2. Experimental set-up

Experiments were carried out in a recirculating free-surface water channel with a 150 cm $\times$ 38 cm $\times$ 50 cm (length $\times$ width $\times$ height) test section. Wing-tip vortices were generated using two rectangular half-wing models with rounded tips, mounted vertically (figure 1). One wing had a NACA 0012 cross-section, aspect ratio $ \textit{AR} = 2.5$ and chord length $c = 9.8$ cm; the other had an SD 7003 cross-section (Selig–Donovan, Selig, Donovan & Fraser Reference Selig, Donovan and Fraser1989), $ \textit{AR} = 3$ and $c = 10$ cm. Bubbles were generated by temporarily injecting air into the vortex core through an L-shaped cannula (figure 1 $b$ ). The control parameters are the angle of attack $\alpha$ of the wing and the chord-based Reynolds number $ \textit{Re} = \textit{Uc} / \nu$ , where $U$ is the free-stream velocity and $\nu$ the kinematic viscosity of water.

Figure 1.

Experimental set-up. The PIV plane is located 19 cm downstream of the wing trailing edge.

Bubbles were recorded with a digital camera (Nikon Coolpix P7800) at 25 Hz. The camera was placed at the height of the wing tip, 1 m from the test section, viewing the flow normal to the $y$ – $z$ plane shown in figure 1. Each frame was processed to identify the bubble contour and extract its position, radius, length and estimated volume. In addition, a monochrome high-speed camera (Phantom VEO 410L), positioned at the same location, recorded videos at 1000 Hz to resolve unsteady deformations of the bubble surface.

The vortex flow was characterised by stereoscopic particle image velocimetry (PIV), using a laser and image acquisition system from Dantec Dynamics. Their Dynamic Studio 8 software was used to obtain the three fluid velocity components in a plane perpendicular to the free stream located 19 cm from the wing trailing edge. Measurements of the flow without a bubble were performed over a range of $\alpha$ and $ \textit{Re}$ . To quantify how the bubble modifies the vortex, additional PIV measurements were made with a bubble located upstream of the measurement plane. These cases corresponded to conditions of maximum bubble stability, for which the bubble remained near its equilibrium position for extended times.

3. Two-phase vortex breakdown

Figure 2 shows snapshot visualisations of the new vortex breakdown phenomenon, using two different exposure times; corresponding video sequences can be found in the supplementary material. The images show an elongated air bubble, located in the core of the wing-tip vortex at some distance from the trailing edge. This flow structure is generated by temporarily injecting air near the tip vortex centre using a cannula, with the orifice facing upstream. After stopping the air injection and removing the cannula, the bubble does not rise due to buoyancy or drift away with the free stream, but it migrates along the core to a particular position, where it remains in a quasi-stationary state. For the particular parameter set of figure 2, this can last for several minutes. The short-exposure image in figure 2( $b$ ) shows that the bubble surface is actually unsteady, exhibiting random and helical perturbations, likely caused by the turbulent flow outside the vortex core. The bubble slowly loses air through a continuous stream of microbubbles leaking from its rear end, and occasional large turbulent fluctuations can also remove larger fragments of air. This leads to a steady decrease in volume, making the bubble increasingly susceptible to turbulent breakup, until it eventually disintegrates completely. The bubbles do not form spontaneously, e.g. through cavitation, since the vapour pressure of water is only a small fraction of the lowest pressure in the vortex core, which can be estimated from the measured velocity profiles (see figure 6).

Figure 2.

Bubble images for the NACA 0012 wing at $\alpha =12^\circ$ and $ \textit{Re}=80\,000$ . Exposure: (a) 40 ms, (b) 1 ms. Here, $g$ is the acceleration of gravity. Movies 1 and 2 in the supplementary material show video sequences of these visualisations.

Depending on the angle of attack and the Reynolds number, several regimes can be identified. At low $\alpha$ and/or $ \textit{Re}$ , bubbles cannot form. Either the injected air immediately rises to the water channel surface, when the pressure inside the vortex core is not low enough to overcome the bubble buoyancy, or the bubbles are unstable and break within a few seconds after the air injection is stopped. Above a certain threshold in $\alpha$ and $ \textit{Re}$ , a stable bubble (with a lifetime of at least 10 s) is generated, which migrates along the vortex core to an equilibrium position, generally within two chords from the trailing edge, where it remains and slowly dissipates, as described above. For higher values of $\alpha$ and $ \textit{Re}$ , a bubble is still formed; however, it moves upstream within the vortex core until it gets too close to the wing and breaks up. For the SD 7003 wing only, an additional regime is observed, where a bubble forms, but then continuously drifts downstream until it eventually breaks. Figure 3 shows representative time sequences for three of these regimes: stable bubbles, bubbles reaching the wing and unstable bubbles. Video sequences for all regimes can be found in the supplementary material. Figure 4 presents a regime map for each wing model, summarising the observations across the parameter space considered here: $\alpha =6^\circ$ – $19^\circ$ and $ \textit{Re}=50\,000$ – $100\,000$ .

Figure 3.

