Influence mechanism of tip clearance on flutter stability with different aerodynamic coupling effects for transonic compressor

Flutter is a highly destructive aeroelastic problem in modern compressors, which hinders the improvements of aero-engine performance and reliability. Because the aerodynamic work is mainly concentrated in the blade tip region, the tip clearance which is associated to the complex tip clearance flow has a considerable effect on the flutter stability. In this paper, in order to investigate the influence mechanism of the tip clearance on the flutter stability at different nodal diameters (ND), a series of compressor models were established with different tip clearances based on a transonic compressor rotor. The aerodynamic damping at each ND is obtained by influence coefficient method (ICM), while phase-shifted boundary method (PSB) is adopted to analyze the influence of the tip clearance on the flutter stability at different NDs. The results indicate the worst flutter stability for each tip clearance always appears at ND=1, where the aerodynamic damping exhibits a nonmonotonic trend of increasing first and decreasing thereafter along with the rising tip clearance. Two kinds of action are exerted by the tip clearance flow on the flow structures to alter the flutter stability: one is the tip clearance vortices which is generated on the suction side and impinge on the pressure side; the other is the interference of tip clearance vortices to the shock wave and to the flow separation. Besides, the influence of the changing aerodynamic coupling effect makes the above action bring about more drastic fluctuations to the unsteady pressure. At larger NDs, the unsteady pressure amplitude and phase fluctuate more sharply for larger tip clearances. Therefore, diverse change trends of the aerodynamic damping with the increasing tip clearance appear at different NDs, while the impingement of tip clearance vortices on the adjacent pressure side has a constantly stabilizing effect at different NDs.


Introduction
The compressor design has an increasing trend towards higher blade loading and thinner blade shape.However, this also increases the occurrence probability of aeroelastic instability which can be mainly divided into forced response, nonsynchronous vibration, acoustic resonance and flutter [1,2].The former three types rely on the unsteady aerodynamic forces generated by external disturbances.In contrast, flutter is an essentially self-excited blade vibration, which is induced by the unsteady aerodynamic forces derived from the structural motion of the blade itself.As a typical positive feedback process, once 2 flutter occurs, the blade vibration amplitude will increase sharply in a short period of time and ultimately lead to blade failure.
The tip clearance is one of the most important structural parameters in compressor design.The intensity of tip clearance flow and the formation of tip clearance vortices [3,4] are strongly affected by the tip clearance variation.Besides, because the most part of aerodynamic work is concentrated in the blade tip region [5], the tip clearance which has close ties with complex tip clearance flow also has significant effects on the flutter stability.Nevertheless, researches about the effects of the tip clearance on the flutter stability have not reached an agreement.Huang et al. [6] found that the tip clearance flow in small tip clearances (1.25-2.5% chord) improves flutter stability, while the stability is reduced by a fully developed tip clearance vortex in a large clearance (5% chord) in linear turbine cascades.Möller et al. [7] numerically investigated the flutter stability in a 1.5 stage transonic compressor, indicating that the tip clearance flow impingement on the blade pressure side generates the negative aerodynamic damping.Besem and Kielb [8] studied the flutter behavior in a second stage rotor of a 3.5-stage axial compressor with different tip clearance sizes.At each IBPA, their numerical results showed that the aerodynamic damping first increases with tip clearance and then decreases.Fu et al. [9] carried out a similar research about the flutter stability of an axial compressor rotor.But they reached a contrary conclusion that the aerodynamic damping presents a non-monotonic trend of decreasing first and then increasing with the tip clearance increment at the most unstable ND.Dong et al. [10] conducted a research on blade flutter stability in a wide-chord fan with various tip clearances.Their results, illustrated that the aerodynamic damping presents a monotonic decreasing trend at the most unstable ND when the tip clearance varies from 0.25 mm to 2 mm in the first bending mode.
Due to the close arrangement of compressor rotor blades, the flow field and the unsteady aerodynamic forces on the surfaces of vibrating blades are affected by the motion of adjacent blades, which is called aerodynamic coupling effect.When the blades vibrate at different NDs or inter-blade phase angles (IBPA), this effect will be changed, further leading to variations in blade flutter stability [11,12].In addition, some researches [5,9,13] also found that the effects of tip clearance variations on the flutter stability are distinct at different NDs or IBPAs.
Therefore, these inconsistent conclusions among the above researches indicate that the influence mechanism of tip clearances on the flutter stability needs to be further studied, especially the effects of tip clearances on the flutter stability at different NDs which have not been comprehensively investigated so far.These will provide great support to the development and optimization of flutter suppression methods in the future.In this paper, the influence mechanism of the tip clearance on the flutter stability for a transonic compressor rotor is deeply analyzed, and the ND variation influence is also taken into account.Firstly, the aerodynamic performance and internal flow characteristics are studied with different clearance models.Then, the flutter stability of different tip clearance models is predicted by calculating the aerodynamic damping at different NDs with the ICM and the PSB methods.The influence of the tip clearance on the aerodynamic damping at the most unstable ND is discussed in detail.Finally, the underlying mechanism of the ND variation acting on the relationship between the aerodynamic damping and the tip clearance, is revealed by comparing the unsteady pressure distributions of different tip clearance models at various NDs.

