The acoustic approach is based on the FW&H analogy [5] and is used as the reference approach. The subsonic regime is considered in this paper. The finite thickness of the blades is neglected; only loading noise ( p L ) generated by the fluctuating wall pressure on the blade is considered in the following example.

To alleviate the computation time problem, Formulation 1A proposed by Farassat [34] can be used. In this approach, the receiver time derivative in Formulation 1 [34] is transformed into a retarded time derivative. This has also the great advantage of permuting the time derivative and the integral. Moreover, no derivation of the integral is needed. For the case in which the source and the receiver are stationary and the propagation medium is at rest, the Farassat Formulation 1A [34] in the far field becomes:

p L ( x → , t ) = 1 4 π c ∫ S y → [ l ˙ r r ] τ d S y → (1)

where S y is the source surface, r = ‖ r ‖ = ‖ y − x ‖ is the distance between the source position y → on the blade’s surface and the receiver position x → , c is the speed of sound of the acoustic medium at rest, l r = − p n → ⋅ r → / r with n → is the unit normal vector to the source’s surface, and p is the wall pressure fluctuation of the blade’s surface obtained by CFD calculation. M is the Mach vector number and M r = M → ⋅ r → / r and [ • ] τ indicates that all the integrands should be evaluated at the retarded time τ = t − r / c , with receiver time t.

As mentioned previously, Formulation 1, proposed by Farassat, has the main advantage of avoiding the spatial derivatives; however, the receiver time derivative is maintained on l r and M r . Implementing this operation is complex, and the computation time increases. To evaluate Equation (1), two computational approaches are available in the literature [35] [36] using either retarded time or advanced time. The latter is usually chosen when the aerodynamic data comes from CFD computations, as in this study. The received acoustic pressure must then be determined while an irregular receiving time discretization appears despite the regular emission time. An interpolation is thus necessary to obtain the received sound pressure at regular time.

The advanced time approach and the Lagrange interpolation are used in this paper. The calculation of the sound pressure at receiver x → in the far field and in a free medium allows for the calculation of root mean square sound pressure. When considering the loading noise (Equation (1)), the mean square acoustic pressure reads:

p L 2 ( x → ) = 〈 p L ( x → , t ) p L ( x → , t ) 〉 T 0 (2)

where 〈   〉 T 0 is the temporal average over the time period T 0 . Thus, the radiated sound power estimated from far field microphones (i.e., receivers) located on a spherical surface encompassing the source can be written:

P = ∫ S x → p L 2 ( x → ) ρ c d S x → ≃ 1 ρ c ∑ x → p L 2 ( x → ) Δ S x → (3)

where S x → is the receiving surface, Δ S x → is the elementary surface associated with receiver x → , and ρ is the density of the surrounding fluid medium.

Generally, POD used in aeroacoustics is not based on acoustic analogies. The approach developed in this paper is a combination of the dipole term (loading noise) of the FW&H analogy and the POD theory. If one considers a dipole located at y → i , the sound pressure received at point x → at time t according to Equation (1) can be written:

p L ( x → , y → i , t ) = 1 4 π c [ l ˙ r r ] τ Δ S y → i (4)

where Δ S y → i is the i-th elementary surface of the source located at y → i . Here, this elementary surface comes from the discretization of the source’s surface for the LES calculation. Since the sound pressure received at one point corresponds to the contribution of all the dipoles located on the wall surface, two matrices A and W o b s are defined by

W o b s = 1 N A A T (5)

with

A = ( p L ( x → , y → 0 , t e 0 ) p L ( x → , y → 0 , t e 1 ) ⋯ p L ( x → , y → 0 , t e N − 1 ) p L ( x → , y → 1 , t e 0 ) p L ( x → , y → 1 , t e 1 ) ⋯ p L ( x → , y → 1 , t e N − 1 ) ⋮ ⋮ ⋱ ⋮ p L ( x → , y → m − 1 , t e 0 ) p L ( x → , y → m − 1 , t e 1 ) ⋯ p L ( x → , y → m − 1 , t e N − 1 ) )

where t e j represents the emission time for the dipole source located at y → i ( i = 0 , 1 , ⋯ , m − 1 ) at time step j ( j = 0 , 1 , ⋯ , N − 1 ) . Each column vector of the matrix A is the sound contribution of all dipoles at a given reception time, while each row vector represents the sound pressure of one single dipole source along the receiving time. Matrix W o b s in Equation (5) is the correlation matrix of the sources for receiver x → . It is symmetric, real, positive definite, and spatial. Its eigenvalues (modes) are thus real, positive, and space-dependent. When developing a modal basis for matrix W o b s , one defines λ = diag ( λ 0 , λ 1 , ⋯ , λ m − 1 ) and ϕ → = [ ϕ → 0 , ϕ → 1 , ⋯ , ϕ → m − 1 ] , the diagonal matrix of eigenvalues and the matrix of eigenvectors at receiver x → respectively. Each column ϕ → i is the eigenvector associated with the eigenvalue λ i at receiver x → . Then for every receiver x → , the problem to be solved is the following eigenvalue problem:

