Non-synchronous blade vibration analysis of a transonic fan

Yun ZHENG, Qingzhe GAO, Hui YANG,b,*

a School of Energy and Power Engineering, Beihang University, Beijing 100191, China

b Jiangxi Research Institute, Beihang University, Nanchang 330096, China

KEYWORDS Aerodynamic disturbance;Aliasing;Blade vibration;Circumferential vortical propagation;Traveling wave

Abstract This study numerically investigates the aeromechanic behavior of a transonic fan model with a flat tip-leading-edge on the NASA rotor 67 test case.Single-passage unsteady calculations at a near stall operating point of 82% design speed show that the dominant frequencies of mass flow were not the harmonics of the rotor rotational frequency.A full-annulus fluid-structure interaction analysis was subsequently carried out to examine the unsteady flows and their interactions with blade vibrations. The results show that the modal displacement of the backward traveling seventh nodal diameter of the second torsion mode grew exponentially, which reveals that the blade vibration was non-synchronous. The vibration pattern indicates that the aerodynamic mode was resonant with the structural vibration mode. Around the rotor tip, the circumferential vortical propagation induced by interactions among the main flow, tip leakage flow, and tip clearance vortex was the source of aerodynamic excitation. To clarify the mechanism of the non-synchronous vibration, the coupling between aerodynamic disturbance and structural response, i.e., aliasing,was summarized.The frequency spectra of the fluctuating pressure show that an aerodynamic Backward Traveling Wave(BTW)was co-aliased to a structural BTW due to the propagation of the circumferential vortex. The correlation between the frequency and free convective speed of the aerodynamic disturbance determined the directions of aliasing.

1. Introduction

Aeromechanical problems in turbomachinery compromise the safe operation of engines.1-3Traditional problems related to blade vibrations include forced response and flutter. In recent decades, a novel problem called Non-Synchronous Vibration(NSV) has emerged as a focus of research. Unlike forced response, the frequencies of NSV are not in resonance with the harmonics of rotor rotational frequency. Different from flutter in that NSV usually occurs at a high reduced frequency,this typical flutter parameter is well within historically observed stable regions. Furthermore, at the onset of NSV,aerodynamic disturbance exists without blade vibration, and can be considered a non-synchronous forced response. When the resonance condition is attained, the aerodynamic disturbance locks to the structural vibration,and NSV appears more like a flutter problem hereafter. To better understand this aeromechanical problem, extensive research has been carried out on the mechanism of NSV that can be summarized as below.

Nomenclature C wave speed E total energy per unit volume fe external force vector g gradient H total enthalpy k thermal diffusivity coefficient n unit normal vector nx component of the unit normal vector in the x direction ny component of the unit normal vector in the y direction nz component of the unit normal vector in the z direction p pressure ˙qh time rate of heat transfer per unit mass T’ temperature t temporal coordinate u velocity component in the x direction v velocity component in the y direction v velocity vector with the components u, v and w w velocity component in the z direction x, y, z Cartesian axis τij components of shear stress tensor ε discretization error Ω shaft rotation speed Ωr rotor tip speed Subscriptsawn aerodynamic wave number i time index of pressure disturb disturbance n integer related to determining the specified frequency nd nodal diameter prop propagation re real rel relative frame stat stationary frame struct structure t tangential direction th theory α aerodynamic or acoustic disturbance x, y, z components in the x-, y-, z-direction 0 reference

1.1. Acoustic wave propagation

Several studies have shown that a propagating acoustic wave can interact with unsteady aerodynamics and structure.4-10In the circumferential direction, convective vortical structures generated by tip clearance flows can lock-on to the frequency of the reflected acoustic wave induced by a vibrating blade if the acoustic wavelength is equal to a multiple of the blade pitch. In this resonance condition, the amplitude of blade vibration is significantly amplified and discrete peaks associated with NSV are visible in the frequency spectra from the strain gauge signal. Jet Core Feedback Theory (JCFB)5-7is proposed to account for this phenomenon. Based on this, the speed of the acoustic wave is about 1.5 times the rotor speed in the stationary frame for a single rotor.

