This section describes the simulations of some mixture multiphase free-surface flows carried out with SPHERA v 8.0. All the simulations performed concern the analysis of the fast dynamics of free surface flows involving the interaction between the water and non-cohesive sediment.
3.1. Two-Dimensional Erosive Dam Break
The two-dimensional erosive dam break of a water column collapsing over an erodible bed has been performed with SPHERA to evaluate the threshold viscosity and to assess the optimum value for the limiting viscosity.
The problem is illustrated in
Figure 2 showing on the left-hand panel the water and solid bed distribution at the initial time; this demonstrative test represents a modification of an erosional dam break tutorial available in SPHERA v8.0 package [
35] and shares some relevant features with similar laboratory tests that have been adopted for numerical model validation, including weakly compressible SPH [
57,
58] and incompressible SPH [
29].
In the performed test, the initial depth of the water column is much greater than the thickness of the bed sediment, resulting in a higher and faster excavation at the wave front, as shown by the representative frames on the right-hand side of
Figure 2. Moreover, the horizontal dimension of the computational domain is rather limited and therefore the generated wave suddenly impacts against the downstream wall originating a vertical jet.
The initial length and height of the water column are Lw = 1.50 m and Hw = 1.30 m, respectively. The density and bulk modulus of the liquid component are ρf = 1000 kg/m3 and Kf = 1.30 × 106 Pa. The non-cohesive granular bed at initial time has uniform thickness of Hs = 0.20 m and the longitudinal length is Ls = 3.00 m. The geotechnical parameters of the solid component are: median diameter d50 = 1.0 mm, density ρs = 1500 kg/m3, bulk modulus Ks = 1.95 × 106 Pa, angle of internal friction ϕ = 20° and porosity εf = 0.5. The saturated mixture density is obtained through Equation (5) ρ = 1250 kg/m3.
The adopted spatial resolution is dx = 0.005 m while the smoothing length is assumed h = 1.3 dx. The total number of particles in all dam break simulation is 11,300. The coefficient of artificial viscosity is set at αM = 0.1, while the speed of sound for both components of the mixture is cs = 36.1 m/s.
The water column, initially at rest, is confined on the right-hand side by the vertical wall of the tank, while the vertical left-hand boundary of the water is free to collapse. A shallow water wave propagates from the toe of the column toward the left-hand vertical wall of the tank; the mean effective stress represents the main cause of mobilization of the solid grains (as observed also in [
36]) and formation of a bed-load transport layer that moves along with the water wave. The shape of the bed-load transport layer is such that a sloping surface develops below the front of the water wave that is therefore pushed upward. A concentric plunging wave, made of an outer water coating and an inner sediment layer, develops and subsequently impacts the vertical downstream wall at about
t = 0.29 s. After the impact, a vertical water jet climbs on the left-hand wall of the tank while a sediment wedge accumulates at the left corner below the water jet.
The first set of simulations has been carried out to evaluate the convergence of the numerical results with respect to the threshold viscosity.
Table 1 summarizes this set of runs showing the value of threshold viscosity and limiting viscosity adopted in each run. The threshold viscosity is progressively increased from 10 kPa s to 80 kPa s. The limiting viscosity is excluded by assuming in each run
µ0 >
µthr. The last column of
Table 1 shows the total elapsed time in every run.
The time evolution of the water free surface and of the interface between water and the bed load transport layer have been monitored for quantitative comparison among the different runs to obtain a measure of the influence of µthr on the mixture dynamics and to evaluate numerical convergence of results.
During the early phase of the water collapse the differences between the results of the simulations in
Table 1 are almost negligible; this is reasonably because the mixture particles are interested by an elevated rate of deformation, far from the elastic–plastic regime. On the contrary, the results become different after the impact on the downstream wall because stagnation occurs; some solid grains at the lower left-hand corner of the tank enter the elastic–plastic regime owing to their low rate of deformation thus becoming fixed.
For the above reasons, the comparison of results from the runs is made at a reference time of
t = 0.5 s.
Figure 3 compares the free surface (fs) and the mixture–water interface (bls) close to the impact side for all the runs in
Table 1. It can be seen that, as the threshold viscosity approaches the maximum value of 80 kPa s, the maximum rise of the bed-load sediment layer reduces and stabilizes at 0.3 m. The slope of the bottom sediment layer is less affected by changes in the threshold viscosity, as there are minimal differences between the four mixture–water interfaces at
x > 0.05 m.
