OPEN ACCESS
The existence of sediment in water causes excessive abrasion of turbine blades and affects the performance of the turbine. It is important to predict the influence of solid volume fractions on turbine performance. The solidliquid twophase turbulent flow in a Francis turbine was numerically simulated by establishing a mathematical model for the flow passage of the turbine on the basis of the timeaveraged basic equations and κε equations. The turbulent flow in this turbine was calculated on the design point in clear water, as well as in sandy water with the average solid volume fractions of 0.5 % and 5 % at the inlet of the spiral case. The pressure distributions and sand concentrations on the leading side and suction side of the runner blades, as well as the velocities, turbulent kinetic energies and their dissipation rates on the horizontal section of runner were compared under these three conditions to illustrate the influences of the solid volume fractions on the turbine performance. The results show that with increasing solid volume fraction, the pressure difference on the leading side and suction side of the blade increased; the sand erosion on the leading side was much worse than that on the suction side, especially at the outlet area near the runner band; the velocity changes of sand particles and water were small at the outlet area, and the velocity of sand particles became smaller than that of water; the turbulent kinetic energy and its dissipation rate were slightly affected by the solid volume fractions. All of these factors lead to serious cavitation on the suction side and sand erosion on the leading side. The joint effect of cavitation and sand erosion will be aggravated with increasing solid volume fractions which will further reduce the performance of the turbine.
Francis turbine, Pressure distribution, Solid volume fraction, Turbulent flow, Velocity distribution.
In recent years, the solid volume fraction of most rivers has become higher and higher due to the deterioration of the ecological environment. Hence, it has been necessary to use water resources with large solid volume fractions to generate power. The presence of sand particles in water often causes excessive abrasion of turbine components exposed to the flow [1]. In particular, the flow components of Francis turbines are the most vulnerable to abrasive damage [2]. The abrasion of these components may result in considerable reduction of the performance of the entire power plant [3]. It can also lead to a reduced service life which in turn reduces the time between overhauls. The higher frequency of overhaul also decreases the total energy production. Most of the engineering designs are based on the design theory in clear water, and then corrected to the sandy water conditions. However, the performance of the turbine cannot be guaranteed in water with large solid volume fractions. A prototype experiment can solve this problem but with a great cost. Therefore, the numerical simulation [4] of solidliquid twophase flow in Francis turbines has become an important basis for the study of sand abrasion of the turbine [5, 6]. This method has been proved reliable and accurate [7]. Due to the complicated flow state in a Francis turbine, especially the multiphase turbulent flow, it is necessary to carry out indepth study of the flow in Francis turbines to find out what is going on inside the turbine, and to predict the performance of the turbine running on rivers with large solid volume fractions. Presently, the CFD method has been used to study the threedimensional turbulent flow field in hydraulic machinery, and scholars [8, 9] have used this method to calculate the solid phase flow in hydraulic machinery. Takagi et al. [10] reported on the hydraulic performance tests on a Francis turbine model with sediment laden flow, conducted in