# Analysis of Vorticity and Velocity Fields of Jets from Gas Injector Using PIV
## 4 Results and Discussion
### Vorticity Intensity  
Considering any point traveling along with the flow under investigation, the vorticity intensity is defined as the pseudovector field which describes the local spinning motion of such point. Mathematically, it can be expressed by the equation,  
\[\omega\equiv\
abla*v \tag{7}\]  
where \(\
abla\) is the nabla operator and \(v\) the flow velocity. In a two-dimensional analysis, Equation (7) becomes  
\[\omega=\left(\frac{\partial v_{y}}{\partial x}-\frac{\partial v_{x}}{\partial y }\right)*e_{z} \tag{8}\]  
where \(e_{z}\) is the versor along the zeta-axis. During the post-processing of the images the vector field was observed not to change in a significant way, in terms of development pattern, with the pressure ratio. These results, coupled with the large amount of data recorded, led to the decision to illustrate in this section solely the images related to the operative condition at PR4. Specifically, graphs shown in Figures 8-10 were drawn from an ensemble of 20 pairs of frames. Figure 8 shows the vector plots and corresponding vorticity distributions produced using the first nozzle tip geometry. At early stages, two vortexes were visualized on the head section of the jet and the instantaneous center of these rings corresponds to the position of the vorticity peaks. Here, the vorticity intensity reaches values of approximately 85,000 1/s. In a 3D perspective, these two vortices can be assumed to be connected by a toroidal ring surrounding the jet head. The acceleration caused by the continuous supply of helium causes the region downstream the nozzle tip to be characterized by increased vorticity during the entire duration of the injection. This region was noted to extend as pressure ratio increased, from just over 20 mm to almost the entire extent of the jet (\(\approx\)70 mm). After 4.0 ms, two discontinuities separating the steady state region from the head region of the jet were visualized. In these regions the air is dragged inside the jet with an oblique direction that deviates the helium towards the center. This phenomenon leads to the flow \"disconnection\" between the head section and the rear area. Without the supply of helium, the two vortices positioned on the head therefore become unable to sustain the rotative motion, rapidly dissipating their vorticity intensity. At 5.0 ms aSOE, the only relevant regions for the interaction between gas-air remain that close to the nozzle tip. There, the constant supply of helium contributes to the establishment of small eddies which size is lower than the resolution applied for the present study (\(\approx\)0.8 \(\times\) 0.8 mm), thus impossible to be individually recognized. It is worth noting that the first geometry is the least-oriented one and that in agreement with jet models present in the literature, i.e., [34; 35], it clearly shows the presence of two vortices surrounding the jet head from the early stages. These vortices trigger air entrainment from the side regions and gradually lose intensity as the jet propagates. Figure 8: GEO–1, vector plot (**left**) and vorticity distribution (**right**) at 3.4, 4.0 and 5.0 ms aSOE. PR4, averaged frames. Figure 9: GEO–2, vector plot (**left**) and vorticity distribution (**right**) at 3.4, 4.0 and 5.0 ms aSOE. PR4, averaged frames. Figure 10: GEO–3, vector plot (**left**) and vorticity distribution (**right**) at 3.4, 4.0 and 5.0 ms aSOE. PR4, averaged frames. Figure 9 provides a closer look at the jet produced for the second geometry. Since early stages, the two jets coming from the inner and external orifices work differently in terms of interaction with air. The external orifice, which is the most oriented one (18\({}^{\circ}\)), generates an oblique jet able to keep momentum better than inner component, resulting in a greater extension. In contrast, on the inner side the vector plot shows the presence of a complete vortex. At 4.0 ms aSOE, the inner ring starts losing intensity for similar reasons to that seen for the first geometry.