# Orifice versus Converging-Nozzle Grid Turbulence: A Wavelet Perspective
## 4 Results
### Wavelet Analysis
#### 4.4.2 Continuous Wavelet Transform  
Continuous wavelet transforms of the fluctuating velocity signal for both perforated plates are obtained at the central measurement point, utilising the Morlet wavelet function as the mother wavelet. The wavelet coefficients thus obtained are used to obtain the power spectrum from Equation (9). Here, it is worth mentioning that continuous wavelet analysis is crucial for scrutinizing coherent structures present which might have been overlooked while employing discrete wavelet transforms due to its dyadic nature. Figure 8 shows the continuous wavelet spectrum at \(\mathrm{X}\)/D = 10 over time and frequency (scale) domains obtained using MATLAB (2023a). The signal was studied for the entire 12.5 s time duration but only the 0.82 s time width is shown here to enhance the visibility. It should be mentioned that the scale in Figure 8 corresponds to the wavelet scaling as described by Equations (7) and (8), where increasing scale corresponds to a decreasing frequency. The coloration depicts the continuous wavelet energy spectrum as defined by Equation (9). Smaller wavelet energy magnitudes are depicted using dark blue regions in the spectrogram. The bright red regions correspond to higher wavelet energy, indicating intense turbulent events at that scale and position. Additionally, wider spectral bands along the time axis indicate large-scale coherent structure. Figure 7: Relative energy contributions of different frequencies at the centre point of the measurement grid at 10D downstream of the perforated plate for (**a**) lower wind speed of 7.4–7.5 m/s (or lower \(\mathrm{Rep}_{\mathrm{D}}\)) and (**b**) higher wind speed of 11.1–11.9 m/s (or higher \(\mathrm{Rep}_{\mathrm{D}}\)). Figure 8: Continuous wavelet spectrum for the velocity fluctuation signal at centre point of the measurement grid with X/D = 10 for OPP with (**a**) \(\overline{\text{U}}\) = 7.38 m/s and (**b**) \(\overline{\text{U}}\) = 11.04 m/s, and for CNPP with (**c**) \(\overline{\text{U}}\) = 7.49 m/s and (**d**) \(\overline{\text{U}}\) = 11.94 m/s. At the lower studied \(\mathrm{Re_{D}}\), the OPP shows more brighter regions with a few very dominant peaks at \(f_{\mathrm{a}}\approx 16.26\) Hz and \(t\approx 2.23\)-2.78 s, as opposed to the CNPP, where the number of peaks is relatively less and apparently no dominant peaks are observed, as shown in Figure 8a,c. The magnitude of wavelet energy is higher for the OPP, indicating more intense coherent structures. For the OPP, most of the high-wavelet-energy region lies in the frequency range \(12.1\) Hz \(\lesssim\)\(f_{\mathrm{a}}\)\(\lesssim\) 28.9 Hz, showing that the large-scale coherent structures possess most of the energy. A high-energy region for the CNPP is observed at a very low-frequency value, \(f_{\mathrm{a}}\)\(\lesssim\) 4.2 Hz, and this mainly corresponds to the cone of influence region of the spectrogram (i.e., region of the wavelet spectrum in which the edge effect is significant and calculated values are not reliable). The turbulent bursts for both plates are regularly distributed along the time axis, which means that the coherent structures are quasi-static and mostly non-intermittent in nature. At the higher studied \(\mathrm{Re_{D}}\), the number of brighter (high-energy) regions is significantly higher for both plates, as shown in Figure 8b,d, implying that the flow contains numerous localised coherent structures. The wavelet energy is distributed across the scale, indicating suppression of turbulence decay. A few very high-energy regions can be observed for the OPP at \(17.12\) Hz \(\lesssim\)\(f_{\mathrm{a}}\)\(\lesssim\) 30.34 Hz, \(t\approx 5.85\)-6.23 s, and at \(56.88\) Hz \(\lesssim\)\(f_{\mathrm{a}}\)\(\lesssim\) 103.2 Hz, \(t\approx 5.85\)-6.23 s. This shows that significant energy is possessed by moderate-to-large-scale structures. The number of high-energy regions in the higher frequency region increases for CNPP due to small-to-moderate-scale turbulent bursts.