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AEROACOUSTICS

© von Karman Institute for Fluid Dynamics

Beamforming in Non-ideal Acoustic Environments( Grid Adaptation for L1 Norm Generalized Inverse Beamforming)

Peter Toth

Environmental and Applied Fluid Mechanics Department, von Karman Institute for Fluid Dynamics, Belgium, [email protected]

Fluid Mechanics Department of Budapest University of Technology and Economics, [email protected]

Supervisor: Christophe Schram

Associate Professor, Environmental and Applied Fluid Dynamics Department, Aeronautics and Aerospace Department,

von Karman Institute for Fluid Dynamics, Belgium, [email protected]

University Supervisor: Mate Marton Lohasz

Assistant Professor, Department of Fluid Mechanics, Budapest University of Technology and Economics, Hungary, [email protected]

Abstract

This paper presents some advancement of the acoustics beamforming technique developed in the VKI. The devel-opment is focusing on the L1 norm generalized inverse beamforming methodology. The algorithms are improvedand the present study introduces an adaptive methodology to refine the source grid. The refinement gives a non-uniform, unstructured source plane grid. The method helps to achieve similar results compared to the uniformgrid solution with reduced number of grid points. This is useful when the searching zone and the grid is relativelylarge compared to the extent of the true sources. With this method the required computational resources for theinversion procedure can be decreased. Results are promising, but still improvement of the adaptive refinementalgorithm is advisable. The performance of the algorithm is presented in this paper on a numerical test case forthe generalized inverse beamforming algorithm.

Keywords: acoustics, generalized inverse, beamforming, adaptive grid, condition number

1. Introduction

There is an increasing desire for acoustic noise re-duction of machines, those principles of fluid mechan-ics nature. Along with the numerical aeroacoustic ad-vancements of understanding the principles of aerody-namic noise sources the need for experimental testingis increased to quantify complex sources or providevalidation database.

The aim of this PhD project is to develop quanti-tative acoustics measurement configuration for aeroa-coustic source localization in non-ideal acoustics en-vironments. In several cases it is desirable to pre-form aeroacoustic measurements in non-anechoic en-vironments, such as aerodynamic wind tunnels or insitu industrial measurements. The beamforming tech-nique would extend the use of advanced aeroacoustictesting in a cost efficient way in such environments.The beamforming method was first used in radio as-

tronomy and radar technology to obtain a direction-ally sensitive antenna for the detection of the incom-ing signal direction [1]. The principle of this tech-nique is to utilize the signal of several microphones insuch a way that the directional sensitivity of device isadjustable. This is usually achieved with digital sig-nal processing techniques on the simultaneously ac-quired data of the microphones. The acoustics fieldcan be scanned point by point in order to find the un-known source positions by ‘steering’ the array sensi-tive ‘spot’ (main lobe) into different spatial directions,at a given frequency. The result is the so called thebeamforming map, shows the source amplitude dis-tribution in the spatial region that have been scanned.Today’s this technique is widely used in different con-figuration for acoustics source identification. Manyapplication dependent microphone arrangement exist,using advanced data processing algorithms, in order


to obtain high resolution and clean source maps [2].One of the method is using the mathematical conceptof generalized inverse of matrices [3]. This is the tech-nique used in the VKI beamforming project.

After the description of the motivation of the study,first the theoretical background of the used general-ized inverse method will be presented in the first sec-tion. This is followed by a description of the gridadaptation technique and the used numerical sourcemodel. Then the adapted grid results will be showncompared to a conventional uniform grid. This is fol-lowed by the discussion of the results. Finally thestudy is concluded in the last section.

1.1. The motivation

The previous investigations in the VKI of the gen-eralized inverse beamforming (GIBF) technique hasbeen showed that the method is sensitive to the sourcemodeling error introduced by a source grid that is notincludes all the significant sources of the measure-ment. This is often a situation when uniform sourcegrid is used and the measurement contains disturbingsources far from the investigated model. This is a typ-ical situation of a measurement in non-ideal acous-tic environment. These errors decrease the robustnessof the method. The uniform grid is usually extendedonly with a small amount beyond the expected loca-tion of the sources in order to reduce the computationresources and in this way this type if grid is not suit-able for such measurement cases.

The current study gives an attempt to solve theproblem by automatically adjust the model source gridin order to place many grid points close to the approx-imate location of the sources and remove the modelsources where no source is expected. The approxi-mate location of the sources is detected by the firstiteration on a uniform coarse grid that can be chosento include not only the investigated sources but otherdisturbing sources as well. The adaptation of the gridlater gives sufficient source resolution for the final it-erations.

2. Theory

Many different beam-forming methods have beendeveloped in the past few years for aeroacoustic ap-plications [2]. In this section the generalized in-verse beamforming method is introduced based onthe formulation of [3] with iteratively reweighed leastsquares (IRLS) method for the solution of the L1 normproblem.

2.1. The microphone array data model

The microphone array data represents the soundfield generated by physical sound sources. Thesesources are modeled with an appropriate source modeland then this model problem is solved to retrieve theunknown model source amplitudes those are supposedto give a good estimation of the true physical sources.In this paper monopole representation of the soundsources is assumed. The transfer matrix involves eachmodel source and each microphone can be written as:

Vk,l =e−ik|r(ym

k ,ysl )|

4π|r(ymk , y

sl )|

(1)

where the transfer matrix V is called the array man-ifold matrix in beamforming terminology. The dis-tance between the microphone position ym

k , k = 1..Nm,and source position ys

l , l = 1..Ns is r = ymk − ys

l . Thewave number is k = c/ω at a given circular frequencyω. The number of microphones and model sourcesdenoted by Nm and Ns respectively.

The model sources can be represented by their crosscorrelation matrix Css of size Ns × Ns formed by theunknown complex source amplitudes s. The sourcecross spectrum matrix is diagonal if the model sourcesare perfectly uncorrelated (they are incoherent) givinga full rank matrix. The matrix is singular with therank of one, if the model sources are perfectly cor-related (coherent sources). Finally, in case of partlycorrelated model sources the matrix has a full rank,with non-zero off diagonal elements.

During the measurement, the microphone arraycross spectrum matrix Cxx can be computed from themeasurement time data with the help of fast Fouriertransformation. The elements of Cxx represents theamplitude and phase correlation between the acousticssignal sampled at different spatial points. The rank ofthe array cross spectrum matrix indicates the numberof incoherent sources. The array cross spectrum ma-trix Cxx can be represented by its eigendecompositionsince it is Hermitian.

Cxx = UΛU† (2)

Introducing the previous source reconstruction ideathat the model sources with proper amplitude distri-bution can describe measured cross spectrum the fol-lowing relation can be written:

UΛU† = VCssV† (3)

This equation is the basis of the GIBF formulation.


2.2. Generalized Inverse Beamforming algorithm

The GIBF algorithms are based on the inversion ofthe wave propagation problem relating the assumedpossible source distribution and the measured acous-tic field. The idea of the reconstruction is to treat onecoherent source distribution at a time, meaning thata microphone array cross spectrum matrix Cxx canbe constructed from the summation of Cn

xx for eacheigenvalue λn of Cxx, where n = 1..N. The problem issolved for the desired number of eigenvalues and thefinal source map is the squared sum of the resultingreconstructed source maps for each n. The problem ofEqn. 3 can be simplified with this idea as follows:

un√λn = Vsn (4)

Introducing the notation bn = un √λn for the eigen-mode n the following inverse problem has to be solvedto retrieve coherent source amplitudes sn:

sn = V−1bn (5)

In order to retrieve the source amplitudes general-ized inverse method has to be used, because of thepropagation model can contain more sources than mi-crophones or vice verse, therefore the coefficient ma-trix of the problem is non-square.

Non-square coefficient matrices represent underde-termined problems when the number of unknownsis larger than the number of available observationsNs > Nm. Overdetermined system is also possiblewhen the number of unknowns is less than the num-ber of observations Ns < Nm. The non square matri-ces arising in the beamforming problems are usuallyunderdetermined since the number of measurementpoints are limited by the applied hardware (the num-ber of microphones and acquisition channels) and thenumber of unknown model source amplitudes are userdetermined by taking into considerations the resolu-tion of the algorithm (that depends on the frequency ofinterest and the microphone array size) and the scan-ning zone size.

