Mean force and fluctuations on a wall immersed in a sheared granular flow

11 In a sheared and conﬁned granular ﬂow, the mean force and the force ﬂuctuations on a rigid wall 12 are studied by means of numerical simulations based on the discrete element method. An original 13 periodic immersed-wall system is designed to investigate a wide range of conﬁnement pressure and 14 shearing velocity imposed at the top of the ﬂow, considering diﬀerent obstacle heights. The mean 15 pressure on the wall relative to the conﬁnement pressure is found to be a monotonic function of the 16 boundary macroscopic inertial number which encapsulates the conﬁnement pressure, the shearing 17 velocity and the thickness of the sheared layer above the wall. The one-to-one relation is slightly 18 aﬀected by the length of the granular system. The force ﬂuctuations on the wall are quantiﬁed 19 through the analysis of both the distributions of grain-wall contact forces and the autocorrelation of 20 force time series. The distributions narrow as the boundary macroscopic inertial number decreases, 21 moving from asymetric log-normal shape to nearly Gaussian-like shape. That evolution of the 22 grain-wall force distributions is accompanied at the lowest inertial numbers by the occurrence of 23 a system memory in terms of the force transmitted to the wall, provided that the system length 24 is not too large. Moreover, the distributions of grain-wall contact forces are unchanged when the 25 inertial number is increased above a critical value. All those results allow to clearly identify the 26 transitions from quasi-static to dense inertial, and from dense inertial to collisional, granular ﬂow 27 regimes.


I. INTRODUCTION
the mean f : they decrease exponentially up to the highest values of f (see for instance 60 [11, 18,19]). This behaviour is sensitive to the shear velocity (or the deviatoric stress tensor 61 in static cases), as the inertia tends to broaden the distributions (see [9,11,16]). For to focus on the grain-wall interactions, an innovative planar system is designed. While the 82 spherical grains are trapped across the direction perpendicular to the mean flow between two 83 rough walls, a periodic boundary condition is used along the mean flow (shearing) direction. 84 The obstacle is a wall which is fully immersed in the granular bulk and orthogonal to the 85 shearing direction. Note that one initial motivation of the present study was to mimick the 86 problem of gravity-driven free-surface granular flows passing over a rigid wall [24][25][26]. First, both confinement pressure P and shear velocity U : (2) 123 The particle density was taken equal to ρ P = 2500 kg m −3 and the gravity acceleration 124 is g = 9.81 m s −2 . The mean grain diameter d was taken equal to 1 mm and H/d was 125 kept constant H/d = 30. A slight polydispersity was introduced by picking randomly the 126 grain diameters between 0.85d and 1.15d, in order to avoid crystallisation effects on the one 127 side (no polydispersity) and migration/segregation processes on the other side (too high 128 polydispersity). An arbitrary constant macroscopic volume fraction Φ = 0.6 is considered 129 here, which roughly corresponds to the random close packing of a three-dimensional assembly 130 of spheres. We recall here that the numerical simulations presented in this study use spherical 131 particles whose centers stay, by construction, on a planar surface [plane (x,y) on Fig. 1 ].

132
The choice of defining such dimensionless numbers N P and N U based on g is purely arbitrary, 133 has no influence on the results presented in the following, and was essentially motivated by 134 advancing knowledge on the problem of the mean force and force fluctuations exerced by free-135 surface gravity-driven granular flows on rigid walls, as already discussed in the introduction. 136 Moreover, it followed the choice already made in a previous study on a different systemnamely the granular lid-driven cavity-but for which very similar measurements and analysis 138 were made [28]. Considering those definitions thus allows a direct comparison between the 139 two systems (see discussion in Sec. VI B).

140
A boundary macroscopic inertial number I M can be defined from the typical time asso-141 ciated with the top confinement pressure t P = d ρ/P and the typical time equal to the 142 inverse of the macroscopic shear rate t U = H/U : shearing the granular sample (H = 0.03 m and Φ 0.6 for a 3D assembly of spheres).

148
Actually, the numerical system studied here is by construction volume-free and therefore 149 some slight variation of the volume fraction is possible. The mean volume fraction of the 150 granular sample which was actually measured in the DEM simulations will be discussed in 151 the concluding section of the paper (see Sec. VI A).

