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2011,23(3):348-352 DOI: 10.1016/S1001-6058(10)60122-9
EXPERIMENTAL INVESTIGATION OF CAVITATION IN A SUDDEN EXPANSION PIPE* ZHANG Jian-min, YANG Qing, WANG Yu-rong, XU Wei-lin, CHEN Jian-gang State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China, E-mail:
[email protected] (Received October 25, 2010, Revised March 27, 2011) Abstract: For sudden expansion pipes, experiments were carried out to study the cavitation inception for various enlargement ratios in high speed flows. The flow velocity of the prototype reaches 50 m/s in laboratory. The relationship between the expansion ratio and the incipient cavitation number is obtained. The scale and velocity effects are revealed. It is shown that Keller’s revised formula should be modified to calculate the incipient cavitation number when the forecasted velocity of the flows in the prototype exceeds the experimental velocity. Key words: high speed flows, sudden expansion pipe, cavitation incipient, scale effect
Introduction In liquid flows, cavitation generally occurs if the pressure in certain locations drops below the vapor pressure and consequently the negative pressures are relieved by forming cavities filled with gas and vapor. Cavitation can be observed in a wide variety of hydraulic projects[1,2], and it is well known that cavitation flow is usually related with a lot of unpleasant results. In the area of hydropower engineering and hydromachine, studies were carried out to understand the essence of cavitation with the aim to treat it reasonably and properly[3,4]. In these studies, the most important issue is the incipient cavitation, because when the cavitation number of the flow is less than the incipient cavitation number, the flow will become the cavitation flow with various unexpected damages. The incipient cavitation * Project supported by the National Key Basic Research and Development Program of China (973 Program, Grant No. 2007CB714105), the National Science and Technology Pillar Program of China (Grant No. 2008BAB29B04), the Science Foundation of Ministry of Education of China (Grant No. 2008108111) and the Program for New Century Excellent Talents in University (Grant No. NCET-08-0378). Biography: ZHANG Jian-min (1972-), Male, Ph. D., Professor Corresponding author: WANG Yu-rong, E-mail:
[email protected]
occurs within the region of the separated flow, and the point of the laminar separation and the leading edge of the cavity are closely correlated. Keller[5-7] obtained an experiential but very useful formula for the cavitation inception and pointed out the influence of the scale of velocity. Ni[8] also proposed a similar formula based on the air bubble dynamics and pointed out that there exists the effect of the scale and the velocity for the cavitation inception. Yang[9,10] obtained similar results from a series of tests. All these studies show that the scale effect exists and should not be ignored. However, the general methods to study the incipient cavitation number are mainly by the means of experiments in water tunnel or in depression tank, and thus the scale effect needs to be properly included to predict the flow behavior in prototype projects. Dong et al.[11] investigated the aerated behaviors in the cavitation region of high velocity flows through the non-circulating water tunnel by using the advanced experimental facilities, they proposed relations between the smallest air concentration without the cavitation erosion and the flow velocity and compared the cavitation numbers with and without aeration in the cavitation region. Han et al.[12] studied the cavitation structures of semi-cylindrical irregularity by using high speed photography, and revealed the cavitation structures of semi-cylindrical irregularity and the interaction between aeration bubbles and
