International Journal of Advanced Biotechnology and Research (IJBR) ISSN 0976-2612, Online ISSN 2278–599X, Vol-7, Special Issue-Number5-July, 2016, pp1239-1245 http://www.bipublication.com Research Article
A Study on Experimental Model of Dam Break Problem and Comparison Experimental Results with Analytical Solution of Saint-Venant Equations
Mostafa Javadian*, Roja Kaveh and Faride Mahmoodinasab Department of Civil Engineering, Sharif University of Technology, Tehran, Iran. Corresponding Author: Email:
[email protected], Mobile No:+989354596433
ABSTRACT Due to irreparable losses, the dam break is such a very important phenomenon in civil engineering. In this research by removing the gate in front of the water flow in a glass flume that one side is completely dry, dam break phenomenon has been modeled experimentally. Position versus time graph, frontal wave velocity and volume of displaced water is drawn at different times and are compared with theoretical results obtained from characteristic method of analytical solution of Saint-Venant equations. The results indicate that the theoretical and experimental responses have close agreement, but due to the assumptions considered in characteristics method for Saint-Venant equations analysis, the experimental and theoretical approach has the inherent differences. These differences made a small disagreement in the results of two methods that the disagreement is about 20 percent, and independent of the experimental errors. Key words: Dam break, Experimental modeling, Saint-Venant, Characteristics method, Frontal wave velocity
1 INTRODUCTION Nowadays, dams are of great importance in terms of their economic, social, recreational benefit, control of the devastating floods, providing the needs to the water and electricity, waste management and environmental sustainability. Although breaking dams rarely happens but danger is devastating, and is not negligible, the danger which might threaten lives of million people and wildlife. The principles of the construction of the dam and its management has attracted civil engineers; Thus, the successful construction of the dam requires an understanding of the factors influencing their failure and evaluation of different ways to reduce the risk of dams failure. One of the ways to study this phenomenon, is modeling in the laboratory which makes the investigation simpler by creating this
phenomenon on a smaller scale. Ritter (1892) [1] is the first researcher on dam break and its induced wave, who used Saint-Venant equations. Vitham (1995) [2] investigated analytically the turbulent wave caused by dam break. Hant (1994) [3] analytically represented dam break on sloped bed based on kinematic wave.In the study conducted by MohammadNejad et al. (2014 ) [4] dam failure, and its induced wave was simulated two-dimensionally using a numerical method called finite volume and the results of the numerical modeling compared with experimental results and evaluated. Dam break is modeled in two states of dry and wet bed and on the basis of different mesh sizes, various turbulent models of k - e Standard, k - e RNG , RSM, ωk and k - e Realizable, and calculations with first order and
A Study on Experimental Model of Dam Break Problem and Comparison Experimental Results
second order forward schemes, Quick and Power Law. After testing the accuracy of the model, various bed slopes and roughness are modeled and the results were analyzed. The results showed that the numerical model used, has the ability to simulate the dam break in two states of dry and wet bed and represent acceptable results. If the phenomenon of failure and its effects is predicted before construction of the dam, damages caused by the dam break decreases in the future. Saifi et al. (2014) [5] investigated the Polroud dam break due to overtopping using the GUI BREACH model. Considering lagrangian approach, Maghsoudi et al. (2014) [6] used isogeometric method for modeling the dam break. The governing equations are mass and momentum conservation which have been solved in lagrangian form using pressure correction. Ordinary least squares method is used for the spatial discretization. Heshmatifar et al. (2014) [7] conducted