2.1. Scouring and Sediment Transport
An important aspect of the design and maintenance of the bottom-founded submerged structure is non-cohesive sediment scouring. Bottom sediment erosion around the structure is caused by a complex flow pattern induced by the presence of the structure that strongly modifies the upstream undisturbed flow field [
33]. The scouring evolution over time must be properly analyzed and mitigated in order to avoid worsening of structural stability due to foundation exposure.
This topic is widely investigated in fluvial hydraulics, in coastal areas, and in the marine environment. When dealing with a safety assessment of a hydraulic structure placed in the riverbed (e.g., bridges, barrages, etc.), the erosive action of the river stream must be reliably evaluated [
34]. Several empirical formulas to predict final scouring are available, but the phenomenon is time-dependent and is affected by several uncertainties (related to both sediment and flow characteristics) as well as stochastic factors influencing the flow evolution such as transport and deposition of wood debris [
5,
35,
36].
Additionally, in the design of highly demanding marine structures, the scour around the foundations should be dependably predicted [
37,
38]. For instance, a continuous one-way current due to the overtopping flow of an extreme tsunami wave may cause long-term erosion of the foundation on the rear side of the coastal protection structures [
39,
40]. Scour pit evolution behind the seawall induces the formation of eddies with increasing size to adapt the growing dimension of the scour cavity. Excessive sediment erosion directly results in the instability, and even destruction, of these hydraulic structures [
41,
42,
43,
44]. Most empirical relations have been established to predict the bulk and time-averaged sediment transport rate, but their application requires experimental data for calibration. Furthermore, these empirical relations were obtained from small scale laboratory experiments and their extension to a real scale problem may lead to significant erosion estimate errors [
45].
An assessment of the mechanics of detailed temporal erosion processes as well as for the reliable design and assessment of these structures requires both physical investigations and advanced numerical modeling that accounts for the effects of the stochastic variables on the scouring process. Of course, numerical modeling is generally less expensive than physical experimentation, but, in some cases, experimental data may be available from technical literature. As with all engineering problems, any numerical tool must be properly validated against experimental data. After the model has been validated, its parameters require tuning based on the model/experiment results.
Many sediment modeling techniques use mesh-based approaches such as finite difference method and finite volume method to simulate the erosion and fluid-sediment coupled dynamics. However, these mesh-based methods suffer from some intrinsic limitations due to the fixed grid system, which leads to some difficulties in effectively simulating the bed-grain interactions, fluid-sediment momentum transmissions, and the dynamics within the deposits because of their fixed grid system. Furthermore, accurately tracking the free surface and fluid-sediment interface is also a big challenge for these approaches. Lagrangian meshfree particle methods overcome these limitations by intrinsically capturing the free-surface and tracking the particles. These methods have been widely used in recent years for the analysis of erosion processes.
The SPH technique has been proven to be effective in complex multiphase applications [
46] such as water–gas flows or bubbly flow simulations [
47,
48]. In addition, when simulating non-cohesive bottom sediment scouring by rapid water flow, both weakly compressible (WCSPH) and incompressible (ISPH) formulations allow for a reliable description to be obtained. The basic idea is to model the sediment dynamics, likewise a pseudo-fluid, once the onset of sediment particle motion is attained. According to this approach, a criterion should be defined that represents the critical condition for the incipient motion of sediment; this can be done in terms of either critical velocity [
49,
50], or critical shear stress. The first approach may lead to some problems if critical velocity is evaluated as the depth-averaged value that in some cases may not be representative of the scouring, as for the continuous tsunami overflow behind a seawall that induces rapid water depth variation inside the scour pit [
2]. The second approach based on the critical shear stress has been widely adopted to analyze non-cohesive sediment scouring by rapid water flows.
The authors in [
7] implemented two different criteria in a WCSPH model to simulate the erosion process of non-cohesive bed sediment due to constant bottom water discharge from 2D tank (flushing). These criteria are based on Mohr–Coulomb (M–C) yielding theory and on Shields (SH) theory, respectively. The M–C critical condition defining sediment failure and the onset of erosion requires the introduction of a maximum viscosity as the product between sediment viscosity,
μs, 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
μs (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,
μs, 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.
