Anne-Lise Fiquet
email: anne-lise.fiquet@doctorant.ec-lyon.fr

St Éphane Aubert
email: stephane.aubert@ec-lyon.fr

Christoph Brandstetter
email: christoph.brandstetter@ec-lyon.fr

Nicolas Buffaz
email: nicolas.buffaz@safrangroup.comagathe.vercoutter@safrangroup.com

Agathe Vercoutter

ACOUSTIC RESONANCE IN AN AXIAL MULTISTAGE COMPRESSOR LEADING TO NON-SYNCHRONOUS BLADE VIBRATION

Significant non-synchronous blade vibrations (NSV) have been observed in an experimental three-stage high-speed compressor at part-speed conditions. High amplitude acoustic modes, propagating around the circumference and originating in the highly loaded Stage-3 have been observed in coherence with the structural vibration mode.

In order to understand the occurring phenomena, a detailed numerical study has been carried out to reproduce the mechanism. Unsteady full annulus RANS simulations of the whole setup have been performed using the solver elsA. The results revealed the development of propagating acoustic modes which are partially trapped in the annulus and are in resonance with an aerodynamic disturbance in Rotor-3. The aerodynamic disturbance is identified as an unsteady separation of the blade boundary layer in Rotor-3. The results indicate that the frequency and phase of the separation adapt to match those of the acoustic wave, and are therefore governed by acoustic propagation conditions. Furthermore, the simulations clearly show the modulation of the propagating wave with the rotor blades, leading to a change of circumferential wave numbers while passing the blade row. To analyze if the effect is self-induced by the blade vibration, a noncoherent structural mode has been imposed in the simulations. Even at high vibration amplitude the formerly observed acoustic mode did not change its circumferential wave number. This phenomenon is highly relevant to modern compressor designs, since the appearance of the axially propagating acoustic waves can excite blade vibrations if they coincide with a structural eigenmode, as observed in the presented experiments.

INTRODUCTION

The operating range of modern designs in turbomachinery depends intrinsically on unsteady aerodynamics and their interaction with structural vibrations. During the past decades, research on the interdependency between aerodynamics, mechanics and acoustics has been carried out. The coupling of multiphysical effects is difficult to predict even with advanced simulation approaches. It is necessary to conduct complex experiments with synchronized aerodynamic and blade vibration instrumentation to resolve the transient phenomena and interaction mechanisms.

Test campaigns have been carried out on a research multistage axial compressor by Safran Helicopter Engines. During transient experiments, significant oscillations of the wall pressure and vibration of the rotor blades were measured at different operating points at a part-speedline. The frequency of these oscillations did not represent a multiple of the shaft rotation frequency and are hence denoted as non-synchronous.

Studies on non-synchronous vibrations (NSV) in low-speed and high-speed axial compressors have been reported extensively in literature [START_REF] Baumgartner | Non-engine order blade vibration in a high pressure compressor[END_REF][START_REF] Kielb | Blade Excitation by Aerodynamic Instabilities: A Compressor Blade Study[END_REF][START_REF] Parker | Acoustic resonances and blade vibration in axial flow compressors[END_REF][START_REF] Brandstetter | Acoustic and Convective Mechanisms Contributing to Non-Synchronous-Vibrations in a Multistage Compressor[END_REF]. For the case analyzed in the present study, available explanations do not sufficiently describe the observations. To explain how the presented case differs from reported cases of NSV, the main source of NSV are briefly reviewed in this section. Generally, a distinction has to be made between self-excited mechanisms (Flutter) and mechanisms which rely on an aerodynamic disturbance which excites specific modes (NSV). The effects observed in the compressor under investigation were recorded with and without blade vibration, such that a self-excited mechanism can be excluded [START_REF] Fiquet | Non-Synchronous Aeroacoustic Interaction in an Axial Multi-Stage Compressor[END_REF].

Convective phenomena / NSV In modern high-speed compressors, Non-Synchronous Vi-1 FIQUET-TURBO-20-1331 bration (NSV) is typically driven by small scale aerodynamic disturbances, which are convected around the circumference and may adapt their phase to blade oscillation. The most important characteristic of this phenomenon is the convective circumferential phase velocity which depends solely on the steady flow field. In all known publications, this speed is found to be between approximately 30% and 80% of the rotor speed, hence traveling backwards in the relative frame of reference [START_REF] Cumpsty | Discussion: "Rotating Instabilities in an Axial Compressor Originating From the Fluctuating Blade Tip Vortex[END_REF][START_REF] Mailach | Rotating Instabilities in an Axial Compressor Originating From the Fluctuating Blade Tip Vortex[END_REF][START_REF] Möller | Mechanism of Nonsynchronous Blade Vibration in a Transonic Compressor Rig[END_REF]. Brandstetter et al [START_REF] Brandstetter | Measurements of Radial Vortices, Spill Forward, and Vortex Breakdown in a Transonic Compressor[END_REF] have shown in experiments that this circumferential phase velocity is already fixed before the onset of blade vibration and is only slightly altered once NSV establishes. The disturbances were identified as small radial vortices which propagate upstream of the blockage zone created by the tip leakage flow close to the casing. Through random excitation of the blades by passing radial vortices, a dominant structural vibration pattern (Nodal Diameter) can establish. It may enforce a phase lock-in of the initially randomly distributed aerodynamic disturbances.

