#### 4.1. Distributions of Impact Conditions on the Bucket Surface

Figure 3 presents the results of the average sediment impact velocity and angle, where

${90}^{\circ}$ corresponds to a perpendicular impact, on the bucket surface and on a slice corresponding to the pitch diameter position. The impact velocity was about 0.5

${C}_{\circ}$ exactly at the splitter center, due to the acceleration felt by the sediments as a consequence of the low-velocity stagnation zone, although on the splitter edges it was as high as 0.7

${C}_{\circ}$. Throughout most of the bucket, the impact velocity was about 0.5

${C}_{\circ}$, with increasing values towards the bucket outlet, reaching a maximum average impact velocity of 0.8

${C}_{\circ}$. On the other hand, the average impact angle was fairly uniform throughout the bucket, with a value of about 5

${}^{\circ}$; except for the splitter, where the impacts occurred at about 25

${}^{\circ}$. The fact that the sediments did not impact at higher angles on the splitter indicates that they were fairly responsive to the fluid acceleration. These impact condition results are very similar to the 2D results [

32] of the same test case.

#### 4.2. Erosion Distribution of the Bucket Surface

Figure 4 presents the distribution of erodent mass that has impacted and eroded mass on the bucket surface, and on a slice corresponding to the pitch diameter position. The distribution of erodent mass reveals that most of the sediment impacts occurred on the bucket depth, which corresponds to the zone of maximum bucket curvature. These results highlight the fact that one of the driving mechanisms for sediment flux against the wall, the sediment inertia, is directly linked to the surface curvature. The second hot-spot for sediment impacts was the splitter. The sum of erodent mass impacted on the surface was equal to 2.66 times the sum of sediment mass injected, highlighting the fact that sediments tended to impact several times along their path through the bucket.

The superposition of the aforementioned distributions (i.e., where the sediments tended to impact and at what angle and velocity) determines the distribution of eroded mass presented in

Figure 4. The amount of eroded mass per unit area had a peak at the splitter and almost vanished immediately downstream; it gradually increased again, towards the bucket outlet. It is noteworthy that the distribution of erosion did not only depend on the flux of the erodent towards the wall: Even though the sediment mass impacted was greatest at the bucket depth, the eroded mass was greatest at the splitter because of the higher average impact angle at that location. Similarly, in spite of the relatively low erodent mass impacted towards the bucket outlet, the eroded mass was maintained due to the higher impact velocity at that location. Yet again, these erosion distribution results are similar to the 2D results [

32] of the same test case.

The global erosion ratio computed was equal to 4.04 mg kg${}^{-1}$. In other words, at these sediment and hydrodynamic conditions, the bucket lost about 4.0 kg of material for every 1000 tons of sediment injected. According to the following validation, this value predicted by the multiscale simulation was quite close to its experimental counterpart.

#### 4.3. Erosion Depth Validation

The multiscale model described bridges the scale separation between the sediment impact dynamics and the sediment transport hydrodynamics. A projective integration extension has been presented [

32] that allows simulation of the long-term erosion process by including the surface evolution in time, thus bridging the gap between the hydrodynamics and the surface erosion accumulation. The projective integration approach involves running the multiscale model for successive states of the eroded surface, whose evolution is determined by the erosion rate distribution calculated on each of the multiscale model instances. This approach is able to capture the effect the surface erosion has on subsequent erosion rates.

However, the projective integration approach is too computationally expensive to apply to this 3D test case, as it involves several successive macroscale simulations. In consequence, the long-term erosion distribution is estimated by extrapolating the erosion rate distribution obtained on the original bucket, not including the surface alteration induced by the process. Note that the incubation period (the initial delay in erosion due to the preliminary accumulation of damage in the surface) is considered in the microscale model by performing enough sediment impacts to reach the steady-state erosion ratio; what is neglected in the aforementioned extrapolation is the effect of the macroscopic surface alteration, such as wavy patterns and increased roughness, and the resulting modification of the hydrodynamics and, therefore, of the erosion rate.

The total amount of sediments injected in the simulation was equal to 3.43 g along a physical time of about 0.03 s. Rai et al. [

7] reported on the erosion depth distribution after 3180 h of operation, after which 12,540 tons of sediment had traversed the turbine (i.e., 737.65 tons through each one of the 17 buckets). Therefore, the factor used to linearly extrapolate the simulation results to the time frame reported in the experiments was

${f}_{\mathrm{extrap}.}=\frac{\mathrm{737,650}\phantom{\rule{3.33333pt}{0ex}}\mathrm{kg}}{0.00343\phantom{\rule{3.33333pt}{0ex}}\mathrm{kg}}=2.15\times {10}^{8}$.

The eroded depth, after 3180 h, is calculated as

where

$\rho $ is the base material density,

${m}_{e}$ is the total eroded mass accumulated on each FVPM particle on the bucket surface, and

${A}_{i}$ is the surface area of each one of the particles. That is, the eroded mass distribution presented in

Figure 4 can be directly linked to a distribution of eroded depth, that is then extrapolated to the time frame of the experiment by means of

${f}_{\mathrm{extrap}.}$.

Figure 5 presents the multiscale simulation prediction of the erosion depth distribution on the pitch diameter position after 3180 h of operation, as well as the corresponding measurements reported by Rai et al. [

7] on two different buckets. The remarkable agreement between the simulation and the experiment is highlighted, considering the complexity and astounding range of time and length scales involved. Notice that the differences between the two experimental buckets, and even between the two sides of each bucket, is of comparable magnitude to the difference between the experimental data and the simulation results. There is a considerable amount of noise in the numerical results because of the short simulation duration and the corresponding low number of sediments tracked. The smoothness of the simulated erosion profiles is bound to increase if a higher number of sediment particless are injected, as with any process influenced by randomness; in this case, the sediment trajectories were affected by the turbulent fluctuations, and the signal-to-noise ratio would increase by having more samples.

The multiscale simulation prediction of the erosion depth distribution on the bucket splitter is presented in

Figure 6, together with the corresponding measurements reported by Rai et al. [

7] on two different buckets. Whereas the experimental erosion was widely distributed along the splitter, the simulated erosion was concentrated towards its center. Indeed, as the experimental bucket rotates, the jet is allowed to impinge at different locations along the splitter, whereas, in the simulated static bucket, it only impinged at its center. Apart from this discrepancy and the overall lack of smoothness evidenced in the simulated erosion depth, the results agree to a significant extent.