Distribution of void fraction in bubbly flow through a horizontal channel: Bubbly boundary layer flow, 2nd report

A method of enveloping the hull with a sheet of microbubbles is discussed. It forms part of a study on means of reducing the skin friction acting on a ship's hull. In this report, a bubble traveling through a horizontal channel is regarded as a diffusive particle. Based on this assumption, an equation based on flow flux balance is derived for determining the void fraction in approximation. The equation thus derived is used for calculation, and the calculation results are compared with reported experimental data. The equation is further manipulated to make it compatible with a mixing length model that takes into account the presence of bubbles in the liquid stream. Among the factors contained in the equation thus derived, those affected by the presence of bubbles are the change of mixing length and the difference in the ratio of skin friction between cases with and without bubbles. These factors can be calculated using the mean void fraction in the boundary layer determined by the rate of air supply into the flow field. It is suggested that the ratio between boundary layer thickness and bubble diameter could constitute a significant parameter to replace the scale effect in estimating values applicable to actual ships from corresponding data obtained in model experiments.List of symbols a ₁ proportionality constant indicating directionality of turbulence - B law-of-the-wall constant - C _f local skin-friction coefficient in the presence of bubbles - C _f0 local skin-friction coefficient in the absence of bubbles - d _b bubble diameter (m) - g acceleration of gravity (m/s²) - j _g flow flux of gas phase accountable to buoyancy (m/s) - j _t flow flux of gas phase accountable to turbulence (m/s) - k ₄ constant relating reduction of liquid shear stress by bubble presence to decrease of force imparted to bubble by its displacement due to turbulence - l _b mixing length of gas phase (m) - l _m mixing length of liquid phase (m) - l _mb diminution of liquid phase mixing length by bubble presence (m) - Q _G rate of air supply to liquid stream (l/min) - q _/g velocity of bubble rise (m/s) - 2R height of horizontal channel (m) - T _* integral time scale (s) - U _m mean stream velocity in channel (m/s) - U friction velocity in channel (m/s) - V volume of a bubble (m³) - u, ¯ v time-averaged stream velocities inx- andy-directions, respectively (m/s) - u, v turbulent velocity components inx- andy-directions, respectively (m/s) - v root mean square of turbulence component in they-direction (m/s) - root mean square of bubble displacement iny-direction with reference to turbulent liquid phase velocity (m) - y displacement from ceiling (m) - local void fraction - _m mean void fraction in boundary layer - _m constant relating local void fraction to law-of-the-wall constant - _t reduction of turbulent stress (N/m²) - law-of-the-wall constant in turbulent liquid region in absence of bubbles - ₁ law-of-the-wall constant in turbulent liquid region in presence of bubbles - ₂ law-of-the-wall constant in gas phase - _m constant indicating representative turbulence scale (m) - viscosity (Pa × s) - v kinematic viscosity (m²/s) - density (kg/m³) Suffixes G gas - L liquid - 0 absence of bubbles