# On the Link between Diesel Spray Asymmetry and Off-Axis Needle Displacement
## Materials and Methods
### Cavitation Model Assessment  
The cavitation model adopted for the current investigation and the related settings have been assessed simulating two reference cases, i.e., the experiment proposed by [PERSON] et al. [46] and the more recent experiment proposed by [PERSON] et al. [47]. As recently reported in [19], many other contributors [48; 49] have considered the experiment of [PERSON] as a significant reference case. This experiment is based on the investigation of an optically accessible channel (inlet section 0.300 \(\times\) 0.301 mm; outlet section 0.284 \(\times\) 0.301 mm; 1 mm length, 0.02 mm inlet curvature radius). During the experimental research, the inlet pressure was fixed (10 MPa), at a uniform fuel temperature of 300 K, whereas the outlet pressure was adjusted in order to obtain the required flow rate. The same technique has been applied in the simulations. Figure 6 (left) shows the computational grid used in the cavitation model assessment; it represents 1/4 th of the real channel geometry, thanks to symmetry of domain. The flow behavior of the throttle has been positively tested in several operating points. On one side, the hydraulic prediction has been found to be good agreement with experiments, as visible in Figure 6 (right) and in Table 2, which reports the details of mass flow rate in the conditions evidenced by [46] (cavitation incipience--Point A, critical cavitation--Point B and super cavitation--Point C). On the other side, the cavitation patterns related to the three fundamental conditions, as evidenced in [46], have been consistently reproduced, as reported in Figure 7; it has to be explicitly mentioned that the cavitation patterns observed by [PERSON] were fluctuating, due to the highly turbulent flow environment and they have been obtained by an ensemble averaging process of a set of twenty images taken under the same operating conditions. Moving to the second assessment phase, the experiment of [PERSON] [47] has been reproduced by simulation. The experiment consists of a flow through a transparent channel allowing the visualization \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Conditions**} & \multicolumn{2}{c}{**Cavitation Start (A)**} & \multicolumn{2}{c}{**Critical Cavitation (B)**} & \multicolumn{2}{c}{**Super Cavitation (C)**} \\ \cline{2-7}  & **Experiment** & **Model** & **Experiment** & **Model** & **Experiment** & **Model** \\ \hline Pressure & 57.0 & 57.0 & 65.0 & 65.0 & 80 & 80 \\ difference (bar) & 7.21 & 7.1 & 7.72 & 7.60 & 7.72 & 7.63 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mass flow conditions in throttle channel. Figure 6: Computational grid used to model the experiment of [PERSON] et al. [46] (**left**); mass flow within the throttle channel (**right**). Figure 7: Cavitation at throttle flow conditions (**A**, **B**, **C**); simulated cavitation regions (**left**); experiment of [PERSON] et al. [46] (**right**); CS: cavitation start; CC: critical cavitation. (based on Laser Doppler Velocimetry) of cavitation incipience and development. In the experiments, a water flow is produced. The discharge ambient is kept at a constant pressure (ambient pressure) and the inflow condition is varied to realize different flow regimes, represented by the mean liquid velocity (\(V_{\rm n}\)) in the channel. The incipience and the development of cavitation is well reproduced by the simulation, as indicated by the contours of liquid mass fraction shown in Figure 8. The quantitative comparison between the experiment and the simulation is reported in Figure 9, where the available data on velocity profile provided by [PERSON] (2016) are compared with the simulations. The trends are referred to three different axial locations along the throttle axis, as evidenced by the sketch of throttle layout of Figure 9. Good agreement has been found in all of the cases. Figure 8: Cavitation at throttle flow conditions; simulation LVF (-) (liquid volume fraction) (**right**); Experiment of [PERSON] et al.