Lock-in phenomenon of tip clearance flow and its influence on aerodynamic damping under specified vibration on an axial transonic compressor rotor

Le HAN, Dasheng WEI, Yanrong WANG, Mingchang FANG

a School of Energy and Power Engineering, Beihang University, Beijing 100083, China

b Collaborative Innovation Center for Advanced Aero-Engine, Beijing 100083, China

KEYWORDS Aerodynamic damping;Inter blade phase angle;Lock-in;Specified blade vibration;Tip clearance flow instabilities

Abstract In this study, the lock-in phenomenon of Tip Clearance Flow (TCF) instabilities and their relationship to blade vibration are investigated numerically on an axial transonic rotor with a large tip clearance.The capabilities of simulating instability flow and lock-in phenomenon are verified on a transonic rotor and a NACA0012 airfoil by comparing with the test data, respectively.The lock-in phenomenon is first numerically confirmed that may occur to TCF instabilities when its frequency is close to the blade vibration frequency. The lock-in region becomes wider with the vibration amplitude increasing,and it is also affected by modal shapes.For the rotor at the simulation conditions in this study, the bending mode results in a wider lock-in region than the torsional mode. In the lock-in region, the phase difference between the Tip Clearance Vortex (TCV)and the blade vibration changes with the flow condition and the frequency ratio of the blade vibration and the TCV instabilities. The frequency of the TCV instabilities reduces with the mass flow decreasing. Therefore, reducing mass flow and increasing frequency ratio have similar effects on the TCV phase,which causes a significant variation on the unsteady pressure amplitude in the blade tip area. Thus, the aerodynamic damping changes significantly with the TCV phase. The aerodynamic damping displays a nonlinear relationship with the vibration amplitude,and it changes from negative to positive with the vibration amplitude increasing at the same frequency ratio. The negative damping is mainly provided by the tip area of the blade.For unlocked conditions,the period of the TCF instabilities fluctuates over time, and it cannot be directly separated by their frequency features.Inter Blade Phase Angle(IBPA)also has an important influence on the feature of the TCV instabilities.The occurrence of frequency lock-in also requires‘‘appropriate”IBPA.For the examined working conditions,the frequency lock-in occurs under 0 ND(Nodal Diameter),but not under 8 ND. However, no matter 0 ND or 8 ND, the phase of TCV always locks onto the IBPA at the examined conditions.

1. Introduction

Nonsynchronous Vibration (NSV) is an aeroelastic phenomenon different from flutter and forced response. It can happen within a single row. For a tuning system, the blades in a row have similar natural frequencies. When NSV occurs,the blades usually lock onto a certain frequency and phase difference.However,this physical process is difficult to reproduce by numerical simulation.

The excitations of NSV generally come from the separation flow or Tip Clearance Flow (TCF) instabilities. When NSV occurs, the frequency of flow instabilities is usually close to the blade’s natural frequency. Owing to the interaction of the flow instabilities and the blade vibration, the frequency of flow instabilities may step change into the blade’s natural frequency. Within a certain range of operating conditions,the flow instabilities frequency locks onto the structure frequency. This is an important feature of NSV and called lock-in phenomenon.Another kind of frequency lock-in refers to the lock-in between blades in a row. The lock-in between blades in a row is usually a typical feature of flutter. Owing to the geometric mistuning and material defects, the natural frequencies of blades in a row are not exactly the same and the maximum frequency difference can exceed 1%.When flutter occurs,all the blades in a row lock onto the same frequency and vibrate as a whole. The lock-in phenomenon we investigated in this study is the interaction between the flow instabilities and the blade vibration, not the lock-in between blades.

The NSV caused by flow instabilities is widely investigated.Baumgartner et al.identified the excitation as shedding vortices in the blade tip area, which is not in resonance with engine orders.This is also verified numerically and experimentally by Kielb et al.The results showed that the vibration frequency shifts little (about 2%) with the rotating speed increasing and the amplitude of unsteady pressure is significantly higher than that caused by Inlet Guide Vanes (IGV)passing in the tip area. The same compressor was simulated by Im and Zhato analyze the aerodynamic excitation of NSV. They found that the structural vibration amplitude has little effect on the aerodynamic force. NSV is induced by the circumferentially traveling vortices. Mailach,Maerz,Vo,Pardowitzand Taghavi Zenouzet al.investigated the aerodynamic excitation of NSV at subsonic conditions. They all found that the unsteady pressure is caused by the motion of TCF, which only exists in the tip area. Holzinger et al.observed blade vibrations after several periods of unsteady pressure fluctuations. This indicates that the vibration phenomenon in their study is NSV rather than flutter.Moellerconsidered NSV is a phenomenon of self-excited vibration caused by unsteady TCF. Therefore, they specified the blade vibrating at a certain mode and estimated the instabilities of a rotor system with aerodynamic damping. The results show that the unsteady TCF and the blockage at the rotor tip have a significant effect on the aerodynamic damping. With the tip clearance and the tip blockage reducing, the aeroelastic stability increases.Thomassinand Droletet al.suggested that NSV is caused by the unsteady TCF resonating with its feedback pressure wave.

