The cross-spectrum of two sound pressure measurement points at a single frequency as shown as Figure 1 can be expressed as S p p ’ = < p ( x , r , ϕ   ) p ∗ ( x ’ , r ’ , ϕ ’ ) > (1)

In the formula, the superscripted ∗ represents the complex conjugate. Under the linear assumption, the measured sound pressure p is related to the vector a composed of the modal amplitude A m n ± : a = [ a 1   a 2   a 3   ⋯   a K − 1   a K ] T (2)

In the formula, K represents the length of the modal vector, and its value is equal to twice the number of truncated modes at this frequency; the measured sound pressure signal can be expressed as the product of the coefficient matrix Φ and radial modal amplitude vector a : p = Φ a (3) where p ∈ ℂ K represents the complex sound pressure vector at the measured frequency; a ∈ ℂ L represents the complex amplitude vector of the mode, the modal amplitude is A m n , and the size of L depends on the total number of cut-on modes in the duct, the value of matrix Φ ∈ ℂ K × L is determined by the microphone array position and the mode order. The cross-spectrum of two sound pressure signals can be defined as S p p ≜ E ( p p H ) , where E represents the expectation. According to Eq. 3, the cross-spectrum of two sound pressure signals can be written as S p p = Φ S a a Φ H (4)

Therefore, the cross-spectrum of the modal amplitude can be written as S a a = Φ † S p p ( Φ † ) H (5)

In the formula, † represents the pseudo-inverse of the matrix. In the actual experimental measurement, the measurement point number K of the microphone array is usually greater than the magnitude of the modal amplitude vector L, and the coefficient matrix may be singular. When solving such a matrix with a singular coefficient matrix and overdetermined equation, singular value decomposition (Nelson and Yoon, 2000; Kim and Nelson, 2004) is used in this paper. G + p = a ^ ⇔ U ⋅ W ⋅ V T p = a ^ (6)

In the formula, U and V are both transformation and orthogonal matrices, satisfying U ⋅ U T = 1 and V ⋅ V T = 1 , but W is a diagonal matrix, consisting of positive elements and singular values. It can improve the accuracy of the algorithm by discarding smaller singular values appropriately. The best estimate can be obtained by a ^ = V ⋅   [ diag     ( 1 w j ) ]   ⋅     U T (7)

The modal amplitude and cross-term between them can be obtained by formula (5), which is defined according to the modal coherence coefficient between the mode ( m , n ) and ( μ , ν ) : C   ( m , n )   ( μ , ν ) = | 〈 A m n A μ ν ∗ 〉 | 2 〈 | A m n | 2 | A μ ν | 2 〉 (8)

The coherence characteristics between the cut-on modes in the duct can be obtained. It should be noted that the modal amplitude in formula (8) represents the statistical average of the modal amplitude. The sound power carried by each individual mode can be calculated by P W L m n ± ( ω ) = π ( r D 2 − r H 2 ) ρ c α m n ( ω ) ( 1 − M a 2 ) 2 ( 1 ∓ α m n ( ω ) M a ) 2 | A m n ± ( ω ) | 2 (9) with α m n ( ω ) = ( 1 − ( 1 − M a 2 ) ( σ m n / ( k m n R ) ) 2 ) 1 / 2 , R is the duct radius, c is the sound speed, ρ is the flow density, and M a is the Mach number in the axial direction.

