Practical Ship Hydrodynamics

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124 Practical Ship Hydrodynamics solutions based on multipole methods. Today, usually two-dimensional panel methods are preferred due to their (slightly) higher accuracy for realistic ship geometries. These two-dimensional panel methods can be based on GFM or RSM, see previous chapter. The flow and thus the pressure distribution depends on ž for the radiation problem: hull shape, frequency ωe, and direction of the motion (vertical, horizontal, rotational) ž for the diffraction problem: hull shape, wave frequency ω, and encounter angle For the radiation problem, we compute the pressure distributions for unit amplitude motions in one degree of freedom and set all other motions to zero and omit the incident wave. For the diffraction problem, we set all motions to zero and consider only the incident wave and its diffraction. We denote the resulting pressures by: ˆp2 for horizontal unit motion of the cylinder ˆp3 for vertical unit motion of the cylinder ˆp4 for rotational unit motion of the cylinder around the x axis ˆp0 for the fixed cylinder in waves (only the pressure in the undisturbed wave) ˆp7 for the fixed cylinder in waves (only the disturbance of the pressure due to the body) Let the actual motions of the cylinder in a wave of amplitude Ohx be described by the complex amplitudes Ou2,0x, Ou3,0x, Ou4,0x. Then the complex amplitude of the harmonic pressure is: Opi DOp2 Ou2,0x COp3 Ou3,0x COp4 Ou4,0x C ⊲ Op0 COp7⊳Ohx The amplitudes of the forces per length on the cylinder are obtained by integrating the pressure over the wetted surface of a cross-section (wetted circumference): ⎧ ⎫ ⎨ Of2 ⎬ � � � l Of3 ⎩ ⎭ Of4 D D 0 � l 0 n2 n3 yn3 zn2 ÐOpi dℓ � n2 ⎧ � ⎪⎨ Ð ⊲ Op2, Op3, Op4, Op0 COp7⊳ dℓ Ð ⎪⎩ n3 yn3 zn2 Ou2,0x Ou3,0x Ou4,0x Ohx f0,n2,n3g is here in the inward unit normal on the cylinder surface. The index x in the last vector indicates that all quantities are taken at the longitudinal coordinate x at the ship, i.e. the position of the strip under consideration. ℓ is the circumferential length coordinate of the wetted contour. We can write the above equation in the form: OEf D OH ÐfOu2,0x, Ou3,0x, Ou4,0x, Ohxg T ⎫ ⎪⎬ ⎪⎭
Ship seakeeping 125 The elements of the matrix OH, obtained by the integrals over the wetted surface in the above original equation, can be interpreted as added masses aij, damping nij and exciting forces per wave amplitude Ofei: ⎡ ω OH D ⎣ 2 ea22 iωen22 0 ω2 ea24 iωen24 Ofe2 0 ω2 ea33 iωen33 0 Ofe3 ω2 ea42 iωen42 0 ω2 ea44 ⎤ ⎦ iωen44 Ofe4 For example, a22 is the added mass per cylinder length for horizontal motion. The added mass tends towards infinity as the frequency goes to zero. However, the effect of the added mass also goes to zero for small frequencies, as the added mass is multiplied by the square of the frequency. The forces on the total ship are obtained by integrating the forces per length (obtained for the strips) over the ship length. For forward speed, the harmonic pressure according to the linearized Bernoulli equation contains also a product of the constant ship speed V and the harmonic velocity component in the x direction. Also, the strip motions denoted by index x have to be converted to global ship motions in 6 degrees of freedom. This results in the global equation of motion: [S ω 2 e ⊲M C OB⊳]Ou D OEh OB is a complex matrix. Its real part is the added mass matrix A. Its imaginary part is the damping matrix N: ω 2 e OB D ω 2 eA iωeN � � D V⊲x⊳ Ð 1 C iV � ∂ ⊲ OHB Ð W⊲x⊳⊳ dx ∂x L This equation can be used directly to compute OB, e.g. using the trapezoidal rule for the integrals and numerical difference schemes for the differentiation in x. Alternatively, partial integration can remove the x derivatives. The new quantities in the above equations are defined as: OEFE OE D h D � � V⊲x⊳ OHE C L iV � ∂ OHE7 e ω ∂x ikx cos dx � � 0 1 0 tx 0 x V/⊲iωe⊳ W⊲x⊳ D 0 0 1 0 x C V/⊲iωe⊳ 0 0 0 0 1 0 0 tx is the z coordinate (in the global ship system) of the origin of the reference system for a strip. (Often a strip reference system is chosen with origin in the waterline, while the global ship coordinate system may have its origin on the keel.) ⎡ 1 0 0 0 0 ⎤ ⎢ 0 1 0 0 0⎥ ⎢ V⊲x⊳ D ⎢ 0 0 1 0 0 ⎥ ⎢ ⎣ 0 tx 0 1 0⎥ ⎦ tx 0 x 0 1 0 x 0 0 0 ωe
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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
Page 83 and 84: Table 3.1 Recommended values for CA
Page 85 and 86: Resistance and propulsion 75 3.2.6
Page 87 and 88: P B ⋅ 10 3 (kW) 40 30 20 10 0 n P
Page 89 and 90: Resistance and propulsion 79 t is t
Page 91 and 92: Resistance and propulsion 81 are no
Page 93 and 94: 3.4 Simple design approaches Resist
Page 95 and 96: Resistance and propulsion 85 of the
Page 97 and 98: Resistance and propulsion 87 non-li
Page 99 and 100: Resistance and propulsion 89 resist
Page 101 and 102: Resistance and propulsion 91 (doubl
Page 103 and 104: Resistance and propulsion 93 only b
Page 105 and 106: Resistance and propulsion 95 resist
Page 107 and 108: Resistance and propulsion 97 1°. S
Page 109 and 110: Ship seakeeping 99 3. Addition of t
Page 111 and 112: Ship seakeeping 101 1. No recording
Page 113 and 114: Ship seakeeping 103 Re denotes the
Page 115 and 116: Ship seakeeping 105 the ship axis x
Page 117 and 118: Ship seakeeping 107 The procedure t
Page 119 and 120: Ship seakeeping 109 If several ω r
Page 121 and 122: a 0.015 0.010 0.005 0 1 2 3 4 5 Uc/
Page 123 and 124: Ship seakeeping 113 The (only stati
Page 125 and 126: Ship seakeeping 115 height H1/3 for
