Environmental Engineering Reference
In-Depth Information
We can see from this that with constant coefficients c
D
and c
M
, the drag force for large D/
l
values or a small wave steepness H/
l
or also a small H/D ratio is small compared with the
inertia force.
The extent to which the wave force depends on the D/
ratio can be calculated by
solving the diffraction problem. The result according to the diffraction theory of
MacCamy and Fuchs [29] allows us to derive the following rules for vertical cylinders
(see Sections 2.6.2, 2.6.5 and Figure 2.42):
- The Morison formula applies to bodies that are narrow (slender) in hydrodynamic
terms (D/
l
1/5).
- Potential theory can be expected to supply reliable results for large-volume, compact
structures (D/
l>
1/5). This situation corresponds to N
KC
<
l<
2, see above [24].
The orbital velocity at the still water level u (z
¼
0) gives us the Keulegan-
Carpenter number:
N
KC
ðÞ¼
uz
¼
0
ð
Þ
T
H
2
cosh k
d
½
T
D
¼ p
H
D
1
tanh k
d
¼ v
Þ
D
sinh k
d
ð
ð
Þ
For N
KC
<
2 it follows that
H
D
<
2
p
tanh k
d
ð
Þ
0
:
5
In principle, the Morison formula may be used to calculate the forces caused by waves
of finite steepness as well. In such cases the substantial acceleration (Du/dt) should be
used instead of the local acceleration (
t), and the integration of the resulting
actions should be carried out over the momentarily wetted surface of the structural
component.
In many cases the Morison formula must be applied to cylindrical structural members
inclined at an angle. To do this, a vectorial formulation of the Morison formula is
necessary:
@
u
=@
f
¼
c
M
r
p
D
2
4
~
1
2
r
D
~
_
v
N
þ
c
D
v
N
v
N
where
f is the force vector,
the velocity vector and
_
v
N
the acceleration vector,
all perpendicular to the axis of the structural member. The following relationships
apply [30]:
~
v
N
_
e
_
~
v
N
¼~
e
~
v
~
e
Þ
and
v
N
¼~
v
~
e
where
orbital velocity vector
v
¼
u
x
;
u
y
;
w
orbital acceleration vector (Figure 2.35)
v
¼
u
x
;
u
y
;
w
unit vector in direction of member axis (Figure 2.35)
e
¼
e
x
;
e
y
;
e
z
