A fast iterative method to compute the flow around a submerged body

Malmliden, Johan; Petersson, N.

doi:10.1090/S0025-5718-96-00715-6

A fast iterative method to compute the flow around a submerged body
HTML articles powered by AMS MathViewer

by Johan F. Malmliden and N. Anders Petersson PDF

Math. Comp. 65 (1996), 1067-1083 Request permission

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

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.
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.
Jacques Cahouet and Marc Lenoir, Résolution numérique du problème non linéaire de la résistance de vagues bidimensionnelle, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 297 (1983), no. 12, 819–822 (French, with English summary). MR 797456
G. Chesshire and W. D. Hanshaw, Composite overlapping meshes for the solution of partial differential equations, J. Comput. Phys. 90 (1990), no. 1, 1–64. MR 1070471, DOI 10.1016/0021-9991(90)90196-8
R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. (Vol. II by R. Courant.). MR 0140802
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.
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.
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.
P. S. Jensen, On the numerical radiation condition in the steady state ship wave problem. J. Ship Res., 31:14–22, 1987.
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 (1988), no. 4, 729–752. MR 954784, DOI 10.1137/0725044
W. Lindemuth, T. J. Ratcliffe, and A. M. Reed, SHD 1260-1, David W. Taylor Naval Ship Research & Development Center, 1988.
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.
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.
D. E. Nakos and P. D. Sclavounos, On steady and unsteady ship wave patterns, J. Fluid Mech. 215 (1990), 263–288. MR 1061501, DOI 10.1017/S0022112090002646
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.
N. Anders Petersson, A numerical method to calculate the two-dimensional flow around an underwater obstacle, SIAM J. Numer. Anal. 29 (1992), no. 1, 20–31. MR 1149082, DOI 10.1137/0729002
N. A. Petersson and J. F. Malmliden, Computing the flow around a submerged body using composite grids. J. Comput. Phys., 105:47–57, 1993.
G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954
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.
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.