# Void waves propagating in the bubbly two-phase turbulent boundary layer beneath a flat-bottom model ship during drag reduction

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## Abstract

The injection of bubbles into a turbulent boundary layer can reduce the skin friction of a wall. Conventionally, the drag reduction rate is evaluated using time-averaged quantities of the mean gas flow rate or mean void fraction. Actually, as bubbles are subject to strong shear stresses near the wall, void waves and local bubble clusters appear. For pipe and channel flows, such wave-like behavior of the dispersed phase has been investigated intensely as an internal two-phase flow problem. We investigate how this wavy structure forms within the boundary layer as an external spatially developing two-phase flow along a horizontal flat plate. We describe how our model ship is designed to meet that purpose and report bubble-traveling behavior that accompanies unexpectedly strong wavy oscillations in the streamwise direction. A theoretical explanation based on a simplified two-fluid model is given to support this experimental fact, which suggests that void waves naturally stand out when drag reduction is enhanced through the local spatial gradient of the void fraction.

## Keywords

Wall Shear Stress Turbulent Boundary Layer Void Fraction Drag Reduction Bubble Size## List of symbols

*C*Void wave propagation speed (m/s)

*Ca*Cavitation number (dimensionless)

*C*_{f}Frictional coefficient (dimensionless)

*d*_{cb}Distance between the closest pair of bubbles (m)

*d*_{e}Equivalent diameter of bubble (m)

*Fr*Froude number (dimensionless)

*f*Local instantaneous volume fraction of liquid phase (dimensionless)

*f*_{G}Local instantaneous void fraction (dimensionless)

*f*_{void}Frequency of voidage wave (Hz)

*G*_{1}Impact factor of drag reduction to void fraction in the boundary layer (dimensionless)

*G*_{2}Impact factor of drag reduction to local gradient of void fraction (m)

*g*Acceleration due to gravity (m/s

^{2})*H*Height of the model ship (m)

*h*Thickness of the reflective index matching material (m)

*L*Length of the model ship (m)

*l*_{τ}Friction length (m)

*p*Local pressure of the water (Pa)

*p*_{v}Vapor pressure of the water (Pa)

*Q*_{g}Injected gas flow rate (m

^{3}/s)*Q*_{l}Liquid flow rate in the boundary layer (m

^{3}/s)*Re*_{x}Reynolds number on a flat plate (dimensionless)

*t*Time (s)

*t*_{g}Apparent air layer thickness (m)

*U*_{main}Main flow velocity (equivalent to towing speed) (m/s)

*U*_{δ}Averaged flow velocity in the boundary layer (m/s)

*u*,*v*,*w*Velocity components in

*x*,*y*,*z*directions (m/s)*u*_{b}Averaged advection velocity of bubbles (m/s)

*u*_{y}Averaged streamwise velocity at each depth (m/s)

*V*Averaged downward velocity of liquid phase on the border of the boundary layer (m/s)

*W*Width of the model ship (m)

*x*,*y*,*z*Cartesian coordinates of the model ship (m)

*α*_{δ}Void fraction in the boundary layer (dimensionless)

*δ*99% thickness of the boundary layer (m)

*δ*_{g}Superficial air layer thickness (m)

*μ*Viscosity of water (kg/m s)

*ν*Kinematic viscosity of water (m

^{2}/s)*ρ*Density of water (kg/m

^{3})*τ*_{w}Wall shear stress (Pa)

*τ*_{w0}Wall shear stress in single-phase flow (Pa)

## Notes

### Acknowledgements

This work was supported by the Fundamental Research Developing Association for Shipbuilding and Offshore (REDAS), Grant-in-Aid for JSPS Fellows No. 15J00147, JSPS KAKENHI Grant Nos. 24246033 and 23760143, and Grant-in-Aid for Young Scientists (B) No. 16K18006. The authors express their appreciation for all the support. Also, the authors express thanks to Prof. Yasukawa of Hiroshima University for his full support during the towing experiments at the Graduate School of Engineering.

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