In order to overcome this limitation, [
8] proposed a WCSPH formulation of a mixture model for the analysis of dense granular flows consistent with the kinetic theory of granular flow (KTGF). This mixture model, which avoids the use of an erosion criterion, has been integrated into the FOSS code SPHERA v.9.0.0 (RSE SpA). The relevant physical properties (i.e., density and velocity) of the mixture of pure fluid and granular material are expressed as a function of the volume fraction
ε, occupied by each phase at a material point. The balance equations for the mixture are defined accordingly and discretized consistently with the WCSPH approach. The frictional regime of the mixture dynamics is represented under the packing limit of the KTGF, which holds for the volume fraction
εs 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
μf, and the frictional viscosity
μfr, the latter being evaluated on the basis of the mean effective stress
σ’
m, angle of internal friction
φ, and the second invariant
I2 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
μmax 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
μmax, 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.
In [
8], some experiments of erosional dam breaks adopting the physical values for the mechanical parameters of the sediments including the angle of internal friction were simulated with good reliability. The 2D experimental tests, involving rapidly varied mixture flows with erosion and the transport of bed sediment, were simulated in [
7] and [
8]. Moreover, in [
6], mixture flow computation was performed with the open-source DualSPHysics solver [
52] accelerated with a graphic processing unit (GPU). The adopted WCSPH algorithm, representing the improvement of the model in [
53], reproduces the dynamics of the bottom sediment phase combining two erosion criteria with the non-Newtonian rheological model. The adopted rheological model is based on the Bingham-type Herschel–Bulkley–Papanastasiou (HBP) model providing the apparent viscosity
μHBP of the sediment:
This model allows for the transition between un-yielded (fixed) and yielded (mobile) granular material to be described without introducing a maximum value of the viscosity for the solid phase. Proper selection of the HBP model’s parameters,
m and
n, allows for the reproduction of the required rheological behavior as the Newtonian or the pseudoplastic model. The yield stress parameter
τc 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.
If a solid particle is identified as being near the sediment–water interface by means of the criteria described in [
6], then the Shields criterion determines the onset of bed erosion at the sediment–water interface and provides the critical bed shear stress
τbcr, replacing the yield stress parameter
τc, in the HBP model.
If the particle is near the water–sediment interface, but Shields erosion criterion is not satisfied or if the particle is far from the water–sediment interface, then the yielding mechanism is modeled through the Drucker–Prager criterion that allows the critical shear stress τy, replacing the yield stress parameter, τc, in the HBP model to be defined to determine the apparent viscosity for the sediment.
As a result, three distinct regions may be defined, starting from the water–sediment interface and going downward: (a) eroded sediment exceeding the Shields critical bed shear stress (bed load transport); (b) yielded sediment (plastic deformation with slow kinematics); and (c) un-yielded sediment (high viscosity, static condition).
The model proposed in [
6] also accounts for suspended transport. Following [
51], the identification of the suspension layer is obtained through a non-dimensional concentration,
cv, 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
cv 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.
The work in [
2] investigated by means of ISPH the water-induced 2D sediment scouring where the erosion process is mainly related to loose sediment particles suspended in the water flow, as in the case of scouring behind a seawall produced by continuous tsunami overtopping. In contrast to the physics of sediment flushing, where sediment moves as bed load at very high density, in the case of overtopping erosion, the solid particles move more loosely and the density of turbid water is significantly lower than the sediment density. In such situations, the erosion dynamics is controlled by the balance between the suspension effect due to turbulent mixing and settling of suspended solid particles owing to gravity force. This process is strongly affected by the size of numerical particles, which is usually far bigger than the size of a real sediment grain in practical problems. For the reasons explained above, adoption of real sediment density in the computation may lead to an unreliable representation of the erosion. The model proposed in [
2] introduces a simplification due to the size (and hence number) of the particles required to properly model the turbulent mixing. This model is based on the concepts of numerical turbid water particle (TWP) and clear water particle (CWP) to simulate the sediment-entrained flow in cases where the granular particles of sediment move more loosely and sediment is washed away, mainly in the form of a suspended load. Due to the size considerations discussed above, numerical particles should be treated as a combination of clear water and turbid water particles if a suspended load is simulated; therefore, eroded solid particles represent a water–sediment mixture whose density is calculated based on the integral interpolation theory, thus accounting for density reduction as solid particles are suspended and mixed with water. The value of 1250 kg/m
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].
