# Effect of Rotor-Stator Spacing on Compressor Performance at Variable Operating Conditions
## 4 Results and Discussions
### Wake Recovery  
In the analysis of Section 4.1, we demonstrate that the stator loss near the 0.4 span in the \"close\" case decreases. The flow angle comparison in view 4 of Figure 11 highlights that the time-averaged flow angle difference at B and C is almost negligible, and the non-transitional boundary layer response affords \"close\", giving more loss because of the existing deep wake just before the stator contributes to the loss [13]. As mentioned in the introduction, [PERSON] [8] used the unsteady kinetic energy flux defined in Equation (4) to describe the flow unsteadiness, where \(u_{1}\) and \(v_{1}\) represent the axial and tangential velocity, as illustrated in Figure 17. The recovery parameter \(R\) is calculated from Equation (5). [PERSON] [9] removed the pitch-wise operator \(A_{y}\) and solely employed flow data at the mid-pitch, called disturbance kinetic energy(DKE), a method adopted in this paper. \[K=\ 0.5U_{0}A_{y}A_{t}\Big{(}u_{1}^{2}+v_{1}^{2}\Big{)} \tag{4}\]  
Figure 16: Transient pressure distribution of the 0.05 span at C condition. (5)  
\[K_{in}-K_{out}=K_{in}R \tag{6}\]  
We set monitoring points in the two-dimensional cylindrical surface of the 0.4 span (Figure 6), which are extended forward of the rotor's TE so that the last location in the axial direction is the stator's TE. In total, one point in the circumferential direction at the mid-pitch and twenty-one points, including the LE and the TE in the axial direction, are set in the blade passage to monitor three velocity components. There are ten and thirty for \"close\" and \"far\" in the axial spacing, respectively. By analyzing each point's data, the ratio of radial kinetic energy to the total kinetic energy in the non-near blade surface area of the stator does not exceed 1.5%, revealing that the two-dimensional wake recovery analysis using Equation (6) at the span is practical. Figure 18 highlights that the axial variation of DKE from the rotor TE to the stator TE, in which DKE is non-dimensionally treated, along with the maximum value and axial position, is normalized with the stator's axial chord length. The DKE at the rotor TE gradually decreases as the load increases, indicating that the wake's velocity deficit decreases. To compare the two spacing, the bigger the DKE entering the stator, the greater the value of \(R\). For the cases with a far spacing, the wake's DKE almost decayed entirely before entering the stator; thus, the stator's performance is regarded as steady. At the same condition, the difference of \(R\) is sorted as B \(>\) A \(>\) C, but this is inconsistent with the total pressure loss comparison presented in Figure 12. Moreover, the wake decay is caused by wake recovery and dissipation, so we use the integration of turbulence eddy dissipation in a thin cylinder at the 0.4 span (Equation (7)) to quantify dynamic pressure loss. Additionally, we use the vorticity criterion to identify the edge of the boundary layer (Figure 19). Moreover, we define \(\omega_{potential}\) (Equation (8)) to characterize potential loss due to non-uniform freestream, highlighting its significance when used as a relative quantity. \[\omega_{t}=\oiint_{\Delta}\rho ed\Omega/(\frac{1}{2}\dot{m}_{in}U_{in}^{2}) \tag{7}\]  
\[\omega_{potential}=\text{DKE}_{in}/(\frac{1}{2}U_{in}^{3}) \tag{8}\]  
The interface is used as the starting position in the detailed loss analysis. At every condition, \(\omega_{potential}\) of \"far\" is more extensive, and the difference is 0.0014, 0.0006, and 0.0010, respectively, which is small enough to be ignored, and thus the detailed loss comparison is reasonable. As illustrated in Figure 19, the total loss of \"far\" is more extensive in every case where the loss in the spacing contributes the most. As load increases, the proportion of the loss in the spacing decreases, although the increased stator load will aggravate the wake mixing in the gap, showing that the wake's reduced velocity deficit has the greatest impact on the loss in the spacing.