Effect of the Vortex on the Movement Law of Sand Particles in the Hump Region of Pump-Turbine

Abstract

The pump turbine, as the core equipment of a pumped storage power plant, is most likely to operate in the hump zone between condition changes, which has a great impact on the stable operation of the power plant, and the high sedimentation of a natural river will lead to wear and tear in the overflow components of the equipment. Therefore, this paper is based on the Euler–Lagrange model, and seeks to investigate the distribution of vortices in the hump zone of the pump turbine and its effect on the movement of sand particles. The study shows that as the flow rate increases, the strip vortex in the straight cone section of the draft tube becomes elongated, and the cluster vortex in the elbow tube section gradually decreases. The strip vortex encourages the sand particles to move along its surface, while the cluster vortex hinders the movement of the sand particles. The accumulation areas of the sand particles in the straight cone section and the elbow tube section increase axially and laterally, respectively. The blade vortex in the runner gradually occupies the flow channel as the flow rate increases, and the blade vortex near the pressure surface encourages the sand particles to move towards the suction surface, resulting in the serious accumulation of sand particles on the suction surface. As the flow rate increases, the number of blades where sand particles accumulate increases and the accumulation area moves towards the cover plate and the outlet. The flow separation vortex in the double-row cascade decreases as the flow rate increases, which drives the sand movement in the middle and lower sections of the vanes. The area of sand accumulation in the stay vane decreases with increasing flow rate, but the area of sand accumulation between the guide vanes increases and then decreases. The vortex on the wall surface of the volute gradually decreases with the flow rate, and the vortex zone at the outlet first decreases, then disappears, and finally reappears. The vortex at the wall surface suppresses the sand movement, and its sand accumulation area changes from elongated to lumpy and finally to elongated due to the increase in flow. The results of the study provide an important theoretical reference for reducing the wear of pump turbine overflow components.

1. Introduction

In various hydroelectric power plants, the problem of sediment abrasion has always been a difficult engineering problem that disrupts the stable operation of hydroelectric power plant units, which is caused by hydraulic design, changes in operating conditions, the processing quality of overflow components, and so on. Therefore, a large number of scholars have carried out in-depth research on solid–liquid two-phase flow in the hydropower turbine [1,2] and a large number of results have been obtained, but the problem of sediment abrasion has never been completely solved. However, with the development of pumped storage technology [3,4,5], there are more and more pumped storage units on natural rivers, which also leads to the problem of sediment abrasion on the overflow parts of pump turbines, so it is necessary not only to seek a solution to the problem of sediment abrasion on conventional hydropower units, but also to carry out a large amount of research work on the mechanism of solid–liquid two-phase flow in the pump.

At present, the research on the flow mechanism in the pump turbine mainly focuses on the operational stability, and Lai et al. [6] found that the flow in the bladeless zone and the runner is unstable under different operating conditions, while the pulsation amplitude is large. Hu et al. [7] found that the hump characteristic occurs in the pump turbine under pumping conditions, which leads to the unstable operation of the unit. Some scholars have studied the mechanism of hump zone in depth and found that the flow in the hump zone of the pump turbine is often related to vortex and cavitation [8,9,10]. For example, Liu et al. [11] found that the vane stall is a necessary condition for the formation of the hump based on the quantitative identification method of the second-order derivatives of the external features. Liu et al. [12] clarified the reasons for the appearance of the double humps through the results of the PIV test, and found that the emergence of the first hump is related to the vortex in the vaneless zone and the strong pressure pulsation, and the emergence of the second hump is associated with the vortex shedding of the leading edge of the runner blade. Li et al. [13] selected the working point where cavitation just happened to occur in order to investigate the cavitation and hump region. The hump characteristics are related to the cavitation erosion on the front edge of the runner blade suction. There are also scholars who found that the hump characteristic has a hysteresis effect [14], such as Li et al. [15], who, through the pressure fluctuation test of the pump turbine under different guide vane openings, found that low-frequency pressure pulsation is related to high hydraulic loss. They also clarified that the hysteresis of the hump area is related to head reduction and loss increase.

