RANS Based Prediction of Airfoil Trailing Edge

Far-Field Noise: Impact of Isotropic

& Anisotropic Turbulence

M. Kamruzzaman∗, Th. Lutz †, A. Herrig ‡ and E. Kramer§

Universitat Stuttgart

Institute of Aerodynamics and Gas Dynamics (IAG)

Pfaffenwaldring 21, 70550 Stuttgart, Germany

A RANS based CFD flow solver (FLOWer1) is coupled with an airfoil Turbulent Bound-ary Layer Trailing-Edge Interaction (TBL-TE) far-field noise prediction scheme (Rnoise)for the application of the combined aerodynamic and aeroacoustic airfoil design process.The final form of the noise prediction model is expressed as an integral of the turbulencesources (e.g. Reynolds stress component, integral correlation length scale and velocityspectra in normal direction) over the boundary layer height and another integral in thewave number direction.2,3 The main features of the present RANS based implementationare direct derivation of the required turbulence noise source properties by means of differ-ent turbulence model i.e. one/two equation (isotropic) and Explicit Algebric Stress Model(EARSM). Previous investigations2–4 shows that for the current noise prediction method,the turbulence anisotropy have decisive impact on the predicted noise emission. Thus, theanisotropy features of the flow in the derivation of the noise source parameter are consid-ered by different approaches, and predicted turbulence parameters and noise spectra arevalidated with the experimental data. Encouraging results are obtained. It is concludedthat, one may consider anisotropy by a scaling law based on a set of measurement data interms of quantities that are more easily determined or available from a isotropic RANSdata without additional complexity and computational cost.

Nomenclature

Cf Skin friction coefficientc0 Speed of soundE(k) Kinetic energy density spectrumk1, k2, k3 Wave number in x1, y2 and x3 coordinate directionke Wavenumber of the energy containing eddieskT Turbulent kinetic energyl Isotropic Turbulence length scaleL Wetted length of the trailing edgep Fluctuation pressurep Fourier transform of pRij , Bij Velocity spatial correlation tensorR Distance to the observer from source pointUi, ui Mean and fluctuation velocity componentsU∞, Uc Reference and convective velocityxi, x, y, z Cartesian coordinates

∗Research engineer, Aircraft Aerodynamics, AIAA member.†Senior Researcher, Aircraft Aerodynamics, AIAA member.‡Research engineer, LWT, AIAA member.§Professor, Head of the Institute

1 of 15

American Institute of Aeronautics and Astronautics

14th AIAA/CEAS Aeroacoustics Conference (29th AIAA Aeroacoustics Conference)5 - 7 May 2008, Vancouver, British Columbia Canada

AIAA 2008-2867

Copyright © 2008 by 2008. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

ρ Densityk Wavenumber vectorω Angular frequencyΦij Spectrum tensor of velocity fluctuationΦm Moving axis spectrumΛ2 Vertical integral length scaleε Turbulent dissipation rateδ Boundary layer thicknessδ1 Boundary layer displacement thicknessδ2 Boundary layer momentum thickness

I. Introduction

It is more and more important for modern industries to use noise reduction concepts during the designprocess to make their products comply with noise regulations, and succeed in market competition. Noisereduction concepts, such as the geometrical shape optimization of the aircraft airframe or wind turbine blade,are based on an accurate prediction of the aeroacoustic noise. Beside aircraft noise, noise emission is one ofthe major obstacles for further spread of onshore wind turbines and significantly limits public acceptance.Among other flow induced noise sources field tests5 showed that Turbulent Boundary Layer Trailing-EdgeInteraction (TBL-TE) noise remains the most dominant noise source for modern wind turbines. This TBL-TE noise emission also plays an important role in aircraft airframe noise source, especially for a clean wingconfiguration. The first step for most of the flow induced noise prediction methods used today i.e. simplifiedanalytical,2–4 semi-empirical,6, 7 Computational Aeroacoustics (CAA)8–11 or Stochastic Noise Generation& Radiation (SNGR)12 approaches is an accurate prediction of the turbulence noise source data. Thisdata can be obtained either from a semi-empirical post-processing of data from an integral Boundary Layer(BL) procedure, from a detailed RANS/LES simulation, or from measurements depending on the methodemployed.