The resulting Nodal Diameter can hence be predicted for known blade eigenmodes and specific convective circumferential phase velocities. It is possible that backward traveling aerodynamic disturbances with high circumferential wave number alias on forward traveling structural modes [START_REF] Brandstetter | Measurements of Radial Vortices, Spill Forward, and Vortex Breakdown in a Transonic Compressor[END_REF]. For the case observed in this study, the circumferential phase velocity of the disturbance was found to exceed the rotor speed which is physically not possible for a convective mechanism but only for acoustic modes.

(Aero) Acoustic Resonance / Duct Modes Acoustic resonance is a non-synchronous effect that has been observed in several types of compressors. Typically, helically propagating (spinning) modes of zeroth radial order originate near one compressor stage and propagate only in one axial direction. The change of acoustic propagation conditions between upstream and downstream of a blade row enables the establishment of a series of resonant modes with specific circumferential order and circumferential phase velocity. Unstable aerodynamic flow structures (boundary layer separations / periodic vortex shedding) in a blade row can phase-lock to the acoustic mode and hence serve as periodic energy source. The phenomenon does neither require a circumferential convection nor the presence of blade vibrations to establish but can cause damaging effects if an occurring frequency coincides with a structural mode. Observations in multistage compressors have been reported at near stall conditions [START_REF] Parker | Acoustic resonances and blade vibration in axial flow compressors[END_REF][START_REF] Brandstetter | Acoustic and Convective Mechanisms Contributing to Non-Synchronous-Vibrations in a Multistage Compressor[END_REF][START_REF] Kameier | Rotating blade flow instability as a source of noise in axial turbomachines[END_REF][START_REF] Camp | A Study of Acoustic Resonance in a Low-Speed Multistage Compressor[END_REF][START_REF] Hellmich | Causes of Acoustic Resonance in a High-Speed Axial Compressor[END_REF] or at part-load conditions with mismatched compressor stages. In reference [START_REF] Holzinger | Self-Excited Blade Vibration Experimentally Investigated in Transonic Compressors: Acoustic Resonance[END_REF] an experimental observation for a single stage compressor is reported. A comprehensive study on the development of acoustic modes in the flow channel of turbomachines has been presented by Tyler and Sofrin [START_REF] Tyler | Axial flow compressor noise studies[END_REF] and by Cooper and Peake [START_REF] Cooper | Trapped acoustic modes in aeroengine intakes with swirling flow[END_REF]. In a duct without flow, the condition for a mode to be propagative (cut-on) is that its circumferential propagating speed exceeds the speed of sound, otherwise it is cut-off and will decay exponentially in axial direction [START_REF] Tyler | Axial flow compressor noise studies[END_REF]. The circumferential phase velocity rises linearly with the frequency if the circumferential wave number (N a ) is fixed. It decreases linearly with N a , if the frequency is fixed. For narrow ducts with homogeneous flow, the cut-on frequency can be evaluated by calculating the eigenvalues of the homoge- neous wave equation [START_REF] Camp | A Study of Acoustic Resonance in a Low-Speed Multistage Compressor[END_REF].

As will be shown in this paper, modes observed in the experiment and the numerical simulations are cut-on only in one axial direction (upstream or downstream). For one specific experiment presented here, acoustic modes coincide with a structural vibration pattern and lead to significant structure vibrations.

In a previous study [START_REF] Fiquet | Non-Synchronous Aeroacoustic Interaction in an Axial Multi-Stage Compressor[END_REF] on the presented case, the experimental results have been analyzed in detail. However it was not possible to understand the source of the acoustic mode. In the present study, full annulus unsteady simulations of the whole compressor which show the development of acoustic duct modes have been carried out. Based on the results, an interpretation of the excitation mechanism is derived.

The paper is structured as follows:

1. A brief description of the test setup, the experimental methods and the main results are presented. 2. The numerical setup for full-annulus URANS simulation is explained and validation based on performance data is shown. 3. For an operating point close to peak efficiency (OP1), where first traces of non-synchronous activity have been observed in the experiment, a detailed analysis of numerical simulations concerning unsteady flow structures and waveforms is presented. 4. A further throttled operating point (OP2) is analyzed, which is in the zone where intense pressure fluctuations and blade vibration occurred in the experiment. 5. Based on a comparison of both simulations with the experiment the interaction mechanism is discussed.

TEST CASE AND SETUP Experimental setup, methods and results

The research compressor rig designed by Safran Helicopter Engines is composed of struts, variable inlet guide vanes (IGVs) and three axial compressor stages, as illustrated in Fig. 1. The stagger angle of IGVs and Stators-1 are varied according to the shaft speed. The presented experimental data are measured at constant part-speedline (87% of design speed). The order of the blade aspect ratio for all rotors lies between 0.8 and 1.0, the hub to tip ratio evolves between 0.6 and 0.8. The tip clearance of all rotors is lower than 1% of chord. The stagger angles of the two rows (IGV and Stator-1) are held at a fixed position during the measurement.

