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This paper examines the use of proper orthogonal decomposition (POD) and singular value decomposition (SVD) to identify zones on the surface of the source that contribute the most to the sound power the source radiates. First, computational fluid dynamics (CFD) is used to obtain the pressure field at the surface of the blade in a subsonic regime. Then the fluctuation of this pressure field is used as the input for the loading noise in the Ffowcs Williams and Hawkings (FW&H) acoustic analogy. The FW&H analogy is used to calculate the sound power that is radiated by the blade. Secondly, the most important acoustic modes of POD and SVD are used to reconstruct the radiated sound power. The results obtained through POD and SVD are similar to the acoustic power directly obtained with the FW&H analogy. It was observed that the importance of the modes to the radiated sound power is not necessarily in ascending order (for the studied case, the seventh mode was the main contributor). Finally, maps of the most contributing POD and SVD modes have been produced. These maps show the zones on the surface of the blade, where the dipolar aeroacoustic sources contribute the most to the radiated sound power. These identifications are expected to be used as a guide to design and shape the blade surface in order to reduce its radiated noise.

Reducing the noise produced by the interaction between a turbulent subsonic flow and a solid’s surface can be difficult, especially in industrial configurations where the noise level is a quality and selection criterion. When considering subsonic turbomachines, this noise can be reduced if the geometry of the profile is appropriate. However, there is no general solution, and each fan configuration needs a specific design. Therefore, most of the time acoustic engineers use a case by case approach, either experimentally or numerically [

This paper proposes a method of identifying and analyzing these zones based on the combination of Ffowcs Williams and Hawkings’ acoustic analogy (FW&H) [

The pressure fluctuations obtained from a CFD approach are used as input data for the FW&H acoustic analogy [

POD was first introduced by Lumley in 1967 [

SVD is generally used to search for the propagation operators such as the Green function [

In the light of previous works, this study applies the POD and SVD methods to the problem of aeroacoustic noise generated by the interaction between a stationary blade and a turbulent flow in a channel. The objective is to understand the link between the decomposition modes of POD or SVD and the noisiest zones of the blade’s surface. The methodology relies on a three-step methodology: 1) The internal flows of the centrifugal fan are modeled using LES method [

The acoustic approach is based on the FW&H analogy [

To alleviate the computation time problem, Formulation 1A proposed by Farassat [

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 [

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 [

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 [

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 .

In

outlet (^{−}^{5} Pa・s). An airflow of U = 70 m / s is imposed on the inlet. The periodicity boundary conditions are applied to the periodic face and the no slip boundary conditions are imposed on the walls (blade, top and bottom). The purpose here is to investigate the possibility of both the POD and SVD approaches for identifying zones of the blade that are responsible for noise generation. Only frequencies in the 50 Hz to 10 kHz range are considered.

The calculation is performed on a hybrid spatial discretization realized with the free software Salome [

The large eddy simulation method is used to simulate the internal flow channel with the free software OpenFOAM [

The second invariant of the velocity gradient tensor, named Q-criterion, was introduced by Hunt et al. [

Once the calculation is converged, the wall pressure fluctuations on the blade are saved at time interval Δ t a c o u s t = 10 Δ t C F D . N = 2500 samples of the pressure fluctuations are then the input data of both POD and SVD methods. The sound pressure, correlation matrix and modes are calculated for receivers placed on a sphere of radius R around the source, according to ISO 3745. The distance R is such that the far field hypothesis is verified (i.e. k R ≫ 1 ≫ k l where l is a characteristic length of the source and k = 2 π f c / c is the wave number based on the smallest cut-off frequency that is f c = 50 Hz ). For this investigation, N o b s = 20 receivers were placed on the sphere of radius R = 6 m encompassing the source (

sphere. The circle is in the plane perpendicular to the chord of the blade (i.e. in the plane ( y → , z → ) . The 50 receivers are identified by their polar coordinates ( R , φ ) . The angular origin is from the z → axis. The positions of 4 of the 50 receivers are illustrated in

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,

For example,

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 (

Consider the receivers in

In addition, the two curves of the mean quadratic sound pressure are similar, since the relative error remains less than 3% (

only accumulates 23% (

One considers the reconstructed sound power (Equation (14)) with q = 10 and q = 5 . As shown in

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 (

These previous observations are in agreement with the Q-criterion distribution (

Receivers | Eigen Values | |||
---|---|---|---|---|

φ = 0 ∘ | 1 | 2 | 0 | 3 |

φ = 28 ∘ | 0 | 3 | 1 | 2 |

φ = 56 ∘ | 1 | 2 | 0 | 4 |

φ = 91 ∘ | 1 | 2 | 0 | 9 |

Equation (3) | Equation (14) | ||
---|---|---|---|

q = m | q = 5 | q = 10 | |

P ( dBref 10 − 12 W ) | 54.4367 | 54.271 | 54.3489 |

Error (%) | 0 | 0.3 | 0.16 |

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.

Equation (3) | Equation (17) | |
---|---|---|

q = m | q = 10 | |

P ( dBref 10 − 12 W ) | 54.4367 | 54.2032 |

Error (%) | 0 | 0.43 |

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 (

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

Q-criterion (

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.

A CFD calculation was performed using OpenFOAM in order to estimate the wall pressure fluctuations on the surface of a blade located in a channel. The loading noise was then evaluated using the FW&H acoustic analogy and decomposed using both the POD and SVD methods. It was observed that the sound power and the radiated pressure were recovered using these two methods, even if only a few modes were considered. In the configuration studied here, 10 POD modes or 1 SVD mode can be sufficient to predict the radiated sound pressure. Whatever the approach, the trailing edge of the blade is distinguished. An acoustic treatment or a geometrical modification of this zone can therefore influence the blade’s acoustic radiation.

It was shown that the POD approach provides information about the directivity of the source, which is not the case when using the SVD approach. However, the latter approach would be employed when a global noise reduction is sought, no matter the direction of the sound. On the contrary, if a decrease is preferred in a particular direction, the POD method should be used.

This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC). The authors also wish to thank Compute Canada in Sherbrooke, Quebec, for its assistance.

Kone, T.C., Marchesse, Y. and Panneton, R. (2017) A Numerical Approach to Possible Identification of the Noisiest Zones of a Wall Surface with a Flow Interaction. Open Journal of Fluid Dynamics, 7, 525-545. https://doi.org/10.4236/ojfd.2017.74036