The loss in the compressor is of great interest, and it determines the effectiveness of a flow control method. Therefore, the time-averaged total pressure loss at the plane 20% c _x downstream the trailing edge of three configurations are compared in Figure 5. It can be seen that the CS and PS configurations reduce the C _pt by 9.36% and 11.61%, respectively, based on that of the baseline configuration. Compared with the CS, the PS has 24% more ability in reducing the C _pt of the compressor cascade. To have deep knowledge of the flow control mechanism of the PS, the detailed flow fields are compared and analyzed in the following part.

The Q criterion derived from the second invariant of the velocity gradient tensor is used to identify the vortex structures in the compressor flow field, and the function used to calculate the Q value is given below (Hunt et al., 1988), Q = 1 2 ( Ω i j Ω i j − S i j S i j ) (2) where Ω _ij is an anti-symmetric tensor of the rotation rate, and S _ij is a symmetric tensor of the strain rate. The Q is thought to be a balance between the rotation rate and strain rate of the control volume. Ω z = 1 2 ( ∂ u ∂ y − ∂ v ∂ x ) (3)

Figure 6 displays the vortex structures in the compressor cascade identified by the Q iso-surface and the loss contour on the plane 20% c _x downstream of the blade trailing edge of the three configurations. In the baseline in Figure 6A, it is obvious that the passage vortex (PV) arises from the joint point of the suction leg of the horseshoe vortex (HSV) and the pressure one from the adjacent blade, and the PV rolls up and interacts with the vortex structures from the blade suction surface as flowing downstream and leading to further total pressure loss generation. The loss caused by the vortex interaction contributes a considerable part of the total loss, which can be inferred from the loss contour downstream the blade. In Figure 6B of the CS configuration, however, both the suction leg and pressure leg of the HSV are removed by the suction hole, and the arising of the PV is contributed by the remaining part of the transverse flow near the endwall. In this case, the size of the PV is significantly reduced compared with that of the baseline, and the total pressure loss attributed to the PV is decreased accordingly. Besides, the interaction between the PV and the separation flow from the blade suction surface is attenuated, and this leads to a further reduction in the C _pt . Therefore, the corner separation of the CS configuration is alleviated, and the loss downstream of the blade is reduced accordingly. As for the PS configuration in Figure 6C, in addition to the reduction in the PV size, the loss has a significant reduction compared with the CS configuration, which indicates that the PS has more significant potential in flow control of the compressor flow separation than the CS. In the following section, the detailed flow field of the three configurations will be analyzed and compared to clarify how the PS influences the compressor flow field and the controlling mechanism of the PS.

Figure 8 displays the top–down view of the instantaneous vortex structures at five different instantaneous times in a pulse periodicity, and the time definition is given in Figure 7; τ denotes the time duration of a pulse. When t = T in Figure 8A, the suction hole is in a state in which the previous pulse is ended and another pulse is prepared to start, and the size of the PV, as well as its interaction with the separated flow from the blade suction surface, is small under the influence of the previous pulse, as marked by the black rectangle in Figure 8A. When t = T + 1/4τ in Figure 8B, there is no flow bled into the suction hole, and in this case, the pressure leg of the HSV and the suction one converge near the position of the suction hole and give rise to the PV afterward. The PV interacts with the separated flow from the blade suction surface and induces more serious flow separation and higher total pressure loss. The PV and separated flow at the blade suction corner develop and entrain the ambient air into the vortices and result in higher loss as flowing downstream, as displayed in the rectangle colored blue in Figure 8C. When t = T + 3/4τ in Figure 8D, the pressure leg of the HSV, as well as the suction one, are bled into the suction hole, which postpones the arising of the PV. It can be observed that the vortex structures near the suction hole are greatly alleviated, and the vortex size spanwise and pitchwise are restricted; hence, the local flow field can be improved accordingly. In this case, the remaining part of the transverse flow near the endwall accumulates near the blade suction corner and gives rise to the PV, as is shown by the black dotted line. Compared with the position where the PV forms in Figure 8B, it is indicated that the formation of the PV is significantly moving downstream, which reduces the vortex structure size both spanwise and pitchwise, and the interaction between the PV and the separated flow from the blade suction surface is weakened; hence, the corresponding loss generated is reduced under the influence of the PS.