Inoue et al.experimentally investigated a low-speed compressor and found a phenomenon of ‘‘mild stall condition”, which is caused by the radial vortex forming from the separation flow at the blade leading edge. Pullan et al.explained the rotating stall with the motion of radial vortices.Due to high incidence,the incoming flow separates at the blade leading edge,forms radial vortices.Then,they propagate from blade to blade.Brandstetter et al.conducted a detailed experiment on both transonic and subsonic conditions. They found that the radial vortex disturbances are similar for both conditions. With throttling, the motion of the radial vortex causes the vibration of the blade, and the vibration contributes to the forming of the radial vortex in return.

The vibration induced by vortices has been widely investigated on cylindersand flat platesin cross-flow,which were usually studied under low or transition Reynolds number(Re).The vortex shedding frequency changes with incoming flow and the scale of structure perpendicular to the incoming flow.While the vortex shedding frequency is close to the natural frequency of the structure,it will lock onto the natural frequency.Zhang et al.studied the lock-in phenomenon of a cylinder at Re=60, based on a linear dynamics model. Their results proved that the linear model can accurately reproduce the inherent physics of the lock-in phenomenon and further revealed that the lock-in region can be divided into two patterns: resonance-induced lock-in (dominating by the fluid mode) and flutter-induced lock-in (dominating by the structural mode).

The lock-in phenomenon is also found in airfoils and turbomachineries.Zhu et al.simulated a 2D thin cambered blade with enforced motion in pitching at Re=1.0×10. A V-shaped lock-in region was found and the width of the region depends on the pitching amplitude.Spiker et al.investigated a 2D airfoil tip section of a modern front stage compressor blade with a harmonic balance method. They found that the Inter Blade Phase Angle (IBPA) has a significant effect on the range of lock-in and the largest response is at a frequency in the lock-in region other than the natural shedding frequency. Poirel et al.conducted an investigation on the selfsustained pitch oscillations of a NACA0012 airfoil with Re below 1.5×10. They suggest that the laminar separation of the boundary layer is the critical factor of the self-sustained oscillations. Therefore, inhibiting turbulence is helpful to suppress the vortex-induced vibration.Young and Laiand Ashraf et al.described the lock-in phenomenon of plunging airfoils at Re=20000. The results seem to show a different regular to the pitching motion airfoilthat with the plunging amplitude increasing, the vortex shedding process changes from periodic to quasi-periodic, and finally to chaotic. They believe that the random interaction of leading edge vortices is responsible for chaotic behavior. Poirel et al.investigated the coupled pitch and heave motion with initial disturbance.They found that the heave motion is driven by the pitch motion,and the amplitude increases significantly when the frequency ratio of the pitch and the heave closes to 1(Re ranged from 6.5×10to 1.2×10). Besem et al.conducted an experiment on a NACA0012 airfoil at very high angles of attack with the harmonic balance method. The Re range is between 87000 and 500000. They analyzed the aerodynamic damping in the lock-in region and found that the negative damping is always found in the right side of the lock-in region,where the vortex shedding frequency is lower than the enforced vibration frequency. Zhao et al.studied the correlation between the blade vibration and noise in a multistage highpressure compressor. They found the vibration and the sound pressure level of characteristic frequency noise reaches their maximum amplitudes at the same time and the characteristic frequency is locked in a specific range which presents no variation with the rotating speed. Quan et al.numerically investigated the lock-in phenomenon of an airfoil in transonic flow and found that when the lock-in phenomenon occurs, it can maintain until the double unsteady flow frequency approaches the airfoil natural frequency.Gao et al.investigated different types of aeroelastic phenomena and the lock-in phenomenon on a NACA0012 airfoil. By comparing with the coupled Computational Fluid Dynamics/Computational Structural Dynamics (CFD/CSD) simulation results, it is confirmed that the lock-in phenomenon mainly depends on the linear dynamics instead of the nonlinear features, and the instability of the fluid or structural modes is the root cause of different transonic aeroelastic phenomena and the lock-in phenomenon. The vibration amplitude is usually larger when the coupling system is dominated by the instability of the structural mode.

