Eighteenth Symposium on Naval Hydrodynamics (1991) / Chapter Skim
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Numerical Solution of the 'Dawson' Free-Surface Problem Using Havelock Singularities
Numerical Solution of the 'Dawson' Free-Surface Problem Using Havelock Singularities
Pages 259-272
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From page 259... ...
Z(x, y) To INTRODUCTION Beam Wave spectral function Wave resistance coefficient Gravitational constant Green function Draft Wave number Longitudinal wave number Lateral wave number Characteristic wave number = g/U2 Ship length Unit normal vector into the fluid Pressure Wave resistance Hull surface Ship speed Fluid velocity vector Ship-fixed coordinate system, with x forward, y to port, and z upward Free-surface elevation Double-body wave elevation Velocity potential Double-body velocity potential Perturbation potential Fluid density Havelock source density In 1977, Dawson [1]
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From page 260... ...
where ho is the characteristic wave number defined by o u2 (9) To show that the boundary condition can be applied at the position of the mean free surface, one can expand the potential in a Taylor series about z = 0, and assume that the free-surface elevation Z is of the same order as the perturbation potential.
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From page 261... ...
Since the Havelock singularity automatically satisfies the Kelvin free-surface boundary condition, the source density required to satisfy the Dawson condition should smoothly tend toward zero at points away from the hull where the Dawson condition limits to the Kelvin condition. To evaluate the total potential A, it is necessary to include a term corresponding to the free-stream velocity, N =-US + Eli | dsG(x,y,z;t,,rl,¢,)
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From page 262... ...
The N unknown source strengths on the hull surface panels and the M unknown source strengths on the free-surface panels are obtained by solving a set of independent linear equations composed of the N hull boundary conditions and the M freesurface boundary condition. Consequently, we have eliminated the finite differencing schemes usually employed in the solution to the Dawson problem.
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From page 263... ...
The solutions had more well deRned singularity densities but the resulting wavefields and wave resistances did not change significantly. RESULTS- WIGLEY HULL Our initial attempts to solve the Dawson problem using distributed Havelock singularities employed the submerged body described above.
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From page 264... ...
We set out to demonstrate that the use of Havelock singularities distributed over the free surface as well as on the hull surface, rather than Rankine singularities, could result in a solution to the Dawson problem which was free of the wave reflections often caused by the boundaries of the computational domain and free of the wave attenuations introduced by numerical damping schemes. This Havelock/Dawson solution could be extended to an arbitrary distance in the far-field.
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From page 265... ...
David Taylor Research Center Report DTRC/SHD1260-01, May 1988.
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From page 266... ...
Solid lines are positive contours, dashed lines are negative. SOURCE STRENGTHS Hull SLnguLarLtLes ~ Perturbation Potential ~_..
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From page 267... ...
00 lo cot O At: cat _ 0 o is lo I_ G ~ _ ha r~ o -4.0 -5.0 -6.0 -7.0 -8.0 -9.0 -10.0 -li .o -12.0 LONG I TUD I NRL D I STRNCE / SH I P LENGTH Figure 7. Comparison of far-field wave elevations obtained from the solution to the perturbation potential Dawson problem (solid line)
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From page 268... ...
Source strengths on hull surface panels resulting from a solution to the perturbation potential at Fn=0.40. Figure 11.
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From page 269... ...
Source density on the free-surface resulting from a solution to the perturbation potential for the Wigley hull at Fn=0.40.
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~-o.s Figure 17. Source strengths on the hull-surface panels of Wigley hull resulting from a solution to the perturbation potential.
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From page 271... ...
SOURCE DENSITY ~- ~1 1.45 Fi~re 22. Source denshy on tbe he~sur~ce resuldug hom ~lution to tbe tots1 p~enti~ ~r tbe Wig~y bull ~ Fn=0.40.
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