# Control of Corner Separation with Plasma Actuation in a High-Speed Compressor Cascade
## 3 Results  
To evaluate the control effects of plasma actuation in suppressing high-speed compressor cascade corner separations, the total pressure loss coefficient \(\omega\), static pressure rise coefficient, \(Cp\), and the blockage coefficient, \(\delta\), are defined. The total pressure loss coefficient \(\omega\) is used to evaluate the flow loss and is defined as:  
Figure 4: Distribution of induced velocity by six plasma actuators in quiescent air with different body forces of plasma actuators. \[\omega=\frac{p_{1}-p_{2}}{p_{1}-p_{1s}} \tag{4}\]  
where, \(p_{1}\) and \(p_{1s}\) are the total pressure and static pressure of the incoming flow, respectively, and \(p_{2}\) is the local total pressure. The static pressure coefficient, \(Cp\), is used to evaluate the pressure rise ability of the compressor cascade and is defined as:  
\[Cp=\frac{p_{2s}-p_{1s}}{p_{1}-p_{1s}} \tag{5}\]  
where \(p_{2s}\) is the local static pressure. The blockage coefficient, \(\delta\), is used to evaluate the flow blockage of the compressor cascade and is defined as:  
\[\delta=1-\frac{v_{x}}{v_{\mathrm{mx}}} \tag{6}\]  
where \(v_{\mathrm{mx}}\) is the axial velocity of the main flow stream. To evaluate the overall flow loss in the cascade flow passage, the outlet plane mass averaged total pressure loss coefficient, \(\omega_{\mathrm{s}}\), is defined as:  
\[\omega_{\mathrm{s}}=\frac{\iint\omega\rho v_{x}ds}{\iint\rho v_{x}ds} \tag{7}\]  
where \(s\) is the area of the chosen plane and \(v_{\mathrm{x}}\) is the axial velocity. Similarly, the outlet plane area averaged static pressure rise, \(Cps\), and the blockage coefficient, \(\delta_{\mathrm{s}}\), are defined as:  
\[Cp_{\mathrm{s}}=\frac{\iint Cpds}{\iint ds} \tag{8}\]  
\[\delta_{\mathrm{s}}=\frac{\iint\delta ds}{\iint ds} \tag{9}\]  
To evaluate the flow blockage at different blade heights, the pitch averaged flow blockage coefficient, \(\delta_{\mathrm{t}}\), is defined as:  
\[\delta_{\mathrm{t}}=\sum_{i=1}^{n}\left(\delta_{i}\cdot\Delta t_{i}\right)/t \tag{10}\]  
where \(t\) is the blade pitch. In this paper, \(\omega_{\mathrm{s}}\) and \(Cps\) as well as \(\delta_{\mathrm{s}}\) and \(\delta_{\mathrm{t}}\) are all calculated at the plane 20% chord length downstream of the blade's trailing edge. Flow Structures within the High-Speed Compressor Cascade Corner Separations and Development of the Corresponding Flow Loss  
In this part, ten planes, p0-p9, which are vertical to the blade suction surface, are used to analyze the flow structures and the total pressure loss within the high-speed compressor cascade passage. With the distributions of streamwise vorticity on planes p1-p9, Figure 5 shows the main flow structures within the high-speed compressor cascade corner separation and the development of the corresponding flow loss at the incidence angle of 0\({}^{\circ}\). It can be found that the high-speed compressor cascade corner separation mainly consists of the passage vortex (PV), wall vortex (WV), free vortex (FV), corner vortex (CV), and trailing edge shedding vortex (TSV), and the development of the high-speed compressor cascade flow loss is closely related to the corner separation. As the endwall boundary layer moves to the suction surface, the PV arrives at the suction surface/endwall corner at plane p4. Then, the PV further extends to the midspan along the spanwise direction and increases in size, while moving downstream. In this procedure, the suction surface boundary layer is extruded by the PV, and the WV is formed resultantly. While moving downstream, the WV directly interacts with the main flow and determines the boundary of the high-speed compressor cascade corner separation. Meanwhile, it is pushed further and further away from the endwall by the PV and increases in size too. Under the influence of the WV, the FV is aroused from the main flow boundary layer, but it dissipates very quickly, while moving downstream.