Competitive dynamics of two erosion patterns around a cylinder

We investigate experimentally the local erosion of a granular bed near a fixed vertical cylinder that emerges from the bed. The onset of erosion arising at the base of the cylinder and usually ascribed to the wrapping horseshoe vortex is determined and rationalized by a flow contraction effect. We report another erosion pattern visible downstream of the cylinder that consists of two side-by-side elongated holes. This pattern is observed for flow regimes close to the horseshoe scour onset, whose growth usually inhibits its spatiotemporal development.

Figure 1

Schematic of the experimental setup (top view).

Figure 2

Pictures of the granular bed in the vicinity of the cylinder of diameter $D=20$ mm for (a) $\text{Sh}/{\text{Sh}}_c\simeq 0.10, {\text{Re}}_D\simeq 1000$, and $t\simeq 3.0$ h; (b) $\text{Sh}/{\text{Sh}}_c\simeq 0.37, {\text{Re}}_D\simeq 2100$, and $t\simeq 6.9$ h; (c) $\text{Sh}/{\text{Sh}}_c\simeq 0.85, {\text{Re}}_D\simeq 3200$, and $t\simeq 0.72$ h, with a rigid base at the bottom; and (d) $\text{Sh}/{\text{Sh}}_c\simeq 0.85, {\text{Re}}_D\simeq 3200$, and $t\simeq 0.17$ h. Also shown is the topography $\xi (x,y)$ of the (e) wake scour and (f) horseshoe scour patterns for the same parameters as in (b) and (d), respectively.

Figure 3

Average vorticity field $\omega$ downstream of a cylinder of diameter $D=20$ mm. Particle image velocimetry measurements are performed in a horizontal plane located 5 mm above a rough nonerodible bed, for $\text{Sh}/{\text{Sh}}_c\simeq 0.22$ and ${\text{Re}}_D\simeq 1600$.

Figure 4

Critical values of the reduced Shields number $\text{Sh}/{\text{Sh}}_c$ for the scour onset as a function of the cylinder to grain size ratio $D/d$ and cylinder to channel size ratio $D/W$ for the two observed patterns: HSS ($\circ$) and WS ($▾$). The horizontal dashed line $\text{Sh}/{\text{Sh}}_c=0$ corresponds to the onset of erosion without any cylinder. The curved dashed and solid lines correspond to power laws $\text{Sh}/{\text{Sh}}_c=\alpha {(D/d)}^{\beta }$ with the best-fitting values $\alpha =0.73$ and $\beta =-0.23$ for HSS (- - -) and $\alpha =1.60$ and $\beta =-0.58\pm 0.1$ for WS (—).

Figure 5

Dimensionless scour depth ${\xi }_m/D$ close to a cylinder of diameter $D=10$ mm as a function of the dimensionless time $t^{*}=t{(\rho d/\mathrm{\Delta }\rho g)}^{-1/2}{(D/d)}^{-2}$ for the HSS pattern at $\text{Sh}/{\text{Sh}}_c\simeq 0.69$ ($\circ$) and the WS pattern at $\text{Sh}/{\text{Sh}}_c\simeq 0.40$ ($▾$) and $\text{Sh}/{\text{Sh}}_c\simeq 0.69$ ($\blacksquare$) by using a rigid bed at the bottom of the cylinder.

Figure 6

Evolution of (a) the dimensionless long-time scour depth $|{\xi }_{\infty }|/D$ and (b) the inverse of the dimensionless characteristic time evolution of scouring $t^{*}/\tau$ in the vicinity of a cylinder of diameter $D=10$ mm as a function of the reduced Shields number $\text{Sh}/{\text{Sh}}_c$ for the HSS pattern ($\circ$) and the WS pattern without ($▾$) or with ($\blacksquare$) a nonerodible base at the bottom of the cylinder.