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102
Rodney J. Sobey
Lateral structure of uniform flow
Journal of Hydroinformatics
06.2
2004
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Figure 1
Definition sketch for narrow channel.
viscosity. Illustrative predictions are provided for a
zQzQ
t0 f r
(3)
rectangular and a natural channel.
A2
8
in which f is the DarcyWeisbach friction factor. The
alternative Chezy or Manning models can be substituted
`
without any fundamental change in the analysis. These
details are given subsequently.
CROSS-SECTION-INTEGRATED DESCRIPTION
The special case of steady flow has received consider-
able attention in open channel flow. Omitting the time-
Most analyses of nearly horizontal flow in natural
dependent terms, the long wave equations reduce to the
channels adopt a cross-section-integrated description
gradually varied flow equations:
(see Figure 1). The independent variables are longitudinal
position x and time t, the dependent variables are h(x,t)
the local water surface elevation to a fixed horizontal
Q constant
SD
datum and Q(x,t) the local discharge or cross-section-
Q2
dh f uQuQ
d
52gA
P
(4)
2
integrated flow.
dx 8 A2
dx
A
The cross-section-integrated conservation equations
are
which describe the steady-state flow on which flood and
tidal flows are imposed (Henderson 1966). For uniform
Q
∂h
flow, dA/dx is zero, so that dh/dx ( = dzb/dx = - S0), A
∂
0
b
∂t ∂x
and P are also constant, and
SD
Q2
∂Q
∂h
0
gA
∂
t P
(1)
∂t
∂x
A
∂x r
Q2
0 gAS0 f 2 P.
(5)
8A
in which A(x,t) is the local flow cross section, P(x,t) is the
local wetted perimeter,
For a flat-bottom channel (rectangular, trapezoidal, etc), a
uniform (h - zb)n = dn depth (the normal depth) can be
OE
D
2
dzb
established by solution of implicit algebraic Equation (5).
*
1 S
A * h zb dy, P
(2)
dy
dy
But note that this requires the additional assumption that
Bed
A
the lateral water surface profile is horizontal.
zb(x,y) is the local bed elevation, b(x,t) is the local surface
For a natural channel, the depth varies across the
channel and the concept of a normal depth is not
boundary shear. The quadratic DarcyWeisbach friction
especially satisfactory. But the concept of uniform flow
model is adopted, with
remains appropriate. It would be more useful to
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