For the multistage compressors, a partially trapped acoustic mode can be in resonance with a structural mode.8In the measured spectrum, discrete non-synchronous peaks are visible in both the stationary and the relative frames. The propagation characteristics of acoustic duct modes are cut-on upstream and cut-off downstream in the axial direction. A lock-in of periodic vortex shedding in one row of blades is believed to provide energy for the acoustic mode. The propagation speed of the acoustic mode in the Ref.8is 1.27 times the rotor rotational speed in the stationary frame.

1.2. Circumferential vortex propagation

Rotating Instabilities(RIs)observed in several experiments on fans and compressors are considered to be the main cause of NSV.11-19RIs occur because flow instabilities with high mode orders rotate circumferentially in the tip region of the blade,and the pressure spectra exhibit non-synchronous peaks associated with their signatures. Observations have revealed that the flow instabilities are considered tip clearance vortices/tip tornado vortices20-21separated near the Suction Surface (SS)of the blade Leading Edge (LE). Because of the large area of blockage in the tip region, these flow instabilities impinge on the Pressure Surface (PS) of the blade in the circumferential direction and interact with the blade vibrations.14-15,19Several low-speed experimental studies indicate that the disturbances in the periodic vortices are weakly coupled with the blade vibrations and have broadband spectral characteristics in the stationary frame.12-15The circumferential propagation velocity of RIs in different machines is 0.2-0.9 times the rotor speed in the stationary frame.14This instability is more likely to occur in machines with a relatively large tip gap.14-15

In a high-speed compressor test with a regular tip gap, the disturbances in Radial Vortices(RVs)propagating around the circumference are identified as the major cause of NSV.22Circumferential RVs traveling causes a forward spill of the fluid around the LE and a large vortex breakdown in the blade passage. Experimental and numerical observations show that the flow structures of the RV do not seem to be fundamentally different from those of the RI.14-15,21-22Only the intensities and symptoms of the vortices vary during the transient processes.The propagation velocity of RVs is 0.42 to 0.45 times the rotor speed in the stationary frame, and is within the range of RIs.14,22The propagation of the RI or RV is represented as circumferential vortex propagation in this paper.

The frequency and phase of the disturbance in the circumferential vortex propagation can lock-on to structural modes of a blade to cause a high level of vibrations.16-19,22-25In resonant conditions, the aerodynamic disturbance dominated by a single mode and structural vibrations are accompanied by a rise in the discrete spike response in the relative frame. This process is considered a lock-in mechanism that leads to NSV,and the propagation speed of the disturbance of the circumferential vortex propagation can be characterized by the free convective speed and locked-in convective speed. The free convective speed is that of the circumferential vortex propagation,24whereas the locked-in propagation speed is the speed of a specified aerodynamic wave coupled with the blade vibrations.In general,the speed of the lock-in wave does not deviate much from free convection.25

The two phenomena described above share similarities in the frequency of pressure fluctuations and the unsteady flowfield leading to NSV.However,the speed of the acoustic wave is higher than the circumferential vortex propagation in the stationary frame, and can be used to distinguish between and identify the two phenomena. Although many studies have examined the mechanism of NSV, the coupling between the aerodynamic disturbance and the structural response, namely,aliasing, is still not well understood. This paper examines the aliasing process leading to NSV using a numerical method.A transonic fan model with flat tip-leading-edge of the NASA rotor 67 is tested near stall at 82 percent of design speed. The blade vibrations are analyzed and the occurrence of NSV is identified.The unsteady flowfield near the tip region is characterized by the pressure spectra, streamlines, entropy distributions and contours of pressure fluctuations, and circumferential vortex propagation is identified as the aerodynamic excitation source of NSV. The aliasing process, as the coupling between the aerodynamic disturbance and the structural response, is investigated, and the mechanism of NSV is detailed.