Since last decades several efficient TBL-TE noise prediction schemes have been developed in the Instituteof Aerodynamics & Gas Dynamics (IAG), University of Stuttgart (See Figure (1) in Ref.4). Namely thesemethods are Xnoise, XEnoise & Rnoise (current approach). The name convention came from the aerodynamicmethod applied to the noise prediction scheme. For example, Xnoise method is based on airfoil analysis codeXFOIL,13 and XEnoise is based on coupling of XFOIL and finite difference BL code EDDYBL. The XEnoiseprediction scheme is successfully applied in the frame work of several EU projects e.g. SIROCCO (SIlentROtors by aCoustiC Optimization)14 to design new, less-noisy airfoils for the outer blade region of threedifferent wind turbines in the MW class. The designed airfoils were successfully wind-tunnel tested w.r.t.their aerodynamics and aeroacoustic characteristics and full-size tests showed noise reductions.15

Beside Xnoise and XEnoise, the current approach Rnoise uses a RANS flow solver FLOWer1 to calculatethe noise source parameters for the noise prediction model. It is well known that most of the RANS basedflow solvers use the isotropic two-equation turbulence models due to its simplicity in implementation, lesscomputational time and reliable performance. But nearly all existent eddy viscosity type two-equation modelsdo not consider the Reynolds stress anisotropy feature of the flow. This weakness of the linear constitutiverelationship (eddy viscosity type two equation model) for the Reynolds stress tensor can be potentiallyremoved using the so-called anisotropic eddy viscosity models e.g. EARSM. The idea of these modelsis to extend stress-strain relationship (linear constitutive relationship) by adding some nonlinear elementsconstructed from the mean velocity gradient tensor. However, one can also use a full RSM to consideranisotropic effect, but it is still inefficient due to huge computational time and numerical complexity. Theprimary interest of the above discussions and present study is efficient prediction of normal Reynolds stressesand the turbulence integral length scale considering anisotropy from a RANS two-equation model turbulencedata without additional complexity and computational cost for a subsequent computation of a TurbulentBoundary Layer Trailing-Edge Interaction (TBL-TE) noise prediction scheme.4

The most important input parameters for the current noise prediction method2–4 where turbulenceanisotropy needs to be considered are normal Reynolds stress component and anisotropic turbulence in-tegral correlation length scale. In practice, the experimentaly determination of these turbulence noise sourceparameters is always a challenging task. It is therefore advantageous to derive a scaling law to consideranisotropy effects based on a set of measurement data in terms of quantities that are more easily determined

2 of 15

American Institute of Aeronautics and Astronautics

or available from a isotropic RANS data. For example, Figure (2) shows the predicted and measured distri-butions for the vertical turbulent velocity fluctuations

⟨

u22

⟩

, integral correlation length scale, and associatednoise spectra computed by different turbulence models for the airfoil VTE kav16 at x/c = 0.995. Clearly, thiscomputational results (as depicted by the Figure (2)) suggest that a scaling law can improve the accuracyand efficiency of the noise prediction method without additional complexity and computational cost. Fora RANS based method the derivation of such two parameters (e.g. integral correlation length scale andnormal Reynolds stress component) exclusively depends on the type of turbulence model employed. Forexample, a method based on the EARSM or RSM provided all the anisotropic Reynolds stress components,whereas isotropic two-equation turbulence models do not. Thus, for isotropic two-equation turbulence mod-els one needs to use different semi-empirical scaling law to derive an expression for the anisotropic Reynoldsstress components. For the present purpose, three different approaches are considered, namely isotropic⟨

u22

⟩

|kT= 2

3kT , scaling based on⟨

u21

⟩

:⟨

u22

⟩

:⟨

u23

⟩

= 4 : 2 : 3,⟨

u22

⟩

|bl = 89 ·kT and

⟨

u22

⟩

|sc = faniso ·⟨

u22

⟩

iso.

Also if a RANS computation is performed then⟨

u22

⟩

|Bo can be found from linear (e.g. extended Boussinesqhypothesis) or nonlinear constitutive relation depending on the turbulence model applied.

The second important source input parameter, the vertical turbulence integral correlation length scale Λ2

is defined as the integral of the normalized spatial two-point correlation coefficient R22 of the vertical velocityfluctuations

⟨

u22

⟩

. The length scale Λ2, however, is not provided by any established turbulence model orboundary-layer procedure. To derive Λ2 from known quantities, usually, a calculated turbulence length scalel or the mixing length lmix is multiplied by an empirical scaling factor. If a RANS computation is performedwith a dedicated turbulence model, then different approaches can be applied to derive an expression for Λ2.For the present case, various methods are considered i.e. Λ2 as a function of ke or isotropic l, Λ2 based onanisotropic

⟨

u22

⟩

and finally Λ2 depending on the anisotropic scaling factor as derived from a semi-empiricalfunction by fitting experimental data.

