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Torsten
Dose
Section of Hydraulic Engineering, Department of Civil Engineering,
University of Wuppertal, Germany
Torsten Schlurmann
Section
of Hydraulic Engineering, Department of Civil Engineering,
University
of Wuppertal, Germany
IGAW,
Pauluskirchstr. 7, 42285 Wuppertal, Germany
Tel.:
+49 202 / 439-4194, Fax: +49 202 / 439-4196, E-mail: dose@uni-wuppertal.de
Abstract: A technique based on measuring the water surface slope to determine the discharge in regulated rivers is adopted to reaches characterised by free flow conditions. To substitute measurement data a numerical modelling is performed.Although the benefits are not so significant as it has been attested for the application in regulated reaches, the author has concluded that with very little calculation effort and a simple, but to some extend exclusive measurement set up, it is possible to achieve precise results of the actual discharge especially at flooding events.
Keyword: discharge
measurement, looped rating curve, hydrograph, stage gauge
The
management of hydro-systems requires to know the actual discharge of a river at
selected cross-sections. The most common method is to measure the water level
and to determine the discharge using a rating curve. This procedure is suitable
for rivers without any obstructions like weirs and with nearly steady state flow
conditions.
By
means of a special system, which has been investigated at the University of
Wuppertal, it is possible to measure the discharge in regulated reaches without
a permanently installed flow meter. It consists of two stage gauges at
reasonable distance. The system has already proven its reliability and accuracy
[Dose, still unpublished].
Is this paper the same technique is used for an unregulated reach. Although the method seems to be applicable, it had to be figured out, that the benefits are significant to accept the effort of measuring the water surface slope under free flow conditions as well. The discharge determined by using the rating curve is compared with the results of the new technique using the measured water surface slope.
As a basis for the following analysis a numerical simulation is performed, solving the Saint-Venant differential equations, here presented without lateral inflow (equation 1). The algorithm is based on the method of implicit characteristics, which is very stable and accurate. In view of the following simple problem it is the best choice compared with alternatives like finite elements or the difference method.
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We consider a
virtual stream with a rectangular section and a width of B = 30 m. The
reach length is 5 km with a slope of 0.001, the Manning´s friction
coefficient is n = 0.04. The boundary condition at the downstream end
of the reach is adopted to uniform flow.
As upstream
boundary condition we define the discharge to be constant for one hour at Q = 10
m³/s , than increase up to 70 m³/s and reduce back to the original flow with
the shape of a sinus curve and a time period of 2 hours, sequencing an outbound
of another one hour.
The algorithm
solves the backwater curve problem and—to consider unsteady influences
—computes the speed of long waves on still water with equation 2 in time steps
of one minute. The quotient of the flow area A and the width of the water surface B calculates the mean water level.
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From the total
results the discharge Q and the water
level h at two cross-sections in a
distance at 50 m are selected. Additionally calculating the mean value of the
discharge and the difference of the two water levels the resulting data set are
shown in figure 1.
The conventional
method to determine the discharge at a stage gauge is based on the rating curve,
which is determined by direct flow measurement. It can be substituted by e. g.
the most commonly used friction law, the Manning´s formula in equation 3.
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It encloses an
roughness coefficient n, two geometric
quantities – the hydraulic radius R
as well as the flow area A—and the
slope I, which is normally assumed as
the river bed slope IS and
in this sense is also a geometric constant. It is well known, that the energy
slope IE is the substantial
quantity to calculate the mean velocity and resulting the discharge. Because the
energy slope is not direct accessible, it is replaced by the water surface slope
IW (equation 4) as best
approach.
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Taking this into
account, the generated data are analysed in two ways: Figure 1 shows the results
of the computation with the constant river bed slope IS;
the water surface slope IW
is taken into account in figure 2. In both figures the dotted curves depicts the
original calculated data. It is cognisable, that the results of figure 1 are
matching precisely although no time derivative terms are taken into account,
while the calculation of the discharge using the river bed slope results in
underestimating the peak. Also a shift in time is observed.
Comparing the
two figures 2 and 3, it is obvious to prefer the method based on the water
surface slope. The expenditure is higher to obtain the required measuring data
particular depending on the water surface slope, although it could be planned
ahead to build up a reliable system at new sites.
On the one hand
a very precise system is achievable, on the other hand there are calculation
methods [Fenton, 1999], which give good—but still not accurate—results with
only one measured water level. The conclusion is: it depends on the required
precision which of the mentioned systems will be chosen.
Acknowledgements
The algorithm
for the numerical simulation was developed by G. Schmitz at the University of
Munich and is named “IMOC”. Thanks to Dr. Zunic, University of Munich, who
was so kind to support the author operating the program and to assist by the
additional programming.
References
Press,
H., Schroeder, R., “Hydromechanik im Wasserbau”, Verlag Wilhelm Ernst &
Sohn, 1966, Berlin/München, Germany.
US
Army Corps of Engineers, “HEC-RAS hydraulic reference manual”, Dodson &
Association, 1997, Houston, USA.
DVWK Guidelines, “Manual for Water Level Gauging and Discharge
Measurements”, 1990, Verlag Paul Parey, Hamburg/Berlin, Germany.
Schmitz,
G, “Instationäre Eichung mathematischer
Hochwasserablauf-Modelle auf der Grundlage eines neuen Lösungsprinzips für
hyperbolische Differentialgleichungs-Systeme”,
report no. 46, 1981, TU München, Germany.
Fenton,
J.D., “Calculating Hydrographs from Stage Records”, proceedings of XXVII
IAHR congress, 1999, Graz, Austria.
Fig. 1 Generated data
Fig. 2 Discharge calculated with Manning´s formula with river bed slope
Fig. 3 Discharge calculated with Manning´s formula with water surface slope