Computation of Flow Induced Noise by Large Eddy Simulation with PowellVortex Sound Theory and FW-H Acoustic AnalogyZHANG Nan, ZHANG Xiaolong, WANG Xing, WU Baoshan, XIE HuaNational Key Laboratory of Hydrodynamics, China Ship Scientific Research Center, Wuxi214082, China,

Abstract: The sound generated by an NACA0012 airfoil in the wake of a rod is numerically simulated by twoapproaches, one is LES with FW-H acoustic analogy and the other is LES with Powell vortex sound theory, inorder to compare the prediction accuracy of them. The vortex structures around rod-airfoil were computed byLES and captured by vortex identification(Q). Acoustic predictions were validated by measurement. It shows thecomputed results by two methods are very similar to each other and both of them are proven to be satisfactory inpredicting the noise generated by an unsteady flow. Subsequently, a numerical simulation of wall pressurefluctuations and flow-induced noise of a NACA0015 airfoil were presented by LES with FW-H equation. At twoangles of attack (0°and 8°), the wall pressure fluctuations of the NACA0015 airfoil were computed. Theobtained power spectra of wall pressure fluctuations were analyzed and compared with measured data. And thevortex structures around the airfoil at two angles of attack were simulated and analyzed. After that, the flowinduced noises of the NACA0015 airfoil at two angles of attack were predicted. The radiated sound spectrumwere analyzed and compared with experimental data. Comparison shows that the numerical approach is robustand credible. All the numerical simulations were carried out by the parallel processing in Wuxi supercomputingcenter.Key words: flow induced noise; large eddy simulation(LES); Powell vortex sound theory; FW-H acousticanalogy; rod-airfoil; airfoil

1.1 Introduction

The unsteady flow, wall pressure fluctuations and flow induced noise are three interesting but difficult tasks inthe field of aero-hydroacoustics. Flow induced noise is a serious problem in many engineering applications. Itcan cause human discomfort and influence the stealth operation of military vehicle. In industrial applications, thenoise generated by turbine, propeller, hydrofoil, and even transitional and turbulent boundary layer on thesurface of vehicle are serious concerns.

The mechanism of aerodynamic noise generated by a turbulent shear flow has been a subject of great interestsince Lighthill (1952) first formulated a general theory (acoustic analogy) for the aeroacoustics [1]. In the theory,sound is induced by unsteady flow through the nonlinear interaction of velocity fluctuations, entropyfluctuations, as well as viscous stress. It is possible to predict sound by separate computations of the source flowfield and resulting acoustical field.When a flow encounter a solid body, Curle’s integral solution (1955) to theLighthill equation provides a theoretical framework for predicting the noise generated by the flow-bodyinteraction [2]. The presence of solid boundaries, generally makes the sound radiation more efficient. The mostgeneral form of Lighthill’s analogy is the extension developed by Ffowcs Williams and Hawkings (1969), whichincorporates the effect of surfaces in arbitrary motion [3]. They utilized the powerful technique of generalizedfunction theory to develop the equation that has become associated with their names. The FW-H equation is anexact rearrangement of the continuity equation and the Navier-Stokes equations into the form of aninhomogeneous wave equation with two surface source terms (monopole and dipole) and a volume source term(quadrupole). The method of FW-H acoustic analogy is efficient and physically illuminating. Today, the FW-Hmethod has become a viable approach for the aero-hydroacoustic noise problem (Wang 2006) [4].

One of the drawbacks of the acoustic analogy of Lighthill is that the source term is spatially quite extended. Asobserved by Powell, the sound production in subsonic homentropic flow is associated to the dynamics ofvortices. Vorticity appears to be spatially less extended than the Reynolds stress. Around vortices there is a large

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region of potential flow that actually does not produce any sound. Powell (1964) applied the vortex sound theoryto free-field conditions at low Mach numbers [5]. It was only a special form of Lighthill’s analogy stressing therole of vorticity, in which the role of vorticity becomes explicit. Various modifications of the theory of Powellhave been proposed that more explicitly impose the conservation of momentum and energy on the flow in thesource region (Möhring 1978; Schram and Hirschberg 2003) [6][7]. This also improves the performance of thetheory.

Computational fluid dynamics (CFD) has reached maturity to solve many industrial problems routinely.Consequently, there is now a large and still growing group of experts engaged in the field of fluid-acousticscoupling. For a long time their work was mostly based on analytical and experimental studies, but theastonishing advances in computer technology have made a numerical approach feasible. The most generalmethodology for the prediction of flow induced noise, which is defined as hybrid approach, is to compute thenear-field unsteady flow field using LES with an acoustic analogy for far-field sound.

