2.1. Experimental set-up

A schematic of the experimental set-up is shown in figure 1. This set-up has been used in previous studies of fluid flows around a translating disk (Steiner et al. Reference Steiner, Morize, Delbende, Sauret and Gondret2023, Reference Steiner, Morize, Delbende, Sauret and Gondret2025). In summary, the experiments were performed in a square glass tank (side $40\,\textrm {cm}$ and height $60\,\textrm {cm}$ ) with the water level set to $40\;\mathrm{cm}$ . A transparent acrylic lid was placed a few millimetres beneath the free surface to suppress surface waves. At the bottom, a $1\,\textrm {cm}$ thick, horizontally levelled bed of glass beads (mean diameter $d\simeq 250 \pm 50\,{\unicode{x03BC}} \textrm {m}$ , density $\rho _{p}=2500\,{\textrm {kg}\,\textrm {m}^{-3}}$ ) was prepared. Before each erosion experiment, the surface was smoothed with a straight edge to ensure a repeatable packing and a flat surface. While we do not measure the exact packing fraction for each preparation, we estimated the practical reproducibility of the bed preparation by repeating some threshold measurements under the same control parameters, and the resulting standard deviation of the measured critical velocity of the disk at the onset of erosion is typically 5 %–10 %, which includes the variability due to bed preparation.

Figure 1.

Schematic of the experimental set-up: 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.

A circular rigid PVC disk with a diameter $D \in [5,\, 15]\,\textrm {cm}$ , small enough compared with the tank dimensions so that lateral wall effects remain negligible and with a small thickness $e=2\,\textrm {mm}$ ( $e/D \leqslant 0.04$ ), was attached to a vertical shaft (1 cm diameter) driven by an AC servomotor (ECMA-C20807RS, Delta Electronics). An eccentric cam converts the continuous rotation of the motor into a sinusoidal translation of the vertical shaft. The stroke length $L$ has been varied here in the range $2\leqslant L \leqslant 5.2\,\textrm {cm}$ , and the minimum distance $b$ between the disk and the granular bed in the range $0.2 \leqslant b \leqslant 2\,\textrm {cm}$ .

Throughout this study, the motion of the disk is a single translation over the time $\tau$ , during which the servomotor performs half a revolution at constant angular velocity $\pi /\tau$ , so that the disk accelerates and decelerates sinusoidally. The disk therefore travels the prescribed distance $L$ in a time $\tau$ and then comes to rest. Introducing the dimensionless time $t^{\ast }=t/\tau$ , the prescribed vertical position $h$ and velocity $V$ of the disk are

(2.1) \begin{equation} h(t^{\ast })=b+\frac {L}{2}\!\left [1\mp \cos (\pi t^{\ast })\right ]\!,\quad V(t^{\ast })=\pm V_{m}\sin (\pi t^{\ast }), \quad 0\leqslant t^{\ast }\leqslant 1, \end{equation}

with $V_{m}=\pi L /(2\tau )$ the maximum translational velocity at mid-stroke $L/2$ and mid-time $t^{\ast }=1/2$ . The upper (respectively lower) sign applies when the disk starts from the bottom (respectively top) position.

The parameter ranges explored across all experiments are summarised in table 1, along with the corresponding data symbols used in the figures. The first five sets of experiments correspond to the same disk diameter, $D = 10$ cm, with different stroke lengths, $L$ , and the data will be represented by different-coloured circles. The last four sets of experiments, together with the second set, correspond to different $D$ values for the same $L = 2.8$ cm, and the data will appear as blue symbols of different shapes.

Table 1.

Ranges of the parameters $L$ and $D$ explored in the different sets of experiments together with the corresponding symbols used in the figures.