an experiment and simulation on dam break to investigate the dam break induced wave. They compare the results with the theoretical analysis based on SiantVenant equations. The highest damage occurs in failure peak discharge (Qp) and if is forecasted accurately, the downstream flooding can be zoned with hydraulic methods and also the induced damages can be reduced. Nouri et al. (2015) [8] stated the goal of their paper to estimate Qp accurately as well. Considering statistical measures such as the initial tests, identifying errors and efficiency are innovations of this investigation. So far, multiple linear and nonlinear patterns have been fitted to data. Mesh less methods (Lagrangian methods) such as Moving Particle Semi-implicit (MPS) and hydrodynamic smoothed particle hydrodynamics (SPH), are the latest generation of these methods in the field of computational fluid dynamics which have attracted researchers in practical issues in which the large strain and flow discontinuity exists. Hosseini et al. (2015) [9] developed and improved the simulation of the free surface flow using new
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model called Weakly Compressible Moving Particle Semi-implicit with (WC-MPS). Fondeli, Andrini and Feschesni (2015) [10] conducted their research by numerical modeling of dam break. They investigated the VOF model of the dam break three dimensionally, using mesh adaptation approach 2 MATERIAL AND METHODS 2.1 Investigation and solving the Saint-Venant equation using method of characteristics for dam break phenomenon In the dam break event, an unsteady flow with Saint-Venant and Navier-Stokes equations is dominant. Generally in this case, the flow is evaluated one dimensionally in which a series of equations known as Saint-Venant equations are used. The global form of these equations is shown in Eq.(1) and (2). Q A q0 X t Q Q2 y ( ) gA ( S f S 0 ) 0 t X A x
(1) (2)
Where Q is the flow rate, A the cross-section of stream, X the distance of two sections, g acceleration of gravity, y the flow depth, Sf the friction slope, S0 the channel bed slope, and t is the time. Assumptions in deriving Saint-Venant equations include hydrostatic pressure distribution, uniform flow velocity in the channel cross section and low slope of the channel bed. Ignoring the input and output flow (q0), and applying method of characteristics Eqs. (1) and (2) can be turned into the following equation: D (v 2c ) g (S 0 -Sf ) Dt D (v 2c ) g (S 0 -Sf ) Dt
(3) (4)
where S0 is the channel bed slope, Sf the friction slope (slope of the energy line) and C is the wave velocity. Equation (3) and (4) show the forward characteristic of the flow (C1) and backward characteristics (C2), respectively. If the bed slope is ignored , then Eqs. (3) and (4) can be rewritten as follows:
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A Study on Experimental Model of Dam Break Problem and Comparison Experimental Results
dx v c dt dx v c dt
(5) (6)
Eq. (4) implies that the value of (v-2c) regardless of the bed slope is constant on the characteristics of C2. Also considering these assumptions in the method of characteristics, all the characteristics of C1 are the straight line and their slope are as follows: V 2C V 0 2C 0
(7)
1 C (V V 0 ) C 0 2 dx V C 3C 2C 0 dt
(8) (9)
where V0 and C0 are the initial flow and wave velocity respectively. If we consider the gate position as x=0, according to equation (9), the water surface profile equation for any initial conditions (y0), and at any time (t) is obtained as follows: x 3 t 1 y= 9g
gy 2 gy 0 (
x +2 gy 0 ) t
(10) 2
(11)
Where y0 is the initial height of water behind the dam gate before dam break. The velocity of wave front is obtained by substituting zero into the elevation (y) in equation (11) as follows: V front
1 0 4 ( 2 gy 0 ) 2 y 0 9g t 9 2 V=2( gy gy 0 ) gy 0 3 1 3 8 q V y ( g 2 )( y 02 ) 27 1 3 8 displaced Water q .b .t b .t .( g 2 )( y 02 ) 27 y
x 2 gy 0 2c 0 t
(12)
Where Vfront is the velocity of the wave front. At the gate position (x=0), the values of height, velocity, discharge per unit width and volume of water displaced in time t are obtained using Eqs. (13), (14), (15) and (16) respectively.
