4.2 Wave Parameters Based on Small Amplitude Wave Theory
141
Let us consider a typical wave period, T = 10 s, and maximum horizontal
velocity U m = 1 m/s. Thus, for marine organisms with characteristic lengths
between 1 mm and 10 cm, the Keulegan-Carpenter number, K, varies between
100 and 10 4 . For such large values of K, the oscillatory nature of the flow
does not effect the flow around the organisms, and the drag, lift and inertia
coefficients are nearly independent of K (Sarpkaya and Isaacson, 1981).
Resulting total wave-induced force can be found from Morison's equation
(2.90) for Cd and Cm coefficients being, in general, the functions of the Reynolds
number, Re (see, for example, Fig. 2.24). Because in oscillatory flow, the
direction of flow changes every wave period, the expression for drag force (2.62)
should be:
(4.75)
in which U m is the maximum velocity, w is the frequency of wave motion, i. e.
w = 2'if /T, and S is the projected area of the body on the flow direction.
Symbol I I denotes an absolute value. For example, if the value of x is positive,
then Ixl = x, but if this value is negative, absolute value Ixl = -x. Therefore,
in each case, the Ixl always has a positive value.
Let us represent horizontal acceleration as (see Eq. 4.28):
a(t) = amsin(wt),
(4.76)
where am is the maximum acceleration. From Eq. (4.25) and (4.28) it follows
that am = WU m . Thus, the inertia force in Eq. (2.86) takes a form:
(4.77)
where V is the volume of the body.
Let us now form a ratio of the maximum inertia and drag forces, i.e.:
(4.78)
The ratio V / S is proportional to the characteristic length of the body in the
flow direction, Lb, i. e. V / S = o:Lb, in which 0: is a proportionality coefficient.
For example, for a sphere of diameter D, we have V / S = D /6 (0: = 1/6), and
for a vertical cylinder of diameter D and unit height, V / S = 'if D /4 (0: = 'if /4).
Using this in Eq. (4.78), we obtain:
(4.79)
Equation (4.79) indicates that inertia force is dominant only for very small
values of the Keulegan-Carpenter number K. As Cm , Cd and 0: are all of the
141
Let us consider a typical wave period, T = 10 s, and maximum horizontal
velocity U m = 1 m/s. Thus, for marine organisms with characteristic lengths
between 1 mm and 10 cm, the Keulegan-Carpenter number, K, varies between
100 and 10 4 . For such large values of K, the oscillatory nature of the flow
does not effect the flow around the organisms, and the drag, lift and inertia
coefficients are nearly independent of K (Sarpkaya and Isaacson, 1981).
Resulting total wave-induced force can be found from Morison's equation
(2.90) for Cd and Cm coefficients being, in general, the functions of the Reynolds
number, Re (see, for example, Fig. 2.24). Because in oscillatory flow, the
direction of flow changes every wave period, the expression for drag force (2.62)
should be:
(4.75)
in which U m is the maximum velocity, w is the frequency of wave motion, i. e.
w = 2'if /T, and S is the projected area of the body on the flow direction.
Symbol I I denotes an absolute value. For example, if the value of x is positive,
then Ixl = x, but if this value is negative, absolute value Ixl = -x. Therefore,
in each case, the Ixl always has a positive value.
Let us represent horizontal acceleration as (see Eq. 4.28):
a(t) = amsin(wt),
(4.76)
where am is the maximum acceleration. From Eq. (4.25) and (4.28) it follows
that am = WU m . Thus, the inertia force in Eq. (2.86) takes a form:
(4.77)
where V is the volume of the body.
Let us now form a ratio of the maximum inertia and drag forces, i.e.:
(4.78)
The ratio V / S is proportional to the characteristic length of the body in the
flow direction, Lb, i. e. V / S = o:Lb, in which 0: is a proportionality coefficient.
For example, for a sphere of diameter D, we have V / S = D /6 (0: = 1/6), and
for a vertical cylinder of diameter D and unit height, V / S = 'if D /4 (0: = 'if /4).
Using this in Eq. (4.78), we obtain:
(4.79)
Equation (4.79) indicates that inertia force is dominant only for very small
values of the Keulegan-Carpenter number K. As Cm , Cd and 0: are all of the