2.2. Control parameters and non-dimensional groups

The problem is fully characterised by the geometric and kinematic parameters $(D,\,L,\,b,\,V_{m})$ of the disk motion, the density $\rho$ and dynamic viscosity $\eta$ of the fluid, and the grain parameters $(\rho _{p},d)$ together with the gravitational acceleration $g$ . Thus, the six following independent dimensionless parameters can be considered:

(2.2) \begin{equation} \textit{Re}_D=\frac {\rho \, V_{m}\,D}{\eta },\quad \frac {L}{D},\quad \frac {b}{D},\quad \frac {\Delta \rho }{\rho },\quad {Ar}=\dfrac {\rho \,\Delta \rho \,g\,d^{3}}{\eta ^{2}},\quad \textit{Sh}=\dfrac {\rho V_{m}^{2}}{\Delta \rho \,g\,d}, \end{equation}

where $\Delta \rho = \rho _p - \rho$ is the grain-fluid density difference. In the present study, the density ratio is constant at $\Delta \rho / \rho \simeq 1.5$ with $\rho =10^3 \,{\textrm {kg}\,\textrm {m}^{-3}}$ the density of water. The Archimedes number ${Ar} \simeq 230$ indicates that grain settling occurs in the inertial regime. The remaining dimensionless groups cover the following ranges: with $\eta =10^{-3} \,\textrm {Pa s}$ the dynamic viscosity of water, the Reynolds number $\textit{Re}_D$ spans $10^3 \lesssim \textit{Re}_D\lesssim 10^{5}$ , the relative stroke of the disk lies within $0.19 \lesssim L/D \lesssim 0.56$ , the normalised gap $b/D$ ranges from 0.013 to 0.4, and the Shields number Sh based on the driving fluid inertial forces varies approximately in the range $10^{-2} \lesssim \textit{Sh} \lesssim 10^2$ . The range of $\textit{Re}_D$ ensures that the disk-generated vortical flow is mostly in the inertial regime, while the broad variation of $b/D$ together with the variation of $L/D$ allows us to investigate the respective roles of the different flow features on the erosion threshold.

2.3. Determination of the erosion threshold

Two complementary procedures were used to identify the onset of erosion of the granular bed. First, direct visualisation was employed: for a set of control parameters $(L,\,D,\,b)$ , the disk velocity $V$ was increased by decreasing the translation time $\tau$ . The threshold of erosion corresponds to the critical value $\tau _c$ at which the very first grains were observed to be eroded from the bed. The corresponding maximum velocity $V_{m}^{\,c}=\pi L/(2\tau _{c})$ is taken as the erosion threshold. Each erosion threshold was measured three times in independent experiments, and the resulting standard deviation of the critical velocity was typically approximately 5 %–10 %.

A second method used relied on laser profilometry: the threshold was defined as the smallest $V_{m}$ for which the mean depth change exceeded $10\,\%$ of the initial layer thickness. Although the absolute values obtained by the two methods differed slightly, they are consistent for comparative purposes. Erosion thresholds generated by the radial flow before the disk stops (during both approach and retreat) were determined by direct visualisation in §§ 3.1 and 3.2, whereas the laser-profilometry criterion was applied exclusively in § 4 to the vortex-driven erosion that develops after the disk has stopped, whose strongly unsteady and fluctuating nature precludes a reliable threshold from instantaneous images.

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 $D = 10\,\textrm {cm}$ translates along the stroke length $L = 2.8\,\textrm {cm}$ during the travel time $\tau \simeq 0.21\,\textrm {s}$ down to the minimum/final gap distance $b = 0.5\,\textrm {cm}$ from the granular bed ( $V_m = 21$ cm s^–1 $\simeq 2V_m^c$ ).

2.4. Phenomenology

Figure 2 presents a time-lapse sequence from an experiment performed well above the erosion threshold, illustrating the two distinct mechanisms by which erosion arises when the disk moves downward.

The first mechanism occurs when erosion takes place while the disk is still moving ( $t^{\ast }\lt 1$ ). As the disk approaches the bed, the fluid trapped beneath is expelled radially. The resulting outward radial flow, confined in the thin gap, shears the grains near the edge of the disk where the velocity is the largest (Steiner et al. Reference Steiner, Morize, Delbende, Sauret and Gondret2025). Here, between $t^{\ast } \simeq 0.4$ and $t^{\ast } \simeq 1$ , a small ridge forms at $r\simeq D/2$ (see Box 1 in figure 2 at $t^*=0.84$ ) and drifts outward, revealing that only a narrow annulus of grains is eroded while the disk is in motion. At the same instant, the starting vortex is already visible in the wake of the disk, but its core is sufficiently far from the bed for the outward flow beneath the disk to remain predominantly radial and axisymmetric. Therefore, the first erosion threshold is governed by the wall shear generated by this squeezing flow in the gap.

