Effect of the rotor–stator gap variation on the tonal noise generated by axial-flow fans

Applied Acoustics 94 (2015) 29–38

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Effect of the rotor–stator gap variation on the tonal noise generated by axial-flow fans Edward Canepa, Andrea Cattanei ⇑, Fabio Mazzocut Zecchin DIME – Università di Genova, Genoa I-16145, Italy

a r t i c l e

i n f o

Article history: Received 12 July 2014 Received in revised form 15 January 2015 Accepted 19 January 2015 Available online 27 February 2015 Keywords: Fan noise Rotor–stator aerodynamic interaction Inlet turbulence Axial gap Propagation effects Phase delay

a b s t r a c t The effect of the rotor–stator axial-gap on tonal noise generated by an axial-flow fan employed for automotive cooling systems has been studied. A fan equipped with a 9 blade rotor and a 18 vane stator positioned at several axial-gaps has been tested in a hemi-anechoic chamber. The acoustic measurements have been performed during rotational speed ramps. To analyze the experimental data, the propagation function, obtained by means of the spectral decomposition, has been compared with the velocity-scaled SPL and the phase angle, evaluated at the 1st and 2nd blade passing frequency harmonics. Opposite to what is commonly observed, the SPL due to the rotor–stator aerodynamic interaction does not monotonically decrease with the axial-gap and at the shortest gaps it may not be scaled with a single power of the rotational speed. The listed quantities have been plotted versus frequency, rotational speed, and axial-gap. Their analysis provides a detailed picture of the investigated phenomena and supports the assumption that the observed behavior is due to acoustic effects and not to the aerodynamic noise generating mechanism. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Tonal noise is an important component of the noise generated by axial-flow fans employed in automotive cooling systems. It may be due to the ingestion of large scale inlet turbulent structures, Homicz and George [1], Majumdar and Peake [2], Hanson [3], or to the aerodynamic rotor–stator interaction Lowson [4], Tyler and Sofrin [5], Kaji and Okazaki [6]. Compared to broadband noise, it is often a major cause of annoyance, due to its periodic nature. Basing on established knowledge [4,6], the designer expects that increasing the axial-gap between rotor and stator allows to decrease the tonal noise and hence he seeks for a trade-off between module compactness and noise reduction. Other important aspects are the scaling properties, i.e. the possibility of relying on simple relations expressing the dependence of the SPL on the rotational speed, Neise and Barsikow [7]. Conversely, an unexpected dependence on both rotational speed, Quinlan and Krane [8], and axialgap, Canepa et al. [9], has been observed, resulting in difficulties in deciding how to proceed with the design. This strongly happens to the noise due to rotor–stator interaction since it has a high temporal coherence. Indeed, the tonal components excite the acoustic response of the system (fan assembly ⇑ Corresponding author. Tel.: +39 010 353 2445. E-mail address: [email protected] (A. Cattanei). http://dx.doi.org/10.1016/j.apacoust.2015.01.015 0003-682X/Ó 2015 Elsevier Ltd. All rights reserved.

and test environment) at precise frequencies, possibly resulting in relevant attenuation or amplification if small variations of the rotational speed take place. This phenomenon has been reported by Margetts [10] and Canepa et al. [11]. In real cooling units, the fan often operates at variable speed and the tonal noise peaks sweep the acoustic response function of the system. As a consequence, irregularities in the growth and in the decrease of tonal components during the fan operation may be heard, resulting in an important cause of annoyance. However, such irregularities could also depend on variations in the generating mechanism strength. Due to such a complicated behavior, tests taken at a limited number of axial-gap values, [9], resulted in interesting indications on the features of the phenomenon but could not provide a detailed picture of it. In the present work, only one stator geometry has been considered, the axial-gap has been systematically varied with a reduced step, and attention has been focused on the first and the second BPF harmonics only. This has allowed to perform a deeper and more complete study. Furthermore, installation effects have been discussed more deeply. Experimental techniques based on the employ of simple instrumentation and facilities, i.e. tests with one or few microphones in hemi-anechoic chambers, are usual in the automotive field. This limits the available information and, at the same time, may introduce further undesirable acoustic phenomena, Canepa et al. [11] and Roger [12]. In the present paper, amplitude and phase of the

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Nomenclature a0 c Cn d Dhub Dtip Dn f G He K, K’n lrefl Matip n OASPL Q r R Spp(f,O) SPL SPLn St t utip

speed of sound (m/s) blade chord (m) constant used in the scaling of SPLn (dB) rotor–stator axial-gap (m) rotor hub diameter (m) rotating shroud inner diameter (m) scaled SPL (departure of SPLn from the linear dependence on 10 log10 X) (dB) frequency (Hz) propagation function (dB) Helmholtz number based on Dtip and on a0, = f Dtip/a0 (–) constants used in the SPL scaling (dB) length of the path followed by reflected waves Mach number based on utip (–) multiple of the harmonic order of tonal noise components, n = St / DSt (–) overall sound pressure level (linear, ref. 20 lPa) (dB) volume flow rate (m3/s) distance of the receiver to the source (m) radius of a blade airfoil (m) power spectral density of the received acoustic pressure (Pa2 s) sound pressure level spectrum (ref. 20 lPa, Df = 12.5 Hz) (dB) sound pressure level at the nth BPF harmonic (ref. 20 lPa) (dB) Strouhal number based on the rotational frequency, =60f/X (–) time of reception of the acoustic wave (s) peripheral speed at the blade tip (m/s)

acoustic pressure measured during speed ramps are studied according to the theory described in [11], and the spectral decomposition method, Bongiovì and Cattanei [13], is employed to highlight the acoustic response of the whole system. Finally, the obtained results may provide useful information for more theoretical studies such as the ones of Abid et al. [14] or Trabelsi et al. [15]. 2. Experimental facility, data processing method, and tests organization 2.1. Tested fan The measurements have been taken on a fan whose rotor is made of polyamide with glass fibers, Fig. 1. The rotor is provided with zR = 9 evenly spaced blades and with a cylindrical rotating shroud of external diameter Dtip = 460 mm, the hub has a diameter Dhub = 181 mm, and the blade chord c varies between 65 mm at the hub and 72 mm at the tip. The stator employed in the present work is evenly-spaced and is composed by zS = 18 constant-thickness, cambered vanes of 30 mm chord, Fig. 1. The axial-gap between the rotor blade trailing edge and the stator vane leading edge at the rotor tip, d, has been varied in the range 12–27 mm (d/ctip = 0.167–0.375) with 1 mm steps, resulting in 16 different position. The choice zS = 2zR is not realistic since in actual fans zR and zS are never integer multiples but it enhances the SPL at the low BPF harmonics. Indeed, the stators employed in production units are usually composed of non-evenly spaced vanes and struts, thus resulting in tonal noise components at all BPF harmonics. In the present case, the standard gap is d = 23 mm, which equals to about one axial chord of the blade. Other details of the experimental facility are reported in Canepa et al. [16].

