Erosion induced by a disk translating toward or away from a granular bed

TL;DR

This work quantifies erosion thresholds on a granular bed caused by a disk translating toward or away from the bed under a single vertical stroke. It identifies two erosion mechanisms for approach: an outward gap-squeezing flow during motion and a vortex-dipole interaction after stopping, each with distinct scaling laws; for retreat, erosion is driven by inward suction and the nascent vortex, occurring earlier in the stroke. The study introduces a gap-averaged velocity scaling, V_m^c = U_m^c (4b/D) sqrt(1+L/b), with U_m^c ≈ 21.3 cm s^−1 for approach and U_m^c ≈ 13.4 cm s^−1 for retreat, and defines inertial Shields numbers Sh_τ^c ≈ 0.19 (toward) and ≈ 0.095 (away). These results reveal how transient, vortex-dominated flows govern erosion thresholds and provide a framework for predicting sediment mobilization in impulsive, near-wall flows relevant to biology and geophysics, including applications to burrowing, locomotion, and scour around structures. $V_m^c = U_m^c rac{4b}{D} rac{1}{ oot 2rom ext{ extquotesingle}} \, sqrt{1+rac{L}{b}}$ with $U_m^c$ on the order of tens of cm s^−1, and $Sh_τ^c$ values around 0.1–0.2 depending on direction, characterize the thresholds governing transient erosion.

Abstract

Unsteady flows generated when a body approaches or departs from a granular bed arise in swimming, burrowing, and maneuvering devices. Yet, the threshold for grain motion in such transients remains poorly modeled due to the complexity of the flow. In this study, we report laboratory measurements of the onset of erosion when a rigid circular disk is subjected to a single vertical stroke through quiescent water above a granular bed. The stroke length and travel time were varied independently to determine the critical velocity at which the granular bed is eroded for different minimum distances from the bed. Two erosion mechanisms are observed for disk motion towards the bed: during the stroke, the outward squeezing flow erodes grains near the edge, while after stoppage, the starting vortex or associated secondary vortices impinge on the surface. For motion away from the bed, only the early interaction between the inward suction flow and the nascent vortex entrains grains. The resulting dimensionless thresholds clarify the respective roles of radial flows and vortices in transient, impulsively driven erosion.

Figures (15)

• Figure 1: Schematic of the experimental setup: A rigid disk of diameter $D$ translates vertically over a stroke $L$ in time $\tau$, either descending to the minimum gap $b$ or ascending from it.
• Figure 2: Pictures taken at nine successive dimensionless times $t^{\ast}=t/\tau$ illustrating the erosion of a granular bed induced by the flow generated when a disk of diameter $D = 10\,{\rm cm}$ translates along the stroke length $L = 2.8\,{\rm cm}$ during the travel time $\tau \simeq 0.21\,{\rm s}$ down to the minimum/final gap distance $b = 0.5\,{\rm cm}$ from the granular bed ($V_m = 21$ cm/s $\simeq 2V_m^c$.)
• Figure 3: Critical velocity $V_m^{c}$ of the disk at the onset of erosion measured before the disk stops ($t^{*}<1$) for a disk translating towards the granular bed as a function of (a) the minimal/final distance from the granular bed $b$ for $L = 2.8\,{\rm cm}$ and $D =10\,{\rm cm}$, (b) the disk diameter $D$ for $L = 2.8\,{\rm cm}$ and $b = 1\,{\rm cm}$, and (c) the stroke length $L$ for $D = 10\,{\rm cm}$ and $b = 1\,{\rm cm}$. Dashed lines are the best fits from Eq. \ref{['eq:Vmcrit']} corresponding to (a) $V_m^c = 8 b \sqrt{1+2.8/b}$, (b) $V_m^c = 156 / D$, and (c) $V_m^c = 7.9 \sqrt{1 + L}$ with (a) $U_m^c = 19.75$ cm/s, (b) $U_m^c = 20.0$ cm/s, and (c) $U_m^c = 19.75$ cm/s.
• Figure 4: Critical velocity of the disk at the onset of erosion by the radial outflow before the disk stops ($t^*<1$), $V^{c}_{m}$, as a function of (a) the minimum distance $b$ and (b) the parameter $4b(1+L/b)^{1/2}/D$. The dashed line corresponds to the best fit of the data by Eq. (3.1) with $U_{m}^{c}=21.3\,{\rm cm\,{s}^{-1}}$.
• Figure 5: (a) Velocity profile $u(y)$ across the gap between the disk ($y = h$) and the wall ($y = 0$), sampled at the disk edge ($r=D/2$) at the dimensionless time $t^{\ast}=0.76$ where and when the fluid velocity $U_m^c$ is maximum, for $L = 2.8\,\text{cm}$, $D = 10\,\text{cm}$, $b = 0.5\,\text{cm}$, and $\tau = 1.25\,\text{s}$ (Re $\simeq 3.5 \times 10^3$). The data ($\textcolor{blue}{\circ}$) are obtained from the numerical simulations of Steiner2025. The shaded region corresponds to the location of the disk. The solid line is $u = 73.1 y$ corresponding to $u_{\tau} = 8.5$ mm/s and $\delta_{v}= 1.2 \times 10^{-4}$ m. (b) Corresponding dimensionless velocity profile in wall units $u^+ = u/u_{\tau}$ as a function of $y^+ = y/\delta_{v}$. The solid line is $u^+ = y^+$.