The second mechanism is through vortex-driven erosion after the disk stops ( $t^{\ast }\gt 1$ ). Indeed, once the stroke ends, the starting vortex detaches, flows around the disk and impinges on the bed. The interaction of this vortex with the edge of the disk detaches a counter-rotating stopping vortex. Together, they form a dipole that translates parallel to the surface of the granular bed. In the present example, from $t^{\ast }=1.12$ to $1.68$ , grains are entrained simultaneously outward by the starting vortex and inward by the stopping vortex, producing the bidirectional transport highlighted in the panels at $t^{\ast }=1.40$ and $1.68$ . The amount of suspended granular material grows until the vortices weaken, and by $t^{\ast }=2.52$ , they can no longer sustain the grains in suspension, which then resettle onto the bed. Because the circumferential velocities in the starting and stopping vortex after the disk stops are comparable and larger than the outward flow velocity before the disk stops, the erosion mechanism by the impact of a vortex ring dipole typically imposes a higher wall stress than the squeezing radial flow. It therefore controls a separate and smaller erosion threshold.

Figure 3.

Time lapse showing the erosion of the granular bed produced by the unidirectional upward translation over the stroke length $L = 5.2\ \mathrm{cm}$ and time $\tau \simeq 0.4\ \mathrm{s}$ of a disk of diameter $D = 10\ \mathrm{cm}$ starting from its minimal position $b = 0.5\ \mathrm{cm}$ during the first accelerating phase ( $0 \leqslant t^\ast \leqslant 0.5$ ) up to the disk velocity $V_m = 20$ cm s⁻¹ $\simeq 2.5 V_m^c$ .

When the disk moves away from the granular bed, a single erosion mechanism is observed, as the initial vortex ring forms beneath the translating disk and therefore never impinges directly on the bed. The erosion threshold associated with the resulting radial inflow is nevertheless modified compared with the approaching-disk case, as the confined vortex increases both radial inflow and wall shear stress.

A representative experiment is shown in figure 3, where the disk velocity is set well above the critical value $V_m^c$ , allowing each stage to be clearly visible during the disk motion ( $0 \lt t^\ast \lt 1$ ). At the very early time $t^{*}=0.1$ , a few grains already start to lift off, forming a mound near the granular bed. As time progresses, even more grains are entrained. They move mainly upward and inward, driven by the combined action of the inward radial flow that feeds the widening gap and the starting vortex in the wake of the disk. Because this suction flow is established as soon as the vortex begins to form, erosion is observed very early in the stroke. The deformation of the bed remains almost axisymmetric and relatively shallow compared with the erosion pattern generated when the disk approaches the bed.

In summary, when the disk moves towards the bed, the transient erosion arises first from the shear induced by the outward squeezing flow in the gap and then, if the disk stops sufficiently close to the bed, from the following impact of the starting–stopping vortex pair. When the disk moves away from the granular bed, a single erosion mechanism is observed, due to the radial inward suction flow in the gap, as the initial vortex ring forms beneath the translating disk and therefore never impinges directly on the bed.

In the following, we quantify the different erosion thresholds and describe how they scale with the governing parameters.

3.1. Erosion threshold by the radial squeezing flow of a disk moving towards a granular bed

When the disk moves towards the granular bed, the fluid confined in the narrow gap is expelled radially. The associated shear can erode the granular bed before the disk comes to a stop. We quantify here the critical disk velocity $V_{m}^{c}$ corresponding to the first onset of erosion and demonstrate how it depends on the control parameters.

Figure 4.

