Journal of Engineering Mathematics

, Volume 70, Issue 1–3, pp 43–66 | Cite as

Viscous and inviscid matching of three-dimensional free-surface flows utilizing shell functions

  • J. Andrew Hamilton
  • Ronald W. Yeung
Open Access


A methodology is presented for matching a solution to a three-dimensional free-surface viscous flow in an interior region to an inviscid free-surface flow in an outer region. The outer solution is solved in a general manner in terms of integrals in time and space of a time-dependent free-surface Green function. A cylindrical matching geometry and orthogonal basis functions are exploited to reduce the number of integrals required to characterize the general solution and to eliminate computational difficulties in evaluating singular and highly oscillatory integrals associated with the free-surface Green-function kernel. The resulting outer flow is matched to a solution of the Navier–Stokes equations in the interior region and the matching interface is demonstrated to be transparent to both incoming and outgoing free-surface waves.


Integral equations Open-boundary condition Pseudo-spectral solutions Time-dependent free-surface Green function Viscous-inviscid matching Wave-body interaction 



R.W. Yeung acknowledges partial support for the preparation of this article under Office of Naval Research Grant No. N00014-09-1086 at UC Berkeley. J. A. Hamilton acknowledges support from the David and Lucille Packard foundation.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Lai CH, Cuffe AM, Pericleous KA (1996) A domain decomposition algorithm for viscous/inviscid coupling. Adv Eng Softw 26: 151–159CrossRefGoogle Scholar
  2. 2.
    Drela M, Giles MB (1996) Viscous–inviscid analysis of transonic and low Reynolds number airfoils. AIAA J 25Google Scholar
  3. 3.
    Campana EF, Iafrati A (2001) Unsteady free surface waves by domain decomposition approach. In: Proceedings of the 16th international workshop on water waves and floating bodies, Kuju, Japan, AprilGoogle Scholar
  4. 4.
    Campana E, Di Mascio A, Esposito F, Lalli F (1995) Viscous–inviscid coupling in free surface ship flows. Int J Numer Methods Fluids 21: 699–722zbMATHCrossRefGoogle Scholar
  5. 5.
    Guillerm PE, Alessandrini B (2003) 3D free-surface flow computation using a RANSE/Fourier-Kochin Coupling. Int J Numer Methods Fluids 43(3): 301–318zbMATHCrossRefGoogle Scholar
  6. 6.
    Yeung RW, Hamilton JA (2002) A spectral-shell solution for viscous wave–body interaction. In: Proceedings of the 24th symposium on naval hydrodynamics, Fukuoka, JapanGoogle Scholar
  7. 7.
    Wehausen JV, Laitone EV (1960) Surface waves. In: Flugge S (eds) Handbuch der Physik, vol 9. Springer-Verlag, Berlin, pp 446–778Google Scholar
  8. 8.
    Yeung RW (1982) Transient heaving motions of floating cylinders. J Eng Math 16: 97–119zbMATHCrossRefGoogle Scholar
  9. 9.
    Hamilton JA, Yeung RW (1997) Shell-function solution for 3-D nonlinear body-motion problems. Schiffstechnik, pp 62–70Google Scholar
  10. 10.
    Newman JN (1986) Distribution of sources and normal dipoles over a quadrilateral panel. J Eng Math 20: 113–126CrossRefGoogle Scholar
  11. 11.
    Coakley PS (1995) A high-order B-spline based panel method for unsteady, nonlinear, three-dimensional free-surface flows. PhD thesis, University of California at BerkeleyGoogle Scholar
  12. 12.
    Beck RF, Liapis SJ (1987) Transient motions of floating bodies at zero forward speed. J Ship Res 31(3): 164–176Google Scholar
  13. 13.
    Newman JN (1985) Algorithms for the free-surface Green function. J Eng Math 19: 57–67zbMATHCrossRefGoogle Scholar
  14. 14.
    Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C. The art of scientific computing. Cambridge University Press. ISBN:0-521-43108-5Google Scholar
  15. 15.
    Finkelstein AB (1957) The initial value problems for transient water waves. Commun Pure Appl Math 10: 511–522MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Yeung RW, Yu X (2001) Three-dimensional free surface flow with viscosity—a spectral solution. In: Kashiwagi M (ed) Hydrodynamics in ships and ocean engineering. Research Institute of Applied Mechanics, Kyushu University, Japan, pp 87–114Google Scholar
  17. 17.
    Chorin AJ (1968) Numerical solution of incompressible flow problems. Stud Numer Anal 2: 64–71Google Scholar
  18. 18.
    Yeung RW, Ananthakrishnan P (1992) Oscillation of a floating body in a viscous fluid. J Eng Math 26: 211–230MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Yeung RW, Yu X (1994) Three-dimensional flow around a surface-piercing body. In: Proceedings of the 20th symposium on naval hydrodynamics, Santa Barbara, CAGoogle Scholar
  20. 20.
    Yu X, Yeung RW (1995) Interaction of transient waves with a circular surface-piercing body. J Fluids Eng 117: 382–388CrossRefGoogle Scholar
  21. 21.
    Yu X (1996) Free-surface flow around a vertical strut in a real fluid. PhD thesis, University of California at BerkeleyGoogle Scholar
  22. 22.
    Sarpkaya T (1986) Force on a circular cylinder in viscous oscillatory flow at low Keulegan–Carpenter numbers. J Fluid Mech 165: 61–71ADSCrossRefGoogle Scholar
  23. 23.
    Harada T, Koshizuka S, Kawaguchi Y (2007) Smoothed particle hydrodynamics on GPUs. In: Proceedings of the 25th computer graphics international conference, Petropolis, Brazil, pp 63–70Google Scholar
  24. 24.
    Herault A, Bilotta G, Dalrymple R (2010) SPH on GPU with CUDA. J Hydraul Res 48: 74–79CrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Research and Development DivisionMonterey Bay Aquarium Research InstituteMoss LandingUSA
  2. 2.Department of Mechanical EngineeringUniversity of California at BerkeleyBerkeleyUSA

Personalised recommendations