In summary, the research on the internal flow mechanism of a pump turbine mainly focuses on the pure water condition, and there are few studies on multiphase flow, while in multiphase flow research, scholars mainly use numerical simulation and experimental research methods [16,17,18]. As regards numerical simulation, scholars used the Lagrangian model to track the trajectory of sand particles [19,20,21]. For example, Hong et al. [22] analyzed the effect of particle size on the wear of overflow components of two-stage small mining pumps under three flow conditions, and found that the surface wear of the first-stage guide vane was the largest under the three flows when small particles were conveyed by the particle movement trajectory and wear rate, the surface wear of the first-stage guide vane was small under the high flow when large particles were conveyed, and the surface wear of the second-stage guide vane was the smallest under the other flows. Guo et al. [23] determined the wear of the flow parts within a multiphase mixing pump under the main factors leading to the wear of the flow parts at different particle sizes; the results show that the severity and degree of wear in the impeller area is more significant, and the particle Reynolds number on the pressure surface of the vanes is the main factor affecting the wear, while on the suction surface the concentration of sand particles plays a dominant role in the wear.

Therefore, this paper will make use of multi-phase flow, especially the solid–liquid two-phase flow research method, to study the solid–liquid two-phase flow in the water pump turbine, focusing on the investigation of the influence law of the vortex within the overflow components on the sand grain movement, to provide a theoretical basis and research direction for the study of sediment abrasion in the water pump turbine.

2. Research Object

The object of this paper is a mixed-flow water pump turbine, and the computational domain includes the volute, stay vane, guide vane, runner and draft tube. The geometric model is shown in Figure 1, and the basic parameters of this model pump turbine are shown in

Table 1. Basic parameters of pump turbine.

Parameter and Unit	Symbolic	Value
Runner inlet dia. (mm)	D1	470
Runner outlet dia. (mm)	D2	300
Rated rotational speed (rpm)	Nd	1300
Runner blade number (-)	ZR	9
Stay vane number (-)	ZR	20
Guide vane number (-)	ZG	20

below.

3. Research Methods

3.1. Theoretical Approach

3.1.1. SST k-ω Model

The SST k-ω model combines the advantages of the k-ω model and the k-ε model, which not only accounts for turbulent shear stresses, but also avoids the extreme prediction of eddy viscosity coefficients. Its model expression is given as:

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho k\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho k{u}_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left({\mathrm{\Gamma }}_k{\displaystyle \frac{\partial k}{\partial x_j}}\right)+\overline{G_k}-Y_k+S_k$$

(1)

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho \omega \right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho \omega u_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left({\mathrm{\Gamma }}_{\omega }{\displaystyle \frac{\partial \omega }{\partial x_j}}\right)+G_{\omega }-Y_{\omega }+D_{\omega }+S_{\omega }$$

(2)

where $\rho$ is liquid density; k is the turbulent kinetic energy; ω is the turbulent dissipation rate; ${\mathrm{\Gamma }}_k$ and ${\mathrm{\Gamma }}_{\omega }$ are the effective diffusion terms for turbulent kinetic energy and turbulent dissipation rate; $G_k$ and $G_{\omega }$ are the generating terms for turbulent kinetic energy and turbulent dissipation rate; $Y_k$ and $Y_{\omega }$ are the dissipating terms for turbulent kinetic energy and turbulent dissipation rate; Dω is the orthogonal diffusion term; $S_k$ and $S_{\omega }$ are the source terms for turbulent kinetic energy and turbulent dissipation rate.

3.1.2. Lagrangian Model

Discrete phase particles are solved using the Lagrangian method (i.e., orbital tracking method), which seeks to obtain the trajectories of the particles by integrating their momentum equations. The momentum equation of the particle can be directly given by Newton’s second law.

$$m_p{\displaystyle \frac{d{v}_p}{dt}}=F_d+F_g+F_b+F_v+F_p+F_x$$

(3)

where $t$ is time; $m_p$ is particle mass; $v_p$ is particle velocity; $F_b$ is buoyancy; $F_p$ is the pressure gradient force; $F_x$ is the sum of other external forces; $F_d$ is drag force,

$$F_d={\displaystyle \frac{1}{2}}{C}_d{\rho }_w\left(u-u_p\right)\left|u-u_p\right|r_p^2$$

(4)

where $r_p$ is the particle radius; $C_d$ is the drag coefficient; $u$ and $u_p$ are the liquid velocity and particle velocity; ${\rho }_w$ is the liquid density.

