Rotating coherent flow structures as a source for narrowband tip clearance noise from axial fans

Journal of Sound and Vibration 417 (2018) 198–215

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Rotating coherent flow structures as a source for narrowband tip clearance noise from axial fans Tao Zhu a , Dominic Lallier-Daniels b,* , Marlène Sanjosé b , Stéphane Moreau b , Thomas Carolus a a b

Universität Siegen, 57068, Siegen, Germany Dept. Génie Mécanique Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1, Canada

article info

abstract

Article history: Received 21 July 2017 Revised 13 October 2017 Accepted 9 November 2017 Available online 27 December 2017

Noise from axial fans typically increases significantly as the tip clearance is increased. In addition to the broadband tip clearance noise at the design flow rate, narrowband humps also associated with the tip flow are observed in the far-field acoustic spectra at lower flow rate. In this study, both experimental and numerical methods are used to shed more light on the noise generation mechanism of this narrowband tip clearance noise and provide a unified description of this source. Unsteady aeroacoustic predictions with the Lattice-Boltzmann Method (LBM) are successfully compared with experiment. Such a validation allows using LBM data to conduct a detailed modal analysis of the pressure field for detecting rotating coherent flow structures which might be considered as noise sources. As previously found in ring fans the narrowband humps in the far-field noise spectra are found to be related to the tip clearance noise that is generated by an interaction of coherent flow structures present in the tip region with the leading edge of the impeller blades. The visualization of the coherent structures shows that they are indeed part of the unsteady tip clearance vortex structures. They are hidden in a complex, spatially and temporally inhomogeneous flow field, but can be recovered by means of appropriate filtering techniques. Their pressure trace corresponds to the so-called rotational instability identified in previous turbomachinery studies, which brings a unified picture of this tip-noise phenomenon for the first time. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Aeroacoustics Axial fans Tip gap flow Lattice-Boltzmann method Modal analysis

1. Introduction The aerodynamic and aeroacoustic performance of axial fans are strongly influenced by the tip clearance flow [1–6]. Typically, aerodynamic losses and sound radiation increase significantly with larger tip clearances for a given design. For instance, recent studies based on an axial fan with a classical tip clearance (experimental and numerical investigations by Zhu and Carolus [7,8]) have shown that at the design point, the strong tip clearance vortices induced by a large tip gap (s/Da = 1.0% with s the tip-gap height and Da the duct diameter) interact with the fan blade surfaces, which generally produces broadband tip clearance noise. At lower flow rates, narrowband humps associated with tip clearance noise are also observed in the acoustic spectra measured in the far field. Similar issues regarding the tip clearance flow of an axial fan with a large tip gap were also investigated by Kameier and Neise [9], März [10] and more recently Pardowitz et al. [11,12], who suggested that this kind of narrowband hump in the acoustic spectra could be attributed to a so-called “Rotating Instability” (RI) developing in the blade tip region. Similar observations were reported in the case of an axial ring fan, even at design condition: first on the fan-alone flush-mounted test

* Corresponding author. E-mail address: [email protected] (D. Lallier-Daniels). https://doi.org/10.1016/j.jsv.2017.11.014 0022-460X/© 2017 Elsevier Ltd. All rights reserved.

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Fig. 1. Tested configuration: (a) manufactured impeller; (b) fan installation in duct (dimensions in mm).

rig by Magne et al. [13] and Moreau and Sanjosé [14], and later on a complete engine cooling module by Piellard et al. [15] and Lallier-Daniels et al. [16,17]. All these configurations involved highly loaded fans at the tip with significant tip flow features. The noise mechanism was explained as the interaction between coherent structures forming in the tip region ahead of the fan and originating from the backflow from the ring gap with the fan blades. The object of the present work is to shed more light on the noise generation mechanism for this narrowband tip clearance noise using both experimental and numerical methods. Employing unsteady simulations based on the Lattice-BoltzmannMethod, a comprehensive modal analysis can be carried out to detect rotating coherent flow structures induced from the complex and unsteady tip clearance flow, which are construed as the mechanism for the identified narrowband humps. Preliminary results have been shown by Zhu et al. [18]. 2. Experimental investigation 2.1. Investigated fan setup An axial fan impeller, shown in Fig. 1(a), was designed with an in-house blade element momentum based design code for low pressure axial fans (dAX-LP [19]). Instead of having a free vortex design, the blade loading is 70% at hub and 120% at tip, distributed approximately linearly in the spanwise direction. The additional loading of the blade tip is done intentionally to provoke strong secondary tip flow that is eventually responsible for tip clearance noise. Further design parameters are compiled in Table 1. The corresponding axial Mach number Mx ≡ Vx ∕c0 (Vx mean axial speed and c0 speed of sound) in the duct is 0.01. Its effect on acoustic mode propagation is therefore negligible. Two impellers with different diameters were manufactured, providing a variation of tip clearance; one with a large tip clearance ratio s∕Da = 1.0% (i.e. a clearance of 3 mm) and one with an extremely small gap of s∕Da = 0.1% (i.e. a clearance of 0.3 mm). As shown in Fig. 1(b) thin supporting struts were mounted one duct diameter downstream of the rotor, so that rotor/strut interaction was minimized. 2.2. Overall aerodynamic and aeroacoustic measurements The aerodynamic fan performance was determined on a standard plenum test rig for fans according to the ISO 5801 standard (German DIN 24163). Given that Δpts is the total-to-static pressure rise, V̇ ≡ Vx Sa the volumic flow rate (Sa ≡ 𝜋 D2a ∕4 the duct interior surface) and M the true rotor torque applied, the following non-dimensional fan performance coefficients are used: The flow-rate coefficient

𝜙=

V̇

(1)

𝜋2 3 Da n 4

the total-to-static pressure-rise coefficient

𝜓ts =

𝜋2 2

Δpts

(2)

𝜌D2a n2

Table 1 Important impeller design parameters.