This paper proposes different semi-empirical scaling laws for the efficient prediction of the anisotropicnormal Reynolds stresses and integral correlation length scale from isotropic RANS turbulence data for thenoise prediction scheme. For this purpose, several test cases are considered, and computations are performedby isotropic two-equations and EARSM model to investigate the impact of anisotropy on the turbulence noisesource parameters and finally on the noise spectra. A validation study is also conducted to the predictedand measured noise spectra, and the source parameters (i.e. turbulent boundary layer parameters, verticalfluctuation velocity and integral correlation length scale). The experimental results are obtained from theinstitute’s Laminar Wind Tunnel (LWT) at the University of Stuttgart.

II. Prediction Model

A. Aerodynamic Analysis

A general outline of the noise prediction scheme is given in Figure (1) at Ref.4 Within the Rnoise method thetime-mean flow field and the associated turbulent flow characteristics are obtained by solving the ReynoldsAveraged Navier-Stokes (RANS) equations (e.g. with Wilcox-k − ω, SST, EARSM and RSM turbulencemodel) by means of the flow solver FLOWer,1 and used to predict the spectrum of wall pressure fluctuationsat the trailing edge and its far-field noise spectra. For the mathematical description of turbulent flow theRANS equations in integral and conservation form are considered in FLOWer. For details on the govern-ing equations see references.1, 4 The FLOWer flow solver solves the compressible, two or three dimensionalReynolds (Favre) averaged Navier-Stokes equations in integral form. A cell-centered based finite volumeformulation on one block-structured grids (C-type mesh) was utilised for computations presented here. Theconvective fluxes of the main equations were discretized in space applying a second order central scheme witha blend of second and fourth order artificial damping terms, whereas diffusive fluxes were discretized purelycentral. The turbulence equations were discretized by a flux difference first order upwind scheme. Timeintegration to steady state for the main equations was accomplished by an explicit five stage Runge-Kuttascheme with local time stepping, where convergence was accelerated by a multigrid method on four gridlevels with implicit residual smoothing. The source term dominated turbulence equations were integratedin time using a diagonal dominant alternating direction implicit (DDADI) scheme on the finest grid level atvery high CFL numbers.

It is well known that within the RANS equations the additional second order tensor called Reynolds

3 of 15

American Institute of Aeronautics and Astronautics

stress tensor is further formulated by turbulence modelling. For example, a two-equation isotropic turbulencemodel (k − ω, SST etc.) solves two transport equations for the turbulence kinetic energy k and the specificdissipation rate ω together with eddy viscosity µT = ρk/ω. The two-equation models have been widelyused in industrial CFD applications and are known to be simple in implementation and quite reliable inperformance. The k − ω model is significantly more accurate for two-dimensional boundary layer with bothadverse and favorable pressure gradient. Therefore, for the present work computations are performed by thek − ω Shear-Stress Transport (SST) model of Menter and Wilcox.

Isotropic Turbulence

Most of the two-equation models used today are based on the isotropic assumption, where all compo-nents of the Reynolds stress tensor are modelled by only one scalar function, the turbulence kinetic energyk = 1

2 〈uiui〉. In order to ensure this condition in the framework of the applied two-equation models, theconstitutive relation based on the Boussinesq eddy viscosity hypothesis must be appropriately extended17 as

τij = 2νT

(

Sij −1

3

∂Uk

∂xk

)

− 2

3kδij (1)

The extended Boussinesq hypothesis expressed by equation (1) assumes that the turbulent diffusion isisotropic, so that primary shear stresses will be predicted well, but not secondary shear and normal stresses.As a result the Boussinesq hypothesis may not be suitable for many complex flows involving strong three-dimensional effects. Using the standard Boussinesq eddy viscosity hypothesis it is impossible to properlydescribe all turbulent flows with body force effects arising from a system rotation or streamline curvatureand flow, which structures generated by the Reynolds stress anisotropy.