2. Numerical method

An in-house Computational Fluid Dynamic(CFD)code called HGAE (Hybrid Grid Aeroelasticity Environment) was used for all numerical simulations. A three-dimensional (3D)unstructured finite-volume compressible flow solver was applied to the fluid domain, and a structured dynamic solver,with the modal superimpositions method used for blade motion,was applied to the structure domain.The HGAE code has been developed and used more than 15 years,and has been validated for various aerodynamic and aeroelastic cases.26-27

For the aerodynamic models, the Unsteady Reynolds-Averaged Navier-Stokes(URANS)equations were solved with dynamic deforming grids to model the effects of blade motions:

where Ω is the control volume,?Ω is its boundary,and dS represents the surface area of an element. W is the vector of conservative variables,

where C is a constant corresponding to the gas property(120 K),and μ0is the viscosity coefficient at the reference temperature T0(291.15 K).

The governing equations were discretized with a nodecentered finite volume scheme. Roe’s upwind scheme with Monotone Upwind Scheme for Conservation Law (MUSCL)extrapolation was applied to compute the convective terms and central differences for the diffusion fluxes;28,29thus, the numerical scheme achieved second-order accuracy.Turbulence was simulated using the one-equation Spalart-Allmaras model used in previous computations.30To enhance the accuracy of the time-marching solution of the aeroelasticity problems,Jamerson’s dual time-steeping technique was adopted with 15 sub-iterations.31

For the structural models, the blade vibration was simulated using a linear aeroelasticity model. The structural dynamic equation for the vibrations is given as:

where M, C, K, and Q represent the mass matrix, damping matrix, stiffness matrix and aerodynamic force vectors on the blade surface, and x is the displacement vector. The structural dynamic equation, which is a set of linear differential equations, can be solved with a modal superimposition approach:

where q is the generalized coordinate vector and φ represents the mass-normalized mode shape matrix. We left-multiply the transposed matrix φHin Eq. (13) and substitute Eq. (14)into Eq. (13), as:

Mg,Cg,and Kgare all diagonal matrices;Eq. (15)can thus be decoupled into linearly independent ordinary differential equations. Qgis the generalized aerodynamic force vector.An integrated method was used to solve the processes of fluid-structure coupling. Details corresponding to the aerodynamic models, structural models, and the processes of fluidstructure coupling have been provided in Refs.26-27

3. Case study

3.1. Model description

The numerical simulations were based on a transonic fan model with a Flat tip-Leading-Edge (FLE) of NASA rotor 67.32Based on the flow characteristics of circumferential vortex propagation described in the literature, the shape of blade LE near tip region influenced the flow incidence, and might have had a prominent impact on the formation of tip vortices to induce blade vibration.Therefore,the LE near blade tip was cut from a round to a flat shape, as NSV was more likely to occur owing to the increased flow separation under a higher flow incidence.The cutting position started from the 80%span to the blade tip. The geometric view of this model combined with the original model is shown in Fig. 1(a), where the pink lines show the investigated fan model. For convenience, this model is called the FLE fan. Most design parameters between the FLE fan and the rotor 67 were in common, whereas the rotor tip chord length of the FLE fan was 81.83 mm,10.53 mm shorter than that of rotor 67.