Test case
As a representative modern transonic axial compressor rotor, NASA Rotor 67 is adopted in this paper to study the flutter stability for various tip clearance models at different NDs.Due to the detailed aerodynamic characteristics and geometry data in the experiment conducted by Strazisar et al. [14], Rotor 67 has been widely used in flutter researches [15][16][17], which can provide a certain reference to the flutter analysis in this paper.The meridian plane with aerodynamic survey locations is illustrated in Figure 1.The basic design parameters are shown in Table 1.

Computational fluid dynamics model
The computational fluid dynamics (CFD) simulations are carried out in Ansys CFX solver with the SST turbulence model.Ideal gas axially flows into the inlet, where total pressure of 101325 Pa and total temperature of 288.15K are set.Average static pressure is set at the outlet based on the principle of radial equilibrium, while adiabatic and no-slip walls are adopted at other solid walls.Steady performance and flow characteristics are simulated in a single-passage domain, while a multi-passage domain is used in unsteady flutter simulation whose detailed description is provided in section 3.3.The rotating speed is specified as the designed speed (16043 rpm) and assigned to the fluid domain.
Figure 2 shows the multi-block structured grids with O4H topology generated by NUMECA AutoGrid5.Detailed distributions of grids near the LE and TE are also presented.The first element adjacent to the walls keeps a constant value of , which means the y+ is less than 1, to satisfy the requirements of the SST turbulence model.Meanwhile, the grid independence verification is conducted for the design tip clearance model including grids with 0.85 million, 0.93 million and 1.04 million cells.As shown in Figure 3, grids with 0.93 million cells only bring a small change in mass flow rate and total pressure ratio compared with grids with 1.04 million cells.Hence, the grid with 0.93 million cells is a good compromise between accuracy and computational cost.In addition to the design tip clearance  (1.016 mm), other four tip clearances applied in this paper are represented by 0.25 (0.254 mm), 0.5 (0.508 mm), 1.5 (1.524 mm) and 1.75 (1.778 mm), respectively.Table 2 shows the single-passage grid settings in above tip clearance models, where the number of grid layers set in the tip clearance vary with the clearance size.
The accuracy of aerodynamic simulations will impact the following flutter simulations.Thus, the above computation settings are validated by single-passage steady simulations with the design tip clearance.And a comparation with experimental results is shown in Figure 4, where the mass flow at horizontal coordinate is normalized by the choking mass flow of the experiment [14].The performance maps from numerical simulations match well with the experimental data, indicating that the grid and aerodynamic computation settings can meet the simulation requirements.