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(13) (14) (15) (16)
Where q is the discharge per unit width and b represents the channel width. In this case, the characteristic curves for each point of the flow, is shown in Figure 1. 2.2 Experimental Modeling The dam break experiment were carried out in a glass flume of 200x50x20 cm with a gate inside in the middle of channel with minimum bumps, to experimentally model dam break by abruptly rising it. In order to trap the gate well at its position, using some bumps on flume is inevitable. In the previous studies, theses bums has been very significant, but in this experiment, minimum bump only in lateral parts of the flume have been used to provide the conditions of the theoretical analysis of Saint-Venant equation. Furthermore, it should be noted that the downstream of the gate should be completely dry, which highlights the necessity of sealing at the gate location. After inserting the gate, water mixed with colored powder (for more clarity in the images) is poured into the one side of gate to a certain height and after ensuring no leak in the other side, we wait for the flow behind the gate to be completely laminar and stable to implement assumptions of analytical solutions. Then we suddenly and quickly remove the gate which makes the pressure suddenly to be zero at all flow fronts. Rapid rising of the gate is very important because assuming zero pressure at flow fronts is resulted from the abrupt rising of the gate. We film this phenomenon from an appropriate angle, and then turn the films into the images with proper format using image editing software such as Adobe Premiere with a rate of 24 photos per
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A Study on Experimental Model of Dam Break Problem and Comparison Experimental Results
second. It is to be mentioned that other softwares were tested and the results indicated that they are not accurate enough and must not be recommended for such a phenomenon that occurs in less than a second. Then we import the obtained images into the Plot Digitizer software. In this step, it is necessary to define 3 points with known coordinates in the software for the image to be calibrated. Then, the whole water surface profile is carefully punctuated and the output is obtained in Microsoft Excel software format. Then the water surface profile is drawn in Excel. Figure 2 shows the punctuation on the water surface profile. 3. RESULTS Water surface profiles for each of the three tests was drawn at 8 different times (a total of 24 profiles). As an example of these profiles, the results for the test with initial height of 13 cm at time t = 0.33s is plotted in Figure 3. Position versus time graph for the wavefront using obtained profiles for 3 tests and at 8 moments with the initial height (y0) of 13, 12 and 11 cm are depicted in Figure 4 . The resulted values for velocity of wavefront from experimental modeling and Saint-Venant theoretical solutions using method of characteristics is shown in Table 1. It should be noted that the theoretical velocity of wavefront is obtained by the equation (12). Table 1-Velocity of wavefront in experimental and theoretical modeling y0 (cm)
Experimental velocity (cm/s)
Theoretical Velocity(cm/s)
13
153.43
225.85
12
149.86
216.99
11
152.86
207.75
Figure 5 indicates the displaced volume of water
resulted from numerical integration of profiles obtained at one side of the gate (preferably the left side of the gate because of less turbulence) for three moments of t=0.33s, t=0.5s and t=0.66s. The
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related theoretical value for specific test with known height of y0 at three considered moments is calculated using Eq. (16) To obtain the theoretical value of displaced volume of in a specific time, one can utilize another method. That is to say, first we need to obtain equation of water surface profile at that time using Eq. (11) and then to integrate over the distance from intersection point of initial height with profile to the point of wavefront. Multiplication of this value by flume width results in displaced volume of water. As expected, the theoretical volume of water is the same for both methods. 4. CONCLUSION Comparison of analytical solution of Saint-Venant equations using method of characteristic for the dam break phenomenon with experimental responses indicated that the results have good accuracy. The difference percentage between the experimental and theoretical calculations is shown in Table 2. Table 2- Differences between theoretical and experimental results y0 (cm)
Wavefront velocity diff. (%)
Displaced water diff. (%)
13
32.6
28.4
12
30.4
30.3
11
26.1
33.2
Much of the dispute is due to the assumptions in analytical solutions of Saint-Venant equations using method of characteristic which cannot be achieved in experimental conditions. These assumptions are: - Sudden removing of the gate: No matter how it's trying to be done with greater speed, you cannot provide a situation in which zero pressure occurs at flow front abruptly. This is inevitable and is the reason of turbulences in initial moments of flow which leads to reduction of velocity of wavefront and displaced volume of water per unit time.