3.2. Aerodynamic mesh

A computational structured mesh with an OH4 topology of the FLE fan was generated by the commercial program AUTOGRID. The width of the first wall cell was 10-5m and y+was about 5. There were 73 nodes in the spanwise direction and 17 nodes within the tip-gap region at a tip clearance of 1.016 mm. The single-passage mesh is shown in Fig. 1(b).At the subsonic inlet,the flow angle(0 deg),total pressure(101325 Pa), and total temperature (288.15 K) were specified,whereas the inlet Mach number(0.6 Mach)was entered to initialize the flowfield.At the exit,the static pressure was imposed by a simple radial equilibrium equation. At the shroud, the inlet and outlet of computational domains were located 1.5 and 1.6 axial tip chord length respectively away from the rotor blade. To analyze the discretization errors, including the grid dependence and achieved order of convergence, four meshes composed of 344491 nodes, 491121 nodes, 635005 nodes,and 821777 nodes per passage were examined. The error was estimated using the Richardson Extrapolation(RE)technique,and the solution of the mesh with 821777 nodes was considered the exact solution.33Other details corresponding to the choice of norms are illustrated in Refs.34,35Fig. 2 presents the norms of the density discretization error as the mesh was refined and the position of the monitoring points for density was at the shroud. The results show that the grid independence was reached and the manufactured solutions attained secondorder accuracy, identical to that of the nominal convergence of the numerical method. Considering the computational resources at hand and the reliability of the results, the mesh with 635005 nodes was used in the computations.

Fig. 1 Computational model.

Fig.2 L1 norm of discretization error in density ρ as function of mesh size h.

3.3. Single-passage unsteady calculations

The single-passage unsteady model could identify some disturbances of unsteady aerodynamics leading to NSV, and it was feasible to perform these low-overhead calculations to identify disturbances in all working conditions. Single-passage unsteady calculations were carried out at different rotating speeds. To verify the accuracy of the computational results of the HGAE, simulations of the fan were also performed using NUMECA with the same meshes. The chocking mass flow rate obtained by the HGAE was 34.27 kg/s at 100%speed, and was 34.18 kg/s using NUMECA. Fig. 3 shows the total pressure ratio against normalized mass flow at four rotating speeds. The total pressure ratio and mass flow near stall were obtained by averaging the last two rotor revolutions owing to the fluctuating solutions. The fan characteristics obtained by the HGAE showed good agreement with the results of NUMECA.

Fig. 3 Comparison of fan characteristics between HGAE and NUMECA simulations.

At the 82% rotating speed, the history of mass flow at the outlet for two revolutions at the near-stall point, labeled as‘‘A”in Fig.3,is shown in Fig.4.A Fast Fourier Transformation (FFT) of the periodically oscillating mass flow is presented in Fig. 5. The dominant frequencies associated with strong unsteady disturbances were not the harmonics of the rotor rational frequency,evidenced by the fact that NSV might have occurred in this condition.

3.4. Setup for full-annulus FSI computation

The full-annulus FSI computation,with back pressure of point‘‘A”,began from a full-annulus steady solution.The structural eigen-solutions of the fan blades were obtained from the commercial FEA package ANSYS. It was assumed that the constraint on the fan blade was like that on a cantilevered beam without mechanical damping and centrifugal force was included to consider its effects on blades stiffness.The first five blade modes were considered, and their mode shapes are shown in Fig. 6. The natural frequencies of the modes were 485 Hz (Mode 1), 1165 Hz (Mode 2), 1819 Hz (Mode 3),2459 Hz (Mode 4), and 3024 Hz (Mode 5) respectively. The global time step was 1/50 of the vibration period of the Mode 5(2nd Torsion Mode).The FSI computations were performed by considering all NDs.

4. Results of FSI computation

4.1. Analysis of blade vibration

Histories of blade vibrations corresponding to all Nodal Diameters (NDs) of the five modes were investigated, and the results show that a backward travelling 7th ND assembly mode arising from the Blade Mode5 (BM5) was the most unstable.The mode was labelled as BM5/-7ND or 2nd Torsion Mode (2T)/-7ND for brevity. The -7ND modal displacements of the five modes are shown in Fig.7.The amplitude of the 2T mode grew exponentially and the FFT of the 2T/-7ND modal displacement presents a single peak in the frequency spectrum in Fig.8.This vibration pattern indicates that the aerodynamic mode was in resonance with the structural vibration mode.The location of aeroelastic instability is marked by a red circle in the Campbell diagram in Fig.6,which is between 13 Engine Order(EO)and 14 EO.This indicates that the blade vibrations were not synchronous with rotating speed. The reduced frequency of blade vibrations was 4.4, which was relatively high and beyond the range of the classical flutter. This shows that the aeroelastic phenomenon can be identified as NSV as described in the literatures.

