×

An Eulerian method for transient nonlinear free surface wave problems. (English) Zbl 0586.76012

An Eulerian difference method is developed for the transient potential flow of an incompressible fluid with fully nonlinear free surface conditions. The free surface coordinate \(y=\eta (x,t)\) and the velocity potential \(\phi (x,y=\eta;t)\) on the free surface are recognized as the primary unknowns to be solved as an initial value problem from the pair of nonlinear partial differential equations representing the dynamic and the kinematic conditions of the free surface. The continuity relation \(\nabla^ 2\phi =0\) for the velocity potential \(\phi\) (x,y;t) over the flow field \(\Omega\) below the free surface is recognized as a subsidiary condition to be enforced at all times. The field of computation is transformed into a time invariant cartesian region with the free surface \(\eta\) (x,t) represented by a coordinate line (or surface). The iterative solution for \(\phi\) (x,y;t), p(x;y;t) in this fixed field of computation is facilitated by the use of fast Fourier transform (FFT). The iterative process converges rapidly. In terms of this converged \(\phi\) (x,y;t), the free surface location \(\eta\) (x,t) and its potential \(\phi (x,y=\eta;t)\) are advanced in time. Results from two planar examples are illustrated. The method is equally applicable to problems in three space dimensions, possibly involving interactive matching with neighboring flow fields. If the initial free surface potential \(\phi (x,y=\eta\); \(t=0)\) is unknown, difficulties may be encountered in data specification for securing a well posed problem for solution.

MSC:

76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bailer, G. R.; Meiron, D. I.; Orszag, S. A., J. Fluid Mech., 123, 477-501 (1982)
[2] Schwartz, L. W.; Whitney, A. K., J. Fluid Mech., 107, 147-171 (1981)
[3] Menikoff, R.; Zemach, C., J. Comput. Phys., 51, 28-64 (1983)
[4] Longuet-Higgins, M. S.; Cokelet, E. D., (Proc. Roy. Soc. London A, 364 (1978)), 1-28, and others · Zbl 0346.76006
[5] Harlow, F. H.; Welch, J. E., Phys. Fluids, 8, No. 12 (1965)
[6] Harlow, F. H.; Welch, J. E., Phys. Fluids, 9, No. 5 (1966)
[7] Nichols, B. D.; Hirt, C. W., J. Comput Phys., 8, No. 3 (1971)
[8] Nichols, B. D.; Hirt, C. W., J. Comput. Phys., 12, No. 2 (1973)
[9] Chan, R. K.-C.; Street, R. L., A Computer Study of Finite Amplitude Water Waves: An Evaluation of SUMMAC, (Lecture Notes in Physics, Vol. 8 (1970), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0207.27403
[10] Chan, R. K.-C.; Street, R. L.; Fromm, J. E., J. Comp. Phys., 6, No. 1 (1970)
[11] Taylor, C. J.; France, P. W.; Zienkiewicz, O. C., The Mathematics of Finite Elements and Applications (1973), Academic Press: Academic Press New York · Zbl 0281.76014
[12] Visser, W.; van der Wilt, M., Finite Element Methods in Flow Problems (1974), UAH Press: UAH Press Alabama
[13] Nickell, R. E.; Tanner, R. I.; Caswell, B., J. Fluid. Mech., 65 (1974), pt. 1
[14] Bai, K. J., (Proceedings, First International Conference on Numerical Ship Hydrodynamics. Proceedings, First International Conference on Numerical Ship Hydrodynamics, Gaithersburg, Maryland (October 1975))
[15] Bai, K. J.; Yeung, R. W., (Proceedings, Tenth ONR Symposium on Naval Hydrodynamics. Proceedings, Tenth ONR Symposium on Naval Hydrodynamics, Cambridge, Mass. (June 1974))
[16] Yen, S. M.; Lee, K. D., (Proceedings, Second International Symposium on Finite Element Methods in Flow Problems. Proceedings, Second International Symposium on Finite Element Methods in Flow Problems, Rapallo, Italy (June 1976))
[17] Haussling, H. J.; Van Eseltine, R. T., A Combined Spectral Finite-Difference Method for Linear and Nonlinear Water Wave Problems, Naval Ship R&D Center Report 4580 (November 1974)
[18] Haussling, H. J.; Van Eseltine, R. T., (Proceedings, First International Conference on Numerical Ship Hydrodynamics. Proceedings, First International Conference on Numerical Ship Hydrodynamics, Gaithersburg, Maryland (October 1975))
[19] Haussling, H. J.; Coleman, P. M., (Proceedings, Second International Conference on Numerical Ship Hydrodynamics. Proceedings, Second International Conference on Numerical Ship Hydrodynamics, Berkeley, Calif (1977))
[20] Lebail, R. C., J. Comput. Phys., 9 (1972)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.