importance in various industrial contexts, and its behaviors can be extremely variegated [5]. The rotating drum is in fact considered a benchmark problem of dense granular flow [5]. Great effort to classify the flow behavior of granular materials in rotating cylinders was spent by Henein et al. [6], who proposed the use of “Bed Behavior Diagrams” to conveniently delineate the different flow behaviors. The most thorough schematization of the possible granular flow configurations was proposed by Mellmann [7], who identified several flow regimes and the transitions among them by means of model calculations. Basically the grain motion in a rotating cylinder can be classified into three categories: the slippingmotion, at low filling and low friction, where the grains do not flow and move as a bulk; the rolling motion, at medium rotational velocities, characterized by the onset of a liquid-like flow; a cataracting regime, as the centrifugal acceleration starts to be of the same order of the gravity; and, finally, a centrifugal regime. The rolling motion in rotating cylinders can be further subdivided into three regimes [7]. When the rotation velocity is low, the slumping flow may occur. This flow consists in a series of successive distinct avalanches. As the rotational speed increases, a transition to fully developed rolling takes place, and the discrete sequence of avalanches evolves in a single continuous motion in the upper part of the bed. As the rotational speed further increases, the bed surface begins to arch and cascading sets in. This work is devoted to the study of the fully developedrollingflow of a granular medium in half-filled rotating cylinders through numerical simulations. A continuum mechanical level of description is adopted, with the above mentioned constitutive equation [4] for the stress tensor of the granular material. 15.2 Experimental Evidences in the Rolling Regime Several experimental procedures to study the rolling flow have been devised. One approach consists in looking at the rotating cylinders from one of the lateral transparent wall, measuring the flow depthhand the dynamic angleθ, defined as the average angle of the free surface of the grain phase. Such measurements are usually performed in very narrow cylinders (quasi–2D cylinders), in order to avoid flow in the axial direction. Another approach relies on the use of Magnetic Resonance Imaging (MRI) techniques to investigate the same quantities (h and θ) at the center of wide rotating cylinders. Those approaches are clearly different in many ways. The quasi-2D approach cannot avoid wall effects, while the MRI does not suffer of this limitation. We believe this to be a crucial point: even if is commonly accepted that there is a strong influence of the lateral wall on the flow [8–13], most of the experiments are performed in the worst condition, i.e. by looking at grains near the wall. The MRI drawbacks, on the other hand, lay in the difficulties of the experiment itself, and in that the space and time resolutions of such a technique are quite lower than those of direct measurements. A third approach makes use of the Positron Emission Particle Tracking (PEPT) [14], which again requires very complex apparati. A review of experimental data obtained by investigating the flow close to the lateral wall can be found in GDR MiDi [1] and Pignatel et al. [8]. GDR MiDi [1] found that the depth of the flowing layer scales with the rotational velocity to the power of 0.5, and that is what one would obtain by hypothesizing a constant shear rate in the flowing layer. The dynamic angle increases with increasing rotation velocity and with decreasing cylinder width. Felix et al. [15] and Pignatel et al. [8] also report a monotonically increasing scaling of the dynamic angle and the flow depth with the rotational velocity; they found, however, a wide exponent range for the power law dependency of the flow depth on the rotational velocity, with values from 0.15 to 0.68, depending on the ratio D d P = of the cylinder diameter over the particle diameters. The angle is also found to increase with decreasing width, even if an appropriate parameter to collapse the data is not proposed [8]. Pignatel et al. [8] also point out that the wall friction plays an important role in narrow cylinders. Nakagawa et al. [16] and Yamane et al. [17] used MRI to investigate the rolling flow far away from lateral walls. Their experimental works focused mainly on velocity [18] and density profiles along the depth of the granular material, but reported a single dataset of flow depth versus angular velocity, which scales with a power of 0.4. Yamane et al. [17] also report a linear increase of the dynamic angle with the rotational speed, with the angle being ca. 5 higher near to the wall than at the center of the cylinder at all the angular velocity explored. As already said, the walls exert a strong influence on tumblers dynamics. Dury et al. [12] measured the difference in dynamic angle at the center (via MRI) and at the wall (via direct visualization), finding a constant difference of ca. 4 regardless of the rotational velocity. Furthermore, they performed DEM simulations to access the profile of the dynamic angle along the axis of wide cylinders. In such a way, they were able to quantify the distance from the wall at which the angle becomes that of the center of the cylinder. Such a distance was found be linear in the drum diameter, following the scale L ¼0.14 D. They also found finite size effect for D d P<15 = . Pohlman et al. [10] and Chen et al. [11] studied the 2D velocity profiles on the free surface of wide cylinders both experimentally [10] and numerically [11]. They proved the existence of an axial flow near the walls directed toward the 122 G. De Monaco et al.
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