2.3. Regularized solution methodology

If the problem is underdetermined a robust solutionof Eqn. 5 can be obtained by imposing additional apriori information on the sources. Certain order ofsmoothness of the solution can be such an additionalinformation. In the simplest case a constraint givingpriority to the source distributions with smaller normcan be constructed. In order to allow this side con-strain to have an effect on the solution, the residual of

the solution in some p norm Rp = ||Vsn − bn||p is al-lowed to be higher than zero. These requirements canbe formulated as a minimization of the following costfunction.

Jp = ε||sn||p + ||Vsn − bn||p (6)

The solution of the inverse problem formulated asa minimization problem helps to overcome on theproblem of the often badly conditioned V. The costfunction of Eqn. 6 with p = 2 represents the origi-nal Tikhonov regularization problem where the regu-larization parameter is ε. The regularization param-eter can be considered here as a user definable pa-rameter controlling the balance between the solutionnorm and the residual norm Rp. If ε = 0 the resid-ual norm is minimized leading to the unregularizedsolution. This can give meaning full result in numer-ical test cases when the condition number of the ar-ray manifold matrix V is small and there is no mea-surement error present in bn. When ε = ∞ the ef-fect of the constraint vanishes and only the solutionnorm is minimized. This would essentially lead to thes ≡ 0 solution. Excessive regularization also decreasethe resolution of the source maps, due to suppressingthe effect of the smallest singular values of V [4] .Knowing that the small or zero regularization givesnon-physical results, in real measurement cases theamount of the applied regularization should be care-fully adjusted to the investigated case otherwise quan-titative beamforming or desired resolution is not pos-sible. In order to choose an optimal regularization pa-rameter the L-curve analysis is used in this study, bydefining the optimum at the corner of the curve [5].This is automatically detected by the algorithm in thepresent Lp norm implementation.

The solution of the minimization of the cost func-tion Eqn. 6 with a given ε and p = 2 can be writtenas:

sn = V†(VV† + εI)−1bn (7)

The resolution of this method is only slightly betterthan the least squares beamformer, and as it is pro-posed by Suzuki [6; 3] the issue can be addressed byreplacing the L2 norm with L1 norm in the cost func-tion Eqn.:6 (means p = 1)

This cost function prevents the spreading of the am-plitudes between grid points close to each other. Con-trary to the p = 2 case when direct solution is ex-ist with Eqn. 7 the minimization problem of the costfunction Eqn.: 6 with (p , 2) can be solved only withiterative procedures. One solution, that is proposed


by [3], is to use the iteratively reweighed least squaresapproach to obtain a solution.

snq+1 = WqV†(VWqV† + εI)−1bn (8)

Here Wq = diag(|snq|

2−p) is a weighting diagonal ma-trix, and the iteration counter is q. The iteration isstopped when the solution norm begin to increase, orthe maximum number of iteration is reached, that isqmax = 15 in this study.

During this iteration procedure the number of theunknown model sources can be decreased by remov-ing the smallest amplitude sources at each iteration.This is increasing the speed of the iteration and canalso increase the final resolution of the algorithm.However it is complicating the stopping criteria of thealgorithm since small level sources can be removedby the inappropriate stopping criterion. This proce-dure is proposed to use by [3]. In this paper an alter-native solution methodology is proposed by using anadaptive grid technique to reduce the number of pointstherefore speed up the iteration and allow the properresolution of every sources on the source map. It isworth to note that the problems presented here alwaysfalls into the category of the underdetermined systemof equation.

2.4. Adaptive re-meshing of the model source domain

For the adaptive modification of the model sourcepositions an unstructured grid is created with the helpof the algorithm presented in [7]. The meshing algo-rithm takes the geometry and a distance function asan input and it is initialized with a uniform grid pointdistribution. This initial grid point distribution is de-fined by the user as well as the boundaries of the 2Ddomain where the source model have to be imposed.The distance function is prescribed by the result ofan initial GIBF calculation on the user defined uni-formly meshed domain. In the present case this in-volves only one iteratively reweighed least squares it-erations (q = 1 in Eqn. 8). The resulted source mapis smoothed with a moving average filter in order tohave a smooth size control function and it is used withthe help of a linear interpolation function to obtain therelative distances in the location of every unstructuredgrid points for the meshing algorithm.

The peaks on this map prescribes where the meshshould be fine and the valleys where it can be coarse.In real situations when the aeroacoustic source natureis of broadband type, this grid can be used to calculatesource amplitudes for the close spectral lines also andtherefore it is not needed to be recomputed often.

3. Application on a numerical test

In this section the application of the presented adap-tive model source grid generation and GIBF algorithmis presented on a numerical test.

3.1. The model of the true sources with added noise

The used test case consist of two monopoles placedat x, y coordinates [−2λ,−2λ], and [2λ, 2λ] respec-tively. The wavelength is denoted by λ. The sourceplane is placed 11.66λ away from the plane of the mi-crophone array (corresponds to 1m at f = 4000Hzwith the defined sound speed). Both monopole sourcehas an amplitude of 1 kg/s2. The used microphonearray layout is a multi-arm spiral layout of 32 micro-phone positions, depicted in Fig. 1. The origin of thereference coordinate system is defined in the center ofthe array as it is shown in the picture.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

y [m

]

x [m]

Figure 1: Optimized multi-arm spiral array geometry

In the presented test case the microphone arraycross spectrum matrix Cxx is directly generated fromthe known position, frequency and amplitude of theimposed sources. In order to simulate realistic crossspectrum matrix additional noise is generated and in-troduced to this cross spectrum matrix. The sourcesare considered to be coherent.

3.1.1. Random noise cross spectrum modelThe cross spectrum matrix is a positive semi def-

inite Hermitian matrix. By introducing additionalnoise to simulate the real measurement scenario these


properties has to be respected. The array cross corre-lation is a transformation of the source cross correla-tion matrix with the array manifold matrix that is writ-ten for the true source locations. Since Eqn. 2 holds,each orthogonal eigenmode can be treated separatelyand their cross spectrum can be formed by the signalvectors xn

s .

Cxx =

Nm∑n=1

Cnxx =

Nm∑n=1

1K

K∑m=1

xn,ms xn,m†

s (9)

Averaging process is taken place for K samplesof the microphones signal vector (length Nm) can bewritten:

xns = Vsn + xn

noise (10)

where xnnoise is the additive noise formed by intro-

ducing a signal with random phase and constant am-plitude as a fraction of the monopole source amplitudecomputed for the array center.

xnnoise = β

||sn||

4πLei2πw (11)

where w ∈ U(0, 1) is a vector of Nm elements froma uniform random distribution U(0, 1). The L is thearray source distance (L = 1m in the study), sn is theimposed source amplitude vector and β is a user de-fined parameter controlling the level of the introducednoise. The number of averages K also has an effect onthe generated cross spectrum. As the number of aver-ages increase the noise in the cross spectrum decreasewhile the auto spectra in the diagonal still containsthe averaged level of noise. In this way cross spec-trum matrix with similar white noise representation asof the real measurement scenarios can be build. Thepresent test case β = 0.05 and K = 60.

4. Results

The normalized ordered eigenmodes of the noisymicrophone array cross spectrum matrix is illustratedin Fig. 2. It can be seen that all the eigenmodes arenon-zero due to the added random noise, but the firstone has a significantly larger value than the others.This is the expected coherent mode, corresponds tothe defined sources and the associated noise in thatmode. In the reconstruction of the sources thereforeonly this eigenmode desired to be considered.

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

||bi||1/max(||bi||1)

Eigenmode number (i)

Figure 2: The normalized ordered eigenmodes for the test case

First the reconstruction is presented on a uniformgrid without refinement. The grid size is chosen to be6λ along which 101 grid points are placed in each di-rections. The reconstructed source map is presentedin Fig. 3. Both sources are well retrieved. Thesource map spot size is about 0.3λ computed for thespot half width. The level of the spurious sources issmall. The integrated source amplitude is approxi-mately 0.5[kg/s2] computed for the half width spot.In previous studies linear dependence of the sourceamplitude of the size parameters of integration regionhave been found, therefore the half width integratedvalue (size parameter 0.5) should correspond to ap-proximately of the half of the true source amplitude.One should not forget that the added noise can causesome additional uncertainty. In the light of these con-siderations the reconstructed amplitude correspondswell to the true source amplitude as well as the po-sition of the sources.

