Numerical Simulation of Cavitation and Aeration in Discharge Gated Tunnel of a Dam Based on the VOF Method

Cavitation, usually known as a destructive phenomenon, involves turbulent unsteady two-phase flow. Having such features, cavitating flows have been turned to a challenging topic in numerical studies and many researches are being done for better understanding of bubbly flows and proposing solutions to reduce its consequent destructive effects. Aeration may be regarded as an effective protection against cavitation erosion in many hydraulic structures, like gated tunnels. The paper concerns numerical simulation of flow in discharge gated tunnel of a dam using RNG model coupled with the volume of fluid (VOF) method and the zone which is susceptible of cavitation inception in the tunnel is predicted. Then a vent is considered in the mentioned zone for aeration and the numerical simulation is done again to study the effects of aeration. The results show that aeration is an impressively useful method to exclude cavitation in mentioned tunnels.


Introduction
Since many natural phenomena which relates to fluid dynamics concern with two phase flow, such flows have received growing research attention among fluid dynamics researchers. Cavitating flows are a group of two phase flows which happens when there is a drastic pressure drop in fluid (usually due to high velocity) insofar as it falls to below the local vapor pressure of main fluid. So vapor bubbles will form and move downstream into a region where the pressure would increase and vapor bubbles collapse. This phenomenon causes destructive effects such as erosion, noise and vibration.
Although in some cases cavitating flows are useful, like supercavitation; when the length of the body is less than the cavity length and is useful in drag reduction [1], cavitation causes destruction in many systems like pumps, valves and hydraulic systems. As it is expensive and time consuming to repair the damaged systems [2], it is important to predict the zones in which cavitation incepts, and use some schemes to prevent this phenomenon. Inasmuch as experimental and analytical methods applied by scientists to predict cavitation inception are of high costs and impossible to be done for many systems, numerical schemes have been employed to simulate cavity flows and have led to better understanding of the dynamics of bubbly flows.
Two current approaches available for the numerical calculation of multiphase flows are the Euler-Lagrange approach and the Euler-Euler approach. The volume of fluid (VOF), the mixture model, and Eulerian model are three different Euler-Euler multiphase models which are usually used to model bubbly flows (usually in commercial softwares like ANSYS/FLOTARN). VOF method is used when the position of the interface between two fluids available in the domain is important. In this model a single set of momentum equations are solved for the fluids and the volume fraction of the fluids available in the domain is determined in each computational cell. Applications of the VOF model include free-surface flows [3]; the motion of large bubbles in a liquid and the motion of liquid after a dam break. In mixture model, the mixture momentum equations are solved and relative velocities are determined for available phases. The applications of this model include particle-laden flows with low loading, bubbly flows and sedimentation. In Eulerian model a set of n momentum and continuity equations are solved for each phase. The pressure and interphase exchange coefficients, depending on the type of phases involved, are used for coupling. Applications of the Eulerian multiphase model include bubble columns, risers, particle suspension, and fluidized beds.
One of the hydraulic systems which are damaged due to cavitation, are discharge tunnels of dams [4]. An impressive method applied for preventing bubble forming and reducing noise in many hydraulic systems, is injecting another fluid (like air) and so increasing pressure, this process is known as aeration [5]. In this paper a two-dimensional cross section of a sample tunnel is modeled in commercial software; ANSYS/FLOTRAN [6]. The modeling is done at different gate openings and the zone of cavitation inception is predicted for each model. Using aeration to reduce cavitation erosion in the tunnel, a vent is considered for aeration after the gate and the models are numerically analyzed again. The RNG ε − k model of water-air two-phase flow coupled with the volume of fluid (VOF) method is used in modeling the section both after and before aeration.
It will be shown in results that aeration increases pressure so that the total pressure is more than the amount in which water evaporates and so cavitation is excluded from the system. Fig. 1 shows two-dimensional cross section of the gated tunnel discussed in this paper. The water flow is modeled at different gate openings, so the gate shown in Fig. 1 changes its position in eight steps and moves 1.2m upward. In each step, the flow is numerically simulated and the zone of cavitation inception is predicted. The models are simulated again considering an aerator after the gate. The exact position of the aerator and pressure of aeration is defined using the results which show the place of cavitation inception. The position of aerator is shown in Fig. 1 too.

The Numerical Approach
Two-dimensional model of the gated tunnel and flow direction, dimensions are in meter.

A. Cavitation Modeling
Two phase cavity flow is modeled by using VOF method here. In this method it is assumed that both phases have the same velocity in a cell and a series of equations can be used for the flow field like a single-phase flow [7]. In this method a function (f) is used to determine volume fraction of liquid in a cell and can have three quantities; f=0, 0<f<1, f=1 which respectively shows that the cell is full of air, includes both air and water and is full of water. Apparently the volume fraction of air is 1-f. The transport equation for volume fraction f is: So density and viscosity is calculated via these formulas: The suffix v and l denote vapor and liquid phase. l ρ , v ρ , l µ and v µ are the density(kg/m3) and molecular dynamic viscosity(N·m/s) of vapor and liquid respectively, t is time(s) and u i is the components of velocity in x i directions (m/s).
The mass conservation equation is: And the momentum conservation equation is: , µ c is constant; u i ,u j are the velocities in x i ,x j directions (m/s), and P is pressure (Pa).

