Non-Synchronous Vibration: Characterization of the Aerodynamic Disturbance and Its Dependency on Local Tip Clearance Open Access

Nonsynchronous vibrations (NSV) in fans and compressors pose a major challenge to aircraft engine manufacturers in pursuit of efficient designs, particularly in aerodynamically highly loaded rotor blades and integrated bladed disks (blisks) with negligible mechanical damping. The particular type of NSV investigated in this paper was first described by Kielb et al. [1]. It occurs near the stall boundary when small scale aerodynamic disturbances develop, propagate circumferentially and couple with blade vibration, leading to high vibration amplitudes and potential mechanical failure. Due to its safety-critical nature, NSV must be prevented, but a lack of reliable prediction methods makes it difficult to design against, forcing manufacturers to resort to conservative designs and restrict operating spaces. The present research aims to contribute to the removal of un-necessary design constraints in the long term by delivering new physical insights into the aerodynamic disturbances associated with NSV and its coupling mechanisms with blade vibration.

The occurrence of unsteady flow features, also referred to as disturbances in this paper, and their relevance to stall onset has been studied extensively. A review of this is given by Ref. [2]. Of particular relevance to vibration are the flow features associated with spike-type stall inception and so-called rotating instabilities. In spike-type stall inception small aerodynamic disturbances appear in the rotor tip region which eventually develop into rotating stall. They are of short length-scale and are localized. Recent work has identified these as regions of high radial vorticity, or radial vortices [3,4]. Most of these have been investigated in a purely aerodynamic context. However, experiments on a 1.5 stage transonic compressor blisk rig showed how these disturbances can lock-in with blade vibration and lead to high amplitude NSV [5]. Comparable results have been obtained recently for a power generation compressor [6]. In this lock-in process, fluid-structure coupling modifies the frequency and wavelength of the aerodynamic disturbance. Lock-in has also been demonstrated in multistage compressors, where individual stages operate with fully developed part-span rotating stall and the cell count and propagation speed adjust under the influence of blade vibration [7].

Most of the recent research on NSV thus far has been performed on core compressors [8–11] but NSV has recently also been detected in modern ultrahigh-bypass-ratio (UHBR) fans [12,13] and for the ECL5/CATANA rig fan [14,15]. The latter developed NSV in different modes at all subsonic speedlines, while showing spike-type stall inception without vibration-precursor at transonic speeds.

In specifically designed experimental campaigns on the aforementioned 1.5 stage transonic compressor [16,17] and sensitivity studies using a low order model [18,19] it was shown that the crucial factors determining NSV behavior for a given rotor and vibration mode are:

• The flow structure, or passage blockage, in the tip region

• The propagation speed of the aerodynamic disturbance

However, some important questions regarding these two remain. It is still unknown whether there is an aerodynamic onset criterion, which determines the NSV operating point, and what role the tip leakage flow plays in the unsteady disturbance. Furthermore, the system properties setting the propagation speed remain elusive. Understanding of both, the aerodynamic features driving NSV and the propagation speed, is important for the prediction of NSV, as it enables early identification of critical operating points and critical vibration modes. Knowledge of the latter would make it possible to detune the system to prevent aeroelastic instability.

This paper presents a joint experimental-numerical study to address these gaps in knowledge by investigating the aerodynamic flow structures associated with NSV and their structural coupling mechanisms on the fan ECL5, which was investigated in the European project CATANA.

The test case is a fan rig representative of modern UHBR fans, equipped with extensive aerodynamic and structural instrumentation, enabling a detailed experimental characterization of NSV. A particular feature of this rig is its circumferentially nonuniform running tip clearance [15]. The design tip clearance is 1 mm at the trailing edge and uniform, but due to manufacturing problems individual blades differ by $±$0.1 mm, which exceeds the variability normally expected from manufacturing tolerances. Although unintentional, the nonuniform tip clearance makes the test case ideally suited to characterize the aerodynamic features causing NSV, and their dependence on tip clearance as well as the factors determining propagation speed.

By analyzing the unsteady flow structures with uniform and nonuniform tip clearance, as well as their coupling with blade vibration, this paper aims to answer the following research questions:

1. How can the aerodynamic disturbance be characterized?

2. How does non-uniform tip clearance affect the nature, amplitude and propagation speed of the aerodynamic disturbance?

This is the first time that the influence of tip clearance is studied locally without any associated changes in mean aerodynamic loading.

The test case investigated in this study is the UHBR OPEN-TEST-CASE FAN ECL5/CATANA, which is a 1:4 scale rig representative of modern UHBR fans. A fan blade is shown in Fig. 1(a). The fan stage comprises 16 composite rotor blades and 31 outlet guide vanes. The downstream OGV is constructed conventionally from aluminum vanes. The aerodynamic design point was numerically predicted at a standard mass-flowrate of 36.0 kg/s, the fan-stage total pressure ratio is 1.35 and the isentropic efficiency is 92%. From a total of 48 manufactured blades 16 blades have been selected for the reference configuration based on the ping-test measured eigenfrequencies for the first three eigenmodes to minimize structural mistuning. The resulting frequency deviation from the mean is less than 0.65% for all vibration modes. Key performance parameters are given in Table 1 and details about the design can be found in Ref. [20].