NSV caused by flow instabilities can also be treated as a kind of vortex-induced vibration.However,there is limited literature on the influence of the blade vibration on the flow instabilities (lock-in phenomenon), especially caused by the TCF, and most of the studies on the lock-in phenomenon focus on the 2D flow. Han et al.investigated the influence of the blade vibration on TCF instabilities and the lock-in phenomenon on a transonic rotor,which shows a strong 3D effect.This study also puts efforts on the lock-in phenomenon of the TCF instabilities and further investigated the aerodynamic damping in and out of the lock-in region and the influence of Nodal Diameter (ND) on the lock-in phenomenon. First,the capabilities of simulating instability flow and lock-in are verified on a transonic rotor and a NACA0012 airfoil by the test data, respectively. Then, the process of the TCV instabilities is introduced. The relationship between the Tip Clearance Vortex (TCV) instabilities and the blade vibration is analyzed in the lock-in and unlocked regions. The lock-in phenomenon and the aerodynamic damping are investigated under different mass flow, frequency ratios and vibration amplitudes. At last,the lock-in phenomenon is simulated under different IBPAs.

2. Numerical methods and verification

2.1. Numerical methods

A transonic rotor is numerically investigated here,in which the TCF instabilities phenomenon was observed at large tip clearances. Thus, the rotor is simulated at a large tip clearance to reproduce the TCF instabilities (1.9% tip chord length, C).This rotor has also been investigated in Refs. 38–40 and the geometries and parameters are shown in Table 1.

In order to capture the instability flow, a time-marching method was employed to compute the three-dimensional Reynolds-averaged Navier-Stokes equations. These equations were discretized using a cell-centered control volume method.A high-resolution spatial discretization scheme was adopted,with a special nonlinear recipe at each node.A secondorder backward Euler scheme was adopted for the transient term, and the k-ε two-equation turbulence model with wall functionwas used. The instability flow was simulated under three different time resolutions in Ref. 44, and it is found that about 30 time steps in each period are enough forTCF instabilities simulation. Therefore, 30 time steps were specified for the simulation of the rigid blade. When we calculated the aerodynamic damping in the lock-in region, 60 time steps were specified in the vibration period for the simulation of oscillation flow fields. When we investigated the influence of inter blade phase angle with the full annulus model,40 time steps were specified to save the calculation time.

Table 1 Geometries and parameters of rotor blade.

A structured multi-block grid system is used here. As shown in Fig. 1, full annulus and single-passage models were simulated.They have similar grid structures:H-type grids were generated in the main flow region and up/downstream of the blade, while O-type grids were generated in the tip clearance region and around the blade. For the full annulus model, the computational domain consisted of 141 points streamwise,1767 points pitchwise (each passage contains 57 points and 31 passages in total), and 57 points spanwise, for a total of about 1.36×10points.To satisfy the conditions of the turbulence model,the spacing of the first grid layer on the solid wall was set to 4.6×10m. The grid dependency was also done on single-passage model in Ref. 45, and a total number of 0.44 million grid points were generated in each passage for grid independence.

No-slip and adiabatic boundary conditions were applied to the solid walls, the stagnation pressure, stagnation temperature, and flow angle were specified at the inlet boundary, and the static pressure was adopted at the outlet boundary. The upstream and downstream ducts have axial lengths of 1.47C and 2.61C,respectively.For the cases of single-passage simulation, the periodic boundary condition without phase lag was used directly. For the case of full annulus simulation, the single-passage simulation results with the same inlet and outlet conditions were utilized as the initial field.

2.2. Verification on reproducing instabilities and lock-in

Fig. 1 Compressor geometry and computational grid systems for full annulus and single-passage models.

For the purpose of this study,two aspects of simulation capabilities are very important. The first one is the capability of reproducing instability flow,which was experimentally verified on the full annulus model of a transonic rotor with 15 blades.For the transonic rotor with 15 blades, the instabilities and unsteady pressure fluctuations are generated by the separation flow at the tip leading edge of the blade, just as Pullan et al.described.The instabilities occur at partial rotating speed.The numerical investigation on this rotor mainly focused on the interaction between flow instabilities and blade passing.Therefore,a numerical monitor was placed near the blade tip trailing edge before simulation, where both disturbances are strong.The experiment of this rotor mainly investigated the noise caused by flow instabilities. Therefore, a series of pressure transducers were installed upstream of the blade. Owing to the pressure of simulation and experiment was obtained at different tip positions, their amplitudes were difficult to match with each other. However, the main frequency components in the spectrum should be mutually verifiable, thus we used the experimental results to verify the capability of numerical method on capturing the frequency of instabilities.Comparing the pressure spectrum of the simulation results with the test data (Fig. 2), the frequency of instabilities is numerically captured around 3.50 EO (Engine Order) and is around 3.53 EO from the test data. The simulation results and the test data show a Blade Passing Frequency (BPF) of 15 EO and 15.05 EO,respectively, which is consistent with the theoretical value of 15. Thus, we think the numerical method successfully predicts the frequency of instabilities with an error of about 1%. The details of the rotor and the results can be found in Ref. 46.

