Hydraulic Structures: Fourth Edition

whereNis the number of Fourier components, CLnand (^) nare the nth lift
coefficient and corresponding phase angle respectively.
For Keulegan–Carpenter numbers above 25 Bearman et al.(1984)
developed an equation for the lift force assuming quasi-steady vortex
shedding at a constant Strouhal number of 0.2 based on the instantaneous
flow velocity. The lift force for the half cycle on a unit length of the cylin-
der is:
fy(t)

1

2

CL^ um^2 Dsin^2 tcos[
(1^ cost)] (14.71)

where
KcSandis the phase angle for the lift force depending on the

history of the reverse flow.

ForKc20, an empirical expression of Kao et al.(1984) for the lift

force on horizontal cylinder is:

Fy(t) LCmy  DLCLu^2 Cw Lu^2 t (^)  um^2 
(14.72)
whereCw,and are empirical coefficients. The first two terms on the
right of the above equation are the transverse inertia and lift terms pre-
dicted by the potential theory, the third term is due to vortex influence.
Chioukh and Narayanan (1997) present values of CmyandCLfor potential
flow as functions of G/D. Empirical values of the coefficients Cw,and
expressed as functions of G/DandKcare presented by Kao et al.(1984).
(d) Wave forces on large cylinders
Linear diffraction theory is used for the determination of wave forces on
large cylinders, the diameters of which are greater than 0.2 times the wave-
length: it is only briefly described here. The velocity potential, T, of the
field in the presence of the cylinder is expressed as
Tis
whereiis the incident wave potential and sis the scattered wave poten-
tial.smust satisfy the linearized energy condition on the free surface, the
no-penetration condition on the sea bed and the condition that the veloc-
ity on the surface of the cylinder due to the scattered wave must be equal
and opposite to that of the incident wave. At large distances from the
cylinder, a boundary condition called Sommerfeld’s radiation condition
must be imposed for the scattered wave. The Laplace equation (equation
(14.4)) for sis solved satisfying the kinematic boundary conditions on the

T



2 π

D



2

1



2

∂



∂t

πD^2



4

608 WAVES AND OFFSHORE ENGINEERING