Duct Diameter Hub to Tip Ratio Number of Blades

Da

𝜈 z

[mm] [–] [–]

300 0.45 5

Design Flow Rate Density of Air Rotational Speed

𝜙 𝜌 n

[–] [kg/m3 ] [rpm]

0.195 1.2 3000

200

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Fig. 2. Duct test rig with semi-anechoic chamber (top view, dimensions in mm); (a) fan assembly (center line 1350 mm above reflecting ground), (b) star flow straightener, (c) duct, (d) in-duct microphone, (e) hot-film mass-flow meter, (f) anechoic termination, (g) adjustable throttle, (h) far-field microphones, (i) semi-anechoic chamber, (j) air-inlet grid on ground.

the shaft-power coefficient

𝜆=

2𝜋 nM

𝜋4 8

𝜌D5a n3

(3)

and the total-to-static efficiency

𝜂ts =

V̇ Δpts 2𝜋 nM

(4)

The aeroacoustic investigations were carried out on a standardized acoustic duct test rig for fans according to ISO 5136 standard, as shown in Fig. 2. The impeller takes air from a large semi-anechoic room (4.50 m by 3.50 m by 3.23 m) and exhausts into a circular duct with an anechoic termination. The frequencies of the corresponding first duct modes are 664 and 1102 Hz. The flow rate is controlled by a throttling plate downstream of the termination and is determined by a calibrated hot-film probe in the duct. The far-field noise is measured using three microphones (Brüel & Kjaer, type 4190) placed on a virtual circular arc around the inlet. The suction side sound pressure level Lp5 is obtained by averaging the signals over the three microphones. All time signals were captured with a sampling frequency fs−Exp = 25.6 kHz. All presented spectral analysis rely on the power spectral density (PSD) obtained by the function pwelch (based on Welch’s modified periodogram method [20]) in MATLAB R2012a. The parameters chosen for the periodogram analysis were the use of a Hanning window function with 50 non-overlapping samples taken over a 10 s experimental recording time. This was done to obtain a ΔfExp = 5 Hz spectral resolution in the acoustic spectra to allow for a fair comparison with the LBM results which only have relatively short time signals. All overall levels are the sum of narrow band levels from 100 Hz to 3 kHz (limited by the maximum frequency fmax resolved in the LBM results). Similarly, cross power spectral density (CPSD) and magnitude-squared coherence between two synchronized signals were evaluated using the cpsd and mscohere functions from MATLAB R2012a, which also make use of Welch’s averaged periodogram method. The windowing parameters were set to be the same as with the power spectral density evaluations to maintain the same frequency resolution. 2.3. Wall-pressure measurements In addition to the acoustic measurements, two adjacent blades of the impeller were each instrumented with eight flushmounted miniature pressure transducers to measure wall-pressure fluctuations. The transducers employed are miniature condenser microphones (Knowles Acoustics type FG-3329-P07) with a sensitive diameter of 0.7 mm fitted with a protective mesh. The protective mesh was flush-mounted with the blade surface. One set of sensors is placed on the suction surface (denoted Sxx ) of one blade while the other series is situated on the pressure side (Pxx ) of the preceeding blade as shown in Fig. 3. Note that all instrumented transducers were located in the same blade passage. It was verified that the instrumentation of the transducers had no effect on the aerodynamic and acoustic performance [21]. A slip-ring transducer transferred the signals from the rotating fan to the stationary laboratory system. The random noise from the slip-ring transducer was found to be negligible. The signals from the transducers in the tip region (S01 to S13 and P01 to P11) as well as the three far-field and in-duct microphones were recorded synchronously to achieve some correlation analysis. All the miniature pressure transducers were calibrated in-situ with a white noise excitation signal ranging from 100 Hz to 10 kHz produced by small in-hear headphones, as shown in Fig. 3 (b).

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Fig. 3. Transducer set-up for the fan blade wall-pressure measurements (a) sketch of the miniature pressure transducer positions on the blade suction surface; (b) arrangement for the calibration; (c) the instrumented blades; (d) transducer at the tip of the blade.

