# Numerical Simulation of Hot Jet Detonation with Different Ignition Positions
## 3 Results Analysis and Discussion
### Mechanism of Hot Jet Detonation Initiation  
The hot jet detonation process is divided into three stages to study according to the combustion characteristic and temperature distribution. They are the stage of hot jet formation, vortex-flame interaction and detonation wave formation respectively. The flame propagation in hot jet tubes directly impacts the jet velocity and pressure, which might further affect the following formation of the final detonation wave. Therefore, the hot jet formation stage is discussed firstly. Figure 5 shows the hot jet formation in hot jet tube. Since the hot jet tube is a smooth tube, the flame-wall boundary layer interaction is an important factor affecting the flame propagation [42; 43]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Theoretical** & **Numerical** & **Deviation** \\ \hline \(p/p_{I}\) & 21.15 & 21.87 & 3.40\% \\ \(T/T_{I}\) & 9.94 & 10.08 & 1.41\% \\ \(V_{C\!I}\) (m/s) & 1904.4 & 1909.5 & 0.27\% \\ \hline \hline \end{tabular} Note: \(p\) is detonation front pressure. \(T\) is the detonation front temperature. \(p_{I}\) is unburned pressure. \(T_{I}\) is the unburned temperature. \(V_{C\!I}\) is the detonation wave velocity.  
\end{table} 
Table 1: Comparisons of theoretical and numerical detonation wave parameters. Figure 4: Pressure curves of experiment and numerical simulation. From Figure 5a-c, high-temperature ignition source triggers laminar flame, and an expansion wave is generated during thermal mixture expansion [44]. Due to the obstructive effect of the wall boundary, the expansion waves are continuously superimposed and reflect on the flame. The flame wrinkles and the convex flame front propagates downstream at a laminar flow velocity of about 90m/s. In Figure 5d, since the pressure wave generated by the convex flame pushes unburned mixture to move towards near the wall, the high-density unburned mixture is formed near the wall and accelerates the flame propagation in this area. Then flame propagation speed near the wall increases rapidly, the contact area between flame and unburned mixture begins to decrease. Mass and thermal diffusion will also decrease. Flame front develops into an approximate plane, and flame propagation speed slows down. Figure 6 shows the field of temperature and turbulent kinetic energy at 1.575 ms. The jet flame enters the detonation chamber, so \(t_{TDC}\) is assumed to be 0 ms at this time. To further increase the jet velocity entering chamber, inspired by Ref. [18], the outlet of the hot jet tube has been designed as a sudden shrunken form as shown in Figure 6. Due to the narrow structure, convex flame propagates into the detonation chamber at a velocity of 333m/s. According to turbulent kinetic energy field, it is found that not only the flame front forms turbulence but also there are some high turbulence areas in the detonation chamber. The reason is that the pressure wave enters the detonation chamber ahead of flame and inevitably produces some disturbances in the detonation chamber. The \(Q\) criterion field is used to analyze these disturbances as shown in Figure 7. The \(Q\) criterion is defined by [PERSON] [45]:  
\[Q=(\Omega_{ij}\Omega_{ji}-S_{ij}S_{ji})/2 \tag{1}\]  
\[\Omega_{ij}=(\mu_{ij}-\mu_{ji})/2 \tag{2}\]  
\[S_{ij}=(\mu_{ij}+\mu_{ji})/2 \tag{3}\]  
where \(\Omega_{ij}\) and \(S_{ij}\) are the rotate-rate and the strain-rate tensor of the velocity components, \(\mu_{ij}\) and \(\mu_{ji}\) are the partial derivatives of the velocity in the \(x\) and \(y\)-direction. The \(Q\) criterion can describe the structural characteristics of the vortex in the flow field. Two vortexes (V\({}_{1}\) and V\({}_{2}\)) can be found at the corners of the hot jet tube in Figure 7a. The pressure wave compresses the unburned mixture inside the hot jet tube into the detonation chamber. The velocity of the moving unburned mixture is significantly faster than the surrounding fluid. The intermittent velocity causes fluctuations, and the vortex occurs after the interface of the gas layer is destabilized [46].