Figure 9 displays the evolution of the axial vorticity of the PS configuration, and the results at time T and T + 1/2τ are shown, corresponding to two different states of the suction hole in a pulse periodicity, and the contour of the C _pt can be seen in Figure 10. When t = T in Figure 9A, it can be known that the boundary layer near the endwall from the far upstream, as well as the transverse flow upstream of the suction hole, is removed by the suction hole; hence, the arising of the PV is interrupted. The formation of the PV, which is induced by the accumulation of the transverse flow near the endwall downstream of the suction hole, could be observed and indicated by the black curve. Combined with the contour of the loss evolution in Figure 10A, it can be known that less loss will be generated by the PV in this case, and its interaction with the blade suction surface flow separation will be attenuated accordingly, which contributes to the improvement of the compressor flow field. When t = T + τ in Figure 9B, the pressure leg of the HSV, colored red, and the suction leg of the HSV, colored blue, converge and give rising to the PV, as indicated by the black curve. The PV interacts with the blade suction surface flow at 60% c _x downstream of the leading edge and breakdown, causing a high level of total pressure loss, as shown in Figure 10B.

Figure 11 displays the three-dimensional instantaneous vortex structures in a pulse periodicity of the PS configuration. Combined with the analysis in Figure 8, it can be indicated that the compressor flow field is alternatively dominated by two different kinds of PV formation mechanisms under the influence of the PS. One of the PV formations is induced by the pressure leg and suction leg of the HSV, with large size, and results in a higher loss, as shown in Figures 11B, C. The other one is induced by the remaining part of the transverse flow near the endwall downstream of the suction hole, and due to the removal of most of the inlet boundary layer by the suction hole, the size of the PV is significantly reduced, and this leads to less loss. Though the suction hole can only affect the flow field near it, the formation mechanism of the PV downstream the suction hole is distinct, and in this way, the PS can have a more profound influence on the whole compressor flow field through the alternating transformation of two kinds of the PV formation mechanisms. Besides, the blade suction surface will be influenced by the periodic impingement of the PV, and the periodic perturbation of the PV will promote the momentum transport between the separated flow and the main flow, thus, leading to a further benefit in C _pt .

The instantaneous contour maps of the spanwise vorticity evolution of three configurations are given in Figure 12, and the contour maps at the plane at 5 mm away from the endwall are displayed. In the baseline in Figure 12A, the PV, which is derived from the convergence of the pressure leg (colored in red) and suction leg (colored in black) of the HSV, interacts with the flow on the blade suction surface at 55% c _x downstream the blade leading edge at the plane 5 mm away from the endwall. The vortices breakdown and result in higher loss as it flows downstream afterward. For the CS configuration, however, the interaction of the PV and the blade suction surface flow moving downstream at 73% c _x downstream the blade leading edge at the plane 5 mm away from the endwall. This indicates fewer vortex interactions and less loss generation compared with the baseline. For the PS configuration, there are two distinct flow patterns according to the opening or closing state of the suction hole, and one flow pattern is similar to that of the baseline, and the other one is similar to that of the CS configuration. These two flow patterns are alternately changed as the suction hole opens up and closes off, and this may transfer the excitation frequency of the suction hole to the whole compressor flow field, especially at the blade corner region where the PV and separated flow dominated, and reordered the local flow field; hence, the PS would have a better control performance than the CS at the same suction rate. To have better knowledge of the frequency information of the flow field, especially at the corner separated flow region, the fast Fourier transform (FFT) is applied to the local velocity at point A (marked in Figure 13A), and the results are shown in Figure 13. Figure 13 shows the power spectral density (PSD) distribution along with the frequency of three configurations, and it can be seen that the baseline, as shown in Figure 13B, has a dominant frequency of 85 Hz, and the PSD in this frequency is no more than 1,200 W/Hz. In Figure 13C of CS configuration, the dominant frequency was replaced by 1,172 Hz with a lower PSD of about 700 W/Hz. In PS configuration in Figure 13D, however, there are two frequencies (97.6 and 109.9 Hz) with large PSD, which means these two frequencies dominate the local flow field, and combined with the frequency of the PS, it can be inferred that the PS can reorder the flow field even at far downstream the suction hole through managing two kinds of the PV formation.

