How vertical oscillatory motion above a saturated sand bed leads to heap formation

We show how oscillations in fluid flow over a fluid-saturated and porous sediment bed leads to the development of a bedform. To understand the role of pressure fluctuations on the bed associated with flow oscillations, we analyze how the flow penetrates into and through the bed. We then calculate the corresponding vertical pressure gradients within the bed that tend to expand the bed along the vertical direction. When these pressure gradients are large enough, they facilitate small irreversible rearrangements of the grains within the bed, and so cause granular creep. We conjecture that this granular creep alternates with jamming to produce a granular ratchet that slowly lifts the surface of the bed locally where pressure gradients dominate, and depresses the surface where shear stresses dominate. We observe that the shape of the resulting heap exhibits a constant characteristic width. The height of this heap evolves approximately as the square root of time, in agreement with dimensional arguments predicated on a coarse-grained viscous deformation of the bed. The surface of the heap contracts initially with the square root of time, consistent with an incompressible analysis of the flow of grains within the heap. Near its peak the heap grows due to a dilatation of the bed, to inward radial flux, or to a combination of the two.

Figure 1

Top left: A laser beam (red) reflected off a cylinder to form a vertical light sheet. A plate with width $2W$ oscillated about a mean height $H$ above the bed. Before each experiment, we flattened the bed using an acrylic sheet. Top right: After many oscillations of the plate, a heap developed in the bed. By taking images at regular intervals in time, we quantified the evolution of the bed's shape. Bottom: In this image of the heap from the side, the surface of the heap is marked with a black curve determined using an algorithm. The initial bed height is marked with a dashed white curve and the centerline with a vertical purple line. We measured the radial distance, $r$, as the horizontal distance from the centerline.

Figure 2

Images taken from directly above the bed at different times increasing from left to right. A ring of dark beads on the surface of the bed drew in toward the center over time. A spot of dark beads in the center stayed in the center while also contracting slightly. In both cases the beads stayed at the surface of the bed. The left-most image is of the bed in its initial flat condition, and the subsequent images were take 20 and 60 min after being subjected to oscillations of the plate at a frequency of 20 Hz and an amplitude of 0.4 mm.

Figure 3

Half of the oscillating plate appears as a horizontal bar at $z=H$. Its full width is $2W$ and it oscillated vertically with amplitude $A$. The curves between the plate at $z=H$ and the bed surface at $z=0$ are the streamlines of an inviscid stagnation point flow. This flow applied an oscillating pressure on the surface of the bed, which in turn generated flow through the porous bed modeled by Darcy's law and shown by the blue streamlines and red isolines of pressure. We predict sediment transport primarily under the origin of the coordinate system, and the peak of the observed heap grew up along the $z$ axis at $x=0$.

Figure 4

Blue dye injected into the flow generated by plate oscillations showed a region beneath the plate within which the dye followed the streamlines (black curves) of an inviscid flow associated with a stagnation point. These streamlines are superimposed on the image. In the inviscid theory, the fluid motion is symmetric in time and so flows in both directions along the streamlines. In the experiment and outside the boundary of the plate, the flow curled into a slow clockwise vortex whose turnover time was much longer than the period of plate oscillations. Our analysis of the effects of fluid motion on the bed applies to the region directly under the plate.

Figure 5

Prediction of sediment motion by $\varphi$, Experimental data are represented by circles (blue) and crosses (red): circles when sediment motion occurred and crosses when it did not. The solid lines are curves of constant $\varphi$, defined in Eq. (15). The thickest line corresponds to the value ${\varphi }_{\mathrm{RCP}}=0.024$. We find that $\varphi \approx 0.020$ generally separates conditions that formed heaps from those that did not. Values above this one correspond to integer multiples of ${\varphi }_{\mathrm{RCP}}$ and those below to 1/2, 1/4, and 1/8 of ${\varphi }_{\mathrm{RCP}}$. Note that the predicted onset of sediment motion using the Shields theory [22] corresponds to $A/d$ values of $O\left(1000\right)$.

Figure 6

Left: The profile of the bed, $h(r,t)$, at intervals of 4800 cycles (320 s), resulting from plate oscillations with an amplitude of $A/d=2.72$. Color transitions from yellow to purple indicate increasing time. The profile is normalized by the diameter of the glass beads, $d$, and the radius is normalized by the plate half width, $W=3.8\mathrm{cm}$. Right: The profile of the bed with respect to the initial height of the bed, $h_0\left(r\right)=h(r,0)$. Each profile is normalized by its peak height, $h_p\left(t\right)=h(0,t)$. The profiles are approximately invariant in time after an initial transient. We call the invariant profile $\alpha (r/W)$. The inset shows all profiles, including $h_0$, normalized by the bead diameter, $d$, and the radius normalized by the plate half width.

Figure 7

The growth of the peak height, $h_p$, was consistent with the square root of time (black line) predicted in Eq. (16). From the lower to the upper curve, the normalized plate oscillation amplitudes were $A/d=1.64$, 1.76, 1.92, and 2.04. No heap grew for a normalized oscillation amplitude of less than 1.52. The normalized oscillation frequency for all cases was $f\sqrt{d/g}=0.11$. The data are smoothed using a moving average and error bars indicate the standard error for a few select points.

Figure 8

Top: The radius of a ring, $R\left(t\right)$, on the surface of the heap contracted over time, shown by the major ($\times$) and minor ($+$) axes of an ellipse fit to images of the ring, their mean ($•$), and a square-root function for reference (–). The square-root contraction goes hand-in-hand with the heap growth as described in the text. The initial radius was $R_0=2.8\mathrm{cm}$, and the frequency of the plate oscillations was $f=20\mathrm{Hz}$. Bottom: The logarithmic scale reveals that as the ring approached the peak of the heap, its contraction slowed down relative to the square root of time. The product of the shape factor, $G\left(R\right)$ in Eq. (20) (–$\circ$–), and a 1/2 power law shows that the combination of geometric effects with the slowing dynamics reproduces the observation that the ring radius reached an approximate constant at large times (––).