140
4 How to Determine Wave Parameters
type of models (for example Jonsson, 1966) produce very good results for friction factors, f, corresponding to smaller roughness elements, when r / A < 0.03,
where r is the bed roughness size and A is the horizontal displacement of particles at the sea bottom (see Eq. 4.32).
In another group of horizontally uniform models, empirical expressions for
the velocity distribution are used with close analogy to laminar oscillatory
flow. Nielsen (1992) reviewed the existing literature on this subject.
4.2.8 Forces Induced by Waves
In Sect. 2.6 we examined a variety of forces imposed on submerged bodies by
unidirectional flow. These forces included drag, inertia and lift type of forces.
Because the precise pattern of flow around most submerged objects is not
known, flow-induced forces are assumed to be proportional to some empirical
coefficients Cd, Cm and Ct which, in general, are the functions of the Reynolds
number, Re.
However, when flow becomes oscillatory, dependence of coefficients Cd, Cm
and Cl on Reynolds number, Re, is not sufficient as Re does not incorporate
any factors relating to oscillatory flow. Therefore, for complete specification of
the flow pattern, especially its periodicity, another parameter is required. This
dimensionless number is known as the Keulegan-Carpenter number K, or the
period parameter (Sarpkaya and Isaacson, 1981):
(4.74)
in which U m is the maximum velocity of the oscillation flow, T is the wave
period and Lb is the characteristic length of an object along the direction of
flow. The physical meaning of the Keulegan-Carpenter number, K, is quite
simple. In wave motion, the direction of flow changes every half wave period.
Thus, flow past the object travels in one direction, stops, and then flows back
in the other direction. When the distance that the water moves when flowing
in one direction is greater than the length of the object, the flow pattern is
similar to that for unidirectional flow. In this case, the Reynolds number, Re,
controls fluid separation from the surface, and there is sufficient time for the
formation of a wake of vortices behind the body, which results in a reduction of
pressure (see Sect. 2.6). If, however, the distance travelled by the fluid in one
direction during half of wave period is smaller than the length of the object,
there is not enough time for separation to occur, before the fluid again changes
direction. The distance traveled in one direction is roughly equal to umT, and
it has to be compared with the characteristic length of an object, Lb. This is
the significance of the Keulegan-Carpenter number, K. When K is large, the
fluid moves many times the length L b , and when K > 30, the flow may be
expected to mimic steady flow. For smaller values of K, the flow pattern is
affected by changes in flow direction.
4 How to Determine Wave Parameters
type of models (for example Jonsson, 1966) produce very good results for friction factors, f, corresponding to smaller roughness elements, when r / A < 0.03,
where r is the bed roughness size and A is the horizontal displacement of particles at the sea bottom (see Eq. 4.32).
In another group of horizontally uniform models, empirical expressions for
the velocity distribution are used with close analogy to laminar oscillatory
flow. Nielsen (1992) reviewed the existing literature on this subject.
4.2.8 Forces Induced by Waves
In Sect. 2.6 we examined a variety of forces imposed on submerged bodies by
unidirectional flow. These forces included drag, inertia and lift type of forces.
Because the precise pattern of flow around most submerged objects is not
known, flow-induced forces are assumed to be proportional to some empirical
coefficients Cd, Cm and Ct which, in general, are the functions of the Reynolds
number, Re.
However, when flow becomes oscillatory, dependence of coefficients Cd, Cm
and Cl on Reynolds number, Re, is not sufficient as Re does not incorporate
any factors relating to oscillatory flow. Therefore, for complete specification of
the flow pattern, especially its periodicity, another parameter is required. This
dimensionless number is known as the Keulegan-Carpenter number K, or the
period parameter (Sarpkaya and Isaacson, 1981):
(4.74)
in which U m is the maximum velocity of the oscillation flow, T is the wave
period and Lb is the characteristic length of an object along the direction of
flow. The physical meaning of the Keulegan-Carpenter number, K, is quite
simple. In wave motion, the direction of flow changes every half wave period.
Thus, flow past the object travels in one direction, stops, and then flows back
in the other direction. When the distance that the water moves when flowing
in one direction is greater than the length of the object, the flow pattern is
similar to that for unidirectional flow. In this case, the Reynolds number, Re,
controls fluid separation from the surface, and there is sufficient time for the
formation of a wake of vortices behind the body, which results in a reduction of
pressure (see Sect. 2.6). If, however, the distance travelled by the fluid in one
direction during half of wave period is smaller than the length of the object,
there is not enough time for separation to occur, before the fluid again changes
direction. The distance traveled in one direction is roughly equal to umT, and
it has to be compared with the characteristic length of an object, Lb. This is
the significance of the Keulegan-Carpenter number, K. When K is large, the
fluid moves many times the length L b , and when K > 30, the flow may be
expected to mimic steady flow. For smaller values of K, the flow pattern is
affected by changes in flow direction.