3. Numerical investigation 3.1. Numerical method Lattice-Boltzmann simulations were chosen as a tool to simulate the unsteady flow phenomena linked to the tip gap flow and their sound radiation in the far field. The lattice-Boltzmann method, unlike more conventional CFD methods, is based on the mesoscopic kinetic equations (Boltzmann equation) to solve the macroscopic quantities of fluid flow [22,23], i.e. it attempts to solve the advection equation of the probability density function of fluid particles with a collision source term. Solving the continuous Boltzmann equation, however, is no less daunting than solving the Navier-Stokes equations used by mainstream CFD codes directly. The Boltzmann equation is thus discretized over a finite lattice mesh used with a discrete number of particle velocity paths connecting adjoining lattice blocks [24–26]. The macroscopic flow variables are recovered through the appropriate moments of the lattice particle distributions. In this study the commercial code PowerFlow version 5.0c is used in its Very Large Eddy Simulation (VLES) mode. In order to model the effects of unresolved small scale turbulent fluctuations, the collision source term [27] in the Lattice-Boltzmann equation is extended by replacing its molecular relaxation time scale with an effective turbulent relaxation time scale derived from a systematic Renormalization Group (RNG) procedure [24,25,28,29]. A turbulent wall-model, including the effect of pressure gradients, is also integrated into the solver to prevent having to solve the whole boundary layer. This method has been validated and used for several aeroacoustic problems [14–17,30–33]. The LBM scheme is solved on a grid composed of cubic volumetric elements, also called voxels, using a Voxel Refinement (VR) strategy, where the grid size changes by a factor of two for adjacent resolution regions [26]. For the simulations of flows in domains consisting of rotating and stationary regions, the computational domain is divided into a Local Reference Frame (LRF) moving with the fan and a stationary reference frame connected by an interface [34].

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Fig. 4. Simulation domain (a) including the semi-anechoic room and test-rig duct (left); (b) mesh details in the fan section with LRF-domain.

3.2. Numerical setup The fan set-up presented in Section 2.1 is simulated with the computational domain mimicking the complete full scale acoustic test rig shown in Fig. 2. For simplicity, all inner walls of the semi-anechoic room (i.e. even the floor) are defined as highly absorptive porous media ‘sponge’ zones that serves to absorb acoustic waves and prevent reflections back to the source. The air inlet and outlet boundary conditions also include damping zones to avoid acoustic reflections. All other surfaces (nozzle, duct, fan impeller and fan hub) are specified as perfectly rigid walls. The mass flow rate according to the required fan operating point is specified at the inlet. Free exhaust at ambient atmospheric pressure is assumed at the outlet. The simulation is performed for the fan design point and at a part load operating condition, i.e. flow rate coefficient of 𝜙 = 0.195 and 𝜙 = 0.165 respectively. The grid used has 10 VR regions with a minimum voxel size of 0.5 mm (see Fig. 4). The finest VR includes the rotating domain of the fan. The large tip clearance is thus resolved with 6 voxels. The LRF-domain around the rotating fan impeller is discretized using the finest grid resolution of Δx = 0.5 mm (VR10) corresponding to 600 voxels across the duct diameter. It is shown that the aero-acoustic effect of large tip clearance can still be captured even with this relatively coarse resolution. For a similar lowspeed fan and at similar flow conditions, Moreau and Sanjose found grid independence on the sound power level at this grid refinement [14]. The small tip clearance is replaced by s∕Da = 0.0%, i.e. the tip clearance is completely neglected. The agreement of LBM predictions with the experiment demonstrates that this is an acceptable simplification. The regions around the microphone probes are spatially resolved with Δxmic = 8.0 mm (VR6). Assuming that the required number of points per wavelength to accurately capture acoustic waves is Nppw = 16 [35], the maximum frequency fmax that this grid can resolve should be at least fmax ≈

c0 Nppw Δxmic

(5)

For the configurations with s∕Da = 0.0% at both operating points and with s∕Da = 1.0% at 𝜙 = 0.165, the LBM-results are evaluated using the data captured over an overall time of at least Tsim ≈ 1 s corresponding to approximately 50 impeller revolutions upon arriving at a statistically stable fan operating point, i.e. with settled flow rate and pressure rise. The simulation time step is Δt = 8.2 × 10−7 s in VR10. The data at the monitoring points in the acoustic far field, i.e. at the microphone probe positions, are captured with a sampling frequency fs−Probe = 75822 Hz; the static pressure fluctuations on the blade surface and in an adjacent annular volume are sampled with fs−SurVol = 9478 Hz. All spectra are evaluated from the raw LBM data using the same method described in last section with the appropriate window length yielding the same frequency resolution of ΔfLBM = 5 Hz. The fan overall pressure rise is evaluated using planes approximatively one duct diameter upstream and downstream from the fan impeller, area-averaged in each plane and time-averaged during ten revolutions. A preliminary study showed it was a proper approach to yield the converged aerodynamic performances [8]. The configuration with s∕Da = 1.0% at 𝜙 = 0.165 is simulated for a longer physical time of Tsim ≈ 5 s to get sufficient time resolution for performing the correlation analysis. The pressure probes on the blade surfaces are set at the exact same positions as in the experiment, as shown in Fig. 5. In order to detect the rotating coherent flow structures, a number of probes are placed in the blade passages around the full 360◦ of the fan circumference, spanning from very close to the blade tip (R = 146 mm) down to the middle of the blade (R = 110 mm); the resulting tangential resolution is Δ𝜃 = 12◦ . In the chordwise direction a series of 10 probes are uniformly distributed from the leading edge (C01) to the trailing edge (C10) along the chord line at the blade tip. Key parameters of the probe setup in the LBM simulation are given in Table 2.

