Geoscience Reference
In-Depth Information
formulas. In general, the bank shear stresses are determined by
c
fw
U
x
U
x
+
c
fw
U
z
U
x
+
U
z
,
U
z
τ
xl
=
ρ
τ
zl
=
ρ
(
l
=
1, 2
)
(6.109)
where
c
fw
is the friction coefficient.
If
U
z
U
x
, which is valid for gradually varied flows, the components of bank shear
stresses in the vertical direction are usually ignored, while the longitudinal components
can be related to the total friction force by using Einstein's division of hydraulic radius
(Wu, 1992):
n
b
n
w
3
/
2
g
U
U
|
C
h
χ
|
τ
=
ρ
χ
b
+
χ
(6.110)
x
w
where
U
is the velocity averaged over the cross-section;
are the wetted
perimeters of the bed, banks, and entire cross-section, respectively; and
n
b
and
n
w
are
the Manning roughness coefficients for the bed and banks, respectively.
In the case of
B
χ
b
,
χ
w
, and
χ
1 and without bank shear stresses, Eqs. (6.103)-(6.105) reduce
to the idealized vertical 2-D model equations. Because the width-averaged 2-D model
considers the variations of channel width in the longitudinal and vertical directions,
it is more often used in practice than the idealized vertical 2-D model. On the other
hand, the width-averaged 2-D model is an extension of the idealized vertical 2-D
model; therefore, many numerical techniques developed for the idealized vertical 2-D
model can be applied here.
Note that if the lateral expansion or contraction of channel width is too large (larger
than about 7
◦
), the flow may detach from the two side boundaries and the width-
averaged 2-D model may not be applicable. However, if the side separation zones are
excluded, the width-averaged flow model can still be approximately applied in the
main flow regions.
=
6.4.1.2 Boundary conditions
At the water surface, the kinematic condition is applied:
∂
z
s
∂
U
hx
∂
z
s
+
x
=
U
hz
(6.111)
t
∂
where
U
hx
and
U
hz
are the
x
- and
z
-components of velocity at the water surface.
In the presence of wind, the wind shear force results in a gradient of flow velocity
at the water surface:
z
=
z
s
=
τ
∂
U
s
∂
s
ρν
(6.112)
n
t
where
U
s
is the velocity in the tangential direction of water surface,
n
is the coordinate
along the direction normal to the water surface, and
τ
s
is the streamwise component
of wind shear stress.