The other important capability of the numerical method is reproducing the lock-in phenomenon. During the simulation process,the pressure fluctuation at the monitor is used to judge whether lock-in occurs.When lock-in occurs,there is only one frequency component of the blade vibration existing in the flow field and the TCF instabilities still exist in the tip flow field,which can be identified by the structure of TCV at different time steps(as followed described).The time-domain signal of the lock-in results is similar to the results shown in Fig.3(a).When TCF instabilities unlock to the blade vibration,there are more than one frequency component existing in the flow field and the time-domain signals of the unlocked results are similar to the results shown in Fig. 3(b).

Fig. 2 Comparison of numerically predicting instabilities with test results on a transonic rotor at 55% rotor speed.

Fig. 3 Pressure fluctuation under lock-in and unlocked conditions.

A NACA0012 airfoil was used to verify the capability of reproducing the lock-in phenomenon by comparing with the test data in Ref. 31. The NACA0012 airfoil was simulated at Re=10and attack angle of 40°. The airfoil chord length is 0.25 m. The NACA0012 airfoil was computed at different pitching amplitudes and frequency ratios (f). The results are shown in Fig. 4. The diamonds represent the experimental results and the circles describe the simulation results. The empty shapes mean unlocked and the filled shapes represent the lock-in phenomenon. The black dash line depicts the lock-in region basing on the circles in Fig. 4, the red dash line is the experimental lock-in region and the blue dash line is the lock-in region of the harmonic balance method results in Ref.31. It can be seen that the numerical methods employed here can reproduce the lock-in phenomenon and agree well with the experimental results. The chaotic performances mentioned in Ref. 31 are described as ‘‘an unusual broadband frequency spectrum”. Therefore, they are considered to be unlocked in Fig. 4(b).

3. Tip clearance flow instabilities with rigid blade

The transonic rotor with a tip clearance of 1.90%C was simulated at 100%rotating speed and the performances are given in Fig. 5. The mass flow is normalized by the maximum mass flow. The simulation results agree well with the experimental results. The phenomenon of the TCF instabilities is found at small mass flow conditions (blue star and red dots in Fig. 5)and its feature is analyzed basing on the simulation results of the full annulus model (blue star in Fig. 5). The numerical probes were installed in both the stationary and the rotating frames.They were placed at the same axial and radial position.

A numerical probe(blue dot in Fig.6(a))was installed near the rotor tip trailing edge in the stationary frame,and its FFT results are shown in Fig. 6(b). A broadband with several frequency peaks can be found in the frequency spectrum, which is excited by the TCF instabilities. Some authors suggested that it is an indication of Rotating Instabilities (RIs).In addition, the BPF and the interactive frequency of the BPF and the TCF instabilities can also be observed in the frequency spectrum.

Then, the pressure fluctuation of a probe in the rotating frame(one of the red dots in Fig. 6(a)) is analyzed and shown in Fig. 6(c) that the unsteady pressure caused by the TCF instabilities also contains several major frequency components in the rotating frame, implying that the period of the TCF instabilities fluctuates within a certain range. The average phase difference of TCF between adjacent passages is also shown in Fig. 6(c). The phase difference of the 3701 Hz frequency component is close to 0°, which indicates that the TCF instabilities contain a single-passage periodic feature.

The vortex structure in the tip flow field is described by the second invariant of the velocity gradient tensor and the normalized helicity H.The unsteady process of TCV in the tip flow field is shown in Fig. 7. A radial vortex (in black dot circles) forms in passage and impacts on the tip leading edge of the following blade over an oscillation period. At the same time, with the radial vortex moving circumferentially,part of the blockage falls off the broken TCV, forms a shedding vortex,and then flows downstream along the tip pressure side of the blade.The details and the generation mechanism of the TCV instabilities were investigated and shown in Ref. 44.

Fig. 4 Comparison of lock-in regions.

Fig. 5 Performance of transonic rotor.

Fig. 6 FFT results of unsteady pressure.

4. Tip clearance flow instabilities with enforced blade vibration

The Campbell diagram of the rotor is depicted in Fig. 8. The frequency of every mode is calculated at 100%rotating speed,which we focus on in the study. The frequency of the TCF instabilities will encounter the 8th and 9th modes of the blade.In order to investigate the relationship between the TCF instabilities and the blade vibration, the bending and torsional modes are simulated here. The blade is enforced to vibrate as the 1B (bending) and 8th modal shapes (the 8th mode is selected because it is a torsional mode and is close to the natural frequency of the TCF instabilities). In order to simulate the lock-in phenomenon, the enforced vibration frequency is modified to the TCF instabilities frequency for convenience.The modal shape is also shown in Fig. 8. As the results in Fig.6(c)display that the frequency component of the 0°phase difference is one of the main components, a single-passage model with periodic boundary is employed in this section.

4.1. Feature of TCV instabilities in lock-in region

4.1.1.Lock-in region under different operating conditions,modes and amplitudes

Fig. 7 TCV instabilities process in tip flow field.