The reconstructed source field of the same micro-phone array cross spectrum matrix can be seen inFig. 4 for the adapted grid. The distance functionfor the adaptive mesh generator was produced withone GIBF iteration on a 51 × 51 grid and the resultedsource map is smoothed with a 19 point moving av-erage filter. The final number of grid points is about2930 in this adapted grid, and the model source dis-tribution is plotted in Fig. 5. The model sources arestrongly clustered around the true sources, while inthe other regions of the source map few points areplaced. The grid adaptation algorithm is used 20 retri-angulations in order to obtain the present distributionof the points. The initial uniform distribution of thegrid points for the grid generation algorithm is chosenin order to have approximately the same model source


-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

y \

λ

x \ λ

[4.00kHz], Grid101x101

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

So

uce

str

en

gth

[kg

/s2]

Figure 3: The reconstructed monopole source amplitude on the uni-form source grid 101 × 101

density around the sources as in the 101×101 uniformgrid. The source map spot size is about 0.3λ basedon the spot half width. The spurious sources levelsare higher than in the uniform model source distribu-tion case. The integrated source amplitude is approx-imately 0.46[kg/s2] close to the values found on theuniform grid.

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

y \

λ

x \ λ

[4.00kHz], Grid:Adapt

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

So

uce

str

en

gth

[kg

/s2]

Figure 4: The reconstructed monopole sources on the adapted gridof approximately 2930 model sources. (The colour map is interpo-lated to a 101 × 101 grid)

The convergence of the IRLS iteration is plottedin Fig. 6. The plot shows the continuous decreaseof the absolute value of solution summed for all gridpoints, that is the imposed constrain in the regulariza-tion method.

The effect of the non-uniform grid on the regular-ization method can be measured by comparing the

- 4

- 3

- 2

- 1

0

1

2

3

4

- 4 - 3 - 2 - 1 0 1 2 3 4

y/λ

x/ λ

Figure 5: The adapted model source positions. 101x101

2.2

2.4

2.6

2.8

3

3.2

0 5 10 15 20

||si||p

IRLS Iteration number

Figure 6: The convergence of the GIBF IRLS algorithm on theadapted grid.

singular value distribution of V (the spectrum of V)in case of the uniform and the adapted grid [4]. Theratio of the highest and smallest singular values givesthe condition number of the of the matrix. The valueof the condition number indicates the level of ill-posedness of problem. The higher the condition num-ber the more regularization should be used to over-come the amplification of the errors during the inver-sion process. The singular value distribution for theuniform grid and the adapted grid is shown in Fiq. 7.

The adaptation significantly changed the singularvalue distribution of V. The singular values are signif-icantly decreased. The condition number of the uni-form grid model is 1225 while in the non-uniform gridmodel it is 1990. The adaptation of the grid is in-


creased the condition number, however this increasein not significant and it is not expected that it is chang-ing the behavior of the inverse problem. The conditionnumber of both cases can be considered as relativelysmall and the problem is not severely ill-posed.

0

5

10

15

20

25

0 5 10 15 20 25 30 35

σi

i

UniformAdapted

Figure 7: The distribution of the singular values of V for the twodifferent meshes.

5. Discussion

Comparing the results obtained on the adapted andon the uniform grid both gives very similar sourcemaps. The resolution and the source amplitudes arein good agreement. The uniform grid solution is abit cleaner and the spurious sources are representedwith smaller amplitude than in the case of the adaptedgrid solution. In this respect the uniform grid solu-tion is slightly better than the non-uniform grid so-lution. The actual level of the spurious sources alsodepends on the used noise model. The applied ran-dom noise model significantly changed the distribu-tion of the eigenmodes, that can be observed in Fig. 2.In case of the noiseless cross spectrum matrix onlyone non-zero eigenmode would exist. However in thepresent case all eigenmodes are non-zero and the sec-ond largest eigemode value is about 10% of the largestone. This implies the strong effect of the noise on thereconstruction and on the results. The test case showsthat the algorithm handles well this level of randomnoise. The good resolution of the L1 norm algorithmcan be observed in both cases, that gives a resolu-tion of approximately 0.3λ computed for the spot halfwidth size on the source-map. This is very close tothe λ/4 limit of the resolution. The usability of the L1norm generalized inverse beamforming method on ar-bitrary source map configuration is confirmed by the

test, without the loss of the resolution or source am-plitude prediction.

The convergence of the algorithm is good in thenon-uniform grid also. The convergence characteris-tic shows that several additional iterations could beperformed in order to further increase the accuracy.However it is not expected that the presented resultschanges significantly.

The inspection of the singular values draws an im-portant behavior of the grid adaptation. The distribu-tion of the singular values are significantly changedcompared to the uniform grid, while the conditionnumber, that is the ratio of the highest to smallest sin-gular values, only slightly modified. Considering thefact that the number of model sources decreased by afactor of approximately 3.7 (from 10201 to approxi-mately 2930) and their distribution changed to formtwo dense spots, suggest that the condition numbercannot be significantly changed by varying the modelsource positions and number, in the same area, if theminimum distance between the points kept approxi-mately unchanged. This is an interesting founding ad-ditionally to [8] where they found that the conditionnumber is significantly affected by the microphonespacing, the source distances compared to the wave-length and the source array distance. They found alsothat the condition number of the problem is the small-est when the model source geometry is matching withthe microphone array geometry (i.e. the distributionof the sources on the source plane are similar to thedistribution of the microphones on the array plane).The fact that distributing the sources in an adaptivemanner does not changed the condition number signif-icantly, allows to prepare efficient adaptive grid tech-niques to resolve large source domains with the gen-eralized beamforming method. These findings shouldbe studied more precisely when efficient algorithmsare intended to be created.

Concerning the required CPU time of the algorithmslightly better than linear speed-up have been foundwhen considering the number of grid points used inthe uniform and adapted grids. If the CPU time re-quired for the uniform grid solution is denoted by Tu,then the solution of the same problem with the adap-tive grid method is requires ∼ 0.27Tu, from which there-meshing is ∼ 0.04Tu. The figures are measuredon a 2.5Ghz Intel CPU with 6Mb cache and 2Gb ofRAM. The numbers shows the required computationalresource is reduced, and that can be still slightly im-proved by reducing the time required for the re-buildof the source grid by using a faster implementation ifthe method of by using different technique. The used


method of [7] known to be a bit slow especially onlarge meshes.

The results emphasize that the appearance of thehigher level of spurious sources depends on the modelsource grid. The distribution of the points has to bechosen carefully and low level of spurious sources canbe achieved by using the uniform grid in the expenseof higher computational cost. This results points outthat the method to distribute the model source points iscritical and can be the target of further improvementsof the adaptive source grid beamforming technique.

6. Conclusion

In this paper the adaptive model source grid re-finement technique is presented and evaluated withthe help of a numerical test case. The generalizedinverse beamforming method is used to retrieve thesource positions and amplitudes with the L1 normformulation solved by the iteratively reweighed lestsquares technique. For the regularization parameterchoice the L-curve method is used. The retrievedsource amplitudes and positions agree well with theimposed source parameters. The resolution is closeto the λ/4 limit on the uniform as well as on the re-fined grid. The introduced significant level of randomnoise in the cross spectrum matrix of the microphonearray does not prohibit the good quality reconstruc-tion. On the other hand it causes the appearance ofspurious sources on the source map especially in caseof the adapted source grid. The level of these spuri-ous sources can be possibly decreased with the carefulchoice of the grid size distribution function. This pos-sibility has to be investigated when robust adaptationmethod is searched for.

The solution procedure with the mesh adaptationto the actual source geometry helped to reduce theCPU time of the computation with a factor of approx-imately 3.7. The reason is identified as the reduc-tion of the model source points in the refined mesh.Roughly linear decrease of the computation time withthe amount of source points in the final meshes can beachieved.

The condition number of the model problem is alsoinvestigated. It is observed that the condition numberdoes not changed much between the uniform and re-fined mesh case. This is additionally supports the ideaof using an adaptive model source distribution.

