# Analysis of Vorticity and Velocity Fields of Jets from Gas Injector Using PIV
## 3 Software and Methodology
Figure 2: Image post-processing procedure: raw image (**top-left**); CLAHE and AC algorithm (**top-right**); coordinate center reference system (**bottom-left**) and IA definition (**bottom-right**). However, it is worth noting that the PIV provides an indirect measurement of flow velocity; therefore, this technique is subject to a certain degree of uncertainty, which can be estimated from Taylor series propagation and expressed by the equation:  
\[\left(\frac{U_{u}}{u}\right)^{2}=\left(\frac{U_{AX}}{\Delta X}\right)^{2}+\left( \frac{U_{AI}}{\Delta t}\right)^{2}+\left(\frac{U_{M}}{M}\right)^{2} \tag{2}\]  
where \(u\) is the velocity at a point (x,y), calculated by measuring the displacement \(\Delta\)x during time interval \(\Delta t\), and \(Uu\) represents the corresponding uncertainty. Then, the three variables refer to the uncertainty due to the displacement \(U_{\Delta X}\), laser pulse separation \(U_{\Delta t}\) and magnification U\({}_{\mathrm{M}}\) contribute. The last two parameters depend on laser characteristics (normally \(U_{\Delta t}\) is around 1 ns, leading a negligible errors for most experiments) and correct calibration of the camera with respect to the optical window, that for the present study has been calibrated for adjusting the height of the CVC using spirit levels and monitoring the spatial resolution with respect to the injector tip size. Differently, the first variable depends on the error between real displacement and that measured. Since it is not possible to know a priori the true displacement, the uncertainty quantification is demanded to different theoretical models or by specific experiments in controlled environment [32]. Nevertheless, the accuracy of the PIV can be improved in many ways; among these, the use of multi-pass cross-correlation algorithm has been proven to reduce the error between measured and true displacement. Furthermore, the Stokes number (\(Stk\)) can be used to quantify a particle's ability to follow the flow. It is defined as follows:  
\[Stk=\frac{\rho_{p}d_{p}^{2}U}{18\mu L} \tag{3}\]  
where \(\rho_{p}\) is the seeded particle density (vegetable oil \(\approx\) 900 kg/m\({}^{3}\)), \(d_{p}^{2}\) the diameter (\(\approx\)600 nm), \(U\) the velocity, \(\mu\) the dynamic viscosity (1.98 \(\times\) 10\({}^{-5}\) Pas) and \(L\) the characteristic length, which depends on orifice size. Regarding the speed, considered the condition with a \(\Delta\)P of 19 bar (P\({}_{\mathrm{CVC}}\) 1 bar-P\({}_{\mathrm{INJ}}\) 20 bar), in proximity of the injector tip the flow can be assumed to be in choked conditions (See Equation (5)); then, the equation to calculate the velocity becomes  
\[U=\sqrt{\gamma RT} \tag{4}\]  
where \(\gamma\) is the specific heat ratio of the gas (\(\approx\)1.66 for helium), \(R\) the gas constant (2077 J/kgK) and \(T\) the temperature (300 K). The calculated speed is of approximately 1020 m/s. Since, at the early stages the jets emerging from the orifices are still separated, the characteristic length used in the \(Stk\) formula is set equal to the equivalent diameter of the single orifice. For example, for the first geometry, the characteristic length is 0.3 mm, and the calculated \(Stk\) is greater than 1, suggesting a slow particle response to flow acceleration, which justifies the presence of the weak correlation area. The situation rapidly changes downstream of the tip (5-15 mm depending on PR), where the maximum velocity recorded by the software decreases to 250-300 m/s and the individual jets merge into a single structure with an average diameter ranging from approximately 1.8 to 3.8 mm, depending on the nozzle geometry and operative condition. In particular, at PR20 in correspondence of the peak velocity, the Stokes number falls within a much lower range, from 0.07 to 0.09, indicating the particles' ability to accurately follow the flow under subsonic conditions.