Classwork 8 Laboratory experiences on open channel flow (in collaboration with Dr. Ing. Luca Milanesi)
Why a practical lesson on open channel flow? In the first part of the course we studied open channel flows with both theoretical and numerical approaches and now it s time to apply what we have learnt through simple laboratory experiences. This lesson will be held into the Hydraulics Laboratory of the University of Brescia, where a 10.8 m long flume with adjustable bed slope is installed. In this lesson we ll focus on situations of steady non-linear flow such as the passage under a sluice gate or over a weir and we will use these control situations as pre-calibrated structures for stream gauging. Then we will measure the flowing discharge with a micro-propeller flow meter, finally comparing these results with the output data of the electromagnetic flow meter installed on the laboratory flume. By operating on a sluice gate, we shall induce an S1 profile and we will measure the water surface elevation at fixed distances in order to compare it with the numerical results obtained by a standard step method as shown in classwork 3 and 4. Hydraulic, geometric data and constants The current configuration of the flume can be schematized through 3 stretches with constant slope but rectangular cross sections with different width. The first one (from 0 to 5.40 m) shows a constant width b up = 0.355 m; the second one (from 5.40 to 7.0 m) has linearly variable width from 0.355 m to 0.180 m; the third and last one (from 7.0 to 10.80 m) has a constant width b down = 0.180 m. In the second stretch a lateral weir is installed. The discharge value, that will be kept constant during the whole experience, is 0.004 m 3 /s and is measured continuously by the electromagnetic flow meter in Figure 1 and. The bed slope that will be used in this practical lesson is 0.0031 m/m. Figure 1. The electromagnetic flow meter Figure. The output display of the flow meter During previous calibration tests on the flume, a resistance law was identified of the type: λ = 0. 17586 0. 15185 Re that is a resistance law of smooth turbulent flow and is valid for 0000<Re<50000. This resistance law has been obtained in the field 0.6<Fr<.06. STEP 1 First of all, in order to become familiar with the instrumentation, we will measure the channel slope by hydrostatic levelling (see Figure 3); then, turning on the recirculating system and opening the final sluice gate of the channel, we will measure the normal depth at different sections comparing it with the numerical solution of the Chezy equation.
Figure 3. hydrostatic levelling for computing the flume slope The measure of the water depth will be done through a pointed shaft mounted on a graduated rod (Figure 4) which can be moved vertically through a vernier (precision 0.0 mm Figure 5). Since we cannot establish an absolute reference system, we will provide water depth through differential measures of the thalweg (channel bed) and the free surface. While the measure of the thalweg is easy, the correct reading of the water surface elevation needs great care to be accurate. This last measurement is obtained when the point touches the free surface (Figure 6); the main cause of errors is due to the water meniscus near to the point and to the inevitable fluctuations of the free surface during motion; the contact position may be usefully observed in the reflection of a window or a light. Figure 4. The pointed shaft Figure 5. The vernier Figure 6. The point On the basis of the input geometrical and hydraulic data, we can easily verify that in the experimental conditions the 1 st stretch of the flume is steep; for this reason the normal depth is lower than the critical one and the uniform flow is supercritical (k=.35 cm and y normal =1.0 cm). A preliminary numerical computation of the S profile arising in the first stretch of the channel, shows that the flow goes from the critical depth at the channel entrance down to 1.01*y normal at distance of 4.49 m (see Figure 7 and Figure 8, where the vertical scale, y [m], is exaggerated). Accordingly we can reasonably suppose that the normal depth characterizes most of the first stretch and is not influenced by the contraction of the second stretch. Therefore we can measure the water depth in the centre of three different sections: 3.40 m, 4.0 m and 5.00 m (measured from upstream), taking care to avoid the areas influenced by the presence of some stationary crossing waves caused by small dents on the channel lateral walls.
y [m] 0.0400 0.000 0.0000 0.01800 0.01600 0.01400 0.0100 0.01000 y 1.01*y0 1.05*y0 y0+e-5 measured depth 0 1 3 4 5 x [m] Figure 7. S profile and computational tolerances y [m] 0.01100 0.01090 0.01080 0.01070 0.01060 0.01050 0.01040 0.01030 0.0100 0.01010 0.01000 y 1.01*y0 1.05*y0 y0+e-5.5 3 3.5 4 4.5 5 x [m] Figure 8. Zoom of the sections from to 4.5 STEP Now we will study the passage of the current under a sluice gate (Figure 9), which is located in the third stretch, under the hypothesis of negligible head losses.
