Self-similar velocity profiles and mass transport of grains carried by fluid through a confined channel

Abstract

Confined fluid-driven granular flows are present in a plethora of natural and industrial settings yet even the most fundamental of these are not completely understood. While widely-studied grain flows such as bed load and density-matched Poiseuille flow have been observed to exhibit exponential and Bingham style velocity profiles respectively, this work finds that a fluid-driven bed of non-buoyant grains filling a narrow horizontal channel -confined both from the sides and above -exhibits self-similar Gaussian velocity profiles. As the imposed flow rate is increased and the grain velocity increases, the Gaussian flow profiles penetrate deeper into the packing of the channel. Filling fractions were observed to also be self-similar and qualitatively consistent with granular theory relating to the viscous number I, which at a given position on the self-similar Gaussian curve is found to be generally constant regardless of imposed flow rate or velocity magnitude. An empirical description of the flow is proposed, and local velocity and filling fraction measurements were used to obtain local grain flux and accurately recover a total grain flow rate.

From Darcy to Gaussian to fully mobilised grain flow in a confined channel

“…For all h displayed, z f increases with Q in a manner consistent with a power law of exponent approximately ∼ 1/3. This demonstrates how the flow gradually penetrates the packing of the channel and is consistent with previous findings [9]. Once z f = h and Q ≥ Q 1 all parts of the channel satisfy the condition v(z) > v max /100, hence there is no more packing to mobilise and full channel flow occurs.…”

“…Figure 3 shows typical velocity profiles as a function of channel depth z for a range of flow rates. As recently reported [9], the grains exhibit velocities consistent with Gaussian profiles centred at the upper wall which can be described by…”

“…Conversely, Poiseuille flow of grains typically concerns neutrally buoyant granular material with velocity profiles comparable to those of Bingham-style plug flows [12,13]. However, we recently reported that grains filling a confined horizontal channel, when driven by fluid appear to exhibit self-similar Gaussian velocity profiles that become faster and penetrate deeper into the bed as the flow rate is increased [9]. This contrast to bedload flow was linked to the contact between the granular material and the upper confining wall, not unlike Couette grain flows which also betray decaying Gaussian velocities centred at the boundary.…”

“…Employing a scaling of the same form to that in ref. [9], the inset of Figure 3 shows velocity profiles normalised with respect to v 0 and σ such that v 0 = α( Q Q 0 − 1) 0.64 and σ = β( Q Q 0 − 1) 0.33 where α = 0.22 cm/s and β = 0.033 cm. These constants are similar to those found with a 6 mm channel [9], with slight discrepancy possibly due to differing grain samples.…”

“…Dry silos with a lateral orifice can also adhere to Beverloo's Law, although flow ceases when the silo walls become too thick [7,8]. If driven by fluid however, flow of non-buoyant grains can occur through a lateral orifice and into a horizontal conduit [9].…”