Skip to main content
SpringerLink
Log in

Simple Lagrangian formulation of bubbly flow in a turbulent boundary layer (bubbly boundary layer flow)

  • Original Articles
  • Published:
Journal of Marine Science and Technology Aims and scope Submit manuscript

Abstract

For the theoretical consideration of a system for reducing skin friction, a mathematical model was derived to represent, in a two-phase field, the effect on skin friction of the injection of micro air bubbles into the turbulent boundary layer of a liquid stream. Based on the Lagrangian method, the equation of motion governing a single bubble was derived. The random motion of bubbles in a field initially devoid of bubbles was then traced in three dimensions to estimate void fraction distributions across sections of the flow channel, and to determine local bubble behavior. The liquid phase was modeled on the principle of mixing length. Assuming that the force exerted on the liquid phase was equal to the fluid drag generated by bubble slip, an equation was derived to express the reduction in turbulent shear stress. Corroborating experimental data were obtained from tests using a cavitation tunnel equipped with a slit in the ceiling from which bubbly water was injected. The measurement data provided qualitative substantiation of the trend shown by the calculated results with regard to the skin friction ratio between cases with and without bubble injection as function of the distance downstream from the point of bubble injection.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

B :

law of wall constant

C f :

local coefficient of skin friction

C f0 :

local coefficient of skin friction in the absence of bubbles

d b :

bubble diameter [m]

g :

acceleration of gravity [m/s2]

k 1∼ k4 :

proportional coefficient

k L :

turbulent energy of the liquid phase [m2/s2]

L :

representative length [m]

l b :

mean free path of a bubble [m]

m A :

added mass of a single bubble [kg]

m b :

mass of a single bubble [kg]

N x ,N y ,N z :

force perpendicular to the wall or ceiling exerted on a bubble adhering to that wall or ceiling [N]

P :

absolute pressure [Pa]

Q G :

rate of air supply [ℓ/min]

q ′(i) L :

turbulent velocity at the ith time increment [m/s]

R> ex :

Reynolds number defined by Eq. 32

T *L :

integral time scale of the liquid phase [s]

U :

velocity of the main stream [m/s]

ū,¯v,¯w :

time-averaged velocity components [m/s]

u′,v′,w′ :

turbulent velocity components [m/s]

û L ′,vL′:

root mean square values of liquid phase turbulence components in thex- and y-directions [m/s]

V :

volume of a single bubble [m3]

X,Y,Z :

components of bubble displacement [m]

x s ,y s ,z s :

coordinate of a random point on a sphere of unit diameter centered at the coordinate origin

Ŷ :

root mean square of bubble displacement in they-direction in reference to the turbulent liquid phase velocity [m]

α :

local void fraction

α m :

mean void fraction in a turbulent region

γ :

regular random number

ΔR v :

increment of the horizontal component of the force acting on a single bubble, defined by Eq. 22 [N]

Δt :

time increment [s]

Δɛ1 :

reduction of turbulent stress [N/m2]

ε L :

rate of liquid energy dissipation [m2/s3]

η m :

coefficient defined by Eq. 30

κ :

law of wall constant in the turbulent region in absence of bubbles

κ 1 :

law of wall constant in the turbulent region in presence of bubbles

References

  1. McCormick ME, Bhattacharyya R (1973) Drag reduction of a submersible hull by electrolysis. Nav Eng J 85:11–16

    Google Scholar 

  2. Bogdevich VG, Evseev AR, Malyuga AG et al (1977) Gas-saturation effect on near-wall turbulence characteristics. BHRA Fluid Eng D2:25–37

    Google Scholar 

  3. Merkle CL, Deutsch S (1989) Microbubble drag reduction. Frontiers Exp Fluid Mech 46:291–335

    Google Scholar 

  4. Kato H, Miyanaga M, Haramoto Y et al (1994) Frictional drag reduction by injecting bubbly water into a turbulent boundary layer. ASME Symposium on Cavitation and Gas-Liquid Flow in Fluid Machinery and Devices, Lake Tahoe, FED-vol 190, ASME, New York, pp 184–194

