# Analysis of Pressure Fluctuation of Tubular Turbine under Different Application Heads

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## Abstract

**:**

_{1}. Moreover, in the case of small H/D

_{1}, the amplitude of pressure pulsation in the draft tube is larger, and concentrated high-frequency pressure pulsation occurs. These factors will lead to the occurrence of material fatigue damage, unstable output, and increased vibration in low-head tubular turbines.

## 1. Introduction

_{1}).

_{1}of the above-mentioned power plant model machine is 26.7 times that of the prototype machine. Then, the hydrostatic pressure difference ρgD

_{1m}in the model runner chambers is only 4.5% of the hydrostatic pressure difference ρgD

_{1p}in the prototype runner. This result shows that the model machine and the prototype machine fail to meet similar conditions. Moreover, the tubular turbine has a short distance from the upstream and downstream reservoir areas, which is a typical flow including the open channel part. Therefore, the water flow of the model and the prototype is required to be similar under the action of gravity to satisfy the Froude similarity criterion. In addition, the scale of physical quantities between the model and the prototype is also constrained by this criterion. However, the difference between the gravitational acceleration of the model and the prototype is very small, and g

_{p}is approximately considered to be equal to g

_{m}. If the flow of the model is similar to that of the prototype, then the parameters such as the velocity scale, flow scale, time scale, acceleration scale, and pressure scale must all be a specific value, making it more difficult to implement. In addition, the influence of the fluctuation of the free surface in the reservoir area and the water gravity on the flow characteristics in the turbine cannot be reflected in the model test. Therefore, the severe vibration and noise generated during the operation of the low-head tubular turbine cannot be accurately known by the model machine. With the increase in the size and capacity of the ultra-low-head tubular turbine, the above phenomenon becomes more prominent and becomes an important factor restricting the stable operation of the low-head tubular turbine. At present, the influence of free surface and gravity on the internal flow of the tubular turbine has attracted the attention of relevant scholars. The relevant literature showed that gravity has an important influence on the dynamic stress characteristics of the cavitation performance of the horizontal tubular turbine [22,23]. Furthermore, the dynamic fluctuation of the free surface [24,25] makes the force on the runner blade of the tidal tubular turbine fluctuate greatly.

_{1}. The study also deeply analyzes the internal transient flow characteristics and water pressure pulsation characteristics of the tubular turbine with different H/D

_{1}. Moreover, the influence law of free surface and water gravity on prototype tubular turbines with different H/D

_{1}is revealed. These research results provide a theoretical basis for the hydraulic design and stable operation of the tubular turbine.

## 2. Materials and Methods

- (1)
- Continuity equation

- (2)
- Momentum equation

## 3. Results

_{1}conditions and to explore the relationship between the distribution law of the flow parameters and H/D

_{1}. Thus, different actual hydropower turbines under different water heads and with five-, four-, three-, and two-blade runners are selected as the research objects. Table 2 shows the basic parameters for each actual hydropower turbine.

## 4. Experimental Verification of Numerical Calculation Method

## 5. Analysis and Discussion of Calculation Results

#### 5.1. Analysis of the Internal Flow Field Characteristics of the Tubular Turbine with Different Head Sections

_{1}. In the entire calculation domain, the pressure above the free surface is constant at atmospheric pressure. Below the free surface, the pressure of the runner part fluctuates because of the output power when the runner rotates. Furthermore, the pressure in the rest flow parts increases with the increase inwater depth. The pressure in the upstream channel of the runner is determined by the water depth of the upstream reservoir area. Moreover, the pressure of the downstream channel of the runner is determined by the water depth of the downstream reservoir area. The reason is that the water head of the turbine is the difference between the upstream and downstream water levels. This water level difference is also the energy that the runner needs to convert; thusit also indirectly reflects the pressure difference before and after the runner.When H/D

_{1}is large, the upstream water level is deep, and the downstream water level is shallow, and thusthe pressure changes before and after the runner are evident. When H/D

_{1}is small, a small difference between the upstream and downstream water levels makes the pressure change before and after the runner smaller.

