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

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- Received:
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- Accepted:

## 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.

### 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)

### References

- Amromin E, Karafiath G, Metcalf B (2011) Ship drag reduction by air bottom ventilated cavitation in calm water and in waves. J Ship Res 55:196–207Google Scholar
- Ceccio SL (2010) Frictional drag reduction of external flow with bubble and gas injection. Annu Rev Fluid Mech 42:183–203CrossRefGoogle Scholar
- Foeth EJ, Eggers R, Quadvlieg EHHA (2010) The efficiency of air-bubble lubrication for decreasing friction resistance, Paper no. 12. Prof Int Conf Ship Drag Reduction (SMOOTH-SHIP), Istanbul, TurkeyGoogle Scholar
- Fukuda K, Tokunaga J, Nobunaga T, Nakatani T, Iwasaki T (2000) Frictional drag reduction with air lubricant over a super-water-repellent surface. J Mar Sci Technol 5:123–130CrossRefGoogle Scholar
- Hamilton JM, Kim J, Waleffe F (1995) Regeneration mechanisms of near-wall turbulence structures. J Fluid Mech 287:317–348CrossRefMATHGoogle Scholar
- Harleman MJW, Delfos R, Terwisga TJCV, Westerweel J (2011) Dispersion of bubbles in fully developed channel flow. J Phys Conf Ser 318:052007CrossRefGoogle Scholar
- Hinze JO (1955) Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes. AIChE J 1:289–295CrossRefGoogle Scholar
- Hirayama A, Soejima S, Miyata H, Tatsui T, Kasahara Y, Okamoto Y, Iwasaki Y, Shimoyama N (2003) A study of air lubrication method to reduce frictional resistance of ship—an experimental study using flat plate and 16 m-model. In: West-Japan Society of Naval Architects meeting, vol 105, pp 1–9
**(in Japanese)**Google Scholar - Jang J, Choi SH, Ahn S, Kim B, Seo JS (2014) Experimental investigation of frictional resistance reduction with air layer on the hull bottom of a ship. Int J Nav Archit Ocean Eng 6:363–379CrossRefGoogle Scholar
- Johansen J, Castro AM, Carrica P (2010) Full-scale two-phase flow measurements on Athena research vessel. Int J Multiphase Flow 36:720–737CrossRefGoogle Scholar
- Katsui T, Okamoto Y, Kasahara Y, Shimoyama N, Iwasaki Y, Soejima S (2003) A study of air lubrication method to reduce frictional resistance of ship: experimental investigation by tanker form model ship and estimation of full scale ship performance. J Kansai Soc Nav Archit Jpn 239:45–53
**(in Japanese)**Google Scholar - Kitagawa A, Sugiyama K, Murai Y (2004) Experimental detection of bubble–bubble interactions in a wall-sliding bubble swarm. Int J Multiphase Flow 30:1213–1234CrossRefMATHGoogle Scholar
- Kodama Y, Kakugawa A, Takahashi T, Kawashima H (2000) Experimental study on microbubbles and their applicability to ships for skin friction reduction. Int J Heat Fluid Flow 21:582–588CrossRefGoogle Scholar
- Kumagai I, Takahashi Y, Murai Y (2015) A new power-saving device for air bubble generation using a hydrofoil for reducing ship drag: theory, experiments, and applications to ships. Ocean Eng 95:183–194CrossRefGoogle Scholar
- Lahey RT Jr (1991) Void wave propagation phenomena in two-phase flow. AIChE J 37:123–135CrossRefGoogle Scholar
- Lammers JH, Biesheuvel A (1996) Concentration waves and the instability of bubbly flows. J Fluid Mech 328:67–93CrossRefMATHGoogle Scholar
- Latorre R, Miller A, Philips R (2003) Micro-bubble resistance reduction on a model SES catamaran. Ocean Eng 30:2297–2309CrossRefGoogle Scholar
- Lisseter PE, Fowler AC (1992) Bubbly flow—II: Modelling void fraction waves. Int J Multiphase Flow 18:205–215CrossRefMATHGoogle Scholar
- Madavan NK, Deutsch S, Merkle CL (1984) Reduction of turbulent skin friction by microbubbles. Phys Fluids 27:356–363CrossRefGoogle Scholar
- Mäkiharju SA, Elbing BR, Wiggins A, Schinasi S, Vanden-Broeck JM, Perlin M, Dowling DR, Ceccio SL (2013) On the scaling of air entrainment from a ventilated partial cavity. J Fluid Mech 732:47–76CrossRefMATHGoogle Scholar
