# Air-Water Bubbly Flow by Multiple Vents on a Hydrofoil in a Steady Free-Stream
## 4 Conclusions  
In this work, we characterized bubbles flowing through multiple vents on the surface of an inclined hydrofoil in a free-stream. The effect of buoyancy was evident at low non-dimensional air injection rates, \(C_{q}\), and low Froude numbers, \(Fn\). The variation of the bubble size in pixels was obtained with time by image processing. Based on the bubble size, the deformation rate, \(\epsilon\), and standardization, \(\epsilon_{s}\) was observed to change with time, even as the bubble was advected by the free-stream. By image processing, the packing of the bubbles in a unit area was quantified in terms of spatial density, \(S_{D}\). The spatial density was observed to vary with time at several combinations of the Froude number, \(Fn\), and non-dimensional air injection coefficients, \(C_{q}\). The time-averaged spatial density was observed to increase in general with the increase in the air injection rate at all Froude numbers in a logarithmic manner. We obtained a power-law relation for the time-averaged spatial density with the non-dimensional parameters such as, \(Fn,C_{q}\) and \(Re_{air}\). The thickness, \(t\), of the jet was observed based on the top view images. The thickness of the jet was observed to increase monotonically with the increase in the air injection rates in general, at all free-stream velocities, at all distances downstream of the hydrofoil. A power-law relation was obtained based on the regression analysis for the non-dimensional jet thickness in terms of \(Fn,C_{q},Re_{air}\) and \(x/x_{ref}\). Such a study on bubble dynamics, due to multi-vent injection, is necessary for engineering applications that involve air injection in water.  
Conceptualization and methodology, K.K. and B.-K.A.; investigation, K.K., D.N. and B.-K.A.; writing--original draft preparation, D.N. and K.K.; writing--review and editing, supervision, B.-K.A.; project administration and funding acquisition, C.P., G.-D.K. and I.-S.M. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.2019R1A2C1084306), the Korea Institute of Energy  
Figure 20: The non-dimensional jet thickness, \(t/c\), was plotted against \(f_{t}\), which was a function of Froude number, \(Fn\), and air injection coefficient, \(C_{q}\), Reynolds number, \(Re_{air}\), and non-dimensional streamwise distance, \(x/x_{ref}\), with the colors in red, blue, green, magenta and violet representing \(x/x_{ref}=4,6,8,10\) and \(12\), respectively. The \(R^{2}\) value for the linear curve fit was \(0.94\).  
Technology Evaluation and Planning(KETEP) grant funded by the Korea government(MOTIE) (20214000000090) and Korea Research Institute of Ships and Ocean Engineering (PES3900).  
Not applicable.  
The study did not involve humans.  
The data presented in this study are available on request from the corresponding author.  
We thank the National Research Foundation of Korea (NRF) of the Korean government, the Korea Institute of Energy Technology Evaluation and Planning(KETEP) and the Korea Research Institute of Ships and Ocean Engineering for the funding.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript or in the decision to publish the results.