Geoscience Reference
In-Depth Information
Q
2
A
∂
Q
∂
+
∂
∂
β
gA
∂
z
s
+
x
+
gAS
f
=
q
l
v
x
(5.2)
t
x
∂
where
x
is the spatial coordinate representing the streamwise distance;
A
is the flow
area;
Q
is the flow discharge, defined as
Q
AU
, with
U
being the flow velocity
averaged over the cross-section;
z
s
is the water stage;
=
is the correction factor for
momentum due to the non-uniformity of streamwise velocity over the cross-section;
q
l
is the side flow discharge per unit channel length;
v
x
is the velocity of side flows in
the direction of the
x
-coordinate; and
S
f
is the friction slope:
β
Q
|
Q
|
S
f
=
(5.3)
K
2
AR
2
/
3
where
K
is the conveyance. For a simple cross-section,
K
n
, with
R
being
the hydraulic radius and
n
the Manning roughness coefficient of the channel. For a
compound cross-section, determining
K
or
n
is introduced in Section 5.1.1.4.
Note that “
=
/
”, representing the section-averaged quantities in Eqs. (2.102)-(2.111),
is omitted hereafter, for simplicity.
∧
Diffusion wave model
The diffusion wave model assumes that the local and convective accelerations in the
momentum equation (5.2) are negligible, thus yielding
gA
∂
z
s
x
+
gAS
f
=
q
l
v
x
(5.4)
∂
The continuity equation (5.1) is still used in the diffusion wave model.
The diffusion wave model is more stable than the dynamic wave model, but the
latter is more accurate and can be applied in a wider range of flow conditions. Wu and
Vieira (2002) investigated the errors of the diffusion wave assumption in various cases.
One example was steady flow through a channel contraction, as shown in Fig. 5.1.
The diffusion wave model exhibits errors in the computed water surface profile in the
transition region near the contraction, whereas the two models give identical results
in the upstream and downstream regions with uniform flow. Normally, the relative
errors are less than 10%, if the Froude number is less than 0.5.
Kinematic wave model
For the kinematic wave, the variations in flow velocity and depth are negligible in
comparison with the variation in channel bed elevation, and thus, the momentum
equation (5.2) can be simplified considerably as follows:
S
f
=
S
0
(5.5)
where
S
0
is the channel slope in the longitudinal direction.