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Fig. 5. Probe locations in the LBM simulation (a) on the blade surfaces; (b) in the blade passage.

4. Validation results 4.1. Overall aeroacoustic performances The aerodynamic overall performance characteristics in terms of 𝜓ts , 𝜆 and 𝜂ts as a function of 𝜙 are shown in Fig. 6(a). The effect of the tip clearance on the aerodynamic characteristics is rather obvious: pressure rise and efficiency drop and the onset of stall moves to higher flow rates as the tip clearance is increased. The LBM predictions at both design and part-load operating conditions are rather satisfactory. The acoustic performance characteristics in terms of overall sound pressure level Lp5 (integration between 100 and 3000 Hz) in the far field as a function of 𝜙 are shown in Fig. 6(b). The sudden increase in noise levels correlates with the drop of aerodynamic performances and the onset of stall. Note this is more abrupt around 𝜙 ≈ 0.12 with the smallest tip clearance. The sound pressure as predicted from LBM reproduces the effect of the tip clearance quite well: both LBM and experiment show about a 10 dB increase at the design point (𝜙 = 0.195) and a 15 dB at a lower flow rate (𝜙 = 0.165) because of the large clearance. 4.2. Far-field sound spectra The far-field sound pressure level LP5 at the low flow rate (𝜙 = 0.165) is shown in Fig. 7. Both LBM-predictions and measurement show that the broadband sound is enhanced by increasing the tip clearance. Looking at the broadband component, the agreement between experimental and LBM predicted spectra is satisfactory and similar to what was achieved for a similar configuration [14]. Yet in the experiment, the peaks at the blade passing frequency (BPF) and its harmonics are very distinctly above the broadband level for the small tip clearance which cannot be captured by the present short LBM simulation. Sturm et al. [36] have recently traced these tones in this very axisymmetric configuration to a large-scale inflow distortion observed by oil vapor visualizations. They then numerically showed that this large-scale vortex at the duct inlet was formed by the slow-settling flow in the room. They had to run a coarse LBM simulation over 100 s in order to form this inlet vortex, which cannot be captured by the present relatively short LBM simulation. In the case of the large tip clearance, the narrowband humps are generally well predicted by the LBM simulation. In the experimental spectra the center frequencies of the narrowband humps are at approximately 185, 370, 555 and 730 Hz, which corresponds to frequencies equivalent to 0.74 × BPF, 0.74 × 2 BPF, 0.74 × 3 BPF and 0.74 × 4 BPF respectively. Note that these subharmonic humps are either below (for the first three ones) or in between duct modes (for the last one) and cannot be attributed to some duct resonance. In the LBM predicted spectrum, the first three humps can be observed, the center frequencies Table 2 Distribution of numerical probes in the blade passage.

Direction

Start

End

Spanwise (Radial) Chordwise Tangentially

R = 146 mm (R146) R = 110 mm (R110) Leading Edge (LE, C01) Trailing Edge (TE, C10) 5 rows of probes (C01 − C10)/blade passage

Distribution

ΔR = 4 mm Uniform from LE to TE

Δ𝜃 = 12◦

204

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Fig. 6. Overall performance of the studied impeller (a) aerodynamic characteristics; (b) acoustic characteristics. Dashed (Full): Experiment - s/Da = 0.1% (1.0%); Black dot (cross): LBM - s/Da = 0.0% (1.0%).

Fig. 7. Sound pressure levels at low flow rate, ΔfExp = ΔfLBM = 5 Hz. Dashed black (Full black): Experiment - s/Da = 0.1% (1.0%); Dashed grey (Full grey): LBM - s/Da = 0.0% (1.0%).

of which are slightly shifted to 175, 350, and 525 Hz, which corresponds to frequencies equivalent to 0.70 × BPF, 0.70 × 2 BPF and 0.70 × 3 BPF respectively. The fourth harmonic is not clearly seen in the LBM simulation either. Both discrepancies could be traced to the fact that the tip gap might differ between the experiment and the simulation during operation, as the numerical model does not account for the structural deformation caused by the centrifugal load (fully rigid model) [14]. 4.3. Wall-pressure on the fan blades Another possible comparison between the LBM simulations and experiment consists in wall-pressure spectra as recorded using the flush mounted sensors on the rotor blades (see Section 2.2). The analysis of the surface pressure from a pair of probes on the blade surface, namely S11 on the suction side and P11 on the pressure side, located near the leading edge at the tip of the