With the help of the adaptive model source gridtechnique the source domain size can be increased.Consequently one of the important source of error,

the model error of the generalized inverse beamform-ing can be decreased. The grid adaptation algorithmtherefore helps to extend the possibilities of the beam-forming methodology for non-ideal acoustics environ-ments.

Acknowledgments

This work is supported by the VALIANT projectcoordinated by the VKI.

References

[1] R. P. Dougherty, Noise source imaging by beamforming, in:Society of Automotive Engineers Brazil Noise and Vibration.Conference, Florianopolis SC, 2008.

[2] L. Koop, Beamforming methods in microphone array measure-ments theory, practice and limitations, in: M. L. Riethmuller,M. R. Lema (Eds.), Experimental Aeroacoustics, von KarmanInstitute, 2007.

[3] T. Suzuki, L1 generalized inverse beam-forming algorithm re-solving coherent/incoherent, distributed and multipole sources,Journal of Sound and Vibration doi:10.1016/j.jsv.2011.05.021.

[4] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems,Society for Industrial and Applied Mathematics, 1998.

[5] P. C. Hansen, The L-Curve and its Use in the Numerical Treat-ment of Inverse Problems in Computational Inverse Problemsin Electrocardiology, Advances in Computational Bioengineer-ing, 2000.

[6] T. Suzuki, Generalized inverse beam-forming algorithm re-solving coherent/incoherent, distributed and multipole sources,in: 14th AIAA/CEAS Aeroacoustics Conference (29th AIAAAeroacoustics Conference), Vancouver, British ColumbiaCanada, 2008.

[7] P.-O. Persson, G. Strang, A simple mesh generator in matlab,SIAM Review 46 (2) (2004) 329–345.

[8] P. A. Nelson, S. H. Yoon, Estimation of acoustic sourcestrength by inverse methods: Part i, conditioning of the inverseproblem, Journal of Sound and Vibration 233 (4) (2000) 639–664. doi:10.1006/jsvi.1999.2837.


Prediction of Free and Scattered Acoustic Fields of Low-Speed Fans:Scattering of Tonal Fan Noise by aRigid Corner

Korcan Kucukcoskun

Ecole Centrale de Lyon, Laboratoire de Mecanique des Fluides et d’Acoustique, France, [email protected]


Associate Professor, Environmental and Applied Fluid Dynamics Department, von Karman Institute for Fluid Dynamics, Belgium,

[email protected]

University Supervisor: Michel Roger

Professor, Laboratoire de Mecanique des Fluides et d’Acoustique,Ecole Centrale de Lyon, France, [email protected].

Abstract

The exact analytical solution for the scattering oftonal fan noise by a rigid corner is addressed. The theoryproposed for a monopole is first extended in order to compute the scattered field of a dipole. Combining theextended model with the dipole array approach, the scattered field of tonal fan noise by a corner is computed.Different blade forces and fan orientations are tested. A very good agreement is satisfied for all configurations incomparison with numerical simulations performed with a commercial BEM software.

Keywords: Aeroacoustics, tonal fan noise, scattering by a rigid corner.

1. Introduction

Exact analytical solutions addressing the acousticfree-field radiation from aerodynamicnoise sourcesexist for many aeroacoustic problems. However, dueto installation effects and presence of solid surfaces inradiation field, free-field propagation condition maybecome invalid. Scattering of acoustic waves by ob-stacles therefore needs being taken into account.

This paper deals with the analytical solution for thescattering of aerodynamic noise by a rigid corner andits application to tonal fan noise. One of the possibleapplication areas is the cooling units of locomotivesin parking position. The noise generated by a coolingunit located on the top of a locomotive and its scat-tered field by the corner of the locomotive-body mayresult in annoyance for passengers waiting on the plat-form. Noise generated by a small wind turbine locatedon the roof of a building and its effect on inhabitantscan also be associated as another application area.

For low-frequency problems, numerical techniquessuch as Finite Element Method (FEM) and BoundaryElement Method (BEM) are already in use to com-

pute acoustic scattering by complex geometries [1].However they are knownto be dependent on the meshresolution, such as the assumption of the requirementof 10 elements per wave length,λ [2; 3]. At higherfrequencies, the number of elements required for anaccurate predictionincreases, hence the numerical so-lution becomes computationally demanding. There-fore, it is more convenient to use analytical solutionsfor high frequency problems if possible. For problemsincluding simple geometries such as an infinite plane[4] or a semi-infinite plate [5], exact analytical scat-tering techniques can be used independent from themesh resolution.An analytical method was proposedby MacDonald in order to deal with the acoustic scat-tering by a rigid wedge [6].

2. Scattering by a Rigid Corner

In the proposedanalytical method, the scatteringproblem is treated as a boundary-value problem. AGreen’s function is derived for the Helmholtz equa-tion with the exact boundary conditions. The rigid-wall boundary condition on the wedge surface and


(ro, θo, φo)

(rs, θs, φs)

−→

ց

θ = 0

φ = 0

θ = π/2

Figure 1: Sketch of thewedge, source-observer positions defined inpolar coordinates

Sommerfeld radiation condition at large distancesfrom the source are satisfied in the derivation.

MacDonald derived the Green’s function for theHelmholtz equation around a wedge using polar co-ordinates [6]. The source and observer coordinatesare (rs, θs, φs) and (ro, θo, φo), respectively shown inFigure1. In MacDonald’s formulation, the source isassumed to be located in the mid-plane of therigidwedge, henceθs = π/2.

For any wedge angle 0< φ < 2π, the Green’s func-tion is defined as [6]

G(rs, ro) =2πφ

∞∑

m=0

ε cos(m′φo) cos(m′φs)

∞∑

k=0

2m′ (2m′ + 4k+ 1)e(m′+2k+ 12 ) 1

2πi

r− 1

2o r

− 12

s Jm′+2k+ 12(κr<)Km′+2k+ 1

2(iκr>) cos(kπ)

Π(m′ + k− 1/2)Π(k)Π(−1/2)

P−m′m′+2k(cosθo). (1)

The parameterε is 1 form= 0 and2 for m> 1 [7].P−m′

m′+2k stands for the general Legendre function wherem′ = mπ/φ. Jυ andKυ are the Bessel function of thefirst kind and of orderυ and the modified Bessel func-tion of orderυ, respectively. Corresponding expres-sions withr> and r< interchange with the definitionof the source and observer positions,r> = max(rs, ro)andr< = min(rs, ro). κ is equal to 2π/λ.

Mel’nik and Podlipenko later proposed an expres-sion for the scattering by a wedge containing softwalls for arbitrary source locations, as [8]

G(rs,ro) =iκφ

∞∑

m=1

sin(m′φs) sin(m′φo)

∞∑

n=0

(2n+ 2m′ + 1) jm′+n(κr<)h(1)m′+n(κr>)

Γ(n+ 2m′ + 1)Γ(n+ 1)

P−m′m′+n(cosθs)P

−m′m′+n(cosθo) (2)

where jυ andh(1)υ are the sphericalBessel and Hankel

functions of the first kind and of orderυ, respectively.The spherical functions are related to the cylindricalBessel and Hankel functions via,

jm′+n(κr<) =

√

π

2κr<Jm′+n+1/2(κr<) (3)

h(1)m′+n(κr>) =

√

π

2κr>H(1)

m′+n+1/2(κr>). (4)

Combining formulations(1) and (2), the Green’sfunction of a rigid wedge for arbitrary source andob-server positions then becomes [14]

G(rs, ro) =−πi

4φ√

rsro

∞∑

m=0

ε cos(m′φs) cos(m′φo)

∞∑

k=0

(2m′ + 4k+ 1)Γ(2m′ + 2k+ 1)Γ(2k+ 1)

P−m′m′+2k(cosθs)

P−m′m′+2k(cosθo) Jm′+2k+1/2(κr<) H(1)

m′+2k+1/2(κr>). (5)

The trigonometric expansion of the Legendre func-tion P−m′

m′+k(cosθ) employed in the computations [9] is

P−m′m′+2k(cosθ) =

21−m′ (sinθ)−m′

√πΓ(1/2−m′)

∞∑

n=0

Γ(n+ 1/2−m′)Γ(2k+ n+ 1)Γ(n+ 1)Γ(2k+ n+m′ + 3/2)

sin[(2n+ 1+ k)θ].