2

FIQUET-TURBO-20-1331

Unsteady static pressure measurements are carried out using case-mounted and stator-mounted Kulite transducers which are depicted in Fig. 1. Blade vibrations of the three rotors are also measured, using two different technologies that both measure vibrations in the relative frame of reference:

1. Magnet-coil system, consisting of a small magnet integrated in the tip of one rotor blade whose axial movement is measured via a circumferential coil. Details on this measurement system are given by J. Garnier et al. [START_REF] Garnier | Magnet-Coil measurement technology for analyzing turbomachine blade vibrations[END_REF]. 2. Tip-timing set up, which consists of several optical sensors mounted in the casing that monitor tangential blade tip vibrations. As this technology can resolve the movement of all blades simultaneously, phase relation between blades can be estimated to derive the nodal diameter pattern. Details on the used method are given by A. Bouchain et al. [START_REF] Bouchain | New Possibilities for Analyzing Complex Asynchronous Blade Vibrations From Tip-Timing Data Using a Sparse Spectral Analysis Method[END_REF].

Since only non-synchronous frequencies are relevant for the results presented in this paper, rotor periodic (synchronous) frequencies are removed from experimental data by subtracting a sliding ensemble average of 5 rotations (see also [START_REF] Brandstetter | Compressible modal instability onset in an aerodynamically mistuned transonic fan[END_REF]). The same procedure has been applied to the numerical results.

In every spectrogram, frequencies are normalized by rotational speed frequency, yielding engine order (EO).

Using results measured in both the stationary and rotating frame of reference, it is possible to derive the circumferential organization and the circumferential phase velocity of pressure waves [START_REF] Baumgartner | Non-engine order blade vibration in a high pressure compressor[END_REF]. Knowing the circumferential propagation speed c prop,stat measured in the stationary frame of reference and the engine order EO stat of the pressure fluctuation in the stationary frame of reference it is possible to compute the circumferential wave number |N a | using Eq. 1. The term Ωr corresponds to the blade velocity at radius r.

|N a | = |EO stat Ωr c prop,stat | (1)
In agreement with Eq.1 the circumferential wave number N a can be derived using the engine order EO rel and the circumferential phase velocity measured in the rotor relative frame of reference c prop,rel :

N a = EO rel Ωr c prop,rel (2)
The sign of N a is the same as the sign of the circumferential phase velicoty of the considered wave in the rotor relative frame of reference c prop,rel . Positive values of N a represent co-rotating waves in the relative frame of reference. The relation between frequencies in the stationary and relative frame of reference is given by Eq. 3.

EO stat = EO rel + N a (3)
A structural vibration mode based on a blade eigenfrequency EO blade,rel that can be in resonance with the aerodynamic or acoustic mode is described in the presented work using the term nodal diameter (ND) instead of circumferential wave number (N a ). In contrast to aerodynamic and acoustic modes, the nodal diameter of a structural vibration mode cannot be greater than half the number of blades N b . If the structure is in resonance A detailed description of the non-synchronous aeroacoustic phenomenon observed during repeated test campaigns is given in [START_REF] Fiquet | Non-Synchronous Aeroacoustic Interaction in an Axial Multi-Stage Compressor[END_REF]. The relevant compressor characteristic for the whole compressor rig is presented in Fig. 2. The locations of the unsteady full-annulus simulations OP1 and OP2 are also shown in this figure, a detailed description will be presented later. At the investigated speedline, non-synchronous modes are observed far from the stability limit of the whole compressor during a transient test where the throttle is slowly closed from the operating point OP2 towards the stability limit. Fig. 2 shows the operating conditions during the considered test, where significant nonsynchronous pressure oscillations with a clearly distinguishable circumferential mode order were observed (pointed out by "NSV Zones" in the figure). However, first traces of non-synchronous activity have been already observed between OP1 and OP2. Two dominant acoustic modes were observed during the considered test.

First, starting at a normalized mass-flow rate of 0.85, corotating acoustic modes establish with a circumferential order of N a = +5 and their harmonics (grey operating points in Fig. 2). According to tip-timing data and magnet-coil measurements, Rotor-2 simultaneously develops a coherent vibration mode (ND) based on its first blade eigenmode.

Second, throttling further towards stall, the acoustic mode changes its circumferential order to N a = +4 whereupon the blade vibration amplitude diminishes (light grey operating points in Fig. 2). The location of this mode switch is visible in Fig. 2 timing data derives a dominant nodal diameter of ND = +10, as depicted in Fig. 4. The wall pressure measurements show a dominant non-synchronous frequency of EO stat = 6.36 and its 2 nd harmonic at EO stat = 12.72, which exactly coincides with the frequency of the blade vibration at ND = +10 (in the stationary frame of reference). Thus, the circumferential order of the fundamental mode at EO stat = 6.36 is determined as N a = +5. In the rotor relative frame, an off-resonant excitation is measured at EO rel = 1.34, which coincides with the fundamental mode in the stationary frame of reference using Eq. 3.

Tab. 1 lists the characteristics of the observed modes during the test. Both pressure wave and structural nodal diameter are coherently traveling forward (spinning in the same direction as the shaft) with a speed at the casing higher than the rotor speed. These results prove that the modes are of acoustic nature and a convective phenomenon can be excluded.