Japan, and showed that the turbine’s best efficiency decreased in direct proportion to the increase in solids concentration. Keck et al. [11] presented a study on the utilization of the CFD method to predict the erosion pattern in a hydraulic turbine and compared it with field measurements of the erosion. P. J. Dunstan [12] studied the enhancement of cavitation by silt erosion. Schilling R. [13] simulated the twophase flow in centrifugal pump impellers. S. Chitrakar [14] studied the simultaneous effects of secondary flow and sediment erosion in Francis turbines. Eltvik M. et al. [15] investigated the relationships between sediment erosion and the operating conditions of the turbine, and found that the erosion process was strongly dependent on the operating conditions of the turbines. In this work, a mathematical model was established for the entire flow passage of a Francis turbine, and then the flow conditions in the turbine runner were simulated to understand the influences of the solid volume fraction on turbine performance.
The following assumptions have been made in this study:
The usual approach for describing turbulence transport in the flow is to separate each of the instantaneous variables into a mean part and a fluctuating part. Thus, the timeaveraged continuity equations and momentum equations for solidliquid twophase flows can be written as [16]:
Timeaveraged liquid phase continuity equation:
$\frac{\partial \varphi_{f}}{\partial t}+\frac{\partial}{\partial x_{i}}\left(\varphi_{f} V_{f i}\frac{v_{t}}{\sigma_{\varphi f}} \frac{\partial \varphi_{f}}{\partial x_{i}}\right)=0$(1)
Timeaveraged solid phase continuity equation:
$\frac{\partial \varphi_{s}}{\partial t}+\frac{\partial}{\partial x_{i}}\left(\varphi_{s} V_{s i}\frac{v_{t}}{\sigma_{\varphi s}} \frac{\partial \varphi_{s}}{\partial x_{i}}\right)=0$(2)
Timeaveraged liquid phase momentum equation:
$\frac{\partial }{\partial t}\left( {{\varphi }_{f}}{{V}_{fi}} \right)+\frac{\partial }{\partial {{x}_{j}}}({{\varphi }_{f}}{{V}_{fi}}{{V}_{fj}})$$=\frac{\partial }{\partial {{x}_{j}}}\left[ \frac{{{\nu }_{t}}}{{{\sigma }_{\varphi f}}}\left( {{V}_{fi}}\frac{\partial {{\varphi }_{f}}}{\partial {{x}_{j}}}+{{V}_{fj}}\frac{\partial {{\varphi }_{f}}}{\partial {{x}_{i}}} \right){{\varphi }_{f}}\overline{{{\upsilon }_{fi}}{{\upsilon }_{fj}}} \right]$$\frac{1}{{{\rho }_{f}}}\left( {{\varphi }_{f}}\frac{\partial p}{\partial {{x}_{i}}}+\overline{{{\phi }_{f}}\frac{\partial p'}{\partial {{x}_{i}}}} \right)+{{\nu }_{f}}\frac{\partial }{\partial {{x}_{j}}}\left[ {{\varphi }_{f}}\left( \frac{\partial {{V}_{fi}}}{\partial {{x}_{j}}}+\frac{\partial {{V}_{fj}}}{\partial {{x}_{i}}} \right) \right]$$\frac{B}{{{\rho }_{f}}}\left[ {{\varphi }_{f}}{{\varphi }_{s}}\left( {{V}_{fi}}{{V}_{si}} \right)+\left( {{\varphi }_{s}}{{\varphi }_{f}} \right)\left( \frac{{{\nu }_{t}}}{{{\sigma }_{\varphi s}}}\frac{{{\nu }_{t}}}{{{\sigma }_{\varphi f}}} \right)\frac{\partial {{\varphi }_{f}}}{\partial {{x}_{i}}} \right]+{{\varphi }_{f}}{{g}_{i}}$ (3)
Timeaveraged solid phase momentum equation:
$\frac{\partial}{\partial t}\left(\varphi_{s} V_{s i}\right)+\frac{\partial}{\partial x_{j}}\left(\varphi_{s} V_{s i} V_{s j}\right)$$=\frac{\partial}{\partial x_{j}}\left[\frac{v_{t}}{\sigma_{\varphi s}}\left(V_{s i} \frac{\partial \varphi_{s}}{\partial x_{j}}+V_{s j} \frac{\partial \varphi_{s}}{\partial x_{i}}\right)\varphi_{s} \overline{v_{s i} v_{s j}}\right]$$\frac{1}{\rho_{s}}\left(\varphi_{s} \frac{\partial p}{\partial x_{i}}\overline{\varphi_{f}} \frac{\partial p^{\prime}}{\partial x_{i}}\right)+v_{s} \frac{\partial}{\partial x_{j}}\left[\varphi_{s}\left(\frac{\partial V_{s i}}{\partial x_{j}}+\frac{\partial V_{s j}}{\partial x_{i}}\right)\right]$$\frac{B}{\rho_{s}}\left[\phi_{f} \phi_{s}\left(V_{s i}V_{f i}\right)+\left(\phi_{f}\phi_{s}\right)\left(\frac{v_{t}}{\sigma_{\phi s}}\frac{v_{t}}{\sigma_{\phi f}}\right) \frac{\partial \phi_{s}}{\partial x_{i}}\right]+\phi_{s} g_{i}$(4)
