Eighteenth Symposium on Naval Hydrodynamics (1991) / Chapter Skim
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Nonlinear Free Surface Waves Due to a Ship Moving Near the Critical Speed in a Shallow Water
Nonlinear Free Surface Waves Due to a Ship Moving Near the Critical Speed in a Shallow Water
Pages 173-190
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From page 173... ...
g U W X,y~z xc CY p INTRODUCTION . pressure : source strength : longitudinal distribution of cross sectional area of ship : blockage coefficient : free surface : projection of SF : maximum cross sectional area of ship : ship surface : time : final time : generation period between first two solitons : ship's speed : tank's half width : rectangular coordinates : x-location of the crestline at y = 0 : speed parameter or unwinding parameter : blockage index : slenderness parameter or variational operator : nonlinear parameter : surface elevation : tank width parameter : dispersion parameter : water density A free-surface flow of an ideal fluid caused by a ship translating with a constant speed near the shallow water celerity is described by an initial/ boundary value problem governed by the Laplace equation with the free surface as a part of solution.
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From page 174... ...
Then the KP equation is numerically solved in the two-dimensional horizontal Resurface plane by an explicit finite difference scheme. In the second method, the original problem is replaced by an equivalent variational problem based on Hamilton's principle applied to water waves derived by Miles [8~.
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From page 175... ...
In the far field, the geometry of the tank affects the propagation of waves, but the generation mechanism of the waves is not known. The wave field is described by a homogeneous two-dimensional KdV or KP equation [14]
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From page 176... ...
To do it, a simple explicit finite difference scheme is implemented for the KP equation, in which forward differences are chosen for time derivatives and central differences for spatial derivatives. But at the wall and the centerline of the tank, one-sided differences are used in order to incorporate with boundary conditions.
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From page 177... ...
are less advantageous in computations with respect to the numerical stability. To remedy this difficulty, we introduce the unwinding and local lumping schemes, which are often used in a wide class of computational fluid dynamics.
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From page 178... ...
Throughout the computations, the motion is assumed to start with a prescribed constant speed as a step function. In presenting the two sets of computed results, we denote those obtained by the KP equation with a slender body approximation by KP, and those of the finite element method by FEM.
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From page 179... ...
In this case the pressure distribution is specified to have the same blockage coefficient as the ship model in the earlier experimental condition. The pressure is assumed by a trapezoial distribution in both x- and y- directions and constant along the length of the parallel middle body in the x-direction.
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From page 180... ...
t10 ~ Choi, H.S. and Mei, C.C., Wave Resistance and Squat of a Slender Ship Moving Near the Critical Speed in Restricted Water, Proceedings of the 5th International Conference on Numerical Ship Hydrodynamics, 1989, pp.43 454.
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From page 181... ...
t21 ~ Bai, K.J., Kim, J.W. and Lee, H.S., An Application of the Finite Element Method to a Nonlinear Free Sruface Flow Problem, The ,/_ y = W ~ \ ~ tax \ a} (~////////////////////////////////////~////////~.2 ~ y=-W ~ elf ~ z = -h Fig.1 Definition Sketch z 10 O- ~I ~I 0 5 10 15 Time (See)
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From page 182... ...
(c) Fh= 1.1 Fig.3 Exact and Linear Approximate Wave Resistances by FEM (2W = 2.44m)
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From page 183... ...
(b) FEM Fig.7 Lift Forces for Three Different Tank Widths ~ 1: 1.22m, 2: 2.44m, 3: 4.88m at the Critical Speed 183
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From page 184... ...
Ut/h = 19.65 -2n -1n 1C (b) Ut/h = 39.3 Fig.10 Wave Contour at the Critical Speed (KP, 2W = 4.88m)
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From page 185... ...
i ,if ~ (b) Ut/h = 40.0 Fig.ll Wave Contour at the Critical Speed (FEM, 2W = 4.88m)
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From page 186... ...
' ~' ' ' 1 ' 1 0 5 10 15 20 25 30 35 40 45 50 Time (See) 2 20 25 Fig.14 Wave Resistance for Various Tank Widths with Ut/h = 160.0 Pressure Patch (Fh = 1.0)
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From page 187... ...
t - 39.6 Sec Sec Sec _ ~ ~ ~ r ~ I ~ l ' I " " " ~ I ~ T I I I I l l l l r I I I I I ~ I ~ I I I I I ~ I ~ i I 1 -O 1 2 3 4 Log (x-xc) Fig.16 Plot of the Crestline of the First Upstream Diverging Wave 187
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From page 188... ...
Speed C/ ~W/L=0.8 W/L=1.6 W/L=3.2 KP 1.218 ~1.155 1.078 Fh = 0.9 FEM 1.175 1.100 1.050 EXP 1.170 1.100 1.060 KP 1.293 1.216 1.153 Fh = 1.0 FEM 1.250 1.175 1.125 EXP 1.240 1.190 1.130 KP 1.367 1.304 1.227 Fh = 1.1 FEM 1.300 1.250 1.200 EXP 1.280 1.260 1.210 (c) Generation Period UT,/b W/L=0.8 W/L= 1.6 W/L=3.2 KP 20.04 31.83 41.26 Fh = 0.9 FEM 30.20 37.40 53.60 EXP 32.70 48.10 65.10 KP 24.89 39.29 60.26 Fh = 1.0 FEM 35.40 47.30 88.40 EXP 37.80 49.80 85.20 KP 33.14 54.75 90.78 Fh = 1.1 FEM 44.20 57.40 128.70 EXP 39.00 50.11 103.60 Table 2 The Amplitude and Speed of the First Soliton, and the Generation Period Between First Two Solitons for a Slender Ship (Cb = 0.667)
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From page 189... ...
DISCUSSION John V Wehausen University of California at Berkeley, USA As I understand the authors, the calculations based upon the KP equations for a vertical strut just touching the bottom and with a profile determined by the section-curve of a Wigley hull or of Series 60, CB = 0.80, whereas those based upon the Laplace equation are for a ship with the same overall dimensions as the latter hull but with an altered section-area curve appropriate to a wedge-shaped hull; the blockage coefficients are the same.
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