# Research on Unsteady Hydraulic Features of a Francis Turbine and a Novel Method for Identifying Pressure Pulsation Transmission Path

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## Abstract

**:**

## 1. Introduction

## 2. Basic Theory and Numerical Model

#### 2.1. Basic Equations

_{b}

_{1}= 0.1335, c

_{b}

_{2}= 0.622, σ = 2/3, c

_{ω}

_{1}= c

_{b1}/k

^{2}+ (1 + c

_{b}

_{2})/σ, ${f}_{\omega}=g{\left(\frac{1+{c}_{\omega 3}^{6}}{{g}^{6}+{c}_{\omega 3}^{6}}\right)}^{\frac{1}{6}}$, c

_{ω}

_{3}= 2, g = r + c

_{ω}

_{2}(r

^{6}− r), $r=\frac{\tilde{\upsilon}}{\tilde{S}{k}^{2}{d}^{2}}$, c

_{ω}

_{2}= 0.3, $\tilde{S}=S+\frac{\tilde{\upsilon}}{{k}^{2}{d}^{2}}{f}_{v2}$, $S=\sqrt{2{S}_{ij}{S}_{ij}}$, ${f}_{v2}=1-\frac{\chi}{1+\chi {f}_{v1}}$, $\chi =\frac{\tilde{\upsilon}}{\upsilon}$, ${f}_{v1}=\frac{{\chi}^{3}}{{\chi}^{3}+{c}_{\upsilon 1}^{3}}$, c

_{ν}

_{1}= 7.1.

_{DES}is described in the form of Equation (2):

_{max}is the maximum local grid size, C

_{DES}= 0.65.

^{(k)}) = p(x(1)|x(k)). Note the n is omitted for convenience. Considering the influence of another process y(m)

^{(l)}on these transition probabilities, the expression of coupling influence of both x(n)

^{(k)}and y(n)

^{(l)}on x(n + 1) is p(x(1)|x(k), y(l)). The transfer entropy can be formulated as below [20]:

#### 2.2. Numerical Simulation Model

## 3. Transient Hydraulic Features in Fluid Domain

_{p}. The equation of C

_{p}is as follows:

#### 3.1. Hydraulic Features of the Flow Field in the Spiral Casing and Vaneless Region

_{p}of measuring points F01 at the spiral casing, F02 between the stay and guide vane, and F03 at the runner inlet. The pressure pulsation of the three measuring points fluctuated regularly from 7.2 to 8.0 s. The closer the position of the measuring point to the runner was, the greater the pressure fluctuation amplitude. Here, we define f

_{n}as the blade rotation frequency 1.25 Hz and 15f

_{n}as the blades passing frequency 18.75 Hz. The frequency domain diagram shows that the main frequency components of pressure pulsation were: 15f

_{n}, 30f

_{n}, and 45f

_{n}, which indicated that pressure fluctuation in spiral casing and vaneless region was mainly affected by the blades’ rotation.

#### 3.2. Hydraulic Features of the Flow FIeld in the Runner

_{p}at the measuring point F04 is illustrated in Figure 8. It is clearly seen that the pressure in the runner chamber fluctuated regularly with the higher pulsation amplitude. Frequency domain analysis in Figure 8b shows that the main frequency components were multiple frequencies of the blades passing frequency 15f

_{n}, which indicated the runner chamber was greatly influenced by the blades’ rotation.

#### 3.3. Hydraulic Features of the Flow Field in the Draft Tube

_{p}at the measuring points F05~F09 in the draft tube. The first main frequency of C

_{p}at the measuring points F05, F06, and F07 was 30f

_{n}, twice blades passing frequency, which indicated that the pressure fluctuation in the draft tube was greatly affected by periodically rotating blades. Low frequency pressure pulsation existed at the measuring points F08 and F09 near the outlet with merely small amplitude.

#### 3.4. Comparison with the In-Site Data

## 4. Pressure Pulsation Transmission Path in the Fluid Flow

#### 4.1. Case Study

_{1}t) + cos(2πf

_{2}t), B = μA + 0.1cos(2πf

_{1}t)cos(2πf

_{2}t) − cos(2πf

_{1}t). μ was utilized as an index to measure the correlation degree between the two sequences and was given a value of 0.8 to compare the results. It is obviously inferred from the formulas of the two sequences that A is a component of B. Hence, we can test the sensitivity of time-delayed transfer entropy in terms of directivity.

#### 4.2. Pulsation Transmission Path Identification Based on Original Data

#### 4.3. Pulsation Transmission Path Identification Based on Wavelet Transform Method

_{n}(18.75 Hz) multiple times. Therefore, the time history of the pressure values of a single frequency component 18.75 Hz was extracted from the original calculation data using the wavelet transform method, and then time-delayed transfer entropy was calculated to identify the pulsation transmission direction of the single frequency component between the measuring points.

