## 3 Results and Discussion
### Determining the Gas Velocity
As a first approach, the assumption is made that the axial velocity of the droplets of 4 \(\mathrm{\SIUnitSymbolMicro m}\) diameter \(u_{\mathrm{d,4\SIUnitSymbolMicro m}}(t)\) represents the velocity of the gas phase. The experimental results of the temporal arithmetic mean of \(u_{\mathrm{d,4\SIUnitSymbolMicro m}}(x)\) at the centre of the pipe are illustrated in Figure 12 for the first three measurement positions downstream of the atomiser. However, the initial velocity of the gas and the droplets at the atomiser exit is not available. Furthermore, the evolution of the velocity directly downstream of the nozzle is governed by gas dynamics and the resulting primary atomisation. In this study, the initial velocity of the droplets and the air is estimated as \(\mathrm{Ma}=1\) at the nozzle outlet based on the estimated droplet starting temperature \(T_{\mathrm{d}0}\). The present atomiser is operated at supercritical flow conditions in order to achieve a strong pressure drop at the atomiser outlet. Hence, this assumption is reasonable. Furthermore, the assumption is conservative in terms of the estimation of the measurement accuracy, since expected shock waves may lead to a lower effective initial velocity of droplets after primary atomisation in the near nozzle region. In the next step, a function of the type  
\[u_{\mathrm{g}}(t)=C_{1}e^{-\frac{t}{\tau_{\mathrm{g}}}}+C_{2} \tag{6}\]  
is used to fit the estimated and experimental data in order to derive an estimated evolution of the gas velocity \(u_{\mathrm{g}}\), which is shown in Figure 12. This function is chosen according to Equation (5). Hence, a timescale of the gas phase \(\tau_{\mathrm{g}}\) can be derived and the Stokes number of a droplet with a diameter of \(4\,\mathrm{\SIUnitSymbolMicro m}\) can be calculated:  
\[\mathrm{St_{4\,\mathrm{\SIUnitSymbolMicro m}}}=\frac{\tau_{\mathrm{d}}}{\tau_{ \mathrm{g}}}=0.13. \tag{7}\]  
Since the condition,  
\[\mathrm{St_{4\,\mathrm{\SIUnitSymbolMicro m}}}\ll 1, \tag{8}\]  
is not perfectly fulfilled, the derived evolution of the gas velocity is used to solve a more generalised equation of motion of one droplet,  
\[\frac{\mathrm{d}u_{\mathrm{d}}}{\mathrm{d}t}=-\frac{3}{4}\frac{\rho_{g}}{\rho _{d}}\frac{c_{D}}{D_{d}}|u_{\mathrm{g}}(t)-u_{\mathrm{d}}(t)|\;(u_{\mathrm{g}} (t)-u_{\mathrm{d}}(t)), \tag{9}\]  
in order to estimate the measurement accuracy. The correlation of [PERSON] et al. [44] is used to determine the drag coefficient  
\[c_{D}=0.36+5.48\;Re_{d}^{-0.573}\frac{24}{Re_{d}}. \tag{10}\]  
This correlation is only valid for perfect spheres, which is a reasonable assumption for the small droplets in this study. Most importantly, Equation (10) was validated experimentally for droplet Reynolds numbers  
\[\tau_{\mathrm{d}}=\frac{\rho_{\mathrm{d}}\ D_{\mathrm{d}}^{2}}{18\ \mu_{\mathrm{g}}}, \tag{3}\]0  
Therefore, the main restriction of the [PERSON] theory (\(Re_{d}<1\)) is overcome. The resulting spatial evolution of the velocity of a droplet of a diameter of \(4\,\mathrm{\SIUnitSymbolMicro m}\)\(u_{\mathrm{d,4\,\mathrm{\SIUnitSymbolMicro m}}}(x)\) is illustrated in \[\tau_{\mathrm{d}}=\frac{\rho_{\mathrm{d}}\ D_{\mathrm{d}}^{2}}{18\ \mu_{\mathrm{g}}}, \tag{3}\]1 Obviously, droplets of a diameter of \(4\,\mathrm{\SIUnitSymbolMicro m}\) follow the estimated gas velocity very well. Furthermore, this  
\[\tau_{\mathrm{d}}=\frac{\rho_{\mathrm{d}}\ D_{\mathrm{d}}^{2}}{18\ \mu_{\mathrm{g}}}, \tag{3}\]2 Estimation of the gas velocity \(u_{\mathrm{g}}\) as a function of the distance downstream of the atomiser nozzle \(x\) and the numerically derived response of droplets of \(4,10\) and \(20\,\mathrm{\SIUnitSymbolMicro m}\) in diameter. approach yields directly the estimated measurement error at position 0 at a distance of \(x_{0}=50\,\mathrm{mm}\) downstream of the atomiser:  
\[\tau_{\mathrm{d}}=\frac{\rho_{\mathrm{d}}\ D_{\mathrm{d}}^{2}}{18\ \mu_{\mathrm{g}}}, \tag{3}\]3  
In view of the strong estimated deceleration at the near-nozzle region, the measurement accuracy is acceptable.