The experiments were performed at the open cycle test facility Phare-2 at Ecole Centrale de-Lyon. The machine is driven by an electrical motor with a maximum power of 3 MW. Details on the facility are given in Ref. [21]. A turbulence control screen (TCS) is installed in front of the machine core to ensure homogeneous inflow conditions and reduce large scale turbulence. The machine is instrumented with rakes and a venturi nozzle for performance measurements. For the aeroelastic study strain gauges on each blade, a capacitive tip-timing and tip clearance measurement system, as well as distributed Kulite sensor arrays (Fig. 1(b)) and microphones are used together. During experiments, the machine was stabilized at constant rotation speed, corrected to ISA conditions and the operation point varied through incremental closing of an axisymmetric throttle downstream of the core section. Further details on instrumentation and experimental methods are given in Refs. [15] and [14].

The structural mode shapes of the first three eigenmodes are presented in Fig. 2. Mode-1 is almost a pure bending mode. Modes 2 and 3 have significant torsional component at the tip with a nodal line crossing trailing and leading edge, respectively. The blades have an approximate mass ratio of 70, which makes them sensitive to aeroelastic coupling phenomena.

In the experiments at 80% of design speed all three modes were active with significant amplitude, and the present study focuses on this speed line. The experiment was aborted due to critical limit-cycle vibrations of Mode-2 [14] but the highest deflection amplitude occured in Mode-3.

The unsteady pressure spectrum measured at a highly throttled condition using a Kulite sensor near the leading edge is shown in Fig. 3(a) (taken from Ref. [14]). Figure 3(b) shows the averaged spectrum of the blade vibration amplitude measured using strain gauges in the rotor frame of reference. All frequencies are normalized with the rotor speed $Ωr$⁠. Here, peaks in all three blade eigenmodes are clearly visible. The dashed black lines indicate 25% of the high cycle fatigue limit for the respective mode. Mode 2 (⁠$ωvR=4.52Ωr$⁠) reaches almost 50% of the high cycle fatigue limit. The pressure spectrum exhibits multiple peaks. Analysis based on cross-correlation of circumferentially distributed pressure sensors in Ref. [14] showed that these peaks are associated with different aerodynamic wave numbers of a circumferentially propagating disturbance. The wave numbers $Na$⁠, propagation speeds in the rotating $ΩaR$ and stationary reference frame $ΩaS$⁠, as well as frequencies in the rotating and stationary frames (⁠$ωaR$ and $ωaS$ respectively) for each peak in the pressure spectrum are listed in Table 2. Some of the aerodynamic frequencies are close to the frequency of the eigenmodes, leading to lock-in and NSV. These are highlighted in bold. Notably, all wave numbers/frequencies are associated with propagation speeds between 55% and 63% rotor speed in the stationary frame of reference (⁠$ΩaS/Ωr$⁠). The speed range of the aerodynamic modes (wave numbers) which do lock with vibration is also $0.55<ΩaS/Ωr<0.63$⁠. This indicates that the measured NSV is caused by aerodynamic disturbance with multiple wave numbers but a characteristic propagation velocity.

As already mentioned, the blades were unintentionally manufactured with small variations in span. Hence, the tip clearance is nonuniform in the experiment. The measured tip clearances, which were later confirmed with geometry scans, are shown in Fig. 4 for three chordwise locations. The mean tip clearance near the leading edge is 1 mm (1.2% tip chord) and it varies by $±$ 0.1 mm.

The flow solver used is AU3D, an unsteady time-accurate Reynolds-averaged Navier–Stokes (URANS) solver for aeroelastic analysis developed at Imperial College with the support of Rolls-Royce. The solver has been extensively validated for fan and compressor flows near stall and aeroelastic instabilities [22–25]. The current study uses the Spalart–Allmaras turbulence model with a correction based on the pressure gradient and helicity as described in Ref. [26].

The computational domain is shown in Fig. 5. It comprises a straight inlet duct, rotor and stator domain and a downstream converging diverging nozzle. The blades meshes are structured in the radial direction and unstructured in the azimuthal plane. Total pressure and temperature boundary conditions (ISA Sea Level Conditions) and zero flow angles are prescribed at the inlet. At the outlet, a static pressure is set low enough to choke the nozzle, such that the downstream converging-diverging nozzle is used to control the mass flow. All simulations use wall functions and the near-wall resolution is set such that 95% of all cells feature a $y+$ not less than 25. A mesh convergence study was performed using steady-state performance as the convergence criterion. The selected mesh contains 704 thousand nodes per rotor passage, and 275 thousand nodes per outlet guide vane.