## 5. Conclusions

- (1)
- Unsteady flow features in the spiral casing, vaneless region, and draft tube were analyzed based on FSI calculation in design operation. The velocity and pressure in the spiral casing and vaneless region rendered symmetrical distribution in the circumference. Pressure gradually decreased with the increasing velocity as water flowed into the high-speed rotating runner. Smaller velocity and higher pressure occurred at the outer side of the elbow, and larger velocity and lower pressure occurred at the inner side. In the diffuser section of the draft tube, distribution of velocity and pressure was more uniform. The overall pressure transition was smooth without obvious sudden changes, and the streamline distribution was continuous. Frequency analysis of the pressure of the measuring points showed that the first main frequency was the multiple times blades passing frequency, so the main hydraulic vibration source was the blades’ rotation.
- (2)
- The transmission path of the pressure pulsation in the flow channel was identified based on the time-delayed transfer entropy and wavelet transform method using MATLAB software. The pressure pulsation in the components downstream the high-speed rotating runner propagated along the flow direction, which was clear and easy to identify using time-delayed transfer entropy. For the fluid flow upstream the turbine, the pressure transmission path was more complicated to identify, since the pressure propagated against the flow direction. The wavelet transform method was used to extract the time-varying pressure in terms of the characteristic frequency, and the pressure pulsation transmission law was obtained in which the axisymmetric small-disturbance standing waves transmitted from runner to upstream and downstream, respectively, which illustrates the type of draft tube and the connection with the runner are critical to the design of the hydraulic turbine, as well as verifying the rationality of using time-delayed transfer entropy and the wavelet transform method to identify the transmission path.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**3D structured grids for the fluid domain. (

**a**) Spiral casing, (

**b**) stay and guide vane, (

**c**) runner, (

**d**) draft tube.

**Figure 3.**Cross-section diagram and 3D structured grids for the solid domain. (

**a**) River-along section diagram, (

**b**) 3D structured grids.

**Figure 4.**Measuring points distribution in the fluid flow region during numerical calculation process. (

**a**) Top view section diagram, (

**b**) river-along section diagram.

**Figure 5.**Distribution of velocity, streamline, and pressure in the spiral casing and vaneless region. (

**a**) Velocity and streamlines distribution, (

**b**) pressure contour.

**Figure 6.**Time and frequency analysis of C

_{p}at the measuring points F01~F03. (

**a**) Time domain graph, (

**b**) frequency domain graph.

**Figure 7.**Velocity and pressure distribution in the runner. (

**a**) Velocity contour, (

**b**) pressure contour.

**Figure 8.**Time and frequency analysis of C

_{p}at the measuring point F04. (

**a**) Time domain graph, (

**b**) frequency domain graph.

**Figure 9.**Distribution of velocity, streamline, and pressure in the draft tube. (

**a**) Velocity and streamlines distribution, (

**b**) pressure contour.

**Figure 10.**Time and frequency analysis of C

_{p}at the measuring points F05~F09. (

**a**) Time domain graph, (

**b**) frequency domain graph.

**Figure 11.**Comparison of results between transfer entropy and the mutual information method. (

**a**) Result using the transfer entropy method, (

**b**) result using the mutual information method.

**Figure 12.**Time-delayed transfer entropy curve of the original fluid pressure pulsation. Transfer entropy between measuring points F01 and F02 (

**a**), F02 and F03 (

**b**), F03 and F04 (

**c**), F04 and F05 (

**d**), F05 and F06 (

**e**), F06 and F07 (

**f**), F07 and F08 (

**g**), and F08 and F09 (

**h**).

**Figure 13.**Presentation of pressure pulsation transmission path based on the original fluid pressure pulsation.

**Figure 14.**Time-delayed transfer entropy curve of the frequency of 18.75 Hz fluid pressure pulsation. Transfer entropy between measuring points F01 and F02 (

**a**), F02 and F03 (

**b**), F03 and F04 (

**c**), F04 and F05 (

**d**), F05 and F06 (

**e**), F06 and F07 (

**f**), F07 and F08 (

**g**), and F08 and F09 (

**h**).

**Figure 15.**Presentation of pressure pulsation transmission path based on the 18.75 Hz wavelet transformation.

Location | ΔH/H with 97% Degree of Confidence (%) | Error (%) | |
---|---|---|---|

Simulation Results | In-Site Data | ||

Head cover | 0.26 | 0.31 | 16 |

Vaneless region | 0.85 | 0.82 | 4 |

Bottom ring | 0.40 | 0.41 | 2 |

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**MDPI and ACS Style**

Wang, S.; Zhang, L.; Yin, G.; Guan, C.
Research on Unsteady Hydraulic Features of a Francis Turbine and a Novel Method for Identifying Pressure Pulsation Transmission Path. *Water* **2019**, *11*, 1216.
https://doi.org/10.3390/w11061216

**AMA Style**

Wang S, Zhang L, Yin G, Guan C.
Research on Unsteady Hydraulic Features of a Francis Turbine and a Novel Method for Identifying Pressure Pulsation Transmission Path. *Water*. 2019; 11(6):1216.
https://doi.org/10.3390/w11061216

**Chicago/Turabian Style**

Wang, Shuo, Liaojun Zhang, Guojiang Yin, and Chaonian Guan.
2019. "Research on Unsteady Hydraulic Features of a Francis Turbine and a Novel Method for Identifying Pressure Pulsation Transmission Path" *Water* 11, no. 6: 1216.
https://doi.org/10.3390/w11061216