We report on direct measurements of the basal force components for granular material flowing down a smooth incline. We investigate granular flows for a large range of inclination angles from $\theta =13.4^{\circ }$ to 83.6° and various gate openings of the chute. We find that the effective basal friction coefficient ${\mu }_B$, obtained from the ratio of the longitudinal force to the normal one, exhibits a systematic increase with increasing slope angle and a significant weakening with increasing particle holdup $H$ (the depth-integrated particle volume fraction). At low angles, the basal friction is slightly less than or equal to tanθ. The deviation from tanθ can be interpreted as a contribution from the sidewall to the overall friction. At larger angles, the basal friction ${\mu }_B$ saturates at an asymptotic value that is dependent on the gate opening of the chute. Importantly, our data confirm the outcomes of recent discrete numerical simulations. First, for steady and fully developed flows as well as for moderately accelerated ones, the variation of the basal friction can be captured through a unique dimensionless number, the Froude number Fr, defined as $\mathrm{Fr}=\overline{U}/{\left(gH\cos \theta \right)}^{1/2}$, where $\overline{U}$ is the mean flow velocity. Second, the mean velocity scales with the particle holdup $H$ with a power exponent close to $\frac{1}{4}$, contrasting with the Bagnold scaling $\left(\overline{U}\sim H^{3/2}\right)$.

中文翻译：

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光滑斜面上颗粒溜槽底部有效摩擦的实验评估

我们报告了直接测量颗粒力顺着光滑的坡面流下的基力分量的情况。我们从以下角度研究了大范围倾斜角下的颗粒流$\theta =13。4^{\circ }$到83.6°以及滑道的各种门开口。我们发现有效的基础摩擦系数${\mu }_{乙}$由纵向力与法向力之比得出的，随着倾斜角的增加而呈现出系统性的增加，而随着颗粒滞留率的增加而显着减弱 $H$（深度积分粒子的体积分数）。在低角度下，基础摩擦力略小于或等于tanθ。与tanθ的偏差可以解释为侧壁对整体摩擦的贡献。在较大的角度，基础摩擦${\mu }_{乙}$饱和度的渐近值取决于滑道的闸门打开程度。重要的是，我们的数据证实了最近离散数值模拟的结果。首先，对于稳定和充分发展的流动以及中等加速的流动，可以通过唯一的无量纲数Froude数Fr来捕捉基础摩擦的变化，该无量纲数定义为$r=\overline{ü}/{（GH\cos \theta ）}^{\mathrm{1个}/\mathrm{2个}}$， 在哪里 $\overline{ü}$是平均流速。其次，平均速度随粒子保持率而变化$H$ 幂指数接近 $\frac{\mathrm{1个}}{4}$，与Bagnold缩放比例相反 $（\overline{ü}〜H^{3/\mathrm{2个}}）$。