# Air-Water Bubbly Flow by Multiple Vents on a Hydrofoil in a Steady Free-Stream
## 3 Results and Discussion
### Jet Thickness  
The time-averaged thickness of the jet was observed from the top-view images at the angles of inclination, \(\alpha=3.5^{\circ}\), at various air injection rates, \(Q\), and free-stream velocities, \(V_{\infty}\). The plot of the time-averaged jet thickness, \(t\), at various combinations of \(Q\) and \(V_{\infty}\) values, at the angle of inclination of \(\alpha=3.5^{\circ}\) is shown in Figure 16. The jet thickness increased with the increase in the downstream distance from the trailing edge of the hydrofoil at all free-stream velocities and the air injection coefficient values. It was observed that there was an increase in the jet thickness with the increase in the air injection coefficient at all free-stream velocities and at all non-dimensional streamwise distances from the trailing edge of the hydrofoil (\(x/c=0\)) in Figure 16. Figure 17 shows the variation of the non-dimensional jet thickness, \(t/c\), along the non-dimensional streamwise distance, \(x/c\), for various non-dimensional air injection rates, \(C_{q}\), at the Reynolds number, \(Re_{air}=30.3\). In general, at a given non-dimensional streamwise distance, \(x/c\), it was observed that the non-dimensional jet thickness, \(t/c\), increased with the decrease in the non-dimensional air injection rate, \(C_{q}\), at \(Re_{air}=30.3\). It is to be noted that \(C_{q}\) was inversely proportional to \(Q\), based on our definition in Section 2. Further, at \(Re_{air}=30.3\), the non-dimensional jet thickness, \(t/c\), was observed to increase along the non-dimensional streamwise distance, \(x/c\), at a given non-dimensional air injection rate, \(C_{q}\). Similarly, Figures 18 and 19 show the variation of the non-dimensional jet thickness, \(t/c\), along the non-dimensional streamwise distance, \(x/c\), for various non-dimensional air injection rates, \(C_{q}\), at the Reynolds number, \(Re_{air}=50.5,70.7\), respectively. The characteristic of the jet, namely, the increase in the non-dimensional jet thickness along the streamwise direction with the decrease in the non-dimensional air injection rate, \(C_{q}\), at a fixed non-dimensional distance from the trailing edge, was observed for \(Re=50.5,70.7\) as well in Figures 18 and 19. Similarly, there was an increase in the non-dimensional jet thickness, \(t/c\), along the non-dimensional streamwise distance, \(x/c\), at a given non-dimensional air injection rate, \(C_{q}\), for \(Re=50.5,70.7\) as well in Figures 18 and 19. A regression analysis was performed on the non-dimensional jet thickness, \(t/c\), based on the non-dimensional parameters, Froude number, \(Fn\), and air injection coefficient, \(C_{q}\), Reynolds number, \(Re_{air}\), and non-dimensional streamwise distance, \(x/x_{ref}\). \(x_{ref}\) was taken as the horizontal distance from the leading edge to the position of the maximum thickness on the hydrofoil in order to be conducive for the regression analysis; \(x_{ref}=25\) mm for NACA 0010 hydrofoil having a chord length of 100 mm. The equation for the non-dimensional jet thickness was given by (5) in terms of the non-dimensional parameters \(Fn,C_{q}Re_{air}\) and \(x/x_{ref}\). The straight line in Figure 20 shows a linear fit between the power-law model and experimental observations, with a coefficient of determination, \(R^{2}=0.9\). The observed behavior of the non-dimensional time-averaged jet width, which was an increase with the increase in the air injection rate, \(Q\), at a constant free-stream velocity, \(V_{\infty}\), and a decrease with the increase in the free-stream velocity, \(V_{\infty}\), at a constant air injection rate, \(Q\), as observed in general through Figures 16-19, was evident from the exponents of the non-dimensional independent variables in Equation (5). \[f_{t}=Fn^{0.48}C_{q}^{-0.49}Re_{air}^{-0.42}(x/x_{ref})^{1.24} \tag{5}\]Figure 4: The instantaneous images showing the bubbles at \(Fn=70.7\) at \(Q=6\) to \(20 SLPM\). (**a**) Side view at \(Q=10 SLPM\). (**b**) Top view at \(Q=10 SLPM\). (Multimedia view.)