# Asymmetric Solid–Liquid Two-Phase Flow around a NACA0012 Cascade in Sediment-Laden Flow

^{*}

## Abstract

**:**

_{50}of 82.7 μm, 65.9 μm, and 31.8 μm near the wall of cascades at an impact angle of 10° in a Venturi circuit. The flow characteristics and velocity slip between solid and liquid phases, as well as the effects of particle size and the Reynolds number on velocity slip, were analyzed. The results showed that: (1) the flow is asymmetrically distributed in front of the cascade’s leading edge at a 10° impact angle, and there is strong velocity slip between solid and liquid phases; (2) under the influence of particle inertia, the velocity of the solid phase is higher than that of the liquid phase in the deceleration stage, while the velocity of the solid phase is lower than that of liquid phase in the acceleration stage; (3) in the process of approaching the leading edge, the velocity difference between the solid and liquid phase increases by about 10% and the angle difference increases by about 8.8°.

## 1. Introduction

_{P}model) or directly tracking the movement of solid-phase particles by the Euler–Lagrange method, can predict the location and amount of abrasion [18,19,20,21,22,23,24,25]. However, due to the limitation of the test’s accuracy or the numerical simulation model, the results obtained by these methods have some deviation from the actual motion of the solid phase.

_{s}= 6.0 mm solid particles in a centrifugal pump by adopting a high-speed photography technique and found that larger particles impact blades on the leading edge region at a high impact angle of 60°~90°, while smaller particles impact blades on the middle part and trailing edge region at a low impact angle of 20°~50°. Yang [27] researched the crystallization phenomenon of a liquid–solid two-phase flow in a centrifugal pump using the Particle Image Velocimetry (PIV) method, revealing the process of crystallization in the pump and finding out that the crystal nuclei move towards the pressure side of the blade. Su [28] established a new theoretical model considering the dual role of microparticles, namely, the viscous effect of silt-sized particles (particle size < 50 μm) and the inertial effect of sand-sized particles (particle size > 50 μm) in synergistic cavitation–particle erosion.

## 2. Methodology

#### 2.1. Hydraulic Circuit

#### 2.2. Parameters of the Cascade Flow Passage

_{1}and L

_{2}between the inlet and outlet of the test section and the rotating center of the middle foil (25 mm from the leading edge point) were 575 mm and 700 mm, respectively, which were 5.75 times and 7.0 times the chord of the foil. The test section profile was optimized by the CFD method, and the inlet and outlet widths L

_{3}and L

_{4}were determined to be 195.8 mm and 198.7 mm, respectively. Table 1 shows the geometric parameters of cascades.

#### 2.3. The Sediment

^{3}was selected as the sediment for the test. Three sediment samples with different particle sizes were obtained after screening with 140 mesh, 200 mesh, and 300 mesh screens, which were named 200 mesh(+), 300 mesh(+), and 300 mesh(−), respectively. The particle size and morphology of the sand were analyzed using the dynamic particle size and shape analyzer (QICPIC/R06-MIXCEL). The respective median particle sizes d

_{50}(diameter of a circle of equal projection area, EQPC) of the sediment samples were determined to be 82.7 μm, 65.9 μm, and 31.8 μm.

#### 2.4. Test Method

#### 2.5. Test Condition

^{4}, 1.0 × 10

^{5}, 3.0 × 10

^{5}, and 6.0 × 10

^{5}.

## 3. Results

#### 3.1. Verification of Test Results

- (1)
- The velocity distribution around the cascade is analyzed on the middle symmetry plane (the same plane as PIV test) of the computational domain, and the consistency of the CFD flow field with the solid and liquid flow field measured by the PIV test is compared.
- (2)
- The leading edge point of the cascade at 0° impact angle is selected as the origin to establish the coordinate system. Twenty-one points are taken along the y direction of a single flow passage at x = −50 mm upstream of the middle foil’s leading edge point, and the y range is −15 mm to 15 mm. The average velocity of these points, v
_{ave}, is calculated. The average velocity deviation of solid and liquid phases is compared between CFD results and PIV measurements.

_{s}|

_{1}and |v

_{f}|

_{1}are the relative velocities of the solid phase and liquid phase, respectively; vs. is solid-phase velocity; v

_{f}is liquid-phase velocity; and v

_{ave}is the average velocity of the incoming flow.