Finite element model
The first bending (1B) mode of the rotor blade applied in flutter simulations is calculated by ANSYS Mechanical module.The centrifugal force provided by the rotor rotation at design speed is considered in prestress modal analysis, where the effects of the mechanical damping and aerodynamic force are ignored.And the material of the rigid rotor blade with root fixed is titanium alloy, whose properties are shown in Table 3.
Though tip clearance variation is achieved by cutting the blade tip, the 1B mode shapes of different tip clearance models are basically identical, as shown in Figure 5.And all the frequency deviations in

Flutter simulation method
In this paper, the energy method [18] is adopted to predict the blade flutter behavior.As a unidirectional fluid-structure-interaction method, it allows the blade vibrate with a small amplitude at a given mode and ND.The energy transfer direction between the blade and the surrounding flow field determines the flutter stability.If the fluid transfers energy to the blade during a vibration cycle, which means fluid applies positive aerodynamic work on the blade, the blade is at a risk of flutter.Within a vibration cycle, the aerodynamic work done by the surrounding fluid on the blade can be calculated as: Where p represents the unsteady pressure, v represents the velocity vector on the vibrating blade surface, n represents the normal unit vector on the blade surface, A represents the area of blade surface, and 1/ Tf = represents the vibration period of blade.Another non-dimensional variable named aerodynamic damping is introduced to describe the blade flutter characteristic as well, which can be expressed as: Where  represents the blade natural frequency, and cfd q represents the ratio of the maximum actual amplitude to the maximum modal amplitude of the blade.According to Equation ( 2) and the definition of cycle W , when the aerodynamic damping is negative, the blade flutter will occur.The PSB and the ICM are used to calculate the aerodynamic damping in this paper.Compared to the traditional TWM which needs multiple full-annulus simulations to obtain the aerodynamic dampings at all NDs for NASA Rotor 67, the PSB only requires a double-passage domain in each simulation.On the other hand, the ICM simply needs a one-time calculation in an odd multi-passage domain to obtain the aerodynamic damping at each ND, whose reliability has been confirmed by some researches [10,[19][20][21][22].During the entire computing process of ICM, only the middle blade marked as the reference blade vibrates with a specific mode shape, amplitude and frequency, while the other blades remain stationary.And the unsteady pressure 0 p on the reference blade surface at different NDs can be calculated based on a linear superposition principle as follows: Where k represents the blade index, N represents the number of blades in the muti-passage domain, i represents the imaginary unit, and  represents the IBPA which is defined as follows: represents the number of rotor blades, ND represents the nodal diameter.In order to improve the computational efficiency and reducing computing resources, the ICM is applied to calculated the aerodynamic dampings at all NDs for different tip clearance models.And the PSB is employed to analyze the influence mechanism of the tip clearance on the flutter stability at different NDs in detail.
The blade vibration amplitude and the timesteps within a vibration period can significantly impact the accuracy of flutter simulations.In order to avoid negative grids and nonlinear flow near the blade tip region, a vibration amplitude of 0.3 mm is selected.Considering the computational efficiency and accuracy, 90 timesteps per vibration cycle are ultimately adopted.Besides, the accuracy of ICM strongly depends on the number of passages in the computational domain.To seek a balance between the computational cost and accuracy, a validation test is executed for computational domains containing 7, 9 and 11 passages, respectively.Rotor blades with 0.25 tip clearance are used for this validation.Figure 6 shows the aerodynamic damping results obtained from the ICM with different passages and the PSB with double passages.Eventually, the ICM with an eleven-passage computational domain which has the most consistent results with PSB is adopted for follow-up flutter simulations.