Fig.4 History of outlet mass flow for last two rotor revolutions(82% rotating speed).

Fig. 5 FFT results of outlet mass flow for last two rotor revolutions (82% rotating speed).

Fig. 6 Campbell diagram of fan blade.

Fig. 7 Time histories of modal displacements for first five eigenmodes.

Fig. 8 FFT results of 2 T/-7ND modal displacement.

4.2. Characterization and identification of flow phenomenon underlying NSV

The frequency maps of blade 1, obtained by the FFT of static pressure from numerical probes on the pressure surface, are plotted in Fig.9.The amplitude of static pressure was normalized by reference pressure at inflow. Those maps show the NSV frequency (3024 Hz) and its second harmonics(5958 Hz) at all probes. The maximum excitation occurred near the leading edge at 99% span, and the excitation amplitude decayed with distance down the chord and span.Fig. 10 shows the frequency spectra of 22 blades from probes located at the 10% axial chord of the PS near the tip region.The NSV disturbances were captured at all 22 blades. The alternating distributed high peak of amplitudes among all blades are presented,and show that the flow structures behind NSV were dominated by multi-cells that had multi-passages periodicity in the circumferential direction.

Three-dimensional streamlines in the relative frame are shown near the tip region in Fig. 11(a), where B-1 stands for blade number one,and so on.The tip clearance vortex marked in pink oval dotted lines was located near the LE of the blade suction surface,whereas the shedding vortex labeled in the red dotted circular lines is presented near the LE of the blade pressure surface. Different from the traditional tip leakage vortex,the tip clearance vortex developed in the circumferential direction, broke down near the middle of the blade passage, and formed the zone of axial reverse flow. The streamlines in the red circle presented in Fig.11(b)show that the shedding vortex had a radial structure,and its footprint corresponded to a local drop in pressure.15,22,36The pressure drop Δp was defined as the negative deviations from the mean pressure of the last two rotor revolutions, and only when it exceeded a certain value, the shedding vortex can be identified.22The one-fifth of dynamic pressure Pdyn，inat the inlet was used as a threshold for Δp ＜-Pdyn，in/5,and the locations of the shedding vortices are shown in Fig. 11(c). Fig. 11(d) shows the detailed vortex structure near the tip region, the shedding vortex was originated from a broken tip clearance vortex and different from the commonly referred shedding vortex at blunt trailing edge.

Fig. 9 Frequency maps for pressure-side probes of blade 1.

Fig. 10 Frequency spectra of 22 blades.

Due to the interaction among the main flow and the tip leakage flow, the shedding vortices generated near the middle of the blade passages at the LE locations impinge on the PS of the adjacent blades. The tip blockage with negative axial velocity prompted the shedding vortices to pass through the blade passages against the direction of rotation. Similar flow phenomena have been reported in Refs.15,16. After reaching the PS of the adjacent blade,the shedding vortices propagated circumferentially to the adjacent passage and moved partly axially.The related temporal evolutions at seven instantaneous times are shown in Fig. 12. The entropy distribution of the blades shows that the periodic disturbance of the propagation of the circumferential vortex was 5/3 T (blade passing time),which was consistent with the NSV frequency of 3024 Hz. It required a circumferential vortex of 2T to pass through one blade passage, which indicates that the propagation velocity was nearly 0.5 times the rotor speed against the direction of rotation.

To identify the unsteadiness of vortex propagation,the contours of the Standard Deviation (STD) σ and FFT of static pressure at 99% span are shown in Fig. 13. Numerical data on two rotor revolutions (5-7) were taken as sampling points for STD and FFT analyses. The equations of the two analytical methods are given as:

where A0is the time-averaged static pressure,fnis the specified frequency determined by the length of the FFT time series t-t0and integer n. Anand φnare the amplitude and phase of the pressure fluctuations, respectively, at a specified frequency.