Modification for Anisotropy

These above-discussed weaknesses of the linear constitutive relationship for the Reynolds stress tensor canbe potentially removed using the so-called anisotropic eddy viscosity models. The idea of these models isto extend the stress-strain relationship (1) by adding some nonlinear elements constructed from the meanvelocity gradient tensor. This leads to somehow better approximation of the normal and shear stresses,and therefore, the turbulence anisotropy structure in general. Wilcox17 proposed the following simplifiednonlinear constitutive relation up to second order approximation for their k − ω2 turbulence model

τij = 2νT

(

Sij −1

3

∂Uk

∂xk

)

− 2

3kδij +

8

9

k (SikΩkj + SjkΩki)

Cµω2 + 2SmnSnm. (2)

The term 2SmnSnm in the denominator of the last term is needed to guarantee that⟨

u21

⟩

,⟨

u22

⟩

and⟨

u23

⟩

arealways positive. The coefficient 8/9 is selected to guarantee

⟨

u21

⟩

:⟨

u22

⟩

:⟨

u23

⟩

= 4 : 2 : 3, (3)

for the flat-plate boundary layer. The primary advantage of nonlinear constitutive relations appears to be inpredicting the anisotropy of normal Reynolds stresses,17 which is important in the present noise predictionscheme, and will be seen later.

B. Noise Prediction Model

The noise prediction method considered in the present study follows the spectral solution of the Poissonequation for the surface pressure fluctuations underneath a turbulent boundary layer following Kraichnan,18

Panton and Linebarger,19 Blake,2 and the evaluation of the noise emission from the trailing edge due to thisfluctuating pressure by solving the diffraction problem .20 Following the spectral solution of the Possion’sequation,2, 18, 19 the wave number-frequency spectrum of the wall pressure fluctuation for the source spectrumΦ22(y2, y

′2,k, ω) is given by2

P(k1, k3, ω) = 4ρ2

(

k21

k21 + k2

3

)

∞∫

0

Λ2(y2)

[

dU1(y2)

dy2

]2

· φ22(y2, k1, k3) ·

φm (ω − k1Uc) ·⟨

u22(y2)

⟩

· e−2|k|y2dy2, (4)

4 of 15

American Institute of Aeronautics and Astronautics

where φ22(y2, k1, k3) is the normalized wave number spectra of u2, i.e.

φ22(y2, k1, k3) =Φ22(y2, k1, k3)

〈u22(y2)〉

, (5)

Λ2 represents a vertical integral length scale for the eddy field. And φm(ω−k1Uc) is a moving axis spectrum.

Note that RANS calculations yield⟨

u22(y2)

⟩

and dU1(y2)dy2

, but all the other terms are modelled. The final

form of the far-field pressure density spectra S(ω) can be expressed as,3

S(ω) =L

4πR2

∞∫

−∞

ω

c0k1P(k1, ω)dk1, (6)

where R is the distance to the observer from the trailing edge and L is the wetted length of trailing edge.

III. Turbulence Noise Source Modeling

The turbulent velocity u2 is a stochastic variable. It can be characterised by a 3D wave number-frequencyspectrum for each layer above the wall. Now it is important to model this spectrum Φ22(k) and also theReynolds stress tensor or the vertical velocity fluctuations

⟨

u22

⟩

and integral length scale.

A. Turbulence Energy Spectra & Velocity Correlation

The velocity cross-spectrum tensor (3D spectral density tensor) and the two-point correlation functions forhomogeneous flow form a Fourier transform pair, thus

Φij(k, ω) =1

(2π)4

∫

∞∫

−∞

∫

e−ik·r+ωτ · Bij(r, τ)drdτ, (7)

where Bij ≡ Rij is the two point correlation function separated by the space vector r for homogeneous tur-bulence. The integral of the turbulence energy spectrum E(k, t) over all wave numbers yields the turbulencekinetic energy (per unit mass),21

1

2Bii(0, t) =

1

2

3∑

i=1

⟨

u2i

⟩

=1

2

∫

∞∫

−∞

∫

Φii(k, t)dk =

∞∫

k=0

E(k)dk. (8)

For experimental reasons, a Fourier analysis with respect to one-space coordinate only is sometimes consid-ered. The resulting one-dimensional longitudinal spectrum, F 1

11(k1) and transverse F 122(k1) spectrum can

then be obtained by integration of the 3D spectrum tensor Φij over the k2 − k3 plane

F 111(k1) =

∫

∞∫

−∞

Φ11(k)dk2dk3, F 122(k1) =

∫

∞∫

−∞

Φ22(k)dk2dk3 (9)

If turbulence is isotropic a flexible and useful form for the energy spectrum is22

E(k) =3

B [(µ + 1)/2, ν]

(

σ2

ke

)

(

kke

)µ

(

1 + k2

k2e

)(µ+1)/2+ν, whereB(x, y) =

Γ(x)Γ(y)