Figure 9. The sluice gate (in red) and the induced profiles (in green). The balance energy equation for a rectangular cross section is: y up + αq = ac αq c gb yup gcc b a + where y up [m] is the water depth just upstream the gate, b [m] is the width of the gate and a [m] is the gate opening. Through this balance equation we can easily calculate the flowing discharge just measuring the upstream depth and the gate opening. The upstream water depth can be easily measured, while the gate opening can be read on the ruler on the gate structure and the contraction coefficient may be assumed C c =0.61. Since an electromagnetic flow meter is located on the pipe circuit, the measured flowing discharge is known and we can compare the measured and the calculated discharge. Alternatively, by using the measured discharge, one could compute the upstream water depth or estimating the contraction coefficient. STEP 3 In this third part of the lesson we remove the sluice gate and we place a Bazin weir (Figure 10) at 8.775 m from upstream. As in the previous experience we ll focus on the determination of the discharge through the measurement of the water depth upstream of the weir. In order to reach this goal we need to measure the height of the weir d [m] operating as in the previous situations through differential measurements of the thalweg and of the weir crest. A circular pipe will connect the space under the nappe with the atmosphere in order to maintain there the atmospheric pressure.
Figure 10. Flowing discharge over a Bazin weir The stage-discharge equation for this weir and the discharge is: Q = µb 3 ghup where h up is measured with respect to the elevation of the crest of the weir. Since many calibration tests for this device have been conducted, the discharge coefficient and the formula have been defined through the Rehbock approach: 3 ghe Q = µb [m 3 /s] where h e [m] is the effective head and is defined as: and the discharge coefficient is: h h + 0.0011 [m] e = up µ = 0. 4101+ 0. 0476 It is important to choose correctly the surface measuring section, that must be selected at a distance upstream the weir equal to (3 4)*h max. In our situation, since h max is about 4.9 cm, the section may be 8.595 m from upstream, that means 0.18 m upstream of the weir. Since in our case the bed slope is significant, we have to remark that the contribution of the geodetic drop should be taken into account during the evaluation of the head referred to the weir crest. As in the previous case, the relative error between the calculated discharge and the measured value has to be evaluated. STEP 4 h e d
In order to complete this overview about pre-calibrated structures for stream gauging, we will now analyze a broad crested weir (Bèlanger) (Figure 11). Figure 11. Transcritical passage over a Belanger weir in steep channel and related steady flow profiles In our flume the Belanger weir is created by a sequence of superimposed plates with the upper one with a rounded leading edge. Being a steep channel and the weir significantly high with respect to the specific energy of the flow, a transcritical passage can be observed upon the weir, while an S1 profile can be seen upstream (see Figure 11). The Belanger weir is located at 9.943 m from upstream. The height d [m] of the weir (0.0754 m) can be measured with a differential reading of the pointed shaft. Moving from the energy balance equation, the relationship used to calculate the flowing discharge as a function of the upstream water depth is the following: 3 Q = Cqbhm g where h m [m] is the upstream water depth with respect to the weir crest and is measured in a section ( 3) *h m upstream the weir; in our case it s located about 1 cm upstream the weir. As before, we take into account the contribution of the bed slope in h m. The discharge coefficient is defined as (Gherardelli): C hm = 0. 385 + 0 108 hm + d q. As usual, it is interesting to compare the computed discharge with the output from the electromagnetic flow meter. STEP 5 Now we want to compare the discharge measured by the magnetic flowmeter with all of the results we obtained till now and with the value of discharge that can be computed by using a current flow meter. However, the flow measurement cannot be accomplished operating in normal flow conditions since the normal depth, both in the 1 st and the 3 rd stretch, is very low. Accordingly we need to make the water surface higher by operating, for instance, with a weir located downstream. To this purpose we will put the Bazin weir at the section 10.100 m from upstream. As a general principle for the selection of the measuring cross-section, it should be chosen sufficiently far away from singularities (both geometrical and hydraulic) and such that the velocity on it ranges in the field of acceptability of the instrument. Considering these general guidelines, we will locate the flow meter at the beginning of the 3 rd stretch (section 7.00 m from upstream), where, however, the distribution of velocity is not uniform on the cross section both in the transversal and in the vertical direction (Figure 1). Accordingly we will have to sample the unknown velocity profile at a sufficiently large number of measurement points, both in the vertical and transversal direction (Figure 13).