zR zS

a b

Df Dp DSt h

s /

u w X

rotor blade number (–) stator vane number (–) exponent in the scaling of the SPL at variable X (–) exponent in the scaling of the constant-X SPL spectrum (–) bandwidth employed in the SPL computation static pressure rise (outlet static pressure minus the ambient pressure) (Pa) non-dimensional bandwidth employed in the SPLn computation angular distance between the noise source and the reflecting tape (deg) time delay (s) phase angle (ref. tacho pulse emission) (deg) flow coefficient (–) pressure coefficient (–) rotational speed (r/min)

Subscripts BB related to broadband noise e wave emission filt filtered by means of the propagation function n related to the nth harmonic order orig computed by the spectrum analyzer prop related to propagation ref emission of the tacho pulse (s) source related to the acoustic source T related to tonal noise

The design performance are a flow rate Q = 1.081 m3/s and a static pressure rise Dp = pout  p0 = 320 Pa at X = 2725 r/min with p0 the total pressure at the rotor inlet and air at ambient conditions (T0 = 20 °C and p0 = 101,300 Pa). This results in a flow coefficient udes = Q/(utippDtip2/4) = 0.095 and a pressure coefficient wdes = Dpdes/ (0.5q0utip2) = 0.134, where utip is the blade tip speed. In the present investigation, the fan has been operated at free-discharge conditions, for which u = 0.164 and w = 0. The rotor is driven by a PC-controlled brushless servomotor (Danaher AKM42E-ANCNR-00, rated power 1.14 kW at 640 V) supported by a steel structure which allows a precise positioning of the rotor by means of a 3-axis system. The structure is fixed to a 690 mm  710 mm rectangular wooden panel. The motor is quieter and more stable than the brushless motors employed in the production units. Its noise does not interfere with the aerodynamic one and the only noticeable effect is a sharp peak at f ffi 16 kHz, which does not influence the noise object of the present study. The wooden panel has a central circular hole and is supported by a metal frame which realizes a free-discharge condition. The tipclearance geometry (with 5 mm axial and radial gaps) is obtained inserting different aluminum rings in the hole. In the present test configuration, at free-discharge conditions, the tip-leakage noise is negligible compared to the tonal one, Canepa et al. [16]. The stator has been inserted in the most external ring and no heat exchanger has been mounted. 2.2. Test configuration The measurement campaign has been carried out in the DIME hemi-anechoic chamber. Below 100 Hz, the SPL spectra may be affected by the loss of anechoicness of the test environment, and, at a so low frequency, they must be carefully treated. Usually,

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Fig. 1. The tested fan. (a) The rotor. (b) The stator.

the rotor axis is horizontal and is located at a height of 1 m, with the microphone on-axis 1 m upstream. For measurements on automotive fans this is a very common configuration, but its employ could impair the assumption of free-field conditions due to the possible interference caused by the waves reflected by the chamber floor. The importance of such an effect depends on the features of the source and on its distance from the floor compared to the distance from the microphone and has to be discussed. When dealing with stochastic broadband noise, the random phase usually prevents interference from taking place and the power spectral densities Spp(f) of the direct and reflected radiation simply sum. Let us assume that the source directivity is negligible. Then, the method of the image source, Roger [12] and Moore [17], may be easily employed. As a result, Spp(f) related to the reflected waves is attenuated of a factor equal to about (r/lrefl)2: in the usual configuration, the direct path length is r ffi 1 m and the one of the p path of the reflected waves is lrefl ffi 5 m = 2.35 m. Then, compared to the ideal free-field conditions, the reflected waves increase Spp(f) at the microphone of about 20%, which means an average increase in the SPL spectrum of only 0.8 dB. On the contrary, when dealing with periodic noise, interference may take place and be important. In the present case, such kinds of generating mechanisms are the rotor–stator aerodynamic interaction (in the following RSI), Lowson [4], and the ingestion of large scale turbulent structures (in the following IT), Homicz and George [1], Majumdar and Peake [2], and Hanson [3]. Evaluating the relevance of the reflections from the floor would require to establish a model for the interference pattern and to identify the predicted trends in the experimental ones. Particularly, the simple model based on a point source plus an image one, located in a symmetrical position with respect to the floor, [11], cannot be used. Indeed, the sources, i.e. the blades, are located within the

rotor disk which cannot be considered acoustically compact in the present case, Moore [17]. Furthermore, the microphone is not on the axis of the image rotor, and thus, source directivity and Doppler effect should be taken into account. Even neglecting them, such a model should consider number, location, and temporal coherence of each source, at least. This would surely result in a difficult task. Hence, it has been decided to modify the test configuration in such a way to definitely change the interference pattern. In this way, it should be possible to evaluate the extent up to which interference may affect the measurements. Beyond the usual one, two configurations which should minimize the effect of floor reflections have been considered, Fig. 2: (1) The H-configuration, in which the microphone is on-axis 1 m upstream, and the rotor axis is horizontal at the height of 1.6 m from the floor. Compared to the standard height, the reflected waves are more attenuated since the path is longer. (2) The T-configuration, in which the mounting panel is tilted of 60 deg and the microphone is on the floor, on-axis, and 1 m upstream of the rotor. Consequently, the microphone is not aligned with the rotor axis and the collected acoustic pressure data may suffer from the microphone directionality. In fact, this effect is important at f > 3 kHz, a frequency range largely above the one relevant to the topic dealt with. Positioning the microphone close to a reflecting wall is a solution employed to avoid interference with the reflected waves, Moore [17] and Lewy and Gounet [18]. However, compared to free-field conditions, it results in a uniform 6 dB increase of the SPL. The rotational speed X has been measured by means of an optical tachometer and a stripe of reflecting tape stuck on one rotor

Fig. 2. The test configurations. (a) The H-configuration. (b) The T-configuration.

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E. Canepa et al. / Applied Acoustics 94 (2015) 29–38

blade, thus generating a TTL signal (the tacho signal), with a oneper-revolution pulse. The absence of vibration of the whole structure has been carefully checked before each measurement campaign.