The turbulent kinetic energy equation (k equation) for solidliquid twophase flow is:
$\frac{\partial }{\partial t}\left( {{\varphi }_{f}}k \right)+\frac{\partial }{\partial {{x}_{j}}}({{\varphi }_{f}}{{V}_{fj}}k)=\frac{\partial }{\partial {{x}_{j}}}\left( \frac{{{\nu }_{t}}}{{{\sigma }_{\varphi f}}}{{\varphi }_{f}}\frac{\partial k}{\partial {{x}_{j}}} \right){{\varphi }_{f}}\varepsilon +{{P}_{r}}+{{V}_{Df}}+{{A}_{Ds}}$ (5)
The turbulent energy dissipation rate equation (ε equation) for solidliquid twophase flow is:
$\frac{\partial}{\partial t}\left(\varphi_{f} \varepsilon\right)+\frac{\partial}{\partial x_{j}}\left(\varphi_{f} V_{f j} \varepsilon\right)$$=\frac{\partial}{\partial x_{j}}\left[\left(v_{f}+\frac{v_{t}}{\sigma_{\varepsilon}}\right) \phi_{f} \frac{\partial \varepsilon}{\partial x_{j}}\right]+\left[C_{\varepsilon 1} P_{r}\phi_{f}\left(C_{\varepsilon 2} \varepsilon+C_{\varepsilon 3} A_{D s}\right)\right] \frac{\varepsilon}{k}$(6)
where,
$\overline{\phi_{f}} \frac{\partial p^{\prime}}{\partial x_{i}}=\rho_{f} \frac{\partial}{\partial x_{i}}\left[\left(C_{1} k^{1 / 2}+C_{2} \frac{k^{3 / 2}}{\varepsilon} \frac{\partial V_{f i}}{\partial x_{k}}\right) \frac{\nu_{t}}{\sigma_{\phi f}} \frac{\partial \phi_{f}}{\partial x_{k}}\right]$$+\rho_{f} \frac{\varepsilon}{k}\left[C_{3} \overline{\phi_{f} v_{f i}}+C_{4}\left(\frac{1}{k} \overline{v_{f h} v_{j k}}+\frac{2}{3} \delta_{i k} \frac{v_{t}}{\sigma_{\varphi f}} \frac{\partial \varphi_{f}}{\partial x_{k}}\right)\right]$$\rho_{f}\left(C_{5} \frac{\partial V_{f i}}{\partial x_{k}}+C_{6} \frac{\partial V_{f k}}{\partial x_{i}}\right) \frac{v_{t}}{\sigma_{\varphi f}} \frac{\partial \varphi_{f}}{\partial x_{i}}$
$p_{r}=\frac{\partial}{\partial x_{j}}\left(\frac{v_{t}}{\sigma_{\phi f}} V_{f i} V_{f j} \frac{\partial \phi_{f}}{\partial x_{i}}V_{f i} \overline{\varphi_{f} v_{f i} v_{f j}}V_{f j} \overline{\varphi_{f} v_{f i}^{2}}\right)$$\frac{\partial}{\partial x_{j}}\left(\phi_{f} V_{f i}\right) \overline{v_{f i} v_{f j}}+\frac{1}{\rho_{f}} \frac{v_{t}}{\sigma_{\phi f}} \frac{\partial \phi_{f}}{\partial x_{j}} \frac{\partial p}{\partial x_{j}}$$C_{k} \frac{v_{t}}{\sigma_{\varphi f}} \frac{k}{\varepsilon}\left(V_{f i} \frac{\partial \varphi_{f}}{\partial x_{j}}+V_{f j} \frac{\partial \varphi_{f}}{\partial x_{i}}\right)$
$V_{D f}=v_{f}\left\{\phi_{f} \frac{\partial^{2} k}{\partial x_{j}^{2}}+\frac{\partial \phi_{f}}{\partial x_{j}}\left(\frac{\partial k}{\partial x_{j}}+\frac{\partial \overline{v_{f i} U_{f j}}}{\partial x_{i}}+v_{t} \frac{\partial^{2} V_{f i}}{\partial x_{j}^{2}}\right)\right.$$\frac{\partial}{\partial x_{j}}\left[\frac{v_{t}}{\sigma_{\varphi f}} \frac{\partial \varphi_{f}}{\partial x_{i}}\left(\frac{\partial V_{f i}}{\partial x_{j}}+\frac{\partial V_{f j}}{\partial x_{i}}\right)\right] \}$
$A_{D S}=\frac{B}{\rho_{f}}\left[\varphi_{f} \varphi_{s}\left(\overline{v_{f i} v_{f i}}\overline{v_{f i} v_{s i}}\right)+\frac{v_{t}}{\sigma_{\varphi f}}\left(\varphi_{f}\varphi_{s}\right)\left(V_{f i}V_{s i}\right) \frac{\partial \varphi_{f}}{\partial x_{i}}\right]\frac{v_{t}}{\sigma_{\phi f}} \frac{\partial \phi_{f}}{\partial x_{i}} g_{i}$
$\overline{{{\phi }_{f}}{{\upsilon }_{fi}}{{\upsilon }_{fj}}}={{C}_{k}}\frac{k}{\varepsilon }\left( \overline{{{\upsilon }_{fi}}{{\upsilon }_{fk}}}\frac{\partial }{\partial {{x}_{k}}}\left( \frac{{{\nu }_{t}}}{{{\sigma }_{\varphi f}}}\frac{\partial {{\varphi }_{f}}}{\partial {{x}_{j}}} \right)+\overline{{{\upsilon }_{fj}}{{\upsilon }_{fk}}}\frac{\partial }{\partial {{x}_{k}}}\left( \frac{{{\nu }_{t}}}{{{\sigma }_{\varphi f}}}\frac{\partial {{\varphi }_{f}}}{\partial {{x}_{i}}} \right) \right)$