Besides, the results presented in [START_REF] Fiquet | Non-Synchronous Aeroacoustic Interaction in an Axial Multi-Stage Compressor[END_REF] indicated a partial trapping of the acoustic modes with maximum amplitude between Rotor-2 and Rotor-3. Those results are comparable to the case of acoustic resonance described in [START_REF] Brandstetter | Acoustic and Convective Mechanisms Contributing to Non-Synchronous-Vibrations in a Multistage Compressor[END_REF]. This point will later be

Numerical setup

A numerical investigation is performed to reproduce the acoustic resonance observed in the presented transient test. The whole compressor has to be considered without axial mixing planes in numerical simulations in order to propagate modes between rows. Furthermore, full-annulus simulations are required since a partial sector would artificially set the circumferential wave number of traveling modes. The flow solver used is elsA, an unsteady Reynolds-Average Navier-Stokes solver developed by ONERA [START_REF] Cambier | The Onera elsA CFD software: input from research and feedback from industry[END_REF]. The Roe scheme is used with second-order accuracy in space, as well as the k-l Smith model to calculate turbulence with wall functions applied to all surfaces (y + ≈ 30).

Unsteady full-annulus simulations are performed with the computational domain exposed in Fig. 5 and they are initiated with steady single passage simulations with mixing planes. The domain is meshed using Numeca AutoGrid5 and mesh visualization for Rotor-2 is represented in Fig. 5. The full compressor without struts is modeled with a total number of cells of 133x10 6 . In Rotor-2 and Rotor-3 approximately 1.1x10 6

SIMULATION RESULTS

Unsteady simulations have been performed at two operating points, depicted in Fig. 2 as OP1 and OP2. The operating points OP1 and OP2 are located, respectively near peak efficiency and at a mass-flow where the intense non-synchronous activity was observed in the experiment. OP2 is the last point that reached periodic convergence in the simulations. The performance of the whole compressor is well predicted by the simulations within the experimental measurement uncertainty but shows a tendency towards a reduced total-pressure rise.

In the experiment, strong non-synchronous pressure oscillations and blade vibrations have been observed for the OP2 condition as described previously. Fig. 6 presents the contours of the time-average relative Mach number in the whole compressor for the unsteady results at OP1 and OP2. Both simulations were initialized by steady results. A detailed discussion on the steady flow structure at OP1 can be found in [START_REF] Fiquet | Non-Synchronous Aeroacoustic Interaction in an Axial Multi-Stage Compressor[END_REF]. It can be seen that for both operating points the relative velocity in Rotor-1 is very low due to the IGV setting. Rotor-2 runs transonic and is moderately loaded at OP1. At OP2 the loading of Rotor-2 increases significantly. Rotor-3 is highly loaded at both operating points, with detached shocks and significant traces of flow separation, which will be discussed later.

In the following section unsteady simulation results are analyzed regarding the evolution of non-synchronous phenomena. In order to characterize the modes, the non-synchronous flow field at 99.5% of channel height is decomposed as follows: First, by applying a spatial filter of the circumferential signal for each individual time step depending on the modulus of the circumferential wave number |N a |, and secondly by calculating the temporal Fourier transform of the spatially filtered data over 8 rotations. Applying the detailed decomposition in a rotor, the frequency in the relative frame of reference EO rel is derived for a fixed circumferential wave number.

Based on these results the circumferential phase velocity in the relative frame of reference c prop,rel is derived according to Eq. 2, the frequency in the stationary frame of reference EO stat with Eq. 3, and c prop,stat in the stationary frame of reference with Eq. 1.

To analyze the physical nature of respective modes (convective or acoustic), the circumferential phase velocity relative to the mean flow velocity is considered and compared to the local speed of sound. In order to distinguish between convected and acoustic modes, the entropy signal as well as the unsteady static pressure signal are analyzed [START_REF] Chu | Non-linear interactions in a viscous heat-conducting compressible gas[END_REF].

Operating Point OP1 -near peak efficiency After less than 10 rotations, the simulation develops non-synchronous pressure

5

FIQUET-TURBO-20-1331 In order to further analyze the origin of these nonsynchronous modes, a decomposition of the flow field in Rotor-3 is presented in the following.

During one cycle of the mode with N a = -3, a strong fluctuation of the flow field in Rotor-3 is observed in the simulations. Fig. 9 presents the relative Mach number at two channel heights for different time instants during one cycle. Clearly, a significant flow separation is observed on the blade suction side with a strongly alternating form. At 50% of channel height, the extension of the separation is larger than close to the tip. Furthermore, the shape of the detached shock at 95% of channel height is fluctuating.

The three-dimensional form of this separation is depicted in Fig. 10 To identify the main interaction zone in the rotor for the nonsynchronous mode, local probes are placed at different locations at 80% of channel height as shown in Fig. 11, with probes 1 to 3 located upstream, 4 to 6 near the trailing edge on the pressure side and 7 to 9 on the suction side. The frequency spectra of the axial velocity signal clearly reveal a maximum amplitude at EO rel = 5.3 for probe 7, which is located well inside the separation zone. Furthermore, the first 5 harmonics of this mode are prominent further downstream on the suction side. All other probes do not show this extensive non-synchronous activity. Spectra of non-synchronous entropy fluctuations -which are omitted for brevity -show the same signatures.

