2 Bubbling at the leading edge of an airfoil

Figure 1 shows a rectangular wing of span $b=0.27~\text{m}$ and chord $c=0.3~\text{m}$ composed of NACA 0012 airfoils (see, for example, Abbot & Von Doenhoff (Reference Abbot and Von Doenhoff1959) for details about the definition of the geometry of this common type of standardized lifting surface). The wing is immersed inside a square-section water tunnel of width $w=0.3~\text{m}\simeq b$ (see figure 1 a); therefore, no wing tip vortices are created and the flow field is two-dimensional (i.e. the velocity vectors are contained in planes perpendicular to the span direction). The flow rate is controlled by varying the angular velocity of the impeller of a centrifugal pump and is measured using a particle tracking method. The gas is injected into the liquid from a pressurized reservoir located inside the airfoil to the different orifices placed at the leading edge region through injection tubes of length $l_{t}=6\times 10^{-2}~\text{m}$ and inner diameter $d_{t}=1.6\times 10^{-4}~\text{m}$ ; see figure 1(b). Since the ratio $l_{t}/d_{t}\gg 1$ , the flow resistance is large enough to keep the gas flow rate constant during the bubble formation process (Oguz & Prosperetti Reference Oguz and Prosperetti1993). The gas flow rate, $Q_{g}$ , is controlled using high-precision pressure regulators to fix the value of the gas pressure $p_{g}$ in figure 1(b). Experiments are visualized using a high-speed camera Phantom v710 operated at 10 000 fps. The focal distance is $\simeq 0.11~\text{m}$ and the final spatial resolution of the images captured is $20~\unicode[STIX]{x03BC}\text{m}~\text{pixel}^{-1}$ .

Since bubbles are formed periodically, bubbling frequencies are determined from the analysis of the videos recorded. This is done measuring the time required for 20 bubbles to cross an imaginary line located downstream of the gas injection orifice. The gas flow rate could have been determined as a function of the pressure difference $p_{g}-p_{\infty }$ but, since the Reynolds number $Re_{t}$ characterizing the flow inside the injection tubes is such that $Re_{t}d_{t}/l_{t}=Q_{g}/(\unicode[STIX]{x1D708}_{g}\,l_{t})\sim O(1)$ , with $\unicode[STIX]{x1D708}_{g}=1.5\times 10^{-5}~\text{m}^{2}~\text{s}^{-1}$ the kinematic viscosity of the gas, the gas velocity profile inside the injection tubes is not Poiseuille-like and we found that the most precise way to calculate $Q_{g}$ is to make use of (1.2) with $d_{b}$ also measured from the experimental images. The local liquid velocity and the local pressure gradient at the gas injection orifices, where bubbles are generated, are calculated numerically, using potential flow theory.

Figure 1. (a) Rectangular wing composed by NACA 0012 airfoils located inside the water tunnel at an angle of attack of $\unicode[STIX]{x1D6FC}=10^{\circ }$ . (b) Sketch of the hollow airfoil, where the pressurized chamber, the gas injection tube and the boundaries of the system (2.2) are shown. Here, $h$ indicates the airfoil distance to the wall.

Indeed, since vorticity is confined to thin boundary layers if the flow around the airfoil is not separated (i.e. for angles of attack verifying the condition $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}^{\ast }$ , with $\unicode[STIX]{x1D6FC}^{\ast }\approx 15^{\circ }$ the angle above which the boundary layer is no longer attached), the velocity field can be expressed as

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}, & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D719}$ the velocity potential satisfying the Laplace equation, subjected to the zero normal velocity condition at the airfoil surface $\unicode[STIX]{x1D6F4}_{s}$ and at the wall, this latter boundary condition being satisfied using the method of images as sketched in figure 1(b), and to the boundary condition at infinity, namely,

(2.2a-c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}=0,\quad \unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\boldsymbol{\cdot }\boldsymbol{n}=0\quad \boldsymbol{x}\in \unicode[STIX]{x1D6F4}_{s},\quad \unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\rightarrow \boldsymbol{U}_{\infty }\quad |\boldsymbol{x}|\rightarrow \infty . & \displaystyle\end{eqnarray}$$

The numerical solution to the system (2.2), which also needs to satisfy the so-called Kutta condition – i.e. that the flow cannot turn around the airfoil trailing edge – is found using a standard two-dimensional boundary integral method whose details can be found, for example, in Pozrikidis (Reference Pozrikidis2002).