In the subsequent work in [
1], the ISPH model of [
2] was successfully extended to the simulation of 3D local bed scour induced by clear water stationary flow passing a submerged vertical cylinder of relatively large size. Additional formulations were introduced to account for transversal and longitudinal bed slope. The erosion model was based on the turbidity water particle concept and the sediment motion was initiated when the calculated shear stress on the interface particles exceeded the critical value. The 3D ISPH erosion model was used to simulate the scouring process around a large vertical cylinder with a diameter of 60 cm in
Figure 3, where the vorticity and the shape of the scour pit are illustrated at time
t = 2.5 s and
t = 3.5 s. The numerical results show that the proposed ISPH model could simulate the relevant features of the flow and the scour process (
Figure 3). The detailed comparisons between the numerical and experimental data can be found in [
1] and the scour dynamics were validated with a suitable degree of accuracy for engineering purposes. Even if a suitable representation of the complex scouring process can be obtained, the vorticity field shows some numerical noise (
Figure 3a). Improvement of the calculated vorticity field could be obtained through the approach proposed in [
46].
2.2. Fast Landslides and Dense Granular Flows Interacting with Water
Numerical modeling of dense granular flows and landslides is still a challenging topic, especially when considering the interaction between the sediment and the water that may be both an internal interaction, related to pore water in landslide-prone saturated soil, and an external interaction with stored water in a basin with unstable slopes.
The interaction between pore water and soil matrix is a fundamental aspect that influences the triggering and propagation of shallow landslides induced by intense rainfall events that represent the most common natural hazards in some areas of the world [
14]. Intense rainfall events induce water infiltration at slopes, leading to an increase of the volumetric water content and pore water pressure that worsen the slope stability of the soil layer close to the surface and may cause its failure. Reliable assessment of landslide susceptibility also requires a proper definition of the rainfall characteristics considering recent climate trends affecting rainfall [
55,
56].
If landslide triggering occurs, in the post-failure phase, the pore water content, combined with geo-mechanical properties of the soil, influence the sediment dynamics, and may induce in some cases flow-like fast earth movements classified as complex landslides because their run-out start as shallow rotational-translational failure, but changes into dense granular flows due to the large water content [
15]. In this case, the fast landslide dynamics is more difficult to predict using traditional analytical models and a numerical approach could be helpful for landslide susceptibility assessment and the creation of debris–flow inundation maps.
The work in [
57] proposed a combined triggering–propagation modeling approach for the evaluation of rainfall induced debris flow susceptibility. They adopted the transient rainfall infiltration and grid-based regional slope-stability model (TRIGRS) [
58], which is based on the infinite slope stability approach, to obtain a map of potentially unstable cells within a study catchment under an intense rainfall event. Since not all unstable cells, in general, evolve to a debris flow, an empirical instability-to-debris-flow triggering threshold is calibrated on the basis of observed events and used to identify, among unstable cells, the so-called triggering cells that could most likely contribute to debris flow. The triggering threshold is, therefore, the key element that allows coupling between the TRIGRS slope instability model and the debris flow propagation model FLO-2D [
59], a finite volume model that numerically solves the depth-integrated flow equations. The work in [
57] assumed a zero excess rainfall intensity and a total friction slope depending on the Bingham-type rheological parameters as a function of the sediment concentration. The calibration of the triggering threshold with geo-morphological data of the catchment area represents a crucial step for obtaining reliable susceptibility maps in the nearby areas. Back-analysis of a catastrophic event that occurred on 1 October 2009 in the Peloritani mountain area (Italy) provided fairly good results.
In order to quantify the level of risk addressing the uncertainty inherent in landslide hazard, the susceptibility evaluation represents only one of the relevant issues involved in risk analysis. Additionally, run-out dynamics should be properly assessed in order to provide a quantitative estimate of the landslide hazard and select appropriate protective measures for risk mitigation. In order to reach such a goal, reliable predictive models should be used to obtain quantitative information on the destructive potential of the landslide. This is mainly related to the following characteristics: run-out distance; width of damage corridor; travel velocity; characteristic depth of the moving mass; and characteristic depth of the deposits [
60].
The work in [
61] proposed a 2D depth-integrated, coupled, SPH model for predicting the path, velocity, and depth of flow-like landslides. While the post-failure flow model of [
57] assumes the heterogeneous moving mass as a single-phase continuum, [
61] modeled the dense granular flow as a two-phase mixture composed of a solid skeleton with the voids filled by a liquid phase. Assuming that the shear strength of the liquid phase can be disregarded, the stress tensor within the mixture is composed of pore pressure and effective stress. Then, the mixture dynamics was described by quasi-Lagrangian depth integrated governing equations of mass and momentum balance, and the pore pressure dissipation equation that were discretized according to the standard SPH approach. Depth-integrated equations do not provide information on the vertical flow structure that is needed for evaluating shear stress on the bottom and depth integrated stress tensor. For this reason, [
61] assumed that this vertical structure would be the same as the uniform steady-state flow according to the so-called model of the infinite landslide having constant depth and moving at a constant velocity on a constant slope. The work in [
61] adopted the simple method proposed in [
62] to obtain the bottom shear stress in a non-Newtonian fluid of Bingham-type. The model in [
61] was used to simulate some catastrophic event that occurred on May 1998 in the Campania region (Italy), showing the relevant role of geotechnical parameters (especially the fluid phase and angle of internal friction) for the reliable prediction of the run-out distance, velocity, and height of the landslides. Proper selection of the value assigned to these parameters assured the best agreement with the field observations.