Critical velocity $V_m^{c}$ of the approaching disk at the onset of erosion measured before the disk stops ( $t^{*}\lt 1$ ) for a disk translating towards the granular bed as a function of (a) the minimal/final distance from the granular bed $b$ for $L = 2.8\,\textrm {cm}$ and $D =10\,\textrm {cm}$ , (b) the disk diameter $D$ for $L = 2.8\,\textrm {cm}$ and $b = 1\,\textrm {cm}$ , and (c) the stroke length $L$ for $D = 10\,\textrm {cm}$ and $b = 1\,\textrm {cm}$ . Dashed lines are the best fits from (3.1) corresponding in CGS units to (a) $V_m^c = 8 b \sqrt {1+2.8/b}$ , (b) $V_m^c = 156 / D$ and (c) $V_m^c = 7.9 \sqrt {1 + L}$ , so that (a) $U_m^c = 19.75$ cm s⁻¹, (b) $U_m^c = 20.0$ cm s^-1 and (c) $U_m^c = 19.75$ cm s^-1.

Figure 4(a–c) reports the values of $V_{m}^{c}$ obtained by progressively increasing the imposed stroke velocity until a few grains begin to be eroded and transported radially outwards when the parameters $b$ , $D$ and $L$ are varied separately. The critical velocity $V_{m}^{c}$ is found to increase with the final gap $b$ (figure 4 a), decreases with increasing disk diameter $D$ (figure 4 b) and increases with the stroke length $L$ (figure 4 c). These trends directly reflect the way the radial flow required to conserve mass in the gap varies with the three parameters. To develop a scaling law using the maximum radial velocity, Steiner et al. (Reference Steiner, Morize, Delbende, Sauret and Gondret2025) assumed that the gap thickness $h(t)$ remains small compared with $D$ . Then, the incompressibility condition leads to a gap-averaged radial velocity profile $U$ that is maximum at $r=D/2$ . The maximum value over time is $U_{m}=V_{m} (D/4b ) (1+L/b )^{-1/2}$ , where $V_{m}$ is the maximum disk velocity during the translation of the disk. The shear at the granular bed required for erosion is directly related to the radial flow and thus to $U_{m}$ . Assuming that the critical value $U_{m}^{c}$ for the gap-averaged radial velocity should remain constant for a given size and density of grains, the critical disk velocity $V_{m}^{c}$ can be written as

(3.1) \begin{equation} V_{m}^{c}=U_{m}^{c}\,\frac {4b}{D}\,\sqrt {1+\frac {L}{b}}. \end{equation}

Dashed lines in figure 4 show the $V_m^{c}(b,D,L)$ best fits of the data from (3.1) that give very close fitting values for $U_m^{c}$ . The very good fit of the data for the corresponding set of parameters, shown in figure 4, suggests that the measured erosion thresholds should obey this relationship.

Figure 5.

Critical velocity of the approaching disk at the onset of erosion by the radial outflow before the disk stops ( $t^*\lt 1$ ), $V^{c}_{m}$ , as a function of (a) the minimum distance $b$ and (b) the parameter $4b\sqrt {1+L/b}/D$ . The dashed line corresponds to the best fit of the data by (3.1) with $U_{m}^{c}=21.3\,{\textrm {cm}\,\textrm {s}^{-1}}$ .

The critical disk velocity $V_m^{c}$ required to dislodge the very first grains for all stroke lengths $L$ , disk diameters $D$ and final gaps $b$ explored here with the colour code given in table 1 is shown in figure 5(a). The observed global trend again shows that $V_m^{c}$ increases with $b$ , indicating that a wider gap weakens the outward radial flow and requires a faster stroke for the shear at the granular bed to reach the erosion threshold. To account for the combined influence of $L$ , $D$ and $b$ on the measured thresholds, the critical disk velocity $V_m^{c}$ is now plotted as a function of $4b\sqrt {1+L/b}/D$ in figure 5(b). All data collapse remarkably well onto a linear master curve as predicted from (3.1) with the fitted value $U_m^{\,c}=(21.3 \pm 2.2) \, {\mathrm{cm}\,\textrm {s}^{-1}}$ . Thus, figure 5(b) demonstrates that the first erosion mode – driven solely by the outward squeezing flow before the disk stops – is fully captured by this gap-averaged kinematic model.