(6)

Formulation (5) is a general solution of acousticdiffraction by a wedge. Replacing the wedgeangleφ by 3π/2 results the specific case of a rigid corner.Equation (5) then becomes


−3 −2 −1 0 1 2 3

−2

−1

0

1

2

y / λ

z / λ

Figure 2: The map of acoustic potential of a point monopole nearbya rigid corner atf = 10 kHz. The point monopole islocated at(1.32λ, π/2, π/6.6).

G(rs, ro) =−i

6√

rsro

∞∑

m=0

ε cos(2mφs/3) cos(2mφo/3)

∞∑

k=0

(4m/3+ 4k+ 1)Γ(4m/3+ 2k+ 1)Γ(2k+ 1)

P−2m/32m/3+2k(cosθs)

P−2m/32m/3+2k(cosθo) J2m/3+2k+1/2(κr<) H(1)

2m/3+2k+1/2(κr>).

(7)

Equation (7) is the exact solution for the acousticwaves scattered by a rigid corner.It will then be usedto predict the scattered acoustic field of sources lo-cated in the vicinity of a rigid corner. For simplicity,a point monopole located nearby a corner will be in-vestigated first. The theory will then be extended to apoint dipole and finally it will be applied to a sourcerepresentation of a fan.

2.1. Scattered field of a monopole

A first test is performed putting a point monopole at(1.32λ, π/2, π/6.6) with respect to the origin of a rigidcorner atf = 10 kHz. Figure2 shows the map of theacoustic potential of a point monopole in theθ = π/2plane.Illuminated and shadow zones due to the cornercan be seen clearly.

Once the Green’s function is derived for a particu-lar geometry, the acoustic field can then be computedfor harmonic sources. The total acoustic field of amonopole is equal to

p(ω, r) = G(rs, ro)q(ω) (8)

where q(ω) is the source strength of the pointmonopole.

10

20

30

40

50

30

210

60

240

90

270

120

300

150

330

180 0

dB

Figure 3: Directivity of a monopole located nearbyof a rigid corner;analytical solution (line) and BEM results (symbols) atf = 1 kHz.The monopole is located at (1.32λ, π/2, π/6.6) and observer dis-tance is 3λ.

A second test is performed in order to compare theanalytical solution with numerical simulations. BEMis known to be able to handle scattered field problemsfor unbounded domains [3]. However, since at least10 acoustical elements per wavelength isassumed tobe required in an accurate BEM computation, in orderto solve the BEM problem in a reasonable time, thetests are performed at lower frequencies. The compu-tations are then performed atf = 1 kHz. A harmonicmonopole source is located at (1.32λ, π/2, π/6.6). Thesource strength of the monopole is selected asq(ω) =(0.01+ 0.01i)kg/s2. In order to minimize effects ofthe free edges, a corner containing two 7λ× 7λ flatplates is built. The size of the quadrilateral elements,l ≈ 0.1λ, are satisfying the BEM criteria [2]. The totalnumber of acoustic elements is then equal to 7200.

Figure 3 shows the directivity of the monopole-corner configuration atf = 1 kHz computed with theanalytical solution (line)and with BEM (symbols).The radius of the field point mesh is selected as 3λ.The sound pressure levels are plotted in dBs. A verygood agreement is observed in comparison of the am-plitudes with the analytical and numerical solutions.The difference for all the directions is not more than1 dB. It may be related to the scattering of the wavesfrom the free edges of the acoustic mesh.

It appears that the scattered field of a pointmonopole by a rigid infinite corner can be com-puted accurately with the analytical model. Multipolemonopoles can be summed up if needed. However,for a point dipole a specific development is more ap-propriate which is derived in the following section.


−3 −2 −1 0 1 2 3

−2

−1

0

1

2

y / λ

z / λ

Figure 4: The map of acoustic potential of a point dipole nearbya rigid corner at f = 10 kHz. The point dipole is locatedat(1.32λ, π/2, π/6.6).

2.2. Scattered field of a dipole

The radiation field of a point dipole is related tothe gradient of the Green’s function with respect tothe source coordinates. The acoustic pressure then be-comes

p(ω, r) = ∇G(rs, ro) · F(ω) (9)

whereF(ω) is the source strength of the point dipole.Detailed derivations of the Green’s function of a rigidcorner for the dipole source can be found in the relatedreference [15].

A contour plot in theθ = π/2 plane of the acous-tic potential of a point dipole located nearby a cor-ner is seen in Figure4. The dipole is oriented at thesame position as the monopole described in the previ-ous section applying perpendicular to theφ = 0 plane.The dipole-like behavior, reflected waves and shadowzone are clearly seen in the figure.

The analytical solution is again compared to the nu-merical model. A perpendicular dipole toφ = 0 planeis again tested. The strength of the dipole is selectedasF(ω) = (0.01+ 0.01i)N.

Figure5 shows the directivity of the dipole-cornerconfigurations atf = 1 kHz. The line and symbolsrepresent the resultsobtained with the analytical solu-tion and BEM model, respectively. A good agreementof the analytical and numerical solutions is again ob-served. The difference between the analytical and nu-merical results for a dipole is less than 1 dB.

Once validated for a single dipole, the theory is nowextended to an equivalent distribution of a tonal fannoise source.

30

40

50

60

70

30

210

60

240

90

270

120

300

150

330

180 0

dB

Figure 5: Directivity of a dipole located nearbya rigid corner; ana-lytical solution (line) and BEM results (symbols) atf = 1 kHz. Thedipole is located at (1.32λ, π/2, π/6.6) perpendicular toφ = 0 planeand observer distance is 3λ.

2.3. Scattered field of tonal fan noise

Several closed-form solutions addressing the free-field tonal fan noise exist in literature [11;12; 13].The exact closed-form analytical solution of free-fieldtonal fan noise is obtained by introducing the free-field Green’s function into the solution of the inho-mogeneous wave equation (see the related references[12; 13] for further derivation).

A tonal fan noise formulation taking near-fieldterms intoaccount was derived by Roger [13]. Theacoustic pressure at thenth harmonic of the bladepassingfrequency (BPF) at the observer is given as[13]

p′nB =iknBΩ

4π

∞∑

p=−∞

(

−G(2)nB−pFD

p xsinθ

+G(3)nB−pFR

p xsinθ +G(1)(FTp (ζ3 − xcosθ) − F(R)

p r ′))

(10)

for B equally spacedidentical blades.FR, FD andFT

are the radial (R), drag (D) and thrust (T) forces acting

βd

Figure 6: Equivalent fan sourcemodeling strategies; (left) singlerotating dipole, (right) continuous array of phase shifted dipoles.


on the blade, respectively. The applying forces are pe-riodic with angular frequencyΩ. knB = nBΩ/c0 is thewave number at thenBth harmonic. The geometricalparameters are shown in Figure7.

Due to its periodicity, the sound field canbe ex-panded as a Fourier series.GN

m is themth Fourier com-ponent of the auxiliary functionsGN (N=1,2,3). Theauxiliary functionsG1, G2 andG3 are defined as

G1(t) =e−ik|R|

|R|2

(

1+1

ik|R|

)

G2(t) = sin(Ωt+ ϕ′ − ϕ) G1(t)

G3(t) = cos(Ωt+ ϕ′ − ϕ) G1(t) (11)

with k = Ω/c0.A common denominator of closed-form methods

mentioned above is the consideration of a rotatingdipole as shown in Figure6 (left). However, intro-ducing the gradient of the Green’s function ofa cornerinstead of the free-field one, it is not possible anymoreto obtain an exact analytical solution for the scatteredacoustic field of a fan operating next to a rigid corner.Still, an equivalent fan source can be obtained em-ploying circular distribution of phase shifted dipolesas shown in Figure6 (right). The phase difference ofthe dipoles is linked to the azimuthalposition of thedipole,βd. Using the dipole array, the equivalent free-field acoustic pressure of aB bladed fan at thenthharmonic of BPF then reads

pnB =BNd

∑

Nd

pnB,d (12)

whereNd is the number of dipoles in the azimuthalarray.