Fig. 8 Campbell diagram of rotor blade.

The blade is enforced to vibrate as the first bending and the 8th modal shapes(torsional)with a specified frequency of 3701 Hz(the TCF instabilities frequency), and the maximum vibration amplitudes (A) are specified as 0.75%C and 0.075%C. Then,the operating conditions are adjusted by changing the backpressure.The relationship between the mass flow and instability frequency is listed in Table 2 and depicted in Fig. 9. The results under 1B, 0.75%C (green triangles in Fig. 9) are marked by A, B and 1–5. The backpressure of Case 3 is the same with the operating condition marked by a blue star in Fig. 5. The frequency of the TCF instabilities decreases with the mass flow.Cases 1 to 5 have the same instability frequency and Cases A and B have different frequency components in the flow field. Thus, we know the lock-in phenomenon can also occur on TCV instabilities. The instabilities frequency locks onto the blade enforced frequency when they are closed. The lock-in region at A=0.75%C is wider than 0.02 mass flow rate (0.86–0.88). However, for the cases of A=0.075%C,the lock-in region is very narrow, which indicates the lock-in rarely occurs. That means the lock-in region increases with the vibration amplitude. Then, lock-in regions of 1B and 8th modal shapes are compared under the same vibration amplitude of 0.75%C. It can be seen that the influence of modal shapes on the frequency of the TCF instabilities is quite different. For the rotor here, the 1B mode is more likely to induce the lock-in phenomenon than the 8th mode for the simulated conditions. This indicates that 1B mode has a stronger influence on the TCV instabilities than the 8th mode. Thus, the influence of the blade vibration on the TCV instabilities is investigated based on 1B mode in the subsequent sections.

The responses of the blade to the TCF instabilities were investigated in Ref.45.The results show that when TCF instabilities approach the natural frequency of the blade,the vibration amplitude is magnified. Basing on the feature of lock-in phenomenon and the results in Ref. 45, we can understand the reason for the occurrence of the lock-in. While the blade vibrates at a small amplitude, the lock-in area is almost unobservable. When the frequencies of the flow instabilities approach the blade vibration frequency at some operating conditions, the vibration amplitude is magnified, which is shown in Ref.45.Thus,with the operating condition changing,it will display a wider lock-in region.

4.1.2. Variation of TCV instabilities phase with operating condition

The results in the lock-in region are signed by Cases 1 to 5(1B,0.75%C) in Fig. 9. The processes of the TCV instabilities are described in Fig.10.The blades are in the equilibrium positionat t=0T and t=0.5T (T: the period of blade vibration), in the maximum amplitude of the suction side at t=0.25T,and in the maximum amplitude of the pressure side at t=0.75T. It will return to the equilibrium position at t=1T. The TCV instabilities still exist in the tip flow field when lock-in occurs. The oscillating frequency is the same as the frequency of the blade vibration and the unsteady process under different operating conditions is similar. However, the phase difference between the TCV and the blade vibration changes with the operating conditions in the lock-in region,which are shown in Fig.11.The TCV phase at t=0T of Case 3 is selected as the initial phase(0),and the TCV phases under different operating conditions are identified by the position of the radial vortex in the tip flow field. The phase difference between the TCF and the blade vibration changes by about 0.32π in the lock-in region, and as it approaches the edge of the lock-in region, the phase changes more. In addition, when the blade moves towards its pressure side(t=0.25T–0.75T),a shedding vortex falls off the blade suction surface,and always keeps synchronized with the blade vibration.

Table 2 Lock-in region under different modes and amplitudes.

Fig. 9 Lock-in region under different mass flow.

Fig. 11 Phase difference between TCV and blade vibration under different operating conditions in lock-in region.

The 1st harmonic of the unsteady pressure on blade surfaces are depicted in Fig. 12. The results show that the unsteady pressure amplitude on the pressure surface is much higher than that on the suction surface.The unsteady pressure distribution on the suction surface is similar under different cases, but that on the pressure surface is different, especially in the tip area.That means the changing of the operating condition in the lock-in region mainly causes the variation of unsteady pressure in the blade tip pressure surface.Comparing Fig. 10 and Fig. 12, the influence area of TCV is consistent with the unsteady pressure changing area. This indicates that the differences of unsteady pressure under different cases in the lock-in region are mainly caused by the variation of TCV phase and the influence area of TCV is mainly above 90%spans. This difference has an important effect on the aerodynamic damping, which is analyzed in Section 4.1.4 and deeply investigated in Ref. 47.

