Practical Ship Hydrodynamics

144 Practical Ship Hydrodynamics
For practical purposes, one tries to obtain quasi three-dimensional solutions
based on strip methods or high-speed strip methods. At the University of
Michigan, Troesch developed a three-dimensional boundary element method
for slamming. However, the method needs to simplify the physics of the
process and the geometry of body and free surface and failed to show significant
improvement over simpler strip-method approaches when compared to
experiments.
Limiting oneself to axisymmetric bodies dropping vertically into the water
makes the problem de facto two dimensional. The study of 3-d water impact
started from the simple extension of Wagner’s theory to such cases. The
water impact of a cone with small deadrise angle can then be treated in
analogy to Wagner’s theory as an expanding circular disk. A straightforward
extension of Wagner’s theory by Chuang overpredicts the peak impact
pressure. Subsequent refinements of the theory resulted in a better estimate
of the peak impact pressure:
⎡
⎤
p⊲r⊳ D 1
2 V2
� 2 � 2
⎣ 4cotˇ
�
1 r 2 /c 2
r 2 /c 2
1 r 2 /c 2
⎦
r and c correspond to x and c in Fig. 4.19. This equation gives about 14%
lower peak impact pressures than a straightforward extension of Wagner’s
theory. Experiments confirmed that the impact pressure on a cone is lower
than that on a 2-d wedge of same deadrise angle. So the 3-d effect reduces
the impact pressure at least for convex bodies. This indicates that Wagner’s
theory gives conservative estimates for practical purpose. Since the impact
on a ship hull is usually a very local phenomenon, Wagner’s equation
has been used also for 3-d surfaces using local relative velocity and angle
between ship hull and water surface.
Watanabe (1986) extended his two-dimensional slamming theory to threedimensional
oblique impact of flat-bottomed ships. This theory was validated
in experiments observing three-dimensional bottom slamming with a
high-speed video camera and transparent models. Watanabe classified the
slamming of flat-bottomed ships into three types:
1. Slamming due to inclined re-entry of the bottom. The impact pressure
runs from stern to bow. No air trapping occurs.
2. Slamming due to vertical (orthogonal) re-entry of the bottom to a wave
trough with large-scale air trapping.
3. Slamming due to vertical (orthogonal) re-entry of the bottom to a wave
crest with only small-scale, local air trapping.
Type 1 (typical bottom impact observed for low ship speed) can be treated
by Watanabe’s 3-d theory. Type 3 (typical for short waves and high ship
speed) corresponds to Chuang’s theory for very small deadrise angle. Type 2
(also typical for short waves and high ship speed) corresponds to Bagnolds’
approach, but the air trapping and escaping mechanisms are different to
simple 2-d models.
The three-dimensional treatment of slamming phenomena is still subject
to research. It is reasonable to test and develop first numerical methods for
two-dimensional slamming, before one progresses to computationally more
challenging three-dimensional simulations. Until such methods are available
with appropriate response times on engineering workstations, in practice