While the collapse in figure 5(b) shows that the gap kinematics predicts the erosion threshold through the maximum gap-averaged velocity $U_m$ , this velocity is not the standard quantity used to characterise sediment mobilisation. To connect with the sediment-transport literature, and to compare the approaching and leaving cases on the same basis, we next seek an estimate of the near-bed shear associated with the squeezing flow. Since the onset of particle motion is known to be sensitive to the structure of the near-wall region and to velocity fluctuations (Quibeuf, Charru & Lacaze Reference Quibeuf, Charru and Lacaze2020), we examine the smooth-wall DNS of Steiner et al. (Reference Steiner, Morize, Delbende, Sauret and Gondret2025) to characterise the velocity field near the wall. Figure 6(a) shows the vertical profile of the radial velocity $u(z)$ across the gap $h$ at the disk rim $r=D/2$ and at the dimensionless time $t^\ast =0.76$ , where and when $U_m$ reaches its maximum. The profile exhibits a nearly uniform core flow together with strong velocity gradients confined to thin regions adjacent to the disk and to the wall. This is expected as the corresponding gap Reynolds number, $\textit{Re}_b=\rho U_m b/\eta$ , is much larger than unity, in the range $4\times 10^2 \lesssim \textit{Re}_b \lesssim 4\times 10^3$ at the erosion threshold.

Figure 6.

(a) Velocity profile $u(z)$ across the gap between the approaching disk ( $z = h$ ) and the wall ( $z = 0$ ), 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 () are obtained from the numerical simulations of Steiner et al. (Reference Steiner, Morize, Delbende, Sauret and Gondret2025). The shaded region corresponds to the location of the disk. The solid line is $u = 73.1 z$ corresponding to $u_{\tau } = 8.5$ mm s⁻¹ 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 $z^+ = z/\delta _{v}$ . The solid line is $u^+ = z^+$ .

Figure 7.

(a) Velocity ratio $u_\tau /U_m$ as a function of the dimensionless gap $b/D$ , obtained from the numerical simulations of Steiner et al. (Reference Steiner, Morize, Delbende, Sauret and Gondret2025). The horizontal dashed line corresponds to the mean value $u_\tau /U_m \simeq 0.133$ . (b) Friction-based Shields number $\textit{Sh}^c_{\tau }$ defined from the friction velocity $u_{\tau }$ at the erosion threshold plotted as a function of $b/D$ . The solid line marks the mean value $ \textit{Sh}_{\tau }^{c}\simeq 0.19$ , while the dashed lines indicate a $\pm 25\,\%$ deviation from this mean.

In that reference configuration, the wall shear stress is unambiguously defined from the local velocity gradient at the wall, $\tau _{w}=\eta (\partial u/\partial z)_{z=0}$ , from which we compute the friction velocity $u_{\tau }=(\tau _{w}/\rho )^{1/2}$ . For the case shown in figure 6, this gives $u_{\tau }=8.5\,\mathrm{mm\,s^{-1}}$ and a viscous length $\delta _{v}=\nu /u_{\tau }=0.12\,\mathrm{mm}$ , so that $\delta _{v}/d \simeq 0.5$ . The profile plotted in figure 6(b) further shows a linear variation of $u^{+}$ with $z^{+}$ close to the wall, which is consistent with the presence of a viscous near-wall region in the DNS.

The friction velocity $u_{\tau }$ is compared with the vertically averaged gap velocity $U_m$ for all relative disk–wall gaps $b/D$ in figure 7(a). The ratio $u_{\tau }/U_m$ remains approximately constant, with a mean value $0.13 \pm 0.04$ , which provides a practical conversion between the gap-averaged velocity and the corresponding smooth-wall estimate of the near-bed shear. Since the onset of particle motion is known to be sensitive to the structure of the near-wall region and to velocity fluctuations (Quibeuf et al. Reference Quibeuf, Charru and Lacaze2020), we use $u_{\tau }$ here as a smooth-wall estimate of the near-bed shear associated with the radial squeezing flow. On this basis, we define a friction-based Shields number

(3.2) \begin{equation} \textit{Sh}_{\tau }=\frac {\rho {u_\tau }^{2}}{\Delta \rho g d}. \end{equation}