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blade (see Fig. 3) leads to some insight into the possible noise mechanism associated with the tip gap flow. The data captured from the experiments and LBM simulations are shown in Fig. 8, in terms of wall-pressure level at each probe (top), the coherence between S11∕P11 (middle) and the unwrapped phase (bottom). Similar shapes with side-by-side peaks of the PSD of the wallpressure fluctuations are obtained with higher levels shifted to lower frequencies in the LBM simulation, most likely caused by the lack of grid resolution in the tip region. Both experimental and LBM results also show, in a certain frequency range, high coherence levels and linear phase behavior. The latter suggests a positive convection velocity from one blade suction side to the pressure side of the next in one blade passage (see Eq. (1) and Fig. 13 in Moreau et al. [37] for instance). The former indicates the most likely event and the low frequency large coherent flow structures. Although there is a certain deviation between the experiment and LBM results in terms of the pressure level and the frequency range with high coherence levels, the frequency interval between the side-by-side peaks from both shows a value of approximately 35 Hz. According to the theory introduced by Kameier and Neise [9], this interval is indicative of the so-called Rotating Instability (RI), which rotates at about 70% of the impeller rotational speed (35 Hz/50 Hz) in the opposite direction of the impeller observed in the rotational frame, or 30% of the impeller rotational speed in the same rotating direction as the impeller observed in the stationary frame. The interaction of this RI with the impeller blades might be the source of the narrowband humps observed in the far-field acoustic spectra.

Fig. 8. Correlation analysis: (a) Identification of probes S11 and P11; (b) Surface pressure level (top), coherence (middle) and phase from probe S11 and P11 on the blade surface, comparison between Exp. and LBM, ΔfExp = ΔfLBM = 5 Hz.

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Yet, to the authors’ knowledge, the physical interpretation of RI is still missing and not fully understood. As recently pointed out by Magne et al. [13] and Moreau and Sanjosé [14] on a ring fan, it might be seen as rotating coherent flow structures formed from the tip clearance vortices that interact with the blades and trigger a circumferential pressure mode or a superposition of several circumferential pressure modes, which can be detected and characterized through a modal analysis method introduced in Section 5.1. 5. Detailed analysis 5.1. Modal analysis for detection of coherent flow structures Working under the assumption that coherent structures generated from the tip gap flow and their interaction with the blades are responsible for the narrowband humps observed in the acoustic spectra in Fig. 7, the difficulty thus becomes to properly identify those structures and model their interaction with the blades. To accomplish this, a coherent structure analysis method previously used on centrifugal [38–42] and axial fans [43] was considered as a basis. The method assumes that structures identified with it can be viewed as circumferential wave patterns superimposed on the base flow, rotating at a certain velocity potentially very different from the base rotor speed. These flow patterns, causing periodic disruptions in both pressure and velocity, through their interaction with the fan blades, would cause periodic fluctuations in blade loading and thus produce tones in the far field; adding the contribution of the different modes together would generate the observed narrowband humps in the spectra. These wave patterns can be quantified through two parameters: their mode order m and rotational speed of the mode nmod . From these characteristics, their interaction frequency with the rotor blades can be calculated. It should be noted that the term mode as used in this work refers to the coherent flow structures detected using the current modal analysis method in the aerodynamic near field around the fan impeller; it should not be mixed up with the term mode which is also used to define acoustic modes propagating in a duct. To calculate the properties of the wave patterns, the modal method relies on the use of synchronous measurements of pressure at two or more points along the circumference at a given radial position; the pressure is chosen as the source data because it encompasses all disturbances in the flow, but it should be possible to perform the analysis using velocity signals. Using time signals of two sequential measurement positions, the CPSD Sxy can be computed as well as the coherence Cxy between the two signals. Furthermore, the phase lag 𝛾xy between the two signals can be calculated as

𝛾xy = arctan

Im(Sxy ) Re(Sxy )

(6)

The angular spacing between two measurement points, 𝜃xy being known, the method proposes the estimation of the mode order of the flow structures as m(f ) =

𝛾xy 𝜃xy

(7)

The maximum mode order that can be detected depends on the number of probes around the circumference. Note here that the coherence function is used as a filter to evaluate the phase only for the frequencies for which the coherence is high enough. The resulting phase signal must also be beforehand corrected for multiples of 2𝜋 radians (unwrapped phase). The rotational speed of the mode in the absolute frame of reference can be calculated using stationary measurement points as nmod =

2𝜋 f m

(8)

and in the case of the probes rotating with the fan impeller, as nmod = n −

2𝜋 f m

(9)

where n is the rotational speed of the fan impeller. Using the mode characteristics and knowing the number of blades in the fan impeller design, the interaction frequency fint between the rotor and the identified modes can then be calculated using the following equation: fint = Nint (m, z) × (n − nmod )

(10)

where Nint (m, z) represents the number of independent interactions between a given mode m and the z blades per relative rotation and is determined as the least common multiple (LCM) of m and z. This factor takes into account the fact that the interaction of a mode with a number of lobes that is a multiple of the number of fan blades (or inversely) will produce a tone at an order multiple of the modified (1 − nmod ∕n) BPF fundamental, with several blades contributing at the same time, whereby the blade passing frequency (BPF) is defined as BPF = n × z. The original method developed at Penn State University [38–41] as revisited by Wolfram and Carolus [42] and Sturm et al. [43], however, does not specifically address the fact that a negative phase would indicate the presence of a mode rotating in a direction opposite to the direction of evaluation. In that case, the mode order will be negative and have an associated negative velocity. These modes should not be discarded as they could have a physical explanation. Furthermore, while the