In
Figure 3, it can be seen that the water free surface is poorly affected by changes in
µthr, probably because the changes in the sediment layer profile are small, in percentage, with respect to the average water depth. The evolution of the free surface and of the bottom interface is quite regular in every run, as well as the pressure field that does not show significant instability phenomena at the transition interface between the water and the bed-load layer. The maximum pressure on the impacting area of the downstream vertical wall is around 10 kPa.
Based on the discussed results, it can be concluded that Run 2 allows obtaining an average profile and maximum run-up of the sediment layer which is rather close to the ones obtained in Runs 3 and 4 with higher values of the threshold viscosity; therefore, solution convergence can be obtained by assuming
µthrmin = 20 kPa s and, in addition, this choice allows saving approximately 69% of computational time with respect to Run 4 and about 46% with respect to Run 3 (see
Table 1).
To obtain a quantitative representation of the numerical optimization, the percentage average errors of the mixture–water interfaces are also reported in
Table 1. These values have been obtained by calculating, near the left-hand corner of the tank, the average of the percentage differences between the mixture–water interface computed in the corresponding run and the mixture–water interface of Run 4. The average percentage error reduces about 1/6 from Run 1 to Run 2; indeed, it remains almost the same from Run 2 to Run 3. These results confirm the above conclusion.
A second set of runs was performed to investigate the effects of the limiting viscosity on saving the computational time and preserving numerical convergence of results.
Table 2 shows the details of the second set of runs carried out. Run 2 from previous analysis has been assumed as a reference for comparison. Runs 5 and 6 adopt the same value of the threshold viscosity used in the reference run
µthr = 20 kPa s while their limiting viscosity has been reduced to 15 kPa s and 5 kPa s, respectively, for evaluating the lower value
µ0min of the limiting viscosity that reduces the computational time while preserving the results accuracy with respect to the reference run.
Run 7 was carried out to test an alternative rheological behavior (green line in
Figure 1) with SPH mixture particles held fixed in case their viscosity values are higher than
µ1 = 5 kPa s.
As previously done, the obtained results have been analyzed at the reference time t = 0.5 s at which the maximum discrepancies were detected in first set of runs.
The obtained free surface (fs) and mixture–water interface (bls) profiles are compared in
Figure 4. In this case, the water profile is also poorly affected by the changes in the limiting viscosity: the free surface for Runs 5–7 is practically coincident with the shape obtained in the reference Run 2.
When considering Runs 5 and 6, the interface between the bed-load sediment layer and the water is weakly sensitive to the decrease of the limiting viscosity. In these runs, the maximum height at which the suspended sediment can rise is approximately the same as the maximum height obtained in the reference Run 2. This means that numerical optimization can be achieved by assuming the limiting viscosity equal to 5 kPa s thus obtaining almost the same average percentage error as in Run 5 with a significant decrease of the computational time that for Run 6 is about 71% lower than the value obtained in the reference Run 2 (see
Table 2).
The same benefit of Run 6 in term of saving computational time can be obtained by excluding the effect of the limiting viscosity and setting the viscosity
µ1 equal to 5 kPa s, as it can be seen from Run 7 in
Table 2. The mixture dynamics deviates significantly from the reference run because the maximum run-up of the bed-load sediment layer is considerably underestimated. This can be seen in
Figure 4, where the profile of the bed-load sediment layer remains below the reference profile in the interval
x < 0.1 m.
To obtain clear representation of this behavior,
Figure 5 shows a magnification of the mixture flow around the lower left-hand corner of the tank for both Run 6 (left-hand panel) and Run 7 (right-hand panel).
It can be seen that in Run 7 the run-up of the bed-load plume is smaller than Run 6: such behavior is due to the presence of a cluster of solid particles forced to remain fixed (colored in light brown in
Figure 5b) whose apparent viscosity is higher than
µ1 = 5 kPa s. For this reason, these particles are excluded from the computation and kept fixed, thus neglecting part of the kinetic energy of the bed-load layer that prevents the bed-load plume from reaching the same height of Run 6 where the threshold viscosity is instead much greater to avoid the formation of a cluster of solid particles in the elastic–plastic regime (
Figure 5a).
This result confirms that introduction of limiting viscosity allows obtaining a suitable physical representation of the phenomenon saving computational time. Further reduction of threshold viscosity to obtain similar benefit in term of computational time leads to a less consistent representation of the mixture flow.