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A Study on Experimental Model of Dam Break Problem and Comparison Experimental Results
- Lack of vertical acceleration: Vertical acceleration is ignored in theoretical calculations but in practice in the early moments after removing the gate, the particles close to the gate has vertical acceleration that is one of the key reasons for the velocity of wavefront and displaced volume of water in experiment to be less than analytical solution. - Lack of bed friction: Bed friction is ignored in theoretical calculations, but in practice this friction exists and affects the flow decreasing the velocity of wavefront and displaced volume of water per unit time. It is obvious that the much of the disputes between the theoretical and experimental results are due to the above reasons which are related to their inherent differences. But a little part of that is related to the experimental errors as follows: - Existence of bums for placing the gate in the middle of flume: Although it has been tried to use the minimum bumps in this study, however, a few bums added to the flume affects the flow especially in the early moments. - Perspective error in imaging: This error arises while digitizing images. It means the flow is in a three dimensional bed, but digitizing is done in a two dimensional screen, and should be done on the two dimensional screen with high accuracy . Generally, this study proves that experimental responses agrees well with those obtained by the analytical solution of Saint-Venant equations using method of characteristics, and if the inherent differences between them is taken into account, one can claim that there is always differences of 20 to 30 percent (the total of intrinsic difference and experimental errors) between experimental and theoretical velocity of wavefront and displaced volume of water. Contribution of the inherent differences between these two methods is about 20 per cent.
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5. ACKNOWLEDGMENTS The authors would like to acknowledge Doctor Mirmossaddegh Jamali, Professor of Civil Engineering Department at Sharif University of Technology, for providing us with the hydraulic laboratory of Sharif University of Technology, and Mrs. Eng. Ladan Homayoun for his collaboration in the lab. 6. REFERENCES 1. Ritter, A.,(1892), The propagation of water waves, Ver Deutsch ingenieur zeitschr, , Vol33, pg 947–954. 2. Witham,G.H., (1995),The Effects of Hydraulic Dam-Break Problem, Proc.Roy. Soc. of Landon, vol-227, pg 399-407. 3. Hunt, B., (1994), Newtonian Fluid Mechanics Treatment of Debris Flows and Avalanches, JI of Hyd. Engrg., ASCE, Vol- 120,pg 1350-1363 4. Mohammad Nejad, B. , Fatemi Kia, M., Behmanesh, J. , Monsari, M. (2014), Twodimensional numerical simulation of dam break vertical wave propagation in a, Journal of Civil and Environmental Engineering, University of Tabriz, Vol-44, Issue 3 , pg 7678 5. Seifizadeh, M., Emadi, R., Fazl Avali, R., (2014), Investigation of the Polrood dam break due to overtopping and routing of the downstream flood, Journal of Watershed Management, Vol-4 , Issue 10, pg 9-14. 6. Maghsoudi, R., Amini, R., Zarif Moghadam Basefat, N. (2014), Using isogeometric method in dam break modeling with Lagrangian approach, Journal of Fluid Mechanics and Structures, Vol-4, Issue 3, pg 6-7. 7. Heshmatifar, A., Rouh zamin, A.h., Mahmoudi, d. (2014),experimental modeling of dam break phenomenon and comparing the responses of experimental and theoretical solution using method of characteristics, Ninth Symposium on Advances in science and technology, Mashhad,2014. 8. Nouri, M., Khodashenas, S.R., Rezai Pajand, Hojjat. (2015), Estimating th peak flow rate
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caused by the embankment dam break based on multivariate statistical models, Journal of soil and water, Vol-29, Issue 2,pg 4-7 9. [9] Hossein, Kh., Jafari Nodoushan, A., Mousavi, S.f., Shakibaei Nia, A., Farzin, S., (2015), 10. Simulation of Dam break flows using Weakly compressible moving-particle semi-implicit
method (WC-MPS), Journal of Civil Engineering Modarres, Vol-15, Issue 3, pg 812 11. Fondellia.T, Andreinia.A, Facchinia.B. (2015), Numerical simulation of Dam-Break problem Using an Adaptive Meshing Approach, Journal of Energy and water, Vol-6, issue 3, pg 6-8
Figure 1- Characteristics curves in dam break
Figure 2- Digitizing images in Plot Digitizer software
Figure 3- Water surface profile for experiment with initial height of 13 cm at t=0.33s
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A Study on Experimental Model of Dam Break Problem and Comparison Experimental Results
Figure 4- Wavefront position versus time graph for three tests
Figure 5- Displaced volume of water for three experiments in three moments and their theoretical values
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