# Analysis of the Guide Vane Jet-Vortex Flow and the Induced Noise in a Prototype Pump-Turbine
## 5 Results and Analysis
### Propagation of Frequency  
To understand the propagation of frequency \(f_{b}\), \(f_{gsv}\), and \(f_{vl}\) in the flow passage, the \(18\times 18\) flow pulsation tracing topology, as shown in Figure 21, is set instead of the points P\({}_{1}\) to P\({}_{7}\). It covers the 1/20 rotational-periodic region including one guide vane passage, one stay vane passage, and a part of the volute passage. Points are ordered as \"1-18\" along tangential \(\theta\) direction and as \"AA-RR\" along radial \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Frequency Value** & **Description** \\ \hline \(f_{b}\) & 64.5 (Hz) & Runner blade frequency \\ \(f_{gsv}\) & 9.6 (Hz) & Dominate frequency in guide vane and stay vane. \(f_{gsv}=f_{2}\) \\ \(f_{vl}\) & 3.2 (Hz) & Dominate frequency in volute \(f_{vl}=1/3f_{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The main frequencies of \(C_{k}\) pulsation on P\({}_{1}\) to P\({}_{7}\). Figure 19: Analysis of \(C_{k}\) pulsation on P\({}_{6}\). (**a**) Time-domain plot; (**b**) frequency-domain plot. Figure 20: Analysis of \(C_{k}\) pulsation on P\({}_{7}\). (**a**) Time-domain plot; (**b**) frequency-domain plot. \(R\) direction. \(\Delta\theta\) is set as 1 degree, and \(\Delta R\) is set as 0.1 m. As the same as on P\({}_{1}\) to P\({}_{7}\), the frequencies of \(C_{k}\) are calculated by applying the Fast-Fourier transformation (FFT) method with Hanning window. In this study, the frequency-dominated turbulence kinetic energy coefficient \(C_{k}^{*}\) is defined for \(C_{k}\) to understand the propagation of frequency:  
\[C_{k}^{*} = \frac{C_{kRMS}}{\Delta C_{k}} \tag{16}\]  
where \(C_{kRMS}\) is the RMS amplitude value of specific frequency. In this case, the \(f_{b}\) = 64.5 Hz, \(f_{gsv}\) = 9.6 Hz and \(f_{vl}\) = 3.2 Hz are tracked using the topology. \(\Delta C_{k}\) is the peak-peak value of \(C_{k}\) within a specific period. Totally, five runner revolutions were studied in this case. Hence, the parameter \(C_{k}^{*}\) can exclude the amplitude difference of local flow pulsation and focus on the frequency character. Figure 22 shows the contour of \(C_{k}^{*}\) of \(f_{b}\) = 64.5 Hz, \(f_{gsv}\) = 9.6 Hz, and \(f_{vl}\) = 3.2 Hz. The color from blue to red denotes the amplitude of \(C_{k}^{*}\). The blank sites are guide vane blades and stay vane blades. In Figure 22, parameter \(R\) is the radial position. Four copies are plotted based on the original \(18\times 18\) topology to have a better blade-to-blade view in \(\theta\) = 0-90\({}^{\circ}\). Figure 21: The \(18\times 18\) topology for tracing the frequency propagation. RSI: Rotor-stator interface; RD: Runner rotation direction; GV: Guide vane; SV: Stay vane; 1–18: Order number along tangential direction; AA\(\sim\)RR: Order number along radial direction. Figure 22: _Cont._  
As shown in Figure 22b, the runner blade frequency \(f_{b}\) mainly affects \(C_{k}\)\({}^{*}\) in the region between the runner and guide vane. The highest \(C_{k}\)\({}^{*}\) site locates at the front side of the guide vane blade. In the stay vane and volute, the influence of \(f_{b}\) on \(C_{k}\)\({}^{*}\) is very weak. According to Figure 22c, the dominate frequency in the guide vane and stay vane \(f_{gsv}\) has a high \(C_{k}\)\({}^{*}\) value everywhere in the vane channels. The highest \(C_{k}\)\({}^{*}\) value of \(f_{gsv}\) is in the guide vane jet region, which is between two guide vane blades. A wide high \(C_{k}\)\({}^{*}\) region can be also found in the volute near the stay vane outlet. Based on Figure 22d, the dominate frequency in volute \(f_{vl}\) has very strong influence on \(C_{k}\)\({}^{*}\) in volute. The highest \(C_{k}\)\({}^{*}\) sites are between the two stay vane jets and overlap the volute vortex sites. More high \(C_{k}\)\({}^{*}\) regions of \(f_{vl}\) are on the back side of the stay vane blade. The region in the guide vane jet is also high in \(C_{k}\)\({}^{*}\) under the influence of frequency \(f_{vl}\).