A numerical analysis was first performed in the synthetic sound fields. It is assumed that the duct is hard walled and has non-reflecting terminations in the following. The duct radius is 0.25 m. An optionally superposed axial flow is represented as a uniform-profile flow. In this section, M a is set as 0.10. The sound pressure at position ( x j , r j , φ j ) is excited by the S x ring(s) of monopole sources (each ring has S φ point-like monopole sources) at position ( x i , r i , φ i ) , which are driven with volume velocity, e.g., q 0 ( x i , r i , φ i ) , cf. Figure 2, can be calculated with the appropriate Green’s function: p ( x i , r i , φ i | x j , r j , φ j )       =       q ∑ m , n χ m n f m n ( r i ) f m n ( r j ) e i m ( φ j − φ i ) e i k m n + ( x j − x i ) (10)

The quantity χ m n is definite for each mode (m, n), which contains the frequency-dependent mode cut-on ratio. Referring to the typical Eq. 3 of duct mode expansion, the amplitude of mode (m, n) excited by the source is written as A m n ± ( q , x i , r i , φ i ) = q χ m n f m n ( r j ) e − i m φ i e − i k m n ± x i (11)

In the following, the modal cross-spectral density is formulated for an array consisting of S x rings of point-like monopole sources; each ring is equipped with S φ equidistantly positioned sources. All sources are placed at the outer duct radius r i = R , considering the experimental implementation. If all sources are incoherent and have equal spectral density, the analytical modal cross spectral can be inferred. 〈 A m n A μ ν ∗ 〉 = χ m n χ μ ν ∗ 〈 q q ∗ 〉 ∑ j = 0 N x ∑ l = 0 N φ − 1 Γ m n ( x j , R , φ l ) Γ μ ν ∗ ( x j , R , φ l ) (12)

Here Γ m n ( x , R , φ ) = f m n ( R ) e − i m φ e − i k m n ± x , the analytical mode coherence function can be calculated subsequently.

In the actual test, environment reflections can appear upstream or downstream of the sensor array. The impact of the reflections can be integrated into the Green’s function with a method of mirrored sources. For simplicity, the reflection coefficient r c is assumed to be independent of the mode order and also the frequency; here, it is set as 0.2. The effect of mode scattering from an incident mode into another mode with different azimuthal or radial orders is neglected here. The Green’s function can be modified subsequently to p ( x j , r j , φ j ) = ρ c 4 π R 2 q 0 ∑ m , n f m n ( R ) 2 α m n e i m ( φ j − φ i ) ( e i k m n + ( x j − x i ) + r c e i k m n − ( x j − x i ) ) (13)

In the frame of the numerical study, four substantially different arrangements of point-like monopole sources, denoted by Latin numbers I–IV, are investigated. The detailed information of source configurations is displayed in Table 1. The main difference lies in the number of axial sensor rings and also the total number of point-like monopole sources. At each sensor, the position time series of 700 FFT windows are used in the numerical process.

At high frequency, a large number of modes will be cut on inside the duct according to the Tyler and Sofrin modal theory, which brings great challenges to the experimental test of the modal coherence coefficient. As an alternative strategy, it is necessary to study the modal coherence distribution at a high frequency by pre-assuming the modal amplitude or sound source distribution in the duct. The sound source in the low-speed flow duct is mainly composed of dipole and monopole sound sources. This section will establish three representative sound source models: uniformly distributed monopoles, uniformly distributed dipoles, and equal energy per mode (EEPM) sound source models.

When the incoherent sound sources are uniformly distributed on a certain cross-section in the duct, the non-homogeneous wave equation contains the source term ( 1 c 2 D 2 D t 2 − ∇ 2 ) p = q (14)

The source term can be expressed as q = ( D D t ) ν ( − ∂ ) μ ∂ x i ∂ x j ⋯ q i j ⋯ (15)

in which ν represents the time order of the sound source distribution and μ represents the space order of the sound source distribution. In the follow-up study, the sound source distribution is limited to the axial direction. At this time, the source term can be written as q = ( D D t ) ν ( − ∂ ∂ x ) μ q x x (16)

Considering the single mode generated by a fixed sound source in a semi-infinite duct, Joseph et al. (2003) indicated that the spatial differentiation of the single mode (the second item on the right of Eq. 16) is equivalent to ( − ∂ ∂ x ) μ → ( − i k m n + ) μ (17)