$F_g$ is gravity,

$$F_g={\displaystyle \frac{4}{3}}{\rho }_p{r}_p^{3}g$$

(5)

where: ${\rho }_p$ is the density of the particle; $g$ is the acceleration due to gravity.

$F_v$ is virtual force,

$$F_v={\displaystyle \frac{2}{3}}{\rho }_w{r}_p^2{\displaystyle \frac{d}{dt}}\left(u-u_p\right)$$

(6)

3.1.3. Omega Vortex Identification Method

In 2016, Liu proposed the Ω vortex identification criterion via a new mathematical decomposition of the vortex vector. The Ω criterion accurately identifies both the rotational strength and the axis of rotation, can show both strong and weak vortex structures, and has a small dependence on the threshold value, so it can reasonably characterize the rotation of a localized fluid, providing a new and powerful tool for vortex dynamics and turbulence studies. The criterion is expressed as the magnitude of the rotation vector as a percentage of the total vortex magnitude, i.e.,

$$\mathrm{\Omega }={\displaystyle \frac{{‖B‖}_F^2}{{‖A‖}_F^2+{‖B‖}_F^2+\epsilon }}$$

(7)

where $A$ is the symmetric tensor,

$$A={\displaystyle \frac{1}{2}}(\nabla v+\nabla v^T)=\left[\begin{array}{ccc}{\displaystyle \frac{\partial u}{\partial x}}& {\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial v}{\partial x}})& {\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial u}{\partial z}}+{\displaystyle \frac{\partial w}{\partial x}})\\ {\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial v}{\partial x}})& {\displaystyle \frac{\partial v}{\partial y}}& {\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial w}{\partial y}}+{\displaystyle \frac{\partial v}{\partial z}})\\ {\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial u}{\partial z}}+{\displaystyle \frac{\partial w}{\partial x}})& {\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial w}{\partial y}}+{\displaystyle \frac{\partial v}{\partial z}})& {\displaystyle \frac{\partial w}{\partial z}}\end{array}\right]$$

(8)

$B$ is the antisymmetric tensor,

$$B={\displaystyle \frac{1}{2}}(\nabla v-\nabla v^T)=\left[\begin{array}{ccc}0& {\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial u}{\partial y}}-{\displaystyle \frac{\partial v}{\partial x}})& {\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial u}{\partial z}}-{\displaystyle \frac{\partial w}{\partial x}})\\ -{\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial u}{\partial y}}-{\displaystyle \frac{\partial v}{\partial x}})& 0& {\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial v}{\partial z}}-{\displaystyle \frac{\partial w}{\partial y}})\\ -{\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial u}{\partial z}}-{\displaystyle \frac{\partial w}{\partial x}})& -{\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial v}{\partial z}}-{\displaystyle \frac{\partial w}{\partial y}})& 0\end{array}\right]$$

(9)

$\epsilon$ is set to prevent the denominator from appearing to be zero,

$$\epsilon =0.001\times \left({‖B‖}_F^2-{‖A‖}_F^2\right)$$

(10)

$\mathrm{\Omega }$ takes values in the range [0, 1]. When it tends to 1, it means that the fluid undergoes rigid body rotation. After much research by scholars, 0.52 has been proposed as the recommended value.

3.2. Numerical Methods

3.2.1. Grid Division and Independence Verification

In this paper, ICEM 16.0 and Turbogrid 16.0 software are used to mesh the unit. Due to the complex structure of the volute, unstructured tetrahedral mesh is used in this paper, and hexahedral mesh is used for the rest. In order to better capture the near-wall flow and ensure that the Y+ value can meet the requirements of the turbulence model, the boundary layer of the runner blade and the double-row cascade blade is encrypted, as shown in Figure 2. Figure 3 shows the Y+ value of the runner blade, and the maximum value of Y+ in the runner is 10, while the average value is about 7, which is much lower than the Y+ value required by the SST k-ω model. Figure 4 shows the head change of the pump turbine under different grid numbers, which tends to stabilize with the increase in the number of grids. The head changes of scheme 4 and 5 are small, indicating that the simulation is no longer sensitive to the number of grids, but the denser grids will lead to the unnecessary wastage of computational resources. Therefore, considering the computational hardware configuration and computation time, a grid scheme of 9.8 million is finally selected to carry out the subsequent study.