Time sequences illustrating different bubble regimes for the NACA 0012 wing. (a) $\alpha =12^\circ$ , $ \textit{Re}=80\,000$ ; (b) $\alpha =13^\circ$ , $ \textit{Re}=90\,000$ ; (c) $\alpha =10^\circ$ , $ \textit{Re}=90\,000$ . Movies 1, 3 and 4 in the supplementary material show the corresponding video sequences.

Figure 4.

Flow regimes in the $ \textit{Re}$ – $\alpha$ parameter space; (a) NACA 0012 wing, (b) SD 7003 wing. The red dots indicate the conditions of maximum bubble lifetime. Filled symbols represent stationary bubbles, open symbols migrating or unstable bubbles.

The bubble lifetime in the stable regime is variable for a given flow condition. It is mainly determined by the decrease of bubble volume caused by random turbulent fluctuations, as described above. The mean lifetime depends on the free-stream velocity, angle of attack and wing model, and varies smoothly across the parameter space. Optimal conditions for bubble stability are found around $\alpha = 12^\circ$ and $ \textit{Re} = 80\,000$ for the NACA 0012 wing, and $\alpha = 10^\circ$ and $ \textit{Re} = 80\,000$ for the SD 7003 wing, with corresponding mean lifetimes of approximately 5 and 12 minutes, respectively. Bubble stability is highly sensitive to changes in the parameters, e.g. a variation of only $0.5^\circ$ in $\alpha$ can shift the system from a stable configuration with long-lived bubbles to an unstable one.

4. Flow characteristics

Stereoscopic PIV measurements were made to characterise the tip vortex flow without a bubble for both wings across the parameter space, as well as for the most stable configurations with a bubble.

Figure 5 shows axial vorticity and axial velocity fields for the most stable case of the NACA 0012 profile, at $\alpha =12^\circ$ and $ \textit{Re}=80\,000$ . Without the bubble, the core is very thin with a concentrated vorticity distribution, and the axial flow in the core is in excess of the free-stream velocity. In the presence of the bubble, the core vorticity peak is reduced and the axial flow changes to a wake-like distribution, with a velocity deficit in the centre. In both cases, the flow is not strictly axisymmetric: outside the core, vorticity and axial velocity deficit are concentrated in a layer that is still in the process of rolling up at the measurement position. Figure 6 shows profiles of azimuthally averaged axial and azimuthal velocities, in a cylindrical frame centred on the vortex, without and with the bubble for both wings. Far from the vortex centre, for $r/c \gtrsim 0.12$ , the flow is only weakly affected by the bubble, suggesting that the circulation is conserved and that the reduction in peak vorticity is primarily due to its radial redistribution. This redistribution results in a larger core radius and a lower maximum azimuthal velocity. The axial velocity is also reduced in both cases. Whereas for the NACA 0012 wing it changes sign, it still remains weakly jet-like for the SD 7003 wing, which may be related to the fact that the axial flow without a bubble is larger than for the NACA 0012 wing (see figure 7 $b$ ).

Figure 5.

Streamwise vorticity ( $a$ , $b$ ) and velocity ( $c$ , $d$ ), measured at $z/c$ = 2, for the NACA 0012 wing at $\alpha =12^\circ$ and $ \textit{Re}=80\,000$ (conditions of maximum bubble stability). Left column ( $a$ , $c$ ) without a bubble; right column ( $b$ , $d$ ) with a bubble upstream of the measurement plane (see figure 1 $a$ ).

Figure 6.

Radial profiles of azimuthal ( $u_\theta$ ) and axial ( $u_z$ ) velocity at $z/c=2$ , without and with a bubble, for the respective conditions of maximum bubble stability for the two wings; ( $a$ ) NACA 0012 wing at $\alpha =12^\circ$ and ( $b$ ) SD 7003 wing at $\alpha =10^\circ$ , both at $ \textit{Re}=80\,000$ .

Figure 7.

( $a$ ) Vortex circulation and ( $b$ ) axial velocity deficit/excess in the vortex centre for both wings.

The observed variations in the vortex velocity profiles due to the presence of the bubble may help elucidate the mechanisms that sustain the bubble at its equilibrium position, where the net force acting on it must be zero. The pressure on the vortex axis can be estimated as $p_0 = p_\infty - \rho \int _0^\infty (u_\theta ^2/r)\mathrm{d}r$ , where $p_\infty$ is the free-stream pressure and $\rho$ the fluid density, and the axial momentum flux is given by $2\pi \rho \int _0^\infty u_z^2r\mathrm{d}r$ . The profiles measured without a bubble approximately represent those upstream when a bubble is present. The decrease in axial momentum flux represents an effective drag force of the bubble. The core expansion and reduction in swirl velocity cause a pressure increase behind the bubble, with the resulting force in the axial direction balancing the drag.

Figure 7 presents the tip vortex circulation $\varGamma$ , determined inside a circle of radius 0.15 $c$ around the centre, and the axial velocity excess/deficit $\Delta u_z = u_z(r$ = $0)-U$ , for the two wings (without a bubble). Both show little variation with Reynolds number when non-dimensionalised by the free-stream velocity $U$ and the wing chord $c$ . The non-dimensional velocity profiles also show no significant dependence on $ \textit{Re}$ , and the core size is essentially independent of the angle of attack, too.