2.3. Measurement procedure and data processing method Data have been acquired and processed by means of a spectrum analyzer and a pre-polarized ½00 free-field microphone. The measurements have been taken during linear speed ramps from Xmin = 500 r/min to Xmax = 3000 r/min in 1080 s, resulting in a blade tip Mach number Matip = utip/a0 varying between 0.035 and 0.21, with a0 = 344 m/s the speed of sound. Then, the flow may be considered incompressible. The small angular acceleration and the use of the order analysis algorithm, Herlufsen [19], have allowed to compute SPLn, the SPL at the harmonics of the rotational frequency, and the SPL power spectrum at constant X. In both cases, 100 data records with 50% superposition, acquired during the ramp tests, have been employed. For the constant-X spectra, a von Hann window has been used in order to limit the spectral leakage. Let Spp(f, X) be the one-sided power spectral density of the received acoustic pressure with the fan operated at fixed X. The constant-X SPL spectrum is defined as

" SPLðf k ; XÞ ¼ 10log10

Z

1 p2ref

#

f k þDf =2

Spp ðf ; XÞ df

ð1Þ

f k Df =2

where fk = kDf, X is regarded as a parameter, and pref = 20 lPa is the reference pressure. In the present case, the resolution is Df = 12.5 Hz and k ranges between 1 and 1600, thus resulting in f ranging between 0 and 20 kHz. SPLn is defined as

" SPLn ðXÞ ¼ 10log10

1 p2ref

Z

#

f n þDf =2

Spp ðf ; XÞdf

ð2Þ

f n Df =2

where n = St/DSt = (60f)/(XDSt). St = 60f/X is the Strouhal number based on the rotational frequency, DSt is the resolution, and Df = DSt X/60 is the resulting bandwidth. Then

f n ¼ nDSt

X 60

ð3Þ

is the central frequency of the n-th band. A St value or range is characteristic of a given generating mechanism, no matter X. If a tonal component is contained within the range St  DSt/2, St + DSt/2, then the background broadband noise contribution must be small enough not to affect the tonal one. To this aim, DSt needs being chosen small enough, too. In the present case, DSt = 0.2 has shown to be adequate to study SPLBPF = SPL45 and SPL2BPF = SPL90, the SPL at the first and the second BPF harmonics, respectively. At each X, 400 SPLn(X) values have been computed in the range St = 0–80. 100 different X values have been considered within the range X = 500– 3000 r/min, with a resolution DX = 25 r/min. Opposite to the constant-X SPL spectra, DSt = const results in SPLn computed in bands with Df proportional to X. Basing on some parametric tests, these settings have shown to result in a sufficient precision of the spectral data, i.e. the random fluctuations usually affecting the plots of spectral quantities are sufficiently limited not to complicate their analysis. The phase of the received acoustic pressure, / = /(f, X), should be computed from the constant-X cross-spectrum between the acoustic pressure and a reference signal. In the present work, this is done during a speed ramp, and the tacho pulse is the reference signal.

3. Quantities employed for the tonal noise study 3.1. G(f), the propagation function Let us consider a point source radiating in free-field conditions. The free-field Green’s function applies, and a clear correspondence between Spp(f) and the power spectral density of the source strength is present. In practical situations, perfect free-field conditions are never completely fulfilled since propagation effects (reflection, scattering and diffraction) are always present. If they are important, modifications in the SPL spectrum appear which do not correspond to variations in the source strength spectrum. Basing on the similarity theory, see Blake [20] or Mongeau et al. [21], the SPL spectrum may be expressed as:

SPLðf ; XÞ ¼ K þ 10log10 ðX3þb Þ þ 20log10 FðStÞ þ 20log10 GðHeÞ

ð4Þ

where K is a constant. F(St) is a non-dimensional source spectral distribution function which depends on the fluctuations of both aerodynamic pressure and velocity within the flow region. F(St) is independent of acoustic effects. G(He) is the non-dimensional acoustic frequency response function (in the following the propagation function) which depends on the Green’s function of the system and He = f Dtip/a0 is the Helmholtz number. G(He) is independent of aerodynamic effects, i.e. of X, and, in practice, G(He) G(f) because both Dtip and a0 are fixed. F(St) and G(f) may be separated by means of the spectral decomposition method, which is based on the comparison of SPL spectra related to several X values. In the present work, G(f) is determined by means of the method presented by Bongiovì and Cattanei [13]. In fact, when considering a receiver at a given position, G(f) is related to the Green’s function evaluated at the point where the source is located, Mongeau et al. [21]. Hence, depending on the source position on the blade, local differences may appear in G(f), Neise and Barsikow [7], and different noise generation mechanisms may be related to different G(f). If the sources are distributed on the blade surface, i.e. for non-point sources, some kind of spatial average of the Green’s function may be expected, [11]. Thus, tonal noise may be influenced by a propagation function GT(f) different from GBB(f), the one related to broadband noise. The latter is continuously generated along the whole blade span by all blades and the IT noise is generated by the repeated impingement of the incoming turbulent structures on the rotor blades when they are at random positions. On the contrary, the RSI noise is generated by sources well localized on blades and/or vanes, when the rotor is at given positions. Thus, it may be expected that the IT noise is related to a smoother GT(f) than the one related to the RSI noise. GBB(f) could be even smoother. The spectral decomposition method applies an error minimization technique to constant-X spectra. Hence, most of the employed spectral data are related to the broadband noise and the yielded propagation function is mainly dependent on the broadband part of the SPL spectrum, i.e. it coincides with GBB(f). Prior to the spectral decomposition, both b and K must be assigned. This can be done arbitrarily, Bongiovì and Cattanei [13], but a proper choice is obviously recommended and requires information about the active noise generating mechanisms and the propagation environment. In the present case, b = 1.3 has been chosen, which has shown to provide the best fitting of the broadband part of the original SPL curves with the SPLfilt ones, Cattanei et al. [11]. Then, other noise generating mechanisms, e.g. RSI or IT, might not be properly scaled, thus resulting in a further possible cause of difference between GBB(f) and GT(f). Taking the difference SPL(f, X)  GBB(f) allows to eliminate all the irregularities due to propagation effects and yields a quantity directly related to F(St). Then, information on the underlying