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method allows for the detection and identification of coherent structures as well as their possible interaction frequency from an acoustic standpoint, it only provides an “event counter” approach to the identification i.e. it does not provide any information on the amplitude of the interaction for a given event. This was also noted previously by Wolfram and Carolus [42]. A qualitative comparison between the actual acoustics and structure interaction frequency histograms were later presented by Sturm et al. [43]. It should be noted here that a parallel method using a spatial Fourier method attempting to quantify the RI phenomena was also investigated by Pardowitz et al. [11,12]. 5.2. Coherence analysis using impeller blade sensor positions First of all, in order to demonstrate the modal analysis method, an example is carried out using wall-pressure measurements from the transducers on the impeller blades. The coherence is computed for frequencies ranging from 300 to 900 Hz for the experimental data at the S11∕P11 probe pair. This is done with prior knowledge about the coherent structures already captured with the probes on the blade surfaces (Fig. 8). Using a threshold value of Cxy ≥ 0.3 to identify coherent events, the chosen events show a very good linear trend when looking at the phase, as shown in Fig. 9.

Fig. 9. Correlation analysis of measured signals: (a) Identification of probes S11 and P11; (b) top: coherence of signals from blade pressure probe pair S11∕P11 and microphone in the far field P11/MIC02; middle and bottom: detected mode order and unwrapped phase.

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The mode order m and the rotational speed of the modes (observed in the stationary frame) can be calculated and are shown in Fig. 10. It is found that the detected mode orders spread from m = 9 to m = 22 with a certain distribution. The linear regression of the rotational speed of the modes shows that the modes gather around approximately 26% of the rotational speed of the impeller (nmod ∕n = 0.26), which corresponds to the coherence analysis from the surface pressure on the blades, although variations in a certain range can be also observed. The interaction frequency fint from the detected rotating coherent flow structures and the impeller blades is also shown as an aggregate bar plot at the bottom of Fig. 10. The interaction frequencies indicated by the highest number of events correspond quite well with the narrowband humps in the sound pressure spectrum measured in the far field. Similar results have been obtained for the other probe pairs. The normalized rotational speed of the detected modes nmod ∕n spreads in a small range between 0.25 and 0.3, which causes the hump distribution of the interaction frequencies instead of sharp peaks. Such a statistical distribution of the rotational speed was also evidenced by Piellard et al. [15] in more complex engine cooling modules. Note that, as the interaction number Nint (m, z) defined in Eq. (10) is obtained by the LCM of both the mode order m and blade number z, only the mode orders m < 5 as well as m = 5, 10, 15 and 20 (multiples of the blade number z) should contribute to the interaction frequency fint at the center frequencies of the three narrowband humps at 0.74 × BPF, 0.74 × 2 BPF, 0.74 × 3 BPF and 0.74 × 4 BPF respectively, whereby the other mode orders yield the high frequency interactions. This is highlighted in Fig. 9 by the coherence analysis between the wall-pressure sensor P11 and the far-field microphone MIC02, which shows high coherence between the two signals at frequencies corresponding to mode orders m = 10, 15 and 20. This corresponds very well with the interaction frequency plot in Fig. 10, which shows inference from the identified modes at the 0.74 × 2 BPF, 0.74 × 3 BPF and 0.74 × 4 BPF frequencies, with the first subharmonic hump not being predicted. Moreover, some interaction noise might

Fig. 10. Further results from the correlation analysis of the measured pressure fluctuations (S11∕P11); (a) left: histogram of number of events corresponding to detected mode order; right: rotating speed of detected modes; (b) histogram of number of events corresponding to calculated interaction frequency in comparison with the measured far-field sound power spectrum (in grey).

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Fig. 11. Correlation analysis of simulated signals (S11∕P11); (a) Identification of probes S11 and P11; (b) top: coherence of signals from blade pressure probe pair S11∕P11 and microphone in the far field P11/MIC02; middle and bottom: detected mode order and unwrapped phase.

not be detected in the far field because the corresponding acoustic modes are cut-off. A similar analysis can be carried out using the same probes, but this time using the LBM-data for the 100–700 Hz range, which was shown to be the frequency range associated with the numerical RI pressure trace in Fig. 8. In a similar fashion as before in Fig. 11, the coherence analysis is shown from the same S11∕P11 and P11/MIC02 probe pairs. Using the same threshold value of Cxy ≥ 0.3 to identify the coherent modes, a very linear trend regarding the phase lag between the S11∕P11 signals is obtained as in the experimental case. Noticeably the high coherence peak in the P11/MIC02 corresponding to the m = 20 mode which could be associated with the fourth subharmonic hump has disappeared from the spectra; this is consistent with the acoustic result in Fig. 7 where the fourth hump was not present in the LBM simulation spectra, contrarily to the experimental case. The detected mode characteristics are also presented in Fig. 12. For the mode order bar plot, the range for the LBM simulation is seen to be equivalent to the one observed using the experimental data, with the exception of a lack of higher (m = 15 − 20) order modes. The modal velocity is seen to differ slightly from the experimental case, with the average revolving around 30% of the impeller speed. Finally the histogram of the interaction frequency shows that the fourth subharmonic hump that is not present in the simulated spectra is not predicted by the modal method either using the wall-pressure data from the blades. The frequency of the humps is also shifted compared with the experiment as observed previously in Section 4.2, but this corresponds to the observed difference in the modal velocity.

