The Contribution of Kawada to the Analytical Solution for the Velocity Induced by a Helical Vortex Filament and Modern Applications of Helical Vortices

  • Yasuhide FukumotoEmail author
  • Valery L. Okulov
  • David H. Wood
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 26)


Currently, the analytical form of the velocity field induced by a helical vortex filament is well known as Hardin’s solution (1982). But essentially the same result had been obtained by a Japanese scientist Sandi Kawada, which predates Hardin by as long as 46 years. Kawada (1936) provided a comprehensive treatment of deriving the induced velocity by helical vortices with a view to applying it to the propeller theory. This paper recollects Kawada’s contribution, together with his life devoted to lead the Japanese aeronautical engineering in the time of its dawning.


Wind Turbine Vortex Ring Velocity Potential Vortex Sheet Vortex Filament 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



YF is grateful to Prof. Kojiro Suzuki of University of Tokyo for providing us with documents on Sandi Kawada and Tomijiro Moriya. This work has been carried out with a support of the Danish Council for Strategic Research for the project COMWIND—Center for Computational Wind Turbine Aerodynamics and Atmospheric Turbulence: grant 2104_09_067216/DSF and the Russian Science Foundation (grant no. 14-29-00093). DHW acknowledges the support of the Canadian Natural Science and Engineering Research Council through its Industrial Research Chair program in conjunction with the ENMAX Corporation.


  1. 1.
    Kawada, S.: Induced velocity by helical vortices. J. Aeronaut. Sci. 3, 86–87 (1936)CrossRefGoogle Scholar
  2. 2.
    Kawada, S.: Calculation of induced velocity by helical vortices and its application to propeller theory. Rep. Aeronaut. Res. Inst. Tokyo Imperial Univ. 14, 2–57 (1939)Google Scholar
  3. 3.
    Goldstein, S.: On the vortex theory of screw propellers. Proc. R. Soc. Lond. A 123, 440–465 (1929)CrossRefzbMATHGoogle Scholar
  4. 4.
    Fukumoto, Y., Okulov, V.L., Wood, D.H.: The contribution of Kawada to the analytical solution for the velocity induced by a helical vortex filament. Appl. Mech. Rev. 67, 060801 (2015)CrossRefGoogle Scholar
  5. 5.
    Alekseenko, S.V., Kuibin, P.A., Okulov, V.L.: Theory of Concentrated Vortices: An Introduction. Springer, Berlin (2007)zbMATHGoogle Scholar
  6. 6.
    Hardin, J.C.: The velocity field induced by a helical vortex filament. Phys. Fluids 25, 1949–1952 (1982)CrossRefzbMATHGoogle Scholar
  7. 7.
    Lerbs, H.: Moderately loaded propeller with a finite number of blades and an arbitrary distribution of circulation. Trans. SNAME 60, 73–123 (1952)Google Scholar
  8. 8.
    Moriya, T.: On the induced velocity and characteristics of a propeller. J. Eng. 20, 147–162 (1933). Tokyo Imperial UniversityGoogle Scholar
  9. 9.
    Morgan, B.M., Wrench Jr, J.W.: Some computation aspects of propeller design. Methods Comput. Phys. 4, 301–331 (1965)Google Scholar
  10. 10.
    Tibery, C.L., Wrench Jr, J.W.: Tables of Goldstein factor. Report 1534 Department of Navy Washington, pp. 1–69 (1964)Google Scholar
  11. 11.
    Ricca, R.L.: The effect of torsion on the motion of a helical vortex filament. J. Fluid Mech. 273, 241–259 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kuibin, P.A., Okulov, V.L.: Self-induced motion and asymptotic expansion of the velocity field in the vicinity of helical vortex filament. Phys. Fluids 10, 607–614 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Boersma, J., Wood, D.H.: On the self-induced motion of a helical vortex. J. Fluid Mech. 384, 263–280 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Alekseenko, S.V., Kuibin, P.A., Okulov, V.L.: Helical vortices in swirl flow. J. Fluid Mech. 382, 195–243 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wood, D.H., Boersma, J.: On the motion of multiple helical vortices. J. Fluid Mech. 447, 149–171 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Okulov, V.L.: On the stability of multiple helical vortices. J. Fluid Mech. 521, 319–342 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fukumoto, Y., Okulov, V.L.: The velocity field induced by a helical vortex tube. Phys. Fluids 17, 107101 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Okulov, V.L., Sørensen, J.N.: Stability of helical tip vortices in rotor far wake. J. Fluid Mech. 576, 1–25 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Okulov, V.L., Sørensen, J.N.: Refined Betz limit for rotors with a finite number of blades. Wind Energy 11, 415–426 (2008)CrossRefGoogle Scholar
  20. 20.
    Okulov, V.L., Sørensen, J.N.: Maximum efficiency of wind turbine rotors using Joukowsky and Betz approaches. J. Fluid Mech. 649, 497–508 (2010)CrossRefzbMATHGoogle Scholar
  21. 21.
    Okulov, V.L., Sørensen, J.N.: Applications of 2D helical vortex dynamics. Theor. Comput. Fluid Dyn. 24, 395–401 (2010)CrossRefzbMATHGoogle Scholar
  22. 22.
    von Kármán, T.: Aerodynamics: Selected Topics in the Light of their Historical Development. Dover Publications, Mineola (2004)Google Scholar
  23. 23.
    Hayashi, T.: Obituary of professor Sandi Kawada [in Japanese]. J. Jpn. Soc. Aeronaut. Eng. 200 (1970)Google Scholar
  24. 24.
    Takenaka, K.: Classical airplane museum [in Japanese].

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  • Yasuhide Fukumoto
    • 1
    Email author
  • Valery L. Okulov
    • 2
    • 3
  • David H. Wood
    • 4
  1. 1.Institute of Mathematics for IndustryKyushu UniversityFukuokaJapan
  2. 2.Wind Energy DepartmentTechnical University of DenmarkLyngbyDenmark
  3. 3.Institute of ThermophysicsSiberian Branch of the Russian Academy of SciencesNovosibirskRussia
  4. 4.Schulich School of EngineeringUniversity of CalgaryCalgaryCanada

Personalised recommendations