3.2.2. Boundary Condition Setting

This paper numerically calculates the flow field inside the pump turbine based on the Euler–Lagrange model, with the liquid phase as the continuous phase, the solid phase set as the particulate discrete phase, the particulate coupling as the Two-way Coupling, and the interphase traction model as the Schiller–Naumann model, which includes drag, gravity and virtual mass forces, as well as turbulent diffusion effects. The algebraic equation iterative calculation adopts the mean square error residual value, and the convergence accuracy is set to 10⁻⁶.

Velocity inlet is used for the inlet boundary, the outlet boundary averages the static pressure outlet, a no-slip wall is used for the wall surface, and the frozen rotor is used to connect the dynamic and static interfaces between the runner and the guide vane. The sand particles are injected uniformly from the inlet boundary, and the flow velocity is consistent with the velocity of the liquid phase. The sand density is 2650 kg/m³, and the diameter of the sand particles is 1 mm.

3.3. Experimental Investigation

The numerical simulation results of the reversible pump turbine under the pure water condition with three different guide vane openings (a = 21 mm, a = 27 mm, a = 33 mm) are compared and analyzed with the experimental results, as shown in Figure 5. From Figure 5, we see that the model test data and simulation data under three kinds of active guide vane openings are in good agreement; in particular, in the first and fourth quadrants, the “S” characteristic curve of the pump turbine can be observed. In the third quadrant (i.e., pump condition) with a large opening (a = 33 mm), the agreement is poor, and the agreement of the pump condition in the remaining two openings is high, which is because the loss factor is not considered in the numerical calculation process, and the loss is more serious in the large opening. There are three openings in the full characteristic curve in the first, third, and fourth quadrants of the “S” under pumping conditions, with test and simulation error rates of 5% or less. Overall, the selected turbulence model and mesh quality can meet the accuracy requirements.

4. Results

4.1. Relationship Between Vortex and Sand Distribution in the Draft Tube

Figure 6 shows the distribution of vortices and sand trajectory in the draft tube at different flow rates. It can be observed that the vortex in the draft tube mainly exists in the form of vortex bands on the near wall surface of the elbow section of the straight cone, and in the form of a cluster at the junction of the diffusion section and the elbow section. As the flow rate increases, the radius of the vortex band in the draft tube decreases, and the external manifestation is that the shape of the vortex band changes from thick and long to slender and long; the cluster vortex generated by the accumulation of sand particles in the bypass flow area gradually decreases.

Figure 7 demonstrates the volume fraction of sand particles in the draft tube at different flow rates. Observation shows that sand particles mainly accumulate on the walls of the elbow tube section and the straight cone section. From Figure 8a, we see that, after entering the draft tube with the fluid, some of the sand particles hit the elbow section and the velocity becomes 0 m/s, thus accumulating in the elbow section. There are some sand particles after the elbow pipe section due to the reduction in cross-sectional area, and velocity increases, but this is also due to the effect of the centrifugal force on the wall of the straight cone section. Observing the circled portion of Figure 7 shows that, as the flow increases, the sand accumulation in the straight cone section expands axially upwards, the large sand accumulation in the elbow tube section expands laterally, and the sand accumulation in the diffusion sections of the draft tube gradually decreases. A cursory examination of Figure 6 and Figure 7 reveals an intrinsic relationship between the sand distribution and the vortex distribution.

From the previous paragraph, we can infer an intrinsic relationship between the vortex and the sand distribution, and this paragraph further analyzes the relationship between the two in depth from the perspective of sand movement. Figure 8 shows the trajectory of sand particles and vortex distribution in the draft tube. Fluid-wrapped sand movement includes the movement to the elbow tube section of the sand particles due to the reduction in velocity accumulation in the resulting vortex A; in turn, the vortex impedes the subsequent movement of sand particles to the exit of the draft tube, so that the elbow section shows a larger volume fraction of sand particles (A1 region). After entering the straight cone section, the sand particles are affected by vortex B, which consists of two counter-rotating strip vortices, and spiral along the outer surface of the vortex band towards the draft tube outlet, as a result of which some of the sand particles are wrestled by the counter-rotating vortex band onto the wall of the straight cone section (area A), and thus the sand particles pile up in the middle of the two vortex bands and the volume fraction is increased.