3.2. Validation
This section describes the test case that has been performed for validating the mixture model implemented in SPHERA on a fast landslide. The test reproduces a two-dimensional experiment that was set up in 1968 at the Hydraulic Laboratory of Padua University for the analysis of the huge landslide occurred in 1963 on the left-hand slope of the Vajont artificial basin.
The 2D experimental setup and its functioning scheme are widely explained in [
17] to which the reader is referenced for the details.
The left-hand panel in
Figure 6 replicates the central part of the two-dimensional 1:500 scale model schematically reproducing the geometry of a representative cross section of the Vajont basin close to the dam. As the landslide dynamics was driven mainly by gravity, Froude similitude was adopted to convert both height and time to the full scale. Several experiments were carried out in the post-event campaign by testing: different run-out durations, ranging between 0.7 s and 22.5 s, corresponding to 15.7 s and 503.1 s, respectively, at full scale, different gravel sizes for the landslide, and two strokes of the rigid plate pushing the landslide (i.e., 0.5 and 0.8 m).
In the experimental campaign it was observed that the size of the granular material exerted scarce influence on the maximum run-up as, for a given stroke and run-out duration, the experimental points showed negligible dispersion. The larger the stroke, the higher the maximum run-up obtained for a given run-out duration. The maximum height reached for a given plate stroke increases as the run-out duration decreases.
Owing to the partial lack of information on the Padua experiment, only the toe of the landslide has been reproduced in the 2D numerical model. At the initial time, the still water level corresponds at full scale to a height of 700 m a.m.s.l.
The landslide is pushed into the reservoir by a rigid vertical wall that in the numerical model was assumed to be at the abscissa x = 0.0 m (not shown in the Figures) for simulating the effect of the rigid plate in the experimental facility. The movement of the wall is controlled by imposing its velocity components that follows a linear ramp to prevent incorrect numerical effects due to instantaneous change in the velocity at the boundary of the solid particles. During the simulations it was observed that few solid particles penetrate the wall and accumulate at its back. Even if the percentage of lost particles is so small that the effect on the landslide dynamics can be reasonably neglected, in future studies, an alternative approach available in SPHERA that may avoid particles loss will be tested: it consists of simulating rigid plate through a solid body with prescribed law of motion.
The water wave is generated by the landslide entering the basin, and the maximum run-up on the opposite slope is monitored and compared with the experimental result
Figure 6, right).
The experimental test No. 48 is here simulated with run-out duration of 0.8 s (corresponding to 17.6 s at full scale) and observed maximum run-up of 0.712 m (corresponding to 863 m a.m.s.l.), which are close to the characteristics of the Vajont catastrophic event.
It must be pointed out that reliable prediction of the experimental run-up is obtained with the same run-out time of 0.8 s and reduced stroke of 0.36 m with respect to the test No. 48 where a stroke of 0.5 m was instead used. This may be partly related to the fact that only the toe of the landslide was here modeled owing to the lack of available information; therefore, the plate should accelerate an amount of granular material which is considerably smaller than in the Padua experiment.
Table 3 summarizes the runs performed to evaluate model sensitivity to some relevant parameters. In all runs the same spatial resolution
dx = 0.005 m was adopted and the total particles number was 12,309. The first run denoted as V1 is assumed as reference: the values assigned to geotechnical parameters of the landslide are physically consistent with the characteristics of the poorly graded sand with 3–4 mm diameter adopted in the experimental test number 48; in particular, the angle of internal friction is set equal to 35° following the guidelines on the non-cohesive soils [
59]. In Runs V1–V3, the landslide material was assumed dry, thus neglecting the influence of pore water in the landslide portion below the still water level. This simplification was also assumed in a previous SPH model reproducing the test No. 48 of Padua experiment [
17].
In Run V1 (reference run), the minimum value for the threshold viscosity has been selected by following the procedure described in previous section, resulting in µthrmin = 320 kPa s. The limiting viscosity has been set equal to the value of µ0 = 5 kPa s as in the 2D erosive dam break test.
Run V2 aims at evaluating the effects on the landslide–water coupled dynamics of lowering the angle of internal friction to 25°.
Run V3 shows the influence of increasing the limiting viscosity to µ0 = 10 kPa on the numerical results.
The effect of pore water has been accounted in Run V4, where it has been assumed that landslide portion below the stored water level was fully saturated.
