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A Fast Iterative Method to Compute the Flow Around a Submerged Body

Authors: Johan F. Malmliden and N. Anders Petersson
Journal: Math. Comp. 65 (1996), 1067-1083
MSC (1991): Primary 65N12, 76B20
MathSciNet review: 1333320
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Abstract: We develop an efficient iterative method for computing the steady linearized potential flow around a submerged body moving in a liquid of finite constant depth. In this paper we restrict the presentation to the two-dimensional problem, but the method is readily generalizable to the three-dimensional case, i.e., the flow in a canal. The problem is indefinite, which makes the convergence of most iterative methods unstable. To circumvent this difficulty, we decompose the problem into two more easily solvable subproblems and form a Schwarz--type iteration to solve the original problem. The first subproblem is definite and can therefore be solved by standard iterative methods. The second subproblem is indefinite but has no body. It is therefore easily and efficiently solvable by separation of variables. We prove that the iteration converges for sufficiently small Froude numbers. In addition, we present numerical results for a second-order accurate discretization of the problem. We demonstrate that the iterative method converges rapidly, and that the convergence rate improves when the Froude number decreases. We also verify numerically that the convergence rate is essentially independent of the grid size.

References [Enhancements On Off] (What's this?)

  • 1. D. L. Brown, G. Chesshire, and W. D. Henshaw, Getting started with CMPGRD. Introductory user's guide and reference manual. LA--UR 90-3729, Los Alamos National Laboratory, 1989.
  • 2. D. L. Brown, G. Chesshire, and W. D. Henshaw, An explanation of the CMPGRD composite grid data structure. IBM Research Report RC 14354, IBM Research Division, Yorktown Heights, NY, 1990.
  • 3. J. Cahouet and M. Lenoir, Résolution numérique du problème non linéaire de la résistance de vagues bidimensionelle. C. R. Acad. Sc. Paris II, 297:819--822, 1983. MR 86j:76009
  • 4. G. Chesshire and W. D. Henshaw, Composite overlapping meshes for the solution of partial differential equations. J. Comput. Phys., 90(1):1--64, 1990. MR 91f:76043
  • 5. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol II. Interscience Publishers, New York, 1962. MR 25:4216
  • 6. C. W. Dawson, A practical computer method for solving ship wave problems. In Proceedings of the Second International Conference on Numerical Ship Hydrodynamics, pages 30--38. University of California, Berkeley, 1977.
  • 7. L. J. Doctors and R. F. Beck, Convergence properties of the Neumann--Kelvin problem for a submerged body. J. Ship Res., 31:227--234, 1987.
  • 8. S. Eisenstat, M. Gursky, M. H. Schultz, and A. H. Sherman, The Yale matrix package II: The non--symmetric case. Report 114, Dept. of Computer Science, Yale University, 1977.
  • 9. P. S. Jensen, On the numerical radiation condition in the steady state ship wave problem. J. Ship Res., 31:14--22, 1987.
  • 10. M. Lenoir and A. Tounsi, The localized finite element method and its application to the two-dimensional sea-keeping problem. SIAM J. Numer. Anal., 25:729--752, 1988. MR 89k:65138
  • 11. W. Lindemuth, T. J. Ratcliffe, and A. M. Reed, SHD 1260-1, David W. Taylor Naval Ship Research & Development Center, 1988.
  • 12. J. F. Malmliden, An efficient numerical method for 3-D flow around a submerged body. TRITA--NA 9306, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, 1993.
  • 13. C. C. Mei and H. S. Chen, A hybrid element method for steady linearized free-surface flows. Int. J. Numer. Meth. Eng., 10:1153--1175, 1976.
  • 14. D. E. Nakos and P. D. Sclavounos, On steady and unsteady ship wave patterns. J. Fluid Mech., 215:263--288, 1990. MR 91c:76022
  • 15. J. N. Newman, Evaluation of the wave-resistance Green function: Part 2. The single integral on the centerplane. J. Ship Res., 31(3):145--150, 1987.
  • 16. N. A. Petersson, A numerical method to calculate the two-dimensional flow around an underwater obstacle. SIAM J. Numer. Anal., 29:20--31, 1992. MR 92j:76010
  • 17. N. A. Petersson and J. F. Malmliden, Computing the flow around a submerged body using composite grids. J. Comput. Phys., 105:47--57, 1993.
  • 18. G. B. Whitham, Linear and nonlinear waves. Wiley-Interscience, New York, 1974. MR 58:3905
  • 19. G. X. Wu and R. E. Taylor, Hydrodynamic forces on submerged oscillating cylinders at forward speed. Proc. R. Soc. Lond., A 414:149--170, 1987.
  • 20. F. Xia and L. Larsson, A calculation method for the lifting potential flow around yawed surface piercing 3--D bodies. In Proceedings of the 16'th Symposium on Naval Hydrodynamics, 1986, pp. 583--597.

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Additional Information

Johan F. Malmliden

N. Anders Petersson

Keywords: Schwarz iteration, finite difference approximation, composite overlapping grid, potential flow
Received by editor(s): March 20, 1992
Received by editor(s) in revised form: August 9, 1994
Additional Notes: The first author was partially supported by ONR grant N-00014-90-J-1382 and by the U.S. Department of Energy through Los Alamos National Laboratory.
The second author was supported by ONR grants N-00014-90-J-1695, N-00014-90-J-1382 and by the U.S. Department of Energy through Los Alamos National Laboratory.
Article copyright: © Copyright 1996 American Mathematical Society

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