Steady aerodynamic performance and internal flow characteristics
The characteristics curves of total pressure ratio and adiabatic efficiency of different tip clearance models are shown in Figure 7, where the mass flow at horizontal coordinate is normalized by the choking mass flow of the design tip clearance model.As the tip clearance is increased, the mass flow rate at the same back pressure and the peak efficiency declines obviously, and the stall boundary gradually moves to the right as well.The increment in the flow loss, which is resulted from the enhancement of tip clearance flow, is responsible for the above deterioration [23].Therefore, the operating points (in the black box) with a pressure ratio about 1.6 at the same back pressure are chosen for the flutter simulations.It is noted that the operating point selected for 1.75 tip clearance model is near peak efficiency.The flow structures in the tip region are significantly affected by the tip clearance.Figure 8 shows the contours of Mach number at 98% span in different tip clearance models.Within the 0.25 tip clearance model, the pressure side and suction side are weakly affected by the weak tip clearance flow.Therefore, the lambda shock system with the oblique shock and normal shock (both marked with black line) can be seen clearly.Moreover, the normal shock interacts with the boundary layer, inducing flow separation on the pressure side and suction side.However, the shock-induced separation region on the pressure side is not obvious due to the small curvature, whereas a large open separation region is observed behind shock wave on the suction side with a larger curvature.From tip clearance 0.5 to 1.75 , the stronger interaction is built up between the shock waves and tip clearance vortices (marked with blue dashed line), which lead to the more and more sever deformation of the oblique shock and normal shock in the passage.Meanwhile, this interaction also forms a low-speed region (surrounded by blue line) near the adjacent pressure side.As the tip clearance is increased, the impinging position of the vortices on the adjacent pressure side moves closer to the LE, and the low-speed region also becomes larger and thicker, gradually covering the rear 50% area of the pressure side.In addition, the shock wave on the suction side was pushed towards the TE by the growing tip clearance vortices, which have a certain effect on inhibiting the shock-induced separation.Figure 9 shows the streamlines and pressure distributions on the blade surface, where the shock wave structure, shock-induced separation, and the influence range of tip clearance vortices can be observed clearly.The blade surface pressure is normalized by the inlet total pressure.From the tip clearance of 0.5 to 1.75 , the radial flow region (marked with blue dashed box) on the pressure side in Figure 9(a) constantly expands towards the low-span region and the LE.Combined with Figure 8, this kind of radial flow is caused by the impingement of tip clearance vortices on the pressure side.The normal shock position (marked with black dotted line) on the pressure side is pushed closer to the LE by this growing impingement.Meanwhile, the increasing interference of the vortices causes the intensity reduction and distortion of the normal shock.Moreover, the blockage caused by the large low-speed region for tip clearances of 1.5 and 1.75 makes the normal shock over the 90% span almost invisible.
In Figure 9(b), a lambda shock system consisting of an oblique shock (marked with red dotted line) and a normal shock (marked with black dotted line) is clearly observed on the suction side.As the tip clearance is increased, the intersection of these two shock waves gradually moves up to the blade tip region.Because the normal shock in the tip region shifts backward to the TE, the subsequent shockinduced separation region overing the 65-100% span gradually decreases, as well as the intensity of the radial flow.Besides, the appearance and expansion of the local radial flow in red dotted ellipse on the suction side are caused by the larger and larger tip clearance vortex generated near the LE.

Overall flutter stability for different tip clearances
Figure 10 shows the aerodynamic damping versus the ND in different tip clearance models.These results are simulated by the ICM with an eleven-passage computational domain, whose validation in section 3.3 shows it can simultaneously balance the computational efficiency and accuracy for the aerodynamic damping calculation.In general, the similar overall trends of the aerodynamic damping varying with the ND can be found in different tip clearance models of 0.25 , 0.5 ,  , 1.5 and 1.75 .Moreover, ND=1 is always the most unstable ND, where the corresponding minimum aerodynamic damping in each tip clearance model is positive, indicating the rotor blades with the above different tip clearances are always in stable states in this paper.However, different trends of the aerodynamic damping varying with the tip clearance are presented at different NDs. Figure 11 shows the aerodynamic damping versus tip clearance at ND=1.From tip clearance of 0.25 to 1.75 , the total aerodynamic damping on the blade at this ND shows a nonmonotonic trend of increasing first and then decreasing along with the rising tip clearance, while the maximum aerodynamic damping occurs at the design tip clearance.This result is consistent with the research of Besem et al. [8], where the flutter stability is improved first and worsen thereafter with the increasing tip clearance, indicating there is an optimal tip clearance which holds the best flutter stability.On the other hand, the total aerodynamic damping on the pressure side in each tip clearance model is always positive, as well as the total aerodynamic damping on the suction side.Besides, more than 60% of the overall aerodynamic damping on the blade is contributed by the pressure side, where the aerodynamic damping also has nonlinear relation with the tip clearance size.Therefore, the aerodynamic work done on the pressure side plays a dominant role in the blade flutter stability at ND=1.However, the aerodynamic damping exhibits completely different change trends at ND=-3 and ND=6.As shown in Figure 12, when the tip clearance changes from 0.25 to 1.75 , the aerodynamic damping at ND=-3 declines first and then increases, while the aerodynamic damping at ND=6 shows a monotonic decreasing trend.This suggests that the influence of the tip clearance on the flutter stability is affected by the aerodynamic coupling effect.

