Hydraulic Structures: Fourth Edition

whereumis the maximum velocity. Using the above expressions, equation

(14.63) results (in non-dimensional form) in

CD| sint| sintCM cost. (14.64)

The term umT/Dis known as the Keulegan–Carpenter number, Kc. It may

be shown from equations (14.16) and (14.18) that Kc 2 πX/D. Thus Kc

specifies the relative importance of the distance of travel of the fluid parti-

cles to the diameter of the cylinder. It is clear from equation (14.64) that

for small values of Kc(in practice, less than about 1.0), inertia force domi-

nates drag force. If Kcis large, separation becomes important so the drag

outweighs the inertia force.

The in-line force per unit length, dFi/dy, given by equation (14.63) is

obtained by substituting for ufrom equation (14.16) and du/dt, the time

derivative of u.

Assuming constant values of CDandCMacross the depth, Fiafter

integration of equation (14.63) is

FiCD

32

k

H^2 ^2 D | sint| sint

CM cost. (14.65)

Let

AD



k

2



andA 1 ^2 /k. For the maximum force, Fm, on the cylinder, the time deriv-

ative of Fimust be zero, i.e. dFi/dt0. If ttm, at which Fiis equal to the

maximum value Fim, then

tmarccos 2 π. (14.66)

From equation (14.65)

FimCD

3

2

H^2 DADsin^2 (tm)CM HA 1 cos(tm). (14.67)

For an isolated cylinder in inviscid fluid, CM2.0. However, both CD

andCMare functions of the Reynolds number, the Keulegen–Carpenter

number and the surface roughness. The growth of marine plants and

πD^2



8

A 1 D



ADH

CM



CD

sinh(2kd) 2 kd



sinh^2 (kd)

πD^2 a^2



4 k

sinh(2kd) 2 kd



sinh^2 (kd)

π^2 D



umT

d(2Fi/ u^2 mD^2 )



d(y/D)

604 WAVES AND OFFSHORE ENGINEERING