Table 4 summarizes some relevant results from the performed runs: total elapsed time (for 1.6 s of simulated time on 16 cores @ 2.3 GHz 128 GB RAM), calculated maximum run-up and time at which it is reached. The maximum run-up obtained in each run
N is translated into the corresponding maximum height
reached by the generated wave at full scale (with respect to mean sea level) and then it is compared with the maximum height
Zexp from the experimental test No. 48. The absolute percentage error in the last column is computed for each run
N as:
where
Zstill = 700 m a.m.s.l. is the stored water level.
As it can be seen in
Table 4, the angle of internal friction
ϕ does not affect significantly the computational time: in Run V2 the total elapsed time is about 6% greater than the reference Run V1. On the contrary, the maximum run-up is notably influenced by a reduction of
ϕ from 35° to 25° leading to noticeable increment of the absolute percentage Δ
η % that becomes almost three times the one obtained for Run V1.
Such behavior may be related to the dependence of the apparent viscosity µfr from the sine of ϕ through the second of Equations (10): as the angle of internal friction decreases, also µfr reduces in a non-linear manner and lowers the intensity of the shear stresses in the granular material through the first of Equations (10). Therefore, a reduced part of kinetic energy is dissipated during the run-out, thus allowing the landslide to undergo higher deformation and pushing progressively its front more into the basin as ϕ reduces.
This is confirmed by the analysis of
Figure 7 that compares the landslide and water profiles for both Run V1 and Run V2 at the instant of maximum run-up that is
t = 1.35 s. As the angle of internal friction decreases the landslide front gets closer and touches the opposite slope (dash-dot brown line) providing an increased thrust on the stored water that reaches a height of maximum run-up (dash-dot blue line) significantly greater than Run V1 (continuous blue line).
From the comparison of both water profiles in
Figure 7, it can be seen that the surface in Run V2 (dash-dot blue line) is, on average, above the surface obtained in Run V1 (continuous blue line). This means that the reduction of the angle of internal friction influences not only the shape of the landslide front, but it affects also the soil volume entering the basin: in particular, this volume increases because the displaced water volume is greater (it is worth noting that, at the time of maximum run-up, the velocity modulus is almost zero everywhere in the climbing water tongue, as shown in the following).
Dashed lines refer to experimental data: black line indicates the height of maximum run-up; blue line shows the water surface profile that is well reproduced by the Run V1 at abscissa between 0.7 m and 1.1 m, while above it is slightly overestimated probably because particle resolution cannot reproduce the actual thickness of the water blade.
Figure 8 shows the results from Run V3 (dash-dot lines) that are obtained for an increment of the limiting viscosity to 10 kPa s.
The comparison with the reference Run V1 (continuous lines) shows that this change in the limiting viscosity does not affect significantly the profile of the landslide. It can be also noticed that the water profile of Run V3 (dash-dot blue line) is almost overlapped to the corresponding profile obtained in Run V1 (continuous blue line). Therefore, the water surface profile from Run V3 also matches the experimental one (blue dashed curve) at abscissa between 0.7 m and 1.1 m. Even if the maximum wave run-up from Run V3 is a little bit closer to the experimental one, it seems scarcely affected by the considered change of limiting viscosity because the absolute percentage error Δη % remains of the same order of Run V1.
On the contrary, doubling the limiting viscosity induces a significant increase of the computational time of about 105%.
From these considerations follow that assuming µ0 = 5 kPa s allows optimizing the simulation by halving the required computational time while preserving the accuracy of obtained results regarding landslide dynamics and wave run-up.
Figure 9 shows the comparison between the profiles from Run V1 (continuous lines) and Run V4 (dash-dot lines) at the same instant of maximum run-up (i.e.,
t = 1.35 s). Run V4 differs from Run V1 because in the former it has been assumed saturated condition for that portion of landslide below the still water level, as shown in
Table 3.
The amount of soil penetrated into the reservoir is almost the same as in Run V1: this is confirmed by the fact that the water profiles are, on average, very close each other for both runs and hence the displaced water volume is about the same (the two extreme water volumes comprised between the water profiles of Run V1 and Run V4 balance each other). In the case of Run V4, the water surface profile is rather overlapped to the experimental one (blue dashed curve) below abscissa 1.1 m, including the left-hand front of the water surface that is closer to the experimental profile above the back of the landslide.
The two landslide fronts have different spatial distributions. In particular, the saturated layer below the still water level undergoes a higher deformation, as expected. Therefore, the landslide front in Run V4 (dash-dot brown line) appears more sharpened and touches the opposite side of the valley, while the maximum run-up reached by the generated wave is slightly lower than Run V1.
