(1)
Chen Qun Dai Guangqing Liu Haowu
Hydroelectric institute, Sichuan university
Hydroelectric Institute, Sichuan University, Chengdu, 610065, China
Tel:
028-5406458, E-mail: cq_fq@sina.com
Abstract:
Stepped spillway has increasingly become an effective dissipator. Only if the
hydraulic performance of the overflow is clearly known, the energy dissipation
can be increased. However, so far the study for the stepped spillway overflow is
only based on the model test. In this paper, the k-εturbulence
model is used to simulate the complex turbulenct overflow. In order to treat the
irregular boundary, unstructured grid is used. Meanwhile, the Volume of Fuid (VOF)
method isintroduced to treat the complex free-surface problem. The simulation
results are agreed with the experiment data.
Keywords:
k-εturbulence
model, numerical simulation, free surface, stepped spillway
Because to use stepped spillway can significantly reduce the depth and dimensions needed for the stilling basin at the toe of the dam and leads to a great economic benefit, stepped spillways are increasingly paid more attention by people. But at present, except for some theory formulas for the estimate of energy dissipation, the study for the stepped spillway is only based on the experiment. Furthermore, the focal points of study are mostly on the factor related to the energy dissipation. Sorensen[1], Stephenson[2], Peyras[3], Ru Shuxun[4], Liao Huasheng[5], Rice[6], Pegram[7] etc. had done the model test for the stepped spillway and studied on the effect of the flow pattern, geometry of step and the slope of spillway to the energy dissipation. In 1995, Liao Huasheng etc. [8] using the “residual pressure feedback” method and the potential flow theory solved the free surface of the stepped spillway overflow, and by solving the ω-ψ equation with the average difference scheme simulated the stepped spillway overflow. According to the flow function ψ , the streamline of the flow had been plotted. Besides this, the numerical modeling of the stepped spillway overflow has not been reported. Otherwise, Liao Huasheng used the potential flow theory, but the flow in fact is not a simple potential flow.
So far, almost nobody simulated the stepped spillway overflow by way of turbulence numerical simulation. The main reason is that the boundary conditions are very complex to simulate, and furthermore to solve the curved free water surface is a very difficult problem. . In this paper, the fractional volume of fluid methos that is good and efficient method for treating complicated free surface is used. In order to simulate the irregular boundary, the unctructured grid is introduced to the pre-process of the program.
The stepped spillway tested was a standard WES spillway. The spillway crest was 0.789m above the toe, and the design head of the spillway Hd=0.097m. The standard spillway crest profile is described by equation y=3.6326x1.85, connected by a three radius curve up to the vertical upstream face. Below the tangency point, the spillway profile had a 1V:0.75H slope, completed by a ogee with a radius of 0.28m down to the horizental apron at the toe. There were 13 steps on the whole stepped spillway, which are numbered 1, 2, etc. from the top to bottom. Below the No.5 step, 0.06m by 0.45m uniform steps continued down to the toe, corresponding to the slope value. The first five steps were transition steps and they had a variable height-to-width ratio so that the envelope of the step tips followed the standard spillway profile.
The spillway was 0.3m wide and a 3m long approach channel was provided upstream of the crest. Downstream of the spillway toe, there was a tail channel with a horizontal bottom and with no downstream control. Both the approach channel and the tail channel had the same width of 0.3m as the spillway section tested.
The test discharge of the spillway was 0.02m3/s, which was measured by a triangular weir installed before the approach channel. The flow velocity on the step No.7 was measured using velometer produced by DANTEC Company in Denmark. On the horizontal surface of the odd numbered steps from step No.5 to No.13, five piezometer tubes were fixed on each step surface to measure presures.
