Practical Ship Hydrodynamics

88 Practical Ship Hydrodynamics
from the arc lengths Lj (j D 1toi 3) of the streamline between point Pi
and point Pj:
Lj D
� Pj
Pi
dℓ on the streamline
CAi D ⊲CBi C CCi C CDi⊳
CBi D L 2 i 2 L2 i 3 ⊲Li 3 Li 2⊳⊲Li 3 C Li 2⊳/Di
CCi D L 2 i 1 L2 i 3 ⊲Li 3 Li 1⊳⊲Li 3 C Li 1⊳/Di
CDi D L 2 i 1 L2 i 2 ⊲Li 2 Li 1⊳⊲Li 2 C Li 1⊳/Di
Di D Li 1Li 2Li 3⊲Li 3 Li 1⊳⊲Li 2 Li 1⊳⊲Li 3 Li 2⊳
ð ⊲Li 3 C Li 2 C Li 1⊳
This four-point FD operator dampens the waves to some extent and gives
for usual discretizations (about 10 elements per wave length) wave lengths
which are about 5% too short. Strong point-to-point oscillations of the source
strength occur for very fine grids. Various FD operators have been subsequently
investigated to overcome these disadvantages. Of all these, only the spline
interpolation developed at MIT was really convincing as it overcomes all the
problems of Dawson (Nakos (1990), Nakos and Sclavounos (1990)).
An alternative approach to FD operators involves ‘staggered grids’ as developed
in Hamburg. This technique adds an extra row of source points (or panels)
at the downstream end of the computational domain and an extra row of collocation
points at the upstream end (Fig. 3.11). For equidistant grids this can also
be interpreted as shifting or staggering the grid of collocation points vs. the grid
of source elements. This technique shows absolutely no numerical damping or
distortion of the wave length, but requires all derivatives in the formulation to
be evaluated numerically.
+ + + + + + + + + + + + + + + + + + + + + + + +
v
Figure 3.11 ‘Shifting’ technique (in 2d)
+
Panel (centre marked by dot)
Collocation point
Only part of the water surface can be discretized. This introduces an artificial
boundary of the computational domain. Disturbances created at this artificial
boundary can destroy the whole solution. Methods based on FD operators
use simple two-point operators at the downstream end of the grid which
strongly dampen waves. At the upstream end of the grid, where waves should
not appear, various conditions can be used, e.g. the longitudinal component
of the disturbance velocity is zero. Nakos (1990) has to ensure in his MIT
method (SWAN code) based on spline interpolation that waves do not reach
the side boundary. This leads to relatively broad computational domains. Timedomain
versions of the SWAN code use a ‘numerical beach’. Forthewave