_{1}conditions, the relative velocity C

_{v}is introduced:

_{i}is the instantaneous speed of the water flow.

_{1}. From the figure, in most of the areas of the inlet, the non-uniformity of the flow velocity distribution at the inlet of different H/D

_{1}units is different. This non-uniformity is more evident along the direction of the inlet height. To be more intuitive and quantify this inhomogeneity, the relative velocity from the top to the bottom of the middle position of the inlet section is taken, as shown in Figure 12. From the figure, when H/D

_{1}is large, the uniformity of the flow velocity distribution along the height direction is better. Moreover, when H/D

_{1}= 0.57, the difference between the maximum relative flow velocity and the average value at the middle position of the inlet section is up to 0.163, whereas the remaining three H/D

_{1}cases have only 0.071, 0.052, and 0.061, respectively. The small H/D

_{1}aggravates the non-uniformity of the incoming flow of the unit, and thus the flow entering the unit is more concentrated at the top of the inlet section.

_{1}. It can be seen from the figure that with the decrease in H/D

_{1}, the uniformity of the flow velocity distribution along the height direction of the guide vane is better. Along the circumferential direction, owing to the shunt effect of the bulb body and the upper and lower shafts on the flow, before reaching the movable guide vane, a small flow velocity area still exists near the end of the shaft. In addition, the flow velocity distribution near the bulb body is the most uneven. When passing through the guide vanes, the water flow is further equalized by the guide vanes. The uneven distribution of flow velocity in a small area caused by the vertical shaft before the entrance of the guide vanes has been further improved. However, unevenness still exists, as shown in Figure 14.

#### 5.2. Analysis of the Internal Pressure Pulsation Characteristics of the Tubular Turbine with Different Head Sections

_{1}conditions in detail, four monitoring points on the blade (Figure 15) were selected to conduct the transient state analysis of the blade surface pressure pulsation. Figure 16 shows the pressure coefficient Cp (Cp = P/ρgH) fluctuation on the blade under different H/D

_{1}. From the figure, under different H/D

_{1}conditions, the pressure distribution on the blade surface changes periodically during the operation of the horizontal tubular turbine. The vertical displacement experienced by the part of the blade near the shroud is close to the runner diameter D

_{1}. Thus, the hydrostatic pressure change experienced is ρgD

_{1}. The vertical displacement experienced by the part of the blade near the hub is close to the hub diameter d

_{b}; thusthe hydrostatic pressure change experienced is ρgd

_{b}. Moreover, the pressure change on the blade surface is the joint action of the hydrostatic pressure and the dynamic water pressure. Furthermore, the dynamic water pressure on the blade surface changes less with height, and the change law of the pressure fluctuation is mainly dominated by the hydrostatic pressure. Therefore, the magnitude of the pressure fluctuation on the blade near the shroud is greater than that near the hub. As H/D

_{1}decreases, the submerged depth of the runner increases, and the hydrostatic pressure difference from the top to the bottom of the runner chamber increases. When the blade makes one revolution, the fluctuation amplitude of the pressure at different positions on the blade increases significantly. Tubular turbine blades are prone to vibration under the action of this unbalanced periodic pressure fluctuation, and even material fatigue damage occurs. Particularly in the case of ultra-low H/D

_{1}, the vibration of the tubular turbine is more severe, and this phenomenon is more consistent with the actual operation of the power station.