Introducing Equation (9) into Equation (12), the to-tal acoustic pressure field of the fan scatteredby arigid corner reads

r ′ζ3

FRFT

FDy

Ωt+ϕ′

ez

ey

ex

ϕ

θ

xR=x−y

ez

ey

ex

Figure 7: Source and listener coordinates

Figure 8: Sketches of fan-corner configurations; fans operating par-allel to thexy-plane (left) andparallel to thexz-plane (right).

pnB(ω, r) =BNd

∑

Nd

∇G(rs(βd), ro) · F(ω, βd). (13)

Since the Green’s function of arigid corner is ex-tended for arbitrary positions of the source with re-spect to the wedge origin, it can now be used for anypoint dipole of the circular array. Equation (13) isa general solution for a fan operating out ofa rigidcorner. It can therefore be used for different fan-corner applications. The problem is first simplified toa fan operating parallel to thexy-plane. Figure8 (left)shows the sketch of the simplified problem.A possi-ble application area is the cooling units of locomotivesin parking position as mentioned above. Only the scat-tering from the corner of the locomotive is addressed,neglecting other installation effects. Only one side ofthe locomotive is considered for the illustration.

The model fan employed has 10 equally spacedidentical blades. The fan is rotating with 3000rpmmaking the blade loading frequency equal toN/60 =50 Hz. Artificial blade forces are considered for thetest case [15]. The blades are assumed acousticallycompact and reduced to point dipoles. The radius ofthe fan source is selected as 0.15λfor the first BPF,f = 500 Hz. The center of the fan is located at(0.6λ, π/2, π/12.8). The blade Mach number is thenaround 0.1, satisfying the low Mach number condi-tion. The observers are located in theyz-plane with aradius of 1.5λwith respect to the origin of the corner.

Two low-speed axial fans are tested for comparison.The first test includes only the thrust force compo-nent, whereas the second one contains both drag andthrust force components as in a more realistic applica-tion. The radial forces are neglected for both configu-rations.

The problem is also dealt with the numerical ap-proach combining Equation (10) with the BEM for-mulation. The same acoustic mesh and field points


20

30

40

50

60

70

30

210

60

240

90

270

120

300

150

330

180 0

dB

Figure 9: Directivity of a fan operatingparallel to thexy-plane bya rigid corner; analytical solution (line) and BEM results (symbols)for test1. The fan center is located at (0.6λ, π/2, π/12.8). Fan radiusis 0.15λ. Observers are in theyz-plane at a radius of 1.5λ, at the firstBPF, f = 500 Hz.

are used as in the dipole test case for the BEM model.The BEM problem is solved with LMS software Vir-tual Lab [3].

The scattered acoustic field directivities at the firstBPFare seen for the first and the second tests in Fig-ures9 and10, respectively. Line and symbols repre-sent the analytical and numerical results, respectively.A good agreement is found between analytical and nu-merical solutions for both test fans. It is also seenthat adding the drag component results higher acous-tic pressure levels in the shadow zone.

A second fan orientation is selected where the rota-tion axis is perpendicular to thexy-plane, such as theexample of small axial wind turbines located on the

20

30

40

50

60

70

30

210

60

240

90

270

120

300

150

330

180 0

dB

Figure 10: Directivity of a fan operatingparallel to thexy-plane bya rigid corner; analytical solution (line) and BEM results (symbols)for test2. The fan center is located at (0.6λ, π/2, π/12.8). Fan radiusis 0.15λ. Observers are in theyz-plane at a radius of 1.5λ, at the firstBPF, f = 500 Hz.

20

30

40

50

60

70

30

210

60

240

90

270

120

300

150

330

180 0

dB

Figure 11: Directivity of a fan operatingparallel to thexz-plane bya rigid corner; analytical solution (line) and BEM results (symbols)for test1. The fan center is located at (0.66λ, π/2, π/6.6). Fan radiusis 0.15λ. Observers are in theyz-plane at a radius of 1.5λ, at the firstBPF, f = 500 Hz.

roof of buildings. A sketch of the problem is shownin Figure8 (right). The problem can be dealt with themodel described above only considering a new orien-tation of the fan. Same tests, one with only thrust andone with trust and drag forces, are selected for com-parison. The rotation center is at (0.66λ, π/2, π/6.6)and the fan is operating parallel to thexz-plane.

Figures11and12show the scattered-field directiv-ities of the first and second tests, respectively. Theline and symbols represent the analytical and BEMmethods, respectively. It is seen that adding dragforces affects the acoustic field around the rotationplane. Finally, implementation of the proposed ana-lytical model is validated against numerical simula-

30

40

50

60

70

30

210

60

240

90

270

120

300

150

330

180 0

dB

Figure 12: Directivity of a fan operatingparallel to thexz-plane bya rigid corner; analytical solution (line) and BEM results (symbols)for test2. The fan center is located at (0.66λ, π/2, π/6.6). Fan radiusis 0.15λ. Observers are in theyz-plane at a radius of 1.5λ, at the firstBPF, f = 500 Hz.


tions in all tested configurations.As a result, extending the Green’s functionfor arbi-

trary source positions and combining its gradient withthe dipole array model, the scattered field of a low-speed axial fan by a rigid corner is computed accu-rately. For high frequency problems, where the nu-merical methods are demanding, the analytical solu-tion can be useful irrespective of any mesh resolutionissue.

3. Conclusion

The scattering of low-speed axial fan noise by arigid corner is investigated numerically and analyti-cally. It is seen that the scattered-field can be com-puted accurately with both numerical and analyticaltechniques. The numerical approach can be appliedto more complex geometries for bounded and un-bounded domains. On the other hand, for relativelysimple configurations and where the numerical meth-ods are computationally demanding, analytical solu-tion can be useful independent from mesh resolutionat any frequencies.

Acknowledgements

This work is supported by the FP7 CollaborationProject ECOQUEST (grant agreement no 233541).

References

[1] Desmet, W., Sas, P., “Introduction to numerical acoustics”, InLecture Notes of the ISAAC21 Course. K.U.Leuven, Leuven,Belgium, Sept. 23-24, 2010.

[2] Desmet, W., “Boundary element modeling for acoustics”, InLecture Notes of the ISAAC21 Course. K.U.Leuven, Leuven,Belgium, Sept. 23-24, 2010.

[3] Virtual Lab Rev 9: user manual, LMS International, 2009.[4] Powell, A., “Aerodynamic noise and the plane boundary”, J.

Acoust. Soc. Am. 32 (8), 982:990, 1960.[5] Ffowcs Williams, J., E., Hall, L., H., “Aerodynamic sound

generation by turbulent flow in the vicinity of a scattering halfplane”, J. Fluid Mech. 40 (4), 657:670, 1970.

[6] Macdonald, H., M., “A class of diffraction problems”, Proc.London Math. Soc. 14, 410:427, 1915.

[7] Pierce, A., “Acoustics: An introduction to its physical princi-ples and applications”, Ac. Soc. of America, 1989.

[8] Mel’nik, P., Podlipenko, Yu., K., “On Green’s function for theHelmholtz equation in a wedge”, Ukranian Mathematical Jou.45 (9), 1471:1474, 1993.

[9] Abramowitz, M., Stegun, I., “Handbook of MathematicalFunctions”, Dover Publications, 1972.

[10] Crighton, D., G., Leppington, F., G., “On the scattering ofaerodynamic noise”, J. Fluid Mech. 46 (3), 577:597, 1971.

[11] Ffowcs Williams, J.E., Hawkins, D.L. “Sound generated byturbulence in arbitrary motion”, Royal Society of London 264(1151), 321:342, 1969.

[12] Goldstein, M.E. “Aeroacoustics”, McGraw-Hill InternationalBook Company, 1976.

[13] Roger, M., “Noise from moving surfaces”, In advances inaeroacoustics and applications, VKI Lecture Series 2004-05,Rhode-St-Gense, 2004.

[14] Kucukcoskun, K., Roger, M., “On the scattering of aerody-namic noise by a rigid corner”, AIAA paper, Accepted to the18th AIAA/CEAS Aeroacoustics Conference, 2012.

[15] Kucukcoskun, K., “Prediction of free and scattered acousticfields of low-speed fans”, PhD Dissertation (in review), EcoleCentrale de Lyon, 2012.