Influence of tip clearance on flutter stability
The influence mechanism of the tip clearance on the flutter stability is analyzed based on the flutter simulation results obtained by PSB at ND =1.Both the aerodynamic damping distributions on the pressure side and suction side are shown in Figure 13.In Figure 13(a), the tip clearance variation significantly affects the aerodynamic damping distribution on the pressure side, whose variation is consistent with the flow structures observed in Figure 9(a).The normal shock on the pressure side moving upstream with the increasing tip clearance, makes the corresponding strip region A, which has the positive aerodynamic damping, approach the LE.The growing interference of the vortices shown in Figure 9(a) not only reduces the intensity of the normal shock, but also distorts the shock wave to expand its influence range.Thus, as the tip clearance is increased, the damping intensity of region A gradually decreases, but the corresponding range expands.Adjacent to the left side of region A, there is another strip region with the negative aerodynamic damping, whose intensity reaches minimum at the design tip clearance.This is related to the interference of the growing tip clearance vortices to the ever-weakening shock-induced separation in this region.Furthermore, the positive damping region B which locates upon 90% span and near the TE of the pressure side, is precisely the position where the vortices impinge.The relationship between the impingement of tip clearance vortices and the aerodynamic damping distribution on the adjacent pressure side is shown in Figure 14.This indicates the growing impingement of the vortices has a stabilizing effect on the pressure side.Due to these complicated effects, the total aerodynamic damping of the pressure side first increases and then decreases with the rising tip clearance, while the design tip clearance holds the maximum aerodynamic damping.On the suction side shown in Figure 13(b), the positive damping region F within 0-30% chord in the tip region, whose intensity and the range increase with the tip clearance, is affected by the larger tip clearance vortex generated near the LE (enclosed by the red dotted ellipse shown in Figure 9(b)).This implies the generation of the tip clearance vortex near the LE also plays a stabilizing role on the suction side.Different from the stable effect of the normal shock on the pressure side, a strip region C with the negative aerodynamic damping, is generated by the normal shock on the suction side.Its position moves towards the TE, while the corresponding intensity and range gradually increase with the tip clearance.These variations are consistent with the changes of the normal shock in Figure 9(b).The shock-induced separation region (65-100% span) after the normal shock also reduces the flutter stability.However, the reduction of the radial flow in the region E near the TE, results in its transformation from negative damping to positive damping.Furthermore, it is noted that the positive aerodynamic damping in region D, where the oblique shock locates on the suction side, is immune to the variation of the tip clearance.Compared to the change of the negative aerodynamic damping at the normal shock on the suction side, it is suggested that the type of the shock wave may be a factor affecting the flutter stability.

