In addition to the commercial flow meters described in Chap. 5 a new set-up based on the vortex shedding principle was developed. It was realized that such a vortex shedder can be used to actually resolve pulsations if the vortex shedding frequency is substantially higher than the pulsating frequency. By using wavelet analysis it was found that the instantaneous flow rate could be determined as described in detail in Laurantzon et al. (2010b), Paper 3. For further information of the set-up the reader is referred to that paper. In this chapter some more features of the device are demonstrated.
6.1. Stationary flow
The experiment for the stationary case was carried out for five flow rates. The vortex shedding behind a cylinder with a diameter of 3 mm was detected by means of a 5 µm hot-wire probe. The time signal was then analyzed using power spectrum for each flow rate. A clear peak in the energy spectrum for each flow rate was obtained. In Fig. 6.1, the vortex shedding frequency fvs is plotted versus the bulk velocity ub and a near linear dependence between the shedding frequency and the bulk flow can be observed. The Reynolds number for the lowest velocity (10 m/s) is about 2000 which is high enough for the Strouhal number St= f d/u to be constant.
In Fig. 6.2 the vortex shedding frequency is plotted versus the frequency estimated from the bulk velocity, fb. Here fb is obtained as Stu/d where the Strouhal number is given the generally accepted value for a circular cylinder, namely St= 0.211. As can be seen there is a good agreement between the estimated and measured frequency.
6.2. Pulsating flow
The idea behind using the vortex shedding cylinder for pulsating flows is that it should be able respond to changes in the pulsating frequency as long as the pulsating frequency is much lower than the shedding frequency, and that the relation between the shedding frequency and the bulk flow should adhere to
1This value is confirmed from many laboratory experiments, see for example Kundu & Cohen (2004).
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0 10 20 30 40 0
0.5 1 1.5 2 2.5
fvs[kHz]
ub [m/s]
Figure 6.1. The vortex shedding frequency vs. the bulk ve-locity. The cylinder diameter was 3 mm.
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
fvs[kHz]
fb [kHz]
Figure 6.2. The shedding frequency vs. the estimated fre-quency obtained from the bulk velocity. Based on the same data as in Fig. 6.1
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Laurantzon et al. (2010b), Paper 3. However in that case all measurements were conducted without back flow. This was accomplished by means of mixing the pulsating flow with a constant flow through a bypass branch parallel to the pulse generator. Here additional measurement are presented, where the bypass branch was closed, which gives rise to back flow for certain pulse frequencies and flow rates as was also observed for the measurements presented in Chap. 5.
With this exception, there were no further changes in the set-up as compared to Laurantzon et al. (2010b)
In Fig. 6.3, the velocity obtained from the vortex shedding measurements and the subsequent wavelet analysis, is compared with hot-wire and LDV mea-surements. As can be seen from the LDV measurements, there is a period of back flow during the pulse cycle. During this period the hot-wire sensor regis-ters the absolute value of the velocity, which becomes almost a perfect mirror image of the “true” velocity as obtained from the LDV measurement. The vortex shedding flowmeter does not yield a signal when back flow prevails, but for the rest of the pulse period there is an excellent agreement between the three methods, despite the fact that the hot-wire and LDV measurements are at one point in space. This is probably due to the fact that during pulsating conditions the flow has the before described top-hat profile.
0 60 120 180 240 300 360
−1
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
φ [deg]
u/ub
Figure 6.3. The velocity scaled with the bulk velocity ub, obtained from vortex shedding (dashed), hot-wire (solid) and LDV measurement (dashed-dot). Here, fp= 40 Hz and ˙m = 25 g/s
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frequencies were performed. The result for this is presented in Table 1. The mass flow rate obtained with the vortex flow meter is based on the velocity (from the wavelet analysis), the atmospheric density (ρ = 1.20 kg/s), and the cross section area of the pipe. The case from Fig. 6.3 is shown in the table, and as can be seen, at that frequency the flow rate is overestimated with 13%.
This overestimation is expected, because evidently there is back flow at this pulsating frequency.
Flow rate 0 Hz 20 Hz 40 Hz 60 Hz 25 g/s 1.06 1.01 1.13 0.941
Table 1. Mass flow rate obtained with the vortex shedding flow meter, normalized with the reference flow rate.
In order to verify the assumption that the vortex flow meter registers a velocity close to the bulk flow rate, regardless of the radial position of the hot-wire probe, two measurement series were performed. The first measurement series was conducted with the hot-wire probe located in the wake downstream the cylinder and was traversed in the axial direction of the cylinder. The first measurement point was about 2 mm from the pipe wall and measurements were made in steps of 2 mm from this point to the centerline of the pipe. For the second measurement series, the previous measurements were repeated under the same flow conditions, but with the cylinder removed. Thus providing means to measure the velocity directly with the (calibrated) hot-wire. The comparison between the velocity measurements obtained with the vortex flow meter and the velocity measurements with the hot-wire is shown in Fig. 6.4, where one can see that the velocity determined with the vortex flow meter is virtually the same for each measurement point, whereas for the hot-wire measurements, the lower velocity close to the pipe wall is apparent.
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0 60 120 180 240 300 360 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
φ [deg]
u/ub
Figure 6.4. The phase averaged velocity obtained with HWA (solid lines) and vortex flow meter (dashed lines), scaled with the bulk velocity ub. The brighter the color, the closer the centerline.
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