Figure 9 shows the effects of sand velocity and vorticity on the trajectory of sand particles in the draft tube at different flow rates (sand velocity on the left, vorticity on the right, same below). It is found that there is a large range of low peak change area for sand velocity under a small flow rate, which gradually decreases with the increase in flow rate, further indicating that the accumulation area and the volume fraction decrease after the sand velocity decreases. Combining the velocity and vortex curves, it is found that the region of low peak sand velocity corresponds to the region of high vorticity, mirroring the A1 region.

4.2. Relationship Between Vortex and Sand Distribution in the Runner

Figure 10 shows the distribution of vortices inside the runner at different flow rates. Observation shows that there is a blade vortex, which is close to the pressure surface and trailing edge shedding vortex in the runner. As the flow rate increases, the blade vortex inside the runner gradually occupies the entire flow channel, and the trailing edge shedding vortex decreases.

Figure 11 shows the sand volume fraction on the runner blades at different flow rates. There is no accumulation of sand particles on the pressure surface of the runner, and sand accumulation occurs only at the blade inlet under the two high-flow conditions. This is because the small flow rate of sand particles impacts the blade speed to a small extent, which means the strength is small, and its movement easily changes, meaning it does not easily accumulate in the blade inlet. When the flow rate increases, sand particles with high impact velocity will cause wear and tear on the blade inlet, resulting in sand accumulation. However, the sand accumulation on the suction surface is more serious and mainly accumulates in the blade inlet. In the small flow rate on the suction surface, sand accumulates near the runner under the cover plate. As the flow rate increases, the number of blades with sand accumulation gradually increases, and the area with serious accumulation gradually moves towards the runner outlet and the upper cover plate.

Figure 12 shows the sand trajectory in the runner at different flow rates. It is found that the sand particles are not evenly distributed in the runner inlet, but tend to flow on one side. This is because the trajectory of sand flowing out of the draft tube is biased towards the back side of the wall, which makes the trajectory inside the runner biased towards the upper part. At the same time, the sand movement inside the runner is close to the suction surface of the blades, because the blade vortex between the blades affects the sand movement. When the flow rate increases, the sand particles inside the runner gradually move to the lower right half. Because of the increase in sand particles flowing into the runner under high flow rate, there is a collision between sand particles and sand particles at the inlet, which makes some sand particles flow into the lower half of the runner. At the same time, the number of sand particles in the single flow channel increases, the speed of sand particles increases, and the trajectory of sand particles is gradually dense. As a result, the accumulation of sand particles on the runner blades intensifies under high flow rate.

Figure 13 shows the vortex distribution and sand trajectory in the runner, and it is observed that the pressure difference between the pressure surface of the blades and the suction surface forms a block-shaped vortex close to the pressure surface of the blades, which leads to the movement of sand close to the suction surface in the runner, and further leads to an increase in the volume fraction of the sand particles on the suction surface (area A). The sand particles in the B and C regions are affected by the blade vortex and move close to the suction surface and the upper cover plate. However, combined with the sand volume fraction, it is found that only the sand accumulation phenomenon exists in the B region. This is because the result of sand accumulation is affected by sand velocity and impact angle. The sand velocity in the C region is lower than that in the B region, which leads to a decrease in the sand’s impact on the blades in the C region and a decrease in sand accumulation on the blades.

Figure 14 shows the sand velocity and vorticity on the sand trajectory in the runner at different flow rates, and

Table 2. Average velocity and average vorticity on the sand trajectory in the runner at different flow rates.

	308 kg/s	347 kg/s	407 kg/s	426 kg/s
V (m/s)	14.04	14.15	14.30	14.70
ω (s−1)	3214.9	2143.09	3650.29	3770.5

shows the average velocity and average vorticity on the sand trajectory in the runner at different flow rates. It is found that the variation range of sand velocity is concentrated between 10 m/s and 20 m/s and the average value is concentrated at 14 m/s. With the increase in flow rate, the average velocity value shows a small increase, but the average vorticity decreases first and then increases. According to Figure 10, it is found that the distribution of vortex is less under the flow rate of 347 kg/s, and the blade vortex gradually occupies the inner part of the flow channel under a high flow rate, which increases the vorticity. Combining the sand velocity and the vorticity, it is found that the relationship between the two is still dominated by the high vorticity corresponding to the average velocity (corresponding to the B region in Figure 13).