Some flow-like landslides are characterized by a relatively small average depth if compared with the horizontal linear dimensions, therefore the assumption of a depth-integrated model is strongly consistent with the physics of the phenomenon. Furthermore, there could be rainfall induced landslides where the initial average depth is comparable with the horizontal length and width. In these cases, significant variations of the vertical thickness and the vertical velocity profile may occur along the landslide body in the flow direction. This is illustrated in
Figure 4, showing the preliminary results of an on-going study. In this new study, a narrow landslide that occurred during an intense rainfall event on April 2009 in a hilly area of the Oltrepò Pavese, named the Recoaro Valley, Northern Italy, was reproduced by means of the 2D WCSPH simulation. It can be seen that in the early phase, just after the failure, the mass portion close to the landslide front moved faster than the rear portion. Additionally, just after the impact against the downstream vertical wall of a damaged building, the landslide front began decelerating while the rear mass portion on the steep slope still maintained a relatively high average speed.
In the peculiar case described above, the modeling approach based on the infinite landslide with constant depth and moving at a constant velocity on a constant slope may be less appropriate. Therefore, the solution of the governing equation in the three-dimensional form seems more appropriate than the depth-integrated model. The simulation shown in
Figure 4 was carried out with the code SPHERA v.9.0.0 adopting the mixture model for dense granular flow discussed in [
8]. Even if the code has a 3D formulation, a 2D approach may be conveniently adopted in this case because the landslide is relatively narrow and the flow may be assumed to be identical on the vertical planes along the flow direction.
The work in [
63] used a finite volume approach to simulate mudflows and hyper-concentrated flows characterized by suspended fine material by adopting a Bingham rheological model. Similar to [
64], the Cross model was adopted for modeling the non-Newtonian flow, assuming that the constant parameter
m was equal to 1, resulting in the following formulation of the apparent viscosity:
In Equation (2)
denotes the shear rate defined through the second invariant
I2 of the rate of deformation tensor;
K,
μ0, and
μ∞ are three constant parameters that can be conveniently related to common Bingham rheological parameters, namely the yield stress
τB and viscosity
μB. From a physical point of view,
μ0 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,
μeff is limited to a suitably high threshold value, which is set to
μ0 = 103
μB 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.
Landslides occurring at the slopes of confined water bodies (e.g., artificial basin or river-valley reservoir) or at coastal regions involve complex interactions between the solid and the fluid phase. In the post-failure phase of rapid landslides generated at the slope of a water body, an impulse wave is generally induced, which is usually referred to as tsunami [
66,
67]. The characteristics of the generated water wave are related to the velocity and the shape (e.g., thickness and slope angle) of the landslide front, whose dynamic deformation is in turn affected by the water induced stresses at the interface while the landslide is penetrating the water body. Accurate prediction of the landslide induced wave hazard depends on reliable numerical models that can simulate the coupled dynamics.
The work in [
11] proposed a WCSPH model for the analysis of impulsive wave generated in a basin by a deformable landslide. In order to properly reproduce landslide deformation during the post-failure phase, they adopted an elastic–plastic constitutive model for the soil combined with the Drucker–Prager criterion. To account for the interaction with stored water, occurring at a very small timescale due to the fast dynamics, [
11] introduced a bilateral coupling model consisting of two sequential steps: the interface soil particles were initially considered as the moving deformable boundary whose velocity and position was used for solving the governing equations of the water phase; then the soil constitutive equation was solved with the corrected stress taking into account the water-induced surface force. This coupling model would require, in theory, an iterative procedure to assure the consistency of both stress and deformation at the interface at each time step. However, the sequential approach is quite suitable because the interface deformation is mainly caused by the landslide dynamics and small displacements occur within a time step. The mechanical parameters in the model of [
11] have a physical meaning and the corresponding values can be deduced from conventional soil mechanic experiments. Two experiments were simulated concerning the wave generated by a slow landslide [
12] and a fast landslide [
68], respectively, obtaining in both cases good agreement with the experimental data.
