# Study on the Cavitation Characteristics of Shroud Clearance in Prototype and Model of a Kaplan Turbine

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## Abstract

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## 1. Introduction

## 2. Research Object and Methodology

#### 2.1. Numerical Techniques

_{m}and μ

_{m}can be calculated respectively by Formulas (4) and (5):

_{m}represents the density of the mixture; ρ

_{l}and ρ

_{v}represent the component densities of the water and water vapor, respectively; μ

_{m}represents the dynamic viscosity of the mixture; μ

_{l}and μ

_{v}represent the component dynamic viscosity of the water and water vapor, respectively; α

_{l}represents the component volume fraction of the water; u

_{a}(a = i, j, k; i, j, k = 1, 2, 3) represent the three direction components of the velocity in the Cartesian coordinate system; x

_{a}represent the coordinates; p represents the static pressure; μ

_{t}represents the turbulent viscosity; ${\delta}_{ij}$ represents the Kronecker delta (if i = j, ${\delta}_{ij}=1$; if i ≠ j, ${\delta}_{ij}=0$); and m represents the source terms of cavitation, which is used to control the mass transfer rate between the water vapor phase and the water phase.

#### 2.2. Object of Study

#### 2.3. Computation Domain

_{P}) of the clearance leakage vortex cores. The energy characteristics mainly include flow coefficient (Q*), head coefficient (H*), power coefficient (P*), and efficiency (η). As shown by the energy characteristics, the results obtained from the single-blade channel and the whole passage are very close to each other, indicating that the single-blade channel calculation method can predict the energy characteristics of the runner domain very well. As shown by the minimum pressure coefficient of the clearance leakage vortex cores, the pressure coefficient obtained from the single-blade channel calculation is lower, indicating that it can better capture the cavitation phenomenon of the clearance leakage vortex. Therefore, it is valid to use a single-blade computing domain for shroud clearance cavitation flow calculations in this paper.

_{v}represents the volume flow rate, H

_{r}represents the head of the runner domain, T represents the output torque of the spindle, ρ represents the density of water, g represents the gravity acceleration, p represents the minimum pressure at vortex cores, p

_{in}represents the average pressure at the inlet of the runner, and V

_{tip}is the circumferential velocity of the runner tip, calculated using the product of the tip radius and the rotational angular velocity of the runner.

#### 2.4. Operating Conditions and Boundary Conditions

_{11}) is 1.05 m

^{3}/s, and the unit speed (n

_{11}) is 115.38 r/min. The unit flow rate (Q

_{11}), the unit speed (n

_{11}), and the cavitation coefficient (N*) are the same for the prototype and the model under similar operating conditions. The three parameters are defined as

_{a}, H

_{va}, and H

_{s}are the atmospheric pressure, the liquid vapor pressure, and the suction height of the turbine, respectively, which are expressed in terms of the height of the liquid column.

#### 2.5. Mesh Independence Verification

## 3. Calculation Results

#### 3.1. Analysis of Calculation Results

#### 3.1.1. Energy Characteristics

#### 3.1.2. Clearance Leakage Characteristics

_{leak}) is normalized by the flow rate of the inlet (Q

_{in}). Both the prototype and the model exhibit a gradual decrease in clearance leakage flow rate as the cavitation coefficient (N*) decreases. The clearance leakage flow rate changes are small in both the prototype and the model when the cavitation coefficient (N*) is in the range of 1.15 to 0.7. When the cavitation coefficient (N*) decreases to 0.474, the clearance leakage flow rate of the prototype decreases by 1.66% relative to when the cavitation coefficient (N*) is 0.7, while that of the model decreases by 0.33%. The decrease rate of the clearance leakage flow rate of the prototype is obviously higher than that of the model, which is related to the cavitation state of the clearance region. When the flow channel near the clearance is blocked by the clearance cavitation, the clearance leakage flow rate will be reduced. The clearance leakage flow rate decreases as the flow path becomes more blocked. When the cavitation coefficient (N*) decreases further, the degree of cavitation near the clearance region becomes more severe, and the clearance leakage flow rate also continues to decrease. The clearance leakage flow rates of the prototype and model decrease quickly, with the cavitation coefficient (N*) decreasing from 0.248 to 0.208, resulting in a decrease of 1.29% and 2.53%, respectively. During this process, the energy characteristics of the prototype and the model also decrease rapidly.

#### 3.1.3. Cavitation Distribution near the Blade Tip

_{v}= 0.1). The degree of cavitations in the blade tip region of both the prototype and the model becomes more intense with the decrease in the cavitation coefficient (N*). Specifically, when cavitation occurs, the clearance leakage vortex cavitation starting from the head of the blade occurs earlier than the tip clearance cavitation, which is in the middle and the tail of the blade. With a decreased cavitation coefficient (N*), the clearance leakage vortex cavitation diameter thickens and continues to extend to the outlet side of the blade, while the tip clearance cavitation develops bidirectionally towards the head and the tail of the blade. The interaction between tip clearance cavitation and clearance leakage vortex cavitation occurs when the cavitation coefficient (N*) is 0.248. The tail of the blade was completely covered by the cavitation region. As the cavitation coefficient (N*) continues to decrease, the cavitation degree near the tip intensifies further, resulting in a stronger blocking effect on the flow passage.

#### 3.1.4. Volume Change of the Runner Cavitation

_{RV}, the total cavitation volume in the flow passage was denoted as V

_{cav-total}, the tip cavitation volume was denoted as V

_{cav-tip}, and the other cavitation volumes were denoted as V

_{cav-other}.