This behavior suggests that the way the landslide deforms and the resulting shape of its front influence the amount of kinetic energy transferred to the water and the maximum wave run-up: if the front is sharpened and sloping (as in Run V4) the maximum wave run-up decreases even if the protrusion of the landslide toward the opposite slope increases.
Conversely, if the landslide front is close to the vertical direction, as in the case of Run V2 in
Figure 7, more kinetic energy can be transferred to the stored water resulting in an increased maximum run-up with respect to the reference run.
The results from Run V4 show the influence of pore water in the portion of the landslide below the still water level. As expected, the interstitial water contributes to lessen the effective stresses and to increase the soil susceptibility to deformation; therefore, the lower part of the landslide front is more prominent with respect to the case of Run V1, touching the opposite slope and showing a behavior that is similar to the model described in [
17].
Similar behavior can be obtained by lowering the angle of internal friction (Run V2) but the front shape is less sharpened resulting in an overestimation of the maximum run-up. The maximum water height obtained in Run V4 is instead closer to the experimental result
Zexp; the absolute percentage error Δ
η % is half the one obtained in Run V2 and is rather close to the absolute percentage error of Run V1 (
Table 4).
This suggests that in the present study the angle of internal friction has a physical meaning and does not represent a calibration parameter.
Figure 10 shows the field of velocity magnitude for the reference Run V1 at two representative instants:
t = 0.80 s when the pushing plate comes to a stop (upper frame) and at
t = 1.35 s when the maximum run-up is reached (lower frame).
In the upper panel, a wedge within the landslide which is characterized by zero velocity magnitude can be noticed: the wedge, contained between the sliding surface and the rigid plate, has a sloping side toward the basin forming an angle of about 45° with respect to the horizontal direction; the velocity of its solid particles is set equal to zero because their apparent viscosity exceeds the threshold viscosity µthr. The landslide body beyond this fixed wedge is characterized by a regular distribution of velocity magnitude ranging from zero, in close proximity of the wedge frontier, to the highest value of about 1 m/s at the landslide upper surface. The landslide front pushing the stored water is characterized by lower values of the velocity modulus with a maximum of 0.65 m/s close to the free surface. During this phase, the stored water is pushed by the landslide front entering the basin and is accelerated over the right-hand slope where the tip of the water tongue shows a maximum value of the velocity modulus around 0.9 m/s.
The lower panel of
Figure 10 shows the instant
t = 1.35 s of maximum run-up of the generated wave; this wave is characterized by almost zero velocity magnitude and, after climbing the opposite slope, it reaches the maximum height of 0.718 m in the laboratory frame of reference, corresponding to a height of 870.3 m at the real scale.
At the left-hand side of the free surface, a small wedge of water particles with a velocity magnitude of about 0.23 m/s can be noticed. During the descending phase of the generated wave toward the reservoir, these water particles further increase their velocity magnitude and cover the back of the landslide in accordance with the actual behavior of the Vajont landslide that was described above.
Previous results have been obtained for the reference spatial resolution
dx = 0.005 m and evaluating the corresponding optimized value of the limiting viscosity
µ0 that allows obtaining the reference permitted percentage error Δ
η % of the estimate of the wave maximum run-up (see
Table 4).
In the case where one is interested in optimizing the parameter
µ0 for coarser resolutions (i.e., greater values of
dx), two additional simulations have been carried out to show how limiting viscosity changes by varying the resolution to obtain the allowed absolute percentage error. These simulations are summarized in
Table 5 where the adopted values for the spatial resolution
dx and for the limiting viscosity
µ0 in each run are indicated. All the other parameters of Runs V5 and V6 are the same as Run V4 (see
Table 3).
Figure 11 shows the profiles of the landslide (in brown) and of the water surface (in blue) for Run V5 (continuous lines) and for Run V6 (dotted lines with plus marker). These profiles are almost overlapped and the computed maximum run-up is very close to the one obtained for the higher resolution Run V4 (
Table 4). In addition, the experimental water profile is suitably reproduced with major discrepancy above abscissa 1.1 m for the reason previously discussed.
The calculated values of the absolute percentage error Δ
η % are also reported in
Table 5: for the investigated range of coarser resolutions the results show that, to obtain analogous values of Δ
η % that are of the same order of the permitted error in the estimate of maximum run-up, the parameter
µ0 should be varied almost linearly with respect to the particle resolution
dx.
This result seems consistent with theoretical considerations because, when increasing dx, the limiting viscosity should be higher (i.e., should be less influential) to obtain the reference permitted error because part of the error is already due to the lower spatial resolution.