Using the approximately constant ratio $u_{\tau }/U_m$ obtained from the smooth-wall DNS, the corresponding critical values $\textit{Sh}_{\tau }^{c}$ remain nearly constant around $0.19 \pm 0.05$ over the explored range of $b/D$ , as shown in figure 7(b). This value may be interpreted as an effective threshold for the present transient and spatially non-uniform configuration. For reference, it is of the same order as critical Shields numbers reported for steady homogeneous flows by Loiseleux et al. (Reference Loiseleux, Gondret, Rabaud and Doppler2005) and Ouriemi et al. (Reference Ouriemi, Aussillous, Medale, Peysson and Guazzelli2007), but such a comparison is only qualitative since the present threshold is inferred from smooth-wall DNS and not measured directly over a granular rough bed. This comparison with the classical Shields curve should thus be treated with care. Here, the forcing is transient, spatially non-uniform and strongly unsteady. As a result, the threshold may depend not only on the peak stress, but also on the duration over which it remains, which makes direct comparisons with steady Shields data difficult. These points can likely explain why the values of $\textit{Sh}_\tau ^c$ of order $0.1$ – $0.2$ are slightly higher than in the classical situation.

In summary, the erosion driven by the radial squeezing flow is set by the critical gap-averaged radial velocity $U_{m}^{c}\simeq 21\,$ cm s $^{-1}$ corresponding to the critical friction velocity $u_\tau ^{c} \simeq 2.8\,$ cm s $^{-1}$ for the present glass beads. Introducing the Shields numbers based on the radial squeeze velocity $U_{m}^{c}$ and on the friction velocity $u_\tau ^{c}$ respectively as

(3.3) \begin{equation} \textit{Sh}_{\textit{SQ}}^{c}=\frac {\rho \,{U_{m}^{c}}^2}{\Delta \rho \,g\,d} \quad \textrm {and} \quad \textit{Sh}_{\tau _{\textit{SQ}}}^c= \frac {\rho \,{u_\tau ^{c}}^2}{\Delta \rho \,g\,d}, \end{equation}

we obtain the critical values $\textit{Sh}_{\textit{SQ}}^{c} \simeq 11$ and $ \textit{Sh}_{\tau _{\textit{SQ}}}^c \simeq 0.19$ . These results provide a reliable predictive criterion for design situations where rapid gap compression mobilises sediment.

Figure 8.

Critical disk velocity at the onset of erosion, $V_m^{c}$ , for a disk starting from its lower position as a function of (a) the minimum distance to the granular bed, $b$ ( $L = 2.8\ \mathrm{cm}$ , $D = 10\ \mathrm{cm}$ ), (b) the disk diameter, $D$ ( $L = 2.8\ \mathrm{cm}$ , $b=1\ \mathrm{cm}$ ) and (c) the stroke length, $L$ ( $D = 10\ \mathrm{cm}$ , $b=1\ \mathrm{cm}$ ). Dashed lines are best fits from (3.1) corresponding in CGS units to (a) $V_m^c = 4.9 b \sqrt {1+2.8/b}$ , (b) $V_m^c = 97 / D$ and (c) $V_m^c = 5.1 \sqrt {1 + L}$ , so that (a) $U_m^c = 12.25\ \mathrm{cm\,s^{-1}}$ , (b) $U_m^c = 12.44\ \mathrm{cm\,s^{-1}}$ and (c) $U_m^c = 12.75\ \mathrm{cm\,s^{-1}}$ .

3.2. Erosion threshold by the radial suction flow of a disk moving away from a granular bed

The measured disk velocity $V_m^{c}$ for the onset of erosion for a disk that starts from its lower position $b$ and therefore moves away from the granular bed is reported in figure 8(a–c) when each parameter $b$ , $D$ or $L$ is varied separately. We observe that $V_m^{\,c}$ : (i) increases with the minimum gap distance $b$ (figure 8 a); (ii) decreases with increasing disk diameter $D$ (figure 8 b); and (iii) increases with the stroke length $L$ (figure 8 c). These trends are similar to those obtained when the disk moves towards the bed, yet the absolute values are systematically lower, showing that erosion is more easily triggered when the disk moves away from the granular bed, which may be due to the vortex formation within the gap during the disk motion that contributes to erosion. Dashed lines in figure 8 show the best fits of the data using (3.1). Indeed, the same analysis performed for the squeezing flow, where the disk moves towards the granular bed, can also be applied to the present case of suction flow, where the disk moves away from the granular bed (Steiner et al. Reference Steiner, Morize, Delbende, Sauret and Gondret2025). The good fit of the data with close-fitting values for $U_m^{c}$ for the corresponding set of parameters suggests that the measured erosion thresholds should obey this relationship.