With these results, it is clear that a strong interaction between the acoustic mode and the flow separation zone occurs, which strongly modulates its shape and extension during one oscillation cycle. The presence of multiple harmonics of the fundamental mode indicates a non-linear behavior.

Operating Point OP2 -strong NSV At the operating point OP2 a stronger non-synchronous activity is observed, both in experiments and in simulations. The numerical results at OP2 are analyzed in the same way as the operating point OP1.

As shown in the pressure spectrum in Fig. 12, a domi- N a = -5 emerges in Rotor-2 and is observed in the whole domain downstream. Its circumferential phase velocity in the stationary frame of reference c prop,stat = 0.47Ωr is typical for a convective transport mechanism or rotating stall. Furthermore, a mode with |N a | = 1 is observed with low amplitude in Rotor-2, resulting from a modulation between the two described modes with N a = -5 and N a = +6. Seen from the stationary frame of reference, modes with N a = -5 and N a = +6 at OP2 are propagating in the rotor direction. It can be seen that the mode of N a = +6 is of acoustic nature due to its circumferential phase velocity relative to the flow c prop,statv θ which exceeds the speed of sound.

Considering the modulation of the flow structure in Rotor-3 during one period of the N a = +6 mode in Fig. 14, a much stronger fluctuation of the flow separation zone is observed compared to OP1 (see Fig. 10). In addition to the radial migration which was already observed at OP1, a severe flow separation is present in the downstream part close to the casing. However, this mechanism does not lead to a total breakdown of the compressor performance. The circumferential pattern of mode N a = +6 in Rotor-3 does not propagate with a typical convective speed but rather follows a periodic resonance with the passing acoustic mode. This point is important, since it is effectively a modulation of flow separation zones and not a circumferential propagation in form of stall cells in the sense of Emmons [START_REF] Emmons | Compressor surge and stall propagation[END_REF]. Hence the two modes (acoustic: N a = +6 and convective: N a = -5) are considered as independent.

The same applies to the flow field in Rotor-2, which becomes highly unsteady at OP2. In the same manner as Rotor-3, a flow separation zone develops in Rotor-2 which is modulated with the propagating acoustic mode N a = +6 (not presented here). Additionally, a superimposed signature of rotating stall with the mode order of N a = -5 appears in the simulations but the dominant mode remains at N a = +6 (see Fig. 13).

Comparing the axial Mach Number at 20% chord in Rotor-2 1.02 0.82 and Rotor-3 for the two operating points, Fig. 15 depicts clearly how the undisturbed flow field at OP1 in Rotor-2 is strongly modulated at OP2, revealing N a = +6 at OP2. In Rotor-3, a weak modulation of the separation zone with N a = -3 is observed at OP1. At OP2 the modulation with N a = +6 becomes much stronger and extends towards the casing in both rotors. The highly unsteady flow structure in Rotor-2 prevents further throttling in the simulations. With slightly increased back pressure the rotating stall cells develop rapidly and lead the current numerical simulation to diverge.

COMPARISON AND DISCUSSION

Comparing the two operating points, the axial evolution of the modal amplitude of the acoustic pressure of the respective dominant mode at OP1 (N a = -3, EO stat = 2.3) and OP2 (N a = +6, EO stat = 6.65) is presented in Fig. 16, including the average measurement results at OP2 (N a = +5, EO stat = 6.36 , see Fig. 3(a)). The agreement with the experiment at OP2 of both axial evolution and amplitude is remarkable, particularly considering that the mode order is not the same (Experiment N a = +5

8

FIQUET-TURBO-20-1331 and Simulation N a = +6). Fig. 17 represents the cut-on condition upstream and downstream of Rotor-3 based on the URANS results at OP1 and OP2. The cut-on frequency depends on the local flow conditions, the annulus geometry and the circumferential mode order (N a ) [START_REF] Camp | A Study of Acoustic Resonance in a Low-Speed Multistage Compressor[END_REF]. The cut-on frequency upstream and downstream of Rotor-3 for each URANS simulation is represented by linear curves in the figure . It can be seen that with rising wave numbers, the frequency of a mode must increase to enable propagation. Corotating modes (N a > 0) have a higher cut-on frequency due to the positive swirl of the mean flow. The swirl is higher downstream of the rotor and increases with throttling.

The simulation at OP2 shows a rapid decay of the mode downstream of Rotor-3 with N a = +6 due to its frequency (see Fig. 16), which is below the cut-on condition downstream of the rotor -dashed curve in Fig. 17. For the operating point OP1 the counter-rotating mode with N a = -3 does not propagate upstream of Rotor-3, which can also be explained with the figure. A summary of the modes observed in experiment and simulation is depicted in Fig. 17. It can clearly be seen that all co-rotating modes which are observed in simulations and experiment are at the limit of being propagative only upstream of Rotor-3. As detailed in [START_REF] Fiquet | Non-Synchronous Aeroacoustic Interaction in an Axial Multi-Stage Compressor[END_REF], a change in the circumferential organization of the acoustic mode is observed during the test. Both experimental modes (N a = +5 and N a = +4) satisfy the condition of propagating only in the upstream direction, as depicted in Fig. 17. This is the condition which is necessary for acoustic-resonance as described by Parker [START_REF] Parker | Acoustic resonances and blade vibration in axial flow compressors[END_REF] and Camp [START_REF] Camp | A Study of Acoustic Resonance in a Low-Speed Multistage Compressor[END_REF].