The POD method can decompose the main flow structure in the flow field according to the proportion of the energy, and it not only helps in analyzing a specific mode structure independently but also removes the minor structures through reconstructing the flow field with the first few modes. The POD is based on the snapshots of the instantaneous results in this paper, and the data proceeding process is listed as follows, and detailed information can be found in Shi et al. (2019).

(1) The decomposition of the variables and reorder.

X(x,t) and X(x) represent the transient and time-averaged velocity, and X’(x,t) is the variation value. X _i denotes the velocity vector at t _i , and X is the velocity matrix.

(2) Build a covariance matrix R with the velocity matrix X .

The interaction between the PV and the separated flow from the blade suction surface can be obviously observed on the spanwise plane 10 mm away from the endwall; therefore, the axial velocity of the spanwise plane 10 mm away from the endwall is extracted to perform the POD. Figure 14 shows the energy captured of the POD modes and the contour map of the first five orders of modes for the CS and PS configurations. Compared with the energy captured of the two configurations in Figure 14A, it can be seen that the PS increases the energy percent of the low-order modes, which indicates that the flow structures represented by the low-order modes have more ability to dominate the local flow field; hence, the flow field becomes simpler than that of the CS configurations. Besides, compared with the CS configuration in Figure 14B, it can be seen that the scale of the flow structures of the PS configuration in Figure 14C is larger, and the flow structures represented by the modes are much simpler compared with that of the CS configuration. Besides, it can be inferred that the CS only decreases the size of the vortex structures while keeping the disorder of the flow field unchanged. The PS, however, could also reorder the compressor flow field and make the flow field simpler, which could lead to a further reduction in the C _pt .

The DMD is a method that decomposes the flow field into different dynamic modes with distinct frequencies, and the stability of the modes, as well as the flow structures they represent, can be analyzed according to the eigenvalues. Unlike the POD method, the DMD can decompose the flow field from the order of time, and the calculation process is listed as follows (detailed information can be found in Schmid, 2010):

(1) Define velocity matrix X and Y.

U and V are the singular value decomposition basis vectors of the X and S containing the information of the singular value.

(3) Calculate the approximation matrix A ∼ using the vectors U , S , and V achieved from the last step.

Ω and Λ are the eigenvectors and eigenvalues, respectively.

(5) Calculate the DMD modes

ϕ is the matrix that contains the DMD modes.

The axial velocity of the spanwise plane 10 mm away from the endwall is extracted to perform the DMD. Figure 15 is the frequency spectra and the energy distribution for the CS and PS configurations, and the abscissa is the frequency calculated from the image part of the continuous-time eigenvalues, and the ordinate is the growth or decay rate (positive indicates unstable characteristics and negative indicates stable characteristics) of the characteristics represented by the modes for Figures 15A,B. The modes are divided into stable and unstable ones according to the DMD eigenvalues, and if the real part of the eigenvalue is positive, the corresponding flow structures represented by the mode is unstable, and vice versa. It can be known from Figure 15A that some modes (especially the low-frequency ones) of the CS configuration are unstable (concentrated on the low frequency modes), presented as the positive value of the growth rate. For the PS configuration in Figure 15B, however, all the modes are stable, which indicates the flow field can be stabilized under the effect of the pulsed excitation. For example, the boundary layer on the blade suction surface will be much easier to separate from the solid wall under the influence of a small fluctuation if the flow is in an unstable state. For the same reason, the results of DMD with stable modes, such as the PS results, will lead to less flow separation and a reduction in C _pt . From the energy distribution at each mode in Figures 15C,D, it can be observed that three dominant modes occupied most of the energy in the PS configuration, and the effect of other modes are weakened. The frequencies of the dominant modes are 1,107, 2,097, and 98.9 Hz, respectively, and all these frequencies correspond to the excitation frequency and its frequency multiplication of the excitation frequency. It can be inferred that the flow field in the PS configuration is reordered and stabilized compared with that of the CS configuration; hence, the compressor performance could be improved, and the loss generation is reduced accordingly.