^{5}as an example. It can be seen from the figure that the velocity field calculated by CFD is consistent with the velocity field flow pattern measured by PIV. The flow pattern around the cascade is relatively stable; the solid–liquid phase flows downstream along the foil surface with no flow separation at the tail of the foil. Unstable flow patterns such as vortex or secondary flow do not occur.

#### 3.2. Key Flow Zones

#### 3.3. Flow in the Upstream Zone

^{5}and 5.0 × 10

^{4}. The curves of different X coordinates in the figure are distinguished by the curve’s color. The velocity curves of the solid phase and liquid phase are represented by a marked solid line and dashed line, and marked by s and f, respectively. It can be seen from the figure that:

- (1)
- The velocity shrinks from both sides to the middle along the Y-axis, and the curve of velocity amplitude shows a concave shape. The velocity amplitude changes sharply in the middle (y = 0), but slows down at the two endpoints (y = ±15 mm).
- (2)
- The relative velocity in the main flow area is about 0.96 to 1.02 on the +Y side and about 1.04 to 1.16 on the −Y side. The velocity distribution near S1 and S2 surfaces is asymmetric, which is due to the asymmetric impact of the incoming flow on the foil at an impact angle of 10°.
- (3)
- At the same point, the velocity curves between solid and liquid phases do not coincide, indicating that there is a deviation between the velocities of solid and liquid phases.

#### 3.4. Flow in the Near-Wall Zone

^{5}and 5.0 × 10

^{4}. The definition of the curve in the figure is the same as Section 3.3. It can be seen from the figure that:

- (1)
- On the S1 side of the foil (y > 0), the velocity of the solid phase obviously deviates from that of the liquid phase at the position of y = 15 mm, with a relative velocity difference of 0.05. The relative velocity of the solid phase is about 1.0, which shrinks along the −Y direction to the foil surface. All velocity curves are relatively concentrated at the position of y = 3 mm, with a relative velocity of about 0.7 to 0.8.
- (2)
- On the S2 side of the foil (y < 0), the relative velocity of the solid phase is about 1.0 at the position of y = −15 mm, which shrinks along the −Y direction to the foil surface. All velocity curves are relatively concentrated at the position of y = −3 mm, with a relative velocity of about 0.98 to 1.08.

## 4. Analysis and Discussion

#### 4.1. Velocity Deviation between Solid and Liquid Phases

_{s}and θ

_{f}are the flow angle of the solid phase and liquid phase, respectively, which defines the +X direction as 0°.

^{5}. Some characteristics of the velocity deviation between the two phases can be seen from the figure.

#### 4.1.1. Upstream Zone

- (1)
- Points with the maximum velocity deviation and angle deviation exist on both S1 and S2 sides, respectively, and the deviation value increases as it approaches the leading edge point. The asymmetry of the deviation is obvious; that is, the position of the maximum velocity deviation is in the range of y = 2.0 mm to 4.0 mm on the S1 side, while in the range of y = −2.0 mm to −8.0 mm on the S2 side.
- (2)
- The maximum velocity deviation on the S1 side is about 10%, and the angle deviation is negative, indicating that the solid velocity decreases more slowly than the liquid velocity in the process of deceleration approaching the leading edge point, and the velocity direction is toward the foil surface.
- (3)
- The maximum velocity deviation on the S2 side is about −8.0%, and the angle deviation is also negative. According to the velocity distribution curve, velocity in this region increases along the +X direction; that is, the velocity of the solid phase increases slower than that of the liquid phase, but the velocity direction is also toward the foil surface.

#### 4.1.2. Near-Wall Zone

- (1)
- The velocity deviation is less than 0 in the near-wall zone, indicating that the velocity of the solid phase is less than that of the liquid phase. At the position of y = ±15 mm, the velocity deviation of different curves is very close. When it is close to the S1 wall (y = 3 mm), the deviation on different curves decreases to about −8.0% along the +X direction. When it is close to the S2 wall (y = −3 mm), the maximum relative velocity difference reaches about −12%.
- (2)
- The angle deviation in this zone is less than 0, indicating that the velocity angle of the solid phase is more inclined to the cascade. At the position near the S1 wall (y = 3 mm), the angle deviation increases along the +X direction, and its absolute value decreases. On the contrary, at the position near the S2 wall (y = −3 mm), the angle deviation is positive and decreases along the +X direction. Under asymmetry conditions, the maximum angle deviation on the S1 side is about −7°, while that on the S2 side is about 8.0°.

