Journal of Engineering Mathematics

, Volume 71, Issue 4, pp 367–392 | Cite as

Practical mathematical representation of the flow due to a distribution of sources on a steadily advancing ship hull

  • Francis NoblesseEmail author
  • Gerard Delhommeau
  • Fuxin Huang
  • Chi Yang


A practical mathematical representation of the flow velocity due to a distribution of sources on the mean wetted hull surface and the mean waterline of a ship that steadily advances along a straight path in calm water, of large depth and lateral extent, is presented. A main feature of this flow representation is a simple analytical approximation—valid within the entire flow region—to the local flow component in the expression for the gradient of the Green function associated with the classical Kelvin–Michell linearized free-surface boundary condition. Another notable feature of the flow representation is that the singularity associated with the gradient of the Green function is removed, using a straightforward regularization technique. The flow representation only involves elementary continuous functions (algebraic, exponential and trigonometric) of real arguments. These functions can then be integrated using ordinary Gaussian quadrature rules. Thus, the flow representation is particularly simple and well suited for practical flow calculations. The specific case of a low-order panel method—in which the hull geometry, the source density, and the flow velocity are consistently represented via piecewise linear approximations within flat triangular hull panels or straight waterline segments—is considered.


Free-surface flow Green function Source distribution Steady ship waves 


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  1. 1.
    Noblesse F, Delhommeau G, Guilbaud M, Hendrix D, Yang C (2008) Simple analytical relations for ship bow waves. J Fluid Mech 600: 105–132MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Letcher JS Jr, Marshall JK, Olliver JCIII, Salvesen N (1987) Stars & stripes. Sci Am 257: 34–40CrossRefGoogle Scholar
  3. 3.
    Wyatt DC, Chang PA (1994) Development and assessment of a total resistance optimized bow for the AE 36. Mar Technol 31: 149–160Google Scholar
  4. 4.
    Percival S, Hendrix D, Noblesse F (2001) Hydrodynamic optimization of ship hull forms. Appl Ocean Res 23: 337–355CrossRefGoogle Scholar
  5. 5.
    Yang C, Soto O, Löhner R, Noblesse F (2002) Hydrodynamic optimization of a trimaran. Ship Technol Res 49: 70–92Google Scholar
  6. 6.
    Kim HY, Yang C, Löhner R, Noblesse F (2008) A practical hydrodynamic optimization tool for the design of a monohull ship. In: International conference of Society of Offshore and Polar Engineering (ISOPE-08), Vancouver, CanadaGoogle Scholar
  7. 7.
    Noblesse F (1981) Alternative integral representations for the Green function of the theory of ship wave resistance. J Eng Math 15: 241–265zbMATHCrossRefGoogle Scholar
  8. 8.
    Bessho M (1964) On the fundamental function in the theory of the wavemaking resistance of ships, vol 4. Memoirs Defense Academy, Japan, pp 99–119Google Scholar
  9. 9.
    Noblesse F (1975) The near-field disturbance in the centerplane Havelock source potential. In: 1st International conference on Numerical ship hydrodynamics, Washington DC, pp 481–501Google Scholar
  10. 10.
    Noblesse F (1978) On the fundamental function in the theory of steady motion of ships. J Ship Res 22: 212–215Google Scholar
  11. 11.
    Newman JN (1987) Evaluation of the wave-resistance Green function: part 1—the double integral. J Ship Res 31: 79–90Google Scholar
  12. 12.
    Telste JG, Noblesse F (1989) The nonoscillatory near-field term in the Green function for steady flow about a ship. In: 17th Symposium on Naval hydrodynamics, The Hague, Netherlands, pp 39–52Google Scholar
  13. 13.
    Masson E, DeBayser O, Martin D (1991) Evaluation de la resistance de vagues d’un sous-marin en immersion totale, 3emes Journèes de l’Hydrodynamique. Grenoble, FranceGoogle Scholar
  14. 14.
    Ponizy B, Noblesse F, Ba M, Guilbaud M (1994) Numerical evaluation of free-surface Green functions. J Ship Res 38: 193–202Google Scholar
  15. 15.
    Noblesse F, Delhommeau G, Kim HY, Yang C (2009) Thin-ship theory and influence of rake and flare. J Eng Math 64: 49–80MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Noblesse F, Delhommeau G, Yang C (2009) Practical evaluation of steady flow due to a free-surface pressure patch. J Ship Res 53: 1–14Google Scholar
  17. 17.
    Lyness JN, Jespersen D (1975) Moderate degree symmetric quadrature rules for the triangle. J Inst Math Appl 15: 19–32MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Hendrix D, Noblesse F (1992) Recipes for computing the steady free-surface flow due to a source distribution. J Ship Res 36: 346–359Google Scholar
  19. 19.
    Noblesse F (1983) A slender-ship theory of wave resistance. J Ship Res 27: 13–33Google Scholar
  20. 20.
    Debo H, Yunbo L (1997) Ship wave resistance based on Noblesse’s slender ship theory and wave-steepness restriction. Ship Technol Res 44: 198–202Google Scholar
  21. 21.
    McCarthy JH (1985) Collected experimental resistance and flow data for three surface ship model hulls. David W Taylor Naval Ship Research and Development Center, report DTNSRDC-85/011Google Scholar
  22. 22.
    Cooperative experiments on Wigley parabolic model in Japan. In: 17th ITTC Resistance Committee ReportGoogle Scholar

Copyright information

© US Government 2011

Authors and Affiliations

  • Francis Noblesse
    • 1
    Email author
  • Gerard Delhommeau
    • 2
  • Fuxin Huang
    • 3
  • Chi Yang
    • 3
  1. 1.David Taylor Model BasinNSWCCDWest BethesdaUSA
  2. 2.École Centrale de NantesCNRSNantesFrance
  3. 3.Department of Computational and Data SciencesGeorge Mason UniversityFairfaxUSA

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