Figure 2. Variation of the pressure coefficient with the dimensionless distance to the leading edge, $-C_{p}(x/c)$ , at both sides of the NACA 0012 airfoil sketched in the top part of figure 1(b) for two different values of the angle of attack: (a) $\unicode[STIX]{x1D6FC}=6^{\circ }$ and (b) $\unicode[STIX]{x1D6FC}=12^{\circ }$ . Notice that the maximum values of $-C_{p}$ exhibited by the curves in (a) and (b) are larger than the ones corresponding to the case of isolated airfoils: indeed, for isolated airfoils, the maximum values of $-C_{p}$ for $\unicode[STIX]{x1D6FC}=6^{\circ }$ and $\unicode[STIX]{x1D6FC}=12^{\circ }$ are, respectively, $2.7$ and $8.6$ . Our design takes advantage of the geometrical arrangement in figure 1 to increase the values of both the suction peak and of the pressure gradient at the leading edge of the airfoil.

Figures 2 and 3 show, for exactly the same geometry as that used in experiments (see figure 1), the calculated values of the pressure coefficient

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle C_{p}={\displaystyle \frac{p-p_{\infty }}{\unicode[STIX]{x1D70C}\,U_{\infty }^{2}/2}}=1-\left({\displaystyle \frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D719}}}{\unicode[STIX]{x2202}\,\bar{s}}}\right)^{2}, & \displaystyle\end{eqnarray}$$

and of the dimensionless pressure gradient $-d\,C_{p}/d\bar{s}$ , with $\bar{s}=s/c$ and $\bar{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D719}/(U_{\infty }\,c)$ indicating, respectively, the dimensionless arclength along the airfoil surface and the dimensionless velocity potential; see figure 1(b). The results in figure 3 reveal that the sign and the magnitude of the pressure gradient do strongly depend on the position of the injection orifice: for instance, while the pressure gradient is favourable and increases monotonically with the angle of attack for $\bar{x}=x/c=0$ , the pressure gradient changes sign with $\unicode[STIX]{x1D6FC}$ for $\bar{x}\ll 1$ . The numerical results in figure 3 are confirmed by the experimental evidence in figure 4: here, the process of bubble formation for $\unicode[STIX]{x1D6FC}\simeq 12^{\circ }$ at two neighbouring orifices is illustrated. While bubbles are formed periodically at the injection orifice where the pressure gradient is strongly favourable, $-\text{d}\,C_{p}/\text{d}\bar{s}(\bar{x}=0)\simeq 800$ , no bubbles but a long gas jet is formed at the orifice where the value of the pressure gradient is adverse, $-\text{d}\,C_{p}/\text{d}\bar{s}(\bar{x}=0.0025)\simeq -100$ (see figure 3 b). Indeed, even in the case individual bubbles were formed at the injection orifice at which the pressure gradient is adverse, these bubbles would coalesce because they would be strongly decelerated in the downstream direction as a consequence of the smallness of the gas-to-liquid density ratio.

Figure 3. (a) $C_{p}$ versus $\unicode[STIX]{x1D6FC}$ at the three different locations nearby the leading edge of the airfoil illustrated in figure 4(b). (b) $\text{d}C_{p}/\text{d}\bar{s}$ versus $\unicode[STIX]{x1D6FC}$ at the same spatial locations as in (a).

Figure 4. (a) Production of bubbles at two orifices, located respectively at $\bar{x}_{1}=x_{1}/c=0$ (right) and at $\bar{x}_{2}=x_{2}/c=0.0025$ (left) for $\unicode[STIX]{x1D6FC}=11.41^{\circ }$ , $U_{\infty }=0.72~\text{m}~\text{s}^{-1}$ . The production of bubbles is periodic at the orifice located at $\bar{x}_{1}=0$ , where the pressure gradient is favourable; however, in contrast, no bubbles are formed at $\bar{x}_{2}=0.0025$ , where the pressure gradient is adverse (see figure 3 b). The scale bar indicates 1 mm. (b) Illustration of the spatial locations where the different values of the curves in figure 3 are calculated. This figure shows how close the orifice at the left in panel (a) is located to the leading edge of the airfoil; therefore, the sign and the modulus of the pressure gradient is highly dependent on the location of the gas injection orifice.

Therefore, the strategy followed here to produce monodisperse microbubbles is to inject the gas in a region of the flow where $-C_{p}$ is as large as possible, so that the local pressure $p=p_{\infty }+(\unicode[STIX]{x1D70C}\,U_{\infty }^{2}\,C_{p}(\unicode[STIX]{x1D6FC}))/2$ is as small as possible and hence the overpressures needed to make the gas flow through the pipes are as low as possible. In this way, the energy consumption associated with the injection of the gas is clearly reduced with respect to the case in which the local liquid pressure is $\simeq p_{\infty }$ . In addition, in order to minimize the diameters of the bubbles formed, the gas needs to be injected in a region of the flow where the modulus of the favourable pressure gradient is as large as possible; see equation (1.4). These restrictions, together with the evidence depicted in figures 3 and 4, suggest that the appropriate location to produce the bubbles is the leading edge of the airfoil. Consequently, from now on, we will limit ourselves to analyse the production of bubbles from orifices located at $\bar{x}=0$ for the range of experimental parameters shown in table 1.