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cavitation bubbles. Xu et al.[13] and Ye et al.[14] studied the interaction between the cavitation and air bubbles with a rigid boundary and the effects of the air content on the cavitation and the pressure fluctuations by experiments. Liu et al.[15] simulated the cavitation bubble shedding on ALE 25 and ALE 15 hydrofoils and found that a small bubble appears at the lateral cavitation bubble. The small bubble grows to a certain volume then sheds from the cavitation bubble. At the same time, Chahine[16], Wang et al.[17] and Spinivasan et al.[18] also studied the bubble flow interactions by numerical simulations. Pezzinga[19] presents the results of a research on the use of distributed cavitation quasi-2-D models to reproduce experimental runs of cavitating water hammer flow and the comparison between computed and measured head oscillations show that the model allows a good reproduction of the observed phenomena if a proper calibration of the parameters is made. The growth, oscillation and collapse of vortex cavitation bubbles are examined using both two- and three-dimensional numerical models by Choi et al.[20]. The results of Wienken et al.[21] show that a new method to predict traveling bubble cavitation inception is devised. The crux of the method consists in combining the enhanced predictive capabilities of Large Eddy Simulation (LES) for flow computation with a simple but carefully designned stability criterion for the cavitation nuclei and a wider range of applications will become accessible methods for cavitation prediction based on algebraic stability criteria combined with LES. The influence of nozzle geometry on cavitation and near-nozzle spray behavior under liquid pressurized ambient is studied by Payri and the results showed that pressure conditions for inception of cavitation obtained in the visualization tests differs from those seen for choked flow (5%-8% in terms of cavitation number)[22]. In this article, with the aim to study the incipient cavitation of a high speed flow passing the sudden pipe enlargement, a series of experiments with velocities ranging from 18.0 m/s to 50 m/s were carried out. 1. Experimental set-up and methodology 1.1 The experimental set-up The experimental set-up is shown in Fig.1, including underground pool, pump, valve, test section, glass flume and discharge weir. The main geometrical and hydraulic parameters of the experimental devices are shown in Table 1. The experimental flume is 10 m long, 2.0 m wide, and 2.0 m high. The length of the test section is 0.7 m. φ1 and φ2 are the diameters of the pipe before and after the sudden expansion in the test section, respectively, h = (φ2 − φ1 ) / 2 , is the height of
the step, β is the sudden expansion ratio given by
β = A2 / A1 , namely the ratio of the areas after and before the expansions. Test discharge Q varies from 27.0 L/s to 47.0 L/s and the velocity u0 of the experiment segment ranges from 18.0 m/s to 45.0 m/s, to be controlled by the valve located at the outlet of the pump. The pressure is regulated by the valve at the outlet.
Fig.1 The setups
1.2 Methodology In order to measure the pressure in the test section, two different pressure meters with high precision are used in the small and big tubes, and the maximum ranges are 0.16 MPa and 0.80 MPa and the minimum scales are 0.02 MPa and 0.04 MPa, respectively. Before the test, they are calibrated. The big one is calibrated using the piezometric tube, and the small one is corrected by the former to ensure that the error is less than 0.5%. The noise testing system includes two sections: the hydrophone with operation frequency in the range of 5 kHz-150 kHz and sensitivity greater than 195 dB, and the dynamic signal analysis system. In order to investigate the cavitation under different velocities, 30 groups of experiments are designed as shown in Table 1. The common methods to determine whether the cavitation occurs include the visual method and the acoustic noise method. The former is simple and direct by detecting the cavitation cloud. But this method is not accurate as it involves some subjective factors. So the latter is generally adopted because of its relatively high precision. There are two kinds of acoustic methods currently used. One is based on the maximum sound pressure level difference between the
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Table 1 Geometrical and hydraulic parameters in the tests Parameters
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
φ1 (mm)
40
40
40
40
40
32
52
φ2 (mm)
42
44
46
48
50
42
62
h (mm)
1
2
3
4
5
5
5
1.560
1.720
1.420
V (m/s)
12.7-44.5
Q (m3/s)
27-47
β
1.103
1.210
1.323
1.440
cavitation noise and the reference noise. Cavitation is considered to occur when the difference is larger than a critical value[8]. The other considers whether the energy of sound is greater than a critical value. In this paper, we employ the former one, 10 dB is taken as the critical value to determine the inception of cavitation. Figure 2 is the cavitation cloud observed and the noise spectrum in the test.