Concentrating on the interaction with the acoustic mode, a clear explanation can be derived for the excitation mechanism as follows. Fig. 19 presents an instantaneous distribution of the non-synchronous entropy signal at OP1. Emerging from Rotor-3, entropy waves are convected downstream and interact with Stator-3, forcing highly unsteady loading which is transformed

9

FIQUET-TURBO-20-1331 The analysis of the acoustic propagation conditions in Fig. 17 indicates that the observed counter rotating mode at OP1 of N a = -3, EO stat = 2.3 is cut-on only downstream of Rotor-3 while it is cut-off upstream.

Hence, the acoustic mode generated in Stator-3 can propagate upstream into Rotor-3 where it is cut-off due to the change in swirl velocity. Here, a strong interaction with the flow separation zone occurs and enforces a phase lock-in. Since this flow structure is the source for the entropy wave, which was necessary to produce the unsteady loading on Stator-3, a resonant system establishes. We observe circumferential coupling of flow separation zones in Rotor-3, whose wave characteristic is dominated by the acoustic propagation conditions. Of course, the frequency and phase lock-in mechanism can only establish if the natural frequency of the flow separation zone is close to the frequency of the acoustic mode in the relative frame of reference. Only specific waveforms (frequency, mode order and spin direction), which meet the condition of being cut-on only in one axial direction and fit the responsive frequency in the relative frame can resonate.

Fig. 20 shows an instantaneous distribution of the nonsynchronous entropy signal at 80% of channel height at OP2. The entropy signal is less organized compared to OP1 but the mode N a = +6 is clearly visible in Stator-1 and Stage-2. In Stator-3 and downstream, the convective mode N a = +5 is obvious. The same principal phenomenon observed at OP1 is present but in this case also occurring in Stage-2.

Through increased loading, flow separations in Rotor-2 develop and produce entropy waves which are convected downstream, leading to unsteady loading of Stator-2. In the same way as described above, an acoustic resonance establishes with a dominant mode, which is partially trapped in the compressor. At OP2, the mode is propagative upstream between the leading edge of Stator-1 and the trailing edge of Rotor-3 (see Fig. 16).

Through axial convection of the corresponding entropy waves the mode is observed also in Stator-3 but cannot propagate acoustically out of the stage.

In the analyzed configurations, the source for acoustic modes is identified not to be a typical vortex-shedding mechanism, but rather a flow separation zone which creates unsteady loading on downstream stators. The generated acoustic wave establishes at a circumferential mode order that fulfills the condition of being cut-on only in one direction. Depending on the operating point, the frequency in the relative frame of reference changes as well as the wave number.

It is known that the dynamic flow characteristics of a severe separation zone as was observed for OP2 are very hard to capture and to predict accurately in URANS-simulations due to their sensitivity to modeling parameters (mesh size, turbulence model, etc.).

It is surprising that the amplitude and axial evolution of the acoustic mode simulated at OP2 agree well with the experimental data, even though the predicted wave number is 6 instead of 5. To confirm that the acoustic mode in the experiment was not affected by the blade vibration, a separate simulation with preimposed blade vibration has been carried out. It was not sufficient in the simulations to alter the developed acoustic modes, which indicates that the phenomenon is dominated by the aerodynamic fluctuations and the acoustic propagation conditions.

The reason for the inaccurate prediction of the experimental wave number of the acoustic mode is assumed to be related to different points: First, the model resolution (spatial and temporal) could be increased to improve the numerical accuracy concerning flow separations and acoustic modes. Second, geometrical asymmetry and structural mistuning could lead to a preference of the experimental setup for a specific mode, as long as it satisfies the necessary propagation conditions (see Fig. 17). Finally, inaccuracy concerning the compressor performance affects swirl and temperature and hence the relative cut-on frequencies. As Rotor-2 and Rotor-3 operate near stall at OP2 and only global performance data is available for validation, this point could be significant and will be addressed in further studies.

In the experiment, the wave number of the acoustic mode decreased with further throttling, which could be explained by the rising cut-off frequency [START_REF] Fiquet | Non-Synchronous Aeroacoustic Interaction in an Axial Multi-Stage Compressor[END_REF]. The simulation could not be throttled further due to convergence issues, which will be addressed in future studies.

To investigate the influence of blade vibration on the developped acoustic modes, coupled simulations with prescribed oscillation of the rotor blades was set up. As in the experiment a dominant nodal diameter of ND = +10 in Rotor-2 based on the first eigenmode (EO Rotor-2,rel = 2.72, see Fig. 3 (b)) was detected, this modeshape was used for the coupled simulations. For OP1 periodically converged results were achieved while the simulation at OP2 diverges rapidly.