#### 4.2. Effect of Sediment Characteristics on Velocity Slip

^{5}. In the figure, abscissa d represents the distance from the wall, and the curve colors represent sediment types. The solid line indicates that the position is above surface S1, while the dotted line indicates that the position is below surface S2.

#### 4.2.1. Upstream Zone

- (1)
- The deviation decreases from 200 mesh(+) to 300 mesh(–) except for two points of 200 mesh(+) quartz. The velocity deviation of sand sample 300 mesh(–) on both sides of S1 and S2 is very close to 0, indicating that the sediment with a small particle size has a small velocity slip.
- (2)
- The maximum velocity deviations of 200 mesh(+), 300 mesh(+), and 300 mesh(–) particles are 9.47%, 9.14%, and 2.0%, and the maximum angle deviations are 8.77°, 6.74°, and 3.0°, respectively. Therefore, under the same Reynolds number condition, the velocity and angle deviation decrease with the decrease in sediment particle size. Sediment particles with small particle sizes have a better flow-following ability.

#### 4.2.2. Near-Wall Zone

- (1)
- The velocity deviation of sediment 200 mesh(+) and 300 mesh(+) decreases obviously with the increase in distance d, while that of finest sediment 300 mesh(-) remains at a relatively low level. The absolute value of angle deviation also increases with the particle size. It can be determined from the figure that the deviation in large sediment is higher.
- (2)
- The maximum velocity deviation of 200 mesh(+), 300 mesh(+), and 300 mesh(–) of quartz sand is −11.88%, −4.73%, and −5.58%, and the maximum angle deviation is −3.72°, −3.21°, and −0.49°, respectively. The data show that velocity slip increases with the particle inertia. This result verifies the influence of particle size on wear as follows. When tracking the particle size trajectory in the runner or impeller [26,32,33], it is found that the particle trajectory deviates from that of the fluid particle, and compared with the small-sized particles, the large particles usually contact the blade pressure surface at a further distance from the inlet edge.

#### 4.3. Effect of the Reynolds Number on Velocity Slip

#### 4.3.1. Upstream Zone

- (1)
- The deviation in the velocity and angle increases significantly with the decrease in the distance. When the distance d decreases from 10 mm to 1.6 mm, the maximum velocity deviation increases from 2% to about 12%, and the maximum angle deviation increases from 0.8° to about 9.6°.
- (2)
- Under the five Reynolds number conditions, the curves of velocity deviation and angle deviation have basically the same trend with respect to distance; the velocity deviation value varies within the range of 2.0% to 4.0%, and the angle variation is about 2.5°.

#### 4.3.2. Near-Wall Zone

- (1)
- The velocity deviation on the upper side of S1 decreases from positive to negative from 6.4% to about −7.2% along the +X direction, while that on the lower side of S2 decreases from −9.0% to about −12.0% along the +X direction. Considering that this process is a process of increasing velocity, it can be determined that the velocity growth rate of the solid phase is not as fast as that of the liquid phase.
- (2)
- From the angle deviation point of view, the angle deviation on the S1 side decreases from 6.0° to about 0°, while that on the S2 side increases from −8.0° to about −5.0°. The absolute value of the angle deviation decreases with the velocity.
- (3)
- Under different Reynolds number conditions, all velocity deviation curves on the S1 side are very close, and the variation is about 2.0%, while those on the S2 side are slightly different, and the variation is about 5.0%.

