## 3 Results and Discussion
### Effects of Radial Gap Ratio on Vaned Diffuser Performance  
The Mach number value is one of the important properties of the flow which is a significant factor for the occurrence of shock waves. Figure 16a,b show the absolute Mach numbers of different radial gap ratios for M\({}_{\text{u}}\) = 1.16 at the impeller trailing edge and the diffuser leading edge, respectively. Figures 17 and 18 represent the pitch-averaged spanwise contour of the relative Mach number superimposed with isolines at 50% span for a peak efficiency point of M\({}_{\text{u}}\) = 1.16 (design speed) and M\({}_{\text{u}}\) = 0.69, respectively. Decreasing the radial gap size leads to a reduction in the diffuser throat area. As the Mach number is less than 1, a smaller diffuser throat area leads to an increase in the flow velocity value at the interface. Thus, it can be seen that the Mach number at the trailing edge of the impeller and leading edge of the diffuser increases by reducing the radial gap ratio for both M\({}_{\text{u}}\geq 1\) and M\({}_{\text{u}}\) < 1. It is important to note that the Mach number at trailing edge of the impeller changes slightly. Figure 16: Absolute Mach number at: (**a**) impeller trailing edge; (**b**) diffuser leading edge at the peak efficiency point, M\({}_{\text{u}}\) = 1.16 (design speed). Figure 15: Stable flow ranges (SFR) for five tip-speed Mach numbers and six various radial gap ratios. Figure 19 represents the effect of the radial gap ratio on the shock wave structure on the basis of absolute Mach number contours superimposed with isoclines at the mid-span of the vane diffuser at \(\mathrm{M_{u}=1.16}\). As the flow enters the diffuser, it accelerates to a supersonic speed, which leads to shock waves. These shock waves' structures are changed by varying the radial gap ratio. The shock wave starts occurring from the diffuser vane leading edge, trespassing on the suction side of the diffuser vane. By increasing the intensity of the shock wave, the pressure in the region of the suction side and the pressure side increases and decreases, respectively. Therefore, the flow becomes non-uniform and flow separation occurs. Hence, separation in smaller radial gap sizes happens earlier, and it is bigger than that of higher radial gap size (Figure 20). [PERSON] et al. (2016) also showed that the compressor performance deterioration is the result of this supersonic acceleration and notable shock waves interaction with boundary layers. Figure 17: The pitch-averaged spanwise contour of the relative Mach number superimposed with isolines at 50% span at a peak efficiency point of \(\mathrm{M_{u}=1.16}\) (design speed) for various radial gap ratios. Figure 18: The pitch-averaged spanwise contour of the relative Mach number superimposed with isolines at 50% span at a peak efficiency point of \(\mathrm{M_{u}=0.69}\) for various radial gap ratios. However, according to Figure 20, as the impeller inlet tip-speed Mach number is less than the unity value (M\({}_{\text{u}}\) = 0.69), the flow does not accelerate substantially. Therefore, the absolute Mach number remains below 1 and the shock waves do not occur. Figure 21 also represents the narrow distribution of the absolute Mach number at 50% span for a radial gap ratio of 1.04 compared to a radial gap ratio of 1.14 at M\({}_{\text{u}}\) = 1.16 (design speed) and M\({}_{\text{u}}\) = 0.69. At M\({}_{\text{u}}\) = 1.16, by enlarging the radial gap size, the flow velocity increases and decreases in the region of the suction surface and the rear part of the pressure side, respectively. Regarding the fact that the flow momentum is directly related to the flow velocity in an equal mass flow rate, low momentum flow is weakened in the rear region of pressure side for the smaller radial gap size. The load on the rear part of the pressure surface reduces in a larger radial gap size. Therefore, the diffuser can have superior diffusion [18].