Γ(x + y)(10)

where B is called beta function and Γ(.) is the gamma function. σ2ij = 〈uiuj〉 = Bij(0) is covariance and σ2

is the variance of a single velocity component. Here ke represents the wave-number of the energy containingeddies which defines the maximum of the E(k)-spectrum. The classical von Karman spectrum correspondsto µ = 4, and ν = 1/3. This value for ν produces an inertial-subrange spectrum proportional to k−5/3, in

5 of 15

American Institute of Aeronautics and Astronautics

agreement with Kolmogorov’s scaling arguments for this part of the spectrum. The classical 3D von Karmanspectrum is,

E(k) =55Γ(5

6 )

9√

πΓ(13 )

(

σ2

ke

)

·

(

kke

)4

[

1 + ( kke

)2]

17

6

· (11)

and the functional form of the Kolmogorov E(k) in the inertial subrange becomes

E(k) = c · ε 2

3 · k− 5

3 , (12)

where the Kolmogorov constant c ranges between 1.4 and 1.8, and slightly depends on the Reynolds number.Kristensen (1989)23 proposed an equation for the anisotropic spectral density tensor or velocity spectrumwritten as

Φij(k) =

3∑

m=1

Am(k)

(

δmi −kmki

k2

)(

δmj −kmkj

k2

)

(13)

This equation satisfies the incompressibility condition for the turbulence. Note that A1(k) = A2(k) =A3(k) ≡ A(k), and the equation reduces to the familiar equation for isotropic turbulence21

Φij(k) =E(k)

4πk2

(

δij −kikj

k2

)

, where A(k) =E(k)

4πk2. (14)

Assuming isotropic turbulence using the von Karman spectrum the vertical velocity spectrum being relevantto the noise prediction scheme in the k1 − k3 plane parallel to the surface reads as follows,4

Φ22 (k1, k3) =4

9 π· 1

ke· (k1/ke)

2+ (k3/ke)

2

[

1 + (k1/ke)2

+ (k3/ke)2]7/3

·⟨

u22

⟩

, (15)

where kT = 32

⟨

u22

⟩

= 32σ2 is considered. The vertical velocity fluctuations are computed by following ways

in the Rnoise method,

• Isotropic from kT

⟨

u22

⟩

|kT≡⟨

u22

⟩

= 23 · kT

• Isotropic from kT together with an empirical flat plate relation. Assume that,

⟨

u21

⟩

:⟨

u22

⟩

:⟨

u23

⟩

= 4 : 2 : 3,⟨

u22

⟩

|bl ≡⟨

u22

⟩

=8

9· kT (16)

• Modification for anisotropic turbulent flow by an anisotropy factorThe RANS two equation model use isotropic assumption. In order to account for the anisotropy of theturbulence field a modification of equation

⟨

u22

⟩

= 23 · kT is used for predicting the vertical fluctuating

velocity. i.e.

⟨

u22

⟩

|sc ≡⟨

u22

⟩

=2

3· kT · faniso (17)

where faniso is considered as an anisotropy factor, which is derived from the two-point correlationmeasurement data by fitting a smooth curve as depicted in Figure (1)[left most]. These turbulencecorrelation measurements are conducted in LWT by two vertically shifted hot-wire probes at differentflow condition for NACA0012 airfoils.24

• If one use a two equation turbulence models then it is also possible to consider anisotropy from Boussi-nesq hypothesis

(⟨

u22

⟩

|Bo

)

, as described Eqn. (1)

• For the EARSM turbulence model the non-linear constitutive relation(⟨

u22

⟩

|nc

)

, as described Eqn.(2) can be use. Figure 1 [right most] shows the function for the anisotropy factor derived by theconstitutive relation.

6 of 15

American Institute of Aeronautics and Astronautics

B. The Turbulence Integral Correlation Length Scale

For each of the two point correlation Bij(r) an integral scale can be defined once a direction for the separationvector is chosen. For example, integral scales associated with B11 for the principle directions are

Λ111 ≡ 1

〈u21〉

∞∫

0

B11(r1, 0, 0)dr1, Λ211 ≡ 1

〈u21〉

∞∫

0

B11(0, r2, 0)dr2, Λ311 ≡ 1

〈u21〉

∞∫

0

B11(0, 0, r3)dr3 (18)

Similar integral scales can be defined for the other components of the correlation tensor. In index notationfor i = j

Λnii ≡ Λin =

1

Bii(0)

