# Air-Water Bubbly Flow by Multiple Vents on a Hydrofoil in a Steady Free-Stream
## 1 Introduction  
The two-phase flow of air injection in water in engineering applications affects drag reduction and noise attenuation benefits. [PERSON] [1] performed drag measurements using skin-friction force balances on the surface of a flat plate in a cavitation tunnel and found a linear increase in the drag reduction percentage with the increase in the air injection coefficient. [PERSON] [2] measured the total drag coefficient exerted on a model ship in a towing tank using a load cell transducer with air injection underneath the surface in two different air injection configurations. They demonstrated a decrease in the total drag coefficient upon air injection for the two configurations, when compared against the no-injection case at several Froude numbers. The hydrodynamic performance of the National Advisory Committee for Aeronautics (NACA) 4412 hydrofoil in the free-stream flow of two-phase air and water was studied by [PERSON] [3]. The air-water void fraction distributionin the wall-normal direction around a hydrofoil was quantified at several streamwise distances from the leading edge at a known angle of inclination and free-stream velocity. [PERSON] [4] used an air bubble curtain to reduce the noise emitted from seismic exploration vehicles. They measured a considerable sound transmission loss across the measured sound frequency range at several sound incident angles on the sound curtain. Air bubbles, generated due to the flow past a 2D cylinder in the vicinity of the free surface at different angles of inclination about the horizontal, were studied at various free-stream velocities by [PERSON] [5]. They introduced the winged-air induction pipe (WAIP) that generates bubbles to affect drag reduction in ships. [PERSON] [6] further utilized WAIP to affect the drag reduction in a ship and noted a marked reduction in fuel consumption across the ship speed upon the usage of WAIP. They also noted a marked decrease in fuel consumption per day with an increase in the number of WAIPS installed on the hull surface. [PERSON] [7] performed a force balance measurement of a towed model ship and demonstrated a drag reduction on a model with air injection in the hull region. [PERSON] [8] injected air under the surface of a model ship hull in a towing tank facility and found a drag reduction due to air injection by force measurements. [PERSON] [9] measured the drag force exerted on a flat plate and found considerable drag-reduction benefits due to air injection, when compared without air injection. [PERSON] [10] performed numerical simulations of a micro-bubble drag reduction on a flat plate using a two-fluid model. They showed an upward shift in the non-dimensional velocity, \(u^{+}\), in the non-dimensional wall-normal coordinate, \(y^{+}\) due to air injection, which is conducive for drag-reduction applications. [PERSON] [11] demonstrated a reduction in horse-power required to drive a model hull with an increase in the air layer thickness at several hull speeds. [PERSON] [12] utilized the NACA 0012 hydrofoil and NACA 4412 hydrofoil as bubble generators on a model ship. The potential drag reduction benefit was demonstrated by them with a marked increase in the number of bubbles in the observation field with an increase in the velocity of the towed ship model from high-speed imaging. [PERSON] [13] estimated net energy savings due to the drag reduction upon air injection underneath a conventional ship. The bubbles ensuing from two different hydrofoils at various angles of inclination, free-stream velocities and air injection rates were characterized by a shadow imaging technique in the work of [PERSON] [14]. They quantified the bubble diameter distribution at several combinations of non-dimensional flow parameters. [PERSON] [15] studied the bubble size distribution around a NACA 0015 hydrofoil using a two-fluid mathematical model in a two-phase flow. The optimal range of bubble size at various free-stream velocities which result in drag reduction was identified by [PERSON] [16].