More recently, computations of flow induced noise of curved geometries have been carried out. Cylinders andairfoils have so far been the most-studied curved geometries, due to the large amount of experimental dataavailable and the fundamental academic interest that they present. A number of time-accurate numerical studieshave been performed around airfoils, generally using acoustic analogy to obtain far-field sound. Wang and Moin(2000) studied the turbulent flow around the trailing edge of a strut by LES [8]. The incompressible flow datawere used to calculate far-field acoustic information using an integral formulation of Lighthill’s equation. Theyfound a reasonable agreement between their computed acoustic field and experimental results. Manoha (2002)performed an LES simulation around a NACA 0012 airfoil which is placed at 5°angle of incidence to theincoming flow [9]. A Kirchhoff formulation was used to calculate the acoustic far field. Oberai et al (2002)simulated the incompressible flow around an Eppler 387 airfoil and used the results as an input to a variationalform of Lighthill’s equation for the computation of the acoustic far field [10]. Marsden (2008) computed theflow around a NACA 0012 airfoil at zero incidence [11]. Results obtained in the large eddy simulation showed awell-placed transition zone and turbulence level in the boundary layer. These were in agreement withexperimental data. Furthermore, the radiated acoustic field was determined directly by the large eddy simulation,without the use of an acoustic analogy. Third-octave acoustic spectra were compared favorably withexperimental data. It showed that high-order numerical schemes can successfully be used to perform directacoustic computations of compressible transitional flow on curvilinear grids. Winkler (2008) studied flow over alow-speed highly cambered airfoil at small negative incidence by LES [12]. The LES and RANS (SST k − ω)simulations were also compared with detailed velocity measurements made by a 3-D hot-wire in the wake. TheLES predicted the wake thickness and deficit much better than the RANS and was the only one to yield the flowseparation at the trailing edge. The acoustic predictions from two formulations of Lighthill’s acoustic analogycompared favorably with the anechoic wind tunnel measurements at low and mid-frequencies.

ZHANG et al. (2010) have numerically simulated the cavity flow induced noise by LES and FW-H acousticanalogy [13]. It is valuable for the succeeded researches. In the paper, the hybrid CFD/acoustic methods areadopted for the acoustic field prediction. The sound generated by an NACA0012 airfoil in the wake of a rod isnumerically simulated by two approaches, one is LES with FW-H acoustic analogy and the other is LES withPowell vortex sound theory, in order to compare the prediction accuracy of them. Subsequently, a numericalsimulation of wall pressure fluctuations and flow-induced noise of a NACA0015 airfoil were presented by LESwith FW-H equation. It shows that both of the two numerical approaches are credible, and the numericalsimulations have reached the reasonable accuracy for engineering purpose.

1.2 Computational methods

1.2.1 Large eddy simulation (LES)

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Large Eddy Simulation (LES) is adopted in the prediction of unsteady flow. The basic assumptions of LES are:(1) transport is largely governed by large-scale unsteady flow and these structures can be computationallyresolved; (2) small-scale flow features can be undertaken by using appropriate subgrid scale turbulence models.In LES, the motion is separated into small and large eddies, this separation is achieved by means of grid volumefilter. The filter function, G(x,x’), implied here is then:

otherwisex

VxVxxG

',0',/1

)',((1)

Filtering the impressible Navier-Stokes equations, one obtains

0)(

ii

uxt

(2)

j

ij

ij

ij

jji

ji xx

pxx

uux

ut

)()()((3)

where filtered quantities are denoted by an overbar, ij is the stress tensor due to molecular viscosity and ij is

the subgrid-scale stress. In the paper, dynamic Smagorinsky model is used to simulate subgrid-scale stress. Thedetailed descriptions of dynamic Smagorinsky model could be found in reference [14],[15].

1.2.2 FW-H acoustic analogy equation

Ffowcs Williams and Hawkings utilized the generalized function theory to obtain the classic equation that hasbecome associated with their names. The FW-H equation can be written as the following inhomogeneous waveequation:

2 2

202 2

1 '( , ) '( , ) ( ) ( ) ( ) ( )n i iji i j

p x t p x t U f L f T H fc t t x x x

(4)where the three terms at the right-hand side of equation (1.4) are monopole, dipole and quadrupole sourcesrespectively ( from left to right).