∞∫

0

Bii(rn)drn, (19)

where, the term Λin can be read as the correlation length in the rn direction of the ith component ofvelocity. A special case of this correlation length scale is the classical turbulence integral length scale, that

is, Λi = 1Bii(0)

∞∫

0

Bii(ri)dri. In general, each of these integral scales will be different, unless restrictions

beyond simple homogeneity are places on the process (e.g. like isotropy as discussed). Thus, it is importantto specify precisely which integral scale is being referred to; i.e. which components of the vector quantitiesare being used and in which direction the integration is being performed. The length scale which is importantin the present case is vertical integral correlation length scale (separation in vertical direction ) of the verticalvelocity component Λ22, to be denoted here Λ2. The length scale Λ2 is related to the vertical extent of theturbulent eddies. More precisely, it is defined as the integral of the normalised spatial two-point correlationcoefficient R22 of the vertical velocity fluctuations

Λ2 =

∞∫

0

R22(r2)dr2 =

∞∫

0

〈u2(y2, t) · u2(y2 + r2, t)〉√

〈u22(y2, t)〉 ·

√

〈u22(y2, t)〉

dr2 (20)

Based on the writing of Blake,2 R22 must be understood as the two-point correlation of zero streamwisedisplacement and zero time delay τ = 0 of the two vertical velocity components measured at the verticallyshifted sensors y2 and y2 + r2. The length scale Λ2, however, is not provided by any established turbulencemodel or boundary-layer procedure. To derive Λ2 from known quantities, usually, a calculated turbulencelength scale l or the mixing length lmix is multiplied by an empirical scaling factor. As Λ2 measurements arehardly published a comparison of predicted and measured noise spectra may be used to derive the scalingfactor. Such an approach is proposed by Lutz,3 where the scaling factor is derived depending on the BLdevelopment of the respective flow. There exist different approaches to derive an expression for Λ2 eitherbased on ke or l, but the order of magnitude of l and Λ should be same.21 For the derivation of Λ2 followingapproaches are considered

• Λ2 as function of the isotropic turbulence length Scale lHinze21 proposed a relationship between ke and the longitudinal integral length scale Λf for isotropicturbulence which can be written as

Λ2 ≡ Λf =π

σ2F 1

11(0) =√

πΓ(5

6 )√Γ(1

3 )· 1

ke≈ 0.75

ke(21)

If a RANS computation is performed with a two-equation k − ε turbulence model, then Λ2 can bederived (from ref.21) by the scaling defined as

ke =1.37

lwith the turbulence length scale given as l =

k3/2T

ε(22)

Then comparison to Eqn. (21) yields, Λ2 = 0.75 · l

1.37≈ 0.547 · l (23)

The same relation is also valid for a k − ω type two-equation turbulence model with length scale

l =√

kT

Cµω . It is very important to note that this derivation implies that turbulence is isotropic and Λ2

is nothing but the longitudinal integral length scale Λf derived by assuming σ2 ≡⟨

u21

⟩

= 23kT .

7 of 15

American Institute of Aeronautics and Astronautics

• Λ2 based on wavenumber of the energy containing Eddy, ke

The wave number ke can be determined from the predicted turbulence kinetic energy kT and thedissipation rate ε by comparing the asymptotic behavior of the Karman spectrum Eqn. (11) to theKolmogorov spectrum Eqn. (12) for the inertial subrange. From Eqn. (11) for large k

ke, and using

Kolmogorov spectra Eqn. (12) leads to

ke ≈ 1.0463 · ε

u3, (24)

where u =√

〈u21〉.This relation provide a closer evaluation of the numerical constant const. in relation

ke = 1l = const. ε

u3 proposed by Hinze.21 In terms of the predicted kT the Eqn. (24) can be written as

ke = 1.0463 · ε

(23 )3/2k

3/2T

= 1.922 · ε

k3/2T

, (25)

with Eqn. (21) it finally follows

Λ2 = 0.39 · k3/2T

ε(26)

Because kT and ε are provided by the k − ε turbulence model used in the present prediction schemebased on the RANS results, Eqn. (25) allows to determine Λ2 on a sound basis. In fact, a betteragreement to measured Λ2 distributions could be achieved.

• Modification for anisotropic turbulent flow by an anisotropy factor

It is clear that, consideration of the anisotropy effects within the fluctuating velocity component (aswell as in velocity spectra Eqn. (15)) using any of the method described in previous section changethe length scale equation as,

Λ2 = 0.547 · k3/2T

ε· f3/2

aniso (27)

where faniso is the same function for the correlation factor used in the spectra Eqn. (15). This relationcan be easily derived from Eqns. (9, 21) and (15, 17).