4.3. Relationship Between Vortex and Sand Distribution in the Double-Row Cascade

Figure 15 shows the vortex distribution in the double-row cascade at different flow rates. It can be observed that there is flow separation vortex between the vanes and the vortex attached to the vanes in the double-row cascade. From the velocity streamlines, the flow separation vortex is caused by the low-speed spiral motion or even backflow between the blades after the fluid impacts the vanes. From the circled section, with the increase in flow rate, the flow line gradually stabilizes and the flow separation vortex in the double-row cascade decreases, but the attached vortex on the vanes does not change significantly.

Figure 16 illustrates the volume fraction of sand in the double-row cascade at different flow rates. It is observed that sand particles are accumulated on the vanes near the tongue of the corresponding volute. At a low flow rate, sand particles are mainly accumulated at the bottom of the guide vane and the middle and lower part of the stay vane. With the increase in flow rate, the high-volume fraction range of sand on the guide vane increases first and then decreases, and the high-volume fraction range of sand on the stay vane increases gradually.

Figure 17 shows the sand trajectory in the double-row cascade at different flow rates. It is observed that the sand particles move along the vane under a low flow rate, and when the flow rate reaches 407 kg/s, the sand particles near the tongue are affected by the vortex and the speed decreases abruptly, and the trajectory is shifted. Combined with Figure 18, it is found that sand particles in the guide vane are affected by the separation vortex of some sand particles moving along the vane convex surface (shown in Figure D), and some sand particles are affected by the vortex impacting the neighboring guide vane tails to make the speed decrease and then bypassing the stay vane separation vortex, then flowing into the volute along the convex surface of the vane (shown in Figure E). When the flow rate is further increased, the flow separation vortex decreases, and the sand particles continue to move along the vane.

Figure 19 shows the vortex distribution and sand trajectory in the double-row cascade. It is observed that the flow separation vortex between the vanes drives the sand particles to concentrate in the middle and lower sections of the vanes, as well as intensifying the sand movement close to the blade profiles. For example, it is found that the sand particles have a downward movement trend after passing through the separation vortex in the A1 region. The sand particles at the tail of the stay guide vane in the B area move downward; the sand particles in the C region first impact the head of the guide vane and then move along the convex surfaces of vanes. When passing through the interface of the double-row cascade, they are affected by the separation vortex and impact the head of the stay vane. The sand particles flow out along the concave surface of the vanes after bending.

Figure 20 shows the sand velocity and vorticity on the sand trajectory in the double-row cascade at different flow rates. It is observed that the sand velocity amplitude decreases gradually with the increase in flow rate, which indicates that the sand velocity changes steadily and the gradient changes little in the region of the double-row cascade at a high flow rate. Observation of the vorticity peaks revealed that most of them corresponded to regions of higher sand velocity (regions A and B in Figure 18).

4.4. Relationship Between Vortex and Sand Distribution in the Volute

Figure 21 shows the vortex distribution in the volute at different flow rates. It is observed that the vortex is mainly distributed near the wall surface of the volute, and its shape is affected by the section of the volute and is similar to it, while at the bottom of the extended section of the volute, there is a section of vortex zone extending from the tongue. As the flow rate increases, the vortex distribution on the extended section is dense, then sparse and dense again, while the head of the vortex at the bottom of the extended section first decreases, then disappears, and finally reappears again. The vortex on the circumferential wall of the volute and the vortex in the internal flow channel are gradually reduced by the increase in flow rate.

Figure 22 illustrates the volume fraction of sand particles in the volute at different flow rates. It can be observed that the sand accumulation is mainly concentrated on the wall surface of the volute, at the tongue, and at the outlet of the extension section. As the flow rate increases, the sand accumulation at the tongue gradually increases; the sand accumulation at the extension section gradually shifts to the top. The accumulation of sand particles on the wall surface is first in the form of thin strips, and then in the form of lumps when the accumulation range increases, and finally the accumulation of sand particles on the wall surface is concentrated in the middle of the wall and appears in the form of strips.

Figure 23 illustrates the sand trajectory in the volute at different flow rates. Under the influence of centrifugal force, the sand movement is biased to the outside of the wall. The sand particles in the extended section flow along the wall at a lower velocity, and the sand particles that exist directly in the stay vane impact the wall at a certain velocity. With the increase in flow rate, the velocity of sand particles at the tongue decreases, the number of sand particles flowing out from the stay vane increases, and the overall trajectory does not change.

