Boundary-Layer Meteorology

, Volume 156, Issue 1, pp 91–112 | Cite as

Numerical Simulations of Air–Water Flow of a Non-linear Progressive Wave in an Opposing Wind



We present detailed numerical results for two-dimensional viscous air–water flow of a non-linear progressive water wave when the speed of the opposing wind varies from zero to 1.5 times the wave phase speed. It is revealed that at any speed of the opposing wind there exist two rotating airflows, one anti-clockwise above the wave peak and one clockwise above the wave trough. These rotating airflows form a buffer layer between the main stream of the opposing wind and the wave surface. The thickness of the buffer layer decreases and the strength of rotation increases as the wind speed increases. The profile of the average \(x\)-component of velocity reveals that the water wave behaves as a solid surface producing larger wind stress compared to the following-wind case.


Air–sea interaction Air–water flow Opposing wind  Progressive wave 



We would like to thank the constructive comments of the reviewers.


  1. Cohen JE (1997) Theory of turbulent wind over fast and slow waves. PhD thesis, University of Cambridge, CambridgeGoogle Scholar
  2. Crapper GD (1984) Introduction to water wave Ellis Horwood. Wiley, New York 224 ppGoogle Scholar
  3. Dean RG, Dalrymple RA (1984) Water Wave Mechanics for Engineers and Scientists. Prentice-Hall Inc., Upper Saddle River 353 ppGoogle Scholar
  4. Donelan MA (1990) Air–sea interaction. Sea: Ocean Eng Sci 9:239–292Google Scholar
  5. Donelan MA (1999) Wind-induced growth and attenuation of laboratory waves. In: Sajjadi SG, Thomas NH, Hunt JCR (eds) Wind-over-wave coupling, perspective and prospects. Clarendon Press, Oxford 356 ppGoogle Scholar
  6. Fenton J (1985) A fifth-order Stokes theory for steady waves. Waterw Port Coast Ocean Eng 111(2):216–234CrossRefGoogle Scholar
  7. Grare L, Peirson WL, Branger H, Walker JW, Giovanangeli JP, Makin V (2013) Growth and dissipation of wind-forced, deep-water waves. J Fluid Mech 722:5–50CrossRefGoogle Scholar
  8. Harris JA, Fulton I, Street RL (1995) Decay of waves in an adverse wind. In: Proceedings of the sixth Asian congress of fluid mechanics, May 22–26, SingaporeGoogle Scholar
  9. Hasselmann D, Bosenberg J (1991) Field measurements of wave-induced pressure over wind-sea and swell. J Fluid Mech 230:391–428Google Scholar
  10. Kalvig SM, Manger E, Kverneland R (2013) Structure of air flow separation over wind wave crests method for wave driven wind simulation with CFD. Energy Procedia 35:148–156CrossRefGoogle Scholar
  11. Lamb H (1916) Hydrodynamics. Cambridge University Press, Cambridge 708 ppGoogle Scholar
  12. Li PY, Xu D, Taylor PA (2000) Numerical modelling of turbulent airflow over water waves. Bound-Layer Meteorol 95:397–425CrossRefGoogle Scholar
  13. Mastenbroek C (1996) Wind wave interaction. PhD thesis, Delft Technical UniversityGoogle Scholar
  14. Milne-Thomson LM (1994) Theoretical hydrodynamics. Dover Publications Inc, New York 743 ppGoogle Scholar
  15. Mitsuyasu H, Yoshida Y (2005) Air–sea interactions under the existence of opposing swell. J Oceanogr 61:141–154CrossRefGoogle Scholar
  16. Peirson WL, Garcia AW, Pells SE (2003) Water wave attenuation due to opposing wind. J Fluid Mech 487:345–365CrossRefGoogle Scholar
  17. Snyder RL, Dobson FW, Elliott JA, Long RB (1981) Array measurements of atmospheric pressure fluctuations above surface gravity waves. J Fluid Mech 102:1–59CrossRefGoogle Scholar
  18. Sullivan PP, Edson JB, Hristov T, McWilliams JC (2008) Large-Eddy simulations and observations of atmospheric marine boundary layers above non-equilibrium surface waves. Atmos Sci 65:1225–1245CrossRefGoogle Scholar
  19. Wen X (2012) The analytical expression for the mass flux in the wet/dry areas method. ISRN Applied Mathematics. 2012: Article ID 451693, 15 pages, doi: 10.5402/2012/451693
  20. Wen X (2013) Wet/dry areas method for interfacial (free surface) flows. J Num Methods Fluids 71(3):316–338CrossRefGoogle Scholar
  21. Wen X, Mobbs (2014) Numerical simulations of laminar air–water flow of a non-linear progressive wave at low wind speed. Bound-Layer Meteorol 150:381–398CrossRefGoogle Scholar
  22. Young IR, Sobey RJ (1985) Measurements of the wind-wave energy flux in an opposing wind. J Fluid Mech 151:427–442CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Institute for Climate and Atmospheric Science, School of Earth and Environment, Centre for Computational Fluid DynamicsUniversity of LeedsLeedsUK
  2. 2.National Centre for Atmospheric Science, Centre for Computational Fluid DynamicsUniversity of LeedsLeedsUK

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