Vortex ring refraction at large Froude numbers

We have experimentally studied the impact of an initially planar axisymmetric vortex ring, incident at an oblique angle, upon a gravity-induced interface separating two fluids of differing densities. After impact, the vortex ring was found to exhibit a variety of subsequent trajectories, which we organize according to both the incidence angle, ${\theta }_i$, and the interface strength, defined as the ratio of the Atwood and Froude numbers, $A/F$. For grazing incidence angles $\left({\theta }_i\gtrsim 70°\right)$ vortices either penetrate or reflect from the interface, depending on whether the interface is weak or strong. In some cases, reflected vortices execute damped oscillations before finally disintegrating. For smaller incidence angles $\left({\theta }_i\lesssim 70°\right)$ vortices penetrate the interface. When there is a strong interface, these vortices are observed to curve back up toward the interface. When there is a weak interface, these vortices are observed to refract downward, away from the interface. The critical interface strength below which vortex ring refraction is observed is given by ${\text{log}}_{10}\left(A/F\right)=-2.38\pm 0.05$.

Figure 1

(Color online) Left: a schematic of the experimental configuration. Right: a photograph of the vortex ring launcher. See the text for identification of the labels.

Figure 2

(Color online) (a) The normalized salinity profile of an air/saltwater interface. The measured interface width is 4 mm. The maximum salinity in (a) corresponds to a density of $1.00748\text{g}/\text{cc}$. (b) The normalized salinity profile of a freshwater/saltwater interface immediately after it was established (orange squares), after disruption by vortex launches (blue triangles), and after allowing diffusion for six days (purple lozenges). For the squares and triangles, the measured interface widths are 12 and 25 mm, respectively. The maximum salinity in (b) corresponds to a density of $1.00339\text{g}/\text{cc}$.

Figure 3

(Color online) A montage depicting the position, and orientation, of a vortex ring launched from the upper left corner of the image at an angle of $59.2°$ with respect to the vertical. The horizontal dotted line indicates the position of the center of the fluid density interface. The oblique black line indicates the initial trajectory of the vortex ring. The Atwood number for this interface is $A=0.00428$.

Figure 4

The time dependence of the width, $w$, velocity, $v$, Froude number, $F$, and Reynolds number, $R$, for the launch depicted in Fig. 3. The vertical dashed lines delimit the time interval during which the vortex ring is in contact with the fluid density interface.

Figure 5

(Color online) A montage depicting the position, and orientation, of a vortex ring launched from the upper left corner of the image at an angle of $54.1°$ with respect to the vertical. There is no fluid density interface; the trajectory is linear.

Figure 6

The time dependence of the width, $w$, velocity, $v$, Froude number, $F$, and Reynolds number, $R$, for the launch depicted in Fig. 5.

Figure 7

(Color online) A montage depicting the position of a vortex ring launched from the upper left corner of the image at an angle of $53.2°$ with respect to the vertical. There is no fluid density interface. The vortex ring clearly suffers a change from its initial trajectory, indicated by the oblique black line, due to spontaneous shedding of vorticity (not shown).

Figure 8

(Color online) The time dependence of the width, $w$, velocity, $v$, Froude number, $F$, and Reynolds number, $R$, for the launch depicted in Fig. 7. The time of instabilities are marked by vertical dashed lines. Power law fits for each stable region are shown as oblique solid red lines in the velocity plot.

Figure 9

(Color online) Data for three separate vortex ring launches at an incidence angle of $82°$ with respect to a normal to the fluid density interface, shown as a horizontal dotted line. In the right column, for each launch, is shown the trajectory, as well as a linear fit to the first ten to twenty data points above the interface; in the left column is shown the depth dependence of ${\text{log}}_{10}\left(A/F\right)$.

Figure 10

(Color online) Data for three separate vortex ring launches at an incidence angle of $72°$ with respect to a normal to the fluid density interface, shown as a horizontal dotted line. In the right column, for each launch, is shown the trajectory, as well as a linear fit to the first ten to twenty data points above the interface; in the left column is shown the depth dependence of ${\text{log}}_{10}\left(A/F\right)$.

Figure 11

(Color online) Data for vortex rings launched at incidence angles $35°$ (top row), $45°$ (middle row) and $60°$ (bottom row). In each row, three separate launches are shown, represented from right to left by blue triangles, orange squares, and black circles. The trajectories are shown in the right column, along with linear fits to the data above the interface (solid lines) and below the interface (dashed red lines). The depth dependence of ${\text{log}}_{10}\left(A/F\right)$ is shown in the left column.