Steady state simulations with mixing planes at the blade-row interface are used to map the fan performance and to validate this against experimental measurements. For unsteady simulations the domain (intake, fan, OGV, and nozzle) is expanded to a 360 deg full annulus domain with sliding planes at the interfaces. All unsteady (URANS) simulations are performed at a selected near-stall operating point, which will be shown later, and started from the converged steady-state solutions. Three sets of unsteady simulations are shown in this paper: Aeroelastic simulations with blade vibration and nonuniform tip clearance, and aerodynamic simulations with uniform and nonuniform tip clearance but no blade vibration. The nonuniformity follows the same patterns as the experiment (Fig. 4).

The URANS simulations are used to investigate unsteady features which occur close to the stall boundary. These features are nondeterministic, i.e., they are not related to blade passings or vibration frequencies. Following best practice, the time-step is set to give 200 time steps per rotor blade passing period.

A modal approach is used for the coupled simulations. The mode shapes are obtained from a finite element solver and interpolated onto the CFD mesh. The predicted modal frequencies were found to differ slightly from the experimental ones [14], and the mean of the experimentally measured frequencies was used for all simulations. Frequency mistuning was therefore not considered.

The aeroelastic simulations are bi-directionally coupled and include relevant vibration modes (Modes 1–3). The blades are given an initial displacement and the displacement histories are tracked to identify unstable modes. This is often also referred to as “free vibration” modeling, and it is the approach which most closely resembles the experiment.

Before the unsteady analysis, the steady-state computational model is validated against performance measurements. Figure 6 shows the fan 80% speed characteristic, on which NSV was detected, comparing the experimentally measured characteristic against those predicted by RANS with uniform tip clearance. A good match is seen across the range. The stall mass flow is predicted within 3% and the total pressure ratio within 0.8%. For the unsteady analysis, the last operating point for which a converged steady-state solution could be obtained is selected. This point is labeled NS (near stall) on Fig. 6, the design point DP is also indicated. The pressure ratio prediction improves in the unsteady simulations.

For a more detailed validation of the flow field inside the passage, Fig. 7 shows the static pressure contours at the casing. Particular features of interest are the strength and trajectory of the static pressure drop created by the tip clearance flow, which are captured in the RANS simulations.

All unsteady analysis in the following sections is performed at the 80% Near Stall (NS) operating point shown in Fig. 6.

Following the successful steady-state validation, an unsteady aeroelastic analysis is now performed for the case with nonuniform tip clearance at the selected near stall (NS) operating point. The aeroelastic simulation is started from the converged steady-state solution.

For the aeroelastic simulation all nodal diameters (in Modes 1–3) are excited with a fixed modal displacement amplitude, corresponding to a tip leading edge displacement of 0.8% chord, and random phase, and the displacement histories are tracked to identify unstable modes. Figure 8 shows time histories of the modal displacement amplitudes in all forward traveling assembly modes. The amplitudes of all modes initially decrease but from 10 revolutions, the amplitude in Mode 3 nodal diameter 2 (⁠$Nv=2$⁠, here referred to as M3 ND2), begins to rise. This is later followed by an increase in M3 ND3. The maximum displacement in ND3 remains lower than that in ND2, which reaches an equivalent physical displacement of 1.2% chord at the LE. All backward traveling modes decay and are not shown. A high amplitude in M3 ND2 was also measured in the experiment (see Table 2). Hence, the aeroelastic simulation predicts NSV in the correct mode.

The fluctuating amplitudes of several modes in Fig. 8 indicate that multiple frequencies and wave numbers exist and change over time. This is typical for NSV, where irregular unsteady aerodynamic flow structures (disturbances) couple with blade vibrations. This is further investigated by determining the aerodynamic wave numbers $Na$ present in the simulation. These are determined by performing circumferential Fourier transforms on the unsteady pressure field 5% upstream of the leading edge in the rotating frame of reference. The resulting wave numbers are always positive and give no information about the direction of the aerodynamic disturbance. Figure 9 shows the transient evolution of the aerodynamic wave numbers measured at the casing, 5% chord upstream of the LE. Frequency components associated with the outlet guide vane (EO31) are too weak to be detected at that location, and since this is based on the pressure fluctuations in the rotor frame of reference all wave numbers visible in Fig. 9 are associated with nonsynchronous features.

The frequency associated with the wave number $Na=14$⁠, which grows around revolution 35 in Fig. 9, is hence $ωvR=Na|ΩaR|=5.74Ωr$⁠, which is close to the frequency of Mode 3 (see Table 2). The backward traveling $Na=14$ wave aliases onto a forward traveling $Nv=2$ on the 16 rotor blades. The results thus far are hence indicative of a lock-in of an aerodynamic disturbance with M3 ND2.

