High-speed confined granular flows down smooth inclines: scaling and wall friction laws

Abstract

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

A Layering index

In the undirectional and dense flow regime, the particle volume fraction exhibits strong oscillations. It could be interesting to introduce a layering index $I_L$ to quantify the number of oscillation cycles. It is defined as follows. As the wavelength of the oscillation is of the order of one grain diameter, we look at within each layer of one diameter thickness and parallel to the bottom whether the volume fraction oscillates.

Fig. 16

Layering index $I_L$ as a function of the inclination angles for various particle hold-up: $I_L=(1/N_{max})\sum _i Y\left( \phi _i^{max}-\phi _i^{min}-0.1 \right)$ where Y is the Heaviside function

In each layer i, we thus calculate the maximum and the minimum of the volume fraction, $\phi _{i}^{min}$ and $\phi _{i}^{max}$, respectively. If the amplitude of the oscillation (i.e., $\phi _{i}^{max}-\phi _{i}^{min}$) in a given layer is greater than a critical value $\varDelta \phi$, the layer is associated to an ordered layer of particles and the layering index is incremented by one unit. The layer index $I_L$ is defined as $I_L=(1/N_{max}) \sum _i Y\left( \phi _i^{max}-\phi _i^{min}-\varDelta \phi \right)$ where Y is the Heaviside function and $N_{max}=H/0.6$ is the highest possible number of ordered layers within a uniform and dense flow with a mean volume fraction of 0.6 and particle hold-up H. The renormalization of $I_L$ by $N_{max}$ provides an index which is bounded by 1. The critical value $\varDelta \phi$ used to quantify the layering is taken to be 0.1. This choice is somewhat arbitrary but it is good comprise to capture the oscillation of the packing fraction and eliminate random fluctuations of the packing fraction profile.

The variation of the layering index with the inclination angle is shown in Fig. 16. SFD unidirectional and dense flows exhibit a strong layering with a layering index close to 1, indicating that the whole depth of the flow dense flows) is layered. Upon increasing the inclination angle (from $20^\circ$ to $25^\circ$), the layering index decreases progressively towards zero. Above $25^\circ$ (i.e., in the supported flow regime), the layering index has fallen to a small but finite residual value (below 0.2). This mean that even in the supported regime, there remains one ordered layer which is located at the bottom. This residual dense ordered layer disappears at very large inclination angle. The layering index can thus not be employed to delineate the transition towards the supported flow regime.