In the above equations: subscripts f and s represent the liquid and the solid phases, respectively, and i, j, k tensors; t is the time; V_{i} the timeaveraged velocity components; $v_{i}$ the fluctuating velocity components; x_{i} the components of coordinate; P the timeaveraged pressure; $p^{\prime}$ the fluctuating pressure; $\phi$ the timeaveraged volume fraction (concentration); φ the fluctuating volume fraction; g_{i} the component of gravity acceleration in i direction; ρ the material density; ν the material kinematic viscousity; κ the turbulent kinetic energy of liquid phase where $\kappa=\overline{v_{f i} v_{f i}} / 2$; ε the turbulent energy dissipation rate of liquid phase; B the interphase friction coefficient and B=18(1+B_{0})ρ_{f}ν_{f}/d^{2}; and d the particle diameter of solid phase. B_{0} was introduced in order to consider the effects of the virtual mass force, Saffman force and Magus force etc., except the linear Stokes drag. C_{1}=1.0, C_{2}=1.0, C_{3}=4.3, C_{4}=3.2, C_{5}=0.8, C_{6}=0.2, $\sigma_{\mathcal{E}}$=1.3, $C_{\varepsilon 1}$=1.44, $C_{\varepsilon 2}$=1.92, $C_{\varepsilon 3}$=1.2, and C_{k}≈0.1.
The mean relation equations for the liquid and solid volume fractions are:
$\varphi_{f}+\varphi_{s}=1, \phi_{f}+\phi_{s}=0$(7)
$\nu_{t}$ is the kinematic eddy viscousity, and $v_{t}=C_{\mu} k^{2} / \varepsilon\left(C_{\mu} \approx 0.09\right)$.
$\mathcal{V}_{t}$ is the turbulent Schmidt number of $\phi$ and $\sigma_{\varphi f} \approx 1 . \sigma_{\varphi s}$ is given by Peskin [17], and $\sigma_{\varphi_{S}}=\left(1\frac{T_{L}^{2} \varepsilon}{15 v_{f}} \frac{6 A^{2}}{A+1}\right)^{1}$.
where, A is the ratio of particle response time $\tau_{p}$ [ρ_{s}d^{2}/(18ρ_{f}ν_{f})] to the fluid Largangian integral time scale $T_{L}\left(T_{L}=C_{T} k / \varepsilon, \quad C_{T} \approx 0.200.41\right)$.
The liquid phase Reynolds stress $\overline{v_{f i} v_{f j}}$ and the solid phase Reynolds stress $\overline{v_{s i} v_{s j}}$ are respectively calculated from:
$\overline{{{\upsilon }_{fi}}{{\upsilon }_{fj}}}=\frac{{{\nu }_{t}}}{{{\sigma }_{\varphi f}}}\left( \frac{\partial {{V}_{fi}}}{\partial {{x}_{j}}}+\frac{\partial {{V}_{fj}}}{\partial {{x}_{i}}} \right)+\frac{2}{3}{{\delta }_{ij}}\left( k+\frac{{{\nu }_{t}}}{{{\sigma }_{\varphi f}}}\frac{\partial {{V}_{fk}}}{\partial {{x}_{k}}} \right)$
$\overline{{{\upsilon }_{si}}{{\upsilon }_{sj}}}=\frac{{{\nu }_{t}}}{{{\sigma }_{\varphi s}}}\left( \frac{\partial {{V}_{si}}}{\partial {{x}_{j}}}+\frac{\partial {{V}_{sj}}}{\partial {{x}_{i}}} \right)+\frac{2}{3}{{\delta }_{ij}}\left( {{k}_{s}}+\frac{{{\nu }_{t}}}{{{\sigma }_{\varphi s}}}\frac{\partial {{V}_{sk}}}{\partial {{x}_{k}}} \right)$ (8)
where, $\delta_{i j}$ is Kronecker delta, k_{s} the turbulent kinetic energy of the solid phase, and $k_{s}=\overline{v_{s i} v_{s i}} / 2$. For large flow Reynolds numbers and short particle response time, k_{s} is approximated by k/(A+1). The correlation $\overline{v_{f i} v_{s j}}$is approximated by 2$k /(A+1)$.
This model has been used to simulate the solidliquid twophase turbulent flow in the runner of Francis turbines used in Jinping II Hydropower Station on Yalong River in China, and has been proved reliable [18].
3.1 Geometrical model and mesh generation
The turbine simulated was a vertical metal case Francis turbine. The design parameters and flow calculation parameters of this turbine are listed in Table 1.
Table 1. Design parameters of turbine and flow calculation parameters
Parameters 
Value 
Design head, H_{p} 
220 m 
Design flow, Q_{d} 
13.05 m3.s1 
Speed, n 
1000 r.min1 
Runner diameter, D_{1} 
1080 mm 
Number of runner blades, Z 
19 
Number of stay vanes, Z_{2} 
9 
Number of wicket gates, Z_{1} 
16 
Height of guide vane, b_{0} 
285 mm 
Diameter of distribution circle, D_{0} 
1400 mm 
Type of guide vane 
positive curvature guide vane 
Particle diameter 
0.1 mm 
Sand density 
2650 kg.m3 
Water density 
1000 kg.m3 