)/()/(1 00 iii uvU (5))(ˆ nnijiji vuunPL (6)2

0 0( ) ( )ij i j ij ijT u u p p c (7)

'p is farfield pressure fluctuations, ijT is Lighthill stress tensor, ( )f is Dirac delta function, 0 is the ambient

density, 0p is the ambient pressure, H is Heaviside function, u is fluid velocity, v is body surface velocity, c isvelocity of sound, n is a normal vector that points into the fluid.The far field solution of FW-H equation can be written as the following:

020

( )4 ' ( , )(1 )

n nT f

r ret

U Up x t dSr M

20

2 30

( ( ))(1 )

n r rf

r ret

U rM c M M dSr M

(8)

where M is the Mach number vector of a source point on the integration surface, dots on quantities denote timederivative with respect to the source time τ, and the remaining terms are defined as

iin nUU ˆ ， iin nUU ˆ ， iin nUU ˆ ， iir rMM ˆ ， iir rMM ˆ .

2 2 20 0

14 ' ( , )(1 ) (1 )

r r ML f f

r rret ret

L L Lp x t dS dSc r M r M

2

2 30

( ( ))1(1 )

r r rf

r ret

L rM c M M dSc r M

(9)

where iir rLL ˆ ， iir rLL ˆ ， iiM MLL .

dVrK

crK

rcKtxp

fQ

0 33

22

21),('4

(10)

4

5

2

431 )1(3

)1(3

)1( r

rrr

r

rrrrrr

r

rr

MTM

MTMTM

MTK

(11)

322 )1(24

)1( r

iirMrM

r

ii

MTMTT

MTK r

5

2

4

2

)1()1(6

)1(2)1(3

r

rrr

r

rriiMrrr

MTMM

MTMMTMTM

r

(12)

5

22

4

2

3

2

3 )1()1(3

)1()1(6

)1()1(2

r

rr

r

M

r

iiMM

MTM

MTM

MTMTK r

(13)

where jiijrr rrTT ˆˆ is the double contraction of the Lighthill stress tensor ijT , and the other terms are defined as

jiijMM MMTT ，jiijM rMTT

rˆ ，

jiijM rMTTr

ˆ ，jiijM rMTT

rˆ ，

jiijrr rrTT ˆˆ ，jiijrr rrTT ˆˆ . M is the Mach number

vector of a volume source fixed in the body reference frame. Some detailed descriptions could be found inreference [16], [17]

1.2.3 Powell vortex sound equation

According to the theory of powell, the sound sources are related to vortex motions in the low-Mach-numbercondition. The farfield pressure fluctuations, 'p , generated by the unsteady motions of vorticity can be obtainedby solving the following inhomogeneous equation:

2 2 22 2

2 2

1 ' ' ( ) ( )2 2

p u up u u p p cc t t

(14)

The curl of the velocity, u , is the vorticity vector, .For the impressible and isoentropy flow in low-Mach-number condition, the above-mentioned equation can bereduced into (1.12).

2

22 2

1 ' ' ( )p p uc t

(15)

The three dimensional Green function for free space is:

2

1 ( / )( , )4

t x cG x tc x

It has the listed below relation,0 0( / ) ( / )

i i

t x y c t x y cy x y x x y

(16)

x is the observer position and y is the source position.The density perturbation in far field of Powell vortex equation can be deduced as follows：

0' 2 2 3020

( / )1( , ) ( )4 2

t x y cx t v v d yd

c x y

(17)

0' 020

( / )( , ) ( )

4 ii

t x y cx t v

c y x y

202 3( / )1

2 i i

t x y cv d yd

y y x y

(18)

Equation (16) is adopted, so the (18) can be rewritten as:0' 30

20

( / )( , ) ( )

4 ii

t x y cx t v d yd

c x x y

2

02 3( / )12i i

t x y cv d yd

x x x y

(19)

1.2.4 Numerical Scheme

The differential equations are discretized by finite volume method. Bounded central difference scheme isapplied. The velocity-pressure coupling is based on PISO algorithm and algebraic multigrid method is employedto accelerate the solution convergence. The time step is 1×10-5s. y+≈1on the wall. FFT and Hanning window

5

function are used to get frequency spectrum. The flow computations were carried out by using the commercialsoftware FLUENT, and the two codes for solving FW-H equation and Powell equation were written by authorsfor reaserch purposes. All the numerical simulations were accomplished by parallel processing in Wuxisupercomputing center in China.