W o b s ϕ → = λ ϕ → (6)

If ones multiplies each member of Equation (6) to the right by the transpose of the matrix of eigenvectors and considers normalized modes so that ϕ → ϕ → T = I , the expression for the correlation matrix is given by:

W o b s = ∑ k = 0 m − 1 ( λ k Φ → k ) Φ → k T (7)

The correlation matrix W o b s is then written as a sum of independent matrices defined as spatial autocorrelation patterns with proper modes as components. Since the eigenvectors form an orthonormal basis of the source space, p L ( x → , y → i , t ) can be written as:

p L ( x → , y → i , t ) = ∑ k = 0 m − 1 α k ( t ) Φ k , i (8)

where α k ( t ) = α k ( x → , t ) = ∑ i = 0 m − 1 p L ( x → , y → i , t ) Φ k , i are the temporal modal amplitudes or the projection coefficients on the modal basis. Φ k , i = ϕ k ( x → , y i ) is the i-th component of the k-th eigenvector ( ϕ → k ). Thus, coefficients are the root mean square of the acoustic pressure projected on the Φ k ( y → ) axis in the source space (i.e., the blade). According to Merces’ theory [37] , the projection coefficients form an orthogonal basis of the temporal space, and its root mean square 〈 α k ( t ) α k ( t ) 〉 T 0 corresponds to the eigenvalue λ k . The eigenvalues represent the square of the sound pressure projected on the Φ k ( y → ) axis in the source space. The total sound pressure from the loading noise radiated to the receiver x → at time t is evaluated from Equations 1 and 8. It is given by:

p L ( x → , t ) = ∑ k = 0 m − 1 ( α k ( t ) ∑ ​ i = 0 m − 1 Φ k , i ) (9)

Considering the orthogonal eigenvectors, the quadratic pressure at receiver x → is:

p L 2 ( x → ) = ∑ k = 0 m − 1 ( ∑ ​ i = 0 m − 1 Φ k , i ) 2 λ k (10)

In this proper orthogonal decomposition, each mode does not contribute equally to the total quadratic pressure. The latter could then only be evaluated by taking into account the modes that contribute the most. This is reported by the accumulated acoustic energy of the first q modes, E A c q , divided by the total acoustic energy of all modes:

E A c q = ∑ k = 0 q − 1     λ k ∑ k = 0 m − 1     λ k (11)

Once the modes that contribute the most are identified, summations in Equations (3) and (10) are limited up to q − 1 instead of m − 1 . Thus, Equations (9) and (10) become respectively:

p L ( x → , t ) ≃ ∑ k = 0 q − 1 ( α k ( t ) ∑ ​ i = 0 m − 1 Φ k , i ) (12)

p L 2 ( x → ) ≃ ∑ k = 0 q − 1 ( ∑ ​ i = 0 m − 1 Φ k , i ) 2 λ k (13)

In the previous equations, the spatial eigenvectors Φ k , i give information on the acoustic radiation of all dipole sources distributed over the surface S. Thus, considering Equation (13), the acoustic power defined by Equation (3) becomes:

P ≃ 1 ρ c ∑ l = 0 N o b s − 1 ( Δ S l ∑ ​ k = 0 q − 1 λ l k ( ∑ ​ i = 0 m − 1 Φ l k , i ) 2 ) (14)

where Δ S l = Δ S x → is the elementary surface of the ℓ-th receiver. In this study, Δ S l is a constant. N o b s is the number of receivers, λ l k is the k-th eigenvector of the ℓ-th receiver, and Φ l k , i is the i-th component of the eigenvector associated with the k-th eigenvalue λ l k .