E. Canepa et al. / Applied Acoustics 94 (2015) 29–38

generating mechanism is provided, too. These typically include non-perfect anechoicness of the test environment, and diffraction effects from blade edges and parts of the whole fan assembly. A kind of ‘‘filtered’’ SPL spectrum is then yielded:

SPLfilt ðf ; XÞ ¼ SPLðf ; XÞ  20log10 GBB ðf Þ

ð5Þ

SPLfilt(f, X) is somehow arbitrary due to the choice of both b and K. However, if the employed pair yields an average trend which compares to the one of SPL(f, X), then the analysis of SPLfilt(f, X) may be worth. 3.2. Dn(X), the scaled SPL at BPF harmonics In the present case, the X variation during a speed ramp causes a SPL growth of the order of 40 dB, thus complicating the study the SPLn fluctuations. Therefore, it is convenient to scale SPLn with 10 log10 X4+a = (4 + a)10 log10 X, which represents the average growth. This SPLn scaling is based on the similarity theory, and is formally analogous to the one employed for the spectral decomposition, Eq. (4). The employ of Df = const, Eq. (1), requires scaling SPL(f, X) with 10 log10 X3+b, while the employ of Df proportional to X, Eq. (2), requires scaling SPLn(X) with 10 log10 X4+a. When dealing with the same noise generating mechanism, a = b results. At low Ma, a wide scattering in the a value from 1 to 2 is usually reported, see for instance Neise [22]. The actual a is related to the correlation between the velocity fluctuations which cause the noise generation. In case of complete correlation, a = 2 is expected, while in case of low correlation, a = 1 may be expected, Homicz and George [1], Wright [25]. If both RSI and IT generating mechanisms are simultaneously present, but none of these prevails, a spurious dependence of a on X could even appear, depending on the relative strength of the two mechanisms and on the considered BPF harmonic. However, this scaling allows to extract the fluctuating part of SPLn

Dn ðXÞ ¼ SPLn ðXÞ  ð10log10 X4þa þ C n Þ

ð6Þ

where Cn is an arbitrary constant. When studying the tonal noise generated by a single mechanism, in case of perfect kinematic and dynamic similarity, the source strength depends on X2, and a is constant. Then, a constant Dn(X) results if the actual a is employed, while a linear dependence on 10 log10 (X) results if a different value is used. Thus, if the scaling aims at studying the fluctuations of Dn related to different configurations, an a value different from the actual one may be employed without any loss of information. The meaning of Dn goes beyond the empirical scaling of SPLn, [11], since applying the structure of Eq. (4) yields

Dn ðXÞ ¼ SPLn ðXÞ  10log10 ðX4þa Þ ¼ K þ 20log10 FðStÞ þ 20log10 Gðf Þ

ð7Þ

33

When considering tonal noise at the N-th BPF harmonic, G = GT, n = NzR/DSt and St = NzR = const. As a result, F(St) = const and, considering Eqs. (3) and (7) may be rewritten as

  X Dn ðXÞ ¼ K 0 þ 20log10 GT ðf n Þ ¼ K 0 þ 20log10 GT nDSt 60

ð8Þ

where K0 = K + F(nDSt). Eq. (8) shows that, if the structure given by Eq. (4) applies to tonal noise components, then Dn is directly related to GT. Hence, a departure of Dn from the linear dependence on 10 log10 X may be due to variations in the dependence of the noise generating mechanisms on X and/or to variations of GT in the spanned frequency range. Bumps, dips, or sudden changes in slope of Dn(X), which usually take place within limited X ranges, can be unlikely attributed to variations in the dependence of the generating mechanism strength on X. Rather, variations in the acoustic response within the frequency range swept by a tonal noise peak are quite common, Margetts [10], Canepa et al. [11], Stephens and Morris [23]. For instance, in the present case, fBPF grows from 75 to 450 Hz and f2BPF grows from 150 to 900 Hz, but variations in the acoustic response may take place within a 100 Hz range or less, see Fig. 3. Thus, they are likely the cause for Dn(X) departures from the linear trend. Analogies between GBB(f) and GT(f) are generally expected. Then, in case analogies between GBB(f) and Dn(f), with f = fn, take also place, the Dn fluctuations may be reasonably attributed to propagation effects. Local differences between GBB(f) and GT(f) may be a possible reason for the lack of analogy. In such a case, in order to support or exclude the assumption that the Dn fluctuations are due to propagation effects, it may be useful to consider other quantities, such as /n(O). 3.3. /n(X), the phase of the acoustic pressure at BPF harmonics Phenomena which are related to the angular position of the rotor should result in repeatable values of /. Random phenomena, which are not related to the angular position of the rotor, are expected to yield non-repeatable, i.e. randomly distributed, values of /. Let us consider a rotating point source radiating in free-field conditions. /(f, X) can be related to a time delay s = t  tref where t is the time of reception of the acoustic wave and tref is the time at which the tacho pulse is generated. s is given by the sum of ssource and of sprop. The former is the delay between the tacho pulse and the acoustic wave emission by the source, and the latter is the delay due to propagation from the source to the receiver. If noise is generated by the RSI, then ssource is strictly related to the angular position of the source, i.e. of the blade, at the time at which interaction takes place. This position corresponds to the time te of emission of the acoustic wave and may be directly related to the angular position #e of the reflecting tape when the tacho pulse is generated. #e depends on the properties of the generating mechanism and, if the system operates under kinematic and dynamic

Fig. 3. Effect of the test configuration. (a) SPLorig spectra at X = 1200, 1700, 2200 r/min. (b) SPLfilt spectra at X = 1200, 1700, 2200 r/min and GBB(f).

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E. Canepa et al. / Applied Acoustics 94 (2015) 29–38

similarity conditions, it is independent of X and depends on the axial-gap d. Thus

ssource ¼

1 #e ðdÞ 6 X

ð9Þ

where the factor 1/6 appears when expressing X in r/min and #e in deg. Under free-field conditions

sprop ¼

r a0

ð10Þ

with r the distance of the microphone to the source, which is constant since the microphone is on-axis. Hence, no Doppler effects may be present and

/ðf ; XÞ ¼ 

180 

p





#e ðdÞ r 2pf ðssource þ sprop Þ ¼ 360f þ 6X a0

 ð11Þ

where / has been expressed in deg. When taking measurements during speed ramps at very small angular acceleration, a set of records of samples, e.g. the m-th one, may be considered as related to a constant value X = Xm. The further acquisition, i.e. the (m + 1)-th one, provides a further set which may be considered as related to another independent realization of the same experiment but performed at X = Xm+1. At a given X, f must be expressed by means of Eq. (3), and thus