Page 127 and 128: Ship seakeeping 117 using data of B
Page 129 and 130: Ship seakeeping 119 for each strip
Page 131 and 132: Ship seakeeping 121 between near fi
Page 133: Ship seakeeping 123 wave length of
Page 137 and 138: Ship seakeeping 127 4.4.3 Rankine s
Page 139 and 140: Ship seakeeping 129 to zero. Then t
Page 141 and 142: Ship seakeeping 131 for commercial
Page 143 and 144: Ship seakeeping 133 Since the spect
Page 145 and 146: Ship seakeeping 135 ž The ωj are
Page 147 and 148: Ship seakeeping 137 changing sea sp
Page 149 and 150: Ship seakeeping 139 The wave impact
Page 151 and 152: Ship seakeeping 141 pressure gauges
Page 153 and 154: v(t) h(x,t) y u(x,t) Figure 4.23 On
Page 155 and 156: Ship seakeeping 145 computations wi
Page 157 and 158: Ship seakeeping 147 4. Derive the p
Page 159 and 160: Ship seakeeping 149 The free consta
Page 161 and 162: 5 Ship manoeuvring 5.1 Introduction
Page 163 and 164: Ship manoeuvring 153 ž yaw rate (r
Page 165 and 166: Ship manoeuvring 155 Table 5.2 Non-
Page 167 and 168: Ship manoeuvring 157 but also the i
Page 169 and 170: C y 1.50 1.25 1.00 1.75 0.85 0.80 0
Page 171 and 172: Ship manoeuvring 161 section. The l
Page 173 and 174: Ship manoeuvring 163 methods have b
Page 175 and 176: esistance R is proportional to spee
Page 177 and 178: Ship manoeuvring 167 accuracy, but
Page 179 and 180: Ship manoeuvring 169 the difference
Page 181 and 182: Ship manoeuvring 171 it is importan
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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
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ehind the leading edge (nose) is: c
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Ship manoeuvring 183 the rudder. In
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Ship manoeuvring 185 attack ˛ of n
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V a /V 0.4 0.3 0.2 0.1 V a /V 2.0 1
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Ship manoeuvring 189 ž Rudder with
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average between VA and V1: � �
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C L (V corr ≠ V A ) C L (V corr =
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Ship manoeuvring 195 hull influence
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Ship manoeuvring 197 - Estimate the
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Ship manoeuvring 199 rudder tip vor
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Ship manoeuvring 201 (1993) uses de
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Ship manoeuvring 203 ž Profiles wi
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Ship manoeuvring 205 The ship perfo
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6 Boundary element methods 6.1 Intr
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C ⊲t1n3 C n1t3⊳⊲t3 xxz C s3 x
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xx D ⊲ 3 x⊲x xq⊳ ⊳/r 2 xy D
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The velocity in the x direction is:
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∂ ∂z Boundary element methods 2
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Boundary element methods 217 2. Thr
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and ⊲ , ⊳ is: r D � ⊲x ⊳
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⊲1⊳ y D � d 2 y d r 2 f ⊲c
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Boundary element methods 223 the so
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2 0 8xz D 2 2 0 8xxz D 2 2 0 8xzz D
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Figure 6.9 Velocities induced by po
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It is convenient to write ϕ as:
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Boundary element methods 231 usuall
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Boundary element methods 233 From t
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with: Boundary element methods 235
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Numerical example for BEM 237 S is
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Numerical example for BEM 239 In ea
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Numerical example for BEM 241 Once
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Numerical example for BEM 243 ž Tr
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Numerical example for BEM 245 that
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Numerical example for BEM 247 is ch
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Table 7.1 Sign for derivatives of p
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z y Figure 7.5 Coordinate system us
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the contour (Fig. 7.5): n� iD1 i
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Numerical example for BEM 255 Let E
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Numerical example for BEM 257 At th
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Numerical example for BEM 259 This
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Numerical example for BEM 261 The i
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Numerical example for BEM 263 The e
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References Abbott, I. and Doenhoff,
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References 267 Morgan, W. B. and Li
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Index Actuator disk, 44 Added mass,
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Practical Ship Hydrodynamics

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