# Vibration Characteristics of a Tubular Turbine Prototype at Different Heads with Considering Free Surface and Water Gravity

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Calculation Method

#### 2.1. Geometric Model and Mesh Generation

_{1}= 7.2 m, the number of blades Z

_{b}= 3, the number of guide vanes Z

_{g}= 16, hub ratio ${\overline{b}}_{h}$ = 0.35, rated speed n

_{r}= 75 r/min, rated head H

_{r}= 6.1 m, and rated discharge Q

_{r}= 413 m

^{3}/s. The calculation domain model of the bulb tubular turbine including the upstream and downstream reservoir areas should be selected when establishing the model to conform to the actual situation. The calculation domain selected in this study includes the upstream reservoir area, diversion section, guide vane, runner, draft tube, and downstream reservoir area. The length of the upstream reservoir is 7.5D

_{1}, the length of the downstream reservoir is 8.5D

_{1}, and the width is 11D

_{1}. The specific geometric model is shown in Figure 1.

^{5}, the number of guide vanes is 1.02 × 10

^{6}, runner is 1.04 × 10

^{6}, and the draft tube and downstream reservoir area is 4.14 × 10

^{5}.

#### 2.2. Numerical Calculation Method and Validation

^{3}τ

_{ij}is the shear stress of the fluid, N/m

^{2}; p is the pressure, Pa; u

_{i}and u

_{j}are the velocity components of the fluid, m/s; x

_{i}and x

_{j}are spatial coordinate components, m; and S

_{ij}is the additional source term, N/m

^{3}. Since the flow direction of the horizontal unit water flow is perpendicular to the direction of gravity, the influence of gravity on the internal flow field of the turbine needs to be considered in the numerical calculation, so the source term in the equation is defined as: S

_{ij}= ρg.

_{i}, u

_{j}is the velocity component of the fluid, m/s, α is the volume fraction of the fluid, α = 1 is the liquid phase, and α = 0 is the gas phase.

#### 2.3. Presentation of Boundary Conditions and Selection of Operating Point for Calculation

_{1}–H

_{3}are different head of water. The submergence depth of the runner differs when the tubular turbine is at different working heads, so the influence of hydrostatic pressure on turbine performance is also different. The condition points of different guide vane openings at the same blade angle are selected for the study to study the vibration characteristics of the bulb tubular turbines under different water heads with a consideration of the free surface and water gravity, and to avoid the interference of excessive flow difference on the analysis results. The specific parameters of the condition point are shown in Table 1.

## 3. Result Analysis

#### 3.1. Distribution of Flow Characteristics in Tubular Turbine under Different Operating Conditions

#### 3.2. Vibration Characteristics of Tubular Turbine

## 4. Pressure Pulsation Analysis of Tubular Turbine Prototype

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Calculation of catchment grid: (

**a**) Global domain grid; (

**b**) bulb body; (

**c**) guide vane; (

**d**) runner; (

**e**) draft tube.

**Figure 4.**Model test verification: (

**a**) Simplified model and measuring point layout; (

**b**) comparison between numerical simulation and experiment.

**Figure 6.**Pressure distribution on the middle section of the runner: (

**a**) Operation condition 1; (

**b**) operation condition 2; (

**c**) operation condition 3.

**Figure 7.**Turbulent kinetic energy of the runner: (

**a**) Operation condition 1; (

**b**) operation condition 2; (

**c**) operation condition 3.

**Figure 8.**Vortex distribution in tubular turbines: (

**a**) Operation condition 1; (

**b**) operation condition 2; (

**c**) operation condition 3.

**Figure 10.**Pressure fluctuation distribution under different conditions: (

**a**) Bulb type; (

**b**) guide vane; (

**c**) runner inlet; (

**d**) runner outlet; (

**e**) draft tube inlet; (

**f**) draft tube outlet.

**Figure 11.**Pressure spectrum of different monitoring points under different conditions: (

**a**) Operation condition 1; (

**b**) operation condition 2; (

**c**) operation condition 3; (

**d**) 0.3 fn; (

**e**) fn; (

**f**) 3 fn.

**Figure 12.**Pressure pulsation amplitude curves at different monitoring points under different conditions.

**Figure 13.**Pressure pulsation of tubular turbine prototype. (

**a**) Arrangement of measuring points for real machine pressure pulsation; (

**b**) test pressure fluctuation amplitude curves at different monitoring points under different outputs.

Condition | Working Head | Flow (m^{3}/s) | Guide Vane Opening (°) | Blade Angle (°) |
---|---|---|---|---|

1 | 7.7 m | 315.3123 | 62 | 5.2 |

2 | 6.08 m | 303.2468 | 66 | 5.2 |

3 | 4.67 m | 300.1159 | 70 | 5.2 |

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**MDPI and ACS Style**

Zhao, Y.; Li, Y.; Feng, J.; Dang, M.; Ren, Y.; Luo, X.
Vibration Characteristics of a Tubular Turbine Prototype at Different Heads with Considering Free Surface and Water Gravity. *Water* **2023**, *15*, 791.
https://doi.org/10.3390/w15040791

**AMA Style**

Zhao Y, Li Y, Feng J, Dang M, Ren Y, Luo X.
Vibration Characteristics of a Tubular Turbine Prototype at Different Heads with Considering Free Surface and Water Gravity. *Water*. 2023; 15(4):791.
https://doi.org/10.3390/w15040791

**Chicago/Turabian Style**

Zhao, Yaping, Yanrong Li, Jianjun Feng, Mengfan Dang, Yajing Ren, and Xingqi Luo.
2023. "Vibration Characteristics of a Tubular Turbine Prototype at Different Heads with Considering Free Surface and Water Gravity" *Water* 15, no. 4: 791.
https://doi.org/10.3390/w15040791