/ðf ; XÞ ¼ /n ðXÞ ¼ nDSt#e ðdÞ  6nDSt

Xr a0

ð12Þ

In the present case, /BPF = /45 and /2BPF = /90 are studied. Then, in free-field conditions, for sources correlated with the rotor position, a linear trend of /BPF and /2BPF is expected. When plotting / versus f, the expected slope is 360r/a0 at any BPF harmonic. When plotting it versus X, the expected slope is proportional to the harmonic order, e.g. it equals 108r/a0 at the 2nd BPF one. / is related to the phase of the Green’s function, then for distributed (i.e. non-point) sources, some spatial average is expected, but the basic linear trend is expected to be preserved. In case propagation effects take place, Eq. (10) cannot be used and Eq. (12) must be substituted with

/n ðXÞ ¼ nDSt#e ðdÞ þ /prop ðf n ; dÞ   X ;d ¼ nDSt#e ðdÞ þ /prop nDSt 60

ð13Þ

/prop(f, d) being a non-linear function specific to both test and fan configurations. In case of IT noise, during a speed ramp, the trend of /(X) at any BPF harmonic should be random no matter propagation effects are present or not. On the contrary, RSI noise should result in deterministic trends. Thus, at a given BPF harmonic, the

trend of /n as a function of X (or f) may help distinguish tonal noise due to IT from the one due to RSI. In presence of both kinds of sources, it could also be possible to qualitatively evaluate their relative importance. Furthermore, in case of deterministic sources, simple free-field propagation may be distinguished from more complex patterns basing on the degree of linearity of /. Therefore, a comparison of the Dn(X) and /n(X) trends may provide useful information about the observed phenomena, particularly if any of them depart from the expected linear behavior. 4. Basic features of the obtained spectral data In a numerical study of the RSI noise in a low-speed axial-flow fan, Lu et al. [24] found that the relevant noise sources are distributed on the rotor blades surface. The reason for this was that the fluid velocity in the rotor (the relative velocity w) was much larger than the one in the stator (the absolute velocity v). Indeed, a proportionality of Spp on the 5th to 6th power of the reference velocity is expected. Then, as far as the order of magnitude is concerned, the difference between the SPL related to the rotor and the one related to the stator is DSPL = (5–6)  10 log10 (w/v). In the present case, w 6v, and then, DSPL equals 40 dB or more. Hence, tonal noise should be due to both the potential effect of the stator vanes on the rotor blades and the effect of the inlet turbulence on the latter one. The noise radiated by the stator should be negligible. Nevertheless, the stator may have an acoustical influence on the noise generated by the rotor. As a consequence of interference between waves generated by the rotor blades, IT noise is present at all BPF harmonics, but RSI noise is only present at the even BPF harmonics, Lowson [4]. 4.1. Effect of the mounting panel tilting and preliminary analysis of the spectral data In order to verify the effect of the rotor axis orientation, measurements have been taken on the fan with the stator at d = 12 mm in both the H-configuration and the T-configuration (see Section 2.2). Then, the spectral decomposition has been applied to the original SPL spectra computed by the spectrum analyzer (SPLorig, in the present paragraph), yielding both GBB(f) and SPLfilt (b = 1.3 has been employed), Fig. 3. The reported spectra are related to X = 1200, 1700, and 2200 r/min. These X values are within characteristic parts of the trend of D2BPF, Fig. 4. The curves related to the RA-configuration (rotor alone, i.e. without stator, in the H-configuration with the axis at the height of 1 m) constitute a reference case. Up to about f = 3 kHz, the T-configuration yields SPLorig 6 dB higher than the ones related to the H-configuration, according to the position of

Fig. 4. Effect of the test configuration (a) DBPF, D2BPF and GBB(f). (b) /2BPF. (Partially adapted from Ref. [9].)

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the microphone. Then the SPLorig curves have been lowered of 6 dB in order to allow a better comparison with the others. With the exception of the tonal noise peaks, a good agreement seems to be present. Above f = 3 kHz, the microphone directionality causes the SPLorig curves related to the T-configuration to quickly decrease and to become lower than the ones related to the H- and RA-configuration. However, this frequency range is not relevant to the topic dealt with. In all of the three cases, the broadband parts of SPLfilt present a very good agreement, indicating that the sources related to this part of the spectrum are independent of the configuration, of the rotational speed, and of the presence of the stator, too. The tonal noise peaks show large differences, probably due to differences in GT related to the various configurations. However, in the RA case, only IT noise is present at the even BPF harmonics, and thus the levels of the related peaks cannot be compared with the ones related to the cases with stator. This aspect may be better analyzed by means of DBPF and D2BPF. The 6 dB increase due to floor reflections constitutes a propagation effect and GBB related to the T-configuration has been consequently lowered of 6 dB, thus allowing a better comparison with GBB related to the RA- and the H-configuration. Above 4 kHz, the former starts decreasing, since the spectral decomposition algorithm feels the microphone directionality as a propagation effect because it only depends on frequency. The same happens to the peak at 16 kHz, which is caused by the electric motor. Between 1 and 4 kHz, GBB related to the H- and the T-configuration are similar. This indicates that the two configurations are acoustically equivalent in this frequency range. Below f = 1 kHz, fluctuations of 2 dB or less are present in all curves. This departure from a flat trend is due to the propagation effects affecting the received broadband noise. They may affect the tonal noise, too. Finally, it is interesting to notice that GBB related to the RA-configuration is quite similar to the one related to the H-configuration, with the exception of the range 1–7 kHz. In this range the other two GBB are quite similar, as well as SPLfilt related to all of the three cases. This shows that the stator affects the broadband noise acoustically, but does not influence the generating mechanism. DBPF, D2BPF, and /2BPF, related to the three configuration, are plotted in Fig. 4 together with GBB (a = 2 and 1.4 have been employed for DBPF and D2BPF, respectively and the constants Cn have been arbitrarily chosen in order to ease the comparisons between the curves). The interesting features of the plots are: (1) The similarity between DBPF and GBB, particularly clear in the H-configuration no matter the presence of the stator, and the oscillations in DBPF, D2BPF, and GBB (between 200 and 300 Hz), related to the H-configuration both with and without stator. The former feature suggests that the IT noise and the broadband one are affected by similar propagation functions. The latter one is particularly strong when the stator is present, since the RSI generates waves of high temporal coherence at the 2nd BPF harmonics. The described behaviors are common to different generating mechanisms, and, therefore, they are probably due to the acoustic response of the test environment and fan assembly. (2) Above 300 Hz, in both the H- and T-configurations, the presence of the stator causes strong differences between the trends of D2BPF and the ones of DBPF and GBB, and also of the D2BPF one in the RA-configuration. But, most of all, important analogies are present between the various trends of D2BPF when the stator is present. They consist in a strong decrease between 300 and 750 Hz, i.e. between 1000 and 2500 r/min. For f < 500 Hz, D2BPF decreases, then it reaches a minimum about 8 dB deep, and finally D2BPF increases, for f > 600 Hz. (3) The random trend of /2BPF, in the RA-configuration, according to random nature of D2BPF in absence of the stator.