Table 1. Values of the different parameters explored in the present experimental study.

Figure 5. (a,b): Effect of $p_{g}$ on the diameters of the bubbles generated for $\unicode[STIX]{x1D6FC}=8^{\circ },~U_{\infty }=0.58~\text{m}~\text{s}^{-1}$ . In (a), $p_{g}=2.07\times 10^{4}~\text{Pa}$ , and in (b), $p_{g}=7.13\times 10^{4}~\text{Pa}$ . (c,d): Effect of $U_{\infty }$ on the diameters of the bubbles generated for $\unicode[STIX]{x1D6FC}=9.61^{\circ }$ , $p_{g}=2.07\times 10^{4}~\text{Pa}$ . In (c) $U_{\infty }=0.36~\text{m}~\text{s}^{-1}$ and in (d) $U_{\infty }=0.58~\text{m}~\text{s}^{-1}$ . (e,f): Effect of $\unicode[STIX]{x1D6FC}$ on the diameters of the bubbles generated for $U_{\infty }=0.5~\text{m}~\text{s}^{-1},~p_{g}=2.89\times 10^{4}~\text{Pa}$ . In (e), $\unicode[STIX]{x1D6FC}=4.66^{\circ }$ and in (f), $\unicode[STIX]{x1D6FC}=11.41^{\circ }$ .

Figure 5 shows the influence of $\unicode[STIX]{x1D6FC}$ , $Q_{g}$ and $U_{\infty }$ on the bubble generation process. As is expected from (1.4) and from the results shown in figure 3, figure 5 reveals that the diameters of the bubbles formed decrease for increasing values of $\unicode[STIX]{x1D6FC}$ and $U_{\infty }$ and decreasing values of $Q_{g}$ – or, equivalently, of the gas pressure in the reservoir, $p_{g}$ . The qualitative observations depicted in figure 5 are quantified in figure 6, where both $f_{b}$ and $d_{b}$ are shown for the values of the parameters in table 1. The exhaustive analysis of the experimental data yields values of the PDI below 0.05; therefore, the bubbles produced using our method are monodisperse (Rodríguez-Rodríguez et al. Reference Rodríguez-Rodríguez, Sevilla, Martínez-Bazán and Gordillo2015).

Figure 6. (a,b) Bubbling frequencies for two different values of the incident velocity. (c,d) Diameters of the bubbles generated corresponding to the bubbling frequencies depicted in (a) and (b), respectively.

In order to rationalize the experimental observations in figure 6, where the bubbling frequencies seem to be randomly distributed, notice first that, for the case of the flow around the airfoil, the local pressure gradient is

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle |\unicode[STIX]{x1D735}p|=\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC}){\displaystyle \frac{\unicode[STIX]{x1D70C}\,U_{\infty }^{2}}{c}}\gg {\displaystyle \frac{\unicode[STIX]{x1D70C}\,U_{\infty }^{2}}{c}}, & \displaystyle\end{eqnarray}$$

with

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\text{d}C_{p}}{\text{d}\bar{s}}}(\unicode[STIX]{x1D6FC},x/c=0), & \displaystyle\end{eqnarray}$$

and $-\text{d}C_{p}/\text{d}\bar{s}$ given by the red curve in figure 3(b). Therefore, making use of equations (1.3) and (1.4) we expect that, if condition (1.5) is satisfied, the bubbling frequencies and the bubble diameters can be expressed as a function of the control parameters as

(2.6a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle f_{b}=K_{f1}\left({\displaystyle \frac{U_{\infty }}{\sqrt{c\,d_{b}/\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}}}\right)\quad \text{and}\quad {\displaystyle \frac{d_{b}}{L}}=K_{b1}\left({\displaystyle \frac{Q_{g}}{U_{\infty }\,L^{2}\sqrt{L\,\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})/c}}}\right)^{2/5}, & \displaystyle\end{eqnarray}$$

with $L=10^{-3}~\text{m}$ the same length scale as in Evangelio, Campo-Cortés & Gordillo (Reference Evangelio, Campo-Cortés and Gordillo2015) and $K_{f1}$ and $K_{b1}$ constants. However, if condition (1.5) is not satisfied, namely, if

