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106
Rodney J. Sobey
Lateral structure of uniform flow
Journal of Hydroinformatics
06.2
2004
before for the numerical integration of Equations (23). An
error-correcting adaptive step size ODE code ensures
adequate precision in evaluations of Equations (25). As
the evaluation of the implicit Equations (25) involves
the numerical integration of the ordinary differential
equations (23), numerical precision in the evaluation of
Equations (25) becomes a potentially significant issue. The
use of IEEE standard double (64 bit) precision is import-
ant; this is now implicit in many engineering software
platforms.
It is also recognized that the definition of Z2 to Z4 in
Equations (23) makes it very likely that Z1 to Z4 will be
very different in magnitude, as they also are in dimensions.
Possible numerical precision consequences have been
avoided by non-dimensionalizing all variables by a space
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Figure 2
Rectangular channel.
scale that approximates the width W of the channel, and a
timescale W/U, where U approximates the mean flow
velocity in the channel.
tb = (f/8)rU2, where the cross-section-averaged flow
velocity U is Q/(Wd). W is the channel width and d is the
mean channel depth.
The length scale of the turbulence would be the large
CHANNEL FRICTION
eddy scale, for which the mean channel depth d is a good
estimate.
The bottom friction factors f and f9 and the horizontal
Using these estimates for the velocity and length
eddy viscosity e are independent parameters in the
scales of the turbulence, an order-of-magnitude estimate
numerical algorithm. Their physical relationship is
for the horizontal eddy viscosity is
established through Equation (17).
The cross-section-integrated friction factor f is a given
SD
1/2
f
Q
parameter. The expected magnitude of the depth-
(27)
.
e'
8
W
integrated friction factor f9 would be of the order of f but
rather smaller in magnitude, as resistance in the depth-
This is a suitable initial estimate for e in Equations (25).
integrated description is contributed by both f9 and e.
A simple order-of-magnitude estimate for the hori-
zontal eddy viscosity is provided by a zero equation
turbulence model, in which
APPLICATION
(26)
e'u*l
As an initial example, consider a rectangular channel
(Figure 2(a)) of width 100 m and local bed at elevation
where u* is the velocity scale of the turbulence and l is the
- 5 m. The markers in Figure 2(a) show the discrete (yi,hi)
length scale of the turbulence.
bathymetry pairs that communicate the channel bathym-
The velocity scale of the turbulence is the shear
velocity u* = '(tb/r). From the cross-section-integrated
and the cross-section-integrated flow Q is 1000 m3/sec.
DarcyWeisbach friction model, the boundary shear is
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