(4) The deterministic trend of /2BPF, in both H- and T-configurations, with the stator. The trend consists of two approximately linear parts with a sudden decrease of about 150 deg between 450 and 550 Hz (i.e. between 1500 and 2000 r/min). The reference straight lines have a slope of 1.06 deg/Hz, the value resulting from the term 360r/a0, Eq. (11), evaluated at midspan (R = 160 mm). In the H-configuration, at 200 and 300 Hz, two bumps are present, which approximately correspond to the maxima of the GBB(f) oscillations. Hence, some aspects of the /2BPF behavior may be related to propagation effects. A number of analogies between the trend of GBB and the ones of DBPF, D2BPF, and /2BPF have been shown. Such analogies are an indirect evidence of the acoustic origin of the behavior of DBPF, D2BPF, /2BPF. The coincidence of the slope of /2BPF with the theoretical value predicted by Eq. (11) is a direct evidence. However, the most interesting point is the strong decrease in D2BPF between 300 and 750 Hz when the stator is present. Below 500 Hz, D2BPF decreases and /2BPF has a smooth, linear trend. Between 500 and 600 Hz, D2BPF reaches the minimum and /2BPF has a rather sudden 150 deg decrease. Above 600 Hz, D2BPF regularly increases and /2BPF has again a smooth, linear trend. No analogy with the behavior of GBB appears, nor such trends may be explained by means of a theoretical model. It may only be remarked that they depend on the presence of the stator, and are independent of the test configuration A further analysis is thus worth and is limited to the T-configuration for the sake of brevity. (1) Fig. 5 reports SPL2BPF scaled with both 10 log10 X1.3 and 10 log10 X10. That is, a = 2.7 and a = 6 have been used, resulting in two rather flat parts of the curves at f < 500 Hz and f > 600 Hz, respectively. These parts present a fair analogy with GBB, but the pieces below 300 Hz and above 750 Hz scale with other a values. This shows that the propagation effects affecting the broadband noise seem to affect SPL2BPF, too. But, most of all, strongly different scaling exponents exist depending on X. (2) Assume that the /2BPF decrease of 150 deg, Fig. 4, is related to the generation mechanism. Then, according to Eq. (12), it corresponds to a #e variation of 150 deg/18 = 8.3 deg, which is almost one half of the angular spacing of the stator (20 deg). This means that, during a speed ramp, the source position at the time of emission should displace of almost half a stator pitch between 1500 and 2000 r/min, while, below 1500 and above 2000 r/min, it should remain approximately unvaried. The D2BPF decrease is rather peculiar, and, in order to deepen it, measurements taken at different d are necessary. Although the

Fig. 5. Double-a scaling of SPL2BPF.

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effect of the possible reflections from the floor do not affect such a D2BPF feature, the T-configuration has been employed since it shows more regular trends than the H-configuration. 4.2. Effect of d on the SPL spectra In order to highlight the basic features of the generated tonal noise, the SPL spectra measured at X = 3000 r/min are first analyzed. They are reported in the plots of Fig. 6 as functions of f and d. The tonal noise peaks are clearly present up to the 8th BPF harmonic (f = 3.6 kHz at X = 3000 r/min) and the broadband part of the spectra only shows a slight growth with d in the range 4– 5 kHz. The latter feature may be related to the GBB increase with d, consistently with the discussion of GBB and SPLfilt, Section 4.1, Fig. 3. Namely, it has been shown that the broadband generating mechanisms are independent of the presence of the stator, according to the observed GBB values in the range f = 1–7 kHz, which are larger in the absence of it. The IT tonal noise is due to the ingestion of stretched turbulent structures which are repeatedly encountered by the rotor blades. This yields blade lift fluctuations which are the cause of noise generation, Homicz and George [1], Majumdar and Peake [2], Hanson [3]. Hence, d should not have any relevant effect on the generating mechanism due to IT. According to Lowson [4], the potential effect of the stator on the rotor is expected to decay as 1=d. As a result, as d increases, SPLBPF should remain constant, while SPL2BPF should decrease. On the contrary, as d increases, both SPLBPF and SPL2BPF show important amplitude fluctuations about a constant trend (such a behavior takes place at the other BPF harmonics, too). Particularly, SPL2BPF grows until d = 17 mm, remains approximately constant until d = 25 mm, and then it decreases. The analysis of the same plots related to other X would show a similar behavior.

no apparent influence of d on SPLn is present, and that SPLn shows a good proportionality with 10 log10 X4+a. DBPF varies within a range of about ±3 dB. Beyond the very harsh, random peaks and dips due to the turbulent nature of the noise at the first BPF, another interesting feature are two transverse valleys at about f = 140, and 360 Hz. Compared to DBPF, D2BPF shows a more regular trend and a stronger sensitivity to d. A clear departure from the reference behavior is present at all d, since instead of a flat trend a complicated dependence on both d and f is present. This feature is related to the fluctuations of the peaks at the BPF harmonics already noticed in the plot of Fig. 6. The largest values of D2BPF are at the intermediate d (17 mm < d < 25 mm) and two areas of low values are present at the extreme d and at all frequencies. At d 6 17 mm, D2BPF shows a strong dependence on f, while at the other d the dependence is weaker. (The behavior at d = 12 mm has already been described in Section 4.1, but may help interpret Fig. 7.) At fixed f, from d = 12 mm to d = 17 mm, D2BPF grows and reaches its maximum value. In the area at 17 mm < d < 25 mm and 250 Hz < f < 500 Hz, D2BPF is slightly decreasing, with some local maxima. Compared to the parts at d = 12 and 27 mm, in the same frequency range, these maxima are more than 12 dB higher (this is a rather general feature since

5. Analysis of the effect of d on D2BPF, GBB, and /2BPF

5.1. Analysis of DBPF, D2BPF, and GBB A more quantitative analysis of the effect of X and d on the tonal noise may be performed plotting SPLBPF and SPL2BPF as functions of the pair fBPF, d or f2BPF, d. To highlight the local fluctuations, DBPF and D2BPF are plotted instead (a = 2 and 1.4 have been used, respectively. CBPF = 131 dB and C2BPF  103 dB), Fig. 7. According to the discussion of Section 3.2, a flat trend of Dn would indicate that

Fig. 6. Effect of the rotor - stator axial gap on the SPL (X = 3000 r/min).