Simulations at OP1 have been performed with different vibration amplitudes up to a tip displacement of 2% blade pitch. For the case with maximum amplitude, Fig. 18 shows the spectral analysis of the unsteady pressure. The signature of imposed blade movement is slightly visible at a wave number of 10 FIQUET-TURBO-20-1331 N a = ND = +10. However, the oscillation is only visible in the direct vicinity of Rotor-2 and does not affect the development of the non-synchronous modes observed without blade oscillations. A future study is envisaged to perform the coupled simulations at OP2 with a modified setup.

CONCLUSION

Strong non-synchronous activity has been observed in a research multi-stage axial compressor for helicopter engines at a part-speedline. At moderately throttled operating points, acoustic duct modes developed in a downstream stage in coherence with a structural blade mode. Full annulus simulations of the whole compressor have been performed which develop comparable acoustic modes at two operating points. For the first operating point near peak efficiency, the last rotor (Stage-3) shows flow separation zones which enforce unsteady loading on the downstream stator. This interaction produces an unsteady pressure signature. In a periodically converged state, a propagating acoustic mode develops and couples the phase of the unsteady flow separations in the upstream rotor.

At a higher throttled operating point, Rotor-2 also develops strong flow separations which affect Stator-2 and produce an acoustic mode which is trapped in the compressor. The wavenumber and frequency are very close to those observed in the measurements (Experiment N a = +5, EO stat = 6.36, Simulation N a = +6, EO stat = 6.65). All observed modes in experiment and simulations are propagative only in one axial direction. It indicates that the acoustic propagation conditions drive the occurring wave number instead of a natural vortex shedding frequency.

A simulation with pre-imposed harmonic blade movement is not sufficient to alter the observed wave number of the acoustic mode. Hence, the blade oscillation observed in experiments occurs merely due to coincidence and is not the driving phenomenon.

In the experiment the second harmonic of the mode was in coherence with the vibration pattern, but according to the find-ings in this study, resonance with the fundamental mode could have occurred likewise.

Convective phenomena related to radial vortex transport near the leading edge of a rotor, which are typically responsible for NSV in high-pressure compressors are not relevant for the observed effect. The circumferential coupling of unsteady flow separations is rather produced solely by the propagating acoustic modes. The general problem for the vibration phenomenon is the unstable flow separation which is responsive to acoustic feedback mechanisms. This should hence be focused on when developing countermeasures.

ACKNOWLEDGMENT

The authors thank Safran Helicopter Engines (Safran Group) for permission to publish these results. We are particularly grateful for the technical advice and the comments of J. Demolis of Safran Helicopter Engines during the preparation of this publication. We would also like to thank N. Cumpsty for the helpful discussions on this work.

FIGURE 1 :

 1 FIGURE 1: SCHEMATIC OF SAFRAN HELICOPTER ENGINES TEST COMPRESSOR

FIGURE 2 :

 2 FIGURE 2: FULL COMPRESSOR CHARACTERISTIC FROM EX-PERIMENT; INDICATORS FOR NON-SYNCHRONOUS ACTIVITY IN TRANSIENT EXPERIMENT AND SIMULATION

  at a normalized mass flow rate of 0.72. During establishment of the acoustic mode N a = +5, Fig. 3 (a) shows the frequency spectrum of an unsteady pressure signal measured at the leading edge of Stator-2 and the frequency spectrum of a 'Magnet-Coil' signal (b) measured in Rotor-2, revealing several non-synchronous frequencies. The strongest blade vibrations are observed at a frequency of EO Rotor-2,rel = 2.72, coinciding with the first blade eigenmode. At this frequency, tip

FIGURE 3 :FIGURE 4 :

 34 FIGURE 3: FREQUENCY SPECTRA AT OP2 OF UNSTEADY WALL PRESSURE AT LEADING EDGE OF STATOR-2 IN THE STATIONARY FRAME OF REFERENCE (a) AND MAGNET-COIL SIGNAL IN ROTOR-2 IN THE RELATIVE FRAME OF REFERENCE (b)

FIGURE 5 :

 5 FIGURE 5: COMPUTATIONAL DOMAIN FOR STEADY AND UN-STEADY SIMULATIONS AND DETAIL OF ROTOR-2 MESH

  cells are 4 FIQUET-TURBO-20-1331

FIGURE 6 :

 6 FIGURE 6: RELATIVE MACH NUMBER CONTOURS AT 80% OF CHANNEL HEIGHT FOR WHOLE COMPRESSOR, MEAN AVER-AGED OF URANS FOR OP1 AND OP2, GREY ISOCONTOURS AT LOW SUPERSONIC VELOCITY TO VISUALIZE SHOCK STRUC-TURE

FIGURE 7 :

 7 FIGURE 7: AXIAL EVOLUTION OF THE CIRCUMFERENTIAL WAVE NUMBER (N a ) BASED ON STATIC PRESSURE SIGNAL AT 99.5% OF CHANNEL HEIGHT, URANS OP1

FIGURE 8 :