As the geometry of the turbine body is complicated, it was necessary to divide the body into several parts for solid modeling and mesh generation. Firstly, CAD technology and UG NX software were used in this study to establish the digital geometric models of all flow components of the turbine, as shown in Figure 1. Secondly, the ICEM CFD software was used for mesh generation, and the unstructured tetrahedron meshes were used. The function of Mesh Quality in the ICEM CFD software was used to check the quality of generated meshes (the mesh was qualified if the Quality value was more than 0.2), and the function of Smooth Mesh was used to optimize these meshes. Thirdly, the function of Merge in ICEM CFD was used to connect the meshes of all parts, and then the computational meshes of the whole flow passage were obtained, as shown in Figure 2. The total numbers of computational meshes generated for the whole flow passage of the turbine were 1,219,914, 1,917,481 and 2,193,457, and the later two met the independency requirement of less than 5 %. To save the calculation time, the 1,917,481 value was used. This included 479,516 meshes for the turbine case (including a part of the inlet pipe) with a minimum Quality value of 0.363; 323,396 meshes for the stay vane and wicket gate with a minimum Quality value of 0.291; 485,405 meshes for the runner with a minimum quality value of 0.334; and 629,164 meshes for the draft tube with a minimum quality value of 0.311. The Quality values of all meshes in the various components were more than 0.2, so their qualities met the requirements.
Figure 1. Whole flow passage model of the Francis turbine
Figure 2. Meshes of the whole flow passage
3.2 Numerical method
By comparing Francis turbines with high and medium specific speeds, the parts of the turbine which are severely worn were the outlet tip of the blade near the band and the inner face of the band [19]. So, the velocity distribution and pressure distribution on the runner blades were analyzed in emphasis. This study was based on the timeaveraged basic equations and kε equations. By using the AEA Technology in CFX software and SIMPLEC algorithm, the turbulent flow in the Francis turbine was calculated on the design point (Q_{d}=13.05 m^{3}/s) in clear water, as well as in sandy water with average solid volume fractions of 0.5 % and 5 % at the inlet of the spiral case with solidliquid twophase flow.
3.3 Boundary conditions
The velocity inlet was adopted. It was assumed that the inlet had uniform incoming flow and the inlet velocity was perpendicular to the inlet boundary surface. By using the inlet velocity, we could calculate the values of k and ε. The outlet was provided with the conditions of free development, that is to say, except the outlet pressure, the positive normal gradients of all flow variables were assumed to be zero. The velocity on the solid wall met with the noslip wall conditions, and the standard wall function was adopted for the near wall area. The impact of gravity to the flow field during the calculation was considered and the direction of gravity was reverse to the normal direction of the turbine outlet section.
4.1 In clear water
Figure 3 shows the distributions of pressure on the blade surfaces, as well as the distributions of pressure, velocity, turbulent kinetic energy and its dissipation rate on the horizontal section of the runner at 0.5 times the height of the blade inlet in clear water. It can be seen from Figure 3(a, c) that the pressure on the leading side of the blade gradually reduced from the blade inlet to the outlet, and a small area at the outlet edge near the runner band was of negative pressure. It can be seen from Figure 3(b, c) that the pressure distribution on the suction side of the blade had the same pattern as that of the leading side, but a large area at the outlet edge near the runner band was of negative pressure. The large negative pressure in the cavitating triangle on the suction side of the blade was consistent with the actual conditions. It can be seen from Figure 3(d) that the flow in the blade channel was smooth and stable. There was no secondary flow in the blade channel, no obvious impact on the runner inlet and no obvious flow separation at the outlet, which are consistent with the results from the design point. All of these prove the reliability of the model. It can be seen from Figure 3(e, f) that the turbulent kinetic energy and its dissipation rate near the blade surfaces were large, especially on the blade toe and heel.