The locations of the shock wave and tip clearance vortex are presented in the contours of the mean pressure, labeled as ‘‘A” and ‘‘B” in Fig. 13, respectively. The oscillations of the shock wave and tip clearance vortex were reflected in the STD results, but the spectral analysis associated with the frequency of NSV indicates that the shock wave did not oscillate synchronously. Furthermore, the highest fluctuation and amplitude mainly occurred at the LE of the PS, marked as‘‘C”. Therefore, the circumferential propagating vortex was not the tip clearance vortex, but was a shedding vortex, and could be identified as the aerodynamic excitation source of NSV.

Fig. 11 Sketches of the circumferential propagating vortex in the relative frame.

Fig. 12 Entropy contours at 99% span for seven instantaneous times.

Fig. 13 Static pressure at 99% span.

4.3. Identification of interaction between aerodynamics and structure

The Aerodynamic Wave Number(AWN)calculated by the circumferential DFT of fluctuating pressure at the shroud is plotted in Fig. 14. The fluctuating pressure was that the unsteady pressure minus the time-averaged pressure of the last two rotor revolutions, thus the AWN22 associated with the disturbance of the 22 blades was removed. This spatial flowfield was obtained from the inlet to the outlet of the rotor blade, and the axial locations 0% and 100% represent the position of the LE and TE of the blade, respectively. AWN29, located from axial locations -50% to 140%, was the most dominant,whereas AWN7 and AWN15, located across nearly the entire passage, played a minor role. The aerodynamic wave number in Fig. 14 is the result of the last step of the rotor revolution,while the time-averaged AWN of the two rotor revolutions(5-7)illustrated in Fig.15 had the same distribution as an instantaneous result.

Fig.14 Amplitudes of spatial aerodynamic modes with different wave numbers at the shroud.

Fig. 15 Time-averaged amplitudes of spatial aerodynamic modes with different wave numbers at the shroud.

To further examine the interaction between the aerodynamic wave number and the structural nodal diameter,the frequency spectra of the stationary frame calculated by the FFT of the interpolated unsteady pressure at the shroud near the tip of the blade LE, marked in black line in Fig. 14, and the relative frame obtained by the FFT of the monitoring parameters from the blade probes at 99% span are shown in Fig. 16. The sampling points are the time steps of the two rotational cycles(5-7),and the frequencies normalized with rotor rotational frequency are expressed as EO.

The dominant frequency peak was EO15.27,and the minor peaks were EO6.73 and EO28.73 in the stationary frame(shown in Fig. 16(a)), whereas the dominant frequency peak was EO13.73 in the relative frame (shown in Fig. 16(b)). The frequency spectrum labeled with blue lines was calculated by the FFT of unsteady pressure; thus, EO13.73 marked in blue lines corresponds to the dominant excitation frequency of the circumferential vortex propagation in the relative frame.EO13.73 labeled in pink lines is the result of the FFT of general physical displacement, which matched the frequency of structural response with ND-7, the amplitude of unsteady pressure and displacement in Fig. 16(b) were normalized by their maximum amplitude respectively. According to the frequency spectra in the two frames, AWN was obtained as:

Fig. 16 Frequency spectra of unsteady pressure at the blade LE near the tip region.

The correlation between EO and AWN in the frequency spectra is illustrated in Fig. 16. The highest peak at EO15.27 in the stationary frame corresponded to disturbances of the circumferential vortex propagation, which had a dominant mode order of AWN-29. This is because the high amplitude of the pressure wave with AWN-29 was mainly concentrated near the blade LE in Fig. 14, and was the same as the fluctuating position of the circumferential vortex propagation in Fig. 13. According to the analyses of blade response in Fig. 7, the pressure wave with ND-7/AWN-7 was generated by the structural vibration pattern, and the mode was consistent with the frequency shift between EO6.73 and EO13.73.Ultimately,the pressure wave with AWN 15 had the frequency EO28.73 in the stationary frame,and was induced by the modulation between AWN-29 and the blade number(NB)of rotor 22 (22 × 2-29 = 15, 22 × 2-15.27 = 28.73). From the above analyses,it is clear that the aerodynamic wave EO15.27/AWN-29 was resonant with the structural wave EO13.73/ND-7.

