Environmental Engineering Reference
In-Depth Information
Just as there is a difference in concept between the average free path of air molecules and the mixing
length of turbulent flow, so is there a difference between the eddy viscosity and the dynamic viscosity; the
latter is a property of the fluid, and has nothing to do with location in the flow, whereas the former is not
a constant, but rather varies with both location and local velocity.
In most turbulent flows, the shear stress results mainly from momentum exchange, and Eq. (5.12) can
be written as
d(
U
u
)
WH
m
d
y
in which
2
d
d
u
H
l
(5.14)
m
y
is called the momentum exchange coefficient, which is similar to the kinematical viscosity in the laminar
flow. The momentum exchange coefficient is actually the product of the mixing length and the root-
mean-square of the fluctuation in velocity.
In turbulent open channel flow, if the flow is steady and uniform, the velocity profile follows an empirical
velocity distribution, which is known as the velocity-defect law:
m
uu
1
ln
y
(5.15)
U
N
h
*
is called the shear velocity,
W
is the shear stress at the bed,
s
is bed
slope;
N
is the von Karman constant ( 0.41), and
u
m
is the maximum velocity in the profile and can be
roughly taken as the velocity at the water surface. The formula can be applied to both rough and smooth
boundaries if the flow is turbulent.
For river flow, the boundary is always rough, and the velocity distribution in a region near the bed,
y
/
h
<0.2, is given by
in which
U
WU
0
/
gRs
*
u
1
ln
y
A
(5.16)
r
U
N
k
*
s
where
k
s
is the equivalent sand grain roughness (roughness height), and
A
r
8.5.
In steady open channel flow, the distribution of gravitational shear stress is
0
(1
W W
yh
/
)
(5.17)
By applying the velocity distribution formula the distribution of eddy viscosity can be obtained as
follows:
W
K H
(1
yh
/
)
y
§
·
0
N
Uy
1
¹
(5.18)
¨
¸
t
m
d
d
u
*
h
©
y
Bursting process
—Studies in the 1960s and 1970s indicated that turbulence is not as random as was
initially believed. Space related and time-related orderly motions do exist in turbulent flows. These
motions can be called a quasi-cyclic process. They result from events that are repeated in time and in
space but are not strictly periodical. The two most striking events, which are observed near the boundary,
are (1) the lift-up of low-speed streaks from near the boundary, and (2) the "sweep" of high-speed fluid
toward the boundary.
The intermittently lifted low-speed streaks leave the boundary and penetrate the main flow region.
Figure 5.7 (a) is a sketch of the ejection of low-speed streaks as observed by dye injection. The main part
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