• Anisotropic factor based on linear-nonlinear constitutive relations

Similar to the previous approach one can also derive an anisotropy factor faniso from the linear-nonlinear constitutive relation depending on the turbulence model employed. The following relationcan for example be used as a model of the length scale anisotropy

Λaniso2 = Λf

(

3⟨

u22

⟩

〈u21〉 + 〈u2

2〉 + 〈u23〉

)3/2

≡ Λ2 · f3/2aniso (28)

It should be noted that multiplication by the function faniso with isotropic Λ2 is a direct consequenceof the mathematical definition of the isotropic integral correlation length scale. (See Eqns. (9, 21) and(15, 17). It is very important to note, that determination of the Λ2 in present approach completelydepends on the accurate prediction of the anisotropic normal stress components

⟨

u22

⟩

.

IV. Results and Discussion

The present noise prediction scheme was verified and interesting results were found.4 For this purpose aset of different airfoils considered in the SIROCCO project and within previous investigations were examinedin the LWT applying the new Coherent Particle Velocity (CPV) method.16, 24, 25

A. Outline of the Selected Test Cases

Two airfoils NACA0012 (Case-1) and VTE kav (Case-2) are considered for the present analysis purpose. Itshould be noted that due to the steep pressure rise along the upper surface the VTE kav airfoil features a

8 of 15

American Institute of Aeronautics and Astronautics

rather thick BL and represents a hard test case for any prediction method.16 The aerodynamic measurementswere executed with fixed transition by using turbulators on the pressure and suction side at 5% of the chordlength. Moreover, aeroacoustic measurements were performed by the hot-wire based CPV25 method andturbulence correlation length scale and BL measurements by two vertically shifted X hot-wire probe24 at theLaminar Wind Tunnel (LWT) of the University of Stuttgart.26 The RANS computations for the NACA0012are performed for a Reynolds Number Re = 1.5 · 106, Ma = 0.166 at different lift settings Cl = 0, 0.21, 0.44,and VTE kav case at Re = 3.0 · 106, Ma = 0.178, Cl = 0.7. All RANS computations in present studyare conducted by using isotropic two equation turbulence model, namely Menter (SST) and Wilcox (k − ω)and the Explicit Algebric Reynolds Stress (EARSM) model, and corresponding turbulence properties suchas vertical fluctuation velocity are computed by different approaches discussed in Section (III-A). Also thevertical integral length scale is computed according to the relation of Section (III-B). The chord lengths ofthe airfoils are chosen to c = 0.8m (VTE kav) and c = 0.4m (NACA0012) respectively.

B. Discussion

Comparisons of the RANS data to the results of the XFOIL-EDDYBL turbulent boundary-layer procedure(XEnoise)3 as well as measurement data for Case-2 are analyzed elaborately in Ref.4 It was concluded thatthe Rnoise method provided more promising results than XEnoise and Xnoise approach. Especially withSST turbulence model it shows most efficient results.

The anisotropy on the different Reynolds stress components computed by the constitutive relation Eqn.(1) (for the two equation model) and Eqn. (2) (EARSM model) at x/c = 0.995 are clearly visible from theFigure (3). Reynold stresses as computed by linear constitutive relation shows less anisotropy compared tothe experiment and EARSM data. This may be due to the weakness of the Boussinesq hypothesis as discussedin section (II(A)). The different variables in figure legend denotes the different approaches, e.g.

⟨

u22

⟩

|kT=

23kT ,

⟨

u22

⟩

|bl [Eqn. (16)],⟨

u22

⟩

|sc [Eqn. (17)] and also⟨

u22

⟩

|Bo can be found from linear (e.g. extendedBoussinesq hypothesis) or nonlinear constitutive relation depending on the turbulence model applied [Eqn.(1) and (2)]. Figures (4, 6) and (8) depicted the vertical turbulent fluctuations

⟨

u22

⟩

computed by differentapproaches. It is clear that consideration of anisotropy (either by a function as derived from experimentaldata or from constitutive relation) improve the accuracy of the predicted data compared to the experiment.Especially

⟨

u22

⟩

as derived from nonlinear constitutive relation for the EARSM case (with lightly loaded BL)shows better results than others. On the otherhand, SST/k-ω cases (using linear constitutive relation) alsoprovided significantly good results than others (e.g. from function for anisotropic factor approach Eqn. (16))for almost all flow cases (light and highly loaded boundary layer). The over prediction of

⟨

u22

⟩

|bl approachvalues are due to the empirical relation (see Eqn. 16) which is actually valid for a flat plate turbulenceboundary layer.

