Eighteenth Symposium on Naval Hydrodynamics (1991) / Chapter Skim
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Analysis of Transom Stern flows
Analysis of Transom Stern flows
Pages 207-220
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From page 207... ...
Both of these approaches are applied to the solution for the flow about a high-speed transom stern ship, with encouraging results. C cd CR Cw C' w Cwp Fn NOMENCLATURE Wave spectral function due to sources Wave spectral function due normal dipoles Residuary resistance coefficient, CR = RR/_PSU2 Wave resistance coefficient, Cw = Rw/-pSU2, computed from wave spectral energy Wave resistance coefficient, computed by inte grating predicted pressure over the surface of the hull Wave pattern resistance coefficient derived from measured wave pattern Froude number, Fn = U/~/~ g Gravitational acceleration G Green function i, j, k Unit vectors in the x, y, and z-directions, re spectively k Wavenumber ho Fundamental wavenumber, ho = g/U2 k=, k' Longitudinal wavenumber ky Lateral wavenumber Span Length of panel in the x-direction L Length of ship n Normal vector, taken into the fluid no, ny, nz Components of the normal vector, n, in the x-, y-, and z-directions Neumann-Kelvin Distance from singular point to field point, r = ,/(X _ ()
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From page 208... ...
On the free surface they employed a trailing vortex sheet in the wake of the hull to satisfy the free-surface boundary condition, which in their model was shown to be equivalent to the Kutta condition applied at the trailing edge of a wing. Perhaps the most significant result of their model was the existence of a drag force due to the presence of the trailing vortex sheet.
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From page 209... ...
Therefore, there will exist a region aft of the ship and forward of the breaking stern waves over which the kinematic and dynamic freesurface boundary conditions are valid. Furthermore, if the effects of these breaking waves do not extend forward to the hull itself, then our solution in the region immediately surrounding the ship will be unaffected by an application of the kinematic and dynamic free-surface boundary conditions over the entire free surface.
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From page 210... ...
One approach uses Havelock sources and dipoles which are dis tributed over the hull surface and in the wake. The lin earized free surface boundary condition is implicitly satis fied by the use of Havelock singularities, and therefore no singularities are required on the free surface.
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From page 211... ...
Alternatively, the source strengths could be determined initially by solving the Neumann-Kelvin problem in the absence of the pressure condition, and then solving for the dipole strengths which satisfy both the zero normal flow condition and the pressure condition. The zero pressure boundary condition, Equation (8)
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From page 212... ...
The three options are complicated by the fact that we have already decided to set ¢= and ¢~= at the free-surface panels immediately aft of the transom equal to those values required by Bernoulli's equation. Thus, for the sake of continuity in the boundary conditions, we choose the third option and set ¢= equal to the value required by Equation (8~.
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From page 213... ...
strengths required to satisfy the Kutta condition, Fig. 2b, are driven entirely by the zero pressure condition, in the sense that the dipole strengths would go to zero in the ab 213
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From page 214... ...
The significance of this trailing vorticity can be appreciated by noting that the magnitude of the dipole strengths is comparable to the strengths calculated on an airfoil of equal span, operating at a negative angle of attack of approximately five degrees. The sense of the vorticity will result in an upwelling flow on the centerline of the wake, and a diverging free-surface current across the wake, and may well be the physical mechanism responsible for such flows observed in the centerline wake region of transom stern ships.
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From page 215... ...
The free surface was paneled with 780 panels for the high Froude number case: 10 strips of 62 panels to the side and 8 strips of 20 panels aft of the stern. For the low Froude number case, more panels were deemed necessary because of the smaller wavelengths.
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From page 216... ...
The normal dipole strengths of the wake panels were determined from the linearized Bernoulli equation.
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From page 217... ...
Fn = 0.25. Fig 11 Hull depth and predicted wave height verses longitudinal position near the stern at eight transverse locations.
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From page 218... ...
The Havelock singularity method uses Havelock sources to satisfy the normal flow boundary conditions on the hull, and Havelock dipoles to satisfy tangential flow conditions. This leads to well-behaved solutions to the boundary-value problem.
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From page 219... ...
Flows corresponding to nonzero Froude numbers could be calculated based on linearizing the free-surface boundary conditions on the mean freesurface level outside the hull with its extension and on the surface of the extension.
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