1.3 Results and Discussion

1.3.1 rod-airfoil(NACA0012)

Nowadays, extensions of unsteady CFD techniques to the prediction of noise generated by high Reynoldsnumber flows around complex geometries have to be benchmarked on relevant test cases.The experimentdescribed in the reference[18] is designed for this purpose. The configuration is that of an airfoil embedded inthe wake of a circular rod. A symmetric NACA0012 airfoil (chord c = 0.1 m) and a circular rod (d/c = 0.1) wereplaced in the potential core of a jet. The airfoil was located one chord-length downstream of the rod. Both bodiesextended 30d in the spanwise direction and were supported by rigid smooth plates. The incoming velocity was72 m/s. The corresponding rod diameter based Reynolds number was about 4.8×104, that of the chord length was4.8×105 and the Mach number Ma is approximately 0.2.The rod sheds the well-known Karman vortex streetwhich acts as an oncoming disturbance onto the airfoil. The flow contains both periodic and broadband vorticalfluctuations. The sound spectrum includes two components, one is Strouhal peak tone, the other is broadbandnoise.

The computational domain, boundary conditions and meshes are displayed in Figure 1.1. The computationaldomain extends 28c in the streamwise direction, 24c in the cross-stream direction and 3c in the spanwisedirection.The boundary conditions are composed of the following: inlet boundary, air velocity is prescribed;outlet boundary, aerostatic pressure is set; wall boundary, which is the surface of rod-airfoil, the no slipcondition is set; slip wall boundary, which are two lateral planes of computational domain, zero shear stress isprescribed. The computational domain is discretized by 5.79 million structured cells. The vortex structuresaround rod-airfoil were captured by vortex identification(Q) which is defined by equation (20), Ω is vortextensor and S is strain rate tensor. Q=2000 in the paper. Q isosurfaces are colored by the magnitude of vorticity.

2 212

Q S (20)

Fig. 1.1 Computational domain, boundary conditions and meshes

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Fig.1.2 Computed vortex structures around rod-airfoil modelTab.1.1 Validations of the computed magnitude and frequency of the tone by two approaches

magnitude(dB)

frequency(Hz)

StrouhalNumber(St)

relative error (%)magnitude frequency St

Exp. 91.7 1382.4 0.192 /FW-H 88.9 1474.7 0.205 -3.1 6.7 6.8Powell 87.5 1455.2 0.202 -4.6 5.3 5.2

Fig.1.3 Comparisons between computed flow induced noise and measurement of rod-airfoil by two hybrid approachesThe computed vortex structures around rod-airfoil model, Q isosurfaces are colored by the magnitude ofvorticity, are shown in Fig.1.2. The flow pattern around the rod-airfoil is very complex, but a clear physicalmechanism exists in the flow which can be observed in the figure1.2. There is a fully three dimensional turbulentflow around the rod-airfoil, which includes many complex multi-scale vortices of different shapes. The largescale vortex shedding pattern can be observed. Behind the rod, an obvious Karman vortex flow is formed. Lotsof vortices are shed downstream of the rod. When the vortices impinged on the airfoil, there are many stronginteractions, induce more complex vortex structures. Because the vortex is a coherent structure essentially, andthe Karman vortex is shed on a natural frequency, so the frequency on it the vortices impinge the airfoil is keptconstant, which is the frequency of the tone corresponding to the main Strouhal peak. After the impingement, thevarious vortices are reformed through some modes such as splitting, dissipation, mergence and perturbation.

The validations of the computed magnitude and frequency of the Strouhal peak tone by two approaches are listedin the table1.1. The comparisons between computed flow induced noise and measurement of rod-airfoil by twohybrid approaches are presented in Fig.1.3. The observers are located 185d from the airfoil midpoint and to theairfoil chord in the midspan plane. Table 1.1 shows that the relative errors of predicted magnitude, frequency andSt of the Strouhal peak tone are -3.1%, 6.7% and 6.8% respectively by LES with FW-H acoustic analogyequation (LES&FW-H); -4.6%, 5.3% and 5.2% by LES with Powell vortex sound equation (LES&Powell). Thecomputational accuracies of the two hybrid approaches are similar. Both of them are credible and satisfactory inthe computation of Strouhal peak tone. Figure 1.3 indicates that the computed broadband noise spectra byLES&FW-H and LES&Powell agree well with experimental result, the shape and magnitude are reasonable, andthe prediction error is within 1dB~7dB. In the frequency range 800Hz~1180Hz and 1700Hz~3400Hz, theLES&Powell approach improves the prediction. In the high frequency range (>4000Hz), the computed results byLES&Powell are worse than that by LES&FW-H. The computational accuracies of the two hybrid approachesare similar. Both of them are credible and satisfactory in the computation of broadband noise.

