Recent numerical work has shown that high-speed confined granular flows down smooth inclines exhibit a rich variety of flow patterns, including dense unidirectional flows, flows with longitudinal vortices and supported flows characterized by a dense core surrounded by a dilute hot granular gas [1]. Here, we further analyzed the results obtained in [1]. More precisely, we characterize carefully the transition between the different flow regimes, including unidirectional, roll and supported flow regimes and propose for each transition an appropriate order parameter. Importantly, we also uncover that the effective friction at the basal and side walls can be described as a unique function of a dimensionless number which is the analog of a Froude number: \(Fr=V/\sqrt{gH\cos \theta }\) where V is the particle velocity at the walls, \(\theta\) is the inclination angle and H the particle holdup (defined as the depth-integrated particle volume fraction). This universal function provides a boundary condition for granular flows running on smooth boundaries. Additionally, we show that there exists a similar universal law relating the local friction to a local Froude number \(Fr^{loc}=V^{loc}/\sqrt{P^{loc}/\rho }\) (where \(V^{loc}\) and \(P^{loc}\) are the local velocity and pressure at the boundary, respectively, and \(\rho\) the particle density) and that the latter holds for unsteady flows.

Graphical Abstract

中文翻译：

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高速受限的颗粒状流体顺着光滑的坡度向下流动：结垢和壁摩擦定律

最近的数值研究表明，沿光滑斜面向下的高速密闭颗粒流表现出多种流动模式，包括密集的单向流，具有纵向涡旋的流和以稀疏的热颗粒气体包围的稠密岩心为特征的支撑流[1]。 。在这里，我们进一步分析了[1]中获得的结果。更准确地说，我们仔细地描述了不同流动状态之间的过渡特性，包括单向，滚动和支撑流动状态，并为每个过渡建议了适当的阶次参数。重要的是，我们还发现，在基壁和侧壁的有效摩擦可以描述为无因次数的唯一函数，该因数类似于弗洛德数：\（Fr = V / \ sqrt {gH \ cos \ theta} \）其中V是壁上的粒子速度，\（\ theta \）是倾斜角，H是粒子保持率（定义为深度积分的粒子体积分数）。此通用函数为在平滑边界上运行的粒状流提供了边界条件。此外，我们表明存在类似的通用定律，将局部摩擦力与局部Froude数\（Fr ^ {loc} = V ^ {loc} / \ sqrt {P ^ {loc} / \ rho} \）（其中\（V ^ {loc} \）和\（P ^ {loc} \）分别是边界处的局部速度和压力，以及\（\ rho \）粒子密度），后者适用于非定常流动。

图形概要