The work in [
10] proposed a hybrid model for simulating the coupled dynamics between the landslide and stored water as well as the propagation of generated surge waves. The discrete element method (DEM) was used for simulating the landslide dynamics, while WCSPH was used to solve the governing equations for water. Spurious numerical noise affecting the water pressure field was removed by applying the
δ-SPH technique [
69]. The interaction mechanism between the solid and fluid phase in the hybrid DEM-SPH model was based on the drag force and buoyancy. At each time step, these forces were first calculated considering (i) the initial position and velocity of both fluid and solid particles, (ii) the pressure of the fluid, and (iii) the local soil porosity,
ε, evaluated at the landslide–water interface with a kernel interpolation of the solid particle volume. Based on the calculated drag force and buoyancy, the corresponding new position and velocity of both fluid and solid particles were then calculated to solve the corresponding discretized governing equations. The updated position and the velocity field were subsequently used to recalculate drag force and buoyancy. Therefore, this interaction mechanism requires an iterative process to assure convergence of the calculated forces and consistency of the displacements and velocity field of the involved phases. The hybrid DEM-SPH model was used to simulate the sliding along a 45° sloping plane of a rigid body that mimics a submarine landslide, obtaining a suitable prediction of the experimental time evolution of water surface elevation [
70]. A modified simulation was additionally performed by considering the deformability of the sliding body, showing that smaller and less violent surge waves were generated owing to the landslide deformation.
A similar approach was proposed in [
71] for the analysis of underwater granular collapse by coupling WCSPH (for the fluid phase) with DEM (for the solid incoherent phase). A coupling module was developed for fluid–grain interaction: the force exerted by the fluid on a solid particle was obtained by integrating the contributions on its surface; in turn, the effect of the solid particles on the fluid motion was calculated by including the neighboring DEM particles in the SPH interpolation of the governing equations for the considered liquid particle. Another attempt of coupling SPH, in particular, the DualSPHysics model, with a distributed-contact discrete-element method (DCDEM) was proposed in [
72] to explicitly solve the fluid and solid phases to model a real case of an experimental debris flow. An experimental setup for stony debris flows in a slit check dam was reproduced numerically, where solid material was introduced through a hopper, assuring a constant solid discharge for the considered time interval.
The reference mixture model of [
8] for the dynamics of dense granular flow was modified by introducing a numerical parameter,
μ0, 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
μ0 (lower than
μmax) 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.
The mixture model of [
8], modified with limiting viscosity, was successfully applied in [
9] to the analysis of a fast massive landslide at the slope of an artificial reservoir. The simulated case reproduced a two-dimensional scale laboratory test carried out in 1968 at the University of Padua reproducing in Froude similitude a characteristic cross-section of the Vajont artificial basin. In 1963, a catastrophic landslide, estimated to be about 270 million cubic meters, fell into the Vajont reservoir, generating a tsunami that caused about 2000 casualties [
74].
The experimental campaign of Padua aimed to evaluate the effects of both the material type and the landslide falling time on the maximum run-up of the generated wave over the opposite side of the valley. The landslide velocity was imposed through a rigid plate pulled by an engine through a steel cable. Several tests were carried out considering two types of rounded gravel (3–4 mm and 6–8 mm), crushed stone, and squared tile. For each of these material types, different values of the run-out velocity were tested by varying the plate stroke (in the range 0.5–0.8 m) and its velocity, thus resulting in a landslide falling time ranging approximately between 15 s and 500 s at full scale.
The frames in
Figure 5 show the simulation results obtained with SPHERA v.9.0.0 [
75]; some representative instants were selected during the acceleration phase where the vertical rigid plate pushes the landslide toward the basin, and the subsequent run-out phase of the generated wave climbs the opposite slope. Fairly good results were obtained in terms of the maximum height reached by the rising wave front. The left-hand panels show the velocity field evolution. The right-hand panels show the density field; it can be noticed that the lower landslide layer (light-brown) has a higher density because of the saturation. The effect of pore water allows for the landslide dynamics to be obtained more close to the experimental results because at
t =1.35 s, its front comes into contact with the opposite side of the basin. No tuning of the physical parameters of the model was necessary. The required accuracy was controlled by the proper adoption of the limiting viscosity value,
μ0, 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].
Calculation of the flow-impact induced forces on the submerged structure may be required in order to estimate landslide and water-related hazards. Using the standard WCSPH approach usually leads to high-frequency numerical noise in the pressure field [
29] that may taint the calculation of the load time history acting on the solid structure. To overcome this problem and obtain the correct prediction of the hydrodynamic load for structural stability assessment, several strategies have been successfully introduced [
77].