Environmental Engineering Reference
In-Depth Information
and {x i ,y i ,z i } is the position of the centroid of element “i”. A coefficient a ij describes the
velocity at the centroid {x i ,y i ,z i } of element “i” perpendicular to its surface. This velocity
is induced by a unit source uniformly distributed over element “j”.
If the coefficient matrix {a ij } and the vector on the right {h i } are known, then we can
obtain the unknown singularity densities Q j ¼ Q(
z j ) as solutions to the set of
linear equations. Further, the perturbation potential required (
j j ,
h j ,
w S ) is a linear
combination (Figure 2.36). Powerful commercial computer programs are available for
the numerical analysis.
The load of a compact structure consists of the hydrodynamic pressure acting on the
wetted surface of the body and resulting from the Bernoulli equation [30], see also
Section 2.6.6:
w diff or
"
#
2
2
2
p i ¼ 2
@F i
@
@F i
@
@F i
@
r @F i
@
þ
þ
t r g z
x
y
z
In this equation the first term is the velocity pressure (p v ), the second term the unsteady
pressure (p inst ), the third term the hydrostatic pressure (p stat ) and
F i the total potential at
point {x i ,y i ,z i } on the surface of the body:
Þ e i v t
F i ¼ F 0 x i ;
ð
y i ;
z i ;
t
Þ þ w diff x i ;
ð
y i ;
z i
The velocity field results from the gradient of the total potential to which the steady flow
velocity has been added:
~
v ¼rF 0 x
½
ð
;
y
;
z
;
t
Þ F diff x
ð
;
y
;
z
;
t
Þ
þ~
v C x
ð
;
y
;
z
Þ ~
v 0 þ~
v diff þ~
v C
The acceleration field follows from differentiation with respect to time:
v ¼ @~
v
@
_
t ¼ _
v 0 þ _
v diff
Again, owing to the linear approach for the perturbation potential, only the local component
of the substantial acceleration is considered here.
Where a structure consists of a compact foundation and a more slender substructure
(monotower) that does not have to be analysed according to diffraction theory, then the
hydrodynamic loads can be obtained from a combination of the singularity method and
the Morison formula ([31] and Figure 2.37).
The total potential (
F
) contains not only the (generally) non-linear formulation (
F 0 ) for
the incident wave but also the perturbation potential (
F diff ). This approach enables the
influence of the foundation on the substructure (“blockage effect”) to be taken into
account.
The combined linearised boundary condition at the surface of the water means that the
velocity potential is defined only as far as the still water level. However, the wetted
surface of the structure in the wave and the surface bounded by the still water level are
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