Fig. 7. SPL during speed ramp test. (a) DBPF. (b) D2BPF.

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the largest D2BPF variations are related to d). Then, at d > 25 mm a steeper decrease takes place. Such a strong dependence on d is rather surprising considering that the stator has been displaced within an overall distance of about a single axial chord at the blade tip. Furthermore, at all d, three transverse valleys (at about f = 160, 350, and 550 Hz) are present which are a few dB deep. Two of these valleys correspond to the ones which have been observed in DBPF. At each d, GBB(f), Fig. 8, has been computed employing 5 SPL spectra related to X = 1800, 2100, 2400, 2700, and 3000 r/min with b = 1.3 in the range 0 and 20 kHz (only the range 0–900 Hz is of interest to the present work). GBB results from the composition of a wavy trend (maximum variation 5 dB) dependent on f, and a slightly variable one (maximum variation 2 dB), dependent on d. The relevant aspects are the three transverse valleys at f = 150, 350, and 530 Hz, which approximately correspond to the ones which have been observed in the DBPF and D2BPF plots. This shows that the latter are due to propagation effects. Conversely, no correspondence is present with the large decrease observed in D2BPF between 300 and 750 Hz at d 6 17 mm. 5.2. Analysis of /n /BPF and the quantity D/2BPF, defined by Eq. (14), are represented in the contour plots of Fig. 9, as functions of d and f (with f = fBPF = 9X/60 and f = f2BPF = 18X/60, respectively). The /BPF random trend confirms the turbulent nature of SPLBPF but prevents from obtaining any other useful information. On the contrary, analyzing the effect of d and f on /2BPF may help study the observed phenomena, but the large /2BPF variations complicate the analysis. Then, in order to remove the principal dependence on d and f, the quantity D/2BPF(f, d) is computed:

D/2BPF ðf ; dÞ ¼ /2BPF ðf ; dÞ þ kf f þ kd d  335

ð14Þ

where f = f2BPF, kf = 1.06 deg/Hz and kd = 10 deg/mm. The constant 335 has been subtracted in order to result in D/2BPF ffi 0 deg in the central zone of the plot and the values of kf and kd have been chosen since they yield the flattest trend. The kf value is the average slope given by Eq. (11), term 360r/a0, evaluated at midspan. Since this term expresses the / dependence on f in free-field conditions, oD /2BPF/of = 0 means that no propagation effects should be present. With reference to Eq. (13), the dependence on d may be due either to the generating mechanism delay, term #e(d), or to propagation effects, term /prop(f, d). Neglecting the local, small variations, Fig. 9 shows that, at d > 17 mm, the condition oD/2BPF/of = 0 is well respected and thus /2BPF depends linearly on f only. At d 6 17 mm, the behavior is

Fig. 9. Phase angle as a function of d and f. (a) /BPF. (b) D/2BPF.

more complicated and is similar to the one described in the analysis of Fig. 4: at d = 12 mm, D/2BPF ffi 100 deg for f < 450 Hz, then a rather sudden decrease of 150 deg takes place, and D/2BPF ffi 50 deg for f > 600 Hz. As d increases, the frequency range where the variation takes place becomes wider and the decrease becomes smaller. Such a D/2BPF variation takes place in the same part of the plot where D2BPF grows with d instead of decreasing, see Fig. 7. This is an important point, since it shows that analogies in the non-linear trends of both D2BPF and D/2BPF are not limited to a single distance. Assume that kinematic and dynamic similarity are respected. Then #e(d) is independent of X and, if it were known, its contribution to D/2BPF could be eliminated and /prop(f, d) could be determined by means of Eq. (13). As discussed in Section 4.1, the stator provides a negligible contribution to the noise generation. Then, #e(d) is directly related to the angular position at which the rotor blades radiate the tonal noise, i.e. #e(d) is connected to the tangential position of the potential perturbation upstream of the stator as a function of d. Unfortunately, this is not easily predictable a priori by means of simplified models and then no physical explanation of the employed kd = 10 deg/mm can be provided, nor /prop may be obtained at the present level. 6. Considerations on the observed phenomena

Fig. 8. The propagation function GBB(f).

The analysis of the results has shown that the tonal noise component at the 2nd BPF harmonic has a rather unexpected behavior: both SPL2BPF and /2BPF depend on both d and X (=60f/18) in a