The data in figure 7 can be used to draw isocontours of the vortex Reynolds number $\varGamma /\nu$ , as well as the limit $\Delta u_z =0$ , in the parameter space of figure 4. The result is seen in figure 8 for the two wings. The contours align reasonably well with the regime boundaries for the bubble behaviour. This can be qualitatively understood by the balance between bubble drag, which depends on $U$ , and the axial pressure difference, which depends on $\varGamma$ , as explained above ( $u_\theta \propto \varGamma$ ). The buoyancy of the bubble does not enter the axial force balance for the horizontal tip vortex in our flow. One can expect that for inclined or vertical vortices, the regime diagrams would differ significantly, or it may not even be possible to observe stable bubbles at all. At high $ \textit{Re}$ , the lower bound for bubble stability approaches the limit of zero axial velocity excess. A jet-like core flow appears to be a necessary condition for bubble formation in the wing configurations considered in this study. The results in figure 8 (and figure 4) suggest that the stable-bubble regime may disappear as $ \textit{Re}$ is increased beyond the range accessible in this study, and that injected bubbles may then always migrate upstream towards the wing.

Figure 8.

Isocontours of the vortex Reynolds number $\varGamma /\nu$ in the parameter space of figure 4; ( $a$ ) NACA 0012 wing, ( $b$ ) SD 7003 wing. The regions where the tip vortex core has an axial velocity deficit are also shown.

5. Bubble properties

The bubble dimensions (radius $R$ and length $L$ ) and equilibrium positions $z_b$ were measured from visualisation videos over a range of $\alpha$ and $ \textit{Re}$ . In the following, we analyse how the bubble properties depend on the flow characteristics obtained from PIV in the previous section, focussing primarily on the SD 7003 wing in stable configurations.

For given values of $\alpha$ and $ \textit{Re}$ , bubbles of different sizes can be generated, but both $R$ and $L$ have upper bounds, $R_m$ and $L_m$ , respectively. Figure 9( $a$ ) shows that the maximum radius $R_m$ correlates with the measured vortex circulation, increasing approximately linearly with it in the observed range. An analysis of this dependence is presented in § 6.2. By contrast, the maximum bubble length $L_m$ varies primarily with the chord-based Reynolds number $ \textit{Re}$ (figure 9 $b$ ), the effect of $\alpha$ being small. Furthermore, measurements made on different bubbles obtained under fixed conditions show that the bubble length scales approximately with the square of the radius, $L \propto R^{2}$ , although the proportionality factor varies with the flow condition.

Figure 9.

Maximum bubble radius ( $a$ ) and length ( $b$ ) for stable configurations with the SD 7003 wing. In ( $a$ ), the inset shows qualitatively the pressure contours in a cross-section of the tip vortex, obtained by considering a horizontal potential vortex in a hydrostatic pressure gradient (see § 6.2), and the grey line represents equation (6.3).

Figure 10.

Equilibrium bubble positions for the SD 7003 wing in the stable regime. ( $a$ ) Variation with the vortex Reynolds number for maximum-size bubbles; ( $b$ ) variation with bubble radius for constant flow conditions ( $\alpha =9^\circ$ , $ \textit{Re}=90\,000$ ); ( $c$ ) variation with angle of attack for the same bubble at $ \textit{Re}=80\,000$ .

The equilibrium position $z_b$ of the bubble in the stable regime depends on $\alpha$ , $ \textit{Re}$ and the bubble size. As shown in figure 10( $a$ ), $z_b$ correlates strongly with vortex circulation at fixed $ \textit{Re}$ , with larger $\varGamma$ bringing the bubble closer to the wing. At $ \textit{Re}=80\,000$ , $z_b$ varies approximately linearly with $\varGamma$ , whereas this trend is not observed for other Reynolds numbers, showing a change in the sensitivity of the equilibrium position to circulation. Recordings of the temporal evolution of bubble volume and position for stable cases (not shown) indicate that, as the bubble shrinks due to air leakage, its equilibrium position shifts downstream. This dependence of $z_b$ on bubble size is further illustrated in figure 10( $b$ ), which shows the equilibrium position of bubbles of different sizes under the same flow conditions. The fitted line corresponds to the relation $z_b \propto R^{-2}$ .

The image sequence in figure 10( $c$ ) was obtained for a single bubble at $ \textit{Re}=80\,000$ , decreasing $\alpha$ from $10.5^\circ$ to $8.5^\circ$ in steps of $0.5^\circ$ . As $\alpha$ was reduced, the bubble lost air and its estimated volume dropped by nearly $50\%$ over this sequence, while its radius remained at the maximum value for each $\alpha$ . This indicates that lowering $\alpha$ reduces the maximum admissible bubble volume. Since the vortex circulation also decreases with $\alpha$ , the downstream shift of the bubble position in figure 10( $c$ ) is consistent with the result in figure 10( $a$ ).