## 5. Conclusions

- (1)
- Before contact with the cascade, the sediment particles undergo a velocity change process of deceleration before acceleration. In this process, velocity slip occurs between the solid and liquid phases. In the deceleration stage, the solid velocity is greater than the liquid velocity, and vice versa in the acceleration stage. In the process of approaching the leading edge point, the velocity deviation increases by about 10%, and the angle deviation increases by about 8.8°.
- (2)
- The particle characteristics have a great influence on the velocity slip between solid and liquid phases. Under the same Reynolds number condition, the particles with high inertia have a large velocity deviation. Under the condition of a Reynolds number of 8.0 × 10
^{5}, according to the order of particle sizes from large to small, the velocity deviation of quartz in the deceleration stage is 9.47%, 9.14%, and 2.5%, respectively, and that in the acceleration stage is −11.88%, −4.73%, and −5.58%, respectively. - (3)
- The Reynolds number also has an important influence on the velocity slip between the two phases. Under the conditions of Re = 5.0 × 10
^{4}to 8.0 × 10^{5}, the velocity deviation caused by the change in Reynolds number is about 5.0%.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The Venturi test circuit. (1) 110 kW pump, (2) DN250 pipeline, (3) bend with guide vanes, (4) Venturi test Section, (5) tank, (6) DN150 pipeline, (7) electromagnetic flowmeter, (8) pipe for sediment sampling, and (9) cooling system.

**Figure 3.**Method of simultaneous two-phase PIV test around cascade. (

**a**) Setup of illumination and image acquisition system; (

**b**) photo of cascade illuminated by laser.

**Figure 4.**Comparison of velocity field (Re = 1.0 × 10

^{5}). (

**a**) Liquid phase of PIV test; (

**b**) solid phase of PIV test; (

**c**) CFD result.

**Figure 6.**Velocity distribution in upstream zone (quartz 200 mesh+). (

**a**) Re = 5.0 × 10

^{4}; (

**b**) Re = 8.0 × 10

^{5}.

**Figure 7.**Velocity distribution in the near-wall zone (quartz 200 mesh+). (

**a**) Re = 5.0 × 10

^{4}; (

**b**) Re = 8.0 × 10

^{5}.

**Figure 8.**Velocity deviation and angle deviation between solid and liquid phases (quartz 200 mesh+). (

**a**) Upstream zone; (

**b**) near-wall zone.

**Figure 9.**Effect of sediment characteristics on velocity slip (Re = 8.0 × 10

^{5}). (

**a**) Upstream zone; (

**b**) near-wall zone.

Parameter | Symbol | Units | Value | |
---|---|---|---|---|

Chord | C | mm | 96.8 | |

Pitch | p | mm | 40 | |

Impact angle | α | degree | 10 | |

Length upstream | L_{1} | mm | 575 | |

Length downstream | L_{2} | mm | 700 | |

Width upstream | L_{3} | mm | 195.8 | |

Width downstream | L_{4} | mm | 198.7 |

Case | Units | Re | ||||
---|---|---|---|---|---|---|

5.0 × 10^{4} | 1.0 × 10^{5} | 3.0 × 10^{5} | 6.0 × 10^{5} | 8.0 × 10^{5} | ||

200 mesh(+) | m/s | 0.55 | 1.10 | 3.35 | 6.46 | 8.76 |

300 mesh(+) | m/s | 0.54 | 1.08 | 3.27 | 6.20 | 8.44 |

300 mesh(−) | m/s | 0.54 | 1.07 | 3.31 | 6.29 | 8.57 |

CFD | m/s | 0.56 | 1.11 | 3.23 | 6.40 | 8.50 |

Error | % | 3.57 | 3.6 | 3.72 | 3.12 | 3.05 |

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**MDPI and ACS Style**

Zhu, L.; Zhang, H.; Chen, Y.; Meng, X.; Lu, L. Asymmetric Solid–Liquid Two-Phase Flow around a NACA0012 Cascade in Sediment-Laden Flow. *Symmetry* **2022**, *14*, 540.
https://doi.org/10.3390/sym14030540

**AMA Style**

Zhu L, Zhang H, Chen Y, Meng X, Lu L. Asymmetric Solid–Liquid Two-Phase Flow around a NACA0012 Cascade in Sediment-Laden Flow. *Symmetry*. 2022; 14(3):540.
https://doi.org/10.3390/sym14030540

**Chicago/Turabian Style**

Zhu, Lei, Haiping Zhang, Ying Chen, Xiaochao Meng, and Li Lu. 2022. "Asymmetric Solid–Liquid Two-Phase Flow around a NACA0012 Cascade in Sediment-Laden Flow" *Symmetry* 14, no. 3: 540.
https://doi.org/10.3390/sym14030540