Fig.4 The relationship between β and σ
Fig.2 Cavitation cloud and noise spectrum
Fig.3 The relationship between Vs and σ
2. Results and analysis Figure 3 shows the relationship between the velocity and the incipient cavitation number. For a same enlargement ratio and step height, the incipient cavitation number increases linearly with the velocity. For example, the incipient cavitation number varies from 0.921 to 0.980 under the conditions of β = 1.21 and h = 2 mm , with a difference of 6.4%. For different enlargement ratios and a same step height, the incipient cavitation number varies from 1.340 to 2.081 (see the markers of square, rhombus, and triangle), with a difference up to 55%. Therefore, the enlargement ratio is the primary factor influencing the incipient cavitation number. That is to say, the scale effect of velocity exists objectively, too. However, the results in this article show that the scale effect is smaller than that in Keller’s studies[5-7]. Based on the experimental data, the relationship between the enlargement ratio and the incipient cavitation number is obtained. Figure 4 demonstrates that the relationship is linear approximately. The incipient cavitation number increases with the enlargement ratio. A fitting Eq.(1) can be obtained as
σ = 2.0517 β − 1.5489 ( β > 1.0 )
(1)
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Table 2 The benchmark values σ 0 corresponding to the different expansion ratio β Serial number
Marker
β
Vref (m/s) 1.103
1.210
1.323
1.440
1.56
1.72
1.42
Marker 1
Square
45.0
0.431
0.578
0.707
0.854
1.021
1.041
1.185
Marker 2
Triangle
12.7
0.075
0.101
0.124
0.150
0.178
0.154
0.365
Marker 3
Circle
12.7
0.075
0.165
0.193
0.255
0.279
0.243
0.444
Therefore, for the geometry of the sudden expansion pipe, the incipient cavitation number is affected more by the enlargement ratio than by the step height. Keller[7] proposed a corrected formula through a series of tests about symmetrical objects. The formula is written as follows ª §V σ = σ 0 «1 + ¨ ∞ « ¨© Vref ¬
· ¸¸ ¹
2
º » » ¼
(2)
where σ is the incipient cavitation number, σ 0 is the benchmark value, Vref is the reference velocity which is nearly a constant of value of 12.7 m/s. V∞ is the velocity of the test at the upstream section.
Fig.5 Comparison of measured and calculated σ
However, Keller’s experiments were carried out with velocity less than 12.7 m/s. In order to verify that the formula is applicable for velocity greater than 12.7 m/s, thirty groups of tests were carried out with experimental conditions as shown in Table 2. Figure 5 is the comparison of measured and calculated results. The marker of square shape refers to the reference velocity, Vref of 45 m/s and the marker of triangle shape the reference velocity of 12.7 m/s. In these two cases, σ 0 corresponds to the maximum velocity in the test. The marker of circle means the reference velocity of 12.7 m/s and σ 0 corresponding to the minimum velocity in the test. The benchmark value of
the cavitation number σ 0 corresponding to different heights of the step is shown in Table 2. The errors for these three cases are 18.67%, 28.6% and 41.39%, respectively. It is shown that the error between the calculated value and the experimental data is the smallest for marker 1. Therefore, the maximum velocity in the test ought to be replaced with the reference velocity, the reference velocity needs to be adjusted if the forecasted velocity of the flows in the prototype exceeds the limits of 12.7 m/s. It is well known that the dispersion effect of the incipient cavitation is due to the increase of the flow velocity, as also confirmed by Keller’s experimental investigation. At the same time, the experimental results in this article show that when the practical velocity is much higher than the experimental velocity, the error would be prominent if the maximum velocity in the model is taken as the reference velocity. As a result, it is appropriate in most cases to take the velocity in practical flows as the reference velocity to modify the value of incipient cavitation as is predicted by Eq.(2). Otherwise, the effects of velocity will be magnified. 3. Conclusions (1) The incipient cavitation number increases with the velocity and the scale, which implies that the incipient cavitation number in the prototype (with high velocity) must be corrected when the small scale model results are used. (2) The incipient cavitation number increases linearly with the enlargement ratio for an enlarged geometry. For the same enlargement ratio and step height, the incipient cavitation number increases with the velocity. (3) When the velocity is greater than 12.7 m/s, the reference velocity value 12.7 m/s suggested by Keller does not reflect correctly the scale effect of the velocity. In Keller’s formula, the calculated errors for three cases of reference velocities and benchmark cavitation numbers are 18.67%, 28.6% and 41.39%, respectively. Therefore, it is better to take the forecasted prototype velocity as the reference velocity.
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