- McCormick M, Bhattacharyya R (1973) Drag reduction of a submersible hull by electrolysis. Nav Eng J 85:11–16CrossRefGoogle Scholar
- Mercado JM, Gómez DC, van Gils D, Sun C, Lohse D (2010) On bubble clustering and energy spectra in pseudo-turbulence. J Fluid Mech 650:287–306CrossRefMATHGoogle Scholar
- Mizokami S, Kawakado M, Kawano M, Hasegawa T, Hirakawa I (2013) Implementation of ship energy-saving operations with Mitsubishi air lubrication system. MHI Tech Rev 50:44–49Google Scholar
- Mori K, Imanishi H, Tsuji Y, Hattori T, Matsubara M, Mochizuki S (2009) Direct total skin-friction measurement of a flat plate in zero-pressure-gradient boundary layers. Fluid Dyn Res 41:021406CrossRefMATHGoogle Scholar
- Murai Y (2014) Frictional drag reduction by bubble injection. Exp Fluids 55:1733CrossRefGoogle Scholar
- Murai Y, Matsumoto Y (2000) Numerical study of the three-dimensional structure of a bubble plume. J Fluids Eng 122:754–760CrossRefGoogle Scholar
- Murai Y, Fukuda H, Oishi Y, Kodama Y, Yamamoto F (2007) Skin friction reduction by large air bubbles in a horizontal channel flow. Int J Multiphase Flow 33:147–163CrossRefGoogle Scholar
- Nierhaus T, Vanden Abeele D, Deconinck H (2007) Direct numerical simulation of bubbly flow in the turbulent boundary layer of a horizontal parallel plate electrochemical reactor. Int J Heat Fluid Flow 28:542–551CrossRefGoogle Scholar
- Oishi Y, Murai Y (2014) Horizontal turbulent channel flow interacted by a single large bubble. Exp Thermal Fluid Sci 55:128–139CrossRefGoogle Scholar
- Oishi Y, Murai Y, Tasaka Y, Takeda Y (2009) Frictional drag reduction by wavy advection of deformation bubbles. J Phys Conf Ser 147:012020CrossRefGoogle Scholar
- Park HJ, Tasaka Y, Oishi Y, Murai Y (2015a) Drag reduction promoted by repetitive bubble injection in turbulent channel flows. Int J Multiphase Flow 75:12–25CrossRefGoogle Scholar
- Park HJ, Tasaka Y, Murai Y (2015b) Ultrasonic pulse echography for bubbles traveling in the proximity of a wall. Meas Sci Technol 26:125301CrossRefGoogle Scholar
- Pauchon C, Banerjee S (1988) Interphase momentum interaction effects in the averaged multifield model. Part II: Kinematic wave and interfacial drag in bubbly flows. Int J Multiphase Flow 14:253–264CrossRefGoogle Scholar
- Sanders WC, Winkel ES, Dowling DR, Perlin M, Ceccio SL (2006) Bubble friction drag reduction in a high-Reynolds-number flat-plate turbulent boundary layer. J Fluid Mech 552:353–380CrossRefMATHGoogle Scholar
- Schlichting H (1979) Boundary-layer theory, 7th edn. McGraw-Hill Higher Education, New YorkMATHGoogle Scholar
- Smith CR, Metzler SP (1983) The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer. J Fluid Mech 129:27–54CrossRefGoogle Scholar
- Takagi S, Matsumoto Y (2011) Surfactant effects on bubble motion and bubbly flows. Annu Rev Fluid Mech 43:615–636CrossRefMATHGoogle Scholar
- Takahashi T, Kakugawa A, Makino M, Kodama Y (2003) Experimental study on scale effect of drag reduction by microbubbles using very large flat plate ships. J Kansai Soc Nav Archit Jpn 239:11–20
**(in Japanese)**Google Scholar - Takeuchi T, Kagawa T (2013) Applicability of frequency response test for stability evaluation of gas pressure regulator. Trans Soc Instrum Control Eng 49:747–754
**(in Japanese)**CrossRefGoogle Scholar - Titov I (ed) (1975) Practical problems in ship hydromechanics. Sudostroeniye Publishing Hose, Leningrad (in Russian)Google Scholar
- Tokunaga K (1987) Reduction of frictional resistance of a flat plate by microbubbles. In: West-Japan Society of Naval Architects meeting, vol 73, pp 79–82Google Scholar
- Watanabe O, Masuko A, Shirose Y (1998) Measurements of drag reduction by microbubbles using very long ship models. J Soc Nav Archit Jpn 1998:53–63CrossRefGoogle Scholar
- Yim KT, Kim H (1996) On the variation of resistance components due to air bubble blowing on bulb surface of a ship. Trans SNAK 33:54–64
**(in Korean)**Google Scholar - Zacksenhouse M, Abramovich G, Hetsroni G (2001) Automatic spatial characterization of low-speed streaks from thermal images. Exp Fluids 31:229–239CrossRefGoogle Scholar