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1.3.2 NACA0015 airfoil

The validations of two hybrid approaches are accomplished in section 1.3.1, so LES&FW-H is adopted tocomputing the flow induced noise of NACA0015 airfoil. The three-dimensional NACA0015 airfoil has a 0.61mchord (C) and a 1.8m span (L). All computations presented in this paper were made for a free-stream velocity of30m/s, corresponding to a chord-based Reynolds number of 1.3×106. The boundary condition is composed of thefollowing: inlet boundary, air velocity is prescribed; outlet boundary, aerostatic pressure is set; wall boundary,which is the surface of airfoil, the no slip condition is set; slip wall boundary which are two lateral planes ofcomputational domain, zero shear stress is prescribed. The computational domain and mesh are shown inFig.1.4. The computational domain is discretized by structured cells. The Number of cells in the computation is6.75 million. Experiment was carried out in a wind tunnel with an anechoic system, a detailed description of thesound measurement can be found in reference [19].

Fig.1.4 Computational domain (left), meshes on airfoil surface and mid-span plane (middle) and computed streamline and pressuredistribution of airfoil at attack angle 0°and 8° (right)

The vortical flow and wall pressure fluctuations around the airfoil are two important parameters, both in terms ofaerodynamic characteristics and in terms of acoustic behavior. So the flow patterns of airfoil in two angles ofattack are numerically simulated and the results in mid-span plane are presented in figure.1.4. It is shown thatthere is a weak flow separation in the 8° angle of attack and vortical flow is come into being. Vortexidentification (Q) is adopted to analyze the flow structure. Figure 1.5 shows computed iso-surface of Q withcontour lines of root-mean-square (rms) of pressure fluctuations. At 8° angle of attack, the vortex and the rmspressure fluctuations are stronger and more complex than that at 0° angle of attack.

Fig.1.5 Computed iso-surface of vortex identification (Q) with contour lines of root-mean-square of pressure fluctuations (left: 0°,right: 8°)

In order to demonstrate the capability of LES in predicting unsteady flow characteristics, the power spectra ofwall pressure fluctuations on airfoil surface at two angles of attack are simulated and compared with measureddata. Figure 1.6 shows the comparison. The sensor is put on the position of X/C=0.9 on the midline of airfoilsuction side surface, X is the distance from leading edge. The predicted overall shape and the magnitude of thewall pressure fluctuations spectrum are fairly good compared with experiment data. In the frequency range of100Hz~10KHz, the predicted accuracy is 2～8dB. It shows that the computed results agree better with measureddata in low frequency than that in high frequency. Compared with that of 0° angle of attack, the wall pressurefluctuations is increased about 2~7dB at 8° angle of attack. The characteristics of the unsteady flow aroundairfoil are better captured by LES.

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Fig.1.6 Comparison of measured and computed wall pressure fluctuations spectrum

Fig.1.7 Comparison of computed and measured sound spectrum of NACA0015 airfoil (left: 0°, right: 8°)Unsteady flow past an airfoil may create broadband noise. The formation and behavior of a shear layer and itssubsequent interaction with the airfoil surface drive the noise production. In experiment, only lower frequencyresults (<1KHz) is obtained. The comparison of sound measurement and calculation is presented in Figure 1.7.The trend in computed result is consistent with that in measurement. For sound pressure magnitude, thedifference of computed results and experimental data is 2dB~4dB (100~630Hz). In numerical simulation, thespectrum is decreased rapidly as the frequency increases. The comparison between the prediction and the modelmeasurement shows fairly good agreement. It indicates that the unsteady flow field and flow induced noise canbe well simulated, and the simulation method for flow induced noise through LES&FW-H is established.