Figure 3. Flow conditions in clear water
4.2 In sandy water
Figure 4 and Figure 5 show the distributions of pressure and sand concentration on the blade surface in average solid volume fractions of 0.5 % and 5 % at the case inlet, as well as the distributions of sand velocity, water velocity, turbulent kinetic energy and its dissipation rate on the horizontal section of the runner at 0.5 times the height of the blade inlet.
4.2.1 Pressure distribution on the blade surface
It can be seen from Figure 4(a, b) and Figure 5(a, b) that the changes of pressure distributions on the blade surface in sandy water and clear water were small in general, so the influence of solid volume fraction to pressure distribution was small. However, increasing solid volume fraction caused the pressure difference on the leading side and the suction side of the blade to increase, which aggravates the cavitation tendency.
Figure 4. Flow conditions in sandy water with the average solid volume fraction of 0.5%
4.2.2 Distribution of sand concentration on the blade surface
It can be seen from Figure 4(c, d) and Figure 5(c, d) that the sand concentration on the leading side of the blade was much higher than that on the suction side. On the leading side of the blade, the sand concentration gradually increased from the blade inlet to the outlet, and the sand concentrations in the area near the blade outlet and the runner band were the largest. On the suction side of the blade, the sand concentrations near the runner crown and runner band were small, and the concentration on the toe of the blade was large due to the impact of the sand particles. However, the changes of sand concentrations on the entire surface were small. This also indicates that the abrasion of the blade mainly occurred on the leading side, especially the area near the blade outlet and runner band. Therefore, both cavitation on the suction side and erosion on the leading side can make the outlet area of the blade near the runner band very vulnerable. The joint of effect of cavitation and erosion will reduce the turbine performance drastically.
Figure 5. Flow conditions in sandy water with the average solid volume fraction of 5%
With increasing solid volume fractions, the sand concentrations on the leading side and the suction side of the blade increased simultaneously. Furthermore, the variation trend of sand concentration distribution on the leading side of the blade was basically the same as that of the suction side of the blade. This means that the larger the solid volume fraction is, the worse the erosion of the entire leading side of the blade will be, starting from the outlet area near the runner band.
4.2.3 Velocity distribution in the blade channel
It can be seen from Figure 4(e, f) and Figure 5(e, f) that the velocities of sand particles and water at the blade outlet area were much higher than those at the blade inlet area, and the velocities of sand particles and water in the first half section of the blade near the blade surface were very small. The velocities of sand particles and water flow near the suction side of the blade were higher than those near the leading side of the blade due to the large specific gravity of sand particles under the effect of the Coriolis force. However, the velocity changes of sand particles and water at the outlet area near the leading side and the suction side of the blade were small, which made the sand erosion at the area near the blade outlet and runner band more serious.
