Granular bed consolidation, creep, and armoring under subcritical fluid flow

We show that a freshly sedimented granular bed settles and creeps forward over extended periods of time under an applied hydrodynamic shear stress which is below the critical value for bedload transport. The rearrangements are found to last over a timescale which is millions of times the sedimentation timescale of a grain in the fluid. Compaction occurs uniformly throughout the bed, but creep is observed to decay exponentially with depth and decreases over time. The granular volume fraction in the bed is found to increase logarithmically, saturating at the random close packing value ${\varphi }_{\text{rcp}}\approx 0.64$, while the surface roughness is observed to remain essentially unchanged. We demonstrate that an increasingly higher shear stress is required to erode the bed, after a subcritical shear is applied, which results in an increase in its volume fraction. Thus, we find that bed armoring occurs due to a deep shear-induced relaxation of the bed toward the volume fraction associated with the glass transition.

Figure 1

(a) Schematic of the experimental apparatus which consists of a cylindrical container filled with transparent grains and fluid. The top plate is rotated about its axis to apply shear. (b) Image of the grains inside the bed, which appear dark in contrast with the fluid, which fluoresces bright. (c) Armoring procedure used to apply shear to the bed over time. The grains are initially suspended (step A) and then shear is applied by rotating the top plate at a fixed frequency (step B) after shear is turned off for 10 s. (d) Onset procedure used to find the threshold for eroding a bed which has been initially prepared by suspending the bed (step A) and applying a prescribed shear corresponding to $f$ for a time duration $T$ (step B). Then the driving is turned off before being ramped up linearly to determine the threshold of erosion as a function of $f$ and $T$.

Figure 2

Normalized applied shear stress ${\tau }^{*}$ at the onset of bedload transport as a function of frequency of the top plate normalized by the critical threshold of erosion $f/f_c$. Here ${\tau }^{*}$ is observed to increase linearly before the onset of bedload transport and has been plotted with the data we found with glass beads [10] showing the same behavior. The shear stress corresponding to a flat-bottom (solid) surface and the linear response is also plotted for reference for the acrylic oil. Error bars are smaller than the size of the points.

Figure 3

(a) Measurement of the onset of transient erosion due to preshearing at half the steady-state shear ${\tau }_c^{*}$ needed to erode particles at times ranging from seconds to hours normalized by the the transient erosion threshold with no preparation ${\tau }_0^{*}$. (b) Transient erosion onset for the first 10 min with frequencies (and stresses) that approach the onset of steady-state shear. The error bars represent the rms variation of at least ten measurements in both graphs. The data points circled by the dashed line correspond to the stress reported in (a).

Figure 4

Movement of particles in the bed in the first 90 min of preshear at (a) no shear stress ${\tau }^{*}/{\tau }_c^{*}=0.0$ and (b) ${\tau }^{*}/{\tau }_c^{*}=0.8$, where the color goes from dark to light with increasing particle movement. We see that particles move even at no shear, but there is greater movement at higher shear. Looking more closely in a short segment where we can track all the particles $t=30–90$ seconds are the displacement of particles as a function of distance from the surface in (c) the flow $x$ and (d) gravity $z$ directions. We see an exponential behavior in the flow movement while the bed shifts down linearly with depth compacting uniformly. The inset of (d) is the measure of the strain ${\gamma }_z$ from fitting the slopes of (d).

Figure 5

Displacement of surface particles normalized by the particle diameter during preshear in (a) the flow direction $x$ and (b) the direction of gravity $z$. We see a quick initial increase, but particles continue to move at long times. (c) Plot of $s_x$ versus $s_z$ scattered for ${\tau }^{*}/{\tau }_c^{*}=0.0$. However, $s_x$ increases systematically faster compared with $s_z$ as ${\tau }^{*}/{\tau }_c^{*}$ is increased.

Figure 6

(a) Volume fraction $\varphi$ evolution as a function of depth for ${\tau }^{*}/{\tau }_c^{*}=0.8$ averaged over ten experiments. (b) Average granular volume fraction ${\varphi }_g$ as a function of time $t$ rises increasingly rapidly with ${\tau }^{*}$ before approaching ${\varphi }_{\text{rcp}}$, indicated by a dashed horizontal line. (c) Change in ${\varphi }_g$ scaled with $A$ in the regime where $t/t_s\gg 1$ and ${\varphi }_g<{\varphi }_{\text{rcp}}$. The data are observed to collapse on the line corresponding to Eq. (1). The inset shows the fitted $A$ as a function of ${\tau }^{*}$. This quantity can be correlated with the inset to Fig. 4, where the strain along $z$ is shown. (d) The estimated time $T_{\text{rcp}}$ required to reach ${\varphi }_{\text{rcp}}$ decreases rapidly with increasing applied subcritical shear.

Figure 7

The observed transient erosion shear stress at onset is observed to increase systematically with the granular packing fraction in the bed $\varphi$ regardless of preshearing strength. Error bars indicated correspond to at least ten experimental runs for each condition.

Figure 8

Surface roughness $\xi$ evolution for various ${\tau }^{*}$ shown in Fig. 6. The values corresponding to random and ordered hexagonal packings near the surface are also shown. We see that the surface does not change systematically in either direction, while we saw earlier that shearing for long times changes the onset of erosion.