Numerical probes are placed near the pressure side at 10% axial chord adjacent to every blade and the pressure signal is recorded. Figure 10 shows the location and pressure signals adjacent to three blades as an example. Since the influence of the OGV is too weak to be detected, all fluctuations seen are due to the unsteady disturbance. A dip in pressure is defined as the “impact” of a disturbance on the blade pressure side, and the time for each impact is recorded. As seen in Fig. 10 the static pressure locally drops by over twice the inlet dynamic pressure at each impact. Since there are 16 disturbances, i.e., there is one dip per passage at any point in time, the time between consecutive impact events in adjacent passages is the time it takes the disturbance to cross each passage, $Tpass$⁠. From this, a propagation speed $ΩaR$⁠, i.e., the average speed for the disturbance to cover a distance equal to the pitch, is calculated. (The analysis was repeated at different locations in the passage and the results shown now were found to be independent of the probe location.)

This propagation speed is plotted in Fig. 11 alongside the Mode 3 vibration amplitude. As expected, the rise in Mode 3 vibration amplitude correlates with a rise in relative propagation speed from $ΩaR=−0.41Ωr$ at the beginning of the simulation to $ΩaR=−0.42Ωr$ at peak vibration amplitude. The frequency of the aerodynamic disturbance in the rotating frame of reference with $ΩaR=−0.42Ωr$ is $ωaR=5.88Ωr$ which is in resonance with Mode 3. In other words, the interaction between the blade vibration and aerodynamic disturbance has modified the aerodynamic disturbance such that its frequency and wave number match that of M3 ND2 (⁠$ωv=5.77Ωr→ωv=5.88Ωr$⁠, and $Na=16→Na=14$⁠, exciting $Nv=2$⁠).

To determine the nature of the aerodynamic features which cause this M3 ND2 NSV at the near stall condition, the investigation in the following two sections will be based on purely aerodynamic simulations without any blade vibration.

This section will analyze the unsteady aerodynamic features at the 80% NS operating point identified above for the reference case with uniform tip clearance. A novel method for tracking disturbances will be introduced and some unsteady characteristics, such as the disturbance strength and propagation speed will be validated against experiments.

Figure 12 shows the casing pressure spectra from the numerical simulations. These are obtained by transforming the unsteady pressure signal, recorded at a location 5% upstream of the rotor leading edge, on the rotating rotor casing into the stationary frame of reference and performing a temporal Fourier transform with a sliding window. The frequency is expressed in engine orders, i.e., nondimensionalised with the rotor shaft frequency. To identify the associated circumferential wave number, a circumferential Fourier transform is performed on the reconstructed time signal in a given engine order. The dominant peak in Fig. 12 at frequency $ωaS=9.4Ωr$ in the stationary frame of reference is associated with an aerodynamic wave number $Na=16$⁠, which matches the rotor blade count. The nondimensional propagation speed of the aerodynamic disturbance in the stationary frame of reference is calculated as $ΩaS=ωaS/Na=0.59$ (in the direction of the rotor). The propagation speeds associated with the other peaks in Fig. 12 are summarized in Table 3. Clearly, the propagation speeds for the different peaks are very close to each other, indicating that they are all associated with the same aerodynamic disturbance. The propagation speed matches well with the that of the experiments, which was shown in Table 2.

To visualize the location of this disturbance in the blade passage, the unsteady fluctuations in static pressure in the rotor frame of reference are shown in Fig. 13, for both the experiment and the CFD. These are calculated from phase-locked ensemble-averaged wall pressure measurements in the experiment, and from probes in the rotating frame of reference in the CFD. The influence of the OGV is negligible, and the quantity shown therefore represents unsteadiness associated with the aerodynamic disturbances. It is quantified here as the temporal standard deviation in static pressure $σ(ps)$ normalized by the dynamic head at the inlet. Both CFD and experiment show unsteady fluctuations of the order of dynamic pressure at the inlet. The pressure fluctuations are strongest near the pressure side over the first 10% chord, and in a smaller region near the leading edge suction side.

where $NB$ is the blade count and $ΩaR=1+ΩaS=−0.41$ is the speed in the rotating frame (with $ΩaS=0.59$ from the dominant 9.4EO frequency in Table 3). This characteristic time was already introduced in the aeroelastic analysis in Sec. 4 and will be referred to as the disturbance passing period in the following.

The upper figures demonstrate two features: First, the separation of the boundary layer near the leading edge forms a vortical structure that is convected axially downstream along the blade suction side (F1). Second, a vortical structure emerges from the tip clearance. It remains in the upstream part of the passage and propagates toward the trailing blade's leading edge (F2). The lower figures (looking radially outwards) emphasize this axial separation of features. For $t/Tpass=0.41$ the feature F1 develops close to the suction side and forms a U-shaped structure located in midpassage for $t/Tpass=0.83$⁠. At $t/Tpass=1.07$ feature F2 is detached from the previous blade and impacts the leading edge of the subsequent blade. A part of the disturbance is spilled around the leading edge. The impact of feature F2 on the trailing blade's pressure side causes the unsteady pressure peak observed in Fig. 13.

