Environmental Engineering Reference
In-Depth Information
The external force along the curvilinear domain is defined by the normal current
pressure
p
(
)
·
(
)
˄(
)
·
(
)
s
N
s
and the tangential hydrodynamic friction
s
T
s
F
ext
(
s
)
=
p
(
s
)
·
N
(
s
)
+
˄(
s
)
·
T
(
s
),
(7.6)
where
p
is the tangential friction on the boom
c
.
Neglecting inertial force corresponding to very low acceleration of booms, the
equilibrium equation of the curvilinear domain is
(
s
)
is the pressure and
˄(
s
)
d
ds
F
int
(
s
)
+
F
ext
(
s
)
=
0
,
∀
s
∈]
s
A
,
s
B
[
.
(7.7)
This equation is strongly non-linear and depends on the elastic force
t
(
s
)
and on the
boom position
.
The constraint on the boom section length
L
between
A
and
B
permits to pose the
problem. Generally we use
s
A
=
ʾ
L
.
We will consider mainly moored static booms. It is commonly accepted that the
friction
0 and
s
B
=
of the flow along the boom is negligible. This is not the case for boom
during towing as a flag-like structure (i.e. one end-point is free). In this case, the
tangential friction outweighs the normal pressure.
˄(
s
)
Remark 7.1
The external force
F
ext
has an implicit dependence with the position of
the unknown domain
c
. To simplify the fluid-structure interaction problem we use a
predictor of the boom domain geometry. This predictor can be built for example by
a catenary curve as barrier geometry.
Dirichlet boundary conditions are applied at the end-points of the curvilinear
domain
B
. They represent the mooring positions of a boom
section
AB
. It should be noted that a boom plan can be made of several adjacent
sections.
ʾ(
s
A
)
=
A
and
ʾ(
s
B
)
=
Remark 7.2
The oil pollutant behaviour is not included in the domain equilibrium
because the oil density is near the water density and thus it does not affect the
hydrodynamic pressure.
We do not consider sorbent boomwhich can have a non-linear stress/strain behav-
iour. A non-woven material can be non-elastic. Netting structures are not considered.
7.2.3 The Numerical Model
The finite element method is used to define a set
c
h
H
of rectilinear segments represent-
ing an approximation of the curvilinear domain
c
. The mesh horizontal resolution
is denoted as
h
H
. Each two-node element represents a small elastic cable which
remains straight during any kind of deformation. The tension
t
is considered con-
stant along each cable and it depends on the initial and deformed lengths of the
(
s
)
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