14.12 Vortex-induced oscillations
14.12.1 Vertical cylinders in currents
The fluctuating forces exerted on the piles in the presence of currents can
excite oscillations and lead to failure of the structure; oscillations of piles
during construction are also possible. This subject of flow-induced vibra-
tions is only briefly dealt with in this section. For extensive treatment, the
reader is referred to Hallam et al.(1978), Blevins (1990), Chakrabarti
(1987) and Sumer and Fredsoe (1997).
Consider a rigidly mounted cylinder exposed to a steady two-
dimensional flow normal to its axis. The flow pattern in the wake of the
cylinder is dependent on the Reynolds number of the flow. When the
Reynolds number exceeds about 70, separation of the boundary layer
takes place. The separated shear layers roll up into vortices which shed
alternatively from the cylinder at the extremities of a line perpendicular to
the flow. The frequency of vortex shedding, fv, is expressed in terms of the
Strouhal number, S, defined as fvD/U. For the Reynolds number greater
than about 1000, SfvD/U0.2.
The alternate shedding of vortices is responsible for the periodic in-
line and cross-flow components of force exerted on the cylinder. The
cross-flow excitation is at the same frequency as the vortex shedding fre-
quencyfv. On the other hand, the in-line fluctuations of force are at twice
the frequency fv. The in-line fluctuations of forces are not significant with
respect to flow-induced oscillations in air, but in water they could be
significant. Hence vertical piles in water are susceptible under unfavour-
able circumstances to both in-line and cross-flow oscillations.
The behaviour of a vertical pile is represented by a cantilever system
with the fixed end at the sea bed. Such a single-degree-of-freedom system
under the action of the exciting force and linear damping is represented by
the differential equation
Mx ̈Cx ̇KxF(t) (14.73)
whereMis the mass, Cthe damping coefficient, Kis the stiffness, and F(t)
is the exciting force, which is a function of time. x ̈ and x ̇ are the second and
first derivatives of the displacement xwith respect to time trespectively.
For free oscillations, C0 and F(t)0, and the system has a natural
frequencyfn(1/2π)(K/M)1/2. With only F(t)0 in equation (14.73), the
system performs damped oscillations if CCc(the critical damping):
Cc 4 πMfn. (14.74a)
The decay of the amplitude of damped oscillations is logarithmic. A
612 WAVES AND OFFSHORE ENGINEERING