The feature F2 is similar to stall precursor features described as radial vortices in previous publications [3–5]. This is only accurate for the part attached to the casing and particularly before impact on the subsequent blade. For the present case, the results show that this feature forms in the tip clearance close to the leading edge, and is not directly caused by the suction side separation which occurs at lower channel heights and propagates downstream (F1).

The trajectory and propagation of aerodynamic disturbances observed here is in agreement with results from previous wall-resolved simulations performed using the same geometry at similar operating conditions [27].

The analysis of wall pressure spectra performed above (and frequently for the analysis of stall precursors, stall cells, and nonsynchronous vibration) provides a measure of propagation speed and disturbance amplitude which is effectively averaged over multiple revolutions and all blades. The benefit of this is that it gives a robust value and the same calculation method can be performed in experiments and in CFD. A disadvantage is that it does not provide any information about the local behavior of the disturbance, such as its three-dimensional path and its dependency on individual blade geometry. The snapshots in Fig. 14 suggests that the disturbance is not simply convected past the leading edge but that it impinges on the blade pressure side and triggers the release of the next disturbance. This would imply that there are at least two distinct propagation mechanisms; one for the transport across the passage and one for the transfer across the blade.

To investigate this further, unsteady data recorded during the simulation will be processed to analyze the disturbance in the rotating frame of reference.

First, fluctuations in the three-dimensional vorticity fields are computed for different time steps of the unsteady simulations. Following the analysis of Fig. 14, where structures similar to radial vortices were identified, the radial vorticity component is selected as most representative of the aerodynamic disturbance, and projected onto the casing surface. The results for five snapshots of the disturbance passing period $Tpass$ are shown in Fig. 15. Regions of high vorticity are detected, and at each time frame their position in the rotor frame is noted. The circumferential position is used to build the time-vs-position plot at the bottom of Fig. 15.

The disturbance propagation will now be explained by following the vorticity peak detected on the pressure side of Blade 9 at $t=0$ (blue circles). This peak arrives at the leading edge of Blade 9 at $t/Tpass=0.21$⁠. This is the impact event defined in Sec. 4. At the same time, a new peak emerges on the suction side of Blade 9 (green circles). The original peak then convects along the pressure side of Blade 9 (⁠$0.21<t/Tpass<0.62$⁠), while the newly formed one convects across the passage toward the leading edge of Blade 10. This results in the pairs of lines in the bottom plot. The left line (blue circles) between Blade 9 and 10 leading edge is straight, indicating that vorticity is convected along the pressure side with relatively constant velocity. The right one (green circles) results from the convection of vorticity across the passage, and it smoothly changes slope as the local swirl velocity between Blade 9 and 10 changes. The right line starts a small distance from the suction side near the leading edge, at the same time when the left one reaches the pressure side. This pattern was found to repeat over hundreds of disturbance passing periods, and is evidence that the impact of one disturbance triggers the release of a new one. The convection across the passage (green circles) will be referred to as “free convection” mechanism in the following.

The triggering mechanism, or transfer across the blade, can be better understood by examining the evolution of the relative velocity close to the leading edge at the time of impact. Figure 16 shows the relative velocity vector field at two different time steps of the disturbance passing period, on a slice at constant channel height inside the tip gap. Radial vorticity contours are superimposed to easily identify the vortex cores, which are the same features as the ones seen on the first two frames of Fig. 15 (F2). Both time steps show a local blockage zone, or velocity deficit, downstream of the vortex core in the passage (blue area). When the disturbance arrives at the blade, it has two effects: First, it increases incidence resulting in a larger acceleration of flow around the leading edge (red region at the leading edge at $t/Tpass=0.21$ compared to $t/Tpass=0$⁠). Second, the vorticity disturbance and downstream velocity deficit are passed through the tip gap. Both effects combine to create a new vortical structure on the suction side. This indicates that tip clearance plays an important role in transferring the disturbance from one passage to another.

The disturbance propagation mechanism is schematically illustrated in Fig. 17. Importantly, there are two distinct mechanism of circumferential disturbance propagation. One is simply a convection of vorticity while the other is the impingement and release of new vorticity. The combination of the two results in the circumferential propagation of the disturbance. When the propagation speed is calculated using wall pressure spectra, it is impossible to separate the two mechanisms. By tracking the vorticity, however, it was possible to compute local propagation speeds and analyze the two mechanisms.

