Practical Ship Hydrodynamics

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232 Practical Ship Hydrodynamics 1. Two-dimensional case The potential of a two-dimensional point source is: ϕ D 1 ln r2 4 The integral zero-flow condition for a patch is: V Ð nx Ð l C � i iMi D 0 nx is the x component of the unit normal (from the body into the fluid), l the patch area (length). The flow through a patch is invariant of the coordinate system. Consider a local coordinate system x, z, (Fig. 6.11). The patch extends in this coordinate system from s to s. The flow through the patch is: M D � s s z dx A x B Figure 6.11 Patchin2d z x q,z q A Rankine point source of unit strength induces at x, z the vertical velocity: z D 1 2 z zq ⊲x xq⊳ 2 C ⊲z zq⊳ 2 Since z D 0 on the patch, this yields: M D � s s 1 2 zq ⊲x xq⊳ 2 C z 2 q dx D 1 2 arctan lzq x 2 q C z2 q The local zq transforms from the global coordinates: zq D nx Ð ⊲xq xc⊳ nz Ð ⊲zq zc⊳ xc, zc are the global coordinates of the patch centre, xq, zq of the source. s 2
Boundary element methods 233 From the value of the potential at the corners A and B, the average velocity within the patch is found as: Ev D B A jExB ExAj Ð ExB ExA jExB ExAj i.e. the absolute value of the velocity is: 1 1s D B A jExB ExAj The direction is tangential to the body, the unit tangential is ⊲ExB ExA⊳/ jExB ExAj. The pressure force on the patch is: � 1 Ef DEn p dl DEn 2 � V 2 Ð l � Ev 2 dl Ev is not constant! To evaluate this expression, the velocity within the patch is approximated by: Ev D a C bt C ct 2 t is the tangential coordinate directed from A to B. EvA and EvB are the velocities at the patch corners.The coefficients a, b, andc are determined from the conditions: ž The velocity at t D 0isEvA: a DEvA. ž The velocity at t D 1isEvB: a C b C c DEvB. ž The average velocity (integral over one patch) is Ev: a C 1 2 This yields: a DEvA b D 6Ev 4EvA 2EvB c D 6Ev C 3EvA C 3EvB � 1 b C 3c D Ev Using the above quadratic approximation for Ev, the integral of Ev 2 over the patch area is found after some lengthy algebraic manipulations as: � Ev 2 � 1 dl D l Ev 0 2 � dt D l Ð a 2 C ab C 1 3 ⊲2ac C b2⊳ C 1 1 bc C 2 5 c2 � � D l Ð Thus the force on one patch is 1 Ef D En Ð l Ð � Ev 2 C 2 15 ⊲⊲EvA Ev⊳ C ⊲EvB Ev⊳⊳ 2 1 3 ⊲EvA Ev⊳⊲EvB Ev⊳ � ⊲Ev 2 V 2 ⊳ C 2 15 ⊲⊲EvA Ev⊳ C ⊲EvB Ev⊳⊳ 2 1 3 ⊲⊲EvA Ev⊳⊲EvB � Ev⊳⊳
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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
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Ship manoeuvring 169 the difference
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Ship manoeuvring 171 it is importan
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d, y 20° 10° 0° −10° −20°
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Distance Lateral deviation Length o
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Ship manoeuvring 177 also performed
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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
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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
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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
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Page 229 and 230: and ⊲ , ⊳ is: r D � ⊲x ⊳
Page 231 and 232: ⊲1⊳ y D � d 2 y d r 2 f ⊲c
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: Boundary element methods 231 usuall
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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Practical Ship Hydrodynamics

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