Similarly, the time differential in equation is equivalent to ( D D t ) ν → [ − i ω + U ∂ ∂ x ] ν = [ − i ω ( 1 − α m n M a 1 − M a 2 ) ] ν (18)

Therefore, when the cross-section in the semi-infinite circular duct is uniformly distributed with incoherent sound sources of spatial order μ and time order ν , the modal amplitude value can be expressed in a general form (Joseph and Morfey, 1999): | A m n ( μ , ν ) | 2 ∞ 1 α m n 2 ( M a − α m n 1 − M a 2 ) 2 μ ( 1 − α m n M a 1 − M a 2 ) 2 ν (19)

in which ( μ , ν ) is used to indicate the sound source type, ( μ , ν ) = ( 0 ,   1 ) corresponds to a monopole sound source, and ( μ , ν ) = ( 1 ,   0 ) corresponds to an axial dipole sound source. There is also a type of sound source distribution in the actual research: EEPM, that is to say, the energy carried by each cut-on mode is the same, that is PWL m n = W 0 . The modal amplitude distribution A m n ( e e ) of EEPM can be expressed as | A m n ( e e ) | 2 ∞ 2 ρ c S − 1 α m n − 1 W 0 ( 1 − α m n M a 1 − M a 2 ) 2 (20)

It can be found that the model formula (20) cannot be unified to formula (19). Therefore, the sound source distribution model is integrated as E { | A m n | 2 } = C μ , ν , γ 2 ( M a − α m n 1 − M a 2 ) 2 μ ( 1 − α m n M a 1 − M a 2 ) 2 ν 1 α m n γ (21)

in which E { | A m | 2 } is the expected value of the square of the modal amplitude, and C μ , ν , γ 2 represents the sound source intensity.

The corresponding values of the three models are shown in Table 2; the type of sound source is defined by three Greek letters ( μ , ν , γ ) : Q s 2 ¯ is the mean value of the squared velocity in the monopole sound source distribution; F s 2 ¯ is the mean value of the squared force in the dipole sound source distribution; W 0 is the energy carried by each mode in EEPM. The modal amplitudes of the three sound source models can be expressed as E { | A m n | 2 } = C 012 2 ( 1 − α m n M a 1 − M a 2 ) 2 1 α m n 2 (22) E { | A m n | 2 } = C 102 2 ( M a − α m n 1 − M a 2 ) 2 1 α m n 2 (23) E { | A m n | 2 } = C 011 2 ( 1 − α m n M a 1 − M a 2 ) 2 1 α m n (24)

Eq. 22 represents a uniformly distributed incoherent monopole sound source ( μ , ν , γ ) = ( 0,1,2 ) , Eq. 23 represents a uniformly distributed incoherent dipole sound source ( μ , ν , γ ) = ( 1,0,2 ) , and Eq. 24 represents a modal and equal energy distribution sound source ( μ , ν , γ ) = ( 0,1,1 ) .

One important aspect of this paper is to experimentally investigate the modal coherence distribution with respect to an axial fan test rig. A schematic of the experimental set-up is shown in Figure 4. It was designed with the intention to capture the most relevant acoustic features of an axial fan. On the inlet side there is a short duct section with a bell mouth nozzle and there are no flow straighteners or screens in the inlet duct. The single-stage axial fan consisting of 19 rotor blades and 18 stator vanes was driven at 3,000 rpm with the help of one motor providing 18.5 kW electric power. The diameter is 0.5 m, and the axial Mach number is 0.1. The outlet duct is equipped with an anechoic termination device; this condition strictly conforms to the international standard ISO 5136 for in-duct sound power determination. The number of stator vanes was designed such that the rotor–stator interaction tone at the blade passing frequency (BPF) is acoustically cut on.