Figure 9.

(a) Critical disk velocity at the onset of erosion, $V_m^{c}$ , as a function of $4b(1 + L/b)^{1/2}/D$ for a disk moving away from the granular bed. The dashed line corresponds to the best fit by (3.1) with $U_m^{c} = 13.4\ \mathrm{cm\,s^{-1}}$ . (b) Friction-based Shields number, $\textit{Sh}^{c}_{\tau }$ , as a function of $b/D$ . The solid line is the average value $\overline {\textit{Sh}^{c}_{\tau }} \simeq 0.12$ , while the dotted lines delimit this mean by $\pm 25\,\%$ .

To test if the erosion threshold is only due to the radial suction flow, the measured disk velocity $V_m^c$ at threshold is plotted in figure 9(a) against the theoretical gap-averaged maximum velocity given by (3.1) for all our data. All the data collapse rather well onto a master curve that is close to a linear law with a slope corresponding to $U_m^{\,c}=(13.4 \pm 1.4) \, \mathrm{cm\,s^{-1}}$ , which is indeed significantly smaller than the value $U_m^{\,c}=(21.3 \pm 2.2) \, \mathrm{cm\,s^{-1}}$ found in the case where the disk moves towards the granular bed. As in the previous squeeze case, the characteristic Reynolds number for the gap flow is again much larger than one, in the range $2 \times 10^2 \lesssim \textit{Re}_b \lesssim 2 \times 10^3$ at the erosion threshold, so that we anticipate some possible differences in the velocity profile of the gap flow for the two cases where the disk approaches or moves away from the granular bed, and thus in the onset of erosion.

Following the same procedure as already presented, the friction velocity $u_{\tau }$ in the viscous sublayer is also extracted from the DNS of Steiner et al. (Reference Steiner, Morize, Delbende, Sauret and Gondret2025). Across the full range of disk–wall separations explored, the ratio of this friction velocity to the gap-averaged velocity remains essentially constant with the value $u_{\tau }/ U_{m} \simeq 0.17 \pm 0.04$ , which is a little larger than the value $u_{\tau }/ U_{m} \simeq 0.13 \pm 0.04$ found in the case where the disk moves towards the wall. The particle Reynolds number $\textit{Re}_p=u_\tau d/\nu$ is always larger than unity over the range explored. Using $u_\tau$ as a smooth-wall estimate of the near-bed shear, we again define a friction-based Shields number as $\textit{Sh}_\tau =(\rho u_\tau ^2)/(\Delta \rho \, g\, d).$ The corresponding critical value $\textit{Sh}^{\,c}_{\tau }$ for bed erosion threshold is plotted as a function of the relative minimum distance of the disk to the granular bed $b/D$ in figure 9(b). The mean value for this critical Shields number is $\overline {\textit{Sh}^{\,c}_{\tau }}\simeq 0.12 \pm 0.02$ . When compared with the case where the disk moves towards the granular bed, this mean value is significantly smaller and there is also a significant trend of smaller $\textit{Sh}^{\,c}_{\tau }$ for smaller $b/D,$ which can be observed despite the scattering of the data. This likely reflects that the model neglects the influence of the vortex on the gap-driven inward radial flow. This interpretation is also consistent with the onset times reported in figure 10: when the disk moves away from the bed, erosion can start early in the stroke ( $t_c^*\lt 0.5$ ), i.e. before the disk reaches its maximum velocity. In that situation, a purely gap-driven estimate based on the peak $U_m$ may under-represent the instantaneous near-bed forcing at $t_c^*$ because the developing vortex within the gap contributes to the shear. In addition, figure 10(b) shows that the onset time of erosion increases significantly with $b/D$ . This supports the hypothesis that the vortex contribution depends on the instant when erosion begins: the vortex will be more or less developed and more or less distant from the granular bed.

Figure 10.

