Practical Ship Hydrodynamics

160 Practical Ship Hydrodynamics
Z y / T x
0.50
0
−0.50
−1.00
below
above
0.8
1.0
0.9
0.6
Water line
0 0.25 0.50 0.75 1.00 1.25
T x /B
Figure 5.3 Height coordinate Zy of section added mass m 0
The ‘slender-body contribution’ to X is zero. Several modifications to this
basic formula are necessary or at least advisable:
1. For terms involving ∂/∂t, i.e. for the acceleration dependent parts of
the forces, correction factors k1, k2 should be applied. They consider the
lengthwise flow of water around bow and stern which is initially disregarded
in determining the sectional added mass m0 . The acceleration part of the
above basic formula then becomes:
⎧ ⎫
⎪⎨
X1 ⎪⎬
Y1
⎪⎩ K1 ⎪⎭ D
⎧
� ⎪⎨
0
k1m
L ⎪⎩
0
zym0 ⎫
⎪⎬
⎪⎭ Ð ⊲k1Pv C k2x ÐPr⊳dx
N1
k2xm 0
k1 and k2 are approximated here by regression formulae which were derived
from the results of three-dimensional flow calculations for accelerated
ellipsoids:
k1 D �
1 0.245ε 1.68ε 2
k2 D �
1 0.76ε 4.41ε 2
with ε D 2Tx/L.
2. For parts in the basic formula due to u Ð ∂/∂x, one should distinguish terms
where ∂/∂x is applied to the first factor containing m 0 from terms where the
second factor v C x Ð r is differentiated with respect to x (which results in r).
For the former terms, it was found by comparison with experimental values
that the integral should be extended only over the region where dm 0 /dx is
negative, i.e. over the forebody. This may be understood as the effect of
flow separation in the aftbody. The flow separation causes the water to
retain most of its transverse momentum behind the position of maximum
added mass which for ships without trim may be taken to be the midship