Figure 24 illustrates the vortex distribution and sand trajectory in the volute. It is observed that the vortex distribution at the small section is on the lower side of the wall, and the vortex distribution at the large section is close to the inner side of the wall. Therefore, the sand particles are squeezed by the bottom vortex to bias the movement towards the upper part and influenced by the inner vortex to bias the movement towards the outer part. At the bottom of the extension section, there is a vortex extending from the tongue to the outlet, which directly affects the sand particles flowing out from the stay vane; sand particles can directly pass through the vortex with high radial velocity, but as the velocity decreases, the influence of the vortex on the sand particles increases, and the sand particles gradually flow out of the volute along the vortex.

Figure 25 shows the sand velocity and vorticity on the sand trajectory in the volute at different flow rates. It is observed that the sand velocity inside the volute oscillates between 4 m/s and 13 m/s under the first two flow conditions, and the amplitude decreases significantly with the increase in flow rate. At the same time, the vertical coordinate value of the vorticity on the right side can visualize the decreasing trend of vorticity with the flow rate change. Combining the two curves, it is found that the place with stronger vorticity corresponds to the region with low sand velocity, which is because the sand velocity inside the volute only reaches the maximum value at the tongue, and the distribution of vortex at this place is relatively small; in the rest of the wall of the volute, the velocity holds the average value of the lower region, and the distribution of the vortex is more affected by the flow and the structure.

5. Conclusions

(1)

The vortex in the draft tube mainly occurs in the form of vortex bands on the near wall surface of the straight cone elbow tube section, and in the form of a mass at the junction of the diffusion section and the elbow tube section. As the flow rate increases, the strip vortex becomes thinner and the cluster vortex gradually decreases. Sand particles accumulate mainly on the wall surface of the straight cone section and the elbow pipe section. As the flow rate increases, the sand accumulation in the straight cone section extends upwards, and the sand accumulation in the elbow pipe section extends laterally. Under these four flow conditions, the sand particles in the draft tube spiral along the outer surface of the two counter-rotating strip vortices and are thrown against the wall of the straight cone section; the mass vortex prevents the sand particles from moving to the outlet of the draft tube, resulting in a serious accumulation of sand particles on the wall of the straight cone section and the elbow tube section.

(2)

The vortices in the runner are mainly the blade vortex and trailing edge shedding vortex close to the pressure surface. As the flow rate increases, the blade vortex gradually occupies the flow path, and the trailing edge shedding vortex decreases. Sand particles mainly accumulate at the inlet of the suction surface of the runner. As the flow rate increases, the number of blades where sand particles accumulate increases and moves up the cover. Under these four flow conditions, the blade vortex close to the pressure surface has a greater influence on the sand movement, which encourages the sand particles to move close to the suction surface, which leads to a serious accumulation of sand particles on the suction surface.

(3)

The vortices in the double-row cascade are flow separation vortices and blade attachment vortices between the vanes. As the flow rate increases, the flow separation vortex decreases, and the blade attachment vortex does not change significantly. Sand particles were mainly accumulated at the bottom of the guide vane and the middle of the stay vane. As the flow rate increases, the sand accumulation between the stay vane increases, and the range of sand accumulation between the movable guide vanes first increases and then decreases. The separation vortex between the double-row cascade vanes has a greater influence on the sand movement under the four flow conditions, which encourages the sand movement to be concentrated in the middle and lower sections of the vanes and aggravates the movement of sand close to the blade profile.

(4)

The vortex in the volute is mainly distributed near the wall and in the form of a vortex band on the outlet extension. As the flow rate increases, the vortex at the wall gradually decreases, and the vortex zone at the outlet first decreases, then disappears and finally reappears. Sand particles accumulate on the wall surface of the volute, on the tongue and at the outlet of the extension section. As the flow rate increases, the sand accumulation at the tongue intensifies, the position of the sand accumulation at the outlet moves upward, and the sand accumulation on the wall surface changes from elongated to lumpy to elongated. Under the four flow conditions, the sand particles in the volute are squeezed by the vortex at the bottom of the wall and move upward, the sand particles are moved outward by the vortex inside the wall, and the sand particles flow out of the volute along the outer surface of the vortex band at the outlet of the expansion section.