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle K_{b1}\left({\displaystyle \frac{Q_{g}}{U_{\infty }\,L^{2}\sqrt{L\,\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})/c}}}\right)^{2/5}\lesssim K\left({\displaystyle \frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D735}p\,L^{2}}}\right)^{1/2}, & \displaystyle\end{eqnarray}$$

with $K=0.56$ an experimentally determined constant, $f_{b}$ and $d_{b}$ are expected to be given by (see (1.6))

(2.8a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle f_{b}=K_{f2}{\displaystyle \frac{U_{\infty }(1-C_{p})^{1/2}}{d_{t}}}\quad \text{and}\quad {\displaystyle \frac{d_{b}}{L}}=\left({\displaystyle \frac{6\,d_{t}}{K_{f2}\unicode[STIX]{x03C0}\,L}}\right)^{1/3}\left({\displaystyle \frac{Q_{g}}{U_{\infty }\,L^{2}(1-C_{p})^{1/2}}}\right)^{1/3}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where the local velocity at the spatial location where the gas is injected, $U$ , has been calculated using the definition of the pressure coefficient in (2.3):

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle U=U_{\infty }(1-C_{p})^{1/2}. & \displaystyle\end{eqnarray}$$

The value $K=0.56$ in (2.7) is the one minimizing the dispersion of the experimental data with respect to the values predicted in equations (2.6) and (2.8). Indeed, the results obtained for $K=0.56$ , depicted in figure 7, show that the predictions for $f_{b}$ and $d_{b}/L$ in (2.6), are in fair agreement with the experimental measurements when the values $K_{f1}=1.8$ and $K_{b1}=1.02$ , given in Evangelio et al. (Reference Evangelio, Campo-Cortés and Gordillo2015), are used. For those cases in which the condition (2.7) is verified (i.e. when the value of the local Bond number defined in (1.5) is low enough and, therefore, the role played by capillary stresses in the bubble formation process can no longer be neglected), the inset in figure 7(a) also reveals a fair agreement between the predicted frequency in (2.8) and the experimental data for $K_{f2}=0.1$ , a value which is very close to that reported in Campo-Cortés et al. (Reference Campo-Cortés, Riboux and Gordillo2016). Let us emphasize that the agreement between measurements and the predictions in (2.6)–(2.8) depicted in figure 7 has been obtained using already reported values for the constant in microfluidic geometries and using the values of the pressure gradients and local velocities calculated numerically. These facts indicate that the bubble generator presented here is, in essence, the same as those reported using microfluidics, in spite of the evident geometrical differences.

Figure 7. (a) Comparison between the bubbling frequencies predicted in (2.6) and experiments. The inset represents the comparison between the bubbling frequencies predicted in (2.8) and the few experimental data verifying the condition (2.7). (b) The experimental values of the bubble diameters not verifying (2.7) are in fair agreement with the scaling for $d_{b}/L$ given in (2.6). The diameters of the bubbles corresponding to those experimental conditions for which (2.7) is verified, namely, those experiments for which the role played by interfacial tension stresses in the bubble formation process can no longer be neglected, are not included here.

Equations (2.6) and (2.8) predict the precise way the bubble diameter $d_{b}$ decreases when $Q_{g}$ is decreased or the values of both $U_{\infty }$ and $\unicode[STIX]{x1D6FC}$ are increased. In particular, equations (2.6) and (2.8) indicate that, for fixed values of $\unicode[STIX]{x1D6FC}$ and $Q_{g}$ , $d_{b}$ could have been decreased below the minimum bubble diameter reported in this study ( $\simeq 200~\unicode[STIX]{x03BC}\text{m}$ ), if the maximum liquid velocity in the water tunnel could have been increased above $\simeq 0.7~\text{m}~\text{s}^{-1}$ .

However, the validity of the strategy described above to decrease the diameters of bubbles generated, using the procedure presented here, possesses one clear limitation that prevents increasing $U_{\infty }$ without bound: liquid cavitation. Indeed, the gas is injected in a region of the flow where the pressure is $p\simeq p_{a}+\unicode[STIX]{x1D70C}\,U_{\infty }^{2}\,C_{p}(\unicode[STIX]{x1D6FC})/2$ , with $C_{p}(\unicode[STIX]{x1D6FC})<0$ and $p_{\infty }\simeq p_{a}$ , where $p_{a}\simeq 10^{5}$ Pa is the atmospheric pressure. Therefore, $p$ could be so low for sufficiently large values of $U_{\infty }$ that vapour bubbles could be nucleated. The description of the interaction of the gas bubbles produced with the vapour bubbles that would be nucleated at the leading edge region of the airfoil for sufficiently large values of the incident velocity $U_{\infty }$ is left for a future study.