Additional influence of aerodynamic coupling effect on flutter stability
As mentioned earlier in section 5.1, the aerodynamic damping exhibits different change trends with the increasing tip clearance at different NDs.Therefore, ND=-6, -3, 0, 1, 3 and 6 are chosen in this section to explore the influence of the aerodynamic coupling effect on the flutter stability in different tip clearance models of 0.25 ,  and 1.75 .Located at 98% span, Figures 15(a), 15(b) and 15(c) shows the unsteady pressure amplitude distribution varying with the ND in the above three tip clearance models, respectively.The unsteady pressure amplitude is normalized by the inlet total pressure.Meanwhile, Figure 16 shows the unsteady pressure phase distribution at 98% span, which also changes with the ND.The white area in Figure 16 represents that the pressure fluctuation is in phase with the blade vibration, which leads to the positive aerodynamic damping, while the gray area represents that they are out of phase and causes the negative aerodynamic damping.Both in Figure 15 and Figure 16, the coordinate 0 on the x-axis represents the LE, while coordinate 1 and -1 represent the TE.In addition, the tip clearance flow in 0.25 tip clearance model is so weak that it has little influence on the blade surface (shown in Figure 8(a) and Figure 9).Thus, the unsteady pressure amplitudes and phases in 0.25 tip clearance model at different NDs are used as control groups for the analysis in this section.Figure 15(a) shows the unsteady pressure amplitude distributions in tip clearance model at different NDs.At negative NDs, the unsteady pressure amplitudes on the pressure side and suction side increase with the absolute value of ND, and the same trend is also found at positive NDs.Meanwhile, the unsteady pressure phase varies very regularly with the ND for tip clearance in Figure 16 (a).On the suction side, the normal shock and the shock-induced separation exhibit the negative aerodynamic damping at positive NDs, but the positive damping at negative NDs.The same is true for the 60-100% chord region and the shock-induced separation region on the pressure side.On the contrary, the normal shock on the pressure side shows the positive damping at nonnegative NDs and the negative damping at rest NDs.Within the range of 0-65% chord on the suction side, the aerodynamic damping is always positive at nonnegative NDs, but the negative damping appears in the 45-65% chord region when  The strong interference of the tip clearance vortices, which causes the reduction in the unsteady pressure amplitude in  (Figures 15(b)) and 1.75 (Figure 15(c)) tip clearance models, varies with the aerodynamic coupling effect.On the pressure side in  tip clearance model, the peak positions of the unsteady pressure amplitude, which locate at the normal shock and shock-induced separation, shift with the ND variation.When the ND is nonnegative, the peak amplitude at the normal shock is greater than that at the shock-induced separation, but when the ND is negative, the peak amplitude at the shockinduced separation is larger instead.In the region after 55% chord on the pressure side, which is affected by the tip clearance vortices impingement, the unsteady pressure amplitudes at nonnegative NDs are larger than that at negative NDs.But on the pressure side in 1.75 tip clearance model shown in Figure 15(c), the peak unsteady pressure amplitudes at the normal shock and shock-induced separation, increase with the absolute value of ND.Meanwhile, the peak amplitude difference between the above two regions gradually shrinks with the increasing absolute value of ND.Besides, the peak amplitude position at the normal shock exhibits irregular movement, whereas the peak amplitude position at the shock-induced separation remains stationary.Within the range of 50-100% chord on the pressure side, where the tip 0.25  1.75 clearance vortices impinge, the unsteady pressure amplitudes at ND=-6 and ND=6 are larger than at other NDs.On the suction side, in the region of 0-65% chord in  tip clearance model (Figure 15(b)), where the tip clearance vortices are generated, the amplitudes of unsteady pressure at negative NDs are generally smaller than that at nonnegative NDs.At the subsequent locations of normal shock and shockinduced separation, the amplitudes basically increase with the ND, which is similar to 0.25 tip clearance model.However, due to the stronger action of tip clearance vortices, within the range of 0-65% chord on the suction side in 1.75 tip clearance model (Figure 15(c)), the unsteady pressure amplitude fluctuates more strongly and irregularly with the ND variation.Besides, at the normal shock and shock-induced separation on the suction side in 1.75 tip clearance model, the unsteady pressure amplitudes varying with the ND are very similar to 0.25 and  tip clearance models, but the amplitudes at different NDs have more significant differences.In Figure 16, in contrast to 0.25 tip clearance model (Figure 16(a)), the influence of the changing aerodynamic coupling effect makes the intense tip clearance vortices in  (Figure 16(b)) and 1.75 (Figure 16(c)) tip clearance models bring about more drastic fluctuations to the unsteady pressure phase.On the pressure side in  tip clearance model, both the normal shock and shock-induced separation have stable effects at ND=6, as well as at ND=-3.And at ND=0, 1, and 3, the normal shock shows the positive aerodynamic damping and the shock-induced separation shows the negative aerodynamic damping.But the opposite is true at ND=-6.Besides, the tip clearance vortices impingement region which locates after 55% chord of the pressure side retains the positive damping.When the tip clearance is increased to 1.75 , the changes in the sign of the aerodynamic damping, are generally similar to those for  tip clearance.Particularly, the tip clearance vortices impingement still leads to the positive aerodynamic damping.Therefore, these same results for  and 1.75 tip clearance models indicate that the impingement of tip clearance vortices on the adjacent pressure side always plays a stabilizing role at different NDs.
On the suction side in Figure 16, the normal shock and shock-induced separation in  and 1.75 tip clearance models exhibit negative aerodynamic dampings at positive NDs and positive dampings at negative NDs, which are similar to those in the 0.25 tip clearance model.However, within the range of 0-65% chord on the suction side in  and 1.75 tip clearance models, the combined action of tip clearance vortices and ND variation causes very intense and irregular fluctuations in unsteady pressure phase.In tip clearance model of 1.75 , the tip clearance vortex near the LE even has an unstable effect at ND=6 and -3.