Unsteady Panel Code for Airfoil Turbulence Interaction Computation

Leandro Dantas de Santana

Aeronautics and Aerospace Department, von Karman Institute for Fluid Dynamics, Belgium, [email protected]


Associate Professor, Aeronautics and Aerospace Department, von Karman Institute for Fluid Dynamics, Belgium,

[email protected]

University Supervisor: Wim Desmet

Full Professor, KULeuven - Department of Mechanical Engineering, Belgium, [email protected]

Abstract

This paper describes the theoretical formulation of an unsteady panel code for the prediction of turbulence-airfoilinteraction noise, valid for incompressible turbulent flows. This panel code formulation adopts the vorticity aselementary distributed singularity, in contrast with other approaches that use distributed sources and a constantintensity vortex sheet. Preliminary calculations present the quasi-steady and unsteady capabilities of this codeto predict the response of a thin airfoil subjected to pitch oscillations and incoming vortical disturbances. Theaerodynamic responses are compared with results obtained with the unsteady thin airfoil theory. The panel codeis shown to predict with good accuracy unsteady response of an incompressible sinusoidal gust for reduced fre-quencies up to 8. This panel method advantages, from the traditional methods, for describing complex geometrieswith reasonable accuracy but very low computational cost, being attractive for conceptual design applications.

Keywords: unsteady panel code, incompressible turbulent flow, turbulence interaction noise, airfoil noise

1. Introduction

In turbofan engines and contra-rotating open rotors(CROR), airfoil-turbulence interaction typically con-sidered as the main noise source generation mecha-nism. For an airfoil free of flow separation, the noisesources are mainly localized around the leading andtrailing edges. The first is the main noise source whenthe incoming flow turbulence level exceeds a certainlevel. For lower levels of incoming turbulence, thescattering of self-generated boundary layer turbulenceat the trailing edge dominates. The present work fo-cuses on the incoming turbulence problem, relevantto the interaction between the wakes shed by the up-stream rotor of a CROR and convected across thedownstream rotor plane.

The problem has been already addressed by manyauthors since the seventies, and significant contri-butions to the field consist of semi-analytical ap-proaches, mostly relying on thin-airfoil theory, based

on a frequency-wavenumber representation of the in-coming turbulence.[1; 2; 3; 4; 5] With the recentdevelopment of computational power, Large EddySimulation (LES) and Direct Numerical Simulation(DNS) techniques have been applied to obtain highaccuracy flow descriptions [6; 7; 8] providing valu-able input for prediction schemes based on the aeroa-coustic analogy. These techniques have been provento be powerful for simplified geometries, giving deepinsight into complex physical phenomena. However,they still suffer from limited industrial applicabilitydue to the large CPU effort involved in the flow calcu-lation.

Discrete vortex boundary element methods are aninteresting low-cost alternative to these expensive ap-proaches, while offering more geometrical fidelitythan the thin airfoil theories that linearize the air-foil shape as a flat plate with zero thickness, cam-ber, and small angle of attack. The present workfollows this alternative approach. Previous works in


this manner include Gennaretti et. al [9; 10] whoproposed a unified boundary methodology for heli-copter rotor aerodynamics and aeroacoustic predic-tions. Grace [11] discussed the respective merits ofusing time domain boundary element methods or har-monic gust analysis. Recently Glegg and Deven-port [12] developed an unsteady vortex-based panelmethod and applied Amiet’s theory to compute thefar-field noise of turbulence-airfoil interaction. Theirresults are in good agreement with the flat plate ana-lytical solution, nevertheless further comparison withexperiments or high accuracy simulations of airfoilswith realistic thickness, camber and angle of attackare still needed. Recently Zheng et al. [13] also useda discrete vortex method to predict the noise of a 3-elements high lift device. His panel method formu-lation, closer to the classical methods described byCebeci et al. [14], shows a remarkable aerodynamicagreement with DNS simulations for the same geom-etry, with an angle of attack up to 8 degrees.

To reach its objective the present paper followsthe Glegg and Devenport [12] unsteady vortex basedpanel method to predict the airfoil response to a tur-bulent inflow. In this paper, the aerodynamic unsteadyresponse of this panel code is compared with analyti-cal solutions obtained from the thin airfoil theory.

2. Panel method

Cebeci et al. [14] present a conventional two-dimensional panel method for the unsteady loadingcomputation. This methodology is based on theLaplace’s solution of a potential flow generated bythe superposition of sources and vortex singularities,where the first is localized on each panel applicationpoint and the second is considered as a constant in-tensity sheet, placed along the airfoil chord. The sys-tem of equations is closed when the Kutta conditionis applied to the trailing edge. The wake vortex sheetintensity is determined by Biot-Savart integration ap-plied to the control surface in two dimensions. Con-servation of the total circulation carried by the airfoiland wake is prescribed according to first principles.

While Cebeci’s formulation has been extensivelyused in aeroelastic and wing morphing problems, ithasn’t been as successful for aeroacoustic prediction.A more fruitful boundary element method for aeroa-coustics, which applies the essence of the ideas de-scribed by Cebeci but with a different singularity dis-tribution, has been presented by Glegg et. al [12].In this approach the panel method adopts point vor-tices, instead of sources, as the base singularity. This

singularity is distributed along each panel. This ap-proach has been shown effective for predicting airfoil-turbulence interaction noise for reduced frequenciesup to 10.

To describe the panel method used in this paper,the Howe [15] functional airfoil surface definition isadopted. In this formulation, for a given function fthe region where f (x) = 0 represents the body con-tour, f (x) < 0 defines the airfoil interior region andf (x) > 0 represents the exterior region. Using theHeaviside function H( f (x)) it is possible to specifythe non-penetration boundary condition to the veloc-ity vector v. Based on this definition, Howe proposesthat the incompressible flow over a nonpermeable sur-face can be expressed as:

∇ × ∇ × (H( f )v) = −∇2 (H( f )v) =

= ∇ × (H( f )ω + n× v|∇ f |δ( f )) (1)

with the following solution:

H( f )v(x, t) = ∇ ×

∫D

ω(y, t)4π|x − y|3

dV(y) +

+∇ ×

∫S

n× v(y, t)4π|x − y|3

dS (y) (2)

where D is the volume exterior to the airfoil surfaceS . The first term of Equation (2) represents the veloc-ity induced by the wake vorticity computed by Biot-Savart integration. The second term of Equation (2)is the contribution of the bound vorticity to the airfoilsurface velocity. Considering a right-handed coordi-nate system, where the indices 1, 2 and 3 represent thedirections i, j and k, respectively, the induced velocityby a vortex of intensity γ(y) is represented as:

n× v(y, t) = −γ(y)k (3)

where n is the unit vector normal to the airfoil surface.Substituting Equation (3) in Equation (2) leads to:

H( f )v(x, t) =

∫D

ω(y, t) × (x − y)4π|x − y|3

dV(y) +

−

∫S

γ(y)k × (x − y)4π|x − y|3

dS (y) (4)

Since we consider a two-dimensional problem,Equation (4) can be simplified to:

H( f )v(x, t) =

∫C

∫ ∞

−∞

ω(y, t) × (x − y)4π|x − y|3

dy3dy1dy2 +

−

∮A

∫ ∞

−∞

γ(y)k × (x − y)4π|x − y|3

dy3dS (y) (5)


where C represents the contour of the plane whichcontains the airfoil and its wake vorticity. Integratingin y3 we obtain:

H( f )v(x, t) = −

∫C

ω(y, t) × (x − y)2π|x − y|2

dy1dy2 +

+

∮A

γ(y)k × (x − y)2π|x − y|2

dS (y). (6)

Equation (6) can be solved numerically for thebound vortex sheet strength γ(y) with the impositionof the non-penetration boundary condition on the air-foil surface control and the Kutta condition at the air-foil trailing edge. The time domain is discretized ininterval ∆t and the wake vorticity is represented by asum of elementary vorticity elements as:

ω(y, t) =∑

m

Γmkδ(y1 − y(m)1 (t))δ(y2 − y(m)

2 (t)), (7)

where Γm represents each discretized vortex circula-tion and y(m) its position on the coordinate referencesystem. Equation (6) then simplifies into:

H( f )v(x, t) = −∑

m

Γk × (x − y(n)(t))2π|x − y(n)(t)|2

+

+

∮A

γ(yk × (x − y))2π|x − y|2

dS (y) (8)

which can be numerically solved by a time-marchingroutine and applying the Biot-Savart law.

