Environmental Engineering Reference
In-Depth Information
where
r density of water ( ¼ 1.0 t/m 3 )
u horizontal (orbital) velocity in primary wave
D diameter of vertical cylinder
c M inertia coefficient
c D drag coefficient
The wave theories are only defined as far as the still water level (z ¼ 0). Stretching
methods, for example “Wheeler stretching”, are used to modify the particle kinematics
to account for the momentary displacement of the surface of the water (under the wave
crest and above still water level, see Figure 2.32):
z s ¼ð z zÞ=ð 1 þ z=
d Þ
The total force on the cylinder is given by integrating over its height:
z ¼z
Z
ðÞ
dF x ðÞ
dz
F x ðÞ¼
dz
z ¼ d
Considering just the local (
t) instead of the substantial acceleration (Du/dt) is permis-
sible here because the convective acceleration is negligible with respect to the local
acceleration for the Airy wave investigated initially. In the case of waves of low steepness
(H/
@
u
=@
)] ratio, the
still water level (z ¼ 0) is taken to be the upper bound of the integral. Using the notation j u j u
( ¼ u 2
l
) and a large relative depth of water (d/
l
) or with a low H/d [ ¼ (H/
l
)/(d/
l
sign u) guarantees that the change in direction of the velocity component is taken into
account. The coefficient c D contains both the form drag and the friction resistance.
The inertia force acting on the body is the sum of:
- the pressure gradient force caused by the undisturbed wave field ( ¼ r V @
u
=@
t;
V: displacement), and
- the acceleration drag ( ¼ a xx @
u
=@
t ¼ c a r V @
u
=@
t; c a ¼ a xx /(
r V): hydro-
dynamic mass coefficient).
The following applies for the inertia force acting:
F Ix ¼ r V @
u
@
t þ c a r V @
u
t ¼ð 1 þ c a Þr V @
u
@
t ¼ c M r V @
u
@
@
t
where c M ¼ 1 þ c a
The
t component is the Froude-Kryloff force. Applying the equation for F Ix to a
cylinder element of length “dz”, that is for dV ¼ (
r V @
u
=@
2
D 2 /4) dz, gives us the first term in the
p
Morison formula from the following force:
dF Ix ¼ c M r p D 2
4
dz @
u
@
t
The Morison formula is widely used these days for calculating the hydrodynamic loads on
marine structures with slender cylindrical members (e.g. piles, monopiles, jackets, legs of
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