SVD investigates sound generation (i.e., the eigenvalues and the eigenvectors resulting from the POD) independent of the receiver x → . A global matrix W S V D gathering all the correlation matrices W k ( 0 ≤ k < N o b s ) of the N o b s receivers is built. The global matrix of dimension ( N o b s × m ) × m is defined by:

W S V D = [ W 0 W 1 ⋮ W N o b s − 1 ] (15)

Here, the receivers are distributed over a sphere of radius R around the source. The radius is large enough that the far field assumption is verified. The position of the receivers over the sphere is done according to ISO 3745 [38] . SVD consists of decomposing the W S V D matrix in the following form:

W S V D = U σ V T (16)

where σ is the diagonal matrix of singular eigenvalues, and U and V are the matrices of left and right eigenvectors respectively. U accounts for the difference of acoustic radiation of the sources in the free field of all receivers, and V provides the average information on acoustic radiation of the sources in the free field of all receivers. One can show that U and V can be obtained by applying the POD to matrices W S V D T W S V D and W S V D W S V D T respectively. By retaining the first q energetic modes only, the expression of sound power (Equation (3)) based on the SVD can finally be expressed as:

P ≃ 1 ρ c ∑ k = 0 q − 1 ( σ k ∑ ​ i = 0 m − 1 V k , i ∑ ​ l = 0 N o b s − 1 ( Δ S l ∑ ​ j = m ∗ l m ( l + 1 ) − 1 U k , j ) ) (17)

where σ k is the k-th eigenvalue of the matrix W S V D , V k , i and U k , i respectively represent the right and left i-th component eigenvectors, respectively associated with the eigenvalue σ k .

The POD principle is to search an orthonormal basis of the m elementary sources of the discretized radiating blade’s surface. From this orthogonal basis, the q eigenvectors that contribute the most to the acoustic radiation at a given receiver are identified. The mapping of these q eigenvectors on the blade’s surface allows identifying its most radiant zones. Consequently, the application of the POD method on our stationary blade in the channel aims to minimize the number of modes necessary to understand the most radiant zones of the blade due to its interaction with the turbulent flow.

Indeed, Figure 4 presents the accumulated acoustic energy of the q first modes (Equation (11)) at receivers φ = 0 , φ = 28 ∘ , φ = 56 ∘ , and φ = 91 ∘ . One observes that the first 10 eigenvectors capture 99.60% of the total acoustic energy, whatever the receivers. As a result, these 10 eigenvectors could then be employed for the reconstruction of the total loading noise. This leads to a reduction in the number of modes (from m = 30400 to q = 10 ) to be analyzed in order to identify the noisiest zones.

For example, Figure 5 shows the temporal evolution of the sound pressure of the loading noise reconstructed with q = m (Equation (1)) and (Equation (9)), and q = 10 (Equation (12)) at receivers φ = 0 ∘ , φ = 28 ∘ , φ = 56 ∘ , and φ = 91 ∘ . Acoustic pressures constructed with the first 10 modes (Equation (12)) of each receiver are in accordance with the reference acoustic pressure (Equation (1)). The maximum relative error equals 1.7% at receiver φ = 91 ∘ . The relative

error remains less than 0.1% for the other receivers.

One also observes that the temporal evolution of the sound pressure depends on the receiver location (Figure 5), which characterizes the directivity of radiation. It results in the POD being dependent on the receiver. For example, the maximum amplitude of sound pressure is almost equal to 0 for the φ = 0 ∘ receiver, and is equal to 4 × 10 − 3 Pa for the receiver φ = 91 ∘ . The amplitude of the sound pressure signal is very small for receiver located at 0˚ in comparison with the others. In order to better see this signal a subfigure has been added on Figure 5(a).

Consider the receivers in Figure 3(b). We calculated the mean quadratic sound pressure with all modes (Equations (2) or (10)) and with the first 10 dominant modes (Equation (13)). First, One observes that the acoustic radiation is dipolar (Figure 6). The lowest sound pressure is noticed for receivers located at φ = 0 ∘ and φ = 180 ∘ , while the highest radiations are for receivers at φ = 90 ∘ and φ = 270 ∘ (Figure 6). This observation is in agreement with the directivity defined by Farassat [34] . According to this formulation (Equation (1)) the scalar product between the normal surface of the blade and the direction with the receiver is considered in the loading noise evaluation. Thus this product becomes null for the two former positions (i.e. φ egals 0˚ and 270˚) while equals unity for the two latter (i.e. φ egals 0˚ and 270˚).