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complicated way. Namely, at low d, SPL2BPF does not monotonically decrease with d as it would be expected, Lowson [4] and Kaji and Okazaki [6], the linear dependence of SPL2BPF on 10 log10 X4+a, Homicz and George [1] and Wright [25], is not respected, and /2BPF does not fully respect the linear dependence on 18Xr/a0, typical of free-field propagation. This mainly happens at d 6 17 mm, and may be due to the effect of d and X on the noise generating mechanism and/or on the acoustic response of the whole assembly and of the propagation environment. Unfortunately, the kind of experimental data do not allow to state with certainty what the nature of the observed phenomena is. To this aim, measurements of instantaneous pressure on the blade surface would be necessary and CFD and/or CAA simulations would be surely useful. Nevertheless, the joint analysis of the dependence of SPL2BPF and /2BPF on d, and f2BPF (=18X/60) provides some evidence that the cause of the observed behavior is of acoustic nature. Possibly, that it is caused by the interaction between the stator vanes and the waves radiated by the rotor blades. Indeed, the growth of SPL2BPF with d is hard to justify on an aerodynamic basis. Similarly, discarding the acoustic explanation would lead to the conclusion that, during a speed ramp, kinematic and dynamic similarity do not hold. But the fan operates at freedischarge conditions, at which the blade loading is low and phenomena such as blade stall hardly take place. To this regard, it is worth to notice that an abnormal aerodynamic behavior should result in apparent modifications of the SPL spectrum, but its broadband parts correctly scale with X, even about the tonal noise peaks. This is apparent in the SPL spectra of Fig. 3, which are related to X values at which D2BPF is decreasing, is minimum, and is growing, see Fig. 4. Then, it seems that within this X range the blade profiles work properly and dynamic similarity is respected. As a conclusion, if the acoustic response were unchanged, a rather peculiar aerodynamic behavior should be admitted: at d = 12 mm the strong variation in the dependence of SPL2BPF on X (a = 2.7 at X < 1670 r/min and a = 6 at X > 2000 r/min), Fig. 5, would be due to analogous variations in the strength of the RSI noise generating mechanism, but the remainder of the SPL spectrum would be unmodified. Similarly, the decrease of 150 deg in /2BPF, which would result in a 8.3 deg angular displacement of the source position in a limited X range, should be attributed to the noise generating mechanism. 7. Conclusions The noise generated by an axial-flow fan for automotive cooling systems has been studied in a hemi-anechoic chamber. Particularly, the effect of the rotor–stator axial-gap modification on the tonal noise has been investigated in detail. In order to study the effect of the acoustic response of both the test environment and the fan assembly, the measurements have been taken during low-acceleration rotational speed ramps. The propagation function has been computed by means of the spectral decomposition and it has been compared with the acoustic pressure phase angle and the velocityscaled SPL at the blade passing frequency harmonics. Taking measurements at variable speed has allowed to vary the characteristic frequencies of the noise generating mechanisms, while keeping the characteristic frequencies of the acoustic response unchanged. At the lowest axial-gaps, the SPL peak due to the rotor–stator aerodynamic interaction grows as the axial-gap is increased and its dependence on the rotational speed shows strong variations. This may be attributed to a dependence of the noise generating mechanism on the rotational speed or to the acoustic response of the system. It has not been possible to provide a certain explanation for such a behavior, since to this aim more complicated and time-

consuming studies would have been necessary. Nevertheless, some evidence that the observed phenomena may be due to acoustic effects has been provided. From the practical point of view, it has been shown that the choice of an appropriate rotor–stator axial-gap may result in a non-trivial question due to the simultaneous dependence of the tonal noise on both axial-gap and rotational speed. These findings may help interpret apparently inconsistent experimental results.

Acknowledgments The authors kindly acknowledge Johnson Electric Asti srl for having provided the tested fan, Università di Genova for the financial support to the present work, and the reviewers for the useful comments which have helped improve the quality of the work. References [1] Homicz GF, George AR. Broadband and discrete frequency radiation from subsonic rotors. J Sound Vib 1974;36(2):151–77. [2] Majumdar SJ, Peake N. Noise generation by the interaction between ingested turbulence and a rotating fan. J Fluid Mech 1998;359:181–216. [3] Hanson DB. Spectrum of rotor noise caused by atmospheric turbulence. J Acoust Soc Am 1974;56(1):110–26. [4] Lowson MV. Theoretical analysis of compressor noise. J Acoust Soc Am 1970;47(1):371–85 (Part 2). [5] Tyler J, Sofrin T. Axial flow compressor noise studies. SAE Trans 1962;70:309–32. [6] Kaji S, Okazaki T. Generation of sound by rotor-stator interaction. J Sound Vib 1970;13(3):281–307. [7] Neise W, Barsikow B. Acoustic similarity laws for fans. ASME J Eng Ind 1982;104(2):162–8. [8] Quinlan DA, Krane MH. Aeroacoustic source identification using frequency dependent velocity scaling. In: AIAA paper 1996–1743, 18th AIAA aeroacoustics conference; 1996. [9] Canepa E, Cattanei A, Mazzocut Zecchin FE, Milanese G, Parodi D. Experimental study of the effect of the rotor-stator gap variation on the tonal noise generated by low-speed axial fans. In: AIAA paper 2013–2046, 34th AIAA aeroacoustics conference; 2013. [10] Margetts EJ. A demonstration that an axial fan in a ducted inlet ducted outlet configuration generates predominantly dipole noise. J Sound Vib 1987;117:399–406. [11] Canepa E, Cattanei A, Mazzocut Zecchin F. Installation effects on the tonal noise generated by axial flow fans. J. Sound Vib 2014. http://dx.doi.org/ 10.1016/j.jsv.2014.12.00. [12] Roger M. Near-field fan noise modelling and installation effects due to scattering surfaces. Fan Noise 2007, Lyon, France; 2007. [13] Bongiovì A, Cattanei A. Spectral decomposition of the aerodynamic noise generated by rotating sources. J Sound Vib 2011;330:136–52. [14] Abid M, Trabelsi H, Taktak M, Antoni J, Ville JM, Fakhfakh T, et al. Tonal prediction of a faulty axial fan. Appl Acoust 2012;73:1022–8. [15] Trabelsi H, Abid M, Taktak M, Fakhfakh T, Haddar M. Reconstruction of the unsteady rotating forces of fan’s blade from far-field sound pressure. Appl Acoust 2014;86:126–37. [16] Canepa E, Cattanei A, Mazzocut Zecchin F, Milanese G, Parodi D. Experimental study and velocity scaling of the tip-leakage noise generated by low-speed axial-flow fans. In: AIAA-paper 2014-3347, 35th AIAA aeroacoustics conference; 2014. [17] Moore CJ. A solution to the problem of measuring the sound field of a source in the presence of a ground surface. J Sound Vib 1971;16(2):269–76. [18] Lewy S, Gounet H. Modification of radiated sound directivity due to ground reflection: application to static tests of helicopter turboshaft engines. Noise Control Eng J 1993;41(2):305–12. [19] Herlufsen H. Order analysis using zoom FFT. Brüel& Kjaer application notes, No. 012-81; 1981. . [20] Blake WK. Mechanics of flow-induced sound and vibration. Orlando: Academic Press; 1986. [21] Mongeau L, Thompson DE, McLaughlin DK. Sound generation by rotating stall in centrifugal turbomachines. J Sound Vib 1993;163(1):1–30. [22] Neise W. Application of similarity laws to the blade passage sound of centrifugal fans. J Sound Vib 1975;43(1):61–75. [23] Stephens DB, Morris SC. A method for quantifying the acoustic transfer function of a ducted rotor. J Sound Vib 2008;313:97–112. [24] Lu HZ, Huang L, So RMC, Wang J. A computational study of the interaction noise from a small axial-flow fan. J Acoust Soc Am 2007;122(3):1404–15. [25] Wright SE. The acoustic spectrum of axial flow machines. J Sound Vib 1976;45(2):165–223.