Geoscience Reference
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and a rough bed. The channel width was 1.7m, and the radius of curvature at the
centerline was 4.25m. The flow discharge was 0.19m
3
s
−
1
, and the overall average
flow depth was about 0.2m. The Manning roughness coefficient was 0.024. The
measured secondary flow intensity was determined by least-square fitting of Eq. (6.78)
with the secondary flow velocities measured at cross-section 180
◦
. The helical flow
intensity profile calculated using Eq. (6.81) with
1.5 matches the
general trend of the measurement data. Errors exist in regions close to the sidewalls,
due to the influence of the walls and the possible appearance of the other secondary
flow at the top corner close to the outer wall.
λ
=
3.0 and
β
=
t
I
6.3.2 Dispersion of flow momentum
Jin and Steffler (1993) derived differential equations for dispersion momentum
transports from the 3-D moment-of-momentum equations. However, their equations
are complex and require additional computational effort. The simpler approach is
to use algebraic expressions (e.g., Flokstra, 1977). Wu and Wang (2004a) derived a
general algebraic formulation for the dispersion transports, which is introduced below.
By using the power law (3.27) for the streamwise flow velocity and the linear model
(6.78) for the helical flow velocity, the
x
- and
y
-components of local velocity in a
curved channel are evaluated as
U
s
z
h
1
/
m
+
α
12
U
n
+
b
s
I
2
z
1
u
x
=
α
11
u
s
+
α
12
u
n
=
α
11
m
+
1
h
−
m
(6.83)
U
s
z
h
1
/
m
22
U
n
b
s
I
2
z
1
m
+
1
u
y
=
α
21
u
s
+
α
22
u
n
=
α
+
α
+
h
−
21
m
(6.84)
where
ii
are the coefficients of transformation between the (
x
,
y
) and (
s
,
n
) coordinate
systems shown in Fig. 6.12(a);
U
s
is the depth-averaged velocity in the streamwise
direction;
u
s
is the local streamwise velocity at height
z
; and
m
is usually about 7.
Substituting Eqs. (6.83) and (6.84) into the definition expressions of dispersion
transports in Eqs. (2.82) and (2.83) leads to
α
1
m
12
I
2
(6.85)
b
s
3
α
2
b
s
2
m
11
U
s
D
xx
=−
ρ
)
α
α
+
1
α
α
12
IU
s
+
α
11
11
12
(
m
+
2
+
1
m
22
I
2
b
s
3
α
b
s
2
m
21
U
s
D
xy
=−
ρ
)
α
α
+
1
(α
α
+
α
α
)
IU
s
+
α
11
11
22
12
21
12
(
m
+
2
+
(6.86)