4.1.3. Lock-in region under different frequency ratios and

amplitudes

In this section,the TCV instabilities are investigated under different frequency ratios by adjusting the specified vibration frequency. The operating condition is the same as Case 3 in Fig.9.A V-shape lock-in region is found by changing the specified vibration frequency and the vibration amplitude, which has been widely confirmed on the lock-in phenomenon caused by separation flow. As shown in Fig. 13, lock-in occurs near f=1 and the width of the lock-in region becomes wider with the vibration amplitude increasing. When the amplitude approaches zero, the lock-in frequency close to f=1.005,not the natural frequency f=1. This difference is considered causing by the errors of the time resolution and the data length of FFT.

Fig. 10 TCV instabilities process in lock-in region.

Fig. 12 Unsteady pressure amplitude on blade surfaces.

The vortex structure at t=0T and the unsteady pressure amplitude on the blade surface are described in Fig. 14 (the vibration amplitude is 0.75%C). The vibration frequencies are specified as 3650 Hz (f=0.986) and 3800 Hz(f=1.028), which are marked by the red frame in Fig. 13.As shown in Fig. 14, the phase of the TCV changes with the frequency ratio. Comparing Fig. 14 with Fig. 10 and Fig. 12,it is confirmed that adjusting the specified vibration frequency and changing the backpressure has a similar influence on the phase of TCV, and therefore, has a similar influence on the unsteady pressure distribution. In fact, increasing the backpressure reduces the instability frequency and is equivalent to increasing the frequency ratio.

Fig. 13 Lock-in region under different frequency ratios.

4.1.4. Aerodynamic damping in lock-in region

The aerodynamic damping is further calculated under different mass flow and frequency ratios. The expression of aerodynamic work Wper blade vibration period done to the blade is given in the below equation:

where P is the pressure on the blade surface, v is the vector of the vibration velocity, n is the normal vector of the blade surface and S is the blade surface area.The aerodynamic damping is calculated by

where Aand ω represent the modal amplitude and modal frequency, respectively. The aerodynamic work distribution under different conditions is depicted in Fig. 15. The aerodynamic work changes in the tip area and almost keeps constant in the partial blade spans with the mass flow decreasing and with the frequency ratio increasing. Two things can be confirmed, one is that the phase of TCV does have an effect on the aerodynamic damping, especially in the tip area, and the other is that increasing the backpressure and increasing the frequency ratio has the same effect on the aerodynamic work distribution in the tip area.

Fig. 14 Instantaneous TCV structure at t=0T and unsteady pressure on blade surface under different vibration frequencies (1B).

Fig. 15 Aerodynamic work distribution on blade surface under vibration amplitude of 0.75%C.

Then the aerodynamic work and damping are calculated based on Eqs. (1) and (2), which are listed in Table 3 and Table 4. To obtain the influence of mass flow, frequency ratioand vibration amplitude on the aerodynamic damping intuitively, the variation of the aerodynamic damping is also plotted in Fig. 16. As shown in Fig. 16(a), the aerodynamic damping decreases first and then increases with the mass flow,which is generally consistent with the variation of aerodynamic damping in the tip area in Fig. 15. Fig. 16(b) shows that the aerodynamic damping increases with the frequency ratio,regardless of the variation of aerodynamic damping in the tip area. Increasing the frequency ratio results in the unsteady pressure caused by the blade vibration increasing with the vibration velocity. This implies that the unsteady pressure caused by the blade vibration provides positive aerodynamic damping and has a greater effect on the total aerodynamic damping than the TCV phase, thus the aerodynamic damping increases with the frequency ratio.However,we cannot distinguish the aerodynamic damping of the two components from each other at present. When the frequency ratio is kept at f=1.005, the aerodynamic damping displays a nonlinear relationship with the vibration amplitude,and it changes fromnegative to positive with the vibration amplitude increasing(Fig. 16(c)). By plotting the aerodynamic work distribution,we know that the negative damping is mainly provided by the blade tip area. For detailed investigation can be found in Ref. 47.

Table 3 Aerodynamic work and damping under different mass flow.

Table 4 Aerodynamic work and damping under different frequency ratios and vibration amplitudes.

Fig. 16 Aerodynamic damping in lock-in region.

4.2. Feature of TCV instabilities for unlocked conditions

Case A (1B, 0.75%C) in the unlocked region is analyzed(Marked in Fig.9).The time history of pressure in the rotating frame is plotted in Fig.17(a), and its FFT results are depicted in Fig.17(b),which shows that the TCV instabilities consist of several frequency components at the unlocked condition. Due to the influence of the blade vibration,the period of TCV fluctuates over time.

Fig. 17 Unsteady pressure of numerical probe in rotating frame (1B, 0.75%C).

Fig. 18 Unsteady pressure harmonic components on blade surface (left: suction side, right: pressure side).