 8 FIGURE 8: AXIAL EVOLUTION OF THE CIRCUMFERENTIAL WAVE NUMBER (N a ) BASED ON ENTROPY SIGNAL AT 99.5% OF CHANNEL HEIGHT, URANS OP1

FIGURE 9 :

 9 FIGURE 9: RELATIVE MACH NUMBER IN ROTOR-3 AT TWO CHANNEL HEIGHTS DURING ONE CYCLE OF MODE N A = -3 AT URANS OP1

  at the same 4 time instants as Fig. 9. The isocontour at M rel = 0.5 identified as the separation zone indicates the flow detachment near the leading edge close to the hub, extending radially towards the trailing edge. Instantaneous streamlines are shown to visualize radial migration in the downstream part of the blade. During one period of the N a = -3 mode the shape of the separation fluctuates radically. Close to the casing, the flow 6 FIQUET-TURBO-20-1331

FIGURE 10 :TABLE 2 :

 102 FIGURE 10: FLUCTUATION OF FLOW STRUCTURE ON SUCTION SIDE OF ROTOR-3 DURING ONE CYCLE OF MODE N A = -3, ISO-CONTOURS AT RELATIVE MACH-NUMBER M REL = 0.5; VALUES BLANKED ABOVE 98% OF CHANNEL HEIGHT; INSTANTANEOUS STREAMLINES STARTING NEAR LEADING EDGE AT 50% AND 95% OF CHANNEL HEIGHT, URANS OP1

FIGURE 11 :

 11 FIGURE 11: FREQUENCY SPECTRA IN THE RELATIVE FRAME OF REFERENCE OF AXIAL VELOCITY SIGNAL AROUND ROTOR-3, URANS OP1 AT 80% OF CHANNEL HEIGHT

FIGURE 12 :

 12 FIGURE 12: AXIAL EVOLUTION OF THE CIRCUMFERENTIAL WAVE NUMBER (|N a |) BASED ON STATIC PRESSURE SIGNAL AT 99.5% OF CHANNEL HEIGHT, URANS OP2

FIGURE 13 :

 13 FIGURE 13: AXIAL EVOLUTION OF THE CIRCUMFERENTIAL WAVE NUMBER (|N a |) BASED ON ENTROPY SIGNAL AT 99.5% OF CHANNEL HEIGHT, URANS OP2

FIGURE 14 :FIGURE 15 :

 1415 FIGURE 14: FLUCTUATION OF FLOW STRUCTURE ON SUCTION SIDE OF ROTOR-3 DURING ONE CYCLE OF MODE N A = +6, ISO-CONTOURS AT RELATIVE MACH NUMBER M REL = 0.3; VALUES BLANKED ABOVE 98% OF CHANNEL HEIGHT; INSTANTANEOUS STREAMLINES STARTING NEAR LEADING EDGE AT 50% AND 95% OF CHANNEL HEIGHT, URANS OP2

FIGURE 16 :

 16 FIGURE 16: AXIAL DISTRIBUTION OF THE AMPLITUDE OF NON -SYNCHRONOUS PRESSURE MODE, EXPERIMENTAL RE-SULTS AT OP2, NUMERICAL RESULTS FOR OP1 AND OP2

FIGURE 17 :

 17 FIGURE 17: FREQUENCY AND CIRCUMFERENTIAL WAVE NUMBER OF ACOUSTIC MODES IN EXPERIMENT AND SIM-ULATION, CUT-ON CONDITIONS UPSTREAM AND DOWN-STREAM OF ROTOR-3 (URANS SIMULATIONS)

FIGURE 18 :

 18 FIGURE 18: EVOLUTION OF THE CIRCUMFERENTIAL WAVE NUMBER (N a ) BASED ON STATIC PRESSURE SIGNAL AT 99.5% OF CHANNEL HEIGHT, URANS AT OP1 WITH PRESCRIBED HARMONIC BLADE OSCILLATION OF ROTOR-2 AT ND = +10

NOMENCLATUREc

  Speed of sound [m/s] c prop Circumferential phase velocity [m/s] ∆P s Pressure fluctuation [Pa] ∆S Entropy fluctuation [J/K/kg] M ax Axial Mach Number [-] M rel Relative Mach Number [-] N a Circumferential wave number [-] N b Number of blades [-] ND Nodal diameter [-] Ω Rotational speed [rad/s] r Radius [m] v θ Absolute tangential velocity [m/s] W x Axial velocity [m/s] EO Engine order frequency MC Magnet-coil NSV Non-Synchronous Vibrations IGV Inlet Guide Vane . Fourier coefficient . stat Stationary frame of reference . rel Rotor relative frame of reference

TABLE 1 :

 1 EXPERIMENTALLY

	OBSERVED	NON-

TABLE 3 :

 3 NON-SYNCHRONOUS MODES AT OP2

		Acoustic mode	Convective mode
		Upstream of Rotor-3	Rotor-3
	N a	+6	-5
	EO stat	6.65	2.33
	EO rel	0.65	2.66 (alias |2.33 -5|)
	c prop,stat Ωr	1.1	0.47
	c prop,rel Ωr	0.1	-0.53
	c prop,stat -v θ
	c

FIQUET-TURBO-20-1331