In order to eliminate noise pollution in the experimental measurement, the acoustic environment of the experiment is particularly important. For example, when the standing wave formed by the external enclosed environment radiates into the duct, it will directly affect the experimental measurement of the fan inlet noise and reduce the signal-to-noise ratio of the acoustic signal. The fan inlet and the acoustic measurement section are placed inside the semi-anechoic chamber to reduce the influence of external reflected noise on the fan inlet noise test; perforated acoustic liners are installed on the inner and outer walls of the fan exhaust duct. The phase lock device (shown as a red square in Figure 4) used in this paper is a photoelectric sensor. The pulse signal can be used to capture the circumferential angle of the fan blade and also the real-time fan rotating speed; the accuracy can be guaranteed to be less than 2°. The sampling frequency is 16,384 Hz, and time series of a length of 60 FFT windows were used here, each window comprising 16,384 data samples. In order to meet the requirements of the designed method, the acoustic measurement section (shown as a red box in Figure 4) is divided into two measurement devices: part A and part B. Part A is a combined sensor array that includes a ring of 16 equiangular installed sensors and a row of 15 equidistant microphones. In part B, a rotating measurement system is designed to be driven by a stepper motor; two axial arrays, each consisting of 14 microphones with equal intervals, are installed in the rotating measurement system that can rotate 360° in the circumferential direction in the experiment.

Figure 5 shows the acoustic modal cut-on function of the fan duct. Figure 5A shows the total mode number propagating in the flow duct; the solid line, dash-dotted line, and dashed line indicate the total number of cut-on modes, maximum circumferential mode order, and maximum radial mode order, respectively. At 3,000 Hz, a total of 55 modes are able to propagate in the flow duct. Taking into account the forward and backward propagation of the modes, there will be 110 modal waves propagating in the duct, of which the maximum circumferential order is m = ±   12 and the maximum radial mode order is n = 4 . Figure 5B shows the radial mode details of each circumferential mode. As the frequency increases, more radial modes will be cut on in the duct. At 3,000 Hz, the circumferential mode m = 0 is composed of (0,0), (0,1), (0,2), (0,3), and (0, 4) modes.

In the turbomachinery, the noise generated by rotor blades is a radially distributed sound source term. It is assumed that the ducted broadband sound field consists of fully incoherent modes, which are the important precondition of the broadband mode decomposition method presented by Enghardt (Enghardt et al., 2004). This assumption has not be verified experimentally so far for turbomachinery noise. Indeed, it is reasonable that specific modes are excited with high degree of coherence if certain conditions with respect to the spatial arrangement of sources are fulfilled; the compactness of the sources and the interference of the in-stationary aerodynamic forces also play an important role on modal coherence.

In the analysis of modal coherence function, the modal combinations are divided into two categories: in the first modal combination, the circumferential orders of modes are different from each other ( m ≠ μ ); the second type satisfies the same circumferential mode order but different radial mode order ( m = μ , n ≠ ν ).

The Characteristics of the Synthetized Sound Fields Section and Acoustic Modal Coherence Distribution of the Single-Stage Axial Fan section investigated the modal coherence distribution between modes in synthetized sound fields and a single-stage axial fan and finally came to important conclusions. As the frequency gradually increases, the number of modes will increase exponentially, and the number of sound pressure measurement points required for experimental measurement will also increase exponentially; this requirement results in a great challenge to the acoustic experiment of turbomachines.

In this section, theoretical analysis methods will be used to further expand the research on the acoustic modal coherence in flow duct at high frequencies in the hope of drawing conclusions over a wider frequency range. The modal amplitude distribution function in the duct can be obtained by using three ducted sound source distribution models proposed in the Ducted Sound Source Distribution Models section. Based on the ducted noise propagation model, the sound pressure values at different positions of microphone array measurement can be obtained by complex modal sound pressure superposition; combined with the broadband noise modal coherence analysis method, it can be achieved to further investigate the modal coherence function in a theoretical way. The advantage of this research is that the modal amplitude obtained by the mode decomposition method can be used to verify the reliability of the model based on the preknowledge of the modal amplitude distribution function. The modal coherence is deliberated on the basis of successful verification to make the results more reliable.

