Study on blade vibration of centrifugal compressor - Forced responses caused by casing treatment

A casing treatment of centrifugal compressor for turbochargers is a very important device in aerodynamic performance improvement because it enables to expand an operating range of compressor. However, a rib existing in the circulation flow path of casing treatment is one of the factors causing a resonance problem. In this study, the relationship between a casing treatment and blade vibration was investigated by frequency response analysis that combines computational fluid dynamics and finite element method. As a result, it was clarified that the difference in the amount of vibration response and the excitation mechanism by the existence of rib.


Introduction
Along with CO2 emissions reduction in the global, operating range expansion and higher efficiency of turbomachinery products are continuing to be required.A casing treatment is often applied for compressors as one of the optional devices which can solve these problems.Generally, the casing treatment is constructed with some ribs forming in a circulation flow path, and grooves arranged at upstream and downstream sides against impeller inlet.When the pressure at downstream side is increased higher than that at upstream side, a part of main flow passed the impeller inlet is recirculated from the downstream groove to the upstream groove, and then it flows again toward the impeller inlet.Especially, the recirculation flow rate is increased at the operating condition near surge because the pressure difference between downstream and upstream grooves is high.As a result, the stall is suppressed at the impeller inlet by the improvement of incidence angle and the operating range can be expanded.
Many studies associated with the effect on aerodynamic performance by casing treatment was already reported.Tamaki et al. reported that the recirculation flow rate is strongly affected by the position relationship between the shock wave in the impeller passage and the downstream groove [1].Yokoyama et al. clarified by particle image velocimetry (PIV) method that the swirl component of recirculation flow can be suppressed and the operating range can be expanded by increasing the number of ribs [2].On the other hand, it is well known in the viewpoint of blade vibration that the rib of casing treatment serves as an excitation source and becomes one of the factors causing the resonance problem [3] [4].However, there are few papers investigating the flow mechanism associated with the resonance phenomenon by the rib.
In this study, the resonance phenomenon with the harmonic component higher than the number of ribs was observed in the blade vibration measurement of centrifugal compressor.In order to clarify the reason that the harmonic component was generated, the effect of rib on the blade vibration was investigated by frequency response analysis (FRA).

Compressor specification
The stage of evaluated compressor is consisted of an impeller, casing treatment, vaned diffuser and housing.The impeller is multi-splitter configuration which has three different blade shapes and the number of each blade is seven respectively (7+7+7).The casing treatment has six ribs in the circulation flow path and the vaned diffuser has seventeen vanes.The ribs and vanes are uniformly arranged in the circumferential direction.Figure 1 shows the cross-sectional view of casing treatment and figure 2 shows the ribs in the circulation flow path viewed from inlet side.

Evaluation condition
Figure 3 shows the vibration response result on Campbell diagram of evaluated impeller.Generally, a impeller has many resonance points with high order vibration modes on the line of excitation order (EO) as same as the number of diffuser vanes.In this case, the resonance phenomena were clearly observed not only on EO17 of the number of diffuser vanes, but also on EO12 which was the double component of the number of ribs.Therefore, one resonance point of EO12 expected to be caused by the rib is selected as the evaluation target in this paper.Figure 4 shows the modal displacement contour of vibration mode at the resonance point of EO12 calculated by the modal analysis.This mode is found that the full blade is mainly vibrated.

Numerical procedure
In order to investigate the effect of rib on the blade vibration of impeller, the amount of vibration response is evaluated by FRA.
Figure 5 shows the computational fluid dynamics (CFD) analysis models.Table 1 and table 2 show the CFD analysis setting and the number of mesh nodes.Figure 6 shows the mesh of CFD analysis.The analysis models are consisted of the impeller, casing treatment and vaned diffuser.The cavity at the back disk of impeller is not reproduced."Case 1" is not included the rib in the casing treatment, but the rib shape is reproduced in "Case 2".For the unsteady CFD analysis to calculate the aerodynamic force, the solver is used the commercial software ANSYS CFX (Ver.19.0) and it is calculated as an impeller full-pitch model.A hexahedral mesh is applied to the impeller and vaned diffuser, and a tetrahedral mesh with prism layers is applied to the other domains.The number of mesh nodes are 26 million, 0.7 million and 18 million of the impeller, casing treatment (Case2) and vaned diffuser.The turbulence model is the shear stress transport (SST) k-ω, the y+ value on the blade surface is around 2-5.As the boundary conditions, rotational speed, total pressure, static pressure, total temperature and flow angle are referred from the blade vibration measurement result.Sliding mesh interfaces are applied at the inlet and outlet of impeller respectively.From the unsteady CFD analysis result, the only EO12 component of static pressure variation is extracted as the aerodynamic force.For the vibration analysis, the solver is used the commercial software ANSYS Mechanical (Ver.19.0).A hexahedral mesh with prism layers is applied to the impeller.The number of mesh nodes and elements are 0.29 million and 0.26 million.The material property is applied an aluminum alloy and the damping ratio is applied the value measured by a strain gage in the blade vibration measurement.The aerodynamic force obtained from the unsteady CFD analysis is interpolated to the finite element method (FEM) model for modal analysis.The interpolated area of aerodynamic force is only the blade surface except the hub and back disc surfaces.