Conclusion
In this paper, the influence of the tip clearance on the first bending mode flutter stability at different NDs were investigated based on the transonic compressor rotor NASA Rotor 67 and by using numerical simulations.The ICM with an eleven-passage computational domain is applied to calculate the aerodynamic dampings for different tip clearance models at all NDs.The PSB with a double-passage domain is used to explore the influence mechanism of the tip clearance on the flutter stability at different NDs.The flutter characteristic is described by the aerodynamic damping, as well as the amplitude and phase of the unsteady pressure.Therefore, the main conclusions obtained are as follows: (1) ND=1 is the most unstable ND for each tip clearance model.At this ND, the aerodynamic damping exhibits a nonmonotonic trend of increasing first and then decreasing with the rising tip clearance, indicating the existence of an optimal tip clearance with the best flutter stability.However, there are significant differences in the trends of the aerodynamic damping varying with the tip clearance at different NDs.Overall, larger NDs can give rise to the greater aerodynamic dampings in each tip clearance model.
(2) At the most unstable ND, the tip clearance flow exerts two kinds of action on the flow structures to alter the flutter stability.The first one is the tip clearance vortices which are generated on the suction side and impinge on the pressure side.The growing strength of them can enhance the local positive aerodynamic damping.The second one is the interference of tip clearance vortices to the shock waves and to the flow separation.Especially on the pressure side which plays a dominant role in the overall flutter stability of the blade, this interaction makes the stable effects of the normal shock reach the maximum at the design tip clearance, while the unstable effects of shock-induced separation reach the minimum.
(3) The influence of the changing aerodynamic coupling effect makes the tip clearance vortices generation and impingement, as well as the interaction between the vortices and other flow structures bring about more drastic fluctuations to the unsteady pressure.At larger NDs, the unsteady pressure amplitude and phase fluctuate more sharply for larger tip clearances.Eventually, diverse change trends of the aerodynamic damping with the increasing tip clearance appear at different NDs, while the tip clearance vortices impingement on the adjacent pressure side has a constantly stabilizing effect at different NDs.

Figure 3 .
Figure 3. Grid independence verification for the design tip clearance model.

Figure 4 .
Figure 4. CFD characteristics curves in comparison to the experimental data.

Figure. 4 Table 3 .Figure 5 .
Figure 5. 1B mode shape and vibration frequency f of different tip clearance models.

Figure 7 .
Figure 7. Characteristics curves of different tip clearance models.

Figure 8 .
Figure 8. Mach number distribution at 98% span in different tip clearance models.

Figure 9 .
Figure 9. Pressure distribution and streamlines on the blade surface in different tip clearance models.

Figure 10 .
Figure 10.Aerodynamic damping versus ND of different tip clearance models.

Figure 13 .
Figure 13.Aerodynamic damping distribution on the blade surface at ND=1 in different tip clearance models.

Figure 14 .
Figure 14.Tip clearance vortices impingement and aerodynamic damping distribution on the pressure side in tip clearance model at ND=1.
is negative.Compared to tip clearance, there are strong effects of tip clearance vortices on the both pressure side and suction side in tip clearance models of and .Thus, the amplitude and phase of the unsteady pressure on the blade surface change more significantly with the ND for larger tip clearances.
Figure 1.Meridian plane with aerodynamic survey locations for NASA Rotor 67.

Table 2 .
Grid settings in single passage with different tip clearances.