    Google Scholar 

  5. Kato H, Miyanaga M, Yamaguchi H et al (1995) Frictional drag reduction by injecting bubbly water into a turbulent boundary layer and the effect of plate orientation. Advances in Multiphase Flow, Second International Conference on Multiphase Flow, Kyoto, Japan, Elsevier, Amsterdam, pp 85–96

    Google Scholar 

  6. Tsuji Y, Morikawa Y, Shiomi H (1984) LDV-measurements of air-solid two-phase flow in a vertical pipel. J Fluid Mech 139:417–437

    Google Scholar 

  7. Gore R, Crow CT (1989) Effect of particle size on modulating turbulent intensity. Int J Multiphase Flow 15:279–285

    Google Scholar 

  8. Yarin LP, Hetsroni G (1994) Turbulence intensity in dilute two-phase flow. 3. Int J Multiphase Flow 20:27–44

    Google Scholar 

  9. Sato Y, Sekoguchi K (1975) Investigation of liquid velocity distribution in bubbly flow (in Japanese). Trans Jpn Soc Mech Eng 41-315:3215–3223

    Google Scholar 

  10. Kataoka I, Serizawa A (1995) Modeling and prediction of turbulence in bubbly two-phase flow. 2nd International Conference on Multiphase Flow '95, Kyoto, Japan, vol 2, April 3–7,1995, pp MO2-11–16

    Google Scholar 

  11. Madavan JL, Merkle CL, Deutsch S (1985) Numerical investigation into the mechanisms of microbubble drag reduction. J Fluid Eng 107:370–377

    Google Scholar 

  12. Marie JL (1987) A simple analytical formulation for microbubble drag reduction. Phys Chem Hydrodyn 8:213–220

    Google Scholar 

  13. Masuko A, Shirose Y, Ishida S (1988) Numerical simulation of the viscous flow around ships including bilge vortices. 17th Symposium on Naval Hydrodynamics, The Hague, August 29 September 2,1988, pp 299–314

  14. Masuko A, Ogiwara S (1989) Numerical simulation of viscous flow around practical hull form. 5th International Conference on Numerical Ship Hydrodynamics, Hiroshima, Japan, September 25–28, 1989, pp 211–224

  15. Crow CT, Sharma MP, Stock DE (1977) The particle-source-in cell (PSI-CELL) model for gasroplet flows. J Fluid Eng 99: 325–332

    Google Scholar 

  16. Ishii R, Umeda Y, Yuhi M (1989) Numerical analysis of gasparticle two-phase flows. J Fluid Mech 203:475–515

    Google Scholar 

  17. Clift R, Grace JR, Weber ME (1978) Bubles drops and particles. Academic Press, New York

    Google Scholar 

  18. Saffman PG (1965) The lift on a small sphere in a slow shear flow. J Fluid Mech 22:385–400

    Google Scholar 

  19. Sibuya M (1962) A method for generating uniformly distributed points onN-dimensional spheres. Ann Inst Stat Math 14:81

    Google Scholar 

  20. Guin MM (1995) Studies on frictional drag reduction by microbubbles in turbulent boundary layers. PhD thesis, University of Tokyo

  21. Ikui T, Inoue M (1978) Dynamics of viscous fluid (in Japanese). Rikougakusya, Tokyo

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technology Development Department, Ishikawajima-Harima Heavy Industries Co., Ltd., 2-1-1 Toyosu, Koto-ku, 135, Tokyo, Japan

    Yuki Yoshida & Yoshiaki Takahashi

  2. Department of Naval Architecture and Ocean Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113, Tokyo, Japan

    Hiroharu Kato

  3. Research Institute, Ishikawajima-Harima Heavy Industries Co., Ltd., 1 Shin-nakahara-cho, Isogo-ku, 235, Yokohama, Japan

    Akira Masuko & Osamu Watanabe

Authors
  1. Yuki Yoshida
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yoshiaki Takahashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hiroharu Kato
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Akira Masuko
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Osamu Watanabe
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Yoshida, Y., Takahashi, Y., Kato, H. et al. Simple Lagrangian formulation of bubbly flow in a turbulent boundary layer (bubbly boundary layer flow). J Mar Sci Technol 2, 1–11 (1997). https://doi.org/10.1007/BF01245932

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01245932

Key words

Search

Navigation