Erosion of cohesive grains by an impinging turbulent jet

TL;DR

This study investigates how inter-particle cohesion affects erosion and crater formation when a turbulent air jet impinges on a flat granular bed. Using cohesion-controlled granular material (CCGM), the authors tune the cohesion through a PBS coating and quantify erosion thresholds and crater morphologies, introducing cohesive Shields numbers at both jet and local scales. They find that erosion thresholds shift with cohesion and can be collapsed onto master curves using ${\rm Sh}_{J,c}={\rm Sh_J}/(1+\alpha{\rm Co})$ and ${\rm Sh_{\ell,c}}={\rm Sh_{\ell}}/(1+\alpha{\rm Co})$ with $\alpha\approx0.75$, while crater shapes transition from type II to type I with increasing cohesion. The work provides a transferable framework for predicting erosion and transport of cohesive sediments under turbulent jet forcing, with implications for surface cleaning, soil stability, and aerospace applications.

Abstract

The erosion and transport of particles by an impinging turbulent jet in air is observed in various situations, such as the cleaning of a surface or during the landing of a spacecraft. The presence of inter-particle cohesive forces modifies the erosion threshold, beyond which grains are transported. The cohesion also influences the resulting formation and shape of the crater. In this paper, we characterize the role of the cohesive forces on the erosion of a flat granular bed by an impinging normal turbulent jet in air. We perform experiments using a cohesion-controlled granular material to finely tune the cohesion between particles while keeping the other properties constant. We investigate the effects of the cohesion on the erosion threshold and show that the results can be rationalized by a cohesive Shields number that accounts for the inter-particles cohesion force. Despite the complex nature of a turbulent jet, we can provide a scaling law to correlate the jet erosion threshold, based on the outlet velocity at the nozzle, to a local cohesive Shields number. The presence of cohesion between the grains also modifies the shape of the resulting crater, the transport of grains, and the local erosion process.

Figures (8)

• Figure 1: (a) Schematic of the experimental setup. A turbulent jet exits the nozzle of inner diameter $D$ at the mean velocity $U_{\rm J}$ and impinges the cohesive granular bed placed at a distance $H$. (b) Evolution of the cohesive force $F_{\rm c}$ between two particles of diameter $d=800\,\mu{\rm m}$ as a function of the mean coating layer $b$. The circles are the experimental measurements, the line is equation (\ref{['eq:cohesive_force']}) with $B=230\,{\rm nm}$, and the right axis gives the corresponding value of the cohesion number gans2020cohesion. Inset: Schematic showing a glass bead of diameter $d$, and a PBS coating of thickness $b$. (c)-(d) Mean velocity threshold of the jet at the outlet of the nozzle $U_{\rm J}^*$ as a function of the distance between the nozzle and the granular bed $H$ for glass beads of diameter $d = 800 \,\mu{\rm m}$ and two cohesion numbers: (c) ${\rm Co}=4.4$ corresponding to $b=88.5\,{\rm nm}$, and (d) ${\rm Co}=8.0$ corresponding to $b=200\,{\rm nm}$. The different colors indicate different waiting time, between one minute and ten minutes, before measuring the erosion threshold.
• Figure 2: (a) Velocity of the jet at the outlet of the nozzle $U_{\rm J}^*$ at the onset of erosion as a function of the distance between the nozzle and the granular bed $H$ for glass beads of diameter $d = 800 \,\mu{\rm m}$ and different interparticle cohesion from ${\rm Co}=0$ to ${\rm Co}=10$. The dotted lines show the best fit $U_{\rm J}^*=A\,(H+\lambda)$, where the virtual origin is empirically estimated as $\lambda=-0.66\,{\rm cm}$ for all cohesion numbers considered here. (b) Reynolds number associated with the jet ${\rm Re_{\rm J}^*}$ as a function of the dimensionless height, $H/D$, for the threshold values. An estimation of the laminar regime is marked in light blue and the transition to a turbulent jet occurs around ${\rm Re}_{\rm J} \sim 10^3-2\times 10^3$ as shown by the color gradient. Values of ${\rm Re_{\rm J}^*}$ corresponding to compressible jets are shaded in orange. For a fully developed jet ($H/D>10$), all the experiments are in a turbulent-incompressible regime. The gray shaded region ($H/D<10$) in both figures denotes the distance below which the turbulent jet is not considered to be fully developed.
• Figure 3: (a) Jet cohesionless Shields number, ${\rm Sh_{\rm J}}^*$, as a function of the rescaled distance $H/D$ and varying cohesion numbers ${\rm Co}$ at the onset of erosion. (b) Jet cohesive Shields number $\mathrm{Sh}_{\mathrm{J}, \mathrm{c}}^*$ given by equation (\ref{['eq:cohesive']}) with $\alpha= 0.75$ as a function of $H/D$ for the experiments reported in figure \ref{['fig:Figure3_ShieldsJet']}(a). The solid line corresponds to equation $\mathrm{Sh_{J, c}^*} = C \ (H/D + \lambda/D)^{2}$ with $C = 0.02$ and $\lambda/D=-1.4$. In both figures the grey region denotes the distance below which the turbulent jet is not fully developed ($H/D <10$).
• Figure 4: (a) Evolution of the local erosion threshold values for $\mathrm{Sh}_{\ell}^*$ divided by the cohesionless threshold $\mathrm{Sh}_{\ell, \ {\rm Co=0}}^*$ for $H/D \gtrsim 10$ using the maximum surface velocity (in black) and the maximum shear stress on the surface (in grey). The solid lines show a fit of the form $\rm{Sh}_{\ell} / \rm{Sh}_{\rm {\ell, \ Co=0}} = 1+\alpha\ \mathrm{Co}$ for cohesion numbers $0 \leq \mathrm{Co} \leq 8$ with $\alpha=0.75 \pm 0.04$ (R-square: 0.9944) for the maximum local velocity and $\alpha=0.59 \pm 0.07$ (R-square: 0.9653) using the maximum shear stress. Inset: Threshold values for $\mathrm{Sh}_{\ell}^*$ plateau for $H/D \gtrsim 10$ using the maximum surface velocity, and setting $K=1$. The colored dashed lines display the averages for a given $\mathrm{Co}$ number. (b) Local cohesive Shields number, $\mathrm{Sh^*_{\rm{\ell, \, c}}}$ divided by the cohesionless local Shields number $\mathrm{Sh^*_{\rm{\ell, \ Co=0}}}$ when varying the dimensionless distance $H/D$. The gray shaded area shows $H/D \lesssim 10$, i.e. where the jet is not fully developed.
• Figure 5: Evolution of the shape of the asymptotic crater for $H=6\,{\rm cm}$ ($H/D=12.6$) when varying the jet velocity $U_{\rm J}$ (a) for cohesionless grains, and (b) for cohesive grains with $\mathrm{Co}=8.0$. The jet velocity $U_{\rm J}$ and color code are indicated in each figure. Darker colors indicate larger velocities at the nozzle for the same $H/D$. The inset in (a) shows a picture of crater observed for cohesionless grains.