The path taken by the disturbances varies slightly in time. In other words, not every disturbance follows the exact same path. The range of paths is visible in Fig. 18, where disturbance locations throughout the whole simulation are colored with their local circumferential speed, which is computed from the gradient of the time-vs-position plot in Fig. 15. Averaging the local speeds over hundreds of disturbances gives the local velocity profile shown in Fig. 19. Here, the average local circumferential speed, normalized by the rotor speed, is plotted against pitchwise distance. A local speed can only be easily defined for the free convection region (and not the transfer across the blade). Hence, the speed profile shown in Fig. 19 only extends from about 5% to 95% pitch. There is a huge variation in speed inside the passage, ranging from 20% to 60% of rotor speed. The disturbance is fastest between 20% and 50% of the pitch. The average speed in this free convection region is $ΩaR=−0.39Ωr$⁠, which is very close to the propagation speed value previously determined from pressure spectra (⁠$ΩaR=−0.41Ωr$⁠).

Since the wall pressure and strain gauge spectra analysis showed that this disturbance locks in with the vibration modes (see Table 2), this is further confirmation that the aerodynamic disturbance which drives NSV is a vortical disturbance, which propagates circumferentially. The propagation speed is determined by the convection across the passage and the transfer across the blades. It shows that an accurate prediction of the passage swirl velocities, which determine the circumferential convection speed, is absolutely crucial to correctly predicting NSV.

In this section, the nonuniform tip clearance results are analyzed and a more detailed comparison against experiments is performed. The tip clearance variation causes a change in the time-averaged and unsteady flow in each passage.

Figure 20 shows the time-averaged swirl velocity $u¯θ$ and the passage-to-passage variation in time-averaged swirl velocity $Δu¯θ$⁠, expressed as a fraction of the rotor speed. Two main observations can be made from Fig. 20. First, the magnitude of the deviations is significant, reaching approximately 3% of the rotor speed (or 7% of exit swirl velocity). Second, the region most affected by the change in tip clearance is the suction side where the tip clearance flow mixes with the passage flow. Interestingly, differences are also visible upstream of the leading edge, which indicates that the change in passage flow either causes a change in the potential field ahead of the fan blade or in the way aerodynamic disturbances propagate circumferentially, locally increasing or decreasing leading edge spillage.

As a first investigation of the influence of tip clearance nonuniformity on the unsteady flow, the spectral analysis of the casing static pressure shown in Sec. 5.1 is repeated for the nonuniform case. The results are overlaid on those of the uniform case in Fig. 21. There is no notable difference in the wall pressure spectra between uniform and nonuniform tip clearances. If tip clearance variations modify the aerodynamic disturbance, as many studies on the effect of uniform tip clearance variations on NSV would suggest, these effects become blurred in wall pressure analyses. To investigate the effect of tip clearance more thoroughly, the local disturbance strength and propagation speed are now analyzed for individual blades.

To isolate the influence of an individual blades' tip clearance on the disturbance, the local propagation speed in each passage was determined using the methodology explained in Sec. 5.1.2. This method gives two values: a free convection speed which is measured while the disturbance is just convecting across the passage, and a global propagation speed which combines the free convection and the transfer across the blade, when the disturbance has traveled a single blade pitch. Both are plotted against blade number in Fig. 22(a). The value at a given blade is the speed of the disturbance between the blade and its suction side neighbor.

The values were obtained by averaging over 300 individual disturbance passings per blade. The average of the global propagation speed results in the characteristic propagation velocity $ΩaS=0.59Ωr$ (⁠$ΩaR=−0.41Ωr$⁠) which was identified from the pressure spectra in Sec. 5.1. This confirms that the average propagation speed is not affected by tip clearance nonuniformity. Fig. 22(a) also shows that the free convection speed and global propagation speed appear well correlated. The passage-to-passage variation in propagation speed reaches 6% of rotor speed.

Figure 22(b) shows the change in propagation speed $ΔΩaR$ caused by the transfer of the disturbance across the blade. This was calculated from the time difference between a disturbance arriving at the pressure side and reemerging on the suction side. The sum of $ΔΩaR$ and the average free convection speed gives the global propagation speed in Fig. 22(a). The change in propagation speed $ΔΩaR$ is positive for all blades, i.e., the transfer across the blade is faster than the convection speed, but the change is small (0.5-2% of the rotor speed). Similar to the global propagation speed, there is a noticeable blade-to-blade variation.

Comparing this to the variation in chordwise averaged tip clearance shown in Fig. 22(c), it is clear that some features such as the local drop at Blade 10 and local peaks at Blade 4 and 12 are visible in all three plots.

To explicitly show the relationship between tip clearance and propagation speed, Fig. 23(a) plots the global propagation speed in each passage against tip clearance. There is a clear relationship: A larger tip clearance results in a faster global propagation speed, i.e., a faster combined transfer of the disturbance across the blade and free convection through the passage.

This indicates that one of the effects of tip clearance variations is to accelerate/decelerate the mechanism by which aerodynamic disturbances are transferred across the blade, while another effect modifies the passage flow field and free convection speed. A larger tip clearance results in a faster disturbance transfer across the blade and a faster convection across the passage, hence increasing the propagation speed in the relative frame of reference. The change is in the range of $±$3%$Ωr$⁠.

