Hydraulic Structures: Fourth Edition

the bed and the cylinder, and the gap between the bed and the pipe,
expressed as a ratio of the pipe diameter. The boundary layer on the sea
bed will have an influence too. Experiments carried out by Littlejohns
(1974) in the Severn Estuary (UK) show that for cylinders on the sea bed
the steady drag and lift coefficients are 1.15 and 1.27 respectively.

14.11.2 Wave forces

In estimating the wave forces, the size of the cylinder in relation to the wave-
length is important. If the ratio of the diameter of the cylinder to the wave-
lengthD/Lis less than about 0.2, the viscosity of the fluid and hence the
separation effects become important. In this case the perturbations due to
the presence of the cylinder are local and the wave forces are determined
using the Morison equation (14.63). On the other hand, for D/L0.2, the
excursions of the fluid particles are small relative to the diameter of the cylin-
der and the flow around the body experiences no separation. The waves are
scattered and a diffraction analysis is used to find the pressure distribution on
the cylinder and hence the force. The subject of wave forces is exhaustively
treated by Sarpkaya and Isaacson (1981) and Chakrabarti (1987).

(a) In-line forces on small vertical cylinders

The Morison equation for the wave force exerted on a submerged cylinder
of small diameter considers that the force is simply a sum of the drag and
inertia forces. The latter arise as a consequence of the unsteady nature of
the wave field. The Morison equation for the in-line force, Fi, per unit
length of the cylinder is

CD |u|uDCM (14.63)

whereCDis the drag coefficient, CMis the inertia coefficient, uand du/dt
are respectively the particle velocity and the acceleration normal to the
axis of the cylinder and Dis the cylinder diameter; they are determined
along the axis of the cylinder as though it were absent. The first term on
the right-hand side of equation (14.63) is the drag term, which contains the
modulus of velocity in order for the direction of force to be along the
instantaneous particle velocity vector. The second term is the inertia force
that will arise from the unsteadiness of the flow field even if the fluid is
inviscid (CMis often computed using the inviscid theory).
Placingx0 at the axis of the cylinder, uand du/dtmay be expressed
as

uumsint, du/dtumcost,

du



dt

πD^2



4



2

dFi



dy

FORCES ON CYLINDRICAL STRUCTURES 603