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Fig. 12. Further results from the correlation analysis of the simulated pressure fluctuations (S11∕P11); (a) left: histogram of number of events corresponding to detected mode order; right: rotating speed of detected modes; (b) histogram of number of events corresponding to calculated interaction frequency in comparison with the far-field measured sound power spectrum (in grey).

It should be noted that the angular spacing of the probe pairs on the blade surface, 𝜃xy , is always limited by accessibility and probe size; it is then relatively large in the experiment, i.e. 𝜃xy = 72◦ . Consequently the mode orders that can be detected unambiguously are limited, i.e. theoretically only up to m = 360◦ ∕𝜃xy = 5, whereas modes up to 22 are identified here. Conversely, a virtually unlimited spatial resolution can be obtained by applying the correlation analysis to data from the high resolution unsteady flow field obtained by the LBM simulation. As presented in Fig. 7, the narrowband humps are predicted quite well by the LBM, and even the humps in the pressure fluctuations are captured on the rotating blade surfaces as shown in Fig. 8. The modal analysis is then applied to the LBM volume data, which can detect the higher mode orders for a more extensive spatial range as explained in Section 5.3. 5.3. Coherence analysis using simulation volume data As outlined at the end of Section 5.2, carrying out a modal analysis using a single pair of probes presents a rather localized outlook on the modal patterns in the flow and has potential setbacks regarding the order of the modes that can be detected. To get a statistically significant analysis as well as a more discretized analysis in terms of azimuthal angle, the method has been applied to the fluid probes from the simulation volume shown in Fig. 5(b) around the full annulus for all available spanwise (R110 to R146) and chordwise (C01 to C10, from leading edge to trailing edge) positions. It thus provides a complete overview of the coherent flow structures and their interactions with the blades in the tip region of the impeller. However, the modal analysis shows that results under the spanwise position of R130 mm have such a weak coherence that few coherent structures can be detected by the method in the flow at these positions. The results for the remaining 50 positions are coalesced to provide a comprehensive analysis of the distribution of the coherent structures detected using the modal method, and the contribution of each spatial position is marked by the color of the monitoring points in Fig. 13. Each

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Fig. 13. Color map for the chord- and spanwise distribution of evaluation locations for the full blade passage modal analysis. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 14. Spatial distribution of the characteristics of the coherent flow structures detected in varying chordwise and spanwise positions in the blade passage using rotational frame data from the LBM results.

radius corresponds to a single color and goes from light to dark from the leading edge to the trailing edge respectively. The characteristics of the modes detected from the analysis are presented in Fig. 14, with the abscissa quantifying the event frequency of occurrence of the detected coherent structures captured from different positions in terms of the mode order, rotational speed and interaction frequency with the impeller blades (potential interaction noise). Some interesting points are worth mentioning:

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• All three humps observed in the numerically predicted sound power spectrum can be reconstructed; not only the frequencies but also the relative shapes of these humps match well. Note that the absent fourth hump is not represented by the modal analysis and is also absent in the simulated sound spectra. • The most coherent flow structures are observed in the vicinity of the blade tip (R146-R142, dark blue) and concentrated in the leading edge region of the blades (C01-C05, brightest colors), which is another strong hint at the supposed noise mechanism, which is the interaction of the tip vortices with the blades.

Fig. 15. Tip clearance vortices visualized with iso-surfaces of 𝜆2 criterion with a value of −200 s−2 .

