Flow and air-entrainment around partially submerged vertical cylinders

In this study, a partially submerged vertical cylinder is moved at constant velocity through water, which is initially at rest. During the motion, the wake behind the cylinder induces free-surface deformation. Eleven cylinders, with diameters from $D=1.4$ to 16 cm, were tested under two different conditions: (i) constant immersed height $h$ and (ii) constant $h/D$. The range of translation velocities and diameters are in the regime of turbulent wake with experiments carried out for $4500<\mathrm{Re}<240000$ and $0.2<\mathrm{Fr}<2.4$, where Re and Fr are the Reynolds and Froude numbers based on $D$. The focus here is on drag-force measurements and relatively strong free-surface deformation up to air-entrainment. Specifically, two modes of air-entraiment have been uncovered: (i) in the cavity along the cylinder wall and (ii) in the wake of the cylinder. A scaling for the critical velocity for air-entrainment in the cavity has been observed in agreement with a simple model. Furthermore, for $\mathrm{Fr}>1.2$, the drag force varies linearly with Fr.

Figure 1

Experimental setup. (a) Schematic tilted view of the flume (drawn up to scale) with the following dimensions: $34\times 1.2\times 0.9$ m. (b) Schematic of the arrangement around the cylinder, view from the side above-surface. The shaded red area represents the laser sheet used for the optical method. (c) Typical picture side view. The cylinder is 2.5 cm in diameter and has been translated from left to right with a velocity of 45.8 cm/s, $\mathrm{Re}=11500$, and $\mathrm{Fr}=0.9$. The depth of the cavity $\mathcal{L}$ is represented in the near cylinder wake.

Figure 2

Free-surface height reconstruction for $\mathrm{Re}=8500$. (a) $D=3$ cm and $\mathrm{Fr}=0.5$. (b) $D=1.4$ cm and $\mathrm{Fr}=1.6$. The cylinders are indicated by black circles and the white bands correspond to shadow areas of the cylinders that cannot be analyzed.

Figure 3

Drag coefficients, $C_D$, as a function of the Reynolds number for different cylinder diameters $D$. (a) Constant immersion depth $h=23\pm 0.1$ cm and (b) constant ratio $h/D=2.55\pm 0.05$. Solid symbols represent the situation without air-entrainment and empty symbols represent flow with air-entrainment behind the cylinder. Additional results from the literature are added and the results without free-surface are highlighted using a gray band.

Figure 4

Drag coefficients, $C_D$, as a function of the Froude number, Fr, for different cylinder diameters $D$. Panel (a) corresponds to a constant immersion depth of $h=23\pm 0.1$ cm and panel (b) corresponds to a constant ratio of $h/D=2.55\pm 0.05$. Solid symbols represent the situation without air-entrainment and empty symbols represent flow with air-entrainment in the cavity. For $\mathrm{Fr}>1.2$, $C_D$ is linear and the black line is a linear fit.

Figure 5

View from the side below-surface of the cylinder of 5 cm in diameter. (a) No air-entrainment at $\mathrm{Re}=34670$ or $\mathrm{Fr}=0.97$, see supplementary movie 1 [38]. (b) Air-entrainment in the wake at $\mathrm{Re}=39700$ or $\mathrm{Fr}=1.11$; bubbles encircled in blue, see Supplemental Material, movie 2 [38]. (c) Air-entrainment in the cavity behind the cylinder at $\mathrm{Re}=45680$ or $\mathrm{Fr}=1.3$; bubbles encircled in red, see Supplemental Material, movie 3 [38]. The dashed lines represent bubble trajectories.

Figure 6

View from the side below-surface of the cylinder of 16 cm in diameter. (a) Air-entrainment in the wake at $\mathrm{Re}=122500$ or $\mathrm{Fr}=0.6$; bubbles encircled in blue, see Supplemental Material, movie 4 [38]. (b) Air-entrainment in the cavity behind the cylinder at $\mathrm{Re}=199300$ or $\mathrm{Fr}=1$; bubbles encircled in red, see Supplemental Material, movie 5 [38]. The dashed lines represent observed bubble trajectories.

Figure 7

Flow diagrams Re versus Fr depicting the different air-entrainment regimes: no air-entrainment (solid symbols), air-entrainment in the cavity (partially solid symbols), and air-entrainment in both the cavity and the wake (empty symbols) for different cylinder diameters or Re/Fr. (a) For constant immersion $h=23\pm 0.1$ cm and (b) for constant $h/D=2.55\pm 0.05$. The lines are guides to the eyes delimiting the different air-entrainment regimes.

Figure 8

(a) Nondimensional cavity depth $\mathcal{L}/D$ as a function of ${\mathrm{Fr}}^2\mathrm{Re}$, for different diameters. The inset (b) is a zoom restricted to the regime of no or weak air-entrainment, where the continuous lines are linear fits. Solid symbols represent the situation without air-entrainment and empty symbols represent flow with air-entrainment in the cavity. Panels (c) and (d) are snapshots of the cavity, view from the back, typical of no-air entrainment and air-entrainment, respectively. The continuous vertical red lines represent the emerged cylinder walls and the dashed blue line represents the free-surface at rest.

Figure 9

(a) Critical velocity $U_c$ for air-entrainment in the cavity $\mathcal{L}$ as a function of the cylinder diameter $D$ for constant immersion $h=23\pm 0.1$ cm and constant $h/D\simeq 2.55\pm 0.05$. Results of Benusiglio [20] are plotted with blue crosses. The green dotted line represents the power law $D^{0.22\pm 0.02}$ for the constant $h/D$ case and the red solid line represents the power law $D^{0.20\pm 0.02}$ for $h=23\pm 0.1$ cm. (b) Critical Reynolds number, ${\mathrm{Re}}_c$, as a function of the critical Froude number, ${\mathrm{Fr}}_c$. The black dashed-dotted line represents $\mathrm{Re}/\mathrm{Fr}\simeq 25000$. The shaded region corresponds to no air-entrainment. The inset (c) is the same data in log-log scale.