The influence of solid volume fraction on the velocities of sand particles and water was small, and the variation trend of velocity distribution was basically consistent. However, the velocity of sand particles was slightly less than that of the water. With increasing solid volume fractions, the velocity of sand particles fell even lower than the velocity of water, which will lead to flow separation and reduce the turbine performance.
4.2.4 Turbulent kinetic energy and its dissipation rate distribution in the runner
It can be seen from Figure 3(e, f), Figure 4(g, h) and Figure 5(g, h) that the influence of solid volume fraction on the turbulent kinetic energy and its dissipation rate distribution was small, which made it basically the same in clear water and sandy water.
By establishing a mathematical model, the solidliquid twophase flow conditions in a Francis turbine was simulated to study the influence of solid volume fractions on turbine performance. The following results were obtained.
(1) The influence of solid volume fractions on the pressure distributions on the blade surfaces was small, but the pressure differences on the leading side and suction side increased with solid volume factions, which will increase the cavitation tendency and reduce the turbine’s performance.
(2) The concentration of sand on the leading side was much higher than that of the suction side, so the leading side endures most of the sand erosion. On the leading side, the concentration of sand gradually increased from the inlet to the outlet, so the concentration in the area near the blade outlet and runner band was the largest. This means that the outlet area of the blade on the leading side near the runner band will be eroded first. On the suction side, the change of sand concentration was small. The distribution of sand concentration increased with the solid volume fraction both on the leading side and suction side, so the larger the solid volume fraction, the greater the erosion, and the poorer the stability of the turbine.
(3) The velocities of sand particles and water near the suction side of the blade were higher than those near the leading side of the blade. With the increasing solid volume fraction, the velocities of sand particles and water changed slightly, but the velocity of the sand particles was less than that of the pure water, therefore higher volume fractions inhibit the transfer of energy and lower the efficiency of the turbine.
(4) The turbulent kinetic energy and its dissipation rate barely changed with the increasing solid volume fraction, so the solid volume fraction does not affect the turbulent intensity.
This work was funded by the Open Research Subject of Key Laboratory of Fluid and Power Machinery (Xihua University), Ministry of Education (Grant No. szjj2015026).
B 
dimensionless interphase friction coefficient 

d 
particle diameter of solid phase, mm 

g 
gravitational acceleration, m.s2 

P p 
time averaged pressure, Pa fluctuating pressure, Pa 

V 
time averaged velocity, m.s1 

v 
fluctuating velocity, m.s1 

Greek symbols 

δ 
dimensionless Kronecker delta 

ε 
dimensionless turbulent energy dissipation rate of liquid phase 

κ 
turbulent kinetic energy of the liquid phase, J 

ν 
material kinematic viscousity, kg. m1.s1 

ρ 
material density, kg.m3 

σ 
dimensionless turbulent Schmidt number 

Τ 
fluid Largangian integral time, s 

τ 
particle response time, s 

$\phi$ 
dimensionless solid volume fraction 

φ 
dimensionless time averaged volume fraction 

Subscripts 


f 
liquid phase 

i 
tensor 

j 
tensor 

k 
tensor 

s 
solid phase 

t 
time 
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