1.4 Conclusions

The prediction of flow-induced sound is an important and complex issue in fluid-dynamic acoustics field. Thehybrid CFD/acoustic methods or CAA methods are refined for the acoustic field prediction. The sound generatedby an NACA0012 airfoil in the wake of a rod is numerically simulated by two approaches, one is LES with FW-H acoustic analogy and the other is LES with Powell vortex sound theory, in order to compare the predictionaccuracy of them. It shows that both of the two numerical methods are credible. And the overall performances ofthe presented two hybrid methods have been proven to be satisfactory in the prediction of Strouhal peak noiseand broadband noise. Subsequently, a numerical simulation of wall pressure fluctuations and flow-induced noiseof a NACA0015 airfoil were presented by LES with FW-H equation. It shows that the numerical simulationshave reached the reasonable accuracy for engineering purpose. It is obvious that more work is needed in thisarea. The flow induced noise prediction algorithms must resolve the shear layer behavior well. As expected, theprediction of flow induced sound becomes more feasible with continued advance in computational capabilities.The development of computational models also requires excellent experimental data. The detailed experimentalflow and sound research should be dominated by the studies of near-field flow oscillation and far-field sound. Inthe future, it is necessary to carry out the study on more models for the CFD verification, validation. Furtherwork is also required to clarify the requirements for grid resolution and numerical stability for the more complexsubgrid models.

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1.5 References

[1] Lighthill, M. J (1952) On sound generated aerodynamically. Part I. General Theory. Proc. R. Soc. London Ser. A 211 : 564~587.[2] Curle, N (1955) The influence of solid boundaries on aerodynamic sound. Proc. Roy. London Soc. 231A, 1187, 505~514.[3] Ffowcs Williams J. E, Hawkings D. L (1969) Sound generation by turbulence and surfaces in arbitrary motion. Proc. Roy. Soc.London, 264 A, 321-342.[4] Wang, M., Jonathan B. Freund, and Sanjiva K. Lele (2006) Computational prediction of flow-generated sound. Annu. Rev. FluidMech. 38: 483-512.[5] Powell A (1964) Theory of vortex sound. J. Acoust. Soc. Am. 36(1), 177-95.[6] Möhring,W (1978) On vortex sound at low Mach number, Journal of Fluid Mechanics, 85:685–691.[7] Schram, C., Hirschberg, A. (2003) Application of vortex sound theory to vortex-pairing noise: Sensitivity to errors in flow data,Journal of Sound and Vibration, 266:1079–1098.[8] Wang, M. and Moin, P. (2000) Computation of Trailing-Edge Flow and Noise Using Large-Eddy Simulation, AIAA Journal,Vol. 38, No. 12, pp. 2201–2209.[9] Manoha, E., Herrero, C., Sagaut, P. and Redonnet, S. (2002) Numerical Prediction of Airfoil Aerodynamic Noise, AIAA Paper2002-2573.[10] Oberai, A., Roknaldin, F. and Hughes, Th. (2002) Computation of Trailing-Edge Noise Due to Turbulent Flow over an Airfoil,AIAA Journal, Vol. 40, No. 11, pp. 2206–2216.[11] Olivier Marsden, Christophe Bogey and Christophe Bailly (2008) Direct Noise Computation of the Turbulent Flow Around aZero-Incidence Airfoil, AIAA Journal, Vol. 46, No. 4, pp.874-882.[12] J. Winkler and S. Moreau (2008) LES of the trailing-edge flow and noise of a NACA6512-63 airfoil at zero angle of attack,Center for Turbulence Research, Proceedings of the Summer Program, pp.331-342.[13] Zhang, Nan, Shen, Hong-Cui. (2010) Numerical simulation of cavity flow induced noise by les and FW-H acoustic analogy,Journal of Hydrodynamics B, Vol.22, 242-247.[14] M. Germano, U. Piomelli, P. Moin, W. H. Cabot. (1991) A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A 3(7),1760-1765.[15] D. K. Lilly. (1992) A proposed modification of the Germano Subgrid-scale closure method. Phys. Fluids A 4 (3), 633-635.[16] P. Di Francescantonio. (1997) A new boundary integral formulation for the prediction of sound radiation. Journal of Sound andVibration 202(4), 491-509.[17] D. Casalino. (2003) An advanced time approach for acoustic analogy predictions. Journal of Sound and Vibration 261, 583-612..[18] Marc C. Jacob, Jerome Boudet, Damiano Casalino, MarcMichard (2005) A rod-airfoil experiment as benchmark for broadbandnoise modeling, Theoret. Comput. Fluid Dynamics 19: 171-196.[19] William J.Devenport, JoshuaK.Staubs, StewartA.L.Glegg (2010) Sound radiation from real airfoils in turbulence, Journal ofSound and Vibration 329, pp.3470–3483.