The 1st harmonics of the unsteady pressure relating to the blade vibration and the TCV instabilities are described in Fig. 18. The distribution of the unsteady pressure caused by the TCV instabilities is similar under different modes,but that caused by the blade vibration is decided by the modal shapes.Comparing with the unsteady pressure distribution on the rigid blade,the TCV components on the vibration blade surface are much smaller, especially when the blade vibrates as 1B mode.This is because that the unsteady pressure of TCV and vibration components are not independent of each other. They interact strongly with each other and cannot be directly separated by their frequency features. The TCV component of the 8th mode is much closer to the rigid blade results, which indicates that the motion of bending mode (1B) may have more effect on the TCV instabilities than the torsional mode (8th mode).

The pressure distribution at 98%blade span is plotted over a vibration period (Fig.19(a)). The pressure fluctuation at the pressure surface is much stronger than that at the suction surface and the area with large pressure fluctuation is consistent with the influence area of TCV. However, we cannot confirm here how much of the pressure fluctuation is contributed by the TCV and how much is contributed by the blade vibration.The pressures at the beginning of every blade vibration period are depicted in Fig.19(b).The pressure fluctuation on the suction surface agrees well with each other at different vibration periods and the pressure fluctuation on the pressure surface changes with the vibration period, which means the pressure fluctuation is caused by the TCV instabilities and mainly exists on the tip pressure surface.

Then,two points at 20%C and 80%C positions on the pressure surface are selected and the pressure at the beginning of every blade vitiation period is shown in Fig. 20. The unsteady pressure at 0 s of every vibration period changes irregularly,which means there is no obvious regularity in the periodic changes of TCV. For the point near the blade leading edge(20%C), the frequency of the TCV instabilities is close to the vibration frequency and the frequency fluctuation is small in the first several cycles (1T–10T). However, the unsteady pressure fluctuation at 0 s increases suddenly in the subsequent cycles (10T–20T), which means the frequency fluctuation of the TCV changes significantly. For the point near the blade trailing edge (80%C), the variation of pressure fluctuation is small and not obvious between vibration periods. Comparing Fig. 18, Fig. 19 and Fig. 20, this is because TCV instabilities have less influence near the trailing edge in the tip area.

Fig.20 Pressure of every 0 T at 20%and 80% chord lengths of 98% blade spans.

Fig. 19 Pressure fluctuation at 98% blade span.

Fig. 21 Aerodynamic damping and averaged aerodynamic work distribution on blade surface for unlocked conditions.

In addition, the aerodynamic damping is calculated under 1B mode at the operating conditions of Cases A and B(Marked in Fig. 9). Owing to the period of the TCF instabilities changes with time at unlocked conditions, a long time domain signal is extracted to calculate the aerodynamic damping at different time and to obtain the time-averaged results.The aerodynamic damping changes with the selection of the integration interval. As shown in Fig. 21, it varies from 0.0961%to 0.102%at Case A and from 0.0785%to 0.103%at Case B. The fluctuation of aerodynamic damping at Case B is larger than that at Case A,which is significantly affected by the strength of TCF.The strength of TCF increases with the mass flow rate decreases,which is shown in Fig.10 and can be found in Refs.44,45.The averaged aerodynamic work distribution is also depicted in Fig.21.Compared with the results in the lockin region (Fig. 15), the concentrated area of work in the tip blade disappears, which means that although the TCF causes the fluctuation of aerodynamic damping, it has little contribution to the averaged results for unlocked conditions.

4.3. Influence of inter blade phase angle on lock-in phenomenon

The oscillating flow fields of 0 ND and 8 ND are simulated and analyzed with the full annulus model at the operating condition of Case 3. The relationship between ND and IBPA is described by the formula IBPA=2π∙ND/N. 0 ND and 8 ND correspond to IBPAs of 0 and 0.52π, respectively. The time-varying pressure histories of the monitoring point(Fig. 6(a)) in the rotating frame are depicted in Fig. 22. The pressure fluctuation period obtained at 0 ND is very regular,which is similar to the simple harmonic,and the pressure wave result at 8 ND shows that the disturbance in the flow field is complicated. Although the enforced vibration frequency of the rotor (3701 Hz) is in the natural frequencies range of TCV for the rigid blades (3603–3993 Hz), the lock-in phenomenon does not occur when the rotor vibrates as 8 ND.

Fig. 22 Pressure histories of monitoring point at different IBPAs.

Fig. 23 FFT results of monitoring point at different IBPAs.

Then, the FFT results of the extract time-varying pressure histories are given in Fig. 23. Owing to the contribution of blade vibration, the unsteady pressure amplitudes at 0 ND and 8 ND are much higher than those at the rigid blade.When the blade vibrates as 0 ND,there is only one frequency component of 3701 Hz in the flow field, which is the specified blade vibration frequency and also the natural frequency of the 0 phase TCV component. Thus, it is confirmed that lock-in occurs at 0 ND. The simulation results of the full annulus model at 0 ND are the same as the single-passage results above that the unstable TCV can lock onto the blade vibration frequency. When the blade vibrates as 8 ND, there are two main frequency components of 3701 Hz and 3716 Hz in the flow field. This indicates that the unstable TCV does not lock onto the blade vibration.Comparing the results of 0 ND and 8 ND,it can be found that the IBPA of the blade vibration has an important effect on the lock-in between the TCV instabilities and the blade vibration. Only in the ‘‘appropriate” IBPA,lock-in phenomenon can occur. For the rotor here, 8 ND is not the‘‘appropriate”IBPA when specified the blade vibrating as 3701 Hz.

