Practical Ship Hydrodynamics

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220 Practical Ship Hydrodynamics Combine the expression of s with the above expression for 1/r2 : 1 r 2 ds d D � 1 � 2 2cy C ⊲1 C 2c 2 2 � 1 ⊳ D � 2 2 2cy 2c2 C C r 2 f r 4 f Now the integrands in the expression for r can be evaluated. x D 1 2 D 1 2 D 1 4 � d d � d d ⊲0⊳ [ x ⊲0⊳ x D � d d ⊲x ⊳ r 2 ds d d r 2 f ⊲ ⊲0⊳ C ⊲1⊳ C ⊲2⊳ 2 ⊳⊲x ⊳ 2⊲x ⊳ r 2 f r 2 f ⊲0⊳ C ⊲1⊳ x � xCd d D x d ⊲1⊳ C c ⊲c⊳ x � with r1 D ⊲x C d⊳2 C y2 and r2 D ⊲1⊳ x D � d 2 ⊲x d � xCd ⊳ t⊲x d D 2 x d t⊳ ⊲c⊳ x D � d 4⊲x ⊳y d 2 r 4 f � xCd d D 4y x d ⊲2⊳ x D � d 2⊲x ⊳ d 2 r 2 � xCd d D 2 f x d � 1 r 2 f r 4 f C 2cy 2 r 4 f r 2 f C 2c2 2 ⊲0⊳ C ⊲2⊳ x ⊲ ⊲2⊳ C 2c 2 ⊲0⊳ ⊳] r 2 f 2t t 2 C y 2 dt D[ln⊲t2 C y 2 ⊳] xCd x d D ln⊲r2 1 /r2 2 ⊳ � ⊲x d⊳ 2 C y 2 t 2 ⊲0⊳ ⊲0⊳ dt D x 2 x C y y C y t⊲t x⊳ 2 ⊲t 2 C y 2 ⊲1⊳ dt 2 2 ⊳ t⊲t x⊳ 2 t 2 ⊲1⊳ dt D x 2 x C y 4d � y C ⊲2d⊳3 xy C y ⊲1⊳ y Here the integrals were transformed with the substitution t D ⊲x ⊳. y D 1 2 � d d � d ⊲y ⊳ r 2 ds d d D 1 ⊲ 2 d ⊲0⊳ C ⊲1⊳ C ⊲2⊳ 2 ⊳⊲y c 2 ⊳ � � 2 2 1 2cy 2c2 ð C C d ⊲0⊳ y D D 1 4 � d r 2 f [ ⊲0⊳ y 2y d r 2 f � D 2 arctan t y r 4 f ⊲0⊳ C ⊲1⊳ y � xCd d D 2 x d � xCd x d r 2 f ⊲1⊳ C c ⊲c⊳ y y t 2 dt 2 C y D 2arctan 2dy x 2 C y 2 ⊲0⊳ C ⊲2⊳ y ⊲ ⊲2⊳ C 2c 2 ⊲0⊳ ⊳] d 2 r 2 1 r2 2 d
⊲1⊳ y D � d 2 y d r 2 f ⊲c⊳ y D � d 2 2 4y d r 4 f D 4y 2 � xCd D 2 ⊲1⊳ x x d � xCd d D 2y x d 2 2 r 2 f d ⊲t x⊳ 2 ⊲x t⊳ ⊲t 2 C y 2 dt 2 2 ⊳ 3 x2 4d y 2 r 2 1 r2 2 t 2 ⊲0⊳ dt D x 2 y C y d 2 ⊲2⊳ y D � d 2y d 2 r 2 � xCd d D 2y f x d � xCd x d ⊲t x⊳ 2 Boundary element methods 221 ⊲t x⊳ 2 t 2 dt 2 C y t 2 ⊲1⊳ dt D x 2 y C y y ⊲0⊳ x y ⊲1⊳ x The original formulae for the first derivatives of these higher-order panels were analytically equivalent, but less suited for programming involving more arithmetic operations than the formulae given here. Higher derivatives of the potential are, e.g., given in Bertram (1999). 2. Three-dimensional case The higher-order panels are parabolic in shape with a bi-linear source distribution on each panel. The original procedure of Hess was modified by Hughes and Bertram (1995) to include also higher derivatives of the potential. The complete description of the formulae used to determine the velocity induced by the higher-order panels would be rather lengthy. So only the general procedure is described here. The surface of the ship is divided into panels as in a first-order panel method. However, the surface of each panel is approximated by a parabolic surface, as opposed to a flat surface. The geometry of a panel in the local panel coordinate system is described as: D C C A C B C P 2 C 2Q C R 2 The -axis and -axis lie in the plane tangent to the panel at the panel control point, and the -axis is normal to this plane. This equation can be written in the form: o D P⊲ o⊳ 2 C 2Q⊲ o⊳⊲ o⊳ C R⊲ o⊳ 2 The point ⊲ o, o, o⊳ is used as a collocation point and origin of the local panel coordinate system. In the local panel coordinate system, terms depending on A, B, andC do not appear in the formulae for the velocity induced by a source distribution on the panel. R and P represents the local curvatures of the ship in the two coordinate directions, Q the local ‘twist’ in the ship form. The required input consists of the coordinates of panel corner points lying on the body surface and information concerning how the corner points are connected to form the panels. In our implementation, each panel is allowed to have either three or four sides. The first and third sides of the panel should be (nearly) parallel. Otherwise, the accuracy of the panels deteriorates. For a given panel, the information available to determine the coefficients A...R consists of the three or four panel corner points of the panel and the corner
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Practical Ship Hydrodynamics
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Butterworth-Heinemann Linacre House
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3 Resistance and propulsion .......
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5.4.5 Interaction of rudder and shi
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Preface The first five chapters giv
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1 Introduction Models now in tanks
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Introduction 3 certain investigatio
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Introduction 5 there have been prop
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Introduction 7 is a material consta
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Introduction 9 margin may make a di
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Introduction 11 solution either. Ev