_{1}, two monitoring points above and below the inlet section of the draft tube are selected, as shown in Figure 10. Figure 17 shows the variation curve of the pressure with time at each monitoring point and its frequency spectrum characteristics. From Figure 17, the frequency of pressure pulsation in the draft tube is relatively complex, but for the same unit, the pressure pulsation and frequency spectrum characteristics at the top and bottom of the draft tube have the same law. The pressure difference between the two monitoring points at the same time is the static pressure difference caused by the elevation difference. Moreover, the frequency characteristics are the same except for the difference in amplitude. The draft tube has a typical low-frequency pressure pulsation under different H/D

_{1}with frequencies of 1.87, 1.256, 0.706, and 1.956Hz. When the water head is high (H/D

_{1}= 3.73), the pressure pulsation in the draft tube is mainly dominated by low-frequency pressure pulsation. However, with the decrease inthe water head, when the H/D

_{1}is 1.52, 1.25, and 0.57, respectively, the pressure pulsation is very complex, and the high-frequency pressure pulsation signals with a frequency of 1500 to 2300 Hz appear concentrated. Furthermore, the amplitude of the pressure pulsation of each frequency in the draft tube increases with the decrease inthe H/D

_{1}. The smaller the H/D

_{1}, the more severe the hydraulic vibration caused by the pressure pulsation in the draft tube, which will have a great impact on the stability of the turbine.

#### 5.3. Analysis of Torque Characteristics of the Tubular Turbine with Different Head Sections

_{i}is the torque value of a single blade, and i is the blade number.

_{1}generated by the blades in different positions. The average value of the torque percentage of each blade is used as a reference value. The average value is defined as follows: the torque suffered by the entire runner is recorded as 100%. When the free surface and water gravity are not considered, the torque suffered by each blade is the same, and the average torque percentage of each blade is 100/N (%), where N is the number of blades.

_{1}is different, the blade torque at different positions deviates from the average value. When the blade starts to rotate around the shaft from the bottom of the runner chamber, the change process of torque is as follows: during the rotation of the blade from the bottom to the top of the runner chamber, the torque increases from the average value to the maximum value and then gradually decreases to the average value; when the blade rotates from the top to the bottom of the runner chamber, the torque starts to decrease from the average value to the minimum value and then gradually increases to the average value. The maximum and minimum torque of the blade mainly occurs at the horizontal position, and the position where the torque is close to the average value is the top and bottom of the runner chamber. Therefore, when the runner rotates, the maximum torque always occurs when the blade rotates against the water gravity. Moreover, the minimum value always occurs in the process of the water flow gravity pushing the blade rotation. Each blade bears the fluctuation of the torque when the runner rotates.

_{1}. From the table, the influence of free surface and water gravity on blade torque varies with H/D

_{1}and the number of blades. The smaller the H/D

_{1}is, the lowerthe number of blades is, and the greater the torque fluctuation is, which is not conducive to the safe and stable operation of the turbine. Therefore, when a two-blade runner is used in an ultra-low-head power station, although a small number of blades can maximize the overflow of the turbine, problems such as the increase inblade area and the large fluctuation of torque are unfavorable to the safe and stable operation of the power station.

## 6. Conclusions

_{1}is carried out under the consideration of the free surface in the upstream and downstream reservoir areasand water gravity. The distribution and development law of different hydraulic elements in the prototype tubular turbine during the operation are analyzed; the main conclusions are as follows:

_{1}, the more evidentthis phenomenon is. Although the bulb body and the guide vane have an apparent equalizing effect on the water flow, the axisymmetric of the water flow along the circumferential direction before reaching the runner is still poor.

_{1}is, the submerged depth of the runner increases, and the amplitude of the water pressure fluctuation on the blade increases significantly. Under the action of this unbalanced periodic pressure fluctuation, the blade of the tubular turbine is prone to vibration and even material fatigue damage.

_{1}, a low-frequency, high-amplitude pressure pulsation occurs in the draft tube. The amplitude of this low-frequency pressure pulsation increases with the decrease in H/D

_{1}. Moreover, when H/D

_{1}decreases, the high-frequency pressure pulsation signal with a concentrated frequency appears in the draft tube, which will cause the hydraulic vibration of the low-head turbine.

_{1}, which is not conducive to the stability of the output of the turbine.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Geometric model. (

**a**) Computational domain; (

**b**) diagram of water head (H) and runner diameter (D

_{1}).

