Computation of Steady Viscous Flow Near a Ship’s Stern

  • M. Hoekstra
Part of the Notes on Numerical Fluid Mechanics book series (volume 17)


We consider the problem of computing the steady incompressible viscous flow past the rear part of a ship when free surface effects can be neglected. First some alternative approaches are reviewed which have emanated from different views on how to deal with the pressure in the primitive variable formulation of the Navier-Stokes equations. Then a particular solution method is described. The underlying mathematical model is a slightly reduced form of the Navier-Stokes equations: a main stream direction is identified and diffusion effects in this direction are neglected. The equations are solved in a multiple-sweep space-marching process. Multiple sweeps (global relaxation) are needed to allow the pressure to have influence on the upstream flow field. In each sweep the step-by-step evaluation of the solution is governed by an incomplete factorisation scheme. With this scheme a simultaneous solution is obtained in planes approximately perpendicular to the main stream direction. The performance of the method may be judged from some results of application. The proposed numerical solution to a classical problem in ship hydrodynamics is expected to be of great significance in future ship design studies.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Harlow, F.H. and Welch, J.E.: “Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface”, Physics of Fluids, Vol. 8, pp. 2182–2189 (1965).ADSzbMATHCrossRefGoogle Scholar
  2. [2]
    Chorin, A.J.: “A numerical method for solving incompressible viscous flow problems”, Journal of Comp. Physics, Vol. 2, pp. 12–26 (1967).ADSzbMATHCrossRefGoogle Scholar
  3. [3]
    Rubin, S.G.: “Incompressible Navier-Stokes and parabolised Navier- Stokes formulations and computational techniques”, in Computational Methods in Viscous Flows, ed. W.G. Habashi, Pineridge Press (1984).Google Scholar
  4. [4]
    Raven, H.C. and Hoekstra, M.: “A parabolised Navier-Stokes solution method for ship stern flow calculations”, Second Intern. Symp. on Ship Viscous Resistance, Gotenborg (1985).Google Scholar
  5. [5]
    Hoekstra, M. and Raven, H.C.: “Ship boundary-layer and wake calculation with a parabolised Navier-Stokes solution system”, 4th Intern. Conf. on Numerical Ship Hydrodynamics, Washington D.C. (1985).Google Scholar
  6. [6]
    Hoekstra, M. and Raven, H.C.: “Application of a parabolised Navier- Stokes solution system to ship stern flow computation”, Osaka International Colloquium on Ship Viscous Flow, Osaka (1985).Google Scholar
  7. [7]
    Cebeci, T. and Smith, A.M.O.: “Analysis of turbulent boundary layers”, Academic Press (1974).zbMATHGoogle Scholar
  8. [8]
    Cebeci, T. and Meier, H.U.: “Modelling requirements for the calculation of the turbulent flow around airfoils, wings and bodies of revolution”, AGARD Conference on turbulence boundary-layers, The Hague (1979).Google Scholar
  9. [9]
    Rubin, S.G. and Lin, A.: “Marching with the parabolised Navier-Stokes equations”, Israel Journal of Technology, Vol. 18, pp. 21–31 (1980).ADSzbMATHGoogle Scholar
  10. [10]
    Rubin, S.G. and Reddy, D.R.: “Analysis of global pressure relaxation for flows with strong interaction and separation”, Comp. and Fluids, Vol. 11, pp. 281–306 (1983).zbMATHCrossRefGoogle Scholar
  11. [11]
    Israeli, M. and Lin, A.: “Numerical solution and boundary conditions for boundary layer like flows”, 8th International Conference on Num. Methods in Fluid Dynamics, Aachen (1982).Google Scholar
  12. [12]
    Leonard, B.P.: “A stable and accurate convective modelling procedure based on quadratic upstream interpolation”, Comp. Meth. in Appl. Mech. and Eng., Vol. 19, pp. 59–98 (1979).ADSzbMATHCrossRefGoogle Scholar
  13. [13]
    Vorst, H.A. van der: “Iterative solution methods for certain sparse linear systems with a non-symmetric matrix arising from PDE-prob- lems”, Journal of Comp. Physics, Vol. 44, pp. 1–19 (1981).ADSzbMATHCrossRefGoogle Scholar
  14. [14]
    Larsson, L. ed.: “SSPA-ITTC workshop on ship boundary layers”, Publication No. 90, Swedish Maritime Research Centre SSPA (1981).Google Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1987

Authors and Affiliations

  • M. Hoekstra
    • 1
  1. 1.Maritime Research Institute NetherlandsWageningenThe Netherlands

Personalised recommendations