A further interesting relationship can be seen comparing the variation in speed to the passage-to-passage deviation in swirl velocity in Fig. 20. Blades with a large tip clearance and fast propagation speed, for example, Blade 14 or 15, are associated with a higher-than-average swirl velocity, whereas blades with small tip clearance (e.g., 5 or 10) are associated with lower-than-average swirl velocities. Disturbances propagate within this nonuniform steady field and quantitatively, their $±$3%$Ωr$ speed differences aligns with the variation in swirl velocity. It is hence possible to infer the impact of varying tip clearance on propagation speed from time-averaged/steady flow field analysis.

In addition to the propagation speed, the strength of the disturbance is of interest for stability. To quantify this, the fluctuations in static pressure were tracked in time and the disturbance strength was defined as the static pressure drop on the pressure side “impact” point (the region of high unsteadiness in Fig. 13, with traces also shown in Fig. 10). Figure 23(b) shows the relationship between disturbance strength measured near the pressure side of one blade and the tip clearance of its pressure side neighbor (previous blade), e.g., tip clearance of Blade 1 and strength measured at Blade 2 in Fig. 10. Clearly, a larger tip clearance is associated with a stronger disturbance, i.e., a larger static pressure drop, in the following passage.

Previous work, e.g., [18,24] has shown how important the propagation velocity is for aeroelastic stability. It sets the characteristic interblade phase angle (IBPA) and frequency of the aerodynamics which then lock-in with blade vibration and thus determines the unstable vibration mode and nodal diameter. To test the implications of the detected variations in propagation speed, a linear reduced order model for NSV, which was developed in Ref. [24], is used to determine the sensitivity of aerodynamic damping to propagation speed for this test case. The model assumes a uniform propagation speed, or uniform tip clearance. The result is shown in Fig. 24, where the range of propagation speeds found in the CFD simulations is highlighted in red. The aerodynamic damping values are the minimum values over all nodal diameters, and the most unstable nodal diameter $Nv$ associated with a given speed range is indicated above the plot.

Figure 24 shows that for the range of speeds found in the CFD simulations, damping ratios range from −1.2% to 0.2%. Moreover, the most unstable structural nodal diameter (⁠$Nv$⁠) varies between 1 and 3. The physical explanation for the behavior in Fig. 24 is that the structure responds in the mode that is closest to resonance with the aerodynamic frequency $ωvR=|ΩaR|Na$ and at an interblade phase angle which is close to the aerodynamic phase lag between the blades. This interblade phase angle resonance is given when the structural nodal diameter is equal to the aerodynamic wave number of one of its aliases, such as the $Na=14$ (backward) and $Nv=2$ (forward) condition seen in the aeroelastic analysis in Sec. 4. The aerodynamic wave number can change in response to the blade vibration, as was also shown in Sec. 4, where the initial $Nv=16$ became $Nv=14$ as the vibration amplitude grew. When the propagation speed changes, the system moves in and out of resonance resulting in the repeating pattern in aerodynamic damping. More details on this can be found in Ref. [18].

For these model results, the propagation speed was uniform from blade to blade, but the simple analysis shows how powerful the propagation speed, and hence the tip clearance, is in stabilizing or destabilizing the system. Furthermore, it indicates that tip clearance nonuniformity can mistune the system.

To check if this effect is visible in the experiments, strain gauge data recorded on different blades was used to calculate the interblade phase angle measured between adjacent blades. The IBPAs between a given set of blades fluctuate in time, and hence the number of occurences over 6000 revolutions of a given IBPA was counted. Figure 25 plots the relative number of occurrences of specific interblade phase angles against blade number. The figure shows that there are dominant IBPAs between pairs of adjacent blades, but these are not uniform across the circumference. The range of interblade phase angles seen in the experiment corresponds to a range between nodal diameters $Nv=1$ and $Nv=3$⁠.

This behavior is characteristic of a detuned system, where multiple nodal diameter are active simultaneously. In the experiment, structural mistuning was also present and it is hence not possible to attribute the variation seen in Fig. 25 to the tip clearance alone. However, the range of nodal diameters detected matches that predicted by the reduced order model in Fig. 24 based on the propagation speed range seen in the CFD. Moreover, the blade-to-blade variation in the experimentally measured interblade phase angle (mean IBPA in Fig. 25) resembles the variation in computationally predicted propagation speed, which is repeated in the bottom plot of Fig. 25. There is hence a strong numerical and experimental indication that tip clearance nonuniformity is able to aerodynamically mistune the system.

The mechanisms leading to nonsynchronous vibrations of a state-of-the-art fan rig near stall conditions were investigated using experimental measurements and URANS simulations, with particular attention to the effect of blade-to-blade variations in tip clearance size. The URANS simulations were able to qualitatively and quantitatively predict the unsteady aerodynamic features which lead to Non-Synchronous Vibrations. It was shown that the vibrations are caused by an aerodynamic disturbance which propagates around the circumference at approximately 41% rotor speed in the rotor frame of reference and locks in with vibration in a blade eigenmode with significant torsional component. The disturbance was characterized as an unsteady flow structure with high radial vorticity component, which originates from the tip leakage flow and separation on the blade suction side. While the suction side separation causes predominantly local unsteadiness, the disturbance emerging from the tip gap propagates around the circumference.