The instantaneous tip flow fields at different IBPAs and the schematic diagram of TCV circumferential features are depicted in Fig. 24. The low velocity regions in the tip flow fields are caused by the broken of the TCV and reflect the state of the TCV.When the blade is rigid,the circumferential phase of the TCV displays 1 ND feature overall and contains the features of 0 ND,±1 ND,2 ND,and 3 ND(Fig.6(c)).When the blade vibrates as 0 ND,the TCV not only locks onto the vibration frequency but also is affected by the IBPA that the circumferential phase difference also locks onto 0. This result is consistent with the single-passage results above and proves that the single-passage model with periodic boundary conditions is reasonable to analyze the 0 ND component of the TCV.When the blade vibrates as 8 ND,the tip flow field exhibits a different circumferential feature from that of the rotor with rigid blades or blades vibrating as 0 ND. The TCV displays periodic features every 3–4 passages in the circumferential direction, which is almost consistent with the IBPA. This indicates that although the frequency of TCV does not lock onto the vibration frequency at 8 ND, the circumferential phase of TCV is still affected by the IBPA. In a word, the frequency lock-in occurs at 0 ND, but not at 8 ND. The phase lock-in occurs at both 0 ND and 8 ND.

5. Conclusions

The lock-in phenomenon of the TCV instabilities and their relationship with the blade vibration is investigated numerically on a transonic rotor in this study. The TCV instabilities are obtained by enlarging the tip clearance.(The details about the influence of the tip clearance on the TCV instabilities are described in Ref. 44). As the frequency and behavior of TCV are mystery before simulation, a time marching method is employed and verified on a transonic rotor and a NACA0012 airfoil by comparing it with test data. The main conclusions are as follows.

Fig. 24 Instantaneous tip flow fields at different IBPAs.

(1) The lock-in phenomenon and the lock-in region are confirmed on the TCV instabilities. The occurrence of the lock-in phenomenon requires the frequency of the TCV instabilities to be close enough to the blade vibration frequency. For the rotor at 0 ND, when the frequency difference between the TCV and the blade vibration is less than 1%–1.6%, the lock-in phenomenon can occur. The lock-in region becomes wider with the vibration amplitude, and it is also influenced by the modal shapes.For the rotor at the simulated conditions here, it seems that the bending mode has a greater effect on the TCV instabilities than the torsional mode and causes a wider lock-in region.

(2) The phase difference between the TCV and the blade vibration in the lock-in region changes with the flow condition and the frequency ratio. The frequency of the instability flow decreases with the mass flow decreasing. Thus, increasing the backpressure is equivalent to increasing the frequency ratio, which causes the unsteady pressure amplitude to change significantly in the tip area of the blade pressure surface.

(3) The variation of the TCV phase has an important influence on the aerodynamic damping and the aerodynamic work distribution on the blade surface. The aerodynamic work(distribution)mainly changes in the tip area and is similar at partial blade spans with the variation of the TCV phase. The aerodynamic damping displays a nonlinear relationship with the vibration amplitude,and it changes from negative to positive with the vibration amplitude increasing at the same frequency ratio.The negative damping is mainly provided by the tip area of the blade, where is the influence area of the TCV.

(4) For the unlocked condition, the process of the TCV instabilities is still influenced by the blade vibration and cannot be directly separated by their frequency features. The period of the TCV instabilities becomes fluctuation and its unsteady process is not a simple sine law.Its period keeps close to the period of the blade vibration for several cycles and then fluctuates for the next several cycles. This causes a significant fluctuation of pressure between cycles,especially on the tip region corresponding to the influence area of the TCV.

(5) The IBPA has an important influence on the feature of the TCV instabilities.The occurrence of the lock-in phenomenon requires ‘‘appropriate” IBPA. For this rotor at the examined working conditions, the TCV instabilities lock onto the blade vibration at 0 ND, but not at 8 ND. However, the circumferential periodic feature of TCV is decided by IBPA at the simulated conditions.When the rotor vibrates as 0 ND or 8 ND,the TCV displays circumferential features for every passage (0 ND)or every 3–4 passages (8 ND) in the circumferential direction. Although the frequency lock-in occurs at 0 ND, but not at 8 ND, the phase lock-in occurs at both 0 ND and 8 ND.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The work was supported by the National Natural Science Foundation of China (No. 51475022).