**Figure 3.**Grid distribution of computational domain. (

**a**) The overall grid distribution of the computational domain; (

**b**) grid distribution of the runner with different numbers of the blade.

**Figure 6.**Test model. (

**a**) Pressure pipe; (

**b**) upstream reservoir; (

**c**) distribution of pressure measuring points; (

**d**) draft tube.

**Figure 7.**Pressure distribution at the bottom and top of the pipeline. (

**a**) Top of pressure pipe; (

**b**) bottom of pressurized pipe.

**Table 1.**Parametercomparison between the model machine and prototype machine of a tubular power station.

Model Machine | Prototype Machine | |
---|---|---|

D_{1} (m) | 0.34 | 7.5 |

H (m) | 3 | 2.5 |

H/D_{1} | 8.82 | 0.33 |

Hydrostatic pressure difference in the runner chamber ∆P (Pa) | Ρg × 0.34 | Ρg × 7.5 |

Numberof Blades | Number of Guide Vanes | Hub Ratio | Water Head (m) | Runner Diameter (m) | Rotating Speed (r/min) | H/D_{1} |
---|---|---|---|---|---|---|

5 | 16 | 0.41 | 19 | 5.1 | 115.4 | 3.73 |

4 | 16 | 0.36 | 10.2 | 6.7 | 75 | 1.52 |

3 | 16 | 0.34 | 9 | 7.2 | 75 | 1.25 |

2 | 16 | 0.3 | 4.3 | 7.6 | 60 | 0.57 |

Turbine Flow Parts | Nodes | Grid Numbers |
---|---|---|

Upstream reservoir area and diversion section | 668,275 | 667,168 |

Guide vane | 1,087,760 | 1,017,042 |

Draft tube and downstream reservoir area | 391,208 | 404,160 |

Five-blade runner | 1,847,738 | 1,699,093 |

Four-blade runner | 1,478,185 | 1,359,274 |

Three-blade runner | 1,108,630 | 1,019,456 |

Two-blade runner | 950,086 | 909,637 |

Name | Formula | Value |
---|---|---|

Geometric scale | ${\lambda}_{L}={l}_{p}/{l}_{m}$ | 50 |

Speed scale | ${\lambda}_{V}={\lambda}_{L}^{0.5}$ | 7.071 |

Flow scale | ${\lambda}_{Q}={\lambda}_{L}^{2.5}$ | 17,678 |

Pressure scale | ${\lambda}_{P}={\lambda}_{L}$ | 50 |

Time scale | ${\lambda}_{t}={\lambda}_{L}^{0.5}$ | 7.071 |

Roughness scale | ${\lambda}_{n}={\lambda}_{L}^{1/6}$ | 1.919 |

H/D_{1} | Torque Increase Value | Torque Reduction Value |
---|---|---|

3.73 | +0.83 | −0.85 |

1.52 | +1.293 | −1.706 |

1.25 | +1.802 | −2.056 |

0.57 | +2.75 | −2.75 |

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## Share and Cite

**MDPI and ACS Style**

Zhao, Y.; Feng, J.; Li, Z.; Dang, M.; Luo, X.
Analysis of Pressure Fluctuation of Tubular Turbine under Different Application Heads. *Sustainability* **2022**, *14*, 5133.
https://doi.org/10.3390/su14095133

**AMA Style**

Zhao Y, Feng J, Li Z, Dang M, Luo X.
Analysis of Pressure Fluctuation of Tubular Turbine under Different Application Heads. *Sustainability*. 2022; 14(9):5133.
https://doi.org/10.3390/su14095133

**Chicago/Turabian Style**

Zhao, Yaping, Jianjun Feng, Zhihua Li, Mengfan Dang, and Xingqi Luo.
2022. "Analysis of Pressure Fluctuation of Tubular Turbine under Different Application Heads" *Sustainability* 14, no. 9: 5133.
https://doi.org/10.3390/su14095133