# Bubbly drag reduction accompanied by void wave generation inside turbulent boundary layers

### Abstract

Frictional drag reduction, a technique by which bubbles are injected into the turbulent boundary layer surrounding the hull of a marine vessel, is now at the stage of practical applications. In achieving drag reduction, void waves often stand out naturally, the reason for which still remains unclear. The present study aims at an experimental characterization of void waves along a flat-bottom ship. A 100-m-long water reservoir is used in which a 4-m-long fully transparent experimental model ship, equipped with wall shear stress sensors and cameras, is towed by a train at speeds of up to 3 m/s. From measurements of the transition of the bubble distribution from random to wavy accumulated swarms downstream, the accompanying intrinsic passing frequency of void waves is examined. A 30% drag reduction rate was recorded with the appearance of void waves in the boundary layer at an average void fraction of 4%. This is much greater than the trivial inertia effect from drag reduction. To clarify the characteristics of the measured void waves, we compare the void wave frequency range to those of several flow instabilities that may occur in bubbly two-phase boundary layer flows.

### Graphical abstract

## List of symbols

*a*Propagation speed of gravity wave, m/s

*c*Averaged streamwise velocity in a bubbly flow, m/s

*C*_{f}Friction coefficient, dimensionless

*F*Dimensionless constant when instability comes up, dimensionless

*f*Wave-standing frequency, Hz

*f*_{max}Maximum wave-standing frequency, Hz

*f*_{min}Minimum wave-standing frequency, Hz

*f*_{peak}Peak frequency of void wave, Hz

*Fr*Froude number, dimensionless

*f*_{ref}Referential frequency for normalization, Hz

*f*_{void}Frequency of void wave, Hz

*f*_{0}Referential frequency, Hz

*G*Gain factor of drag reduction to void fraction in the boundary layer, dimensionless

*g*Acceleration of gravity, m/s

^{2}*h*Characteristic length in Richardson number, dimensionless

*k*Wave number, m

^{−1}*Q*_{g}Flow rate of gas injection, m

^{3}/s*Q*_{g,min}Minimum flow rate of gas injection, m

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

^{3}/s- \(R{e_x}\)
Reynolds number on a flat plate, dimensionless

*Ri*Richardson number, dimensionless

*T*Time elapse in the Lagrangian frame, s

*t*Elapsed time, s

*U*Free stream velocity, m/s

- \({U_i}\)
Velocity in fluid with suffixes

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

- \({u_y}\)
Velocity distribution in the downward direction, m/s

*W*Width of the model ship, m

*X*Distance from the bubble injector, m

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

*α*Void fraction in a bubbly flow, dimensionless

- \({\alpha _\delta }\)
Void fraction in a boundary layer, dimensionless

*δ*99% thickness of a boundary layer, m

*λ*Means streamwise renewal distance, m

*ν*Kinematic viscosity of water, m

^{2}/s*ρ*Density of fluid, kg/m

^{3}- \({\rho _i}\)
Density of fluid with suffixes

*i*, kg/m^{3}*τ*_{w}Averaged wall shear stress, Pa

*τ*_{w0}Averaged 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), JSPS KAKENHI Grant no. 17H01245 and Grant-in-Aid for Young Scientists (B) no. 17K14583. The authors express their appreciation for all the support. Also, the authors thank Dr. Yoshihiko Oishi at Muroran Institute of Technology and Prof. Hironori Yasukawa at Hiroshima University for their support during various parts of the towing experiments.

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