Determining the onset of hydrodynamic erosion in turbulent flow

We revisit the longstanding question of the onset of sediment transport driven by a turbulent fluid flow via laboratory measurements. We use particle-tracking velocimetry to quantify the fluid flow as well as the motion of individual grains. As we increase the flow speed above the threshold for sediment transport, we observe that an increasing fraction of grains is transported downstream, although the average downstream velocity of the transported grains remains roughly constant. However, we find that the fraction of mobilized grains does not vanish sharply at a critical flow rate. Additionally, the distribution of the fluctuating velocities of nontransported grains becomes broader with heavier tails, meaning that unambiguously separating mobile and static grains is not possible. As an alternative approach, we quantify the statistics of grain velocities by using a mixture model consisting of two forms for the grain velocities: a decaying-exponential tail, which represents grains transported downstream, and a peaked distribution centered at zero velocity, which represents grains that fluctuate due to the turbulent flow but remain in place. Our results suggest that more sophisticated statistical measures may be required to quantify grain motion near the onset of sediment transport, particularly in the presence of turbulence.

Figure 1

Schematic of the experimental apparatus. The top panel shows a three-dimensional view, including the clear-walled test section with the grains and triangular wedges, and the enclosed, motorized belt drive on the opposite side. The bottom panel shows an overhead view, indicating the clockwise fluid flow direction. Overall, the apparatus is 1.8 m long, 21.25 cm tall, and 50 cm wide, with a 5.08 cm channel width. In the 1.2 m test section, the grain bed height is approximately 5.7 cm, and measurements are taken approximately 90 cm from the wedge at the entrance.

Figure 2

Typical fluid velocity profile, as measured from the streamwise velocities $u$ of tracer particles in the fluid. Each data point (open circles) represents a time average over all tracer particle trajectories for binned values of the height. The error bars show the standard deviation of the velocities in each bin. The dashed line is a fit following the procedure described by Rodríguez-López et al. [26], from which we extract the wall shear stress ${\tau }_{\text{wall}}$ (see the text).

Figure 3

Raw image of the granular bed shown from above. The physical dimensions are 6.8 cm in the horizontal streamwise direction by 2.7 cm in the vertical cross-stream direction, spanning roughly the middle half of the channel width. Selected grain trajectories are overlaid in white. This snapshot shows white circles to indicate the position and radius of a selected subset of tracked grains during a single frame, with trailing lines behind each grain indicating the motion from the beginning of each track. The time interval for each track varies, as the time at which the Lagrangian particle-tracking algorithm began to track each specific grain also varies. We include a 5-mm scale bar on the bottom left and the fluid flows from right to left.

Figure 4

(a) Mean grain velocity $\langle u_g\rangle$ as a function of Shields number $\mathrm{\Theta }$. The error bars show the standard error computed for six trials and the inset shows the same data plotted on semilogarithmic axes. (b) Standard deviation of grain velocity ${\sigma }_g$ as a function of $\mathrm{\Theta }$.

Figure 5

(a) Probability density that a fraction $n_{\mathrm{active}}$ of all tracked grains is moving at any given time for $\mathrm{\Theta }=0.023$, plotted for four different values of the velocity threshold $u_{\mathrm{threshold}}$ defining grain motion: 0.75 cm/s (circles), 1 cm/s (squares), 1.5 cm/s (stars), and 2 cm/s (triangles). These PDFs are highly sensitive to the choice of threshold. (b) Mean fraction of active grains for the same values of $u_{\mathrm{threshold}}$ as a function of $\mathrm{\Theta }$.

Figure 6

The PDFs $P\left(u_g\right)$ of grain velocities $u_g$ (solid curves), for $\mathrm{\Theta }=0.0059$, 0.0232, 0.0283, and 0.0425. Dashed curves show fits of the data to Eq. (3). The inset shows a close-up of the same data near the core of the PDF.

Figure 7

Fit parameters (a) $\sigma$, (b) $\zeta$, (c) $u_g^{*}$, and (d) $B$ from Eq. (3) as a function of Shields number $\mathrm{\Theta }$. In all cases, the error bars are the standard error computed over six different experimental trials. We mark in (c) points (in gray) at which $B<2\times {10}^{-3}$, where our fit for $u_g^{*}$ is no longer physically meaningful. The inset to (d) shows the same data plotted on semilogarithmic axes, with a horizontal line at $B=2\times {10}^{-3}$.