## 3 Results and Discussions
### Small-Amplitude Artificial Perturbations  
Figures 7 and 8 show the numerical smoked foils with different wavelengths of the artificial perturbations. The corresponding cell length variation is shown in Figure 9. Note that the cell length \(L\) is approximately 1.5-2 times the cell size \(\lambda\). In this section, the small amplitude of artificial perturbations is set as 15 K for all cases. When the artificial temperature perturbations are induced, distortion appears on the detonation front only after a very short distance of a few induction lengths, and the transverse waves are also formed with extremely regular cell pattern as shown in Figure 7. It is observed that the cell size approximately equals to the wavelength \(s\) (10 \(\Delta_{\mathrm{I}}\)), indicating the transverse wave evolution is controlled by the artificial perturbations. However, it is also found that the transverse instability starts to amplify itself again later at a position where \(x=400\)\(\Delta_{\mathrm{I}}\). The amplification is finishedat a distance of 700 \(\Delta_{\rm I}\) as a fully-developed cellular detonation is finally formed. It suggests that the artificial perturbations suppress the amplification of the cellular instability for a certain distance, but cannot terminate the process since the wavelength of the artificial perturbations (\(s=10\)\(\Delta_{\rm I}\)) is much smaller than the transverse wave spacing (40-50 \(\Delta_{\rm I}\)).  
Figure 5: Numerical smoked foils without artificial perturbations, (**a**) \(k_{\rm R}=2\) s\({}^{-1}\), (**b**) \(k_{\rm R}=3\) s\({}^{-1}\), and (**c**) \(k_{\rm R}=4\) s\({}^{-1}\).  
As the wavelength increases to \(s=20\)\(\Delta_{\rm I}\), as shown in Figure 8a, a similar process can also be found compared to the case with \(s=10\)\(\Delta_{\rm I}\). The main difference is that the amplification is delayed, indicating the increase of wavelength enhances the suppression of cellular instability. As the wavelength increases to \(s=40\)\(\Delta_{\rm I}\) that is close to transverse wave spacing (approximate \(50\)\(\Delta_{\rm I}\)) controlled by the intrinsic instability in a homogeneous mixture, the cellular instability synchronizes with the artificial perturbations immediately, as shown in Figure 8b. Only at the very end of the channel, the cellular instability appears with the presence of an irregular cellular pattern (or phase variation), which is more obvious in cases with high instabilities. The cell size almost remains the same because of the same wavelength, as shown in Figure 9. However, the increase in the instability factor raises the scattering of the cell length. It can also be found that these processes can be shifted to an early date if the instability factor is increased. Thus, the evolution process is dominated by two competing factors, the intrinsic instability and the wavelength of the artificial perturbations. However, due to a fixed small value of amplitude (\(15\) K), a fully-developed cellular detonation can always be obtained in the channel, and the nature of the detonation almost agrees with that in cases without artificial perturbations.  
Scattering of the pressure oscillation is reduced because of the presence of the artificial perturbations. However, it is found that the frequency is increased, which is also evidenced by the increase of the fluctuation on cell length, as shown in Figure 9. The temperature perturbations can be considered as periodically spaced hot and cold spots. The hot spot promotes the detonation by increasing the reaction rate, while the cold spot prohibited the detonation by reducing the reaction rate. However, the negative effect is stronger than the positive one because the reaction rate has an Arrhenius form. Thus, the presence of the artificial temperature perturbations can prohibit the growing of the front, resulting in a lower pressure oscillation, but also renders the front more discrete (or equivalently the cell size has a large scattering), as shown in Figure 10.