Relationship between rib in circulation flow path of casing treatment and blade vibration
Figure 7 and figure 8 show the amount of vibration response and the distribution of aerodynamic force on the blade surface in each CFD analysis model.Compared with Case 1 and Case 2, it was found that the difference with or without rib in the circulation flow path of casing treatment causes the 7.1 times change in the amount of vibration response.The effect of rib can be also seen on the aerodynamic force distribution.Especially, the aerodynamic force of Case 2 is larger than that of Case 1 at the blade tip near leading edge of full blade where is the maximum displacement region as shown in figure 4.  In addition, the static pressure variation on the blade surface of representative blade during one rotation of impeller was confirmed.Figure 9 shows the evaluation points of static pressure variation and the amount of modal displacement calculated by the modal analysis.The three evaluation points (A, B and C) were chosen as the two large modal displacement positions and the position near the downstream groove at 95% span of full blade.The amount of modal displacement was extracted from the representative nodes on the blade tip surface of full blade.Figure 10 shows the histories about the difference of static pressure ΔPs at each evaluation point during one rotation.ΔPs is calculated with the following formula from the static pressure on the pressure surface Ps_P and suction surface Ps_S of full blade.The latter part of this formula is the average value during one rotation.

∆𝑃 = 𝑃 _ − 𝑃 _ − 𝑃 _ − 𝑃 _ (1)
Although twelve peaks can be seen at all evaluation points in the histories of Case 2, they are not seen in Case 1.The number of peaks is matched with EO observed at the resonance point of vibration response result as shown in figure 3. Therefore, the resonance point of EO12 occurred in the measurement can be considered to be caused by the rib.The shape of peaks at the evaluation point B   near the downstream groove is clear compared with the other evaluation points.It is inferred that the effect of rib propagates with weakening toward the upstream and downstream sides from the downstream groove.The vibration responses at the evaluation points A and C of Case 2 become larger because the aerodynamic force is added at the large modal displacement region.Thus, it can be considered that the modal force defined the product of modal displacement and aerodynamic force is large.From these results, it was clarified that not only EO of aerodynamic force but also the shape of modal displacement and the position of downstream groove had necessary to verify in the resonance risk assessment of impeller with the casing treatment.

Excitation mechanism by ribs
The flow field near the downstream groove was investigated to clarify the excitation mechanism by the rib. Figure 11 shows the history about the difference of static pressure at the evaluation point B of Case 2. The red dot lines show the phase angles of ribs.There are two peaks ((I), (III)) and two bottoms ((II), (IV)) per one rib.Figure 12 shows the static pressure contours on the cross section for the axial direction at the evaluation point B of both cases.This study focused on the flow field near the pressure surface side because it was confirmed that the pressure variation on pressure surface is dominant in the history about the difference of static pressure.
In the Case 1, the pressure change near the pressure surface side is formed by the only impeller blade.Therefore, there is no clear difference on the pressure distribution.The flow field of Case 2 is similar with Case 1 at the first peak (Figure 12     Figure 13 shows the surface streamline colored by the velocity on the cross-section for the axial direction at the evaluation point B. Figure 14 shows the contour of radial velocity on the cross-section for the axial direction at the evaluation point B, and the phase angles is (I) and (II) as shown in figure 11 and figure 12 (b).
The stagnation point is generated on the one side of rib because the swirl flow, which is inflowed from the downstream groove to the casing treatment, collisions to the rib.The back-step flow is formed on the other side.Generally, the recirculation flow from the impeller to the upstream groove via casing treatment is formed in the surge condition, and the bypass flow from the upstream groove to the impeller via casing treatment is formed in the choke condition.The radial outflow and radial inflow are dominant in the recirculation flow and bypass flow, respectively.In the case of near peak condition, the pressure at the upstream groove and the downstream groove are balanced, the radial inflow and the radial outflow are also balanced.As a result, the radial inflow and the radial outflow are formed locally at the downstream groove.The radial inflow is generated near the rib after one blade passed as shown in figure 14 (a).Moreover, the radial inflow is similarly seen between two ribs as show in figure 14 (b).As a result, the pressure on the blade surface is considered to change by the interference between the blade and two vortices near the rib or two radial inflows from the downstream groove to the impeller.

Conclusion
The following conclusions were derived from the present study.
 The effect of rib in the circulation flow path of casing treatment on the blade vibration response of compressor impeller was investigated by using unsteady CFD and FEM analyses.Then, the difference of response amount and the excitation mechanism were clarified. For the vibration mode inferred the resonance caused by the rib from the blade vibration measurement result, the amount of vibration response was calculated by using frequency response analysis.The response amount was increased 7.1 times due to the existence of rib. From the variation of static pressure on the blade surface during one rotation, twelve peaks were confirmed that were matched with the measured excitation order of evaluation mode (EO12).This response was clarified that it was occurred by the resonance due to the rib. The large amount of vibration response generated by the rib was considered that the reason was the increase of modal force because the static pressure variation was added to the large modal displacement region on the blade surface.In conclusion, it was clarified that the vibration response of EO12 was caused by the interference between the moving blade and the six fixed ribs because the blade was close to the rib, but the detail determination of excitation mechanism will be studied as future tasks.

Figure 10 .
Figure 10.Histories about difference of static pressure.

( a )
Evaluation point A. (b) Evaluation point B. (c) Evaluation point C.
(a)-(I), (b)-(I)), but the static pressure near the pressure surface side is seen to decrease as close to the rib (Figure 12 (b)-(II)).And then, the static pressure is recovered as further away from the rib (Figure 12 (b)-(III)).However, the static pressure is decreased again near the middle position between two ribs (Figure12 (b)-(IV)).Therefore, the reason of static pressure variation near the pressure surface side was generated by the interference between the impeller and ribs.

Figure 12 .
Figure 12.Comparison of static pressure contour at evaluation point B.

Figure 11 .
Figure 11.History about difference of static pressure with rib position (Evaluation point B).

Figure 13 .
Figure 13.Surface streamline at evaluation point B.

Figure 14 .
Radial velocity contour at evaluation point B.

Table 2 .
Number of mesh nodes.Mesh for CFD analysis.