# Orifice versus Converging-Nozzle Grid Turbulence: A Wavelet Perspective
## 4 Results
### Wavelet Analysis
#### 4.4.1 Discrete Wavelet Transform  
Discrete wavelet transform was employed for the fluctuating velocity signal measured along the central point of the measurement grid at X/D = 10. The instantaneous fluctuating velocity signal is decomposed into the sum of various wavelet components at different scales using the seventh-order Daubechies wavelet function (db7) as the mother wavelet function. The processed fluctuating signal is first decomposed into a low-frequency approximation (A\({}_{1}\)) and a high-frequency detailed (D\({}_{1}\)) component at level 1. Furthermore, the approximated component A\({}_{1}\) is subsequently decomposed into higher (detailed) and lower (approximated) frequency components at different levels, for a total of 10 levels. This results in the representation of the original signal as the sum of 11 output scales as expressed by Equation (12). The associated range of frequencies for the approximation and detail components of the discrete wavelet analysis were calculated from Equation (13) as summarised in Table 2.  
Figure 6 shows the ten-level wavelet decompositions of the streamwise fluctuating velocity signal. The coefficients for time, \(t<2\) s, are shown for better visibility. The first velocity plot represents the time series of instantaneous fluctuating velocity signals, \begin{table}
\begin{tabular}{c c c c} \hline
**Approximation** & **Frequency (Hz)** & **Detail Component** & **Frequency (Hz)** \\
**Component** & **Frequency (Hz)** & **Detail Component** & **Frequency (Hz)** \\ \hline A\({}_{1}\) & 0–13,846 & D\({}_{1}\) & 13,846–27,692 \\ A\({}_{2}\) & 0–6923 & D\({}_{2}\) & 6923–13,846 \\ A\({}_{3}\) & 0–3462 & D\({}_{3}\) & 3462–6923 \\ A\({}_{4}\) & 0–1731 & D\({}_{4}\) & 1731–3462 \\ A\({}_{5}\) & 0–866 & D\({}_{5}\) & 866–1731 \\ A\({}_{6}\) & 0–433 & D\({}_{6}\) & 433–866 \\ A\({}_{7}\) & 0–217 & D\({}_{7}\) & 217–433 \\ A\({}_{8}\) & 0–109 & D\({}_{8}\) & 109–217 \\ A\({}_{9}\) & 0–54 & D\({}_{9}\) & 54–109 \\ A\({}_{10}\) & 0–27 & D\({}_{10}\) & 27–54 \\ \hline \end{tabular}
\end{table} 
Table 2: Frequency range of different levels of discrete wavelet analysis. following which there are wavelet decomposition coefficients at different frequencies. The last approximation frequency component (depicted by the second line in Figure 6) with frequency \(f_{\rm a}\approx\) 0-27 Hz is able to approximate the main features of the instantaneous velocity corresponding to the local propensity of departure from the mean value with time. The low-frequency undulations depicted by approximation coefficients indicate the presence of dominant large-scale coherent structures in the flow. The higher frequency components, i.e., detail coefficients, demonstrate the features of smaller-scale turbulence structures. The discrete wavelet transform of the fluctuating signal reveals that turbulence energy decreases continuously with increasing frequency from large-scale to small-scale vorticial structures. At moderate-frequency ranges, i.e., \(f_{\rm a}\lesssim 866\) Hz, the detail coefficients show higher fluctuation at a higher \(\rm Re_{D}\) value, clearly indicating low energy loss due to turbulent dissipation. This means that small-scale structures are still active in the flow alongside larger vortices at the higher studied wind speed. It can also be seen that the fluctuations associated with the OPP are relatively more vigorous than those of the CNPP. At very high frequencies, i.e., \(f_{\rm a}\gtrsim 6923\) Hz, the detail coefficients clearly show some significant fluctuations for the higher studied \(\rm Re_{D}\) in the case of the CNPP, whereas the fluctuations for the OPP are hardly observed. This suggests that at the higher studied \(\rm Re_{D}\), the CNPP-induced flow contains considerable number of large to small eddies, and that small-scale structures also possess a significant part of the turbulent energy due to less energy dissipation. However, in case of the OPP-induced flow, large and moderately sized eddies dominate due to the decay of very small-scale vortices.