## 1 Introduction
Due to the non-premixed injection involved in the mixing process and the mixing time being extremely transient because of the short period of RDW propagation in the combustor, it is difficult to achieve the ideal mixing degree for fuel and oxidizer as premixed injection does. As a result, the mixture of fuel and oxidizer is non-homogenous in the fresh gas layer ahead of detonation wave, which makes the RDW structure is not as distinct as that of premixed injection condition and the stability of RDW is also weakened correspondingly. To minimize the above weakness of non-premixed injection in RDC, two significant problems have to be solved: (1) how to make fuel and oxidizer mix sufficiently and rapidly in limited space, (2) how to minimize the pressure loss during mixing process so as to guarantee the performance of RDC. Using H2/air as reactant, [PERSON] et al. [18] numerically investigated three-dimensional RDW with non-premixed injection and compared the parameters of RDW with experimental data. The numerical results indicated that the fresh reactant height in front of RDW is approximately 100 mm\(\sim\)150 mm under their calculated condition, which was favorable according to experimental measurement data [19]. [PERSON] et al. [20; 21] simplified the cylindrical RDC into cuboid-shaped combustor and conducted three-dimensional simulation to explore the mixing characteristics of H2 and air in combustor with different inlet structures. They thought that fuel injected by multiple holes was benefit for increasing mixing quality for hole/slot injection scheme. Taking H2 and air mass flow rate, H2 injection position and back pressure on the outlet into account, [PERSON] et al. [22; 23] investigated the effect of different factors on mixing characteristics in RDC. The results revealed that moving the H2 injecting position into air intake slot and increasing back pressure would enhance the mixing efficiency and quality. Besides, they pointed out that the number of H2 injecting holes exist at an optimum value which made the H2 and air mixing quality be the best when the air injecting area is kept constant. By means of Large Eddy Simulation (LES), [PERSON] et al. [24] numerically investigated an injection element of RDC to study H2/O2 mixing characteristic expecting to reflect the whole mixing situation of RDC. They believed that shear force and collision effect must be taken into account for inlet structure design of combustor to achieve the favorable mixing performance. [PERSON] et al. [25] firstly reported the successful air-breathing premixed RDE. The premixed flow maintained a boundary layer velocity gradient that successfully arrested flashback. Then, they showed the design principles of quenching distance and critical boundary layer velocity gradients and described the safe operation region for a premixed RDE when adjusted for experimental conditions and applied in unison [26]. Hydrogen was widely used in RDC experiments and numerical simulations due to its high detonability and calorific value compared to other fuels. However, as one of the most common fuels, CH\({}_{4}\) is little if anything chosen as the propellant in RDC. Considering CH\({}_{4}\) is abundant in nature, moderate detonability and low explosion hazard compared to H\({}_{2}\), it can possibly become the fuel for RDC in the future. However, one problem has to be faced which is that the critical ignition energy of CH\({}_{4}\) is much higher than H\({}_{2}\). Therefore, how to reduce the critical ignition energy of CH\({}_{4}\) is the first puzzle to be solved. It is well known that the optimum position for ignition is the spot where equivalence ratio is close to 1 in the combustor. This means that favorable mixing characteristic is crucial for successful ignition, and it is also a guarantee for stable propagation of RDW in the first period. Besides, lower loss of the total pressure will have finally some gain in total pressure on the combustor outlet.