The flow condition encountering NSV triggered several pressure waves, and it was necessary to distinguish among them in a simple way. According to the analysis in the Introduction, Section 1, the measured speed of the acoustic wave was higher than that of the disturbance in circumferential vortex propagation. Therefore, the speeds of the pressure waves were calculated by the cross-correlation between probes located at different circumferential positions; the process for obtaining the speed can be found in the Ref.22. Moreover,the speeds were also calculated by using Eq. (21), according to the disturbance frequency in the stationary frame and the corresponding aerodynamic wave number.

The purpose of the comparison was to verify that the source frequency used in the equation for calculating the wave speed in the Ref.11was not necessary.The propagation speeds are summarized in Table 1, where those with respect to the cross-correlation calculations, located in the fourth column of the table, were similar to the results of the formula calculations, in the third column. This proves the feasibility of using Eq. (21) to calculate wave speed. The speed of aerodynamic disturbance with AWN-29 was nearly 0.52Ωrin the stationary frame. As aerodynamic disturbance was resonant with the structural wave with ND-7, the pressure wave with AWN-29 was a locked-in wave type.According to the speed of the pressure waves with AWN 15 and AWN-7,these waves with higher speeds had acoustic signatures.

4.4. Mechanism of NSV

Pictures of the processes leading to NSV are presented in Fig. 17. The speed of propagation of the shedding vortex was -0.5Ωrin the relative frame (0.5Ωrin the stationary frame), as shown in Fig. 17(a), within the range of the speed of circumferential vortex propagation (0.2Ωr-0.9Ωr), and can be called the free convective speed determined by the mean flowfield.24The shedding vortex propagation excited an aerodynamic pressure wave with a mode number of 29, presented in Fig. 17(b), that was in resonance with the structural wave with ND-7 shown in Fig. 17(c). The AWN-29 mode had a locked-in convective speed at -0.48Ωr, and the coupling was triggered by the aliasing process shown in Fig. 17(d).

In general,the direction of circumferential vortex propagation is opposite to that of the rotating speed,because of which the aerodynamic pressure wave corresponding to the disturbance of vortex propagation travels backward.This backward traveling aerodynamic wave induced the blade vibrations if the dominant excitation frequency of the wave was near one of the natural frequencies of the blade.A backward traveling aerodynamic wave can alias to a backward traveling structural wave in the relative frame,and the result is classified as a co-aliasing process. However, the conclusions of Refs.18,19,22demonstrate that the backward travelling aerodynamic wave could alias to the forward traveling structural wave, which was determined to be an anti-aliasing process. To investigate the differences between the processes, the Inter-Blade Phase Angles (IBPA)φ of the aerodynamic wave and the structural wave were determined by.

Table 1 Distinction among pressure waves.

Fig. 17 Graphical representation of the interaction between propagating aerodynamic wave and structural vibration.

The positive sign represents a forward traveling wave in the same direction as the rotation of rotor, whereas the negative sign presents a backward traveling wave. The difference between the anti-aliasing and co-aliasing processes can be expressed as follows:

1) Anti-aliasing process: As has been reported in several studies,18-19,22a backward traveling aerodynamic wave with AWN-13 is in resonance with a forward traveling structural wave with ND8. The sum of the two phases spans an entire circle, 2π,and thus the sum of the mode numbers of the two waves is equal to the number of blades:

2) Co-aliasing process: In this study, a backward traveling aerodynamic wave with a mode number of 29, which was higher than the number of rotor blades, was locked to a backward traveling structural wave with ND-7.The phase difference between the waves was 2π, because of which the difference in mode numbers between them was equal to the number of blades:

