# Effects of Diverging Nozzle Downstream on Flow Field Parameters of Rotating Detonation Combuster
## 2 Numerical Model and Method
### Independence Test and Model Validation  
Before performing the numerical investigation, it was necessary to conduct the independent validations of grid size and time steps. As shown in Figure 2, the multi-block structured grids are generated by using ANSYS ICEM to discretize the computational domain. Taking the RDC with \(\theta=30^{\circ}\) as an example, Tables 1 and 2 compare the variations of RDC key parameters (propagating period of detonation wave trot, average outlet total pressure, and average outlet total temperature) under different grid sizes (ranging from 0.25 mm to 2.0 mm) and time steps (ranging from 0.05 \(\upmu\)s to 0.4 \(\upmu\)s). Additionally, mesh refinements (to 1 mm) were applied in the diverging region to more accurately simulate the flow field characteristics at that location. As proved by the results, they can be independent when grid size is 1.2 mm and time step is 0.2 \(\upmu\)s, and the computational mesh is about 1.98 million grids.  
In this study, considering the characteristics of detonation combustion and the waves, the model validation was conducted to compare the complex waves captured to the results of Innovative Scientific Solutions Inc. (ISSI), and then compare the propagation speed of detonation wave quantitatively to the CJ speed value of the NASA program. This kind of validation method has been used in many literatures [10; 11; 36; 46]. Figure 3 compares the results of ISSI [47] with the results in this study. The results show that the present numerical method can capture the main features of the detonation wave, including the complex waves (such as detonation wave A and oblique shock wave B) and the typical structure of the combustion flow field (such as mixing region C, expandable region D, discontinuous region E between production and premixture, blocking region F, and fresh premixture region G). Additionally, Figure 4 shows the time-varied static pressure at the monitoring point (127.4 mm, 0 mm, 147 mm). According to the steady changing characteristic of the rotating detonation wave, it was calculated that the propagation speed of the detonation wave was about 1795.3 m/s, and the corresponding CJ theory value of the NASA program is 1820 m/s [45], of which the deviation is only 1.36%. In addition, for the RDC using methane-air as fuel, it is important to consider whether the annular width of RDC is reasonable. Many researchers have concluded that the width is related to the detonation cell [30; 48], and pointed out that the critical width is the cell size divided by \(\pi\)[48]. According to the latest research results for the quantitative relationship between the cell size and the initial pressure by Purdue University [49], as is shown in Figure 5, the cell size is greatly affected by the initial pressure. From their results we can see clearly for methane-air, in the initial pressure of 1 MPa, the detonation cell size was reduced to 30 mm, which means the width size of only 10 mm is suitable for the initial pressure of 1 MPa. Furthermore, in this paper, the width of annular 20 mm was large enough for detonation in the initial pressure of 1.209 MPa. Therefore, the present numerical approach is reasonable to investigate the characteristics of rotating detonation wave.  
Figure 3: Comparisons of (**a**) the result of Innovative Scientific Solutions Inc. (ISSI) [47] with (**b**) numerical simulation in this study. (A: detonation wave, B: oblique shock wave, C: mixing region, D: expandable region, E: discontinuous region between detonation product and fresh mixture, F: blocking region, G: fresh mixture region).