## 3 Results and Discussion
### Discussion of Velocity Fluctuations
In literature, high-spatial-resolution PIV is presented as a promising technique for investigation of turbulent fluctuations in dispersed multiphase flows [33]. Subsequently, it will be investigated whether an estimation of the turbulent fluctuations of the gas phase or even the turbulent slip velocity is possible based on the present microscopic imaging data. First of all, the theoretical analysis of particle response in an oscillating gas phase presented by [PERSON] et al. [47] is adapted to turbulent fluctuations. The turbulent gas velocity is simplified by a simple sinusoidal oscillation  
\[u_{\mathrm{g}}=\bar{u}_{\mathrm{g}}+\hat{u}_{\mathrm{g}}\sin\omega t, \tag{13}\]  
which is defined by the amplitude \(\hat{u}_{\mathrm{g}}\) and the angular frequency \(\omega\). The mean velocity \(\hat{u}_{\mathrm{g}}\) is set to \(0\,\mathrm{m}\,\mathrm{s}^{-1}\). Stokes flow is assumed analogous to the analysis of the mean gas velocity. By inserting Equation (13) into Equation (4), the oscillation of the droplet velocity after reaching the steady-state condition can be derived  
\[u_{\mathrm{d}}=\frac{\hat{u}_{\mathrm{g}}}{\sqrt{1+\tau_{\mathrm{d}}^{2} \omega^{2}}}\sin\left(\omega t+\arctan\left(\tau_{\mathrm{d}}\omega\right)\right). \tag{14}\]  
In this study, only the amplitude of the resulting droplet velocity oscillation is of interest. Hence, the function  
\[f(\hat{u}_{g},\omega,C)=\frac{1}{\sqrt{2}}\left(\frac{\hat{u}_{\mathrm{g}}}{ \sqrt{1+\tau_{\mathrm{d}}^{2}\omega^{2}}}+C\right) \tag{15}\]  
Figure 13: Exemplary probability distributions of the radial velocity for different droplet sizes at \(x\leavevmode\
obreak\ =\leavevmode\
obreak\ 50\,\mathrm{mm}\) (**a**) and standard deviation of the radial velocity at different distances downstream of the atomiser nozzle \(x\); (**b**) a curve proportional \(D^{-2}\) is used to fit the data at \(x\leavevmode\
obreak\ =\leavevmode\
obreak\ 50\,\mathrm{mm}\). is employed for fitting the experimental data. The amplitude is scaled by the factor \(\frac{1}{\sqrt{2}}\) in order to enable a direct comparison to the experimentally available standard deviation. Furthermore, the desired dependency of this amplitude function on the droplet diameter is represented by the droplet relaxation time \(\tau_{\mathrm{d}}\), which is affected additionally by the density of the droplets \(\rho_{\mathrm{d}}\) and the dynamic viscosity of the gas phase \(\mu_{\mathrm{g}}\). The constant \(C\) was found to be necessary in order to achieve a sensible fit. This offset is supposed to result from fluctuations of the gas velocity, which are characterised by lower frequencies. Hence, all droplets in the presented size range are assumed to follow those low frequency fluctuations almost perfectly. The fit of \(f(\hat{u}_{\mathrm{g}},\omega,C)\) from Equation (15) is found to be in excellent agreement to the experimental data as shown in Figure 14. The desired information about the underlying oscillation by means of \(\hat{u}_{\mathrm{g}}\) and \(\omega\) can be estimated with reasonable accuracy, which is graphically illustrated by 95% confidence intervals for the fitted function. In addition, the resulting parameters of \(f(\hat{u}_{\mathrm{g}},\omega,C)\) are summarised in Table 3.  
The cut-off Stokes number  
\[St_{\mathrm{cut-off}}=\tau_{\mathrm{d}}\,f=\frac{1}{2\pi} \tag{16}\]  
can be determined using the derived frequency of the oscillation \(f\), as fundamentally introduced in work of [PERSON] et al. [47] for studying the particle response in oscillating flows. This cut-off Stokes number may be used analogous to the cut-off frequency of a first order low pass filter in order to assign different regimes of droplet motion in a turbulent gas flow. In the case under consideration, the droplet diameter of \(4.9\,\mathrm{\SIUnitSymbolMicro m}\) is determined using Equation (16), which is graphically illustrated in Figure 14. Hence, the droplets in this size range can be used as an indicator for turbulent fluctuations of the  
Figure 14: Fit to the standard deviation of the radial velocity.