152
In the present study, the dimensionless numbers N P and N U were varied from 0.01 to 153 100 and from 1 to 20, respectively. This allowed us to investigate a wide range of both

171
where n and s are the unity vectors along the contact normal and shear directions re-172 spectively, k n and k s are the normal and tangential contact stiffnesses, δ n is the normal 173 penetration depth,δ s is the tangential displacement increment, µ is the local friction co-174 efficient, c n is the normal viscosity coefficient and dt is the time-step. The four physical 175 parameters k n , k s , c n and µ, are chosen to fit the behaviour of glass beads. The contact 176 stiffness is reduced to decrease the total time of calculation, but the limit of rigid grains 177 was systematically respected (see details in [28,29]). The coefficient c n is set in the same 178 manner as in [29] with a restitution coefficient e = 0.5, and µ was taken equal to 0.5.

180
The present section investigates the time-averaged dynamics of the system: the stream-181 lines within the granular sample and the vertical velocity profiles along the sample length 182 (Sec. III B), the mean force F on the obstacle-the latter being the wall on the right side of  formed upstream the obstacle (the wall on the right side).

206
The response of the velocity streamlines to the increase of I M is tricky. Figure 2    The presence of a plateau in the center of the curves shown in Fig. 3 demonstrates that the In the above definition, the dead zone is assumed to cover the entire height of the wall.  respectively. The constant I * 0 M = 0.3 is the inertial number for which F /(P hd) = (r 1 + r 2 )/2.

279
It is worth noting that the values of r 1 , r 2 and I * 0 M were obtained for the configuration 280 presented in this paper, and may be influenced by the micromechanical grain parameters (d, 281 µ, etc.), as well as by the system configuration. In particular, the sensitivity of the mean 282 force scaling to doubling L/H will be discussed in Sec. V.

283
It is worthwhile to note that the data saturation (concomitant with some scattering)  For that purpose, the strain (D) and stress (σ) tensors at local (grain) scale were com-297 puted, using a spatial Kernel smoothing method and tesselation techniques. The technical 298 aspects of those calculations can be found in [28]. The pressure p within the granular medium 299 was defined by: The local effective friction was calculated as:

303
where D = D − 1 3 Tr(D)I 3 holds for the deviatoric strain tensor and σ = σ − pI 3 304 is the deviatoric stress tensor. We define I 3 as the identity matrix of size 3 and A = 305 Tr(AA T )/2 for any A. Finally, the local inertial number was computed using the relation: Although the strain field is rather complicated inside the immersed-wall system investi-308 gated here (see the streamlines drawn on Fig. 2), it was generally observed that the strain 309 and the stress tensors were quite well aligned within most of the system volume.  For each position (x, y), the measured µ loc was compared to a µ th (I) derived from the 335 relation proposed by Jop et al. [33], which reads as follows:

337
where µ 1 , µ 2 and I 0 are parameters dependent on the mechanical grain properties. In 338 our study, the best fitting parameters for µ th were found to be (after an iteration process) 339 µ 1 = 0.17, µ 2 = 0.48 and I 0 = 0.18, as shown in Fig. 6. Figure 6 shows the time-averaged

458
where F µ is the scale parameter, F σ is the shape parameter, and the shift (truncation) is 1.

459
S is the normalization factor corresponding to the value at 1 of the survival function of the 460 (untruncated) log-normal PDF. The evolution of these parameters with I M is displayed on 461 Fig. 9(b). distributions, as shown on Fig. 9(a). This reveals that the boundary macroscopic inertial 465 number is the key parameter which controls the distribution of the total force on the wall.

466
Such a remarkable result is further confirmed by the monotonic variation of the two fitting 467 parameters f µ and f σ [see Eq. (12)] with I M , as displayed on Fig. 9(b), regardless of {U, P }.

468
It can be concluded that Eq. (12), fed with the parameters controlled by I M (Fig. 9(b)), 469 provides an empirical model capable of predicting the total force distribution for the granular 470 flow-wall interaction system studied here.

471
Moreover, the DEM simulations show that all distributions of F collapse when I M ≥ 472 6.1 10 −2 , as shown in Fig. 9(a). This observation becomes clearer in Fig. 9(b), as both f µ systems is beyond the scope of the present study and will be the topic of a future study.  The ratioF /(P S o ) typically increases from 1 at low I M to 3 − 4 when I M approaches unity.

652
The empirical relation betweenF /(P S o ) and I * M gives a way to predict the variation of 653 the force on the wall, before reaching the faster inertial regime for which the mean force