Hysteretic growth and decay of a waterspout column

This work explores a model waterspout: a flow of water and sunflower oil driven by the rotating lid in a sealed vertical cylindrical container. The experiments reveal the hysteretic growth and decay of a water column. The numerical simulations uncover vortex breakdown (VB) in the water and oil flows. As the rotation speeds up, (1) a VB water cell emerges near the bottom center, (2) it expands and occupies almost the entire water volume except a thin layer adjusted to the interface, (3) a VB oil cell emerges and disappears above the interface-axis intersection, (4) the interface rises near the axis, descends at the periphery, and shifts from the sidewall to the bottom, (5) the water touches the lid near the axis and forms a column, extending from the bottom up to the lid. As the rotation decelerates, the process reverses, but the flow states differ from those for the direct process at same rotation speeds. It is argued that the hysteresis is a capillary phenomenon and occurs because the interface-wall contact angle differs in the direct and reverse processes.

Figure 1

Schematic of experimental setup.

Figure 2

The laser beam (a) reflects and (b) tangentially touches the interface for measurement the interface hight.

Figure 3

Schematic of the numerical problem.

Figure 4

Velocity on the axis for ${\mathrm{Re}}_o=600$ (a), 700 (b), and 750 (c), calculated with standard (solid curve), improved (dashed curve), and fine (dotted curve) meshes.

Figure 5

Streamline pattern at ${\mathrm{Re}}_o=200$ obtained by (a) HM and (b) FLUENT software.

Figure 6

Clockwise circulation CRW2 emerges as ${\mathrm{Re}}_o$ increases: ${\mathrm{Re}}_o=1$ (a), 100 (b), and 300 (c).

Figure 7

Distribution of velocity $w$ at the axis as ${\mathrm{Re}}_o$ increases: (a) ${\mathrm{Re}}_o=25$ (solid curve), 50 (dashes), and 100 (dots); (b) ${\mathrm{Re}}_o=300$ (solid curve) and 400 (dots).

Figure 8

Dependence on ${\mathrm{Re}}_o$ of difference between interface $z_i$ and separatrix $z_s$ heights at axis.

Figure 9

Interface shapes at ${\mathrm{Re}}_o=0$ (a), 325 (b), 515 (c), 650 (d), 687 (e), 840 (f), 935 (g), and 955 (h), as ${\mathrm{Re}}_o$ increases.

Figure 10

Two different flow states at ${\mathrm{Re}}_o=900$ (a) as ${\mathrm{Re}}_o$ increases and (b) as ${\mathrm{Re}}_o$ decreases after the interface neck breaks.

Figure 11

Two different flow states at ${\mathrm{Re}}_o=650$ as ${\mathrm{Re}}_o$ (a) increases and (b) decreases.

Figure 12

Two different steady states at ${\mathrm{Re}}_o=515$ (a) as ${\mathrm{Re}}_o$ increases and (b) as ${\mathrm{Re}}_o$ decreases.

Figure 13

The interface shapes are similar at ${\mathrm{Re}}_o=325$ for ${\mathrm{Re}}_o$ increasing (a) and decreasing (b).

Figure 14

Dependence of interface height at the axis $h_i$ on the rotation speed as ${\mathrm{Re}}_o$ increases (filled circles) and decreases, starting from ${\mathrm{Re}}_o=875$ (empty circles) and from ${\mathrm{Re}}_o=650$ (crosses).

Figure 15

Comparison of experimental (circles) and numerical (squares and crosses) results for the interface height at the axis $h_i$ vs the rotation speed, as ${\mathrm{Re}}_o$ (a) increases and (b) decreases.

Figure 16

Photos show emergence and disappearance of near-axis oil cell as ${\mathrm{Re}}_o$ increases: ${\mathrm{Re}}_o=325$ (a), 400 (b), 500 (c), 600 (d), 650 (e), and 700 (f).

Figure 17

Azimuthal, $v$, and meridional, $v_t$, velocities at the interface for ${\mathrm{Re}}_o=25$ (dotted curves), 200 (dashed curves), and 400 (solid curves).

Figure 18

Anticlockwise oil circulation CRO2 emerges and disappears as ${\mathrm{Re}}_o$ increases: ${\mathrm{Re}}_o=400$ (a), 600 (b), and 700 (c).

Figure 19

Dependence on ${\mathrm{Re}}_o$ of axial extent, $z_s-z_i$, of oil cell with anticlockwise circulation.

Figure 20

Contours of constant azimuthal vorticity for streamline patterns shown in Fig. 18. ${\mathrm{Re}}_o=400$ (a), 600 (b), and 700 (c).

Figure 21

Comparison of experimental (circles) and numerical (curves) results for velocity distribution along the axis for ${\mathrm{Re}}_o=400$ (a), 600 (b), and 700 (c).

Figure 22

Eigenvalues of $\omega ={\omega }_r+i{\omega }_i$ at $m=1$ and ${\mathrm{Re}}_o$ values shown in the picture.

Figure 23

Energy distribution for the most dangerous disturbance at ${\mathrm{Re}}_o=600$ (a) and 750 (b).

Figure 24

Profiles of swirl $v$ and meridional $v_t$ velocities at the interface at ${\mathrm{Re}}_o=600$ (a) and 750 (b).