Figure 1. Velocity distribution on the cross section for compact shapes; frontal view. Figure 13. Velocity distribution on transversal (upper part) and vertical (lower part) direction. Since the section there is 0.18 m wide, we will measure the velocity on 8 verticals (transects) spaced cm from each other. Each vertical will be studied in 3 point, respectively 0.*y, 0.6*y and 0.8*y from the free surface, where y [m] represents the water depth of the section. The current meter we re going to use is a NIXON Streamflo with a 11.6 mm rotor (Figure 14). The velocity range of the instrument is between 5 to 150 cm/sec with an accuracy of ±1.5% of the true velocity. The propeller is connected to a digital indicator that translates the electrical impulse given by the rotor at each rotation in a conductive fluid into a velocity value through the individual calibration curves of the instrument (Figure 15).
Figure 14. NIXON Streamflo Figure 15. Calibration curve of UniBs Hydraulics Laboratory NIXON Streamflo The indicator reads frequencies over 10 seconds and, since velocity is a fluctuating variable, we will take 6 reading for each point and then we will calculate the mean value through an arithmetic average. The mean velocity on each vertical can be defined as following: V0. + V0. 8 + V V avg = while the global discharge can be calculated as the sum of each unitary contribution: 0. 6
Q Vavg _ i i i = Qi = where A i indicates the influence area of each vertical. An alternative and more speedy procedure allow to measure the punctual velocity of each vertical at the 0.6*y depth only and consider that: V0. 6 = 1. 0 V avg As usual we can compare the flowing discharge measured through the punctual velocity with the output of the electromagnetic flow meter. STEP 6 Finally we want to compare the theoretical reconstruction of a S1 profile induced by closing the sluice gate with the water depth measurements. The sluice gate opening will be a= 0.7 cm. In such a situation, the upstream level increases and some water spills over the lateral weir; then an S1 profile is present that is limited upstream by an hydraulic jump with a non-linear transition to the normal depth upstream. As usual, the measurement of the water depth can be done through a differential reading of the free surface and the thalweg with the pointed shaft. The profile will be studied from section 5. m to section. m because, in the more upstream sections, the turbulence induced by the hydraulic jump doesn t allow accuracy in free surface measurement. The S1 profile can be easily calculated imposing as a downstream boundary condition the measured water depth and using the explicit standard step implemented in the previous classworks,. A comparison between the computed and the measured data is shown in Graph 1. A i Comparison between measured and theoretical water depth Water depth [m] 0.18 0.17 0.16 0.15 0.14 0.13 0.1 0.11 0.1 0.09 3.0%.5%.0% 1.5% 1.0% 0.5% 0.08 0.0%..4.6.8 3 3. 3.4 3.6 3.8 4 4. 4.4 4.6 4.8 5 5. 5.4 Section [m] Measured Theoretical Error % Graph 1. Comparison between measured and theoretical water depth and relative error.