It was shown that the circumferential propagation of the disturbance is caused by a convection of the vortical structure across the passage and its transfer through the tip clearance. The resulting propagation speed and strength of the pressure drop associated with the disturbance are highly dependent on local tip clearance. A larger clearance results in a faster and stronger disturbance. The changes were significant: a $±$0.1 mm clearance variation causes speed changes of $±$3% rotor speed and strength changes of $±$20% inlet dynamic head. Through modification of the propagation speed, the tip clearance nonuniformity results in excitation of multiple structural nodal diameters, which was also detected in experiments. To the authors' knowledge, this is the first time that this relationship has been explicitly demonstrated.

The knowledge gained regarding the influence of tip clearance size on stability has direct implications on the prediction and prevention of NSV in fans and compressors. The correct prediction of propagation speed has been a huge challenge in the prediction of NSV, see, for example [27]. The present results demonstrate that an accurate modeling of tip clearances is crucial to correctly capturing the propagation speed, which ultimately defines the unstable vibration mode. The results also show that the propagation speed of the disturbance can be approximated from the time-averaged/steady swirl velocity. This provides a computationally efficient means of assessing the risk of NSV in fan or compressor design.

The numerical and analytical work presented in this paper was financed through Project Atlantis of the GUIde-7 consortium under coordination of Kenneth Hall at Duke University. The results presented in this paper rely on the contributions of a large research group and the authors gratefully acknowledge the excellent collaboration and support over the past 5 years. The authors particularly thank Xavier Ottavy, Benoit Paoletti, Anne-Lise Fiquet, Alexandra Schneider, Stephane Aubert, Pavel Teboul, Cedric Desbois, Sebastien Goguey, Gilbert Halter, Lionel Pierrard, Laurent Pouilloux, Edouard Salze, Nathalie Grosjean of LMFA, Kevin Billon and Claude Gibert of LTDS for their support and contributions to the experiments.

We are grateful for the continuous collaboration and financial support of SAFRAN Aircraft Engines since the beginning of project CATANA, for which the test module MARLYSA was provided by SAFRAN. Specifically we thank Renaud Daon, Michael Schvallinger and Clara Marty for technical discussions on the presented results. The authors are also grateful to Rolls-Royce plc for permitting the use of its aeroelastic solver AU3D for the work. In addition they would like to thank Jeff Green, Bharat Lad and Mike Meyer of Rolls-Royce for their support and input in technical discussions and Roger Zoepke-Sonntag for support with AU3D and the development of data analysis tools.

The experimental research was financed through the European Union's Clean Sky 2 Joint Undertaking (JU) under Grant Agreement Nos. N864719 and CATANA. The JU receives support from the European Union's Horizon 2020 research and innovation program and the Clean Sky 2 JU members other than the Union. The paper reflects only the author's view and the JU is not responsible for any use that may be made of the information it contains.

Assessment of the test facility was enabled through financial supports of Agence Nationale dela Recherche (ANR, Project d'EquipEx PHARE) and Conseil pour la Recherche Aeronautique Civile (CORAC—Programme CUMIN). Buildings and infrastructure were supported by ECL, instrumentation supported by Institut Carnot (INGENIERIE@LYON—Project MERIT) and SAFRAN Aircraft Engines.

• Cleansky (Award No. N864719; Funder ID: 10.13039/501100000922).

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

EO =

engine order

FEM =

finite element method

LE, TE =

leading edge, trailing edge

OGV =

outlet guide vanes

NSV =

non-synchronous vibration

(U)RANS =

(unsteady) Reynolds-averaged Navier-Stokes

 $Nv$ =

nodal diameter

 $Na$ =

aerodynamic wave number

 $ps$ =

static pressure (bar)

 $pdyn$ =

dynamic pressure (bar)

 $pt$ =

total pressure (bar)

 $Tt$ =

total temperature (K)

 $Tpass$ =

time for aero. disturbance to propagate one pitch (s)

 $ϵ$ =

strain $(μ/m)$

 $Πt$ =

total pressure ratio

 $m˙std$ =

standard mass-flow rate (kg/s)

 $ω$ =

angular frequency (rad/s)

 $Ω$ =

angular velocity (rad/s)

 $· S$ =

stationary frame of reference

 $· R$ =

rotating frame of reference

 $· a$ =

aerodynamic disturbance

 $· v$ =

structural vibration

 $· r$ =

rotor

 $· in$ =

average value at stage input

 $· ¯$ =

temporal average

 $· ̂$ =

temporal maximum