When a small rigid cylinder is free to move in waves, the response equa-
tion is similar to equation (14.73). For the inline oscillations a simple
expression for the time-dependent force is the Morison equation:
Fx(t)CD (u x·)⏐u x·⏐CM. (14.82)
The drag term accounts for the hydrodynamic damping through the use of
the relative velocity between the fluid and structure. For the random wave
fields, it is usual to linearize the drag term to analyse the response of the
structure.
With regard to vertical oscillation of horizontal cylinders due to
waves, equation (14.73) should consider the displacements in the vertical
direction. For smooth cylinders, the lift force Fy(t) on the cylinder may be
expressed by equation (14.71) for Kc25 due to Bearman et al., 1984 or by
equation for Kc20 due to Kao et al., 1984 (see Section 14.11.2(c)).The
peak oscillation of cylinders is found to occur when the ratio of the flow
frequency to the natural frequency of the cylinder assumes integer values
(Bearman and Hall, 1987; Chioukh and Narayanan, 1997). The amplitude
of displacement of the cylinder is of the order of one cylinder diameter
and is affected by the proximity of the cylinder to the channel bed.
Instead of the empirical expressions for the forces Fx(t) and Fy(t),
attempts are being made to obtain these forces by solving hydrodynamic
equations or by discrete modelling (see Sumer and Fredsoe, 1997).
Worked Example 14.1
- At a site off a coast, measurements of wave heights have been made
for 15 min durations at 3 h intervals. For a sample, the wave heights
are as below. Check whether the data fit the Rayleigh distribution.
The number of waves is given within an interval of wave height.
The interval is the wave height below which the same is given minus
the preceding height in the table:
H(m) 0.25 0.5 0.75 1.0 01.2501.50 01.75 2.0 2.25 2.5 2.75 3.0 3.25 3.5
Number
of waves 3 7 7 7 12 11 10 8 9 13 7 2 3 1
The significant wave height for the sample is 2.5 m. The total number
of waves in the sample, N100.
- The highest significant wave on each day has been recorded, to
obtain 365 values for a year. Their distribution is given below:
∂u
∂t
πD^2
4
D
2
618 WAVES AND OFFSHORE ENGINEERING