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Introduction 13 in naval architectu
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Introduction 15 ž Finite differenc
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Introduction 17 makes seakeeping pr
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Introduction 19 ship. Inner flow co
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Introduction 21 performed on workst
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Introduction 23 experts, while boun
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Introduction 25 two more decades be
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Introduction 27 Figure 1.3 A cylind
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Introduction 29 resolved by conside
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Introduction 31 ž LSOR (line succe
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Introduction 33 flow direction. CDS
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Introduction 35 equation, i.e. the
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2 Propellers 2.1 Introduction Ships
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Propellers 39 ž rake iG The face o
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KT KQ h 10 . K Q K T Figure 2.3 Pro
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Propellers 43 methods or panel meth
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Propellers 45 This formula can be i
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Propellers 47 power. The earliest l
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Propellers 49 corrected subsequentl
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Propellers 51 flow computations are
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Propellers 53 occurs earlier, as ca
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Propellers 55 2.5.2 Open-water test
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Propellers 57 the traditional prope
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Propellers 59 (model/full-scale shi
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Propellers 61 represents the cavity
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Resistance and propulsion 63 1. The
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Resistance and propulsion 65 3.1.2
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Figure 3.3 Double-body flow y Wave
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Resistance and propulsion 69 course
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Resistance and propulsion 71 limite
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Table 3.1 Recommended values for CA
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Resistance and propulsion 75 3.2.6
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P B ⋅ 10 3 (kW) 40 30 20 10 0 n P
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Resistance and propulsion 79 t is t
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Resistance and propulsion 81 are no
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3.4 Simple design approaches Resist
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Resistance and propulsion 85 of the
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Resistance and propulsion 87 non-li
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Resistance and propulsion 89 resist
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Resistance and propulsion 91 (doubl
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Resistance and propulsion 93 only b
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Resistance and propulsion 95 resist
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Resistance and propulsion 97 1°. S
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Ship seakeeping 99 3. Addition of t
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Ship seakeeping 101 1. No recording
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Ship seakeeping 103 Re denotes the
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Ship seakeeping 105 the ship axis x
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Ship seakeeping 107 The procedure t
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Ship seakeeping 109 If several ω r
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a 0.015 0.010 0.005 0 1 2 3 4 5 Uc/
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Ship seakeeping 113 The (only stati
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Ship seakeeping 115 height H1/3 for
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Ship seakeeping 117 using data of B
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Ship seakeeping 119 for each strip
