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), 10671083
MSC (1991):
Primary 65N12, 76B20
MathSciNet review:
1333320
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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 twodimensional problem, but the method is readily generalizable to the threedimensional 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 Schwarztype 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 secondorder 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.
 1.
D. L. Brown, G. Chesshire, and W. D. Henshaw, Getting started with CMPGRD. Introductory user's guide and reference manual. LAUR 903729, 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.
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
(86j:76009)
 4.
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
(91f:76043), http://dx.doi.org/10.1016/00219991(90)901968
 5.
R.
Courant and D.
Hilbert, Methods of mathematical physics. Vol. II: Partial
differential equations, (Vol. II by R. Courant.), Interscience
Publishers (a division of John Wiley & Sons), New YorkLon don, 1962.
MR
0140802 (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 3038. University of California, Berkeley, 1977.
 7.
L. J. Doctors and R. F. Beck, Convergence properties of the NeumannKelvin problem for a submerged body. J. Ship Res., 31:227234, 1987.
 8.
S. Eisenstat, M. Gursky, M. H. Schultz, and A. H. Sherman, The Yale matrix package II: The nonsymmetric 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., 31:1422, 1987.
 10.
M.
Lenoir and A.
Tounsi, The localized finite element method and its application to
the twodimensional seakeeping problem, SIAM J. Numer. Anal.
25 (1988), no. 4, 729–752. MR 954784
(89k:65138), http://dx.doi.org/10.1137/0725044
 11.
W. Lindemuth, T. J. Ratcliffe, and A. M. Reed, SHD 12601, David W. Taylor Naval Ship Research & Development Center, 1988.
 12.
J. F. Malmliden, An efficient numerical method for 3D flow around a submerged body. TRITANA 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 freesurface flows. Int. J. Numer. Meth. Eng., 10:11531175, 1976.
 14.
D.
E. Nakos and P.
D. Sclavounos, On steady and unsteady ship wave patterns, J.
Fluid Mech. 215 (1990), 263–288. MR 1061501
(91c:76022), http://dx.doi.org/10.1017/S0022112090002646
 15.
J. N. Newman, Evaluation of the waveresistance Green function: Part 2. The single integral on the centerplane. J. Ship Res., 31(3):145150, 1987.
 16.
N.
Anders Petersson, A numerical method to calculate the
twodimensional flow around an underwater obstacle, SIAM J. Numer.
Anal. 29 (1992), no. 1, 20–31. MR 1149082
(92j:76010), http://dx.doi.org/10.1137/0729002
 17.
N. A. Petersson and J. F. Malmliden, Computing the flow around a submerged body using composite grids. J. Comput. Phys., 105:4757, 1993.
 18.
G.
B. Whitham, Linear and nonlinear waves, WileyInterscience
[John Wiley & Sons], New YorkLondonSydney, 1974. Pure and Applied
Mathematics. MR
0483954 (58 #3905)
 19.
G. X. Wu and R. E. Taylor, Hydrodynamic forces on submerged oscillating cylinders at forward speed. Proc. R. Soc. Lond., A 414:149170, 1987.
 20.
F. Xia and L. Larsson, A calculation method for the lifting potential flow around yawed surface piercing 3D bodies. In Proceedings of the 16'th Symposium on Naval Hydrodynamics, 1986, pp. 583597.
 1.
 D. L. Brown, G. Chesshire, and W. D. Henshaw, Getting started with CMPGRD. Introductory user's guide and reference manual. LAUR 903729, 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, 297:819822, 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):164, 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 3038. University of California, Berkeley, 1977.
 7.
 L. J. Doctors and R. F. Beck, Convergence properties of the NeumannKelvin problem for a submerged body. J. Ship Res., 31:227234, 1987.
 8.
 S. Eisenstat, M. Gursky, M. H. Schultz, and A. H. Sherman, The Yale matrix package II: The nonsymmetric 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., 31:1422, 1987.
 10.
 M. Lenoir and A. Tounsi, The localized finite element method and its application to the twodimensional seakeeping problem. SIAM J. Numer. Anal., 25:729752, 1988. MR 89k:65138
 11.
 W. Lindemuth, T. J. Ratcliffe, and A. M. Reed, SHD 12601, David W. Taylor Naval Ship Research & Development Center, 1988.
 12.
 J. F. Malmliden, An efficient numerical method for 3D flow around a submerged body. TRITANA 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 freesurface flows. Int. J. Numer. Meth. Eng., 10:11531175, 1976.
 14.
 D. E. Nakos and P. D. Sclavounos, On steady and unsteady ship wave patterns. J. Fluid Mech., 215:263288, 1990. MR 91c:76022
 15.
 J. N. Newman, Evaluation of the waveresistance Green function: Part 2. The single integral on the centerplane. J. Ship Res., 31(3):145150, 1987.
 16.
 N. A. Petersson, A numerical method to calculate the twodimensional flow around an underwater obstacle. SIAM J. Numer. Anal., 29:2031, 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., 105:4757, 1993.
 18.
 G. B. Whitham, Linear and nonlinear waves. WileyInterscience, 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 414:149170, 1987.
 20.
 F. Xia and L. Larsson, A calculation method for the lifting potential flow around yawed surface piercing 3D bodies. In Proceedings of the 16'th Symposium on Naval Hydrodynamics, 1986, pp. 583597.
Similar Articles
Retrieve articles in Mathematics of Computation of the American Mathematical Society
with MSC (1991):
65N12,
76B20
Retrieve articles in all journals
with MSC (1991):
65N12,
76B20
Additional Information
Johan F. Malmliden
Email:
johanm@prosolvia.se
N. Anders Petersson
Email:
andersp@na.chalmers.se
DOI:
http://dx.doi.org/10.1090/S0025571896007156
PII:
S 00255718(96)007156
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 N0001490J1382 and by the U.S. Department of Energy through Los Alamos National Laboratory.
The second author was supported by ONR grants N0001490J1695, N0001490J1382 and by the U.S. Department of Energy through Los Alamos National Laboratory.
Article copyright:
© Copyright 1996
American Mathematical Society
