## 4 Results and Discussion
### Vorticity Field during Blow-Off  
To further explain the effect of the hydrogen ratio, the transport equation of enstrophy, defined as \(\Omega=\omega\cdot\omega\), was examined by looking at the different terms shown in [46].  
\[\frac{1}{2}\frac{D\Omega}{Dt}=\omega\cdot[(\omega\cdot\
abla)u]-\omega\cdot[ \omega(\
abla\cdot u)]-\omega\cdot\left[\frac{1}{\rho^{2}}\
abla\rho\times \
abla\rho\right]+\omega\cdot[\
abla\times(\
abla\cdot\tau)/\rho] \tag{9}\]  
The filtered velocity field was used to calculate vorticity. The right-hand side of Equation (9) includes four terms: vortex stretching (term 1), dilatation (term 2), baroclinic torque (term 3), and viscous diffusion (term 4). Figure 12 plots the mean value of enstrophy at \(x/d=1.5\) downstream of the flame for the four different hydrogen ratios considered in this study. The plotted time instance corresponds to the different stages during the blow-off process, as described in Section 4.3. The solid black line represents the stable operating condition, the red represents the moment when the HRR begins to decrease, and the blue represents the point at which the HRR is reduced to 75%; the green dotted lines mean that the flame is blowing-off. An increase in enstrophy over time was observed for the four hydrogen ratio cases during the blow-off sequence. When approaching blow-off, the enstrophy is expected to have a peak value. Similarly, the mean value of enstrophy decreases with the hydrogen blending.  
Figure 13 shows the results of our enstrophy budget analysis at location \(x/d=1.5\). The vortex stretching source term highlights the dynamics between the vorticity vector and strain rate tensor. When the hydrogen ratio is low (i.e., as in the 00H2 and 30H2 cases), the vortex stretching source term has a positive contribution to the enstrophy value at near-blow-off conditions. In contrast, the vortex stretching term suppresses the growth of enstrophy at near-blow-off conditions at a high hydrogen ratio (i.e., for 60H2 and 90H2). It is evident that the vortex stretching source term significantly reduces in magnitude with an increase in the hydrogen ratio, which corresponds to a decrease in the absolute enstrophy value. The dilatation term represents the variable density effects and is a source of vorticity destruction across the flame due to volumetric gas expansion. The baroclinic torque accounts for density changes in the domain and generates additional vorticity across the flame due to the misalignment between the pressure gradient and the normal flame surface. It can be seen that the magnitudes of dilatation and baroclinic torque suddenly decrease near blow-off, which is caused by the drastic change in density. The decreasing values of term 2 and term 3 are the main reason for the enstrophy increase during the blow-off sequence. When the hydrogen ratio increases, the value of each component decreases. This is because the height of the high hydrogen ratio flame is low, and the flame is always at a lower position throughout the blow-off process. Therefore, the impact of each term on the downstream flow field is not significant, and the enstrophy value is correspondingly small.  
Figure 12: Enstrophy at \(x/d=1.5\) during blow-off for different hydrogen ratios.