The influence of different ducted sound source models on modal coherence is discussed first in theoretical analysis. Figure 8 shows the modal coherence results of three sound source models at 5,412 Hz, in which the expected values are calculated with a statistical average process. A total of 170 modes are cut on in the flow duct at 5,412 Hz, including the maximum circumferential order m max = ± 22 and the maximum radial order n max = 7 . The microphone array is composed of 20 rings of measuring points; each ring is equipped with 50 equiangular microphones. The coherence coefficients of modes ( m , n ) and ( μ , ν ) are expressed as C m n μ ν ; the two modes are distinguished by α m n and α μ ν , the range of which is zero to one, referring to Eq. 19. In order to distinguish the forward modes (the modal circumferential order is positive, m > 0) and reverse modes (the modal circumferential order is negative, m < 0), the modal cut-on ratio of reverse modes is set to negative values. Then, all the modes in the duct can be expressed as a single-valued function of the modal cut-on ratio, and the value range becomes −1 to +1. It can be seen in Figure 8 that the coherence coefficients of all modes are lower than 0.1, that is to say the modes are mutually incoherent. By comparing the modal coherence coefficient results of the three ducted sound source models (monopole source model, dipole source model, EEPM source model), it can be found that the coherence results of the three sound source distribution models are very consistent, which shows that it is not necessary to study all the three sound source distribution models, but only one of them can be used to achieve the purpose of further study on modal coherence coefficients.

Figure 9 shows the modal coherence coefficients of different radial modes within the circumferential mode m = 0, m = 1 and m = 2 at 5,412 Hz. Abscissa n and ν represent different radial mode orders, respectively, and ordinate C m n m ν represents the coherence coefficients of ( m , n ) mode and ( m , ν ) mode. According to formula (19), the main diagonal element in Figure 9 corresponds to the autocorrelation coefficient of the mode. The numerical result of the main diagonal is close to 1, which is consistent with the theory. The coherence coefficients between different radial modes are all less than 0.1, which indicates that the different radial modes are mutually incoherent. Compared with the experimental results of acoustic modes in the Coherence Coefficient Spectrum of m = μ Mode Combination section, the modal coherence coefficient becomes more concise in this section; this is because the sound field in the theoretical study is artificially generated by the ducted sound source models, and there is no aerodynamic self-noise in theoretical analysis. The results further illustrate that high degree of coherence appearing between individual modes is due to the fact that the fan rotor blades and stator vanes are radial compact sound sources.

In this section, three kinds of ducted sound source distribution models are established, and the modal coherence characteristics at high frequencies are investigated theoretically for the first time by using the artificially constructed ducted sound fields. The results show that all the modes are incoherent at different frequencies, and the modal coherence characteristics are independent of the sound source distribution models. The coherence characteristics of the three ducted sound source models are approximately the same. At high frequencies, the radial modes within the same circumferential mode are incoherent to each other.

Different modal identification methods (RC-MDM, CSA method) make relevant assumptions and simplifications about the modal coherence; the influence of this simplified treatment on the sound power and modal amplitude distribution of turbomachinery ducted noise field, which we are most concerned about, needs to be further studied. Figure 10 shows the sound power results of incident and reflected sound waves calculated by different modal decomposition methods. The red and blue lines represent the computed sound power results calculated by the RC-MDM and CC-MDM, respectively, and the black line represents the sound power results computed by the CSA method. A series of simplifications of complex modal amplitude cross-terms are carried out in CSA and RC-MDM methods. It can be seen from Figure 10 that the shape of sound power spectral calculated by the three methods are consistent with each other, both in the incident and reflected sound waves.

