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CLC number: O24

On-line Access: 2012-05-03

Received: 2011-10-26

Revision Accepted: 2012-02-09

Crosschecked: 2012-04-09

Cited: 3

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Citations:  Bibtex RefMan EndNote GB/T7714

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Journal of Zhejiang University SCIENCE C 2012 Vol.13 No.5 P.393-402


Numerical solution of potential flow equations with a predictor-corrector finite difference method

Author(s):  Zhi-qiang Luo

Affiliation(s):  Department of System Science and Applied Mathematics, School of Science, Kunming University of Science and Technology, Kunming 650083, China

Corresponding email(s):   zql1009@126.com

Key Words:  Predictor-corrector method, Nonlinear potential flow equations, Finite difference method, Staggered grid, Nested iterative method

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Zhi-qiang Luo. Numerical solution of potential flow equations with a predictor-corrector finite difference method[J]. Journal of Zhejiang University Science C, 2012, 13(5): 393-402.

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%DOI 10.1631/jzus.C1100313

T1 - Numerical solution of potential flow equations with a predictor-corrector finite difference method
A1 - Zhi-qiang Luo
J0 - Journal of Zhejiang University Science C
VL - 13
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SP - 393
EP - 402
%@ 1869-1951
Y1 - 2012
PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.C1100313

We develop a numerical solution algorithm of the nonlinear potential flow equations with the nonlinear free surface boundary condition. A finite difference method with a predictor-corrector method is applied to solve the nonlinear potential flow equations in a two-dimensional (2D) tank. The irregular tank is mapped onto a fixed square domain with rectangular cells through a proper mapping function. A staggered mesh system is adopted in a 2D tank to capture the wave elevation of the transient fluid. The finite difference method with a predictor-corrector scheme is applied to discretize the nonlinear dynamic boundary condition and nonlinear kinematic boundary condition. We present the numerical results of wave elevations from small to large amplitude waves with free oscillation motion, and the numerical solutions of wave elevation with horizontal excited motion. The beating period and the nonlinear phenomenon are very clear. The numerical solutions agree well with the analytical solutions and previously published results.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article


[1]Abbaspour, M., Hassanabad, M.G., 2009. A novel 2D BEM with composed elements to study sloshing phenomenon. J. Appl. Fluid Mech., 2(2):77-83.

[2]Abramson, H.N., 1966. The Dynamic Behavior of Liquids in Moving Containers. Technical Report, SP 106. NASA.

[3]Chainais-Hillairet, C., Peng, Y.J., Violet, I., 2009. Numerical solutions of Euler–Poisson systems for potential flows. Appl. Numer. Math., 59(2):301-315.

[4]Faltinsen, O.M., 1974. A nonliner theory of sloshing in retangular tanks. J. Ship Res., 18(4):224-241.

[5]Faltinsen, O.M., 1978. A numerical non-linear method of sloshing in tanks with two dimensional flow. J. Ship Res., 18(4):224-241.

[6]Faltinsen, O.M., Timokha, A.N., 2002. Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth. J. Fluid Mech., 470:319-357.

[7]Frandsen, J.B., 2004. Sloshing motions in excited tanks. J. Comput. Phys., 196(1):53-87.

[8]Friedrichs, K., 1934. Uber ein minimumproblem fur potentialstromungen mit freiem rande. Math. Ann., 109(1):60-82 (in French).

[9]Hromadka, T.V., Whitley, R.J., 2005. Approximating three-dimensional steady-state potential flow problems using two-dimensional complex polynomials. Eng. Anal. Bound. Elem., 29(2):190-194.

[10]Ikegawa, M., 1974. Finite Element Analysis of Fluid Motion in a Container, Infinite Element Methods in Flow Problems. UAH Press, Huntsville.

[11]Klaseboer, E., Derek, R.M., Chan, Y.C., 2011. BEM simulations of potential flow with viscous effects as applied to a rising bubble. Eng. Anal. Bound. Elem., 35(3):489-494.

[12]Luke, J.C., 1967. A variational principle for a fluid with a free surface. J. Fluid Mech., 27(2):395-397.

[13]Miles, J.W., 1977. On Hamilton’s principle for surface waves. J. Fluid Mech., 83(1):153-158.

[14]Moiseyev, N.N., 1958. On the theory of nonlinear vibrations of a liquid of finite volume. Appl. Math. Mech., 22(5):224-241.

[15]Nakayama, T., Washizu, K., 1981. The boundary element method applied to the analysis of two dimensional nonlinear sloshing problems. Int. J. Numer. Method Eng., 17(11):1631-1646.

[16]Penney, W.G., Price, A.T., 1952. Finite periodic stationary gravity waves in a perfect liquid. Phil. Trans. R. Soc. Lond. A, 224(882):254-284.

[17]Phillips, N.A., 1957. A coordinate system having some special advantages for numerical forecasting. J. Meteorol., 14(2):184-185.

[18]Rocca, M.L., Mele, P., Armenio, V., 1997. Variational approach to the problem of sloshing in a moving container. J. Theoret. Appl. Fluid Mech., 1(4):280-310.

[19]Sarler, B., 2009. Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions. Eng. Anal. Bound. Elem., 33(12):1374-1382.

[20]Tarafder, M.S., Suzuki, K., 2008. Numerical calculation of free surface potential flow around a ship using the modified Rankine source panel method. Ocean Eng., 35(5-6):536-544.

[21]Wang, H., Zhang, H., 2007. Boundary element method for simulating the coupled motion of a fluid and a three-dimensional body. Appl. Math. Comput., 190(2):1328-1343.

[22]Whitham, G.B., 1965. A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid Mech., 22(2):273-283.

[23]Wu, C.H., Chen, B.F., 2009. Sloshing waves and resonance modes of fluid in a 3D tank by a time independent finite difference method. Ocean Eng., 36(6-7):500-510.

[24]Wu, G.X., 2007. Second order resonance of sloshing in a tank. Ocean Eng., 34(17-18):2345-2349.

[25]Zakharov, V.E., 1968. Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prokl. Mekh. Tekh. Fiz., 9(2):190-194.

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