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The correlation analysis of the LBM flow field data clearly shows that the narrowband humps related to tip clearance noise are generated from an interaction of the impeller blades with coherent flow structures being present in the tip region and the leading edge region of the blades. The coherent flow structures can be understood as a mixture of circumferential pressure modes of different mode orders. Conversely, the pressure modes are the trace of these large vortices in the tip gap. The most relevant ones are multiples of the impeller blade number, here m = 5, 10, and 15 as well as possible inference from modes of order m = 1, 2 and 3, as the occurrence of sub-harmonic humps are seen up to the third BPF for the numerical results. 5.4. Flow structures and pressure waves In order to visualize the coherent structures detected in Section 5.3, only the flow field data in an annular region around the blade tips containing the upper half of the fan blade has been recorded synchronously with the probe data used in this section. The vortices in the tip clearance are illustrated in Fig. 15 with iso-surfaces of the 𝜆2 criterion [44] set to an iso-value of −200 s−2 . As already found by Moreau and Sanjosé [14] in a ring fan for similar tip Mach and Reynolds numbers and tip gap, the flow is strongly non uniform and turbulent with many vortices of different scales, which makes the experimental analysis with very few probes in the tip region difficult and unrealistic and explain the limitation noted in Section 5.2. In Fig. 15 (a), the vortical structures are colored by the rotational speed in the inertial frame of reference (nmod ∕n). The range of the color map is set to 0.1 < nmod ∕n < 0.5, which corresponds to the coherent flow structures identified in Fig. 14. Consequently the vortical structures that are not rotating within this rotational speed range are uniformly blue: they mostly correspond to vortices away from the blade tip. This is confirmed in Fig. 15 (b) where the vortices that are not in the rotational speed range 0.2 < nmod ∕n < 0.4 are filtered out. The remaining flow features are the slowly rotating coherent flow structures detected from the correlation analysis, which are concentrated in the blade tip region. The next step is to relate these selected vortical structures with the pressure modes identified by the modal analysis proposed in Section 5.1. In the latter, the most important mode orders m = 10 and m = 15 are found to yield the two dominant narrow-band humps in the computed sound power spectrum at around 0.70 × 2 BPF and 0.70 × 3 BPF (Fig. 12), and to correlate with wall-pressure fluctuations in the frequency range 340–360 Hz and 505–525 Hz respectively (Fig. 11). A threedimensional band-pass filtering of the pressure field is thus applied in these two frequency bands respectively, so that the vortices shown in Fig. 15 (b) can be further decomposed according to filtered pressure fluctuations. Only the structures exhibiting high pressure fluctuations (|p′ | > 10 Pa) are seen as blue and red patterns for the frequency band 340–360 Hz in Fig. 15 (c) and for the frequency band 505–525 Hz in Fig. 15 (d) respectively. Clearly two or three pressure nodes (red and blue patterns) are observed in each blade passage of the fan (with blade number z = 5) according to m = 10 and m = 15 respectively. They are the pressure trace of the coherent vortical structures identified by the 𝜆2 criterion that periodically interact with the blade tip (evidenced by an animation in time of the contours in Fig. 15 (c) and (d)). Note that the current limited grid resolution in the LBM simulation precludes seeing continuous vortices and rather show alternate patches of vortices. The RI identified by Kameier and Neise’s theory [9] can then be, in this case, related to the vortices generated by the tip gap flow. A final evidence of the interaction of the selected vortical structures of mode orders m = 10 and m = 15 with the fan blades is provided by contours of the wall-pressure fluctuations filtered in the frequency bands 340–360 Hz and 505–525 Hz shown in

Fig. 16. Contours of wall-pressure fluctuations power spectral density filtered by frequency range, evaluated over 10 rotor revolutions (a) 340–360 Hz corresponding to m = 10; (b) 505–525 Hz, corresponding to m = 15.

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Fig. 16. For both frequency ranges, the regions exposed to high pressure fluctuations clearly concentrate in the tip region of the blades and at their leading edge, consistently with the model analysis in Fig. 11. 6. Conclusion The present study has focused on the identification of the mechanisms responsible for the generation of narrowband tip clearance noise in the far field in low-speed axial fans. To that end, the full-scale anechoic wind tunnel at the University of Siegen has been simulated by LBM to study the flow features of a typical axial fan with two different tip clearances. The simulations are shown to accurately reproduce the global aerodynamic and aeroacoustic characteristics of the ducted fan for the selected operating points. Noticeably, the large increase of noise with tip clearance is well predicted: both the increase of broadbandnoise levels and the appearance of subharmonic narrowband humps are captured by the LBM simulations for the larger tip clearance. Some near-field aerodynamic features are also well represented which have been previously associated with the generation of narrowband tip clearance noise in axial fans and termed RI [9]. Assuming that coherent structures generated from the tip gap flow and their interaction with the blades are responsible for the narrowband humps observed in the acoustic spectra, a modal analysis based on previous works at Penn State University [38–41] and at the University of Siegen [42,43] has been carried out to identify the modal patterns generated and their interaction with the blades. The correlation analysis of the LBM flow field data clearly shows that the narrowband humps are related to tip clearance noise that is generated by an interaction of coherent flow structures being present in the tip region near the leading edge of the impeller blades. In this modal analysis, the coherent flow structures can be understood as a mixture of circumferential pressure modes of varying mode orders. For this particular flow condition close to design the most relevant ones are multiples of the impeller blade number as found previously by Pardowitz et al. [11,12]. They rotate with approximately 30% of the impeller speed, in the same direction as the impeller. The rotational speed of the different modes is slightly scattered which results in humps rather than sharp peaks in the acoustic far-field spectrum. The visualization of the identified structures shows that these identified modes are indeed the pressure trace of some unsteady tip clearance vortex structures [13–15,17]. The latter are hidden in a complex, spatially and temporally inhomogeneous flow field, but can be recovered by means of appropriate filtering techniques. In summary, a unified picture for subharmonic humps seen in far-field acoustic spectra of low-speed fans with a large tip gap can be proposed. Coherent vortical structures formed in the tip clearance interact with the fan blades, causing periodic fluctuations in the blade loading and thus inducing tonal noise (as well as broadband noise) in the far field. This mechanism is similar to tip-vortex interactions in propellers or contra-rotating open rotors (CRORs), and blade vortex interactions on helicopter rotors. Yet, as these vortices have a range of tangential velocities corresponding to the rotating modes (their pressure trace) identified in the modal analysis of the flow field and previously termed RI, broadband humps are observed as opposed to sharp tonal peaks. Acknowledgements EXA Corp. is gratefully acknowledged for providing PowerFlow licenses and support, and participating in fruitful discussions. The Centre for Research in Sustainable Aviation (CRSA) (Grant number: 414123-2012) is also acknowledged for providing support to D. Lallier-Daniels. 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