This paper examined the co-aliasing process for the mechanism of NSV,which has rarely been reported in literature.The factors that determine the direction of aliasing are worth discussing, thus. According to the correlation between AWN and ND when NSV occurred, as recorded in the literature and this study, a zigzag diagram is shown to represent it in Fig. 18. The positive slopes represent the co-aliasing process and the negative slopes the anti-aliasing process.It is clear that the AWN affected the directions of aliasing;however, it could not be directly obtained very easily in experiments and numerical simulations, nor could it adequately reflect the characteristics of the aerodynamic disturbance. According to Eq. (21),the propagation speed and frequency of the aerodynamic disturbance determine the AWN. Hence, it was necessary to investigate these two parameters.From a‘‘stair case”plot performed by a semi-analytical model in the Ref.24the antialiasing processes were presented at around a free speed of 0.5Ωr,while the cases of co-aliasing appeared at higher speeds.Although the range of speed for the co-aliasing considered in the literature is different from that in this study, the free convective speed was considered to affect the direction of aliasing.Moreover, for a known free convective speed, the time with respect to the circumferential vortex propagating through a blade passage can be obtained, and is represented as Tprop.The perturbation period of circumferential vortices with regard to impinging on the blade LE is the inverse of the frequency of aerodynamic disturbance, given as Tdisturb. As shown in Fig. 12, the correlation between Tpropand Tdisturbdetermined the cell number in a blade passage. For the cases considered in literatures,18-19,22the convective disturbance propagated more quickly than the perturbation period experienced by the blade,resulting in no more one cell in each blade passage for Tprop＞Tdisturb. In this study, the convective disturbance propagated more slowly,and thus more than one cell in a blade passage occurred for Tprop＜Tdisturb.Therefore,the correlation between the frequency and the free convective speed of the aerodynamic disturbance is the key to determining the direction of aliasing.These factors are sensitive to the operating point and Variable Inlet Guide Vane(VIGV)setting.For the single-row model considered in this study, changing the shape of the blade LE may be equivalent to altering the VIGV setting in the multi-stage model.

Fig. 18 Zigzag diagram for an even number of blades.

5. Conclusions

This study used numerical methods to investigate the NSV of a FLE fan blade.Single-passage unsteady calculations were carried out to identify disturbances in NSV,and full-annulus FSI computation was used to investigate unsteady flows and their interaction with the blade vibrations.

(1) The 2T/-7ND modal displacement showed that the blade vibrations were non-synchronous and unstable.The vibration pattern indicated that the aerodynamic mode was resonant with the structural vibration mode.The maximum aerodynamic excitation of NSV had a discrete frequency peak of 3024 Hz located near the leading edge of the blade tip, and frequency spectra of all blades show that NSV was dominated by multiple cells, with multi-passage periodicity in the circumferential direction. The tip flowfield showed that the source of aerodynamic excitation was the circumferential vortex propagation induced by the main flow, tip leakage flow, and tip clearance vortex. The circumferential vortex propagation excited an aerodynamic backward traveling wave in the relative frame, and might have been resonant with the structural backward traveling wave. The pressure waves associated with the modulation between aerodynamic disturbance and the rotor blades were also identified. Pressure waves with aerodynamic and acoustic signatures could be distinguished by the propagation speed.

(2) In most studies in the literature, the coupling between aerodynamic disturbance and structural vibrations has been interpreted as anti-aliasing. This study detailed a co-aliasing process for the mechanism of NSV. The difference between the processes is reflected in the phase relationship between the aerodynamic wave and the structural wave. The correlation between the free convective speed and the frequency of aerodynamic disturbance is believed to determine the direction of aliasing.To illustrate this, the characteristics of NSV at different operating conditions will be studied in future work.

Declaration of Competing Interest

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.

Acknowledgements

This study was co-supported by the National Science and Technology Major Project, China (No. 2017-II-0009-0023).