The cross-terms C m n μ ν of modal complex amplitudes are retained completely in the CC-MDM, which make its calculation results more reliable. However, the need to compute the modal cross-terms makes the calculating process more time-consuming compared with RC-MDM and CSA methods, and this method requires much more sensors to synchronously measure the sound pressure in the experiment. The latter feature becomes unaffordable at higher frequencies because the required measuring points multiply exponentially with increased frequency. RC-MDM and CSA methods have attracted more and more attention due to their stronger practicability. The acoustic sound power of the incident sound wave is nearly 10 dB higher than that of the reflected sound wave. In the sound power spectra of the incident sound wave, the maximum deviation of three methods is only 1 dB. In the sound power spectrum of reflected acoustic waves, although the spectrum trend of the three methods is similar, there are significant differences in the magnitude of sound levels. The calculated results of the RC-MDM (red line) and CSA (black line) are in good agreement. However, there is a deviation of nearly 3 dB compared with the outcome of the CC-MDM. This is because the time-averaged modal cross-terms in RC-MDM and CSA methods are forced to be suppressed. In the calculation process of broadband mode identification, the energy of the modal cross-term is artificially imposed into the square-term of the modal amplitudes. This results in reflected sound waves predicted by the RC-MDM and CSA methods being 3 dB higher than the CC-MDM. Based on the successful application of a fixed sensor array, Figure 10 shows that the modal coherence characteristic has little influence on the prediction of incident sound waves. In order to further investigate the inner mechanism of modal coherence, the rotating sensor arrays shown as part B in Figure 4 were used in turbomachinery broadband noise measurement.

Under the framework of Sino-European cooperation, the NUMECA is the name of an internationally renowned computational fluid dynamics business company, conducted a numerical acoustic simulation on the single-stage axial fan duct device used in this research. As shown in Figure 11, the broadband mode decomposition result calculated by the RC-MDM with rotating sensor arrays is marked as a black solid line; the thick red line is the predicted broadband noise result calculated by NUMECA, using the adaptive spectral reconstruction method. The red circle symbol indicates the predicted tonal noise result. The results of the broadband noise spectrum are in good agreement in the frequency range (50–1,600 Hz), and the overall magnitude is around 68 dB. The broadband noise spectral characteristics are also very consistent at a special frequency, for example, the leap in sound power spectra at 400 Hz. In the tonal noise part, less than 1 dB deviation appears at 1BPF, and less than 0.5 dB arises at 2BPF. At 3BPF, the deviation increases to 1.5 dB. In general, fan noise measurements based on RC-MDM method using the rotating microphone array can be added to the aircraft turbomachinery noise database as standard data, which can provide evaluation and verification criteria for other broadband noise measurement methods or numerical simulation algorithms.

Figure 12 shows the sound power distribution of the incident and reflected modes of fan broadband noise identified by the rotating microphone array. The abscissa is set as the circumferential mode order m, ranging from −12 to +12, and the ordinate is set as the frequency, ranging from 0 to 3,000 Hz. In order to display the sound power results of incident and reflected sound waves more clearly, the legend range of incident sound waves is set to 40–80 dB, while the value of reflected sound waves is set to 35–75 dB, 5 dB lower than incident waves. When the fan operating speed is 3,000 rpm, the 1BPF and 2BPF tones are clearly visible in the incident sound waves; the sound power is mainly concentrated in the −6∼+6 circumferential modes. The 1BPF and 2BPF tonal tones can still be seen in the reflected sound waves, but different from the incident sound wave, the dominant mode in the reflected sound wave is not the rotor–stator interference modes but the plane sound wave. In the whole frequency range of reflected sound waves, the modal acoustic power is mainly concentrated in the −2∼+2 circumferential modes, and is approximately distributed symmetrically with the m = 0 mode as the center. This paper uses an equally spaced axial array to test the fan noise. For large ventilation facilities, this kind of device is often difficult to meet. How to use the obtained modal coherence characteristics to design a new test method needs further research, and the correlation between error characteristics in mode identification and microphone array design aspects also requires further study.