26
STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS
There is a theoretical basis for Morison’s équation, the general form for
which can be deduced by applying the principle of conservation of linear fluid
momentum. This conservation principle is based on Newton s second law as
applied to the fluid occupying a fixed control volume V at any instant of time.
In this case, V is chosen as the imaginary rectangular box surrounding a dise
element of a submerged circular cylinder, as shown in Figure 2.8. Here, V =
Z)2Az, where D is the dise diameter and Az is the dise height. For water of
density p and with a horizontal velocity u along x, the net horizontal shear load
EFX on the dise is given by
/ purfP-r
pu-udAo
Jv
Jv
(2-15)
where dV is an element of the control volume and dAg is the element of area
on the surface of the control volume perpendicular to the flow. The reader is
referred to a standard text such as Munson et al. (1998) for a general dérivation
of équation (2.15).
The terms of équation (2.15) are now evaluated. First, EFX = —q£±z where
q is the loading per unit length. The négative sign is chosen since the net shear
reaction load must oppose the direction of flow. The first intégral on the right
is approximated by
— / pudV ■- — pD2
ïi
(2.16a)
dt Jv
The négative sign is chosen since the fluid decelerates inside the control volume
surrounding the rigid dise. The remaining intégral represents the net momentum
flux along x, or the différence between the out-flowing momentum and the inflowing momentum through area Ag = D &z. Thus
| puudAg= ! pu-udAo
/ pu ■ udAg = 0 — pu ■ \u\ D Az (2.16b)
J net
Jeml
J in
where |u, is the absolute value of u and is used to preserve the sign of q. When
u reverses direction, so does q. With équations (2.16), équation (2.15) becomes
q f pD u |u| + pD2û
(2.17)
To correlate équation (2.17) with experimental results, the coefficients Cp /2
and CmI^ are inserted as multiples of the two respective terms on the right
side of équation (2.17), a resuit that then agréés with Morison’s form, équation
According to the database summarized in Chapter 4, CD and CM both hâve
a range from 0.4 to 2.0. However, based on a multitude of experiments compiled
by the British Ship Research Association (1976) and by Sarpkaya and Isaacson
1 *8111. it is quite apparent that Cp and Cm are not simple constants. In the
case of an imposed pertodic plane flow such as offshore waves of period T, if the
root-mean-square (rms) value is chosen for each over a time that is sufficiently
STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS
There is a theoretical basis for Morison’s équation, the general form for
which can be deduced by applying the principle of conservation of linear fluid
momentum. This conservation principle is based on Newton s second law as
applied to the fluid occupying a fixed control volume V at any instant of time.
In this case, V is chosen as the imaginary rectangular box surrounding a dise
element of a submerged circular cylinder, as shown in Figure 2.8. Here, V =
Z)2Az, where D is the dise diameter and Az is the dise height. For water of
density p and with a horizontal velocity u along x, the net horizontal shear load
EFX on the dise is given by
/ purfP-r
pu-udAo
Jv
Jv
(2-15)
where dV is an element of the control volume and dAg is the element of area
on the surface of the control volume perpendicular to the flow. The reader is
referred to a standard text such as Munson et al. (1998) for a general dérivation
of équation (2.15).
The terms of équation (2.15) are now evaluated. First, EFX = —q£±z where
q is the loading per unit length. The négative sign is chosen since the net shear
reaction load must oppose the direction of flow. The first intégral on the right
is approximated by
— / pudV ■- — pD2
ïi
(2.16a)
dt Jv
The négative sign is chosen since the fluid decelerates inside the control volume
surrounding the rigid dise. The remaining intégral represents the net momentum
flux along x, or the différence between the out-flowing momentum and the inflowing momentum through area Ag = D &z. Thus
| puudAg= ! pu-udAo
/ pu ■ udAg = 0 — pu ■ \u\ D Az (2.16b)
J net
Jeml
J in
where |u, is the absolute value of u and is used to preserve the sign of q. When
u reverses direction, so does q. With équations (2.16), équation (2.15) becomes
q f pD u |u| + pD2û
(2.17)
To correlate équation (2.17) with experimental results, the coefficients Cp /2
and CmI^ are inserted as multiples of the two respective terms on the right
side of équation (2.17), a resuit that then agréés with Morison’s form, équation
According to the database summarized in Chapter 4, CD and CM both hâve
a range from 0.4 to 2.0. However, based on a multitude of experiments compiled
by the British Ship Research Association (1976) and by Sarpkaya and Isaacson
1 *8111. it is quite apparent that Cp and Cm are not simple constants. In the
case of an imposed pertodic plane flow such as offshore waves of period T, if the
root-mean-square (rms) value is chosen for each over a time that is sufficiently