## 4 Results
### Velocity Distribution  
The velocity of the particles inside the drum is a key aspect to compare between simulations and the experiment. For Case A, Figure 8a shows the magnitude of the velocity predicted with DEM during the sticking period of the slumping regime, where the particles move with the walls at the corresponding velocity as it were a rigid body (i.e., \(v=\omega r\)). Figure 8b shows the distribution of the tangential component of the velocity with DEM and experimentally. For these conditions, the DEM exhibits a very high level of agreement in both the instantaneous velocity field and the period of stick and slide observed in the experiment. The CM results are omitted for this case since the general regime cannot be captured.  
Figure 7: Interface between air and the bed of particles for the proposed regimes comparing experiment and simulations.  
Figure 9 shows the simulation predictions of the velocity field magnitude for a transitional regime (Case B), a rolling regime (Case C), and a cascading regime (Case D), respectively.  
In the rolling regime, both approaches show a quasi-planar bed surface where the particles are moving downward at a high velocity near the interface. This layer of high velocity is very thin and the particles rapidly change direction as we move far from the centre of the drum until reaching the rotating wall velocity. In the transitional and cascading regimes, the maximum velocity is reached close to the part of the bed surface that is farthest from the drum cylindrical wall, where the particles move downward in the opposite direction of the rotating wall. Unlike the rolling regime, here the high-velocity layer is broader until it reaches zero tangential velocity. Once again, both models show similar trends.  
Figure 8: Velocity field for Case A: (_a_) Velocity field magnitude with DEM and (_b_) normalized tangential component of the velocity as a function of a radial direction at 35\({}^{\circ}\) from a vertical line measured experimentally and through simulation.  
Figure 10 shows the tangential velocity profile on a radial direction at 35\({}^{\circ}\) from the vertical direction for rolling and cascading regimes. The simulation results are plotted along with the velocity measured experimentally at certain positions. The experimental results are computed based on the relative position of tracer particles inside the bed analyzing each frame of a high-quality video filmed at 25 fps. The error bars amplitude are computed based on the uncertainty of the position measurement of the tracer particles divided by the time span between frames.  
In general, the granular phase velocity is null until the bed surface is reached and then it rapidly becomes maximum and decreases until it changes direction as we move far from the centre. Both computational models are in agreement with the experiment and with each other in the region close to the rotating wall and present more differences close to the bed surface. However, the general trend is in agreement and there is no one technique that stands out from the other in terms of accuracy of the solution in comparison to the experiment. In fact, for the rolling regime, the experiment seems to be close in  
Figure 9: Velocity field magnitude for (_a_) Case B with DEM, (_b_) Case B with CM, (_c_) Case C with DEM, (_d_) Case C with CM, (_e_) Case D with DEM, and (_f_) Case D with CM.  
agreement with the DEM results in the backward flowing region, but it is the opposite for the cascading regime, where the continuum model seems to be closer to the experimental observation. It is worth mentioning that DEM seems to slightly overestimate the tangential velocity in comparison to the continuum approach in the backward flowing region.