The Volume of Fluid (VOF) model was proposed by Hirt[9]. It was designed for two or more immiscible fluids where the position of the interface between the fluids is of interest. The VOF model touch upon multi-phase flow theory. But it is not a multi-fluid model, and the simple single fluid model is introduced in VOF model. Therefore, as for gas and water flow field, a single set of momentum equations is share by the gas and water, and the Volume fraction of each of the fluids in each computational cell is tracked throughout the domain. In each cell, the volume fractions of the air and water sum to unity. So, a additional variable, the volume fraction of the air or water is introduced. If αw denote the volume fraction of the water, then the volume fraction of the air αa can be given as:
(1)
As long as the volume fraction of the air and water is known at each location, the fields for all variables and properties are shared by the air and water and represent volume-averaged values. Thus the variables and properties in any given cell are either purely representative of water or air, or representative of a mixture of them, depending upon the volume fraction values. The tracking of the interface between the air and water accomplished by the solution of a continuity equation having the following form:
(2)
Where, t is time, ui and xi are velocity component and coordinate component respectively.
In the VOF model, because the water and air shared a same velocity and pressure field, the single set of equations can describe the flow field of the air-water two-phase flow such as a single-phase flow. For k-εturbulence model[10], the continuity, momentum equation and the equations for k andεare given as:
(3)
(4)
(5)
(6)
where ρis the volume-fraction-averaged density, P is revised pressure and μis molecular viscosity. μt is turbulence viscosity, and it can be calculated by turbulence kinetic energy k and turbulence dissipation rateεas following:
(7)
Where Cμis a experiential constant, Cμ=0.09. σk andσεare turbulence Prandtl numbers for k andεrespectively, σk =1.0, σε=1.3, C1εand C2εare constants forεequation, C1ε=1.44, C2ε=1.92. G is the generation of turbulent kinetic energy due to the mean velocity gradients. It can be defined as:
(8)
The equation (3), (4), (5) and (6) for the k-εmodel with VOF method are as same as the k-εmodel of the single phase flow except for the expression of the density ρand molecular viscosity μ, which are the volume-fraction-averaged properties, that is, they are a function of fraction of volume αw and are not constants. They can be determined by the presence of the air and water in each cell as following:
(9)
(10)
Where, αw is the volume fraction of water. ρw and ρa are density of the water and air. μw and μa are the molecular viscosity of the water and air respectively. By the iterating solution of the volume fraction of waterαw, the ρand μcan be calculated.
The geometric reconstruction scheme is used to determime the location of the free water surface. It represents the interface using a piecewise-linear approach, that is, the interface between the air and water has a linear slope within each cell and use this linear shape for calculation of the advection of fluid through the cell face. Based on information about the volume fraction and its derivations in the cell, the position of the linear interface relative to the center of each partially-filled cell can be calculated.
The control volume technique is used to convert the governing equations to algebraic equations that can be solved numerically. The integration governing equations on the individual control volumes are linearized using implicit form. The calculation domain is divided into discrete control volumes by the unstructured grid that has a high adaptability as same as the finite element to the complex geometry and boundary. Furthermore, the size and shape of the grid can be changed willfully. So it can treat the complex boundary of the stepped spillway successfully.
The whole calculation domain is shown in Figure 1. The inlet section is at the upstream of the dam and 0.827m away from it, which consists of the inlet of water at bottom and the inlet of air at top. According to the discharge and the depth at water inlet, the velocity at there can be calculated. All the air boundaries are defined as pressure boundary, that is, the pressure is zero at the air boundary. The outlet at the downstream of the dam is also defined as a pressure boundary, so the water can flow out freely. The initial flow field over the stepped spillway is full of air. Through the time-dependent simulation, the water will flow over the dam and form the free surface between the air and water.
The free water surfaces obtained by simulation and measurement are shown in Figure 2. As can be seen from it, the calculated surface is well agreed with that of measurement especially at the upper of the spillway. There is a little discrepency between the simulation and the measurement at the end of the spillway, which mainly because that the air-entrainment occurred on the last several steps, it is quite difficult to measure the water depth accurately.
