## 4 Results and Discussion
### The Mixing Layer Effect with Air Leaking into the Working Fluid  
[PERSON] et al. [38] first investigated the mixing layer effect between primary and the secondary fluids using the CFD method and pointed out that the mixing layer strongly influences the entrainment ratio.  
The mixing layer forming process under the operating conditions of test 2 (see Table 3) is shown in Figure 15 (adapted from [38]). The primary fluid with a larger velocity meets the secondary fluid with a smaller velocity at the nozzle outlet. This forms a mixing layer with varying thickness \(\delta\). The difference in velocity between the primary and secondary fluids leads to a change in the mixing layer thickness. The mixing layer growth rate \(\delta^{\prime}\), defined as \(d\delta/dx\), is an important parameter that indicates that the larger the mixing layer growth rate, the more secondary fluid is drawn into the mixing layer. Therefore, the entrainment ratio also varies with the change in the mixing layer growth rate.  
Many scholarly studies have investigated the mixing layer effect and summarized some empirical correlations for the mixing layer growth rate. The following formulas can calculate the mixing layer growth rate. The growth rate of an incompressible mixing layer, \(\delta^{\prime}_{0}\), can be calculated by this correlation proposed by [PERSON] [39]:  
\[\delta^{\prime}_{0}=C_{\delta}\frac{(1-\phi)(1+\Omega^{1/2})}{2(1+\Omega^{1/2 }\phi)}\left\{1-\frac{(1-\Omega^{1/2})/(1+\Omega^{1/2})}{1+2.9(1+\phi)/(1-\phi )}\right\} \tag{12}\]  
where the \(\Omega=\rho_{2}/\rho_{1}\), and \(\phi=u_{2}/u_{1}\) and \(C_{\delta}=0.35\) is a constant.  
As we adopted the ideal gas model in this study, we could not ignore the effect of turbulent compressibility on the mixing shear layer. The following expression suggested by [PERSON] et al. [40] can calculate the growth rate of a compressible mixing layer:  
\[\delta^{\prime}=(1+\alpha_{0}\Pi_{c}^{2})^{-\beta_{0}}\cdot\delta^{\prime}_{0} \tag{13}\]  
where \(\alpha_{0}=4\) and \(\beta_{0}=0.5\) are approximate values.  
\[\Pi_{c}=\max\left[\frac{\sqrt{\gamma_{i}-1}}{a_{i}}\right]\cdot\Delta u \tag{14}\]  
is a parameter of correlating the compressibility effect. \(\gamma\) is the ratio of specific heats (\(\gamma=1.3\) in this study), \(a\) is the sound speed in the working fluid, and \(i\) is either side of the primary or secondary fluid. Once mixed into the primary or secondary fluid, the air with different mass fractions changed the density, velocity, and sound speed of both fluids, as indicated in Tables 4 and 5, leading to a change in the mixing layer growth rate.  
Figure 15: Velocity vector schematic of mixing layer formation adapted from [38].  
Based on the physical parameters of the primary and secondary fluids in Tables 4 and 5, the mixing layer growth rate is calculated by the above formulas and listed in Table 7. The results showed a positive correlation between the mixing layer growth rate and the entrainment rate. The mixing layer growth rate and entrainment ratio gradually reduced as of \(Y_{1}\) increased, while they increased as \(Y_{2}\) increased. The air leaking into the primary fluid inhibited the mixing effect between the primary and secondary fluids; the larger the air mass fraction in the primary fluid was, the lower the mixing layer growth rate and the entrainment ratio were. However, the opposite results were obtained when air with a larger mass fraction leaked into the secondary fluid. The above findings confirmed by the mixing layer effect were consistent with those acquired by the mass flux. The coefficient distortion (Cd) rate was regarded as the degree of deviation for the measured data from their standard value. The Cd rate of \(Er\) values listed in Table 6 indicate that \(Y_{1}\) has a more significant effect on the entrainment ratio than \(Y_{2}\) at the same air mass fraction.