Through Bernoulli equation, the instantaneous liftper unit span is determined at each time step:

L = −

∮A

pdx1 =12ρ

∮C|v|2dx1 + ρ

∮A

∂φ

∂tdx1 (9)

with φ defined as the velocity potential. The lift canbe calculated from the vortex sheet strength as:

L =12ρ

∮Cγ2dx1 + ρ

∂

∂t

∮A

∫ s

0γ(s′)ds′dx1 (10)

with all integrals performed along the airfoil bladecontour.

3. Panel code validation

3.1. Steady liftThe following step is the unsteady panel code val-

idation considering its static response verification. Inorder to validate its inviscid steady solution the devel-oped code is verified against the XFOIL solution. TheNACA 0012 pressure distribution is compared againstthe reference solution in Figure 1, along the airfoilchord, for angles of attack of 0o, 5o and 10o.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1

0

1

2

3

4

5

6

7NACA 0012

x/c

−cp

AOA = 0 − XFOILAOA = 0 − PanelAOA = 5 − XFOILAOA = 5 − PanelAOA = 10 − XFOILAOA = 10 − Panel

Figure 1: Steady cp comparison with XFOIL for the airfoils NACA0001, NACA 0005, NACA 0012 and NACA 0024.

3.2. Unsteady response - The Wagner FunctionOnce the steady solution is verified, the next step

is to compare the unsteady panel code response witha case with known analytical solution. A first verifi-cation case is the Wagner function. This solution isbased on the linearized airfoil theory, valid for verythick airfoils subjected to small angles of attack. Thisfunction expresses the time domain airfoil lift, whensubjected to an impulsive angle of attack variation.The impulsive change of the angle of attack generatesa strong starting vortex, which by turns, influences theairfoil velocity. Immediately after the angle of attackimpulse, the airfoil lift is considered as half of thesteady lift. As the starting vortex is convected withthe flow speed, its influence over the airfoil is reducedand, after enough time, its influence completely dis-appears and the flow becomes steady.

An approximation to the Wagner function is givenby the equation:

Ψ(t) = 1 − Φ1e−ε1U∞t/b − Φ2e−ε2U∞t/b (11)

where Φ1 = 0.165, Φ2 = 0.335, ε1 = 0.0455 andε2 = 0.3, so that and the time domain lift coefficient isgiven by:

cl(t) = 2παΨ(t) (12)

and the lift force time variation becomes:

l(t) = ρπU2cαΨ(t) = ρπUcwΨ(t) (13)

where w = Uα is the downwash velocity.


Figure 2 presents a comparison of the results ob-tained by the unsteady panel code and the airfoilsNACA 0001, NACA 0005, NACA 0012 and NACA0024. Here the airfoil NACA 0001 is considered asan airfoil which satisfies the thin airfoil theory due toits reduced thickness.

0 10 20 30 40 500.5

0.6

0.7

0.8

0.9

1

1.1Cl x time

2*t/U

cl/c

l stea

dy

Wagner responseNACA 0001NACA 0005NACA 0012

Figure 2: Airfoil lift response to an impulsive variation on angle ofattack.

3.3. Unsteady response - The Sears Function

The Wagner function is an interesting first assess-ment of the unsteady panel code capabilities to pre-dict the time domain unsteady lift, but, impulsive in-creases in angle of attack are not very common andgenerally speaking the the flow ingested by a liftingsurface is not homogeneous, suffering variations ofvelocity module and angle of attack. In principal,these inhomogeneities can be decomposed by a seriesof trigonometric functions, which in turn can be rep-resentative as a sum of sinusoidal form gusts. A directconsequence of considering the airfoil impacting flowas an sum of sinusoidal gusts is that its time domainlift variation F(t) can be directly represented in thefrequency domain as F(ω).

In the Sears theory the flow inhomogeneities areconsidered to travel frozen along the airfoil chord, un-affected by its pressure field, which means that theirdecaying time is large, in comparison with the charac-teristic traveling time along the airfoil chord. Consid-ering that the transverse disturbances are mostly re-sponsible for the unsteady loads and considering thatlinearized airfoil theory is valid (the thickness, camberand angle of attack small) it is possible to affirm thatthe transverse gust component is predominant over thelongitudinal one.

If steady inhomogeneous flow is assumed in aframe of reference moving with the flow, by means ofa Fourier analysis of this flow, it can be decomposedinto a series of transverse sinusoidal gusts, in the senseof unsteady aerodynamics, which can be character-ized by the chord-wise wavenumber k1 and its con-vection speed U∞ as ω = k1U∞. At a given value ofω or k1 the Fourier coefficient F(k1) of the total un-steady lift force is determined by the Sears theory, interms of transverse fluctuation amplitude w(k1). Theconnection between the lift and the upcoming gust isrepresented on Equation 14:

F(k1) = πρcU∞w(k1)T (14)

where T is an additional aerodynamic transfer func-tion.

In this way the local instantaneous lift fluctuation,per unit span, is given by:

l(k1, y1, t) = 2ρ0U0w(k1)

√1 − y∗11 + y∗1

S ∗(k∗1)e−iωt (15)

where y∗1 = 2y1/c is the non-dimensional chord-wisecoordinate, with reference to the mid chord, and k∗1 =

k1c/2 is the non-dimensional aerodynamic wave num-ber of the incident fluctuation, in the chord-wise direc-tion. The theory applies to a given wavenumber andrequires a Fourier analysis of the incident fluctuations.S (k∗1) is the known Sears function which is expressedin terms of Bessel functions as:

S (k∗1) =

2πk∗1

([J0(k∗1) − Y1(k∗1)

[− i

[J1(k∗1) + Y0(k∗1)

[)−1 (16)

For the comparison of the developed code againstthe Sears function the airfoil NACA 0001 is consid-ered as the flat plate, where the linearized airfoil the-ory is applicable. To assess the capabilities of thepresent code to compute the unsteady response ofthicker airfoils to a sinusoidal type gust the NACA0005, NACA 0012 and NACA 0024 are consideredas well. The Figure 3 shows the panel’s code obtainedresponse compared with the analytical Sears functionsolution.


−0.2 0 0.2 0.4 0.6 0.8 1

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Sears Function

Real(S)

Imag

(S)

k=1

k=2

k=3

k=4

k=5k=6

k=7

k=8k=9

k=10

k=11k=12

k=13

Sears FunctionNACA 0001NACA 0005NACA 0012NACA 0024

Figure 3: Airfoil lift response to a sinusoidal form impact gust.

4. Conclusions

This paper describes the aerodynamic part of val-idation of an unsteady panel code to be used forturbulence-airfoil interaction noise. This code is validfor incompressible flows and it is able to describe si-nusoidal gusts with reduced frequency up to 8. Thesteady solution of this code perfectly matches withthe results given by traditional panel codes. With re-gard to the unsteady solutions, for the case of an im-pulsive angle of attack variation, this code gives anovershoot of the lift for the initial time steps mainly,due to the intensity of the starting vortex, which in-fluence is reduced as it is convected with the flowspeed, and consequently the solution smoothly con-verges to the stationary one. The second unsteady test-ing case, where an sinusoidal gust impinges the airfoilsurface it is seen that the thickness effects diverges thepanel code solution from the analytical one. Based onthis two unsteady analysis, it can be concluded thatthis unsteady panel code is suitable for airfoil sur-face forces prediction, which information is requiredfor the turbulence-airfoil interaction noise prediction.Future works intend to present acoustic comparisonagainst analytical solution cases, high precision sim-ulations and experiments. The present results encour-age further studies, because of the presently developedpanel code presents very low computational cost whencompared to traditional aeroacoustic codes, but showspotential to predict the airfoil with the reasonable pre-cision necessary in conceptual design engineering ap-plications.

5. Acknowledgments

Leandro D. Santana acknowledges the BrazilianCoordination for Improvement of Higher EducationPersonnel - CAPES - process number BEX-0520-10-1. The support of the European Commission, pro-vided under the FP7 DINNO-CROR project (GrantAgreement no 255878), is gratefully acknowledged aswell.

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