## 5 Result Analysis  
### Numerical Analysis before and after Optimisation  
As shown in Figure 18a, in the OGVs inlet direction, the airflow is separated at the ventral surface of the guide vane because of the influence of the air admittance attack angle, which leads to obvious separation vortices and backflow, occupying a larger flow space in the vane channel. Figure 18b shows the optimised flow diagram. After optimising the OGVs, it can be seen from the chart that the backflow phenomenon is reduced, and there is no apparent vortex in the ventral part of the optimised OGVs; the flow turbulence is improved, the airflow in the runner is more uniform, and no evident separation flow is produced. Figure 17: Leaf type comparison. Figure 18: Original and optimised streamline. (**a**) Original. (**b**) Optimized. In order to have a better macroscopic probe of the flow separation phenomenon on the blade surface, this section uses the Q-criterion surface to illustrate the flow vortex state of different blades. The Q criterion can be expressed as follows:  
\[Q=\frac{1}{2}\left(\omega_{ij}^{2}-\sigma_{ij}^{2}\right) \tag{13}\]  
\[\omega_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}-\frac{ \partial u_{j}}{\partial x_{i}}\right) \tag{14}\]  
\[\sigma_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}\right) \tag{15}\]  
where \(\sigma_{\mathrm{ij}}\) is the rate of change of strain, \(\omega_{\mathrm{ij}}\) is the vortex amplitude, and \(u_{\mathrm{i}}\) and \(u_{\mathrm{j}}\) are the components of the flow field velocity in the \(i\) and \(j\) directions, respectively. As shown in Figure 19a, due to the turbulent airflow before the optimisation of the OGVs shape, there is a large amount of backflow phenomena in the OGVs duct, and large vortices exist on the ventral surface of the OGVs. In contrast, the vortices shed at the top of the OGVs mix with the vortices leaking from the moving blades, intensifying the development of vortices in the dynamic-static coupling duct, and the vortex is difficult to dissipate as the flow develops. As shown from Figure 19b, after optimisation of the OGVs, the number of vortices in the dynamic-static coupling duct, and the OGVs duct is significantly reduced. According to vortex theory, turbulent vortices are the main contributors to fan noise. For isentropic flow at low Mach numbers, the vortex sound equation is:  
\[\left\{\frac{1}{c_{0}^{2}}\frac{\partial^{2}}{\partial t^{2}}-\
abla^{2} \right\}B=div(\omega\times v) \tag{16}\]  
where \(B\) represents the total enthalpy of the fluid, \(w\) represents the vortex vector of flow in the flow field, and \(v\) represents the yes vector. As the stretching and breaking of vortices is a fundamental factor in the generation of the flow field, the study of vortices is a priority in fan design as unreasonable turbulent vortices can produce unstable pressure pulsations and consequently losses in the flow field. Figure 20 compares the vorticity at each section of the spans before and after optimisation. As shown in Figure 20a,b, at 0.9 span, after the shedding vortex at the trailing edge of the moving blades before optimisation enters the OGVs duct, some air at the top  
Figure 19: Q-Criterion diagram before and after optimisation. **(a)** Original. **(b)** Optimized. of the OGVs flows from the pressure surface to the suction surface and then breaks away from the vane. As shown in Figure 20c,d, at 0.5 span, the vortex at the top of the vane further develops along the OGVs driven by the main stream, forming a stable vortex in the abdomen of the OGVs, resulting in a deterioration of the flow field. As shown in Figure 20e,f, at 0.1 span, the combined effect of the vortex coming off the trailing edge of the leading vane and the incidence of the trailing vane results in significant vortex generation in the dynamic-static coupling duct. The optimised OGVs have a more reasonable air admittance attack angle, and the separation vortices appearing at the OGVs tip are significantly reduced.