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Equations (3) and (4) together with (1) are basic equations used in VOF method. The algorithm used in ANSYS/FLOTRAN for modeling via VOF method is named VFRC (Volume Fraction). In this algorithm a number is devoted to each cell in the grid. This number which is between 0 and 1 shows the percentage of the cell that is full of main fluid (water); if the number is 0 for a region it means that the region is full of air and if that is 1, the region is full of water. In this software, the governing equations are solved numerically by using control-volume based techniques and SIMPLE algorithm [8] was used for pressure correction.

B. Turbulence Modeling
Among Reynolds-averaged Navier-Stokes (RANS) equations approach in turbulence modeling, two-equation models are robust and simple in which two separate transport equations for turbulence variables are solved. One of these models is RNG ε − k model which is based on the theory of "renormalization group" (RNG) as a mathematical technique used to develop standard ε − k model [9].
The transport equations for turbulent kinetic energy, k and its dissipation rate,ε , are as follows: Where G, the generation of turbulent kinetic energy due to mean velocity, is calculated using this formula: In (5), (6), and (7), µ is the molecular dynamic viscosity of the fluid (N·s/m2); k is the turbulence kinetic energy (m 2 /s 2 ); ε is the dissipation rate of k (m 2 /s 2 ). The values of c µ , σ k , σ ε , C 1ε and C 2ε as the constants in the above equations are 0.085, 1.42, 1.68, 1.0 and 1.33, respectively. ρ and µ in the above equations are calculated using (2).

C. Grid Generation and Boundary Conditions
The structured grid generated in the domain is shown in Fig. 2 for minimum opening at discharge of the tunnel. For all eight positions of the gate, the grid is similar to Fig. 2. The mesh is finer near the walls to study near wall condition better. Grid generated in computational domain and boundary conditions for inlet and outlet.
The inlet boundary condition is specified constant pressure (1177.2 kPa, gage pressure). Water is discharged to atmosphere so outlet boundary condition is specified pressure too. Ambient temperature is about 25 o C and saturated pressure of water in this temperature is -97 kPa (gage pressure). Other boundaries are considered as wall with no slip condition. The boundary value for VFRC is specified as 1 at inlet boundary, i.e. inlet section of the tunnel is filled with water. In first step, numerical solution is done to predict the zone in which cavitaion starts and potential zones for cavitation can be observed by studying pressure and VFRC contours shown in Fig. 3.
In the next step an aerator is placed near the predicted place for cavitation inception. The pressure and position of aerator is determined using pressure contours in Fig. 3. Pressure and VFRC contours (Fig. 4) show that minimum pressure in this condition is greater than the local vapor pressure of water (-97 kPa) and cavitation will not occur.

D. Before Aeration
Figs. 3 show pressure and VFRC contours for eight positions of discharge gate when there is no aerator. In this figure minimum pressure in pressure contours is between -140.147kPa to -14.791kPa, so in all cases the pressure is less than -97 kPa and the darkest blue region in pressure contour is potential zone for cavitation inception, except in the last case where the minimum pressure is more than -97kPa and there is no cavitation. VFRC contours in the Fig. 3 show the cells which contain both phases.

E. After Aeration
In this step, an aerator is placed near the predicted place where cavitation happens, so a vent is improvised after and near the gate for entrance of the air. The pressure of injection and exact point of vent is determined using the data shown in Fig. 3. Aeration pressure should be great enough to increase total pressure in a way that minimum pressure in Fig. 3 (-140.147 kPa) exceeds saturated pressure (-97 kPa). So minimum pressure for air injection should be 43.147kPa, but with this pressure, minimum pressure does not exceeds saturated pressure of water in the flow. So by using trial and error final pressure for air injection is attained 600kPa.
The location of the vent is shown in Fig. 1. The gate can be larger or smaller, but it would lead to use more powerful equipments to increase pressure and air flow rate due to changing of vent size. Several parameters affect the length of aeration [10] and exact determination of this length should be done using experiments, but here an initial value is considered and then finalized it by changing aeration pressure simultaneously.
Pressure and VFRC contours for eight positions of discharge gate, considering aerator, are shown in Figs. 4. It can be seen that minimum pressure for all cases is greater than saturated pressure and so cavitation will not occur. VFRC contours also confirm this conclusion.  Figure 4.
Pressure and VFRC contours near the discharge gate in eight positions for gate opening with aeration.

Conclusion
Studying the results and pressure contours, it is concluded that aerating is a useful and practical method used to exclude cavitation in hydraulic systems like gated tunnels of dams. Numerical simulation can effectively be used to determine the position of cavitation inception and also the position and pressure of aeration with high precision and costs less in comparison with experimental methods. VOF method can be effectively useful to study bubbly flows and VFRC contours, as results of this method, are helpful to determine potential zones for cavitation inception in the domain and flow field condition after aeration.

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Mechanical and Aerospace Engineering, ICMAE2011