Eighteenth Symposium on Naval Hydrodynamics (1991) / Chapter Skim
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A Numerical Solution Method for Three-Dimensional Nonlinear Free Surface Problems
A Numerical Solution Method for Three-Dimensional Nonlinear Free Surface Problems
Pages 427-438
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The threedimensional form of the integral equation and the boundary conditions for the time derivative of the potential is derived. By using these formulas, the free surface shape and the equations of motion are calculated simultaneously.
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The nonlinear hydrodynamics of an axisymmetric body beneath the free surface in the time domain were solved by Kang & Troesch (1988~. The free surface shape and the forces acting on a sphere oscillating sinusoidally with large amplitude are calculated and compared with published results.
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Consequently, a Fredholm integral equation of the second kind on the body and of the first kind at the free surface may be solved. Far Field Condition The far field condition is important in the nonlinear free surface problem.
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But since an accurate integration of the singularity requires a higher order quadrature formula, the method following Ferziger (1981) and Dommermuth & Yue (1986)
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= 0 (29) The time derivative of the potential on the free surface,"7, is calculated by using solutions of the integral equation, Eq.
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SB 6. NUMERICAL CALCULATION IIeave motion To demonstrate the usefulness of the technique shown in the previous section, the force acting on a sphere oscillating beneath the free surface is determined.
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The free surface shape and forces acting on a sphere oscillating sinusoidally with large amplitude are calculated and compared with published results. The far field flow away from the body is represented by a three dimensional dipole at the origin of the coordinate system.
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Fig. 3 Time History of the Heave Force Components Acting on the Heaving Sphere (a/R=0.5, h/R=2.0, KR=1.O)
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t=3 2 4T Fig. 7 Wave Profiles for Surge Motion (a/R=0.5, h/R=2.0, KR=1.0)
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6 Time History of the Heave Force Components Acting on the Surging Sphere (a/R=0.5, h/R=2.0, KR=1.0)
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This continuity constraint reduces to requiring continuity of the first and second parametric derivative vectors across the borders of adjacent patches. The B-spline surface is defined by, but does not interpolate, a set of points called control vertices.
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AUTHORS' REPLY Even if the calculation examples in this paper are restricted to the forced motion, this motion can be directly applied to the problems where the body motion is unknown. The calculation method of the forces in this paper is not much complicated because the integral equation for the potential (7)
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