Environmental Engineering Reference
In-Depth Information
fall velocity and initiation velocity of sediment particles are the focal points, which have been studied for
centuries.
The terminal velocity of solid particles settling in liquids, often called the fall velocity, is an important
physical quantity that is used in characterizing sediment transport. The simplest case is a single sphere
falling with a constant velocity in quiescent water of large extent. The force of gravity W that acts on a
sphere with diameter D as it falls in water is
3
D
JJ S
W
(
)
s
6
where
J is the specific weight of sediment particles. The resistance to the motion F is
2
2
S
D
UZ
FC
D
42
in which Z is the fall velocity of the sphere and C D is the drag coefficient.
At the beginning of the settling process, the velocity of the sphere is small, and the force of gravity is
greater than the resistance. Hence, the sphere accelerates, and the resistance to the motion increases with
the velocity. After a certain distance of travel, the resistance equals the force of gravity; and the sphere
then falls with a constant velocity, called its fall velocity. That is, W and F are equal, and the equation for
the fall velocity is
41
3
JJ
2
Z
s
gD
(5.37)
C
J
D
in which the drag coefficient is a function of the particle Reynolds number,
Re p =Z D/ Q.
During the settling process, the motion of a sediment particle causes the surrounding fluid to move
also. If the inertia forces in the fluid are negligible, the Navier-Stokes equations can be linearized and solved.
Early in 1851, Stokes obtained the following relationship in this way (Stokes, 1851):
F PZ
S (5.38)
which is known as Stokes law. In this case, the drag coefficient is inversely proportional to the Reynolds
number,
24
24
Z
Q
C
(5.39)
D
D
Re
p
and it therefore follows a straight line with a slope of -1 as shown in Fig. 5.21. By substituting the Eq. (5.39)
into Eq. (5.37), one obtains the fall velocity of the sphere in the form:
2
JJ
1
18
gD
Z
s
(5.40)
J
Q
The condition for Stokes Law to apply is Re p < 0.4, which in water at normal temperatures, corresponds
to a limiting diameter of 0.08 mm.
For Reynolds numbers larger than 0.4, fluid inertia becomes more and more important as Re increases.
The inertial force and the flow separation make the motion quite different from that characterized by Stokes
Law. If the particle Reynolds number is larger than 1,000, the viscous force at the spherical surface is so
small that compared with the form resistance, it can be neglected. The drag coefficient is then essentially
constant, and, thus, independent of the particle Reynolds number, at C D = 0.45, or
JJ
Z
1.72
s
gD
(5.41)
J
Search WWH ::




Custom Search