Experimental study on internal flow structure and dynamics of dense liquid-particle flow down inclined channel

Abstract

Dense granular flows widely exist in the environment and industry where inter-particle interactions play essential role. Studying the flow behaviour is important for a better understanding and more scientific description of the granular rheology. This paper experimentally investigates liquid-particle mixture dense flows down an inclined channel with bumpy-frictional base. The refractive index matching method is used which permits the determination of the internal flow information, including the velocity, shear rate, granular temperature and solid concentration. It is observed that the wall influence is minor at the observing position. The pressure and shear stress obtained from the integration of the solid concentration matches well with the prediction of the kinetic theory. The particle interaction pattern is analysed from the rheology properties and a coherence length approach. The flow is found to be rheologically stratified, with the near-bottom being collision-dominated and the near-surface being friction-dominated. The bottom pore pressure and stress are also directly measured and analysed in combination with the internal kinetic properties.

Appendices

Appendix A: assumption of fully developed flow

Since the spatial development by measuring the results at different streamwise locations is unavailable for now, we have to assume that the flow is already fully developed at the observing position, i.e. the flow variation along the streamwise direction is negligible. However, two evidences can help support this assumption. Firstly, we have not observed obvious flow height variation spatially along the channel during the plateau stage. Secondly, the experimental setup conditions of the relevant investigations whose granular flows were formed by releasing the upstream mass, similar with the present study, are compared in Table 2. Though the detailed flow condition varies, such as the material properties, velocities, inclinations, here we simply compare the ratio of the observing distance X from the inlet to a characteristic flow height H. This paper sets the observing position at a distance X of 2.4 m away from the channel inlet. The flow height H is generally less than 10d $\approx$ 38 mm, leading to a distance-height ratio of $X/H=2400/38\approx 63$. It can be seen that the ratio adopted in the present study is within a reasonable range of the previous investigations. So the flow is believed to be fully developed at the observing position.

Table 2 Comparison of the observing positon with previous investigations

Appendix B: Granular temperature estimation

The granular temperature is defined as $T=(\overline{u^{\prime 2}}+\overline{w^{\prime 2}})/2$ where the velocity fluctuation is traditionally calculated in a mean-squared (MS) way. However, it is noticed that the MS-based fluctuation results in artifacts due to accumulation of measurement noise from the particle location identification and matching process (Armanini et al. 2005; Gollin et al. 2017; Taylor-Noonan et al. 2021). An alternative estimation is based on the Lagrangian velocity auto-correlation function (ACF) defined as $\left\langle u^\prime (t)u^\prime (t+\delta t)\right\rangle$ (Larcher et al. 2007). It is found theoretically that the function follows an exponential law with $\delta t$ as $\left\langle u^\prime (t)u^\prime (t+\delta t)\right\rangle =\left\langle u^\prime (t)u^\prime (t)\right\rangle \rm{exp}(- |\delta t|/t_0)$ (Armanini et al. 2005). For each statistical period $t_{\rm{s}}=0.1$ s, the particles are tracked for at least ten frames (0.01 s) forward and backward. A typical result of the ACF is given in Fig. 13a. It is found that the ACF generally follows the exponential law and convergences to zero with increasing $\delta t$. In practice, only the first 2–10 data points are used for the exponential fit and then extrapolated back to $\delta t=0$ giving the fluctuation $(\overline{u^{\prime 2}})_{\rm{ACF}}$. It is seen that the fluctuation based on ACF is generally smaller than the MS-based, i.e. the cross at $\delta t=0$ in Fig. 13a. The normal fluctuation $\overline{w^{\prime 2}}$ is obtained in the same way which has comparable magnitude with $\overline{u^{\prime 2}}$. Two estimations of the granular temperature $T_{\rm{ACF}}$ and $T_{\rm{MS}}$ are obtained as given in Fig. 13b. It is obvious that $T_{\rm{ACF}}$ which is believed to be less affected by the measurement noise is smaller than $T_{\rm{MS}}$. For the upmost statistical bin, the increase of granular temperature is actually due to poor particle tracking performance near the flow surface. Though $T_{\rm{MS}}$ have similar trend with $T_{\rm{ACF}}$ along the flow height, it performs badly in predicting the stresses when applied to kinetic theory (see Sect. 5.1). As shown in Fig. 14, $T_{\rm{MS}}$ leads to significantly small results compared with kinetic theory and even wrong direction opposite to the theoretical curves with increasing packing density. So the ACF-based granular temperature $T_{\rm{ACF}}$ is more credible and is adopted in this study.

Fig. 13

a Lagrangian velocity auto-correlation function. The dashed line is the exponential fit of the red crosses. b Comparison of $T_{\rm{ACF}}$ (blue) and $T_{\rm{MS}}$ (red)

Fig. 14

Dimensionless effective normal stress compared with the prediction of kinetic theory for R2S1. The blue and grey markers are the results based on $T_{\rm{ACF}}$ and $T_{\rm{MS}}$ respectively