The velocity vectors on step No.7 obtained by simulation and measurement were plotted in Figure 3 (a) and (b) respectively. As shown in them, there is clear clockwise eddy on each step. Just because of the interaction between the eddy and the skimming flow, the stepped spillway can effectively dissipate the energy. Meanwhile, the rotation direction, magnitude and location of the eddy in the step are agreed with the measured results.
Figure 4 is the pressure isolines in the water domain obtained by simulation. It shows that the pressure distribution on each step is very similar. The maximum pressure is near to the tip and on the horizontal surface of the step. Around there, the pressures reduce gradually. The minimum pressure is on the upper of the vertical surface of the step. The pressures on the water surface are equal to zero.
Figure 5 is the pressure profile on the horizontal surface of step No.5, No.9 and No.11 respectively. As shown in them, the pressure profiles on the step surface obtained by simulation and measurement are similar. Along the horizontal surface, the pressure decreases a little, then increases to the maximum and decreases again at the step tip. The location of the maximum pressure is about 0.8cm apart from the step tip. This maximum pressure is caused by the impact of the flow.
The k-εturbulence model with the VOF method and unstructured grid can simulate the stepped spillway overflow successfully. The VOF model of the air and water can well trace the free water surface based on the time-dependent simulation. The difficult to treat complex boundary of the stepped spillway was overcome by using unstructured grid to discrete the domain.
According to the simulation results, it was obvious that there were clockwise eddy in the corner of the steps, which was verified by the measurement. The flow on the step can be divided to two parts, one is the skimming flow and the other is the eddy in the corner of the step.
The solution of the free water surface has significance in the engineering. As long as the water depth along the spillway is known, the sidewall of the spillway can be determined. In addition, the velocity at the inlet and the end of the spillway can be used to estimate the dissipation rate of the stepped spillway, and the pressure distribution on the step surface has a use for assessing cavitation. Due to the less time and cost of simulation than experiment, the simulation of the stepped spillway overflow has a significant advantage in practice.
References
[1] Sorensen R. M., Stepped spillway hydraulic model investigation. Journal of Hydraulic Engineering, ASCE, 1985, 111(12): 1461~1472.
[2] Stephenson D., Energy dissipation down stepped spillways. Water Power and Dam Construction, Sept. 1991, 27~30.
[3] Peyras L., Royet P. and Degoutte G., Flow and energy dissipation over stepped gabion weirs. Journal of Hydraulic Engineering, ASCE, 1992, 118(5): 707~717.
[4] Ru Shuxun, etc. Stepped dissipator on spillway face. Proceedings of Ninth Congress of Asian and Pacific Division of the International Association for Hydraulic Research. 24~26, August 1994, Singapore, Vol.2: 193~200.
[5] Liao Huasheng, Ru Shuxun and Wu chigong, Test study of hydraulic characteristics of stepped spillway. the Symposium of the Discharge Works and High-Velocity Flow, Supt. 1994.
[6] Rice C. E. and Kadavy K. C., Model study of a roller compacted concrete stepped spillway. Journal of Hydraulic Engineering, June 1996, 292~297.
[7] Pegram G. G. S., Officer A. K. and Mottram S. R., Hydraulics of skimming flow on modeled stepped spillways. Journal of Hydraulic Engineering, May 1999, 500~510.
[8] Liao Huasheng and Wu chigong, Numerical model of stepped spillway overflow. Proceeding of 2nd International Conference on Hydro-Science and Engineering, Beijing, Mar. 1995.
[9] Hirt C. W. and Nichols B. D., Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys., 39: 201~225.
[10] Chen Jingren, Turbulence model and finite analysis method. Shanghai Jiaotong University Press, April 1989.
Fig. 1 The domain and boundary of simulation
Fig.2 The free water surface over the spillway
Fig. 3 The velocity vectors on step No. 7
Fig. 4 The pressure isolines over the spillway
Fig. 5 Pressure profile on the horizontal surface of step No.5, No.9 and No.11