# Analysis of Pressure Effect on Three-Dimensional Flame Surface Density Estimation
## 5 Results and Discussion
### Three-Dimensional Analysis  
To further clarify the isotropic assumption of the flame front, the probability density function (PDF) angles \(\theta\) extracted from the OH-PLIF images, as depicted in Figure 3c, are displayed in Figure 8. For easy comparison, the PDF of the isotropic unit normal vector, i.e., \(p(\theta)=1/(2\pi)\), is also plotted in the Figure. Firstly, the profiles are nearly symmetrical, verifying the turbulent Bunsen flame as being a roughly cylindrical shape. Secondly, there are clearly two peaks in the PDF distribution at \(\theta=-\pi/2\) and \(\theta=\pi/2\), respectively. Figure 6: Two-dimensional flame surface density and its comparison with the evaluation by M1-M4. Upper panel: 0.1 MPa; middle panel: 0.5 MPa; lower panel 1.0 MPa. Figure 7: The variation in two–dimensional flame surface density and the evaluation with the mean progress variable, at 0.5 MPa. Only this case is shown here due to brevity, since the behaviors observed under three pressures are the same. As illustrated in Figure 3c, \(\theta=-\pi/2\) (\(\pi/2\)) corresponds to the flame front propagating outwards from the burner axis. Therefore, this bimodal shape indicates that the flame structure can be viewed as small-scale wrinkles superimposed on large-scale wrinkles, which are then superimposed on a laminar Bunsen flame. At last, and very importantly, it is evident that the PDF profile of \(\theta\) is very different from isotropy assumption and the deviation from isotropy is smaller for flames at higher pressures. This indicates that the wrinkled flame front is closer to being isotropic at high pressures. Thus we expect that the 3D FSD models described in earlier section are more reliable at high pressures. Figure 9 presents the mean components of the normal vectors for different pressures. For all pressures, both \(n_{x}^{2D}\) and \(n_{y}^{2D}\) are not uniform within the flame zone and are both virtually symmetric. Moreover, the mean components decrease with pressure because the flame front is more wrinkled and isotropic [15; 32]. This indicates the flame front is more wrinkled at high pressure. When comparing the two components in Figure 9, we can see that \(n_{y}^{2D}\) is much smaller and more uniform for all pressures. This would be due to the following: (1) Triangle geometry of the Bunsen flame resulting from the high tangential velocity downstream; (2) The fact that the flame fluctuation in \(y\) direction is relative weak compared to that in \(x\) direction. Figure 8: Probability density function (PDF) of angles \(\theta\) obtained from the flame front, as defined in Figure 3c, under different pressures. The PDF of the isotropic unit normal vector \(p(\theta)=1/(2\pi)\) is also plotted in the figure. The surface averaged products \(\langle n_{i}^{2D}n_{i}^{2D}\rangle_{s}\) for different pressures are shown in Figure 10. \(\langle n_{x}^{2D}n_{x}^{2D}\rangle_{s}\) is predominant, indicating that the Bunsen flame can be viewed as cylindrical and the assumption \(\langle n_{z}\rangle_{s}\) = 0 is reasonable. However, \(\langle n_{x}^{2D}n_{x}^{2D}\rangle_{s}\) is close to unity and apparently decreases downstream. This further verifies that (1) The flame wrinkles are not uniform; (2) The flame is less wrinkled near the flame's base due to the non-developed flame-turbulence interaction, which is verified by the results obtained from the high-Karlovitz-number jet flame [22]. Moreover, \(\langle n_{y}^{2D}n_{y}^{2D}\rangle_{s}\) is close to zero since \(n_{y}^{2D}\) is small. The fluctuating components \(\langle m_{i}^{2D}m_{i}^{2D}\rangle_{s}\) under different pressures are shown in Figure 11. According to Equations (16) and (17), \(\langle m_{i}^{2D}m_{i}^{2D}\rangle_{s}\) represents the contribution of the fluctuating in the third direction. \(\langle m_{x}^{2D}m_{x}^{2D}\rangle_{s}\) under ordinary pressure at the flame's tip is larger than other parts of the flame, while \(\langle m_{y}^{2D}m_{y}^{2D}\rangle_{s}\) is predominant nearly within the whole flame region.