## 3 Results and Discussion
### Discussion of Velocity Fluctuations
The resulting parameters of the amplitude function \(f(\hat{u}_{\mathrm{g}},\omega,C)\) are given in Table 3. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Amplitude & \(\hat{u}_{\mathrm{g}}\) & 12.7 \(\pm\)1.0 m s\({}^{-1}\) \\ Angular frequency & \(\omega\) & 27.0 \(\pm\)3.8 kHz \\ Frequency & \(f\) & 4.3 \(\pm\)0.6 kHz \\ Constant offset & \(C\) & 6.6 \(\pm\)0.1 m s\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Estimated properties of the turbulent fluctuations of the gas phase by fitting \(f(\hat{u}_{\mathrm{g}},\omega,C)\) to experimental data. The indicated uncertainty is based on a 95% confidence interval. gas phase. Subsequently, this will be demonstrated based on a more detailed analysis of the spray characteristics at Position 0 (\(x=$50\,\mathrm{mm}$\)). In Figure 15, the Sauter Mean Diameter (\(D_{32}\)) and the diameter at the 90% volumetric quantile (\(D_{\mathrm{V90}}\)) are plotted over the radial position. These results were already discussed in Section 3.1. In particular, the radial positions featuring the strongest average velocity gradient, \(\pm 2\,\mathrm{mm}\) off the centre, were found to correspond to the smallest characteristic diameters. Therefore, shear layer turbulence seems to play a major role during the atomisation process. For a further in depth analysis, the standard deviation of the radial velocity of very small droplets is discussed. The standard deviation \(\sigma_{\mathrm{v,\,5\,\mathrm{\SIUnitSymbolMicro m}}}\) is plotted in Figure 15, which is determined using droplets of a diameter between \(4.5\,\mathrm{\SIUnitSymbolMicro m}\) and \(5.5\,\mathrm{\SIUnitSymbolMicro m}\). The distribution of \(\sigma_{\mathrm{v,\,5\,\mathrm{\SIUnitSymbolMicro m}}}\) is found to peak exactly at \(\pm 2\,\mathrm{mm}\) off the centre, which confirms the statement that these regions correspond to strong velocity fluctuations caused by shear layer turbulence. Furthermore, the largest droplets of the spray, as represented by the \(D_{\mathrm{V90}}\), are strongly correlated to \(\sigma_{\mathrm{v,\,5\,\mathrm{\SIUnitSymbolMicro m}}}\). Obviously, the turbulent fluctuations of the hot gas flow induce the dominating slip velocity, which triggers the atomisation process. In conclusion, the standard deviation of the radial velocity directly yields an estimation of droplet dispersion, which can be utilised to validate the droplet distribution in the pipe flow as a function of the droplet size. Furthermore, information about the underlying turbulent fluctuations can be estimated with reasonable accuracy. Hence, the smallest detectable droplets may be used as an indicator for turbulence intensity. However, the knowledge of the turbulent fluctuations does not directly yield the turbulent slip velocity of the droplets. Actually, the gradient of the mean velocity in an axial direction needs to be taken into account as well, since a small displacement in a radial direction due to turbulent dispersion may induce a high slip velocity in an axial direction as discussed by [PERSON] et al. [31]. Hence, turbulent fluctuations may cause a high slip velocity of relatively small droplets, which are characterised by strong dispersion due to their low inertia. This effect needs to be considered in order to estimate an absolute slip velocity caused by turbulent fluctuations. However, a more detailed analysis of this slip velocity is beyond the scope of this paper. As an outlook, if a high speed camera in combination with a suitable illumination is used, time-dependent droplet data can be recorded, which may give a more detailed insight into the interaction of the spray with turbulent fluctuations of the gas phase. Figure 15: Correlation of characteristics droplet diameters to the standard deviation of the radial velocity \(\sigma_{\mathrm{v,\,5\,\mathrm{\SIUnitSymbolMicro m}}}\) at Position 0 (\(x=$50\,\mathrm{mm}$\)).