Dimensionless time $t_c^\ast$ at which the first grains begin to move at the erosion threshold, with the disk released from its lower position, as a function of (a) the dimensionless time $t_m^\ast =(1/\pi )\textrm {acos}{(1+2b/L)^{-1}}$ at which the maximum gap-averaged radial velocity $U_m$ is predicted to occur by Steiner et al. (Reference Steiner, Morize, Delbende, Sauret and Gondret2025), and as a function of (b) the normalized gap $b/D$ . The solid line corresponds to $t_c^\ast = t_m^\ast$ .

Figures 9(b) and 10(b) show that both the critical Shields number $\textit{Sh}^{\,c}_{\tau }$ and the critical time $t_{\mathrm{c}}^{\ast }$ for erosion increase with $b/D$ . Indeed, at small gaps ( $b/D\lesssim 0.2$ ), the vortex core lies only a few grain diameters above the bed and the circumferential velocity, which decays as $1/r'$ , increases the gap–driven inflow, so grains lift off almost instantly. As the gap widens, the core moves away and its contribution to the near-wall shear decays roughly as $1/b$ . A larger purely gap-driven inflow is therefore needed to reach the same bed stress, which delays erosion (larger $t_{{c}}^{\ast }$ ) and raises the value of the critical Shields number $\textit{Sh}^{c}_{\tau }\propto {U_{m}^c}^{2}$ .

In summary, when the disk moves away from the granular bed, the erosion is triggered during the first accelerating phase of the disk motion by the radial inflow generated between the disk and the granular surface, which is reinforced by the starting vortex ring. The threshold velocity follows the same scaling law of (3.1) as in the approaching-disk case, but with a smaller prefactor. Indeed, the erosion driven by the radial suction flow is set by the critical gap-averaged radial velocity $U_{m}^{c}\simeq 13\,$ cm s $^{-1}$ , which is $40\,\%$ smaller than the critical value $U_{m}^{c}\simeq 21\,$ cm s $^{-1}$ found in the approaching case. However, due to a larger $u_\tau /U_m$ ratio than in the approaching case, the critical friction velocity is $u_\tau ^{c} \simeq 2.3\,$ cm s $^{-1}$ , only $20\,\%$ smaller than the critical value $u_\tau ^{c} \simeq 2.8\,$ cm s $^{-1}$ found in the approaching case. Considering the Shields numbers based on the radial suction velocity $U_m^c$ and on the friction velocity $u_\tau ^c$ respectively as

(3.4) \begin{equation} \textit{Sh}_{\textit{SU}}^{c}=\frac {\rho \,{U_{m}^{c}}^2}{\Delta \rho \,g\,d} \quad \textrm {and} \quad \textit{Sh}_{\tau {\textit{SU}}}^c= \frac {\rho \,{u_\tau ^{c}}^2}{\Delta \rho \,g\,d}, \end{equation}

we obtain the critical values $\textit{Sh}_{\textit{SU}}^{c} \simeq 4$ and $\textrm{Sh}_{\tau _{\textit{SU}}}^c \simeq 0.12$ . These results provide a reliable predictive criterion for design situations where rapid gap expansion mobilises sediment.

The fact that the critical Shields number for erosion is not strictly identical for the squeeze flow and the suction flow is due to the fact that the flow is not strictly identical for two reasons corresponding to unsteadiness. The first reason is that the maximal radial velocity $U_{m}$ of the flow induced by the motion of the disk is identical in both configurations, but not the time $t_{m}$ at which this maximal radial velocity is reached and, thus, not the typical flow acceleration $U_{m}/t_{m}$ . Indeed, the time $t_{m}$ is smaller in the suction flow, for which $0 \lt t_{m} \lt \tau /2$ , than in the squeeze flow, for which $\tau /2 \lt t_{m} \lt \tau$ . As it is typically three times smaller, with a precise value that depends slightly on $b/L$ , the acceleration $U_{m}/t_{m}$ is typically three times larger in the suction flow than in the squeeze flow. This larger acceleration is expected to induce a larger driving drag force on the grains from the added-mass force term, which can explain some of the difference between the two flow configurations. The second reason is the vortex flow that modifies the velocity profile within the gap and induces some additional unsteady effects.