#### 3.1.5. Distribution of Pressure Coefficients on the Blade Surfaces

_{P}) on the suction surfaces of the prototype and model are shown in Figure 10. The pressure coefficient is determined by using Formula (10), with the p here representing the static pressure on the surface of the blade. It can be seen that the distribution of pressure coefficients on the suction surface of the blade in the prototype and model is similar. Due to the influence of the clearance leakage vortex, a strip of low-pressure region similar to the trajectory of the clearance leakage vortex appears at the tip of the blade.

#### 3.1.6. Cavitation Characteristics of the Runner Chamber

## 4. Conclusions

- Under the cavitation conditions, the energy characteristics (head coefficient, power coefficient, and efficiency) of the prototype are higher than those of the model. The operating conditions where the energy characteristics drop rapidly are the same. In addition, the critical cavitation coefficient of the prototype is close to that of the model. To some extent, the critical cavitation coefficient calculated by the model has reference significance for the prototype.
- When the cavitation coefficient is larger than the critical cavitation coefficient, the cavitation characteristics of the blade and the runner chamber in the prototype are more serious than those in the model. When the cavitation coefficient is less than the critical cavitation coefficient, the degree of cavitation in the runner domain of the model is intensified. The cavitation characteristic of the runner chamber of the model is more serious than that of the prototype.
- With the decrease in the cavitation coefficient, the cavitation of the runner chamber in the prototype occurs earlier than that in the model. The runner chamber in the prototype experiences cavitation at the device cavitation coefficient, while the runner chamber in the model does not. When the cavitation coefficient is reduced to close to the critical cavitation coefficient, the energy characteristics of the runner do not change much, but the cavitation significantly intensifies in the prototype and model.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The relation curves between the validation indexes and the mesh number. (

**a**) Efficiency; (

**b**) the minimum pressure coefficient of the clearance leakage vortex cores.

**Figure 4.**Meshes of the calculation domain. (

**a**) Mesh in the single-blade calculation domain; (

**b**) mesh on the blade surface; (

**c**) mesh on the shroud surface.

**Figure 5.**The relationship curves between the energy characteristics and the cavitation coefficients. (

**a**) Head coefficient; (

**b**) power coefficient; (

**c**) efficiency.

**Figure 6.**Clearance leakage characteristics. (

**a**) The streamlines of clearance leakage flow and the clearance leakage vortex. (

**b**) The relationship curves between the clearance leakage flow rate and cavitation coefficients.

**Figure 7.**The tip clearance cavitation and clearance leakage vortex cavitation in the shroud clearance region.

**Figure 8.**The isosurfaces of the vapor volume fraction α

_{v}= 0.1 under different cavitation coefficients. (

**a**) N* = 0.7; (

**b**) N* = 0.474; (

**c**) N* = 0.248; (

**d**) N* = 0.208.

**Figure 9.**The relationship curves between the ratios of different cavitation volumes and cavitation coefficients. (

**a**) V

_{cav-tip}/V

_{cav-total}; (

**b**) V

_{cav-other}/V

_{cav-total}; (

**c**) V

_{cav-tip}/V

_{RV}; (

**d**) V

_{cav-other}/V

_{RV}.

**Figure 10.**Distribution of pressure coefficients on the blade surfaces at N* = 0.474. (

**a**) Prototype; (

**b**) model.

**Figure 11.**Distribution of pressure coefficients on the blade surfaces under different cavitation coefficients. (

**a**) R* = 0.97; (

**b**) R* = 0.5; (

**c**) R* = 0.1.

**Figure 12.**The vapor volume fraction on the shroud surface under different cavitation coefficients. (

**a**) N* = 0.7; (

**b**) N* = 0.474; (

**c**) N* = 0.248; (

**d**) N* = 0.208.

Geometry | Prototype | Model |
---|---|---|

Runner diameter D_{1} | 29.71 D_{m} | D_{m} |

Stay vanes Z_{s} | 25 | 25 |

Guide vanes Z_{g} | 28 | 28 |

Runner blades Z | 6 | 6 |

shroud clearance width ω | 4% D_{m}~4.57% D_{m} | 1.34‰ D_{m}~1.54‰ D_{m} |

Computational Domain | Q* | H* | P* | η | C_{P} |
---|---|---|---|---|---|

Whole passage | 0.545 | 0.266 | 1349 | 95.13% | −0.929 |

Single-blade | 0.541 | 0.263 | 1324 | 95.19% | −1.074 |

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**MDPI and ACS Style**

Zhang, Y.; Luo, W.; Chen, T.; Zhou, L.; Wang, Z.
Study on the Cavitation Characteristics of Shroud Clearance in Prototype and Model of a Kaplan Turbine. *Water* **2023**, *15*, 3960.
https://doi.org/10.3390/w15223960

**AMA Style**

Zhang Y, Luo W, Chen T, Zhou L, Wang Z.
Study on the Cavitation Characteristics of Shroud Clearance in Prototype and Model of a Kaplan Turbine. *Water*. 2023; 15(22):3960.
https://doi.org/10.3390/w15223960

**Chicago/Turabian Style**

Zhang, Yali, Wendong Luo, Tao Chen, Lingjiu Zhou, and Zhengwei Wang.
2023. "Study on the Cavitation Characteristics of Shroud Clearance in Prototype and Model of a Kaplan Turbine" *Water* 15, no. 22: 3960.
https://doi.org/10.3390/w15223960