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Ship seakeeping 121 between near fi
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Ship seakeeping 123 wave length of
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Ship seakeeping 125 The elements of
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Ship seakeeping 127 4.4.3 Rankine s
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Ship seakeeping 129 to zero. Then t
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Ship seakeeping 131 for commercial
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Ship seakeeping 133 Since the spect
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Ship seakeeping 135 ž The ωj are
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Ship seakeeping 137 changing sea sp
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Ship seakeeping 139 The wave impact
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Ship seakeeping 141 pressure gauges
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v(t) h(x,t) y u(x,t) Figure 4.23 On
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Ship seakeeping 145 computations wi
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Ship seakeeping 147 4. Derive the p
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Ship seakeeping 149 The free consta
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5 Ship manoeuvring 5.1 Introduction
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Ship manoeuvring 153 ž yaw rate (r
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Ship manoeuvring 155 Table 5.2 Non-
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Ship manoeuvring 157 but also the i
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C y 1.50 1.25 1.00 1.75 0.85 0.80 0
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Ship manoeuvring 161 section. The l
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Ship manoeuvring 163 methods have b
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esistance R is proportional to spee
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Ship manoeuvring 167 accuracy, but
Page 179 and 180: Ship manoeuvring 169 the difference
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Page 183 and 184: d, y 20° 10° 0° −10° −20°
Page 185 and 186: Distance Lateral deviation Length o
Page 187 and 188: Ship manoeuvring 177 also performed
Page 189 and 190: Ship manoeuvring 179 ž Rudders out
Page 191 and 192: ehind the leading edge (nose) is: c
Page 193 and 194: Ship manoeuvring 183 the rudder. In
Page 195 and 196: Ship manoeuvring 185 attack ˛ of n
Page 197 and 198: V a /V 0.4 0.3 0.2 0.1 V a /V 2.0 1
Page 199 and 200: Ship manoeuvring 189 ž Rudder with
Page 201 and 202: average between VA and V1: � �
Page 203 and 204: C L (V corr ≠ V A ) C L (V corr =
Page 205 and 206: Ship manoeuvring 195 hull influence
Page 207 and 208: Ship manoeuvring 197 - Estimate the
Page 209 and 210: Ship manoeuvring 199 rudder tip vor
Page 211 and 212: Ship manoeuvring 201 (1993) uses de
Page 213 and 214: Ship manoeuvring 203 ž Profiles wi
Page 215 and 216: Ship manoeuvring 205 The ship perfo
Page 217 and 218: 6 Boundary element methods 6.1 Intr
Page 219 and 220: C ⊲t1n3 C n1t3⊳⊲t3 xxz C s3 x
Page 221 and 222: xx D ⊲ 3 x⊲x xq⊳ ⊳/r 2 xy D
Page 223 and 224: The velocity in the x direction is:
Page 225 and 226: ∂ ∂z Boundary element methods 2
Page 227 and 228: Boundary element methods 217 2. Thr
Page 229: and ⊲ , ⊳ is: r D � ⊲x ⊳
Page 233 and 234: Boundary element methods 223 the so
Page 235 and 236: 2 0 8xz D 2 2 0 8xxz D 2 2 0 8xzz D
Page 237 and 238: Figure 6.9 Velocities induced by po
Page 239 and 240: It is convenient to write ϕ as:
Page 241 and 242: Boundary element methods 231 usuall
Page 243 and 244: Boundary element methods 233 From t
Page 245 and 246: with: Boundary element methods 235
Page 247 and 248: Numerical example for BEM 237 S is
Page 249 and 250: Numerical example for BEM 239 In ea
Page 251 and 252: Numerical example for BEM 241 Once
Page 253 and 254: Numerical example for BEM 243 ž Tr
Page 255 and 256: Numerical example for BEM 245 that
Page 257 and 258: Numerical example for BEM 247 is ch
Page 259 and 260: Table 7.1 Sign for derivatives of p
Page 261 and 262: z y Figure 7.5 Coordinate system us
Page 263 and 264: the contour (Fig. 7.5): n� iD1 i
Page 265 and 266: Numerical example for BEM 255 Let E
Page 267 and 268: Numerical example for BEM 257 At th
Page 269 and 270: Numerical example for BEM 259 This
Page 271 and 272: Numerical example for BEM 261 The i
Page 273 and 274: Numerical example for BEM 263 The e
Page 275 and 276: References Abbott, I. and Doenhoff,
Page 277 and 278: References 267 Morgan, W. B. and Li
Page 279 and 280: Index Actuator disk, 44 Added mass,
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