In addition, the two curves of the mean quadratic sound pressure are similar, since the relative error remains less than 3% (Figure 6). This confirms the analysis of the reconstruction of sound pressure with the first 10 most important modes (Figure 5). When considering each term of the mean quadratic sound pressure (Equation (13)). We noted that the contribution of the modes does not necessarily appear in the ascending order (Figures 4-7. For example, considering receiver φ = 91 ∘ , the contribution of mode 1 is greater than the contribution of mode 0, even if the latter accumulates 71% of the energy, while mode 1

only accumulates 23% (Figure 4). This contradiction is due to the phase shift of the components of an even eigenvector in calculating the mean quadratic sound pressure (Equation (13)). As a consequence the modes must be classified in ascending order when calculating either the mean quadratic sound pressure or the acoustic power. Thus, the reclassification of the modes was carried out for each receiver in Table 1.

One considers the reconstructed sound power (Equation (14)) with q = 10 and q = 5 . As shown in Table 2, the relative error of the acoustic power of the first five modes (Equation (14)) with respect to all modes (Equation (3)) is low and even lower with the first 10 modes (Equation (14)). This confirms that all useful acoustic information needed here is contained in the first 10 modes.

It is possible to project the eigenvectors that contribute the most to the acoustic power onto the blade surface, since the quadratic mean pressure is proportional to the sound power (Equations (13) and (14)). For example, when considering receivers φ = 0 , φ = 28 ∘ , φ = 56 ∘ , and φ = 91 ∘ , one observes that the POD method highlights the trailing edge as the region characterized by a great amplitude of the component Φ → k , whatever the side of the blade (Figure 8 and Figure 9). The other regions of the blade do not seem to be affected by such amplitudes. Hence, mode 2 for receivers φ = 0 ∘ , 56˚, and 91˚, and mode 3 for receiver φ = 28 ∘ highly contribute to the acoustic power. The same observation is made for modes 3, 2, 4 and 9 for receivers φ = 0 , φ = 28 ∘ , φ = 56 ∘ , and φ = 91 ∘ respectively (last column of Figure 8 and Figure 9).

These previous observations are in agreement with the Q-criterion distribution (Figure 2), since according to this parameter, the turbulence is high near the trailing edge. The high velocity and pressure fluctuations in this region

would therefore make this region responsible for the loading noise.

It was shown here that both the radiated sound pressure and the sound power can be reconstructed with only a few POD modes. Instead of using all the modes

proposed by the POD method (i.e. m = 30400 ), the first 10 dominant POD modes only were sufficient. In addition, the dipole character of the blade of radiation was noticed. The mapping of the major eigenvectors for several receivers demonstrated that the trailing edge is the region that radiates the most due to the high level of turbulence.

Like the POD, SVD methods based on CFD calculations and acoustic analogies were applied to identify the zones on a stationary blade in a channel that contribute the most to the sound power radiated in a subsonic regime. Using the SVD method, radiated sound power and pressure can be recovered using only the first few modes.

Figure 10 represents the acoustic energy that the q first modes accumulate divided by the total energy (Equation (11)). It emerges that 99.90% of the total acoustic energy is contained within the first 10 modes, which causes a reduction in the number of modes to be analyzed in order to understand the noisiest zones of the blade. The reconstructed sound power (Equation (17)) using the first 10 modes only is given in Table 3 and is compared with the sound power calculated

using all the modes. The relative error equals 0.43%.

As shown previously for the POD method, the most energetic eigenvalues do not necessarily contribute the most to the reconstruction of the acoustic power. Modes 6, 10, 4, and 9 in order of their importance contribute the most (Figure 11).

In addition, the acoustic power estimated with only mode 6 equals 1.15 × 10 − 7   W (i.e., 50.6 dB) representing 93% of the total acoustic power. When adding the contribution of mode 10 to the contribution of mode 6, 96% of the total acoustic power is recovered. Thus, the number of modes to be analyzed to comprehend the most radiant zones of the blade regardless of receiver changes from m = 30400 modes to two modes (modes 6 and 10).

As for the POD method, it is possible to use singular value decomposition to map the right eigenvectors V → k of the four main modes (i.e. modes 6, 10, 4, and 9) on the blade surface as shown in Figure 12. One notices that the amplitude for mode 6 is quite homogenous along the surface. As mentioned previously, this must be due to the turbulence that develops quite homogenously in the blade region; the entire surface therefore contributes equally to the loading noise. When considering the distribution for mode 10, one observes that the trailing edge is characterized by greater amplitudes, which is in accordance with the strong intensification of the turbulence in this region as shown by the

Q-criterion (Figure 2). As a result, this region contributes largely to the radiated acoustic power through mode 10.

As a result, based on this observation, one could expect that the radiated sound power could be reduced by a proper modification of the blade geometry, or surface treatment, in these regions. This conclusion is yet to be validated in future work.