Modeling of the anisotropy effect for the Λ2 are clearly visible in Figures (5, 7) and (9) compared toisotropic results. Clearly modification for anisotropy to the vertical velocity fluctuations (correspondingspectra) has direct impact to the integral length scale. The deviation that appears in the Λ2 distribution forthe EARSM compared to the measurement and SST and k−ω results is significant for isotropic assumption,but scaling by anisotropic factor (using linear-nonlinear constitutive relation) significantly improve thisdeviation. It is clearly visible in Figures (5, 7, 9)[right most] that at the near wall region (corresponding toinner layer) k − ω and SST both cases provide almost similar results but in the outer layer Λ2 distributionfor Wilcox k − ω case grows faster than SST. This result is physically relevant to the criteria of the SSTmodel, because in his SST model, Menter retained the near wall advantages of the Wilcox k−ω model whileeliminating its free-stream sensitivity at the boundary layer edge. For the EARSM case, the overshoot at theouter layer is clearly visible. This effect may be due to the large gradient of the vertical velocity fluctuationat the outer layer as depicted in Figure (1) [right most]. In order to control this overshoot at outer layer anadditional BL damping function

(

[1 − [1 − σ2/(kT )max]2)

is incorporated for all cases. This does not haveany significant effect to the overall noise spectra due to the fact that the dominating noise source of TBL-TEnoise is the near wall turbulence (inner layer).27 The Λ2 data as depicted in all plots are outer layer dampeddata. One can observe old (before damped) and new (after BL damping) results at the right most plot of theFigures (5, 7, 9). In order to investigate the implemented modification of anisotropy, similar computationshave been done for Case-2 at higher lift Cl = 0.7. Predicted and measured length scale results computed bydifferent approaches can be seen in Figure (10). A significant improvement is clearly visible for the Λ2|Bo

approach. The variables in figure legend denote different approaches. i.e. Λ2|iso [derived from isotropicrelation by Eqn. (23)], Λ2|iso2 [derived from isotropic relation by Eqn. (26)], Λ2|sc [applying the function for

9 of 15

American Institute of Aeronautics and Astronautics

anisotropic factor by Eqn. (27)] and finally also Λ2|Bo [applying the function for anisotropic factor by Eqn.(28)]. Among all methods modification of anisotropy employing the linear-nonlinear constitutive relations(Eqns. 1, 2) and (28) shows better results. It should be noted that two different measurement data sets asshown in Figure (10) differ from each other by the normalization procedure of the correlation function R22

[see Eqn. (20)]. The Λ2 distribution as computed by the correlation coefficient normalized by the waitingprobe [similar as Eqn. (20)] and both probes vertical fluctuations velocity component are denoted by theblack empty dots and magenta filled dots respectively. It should be noted that for Case-2 the turbulencecorrelation length scales measurement have been conducted by two vertically separated split-film sensors atposition y2 and y2 + r2.

16

In a next step the accuracy of the noise spectra is verified. The predicted and measured noise spectracomputed by different approaches are presented in Figures (11, 12) and (13), (14). The NACA0012 noisespectra is measured by CPV method by Herrig.24 For all cases the observer distance is R = 1m. It can beseen that the RANS based method using the SST/k − ω together with anisotropic scaling (Eqns. 1, 2) and(28) gives rather good results between the spectra as well as for the shapes of the spectra than others. Thisimprovement of the spectra is actually a direct consequence of the efficient approximation of the anisotropyto⟨

u22

⟩

and Λ2 distribution. Whereas EARSM spectra without scaling (with isotropy in velocity fluctuationbut isotropic Λ2) agree well with measured spectra. This can be attribute as overprediction of the nearwall Λ2 distribution and can be viewed in Figures (5, 7, 9) [third plot from left]. It should be noted thatthe second peak at high frequency (∼ 5-7kHz) is caused by Blunt-Trailing Edge (BTE) noise.28 In thepresent prediction scheme BTE is not considered. Figure (15) shows a comparison plot of the measured andpredicted noise spectra by different methods i.e. Xnoise, XEnoise,3 BPM,6, 7 Rnoise (current approach) forNACA0012 airfoil at α = 0o. The collapse of measured spectra scaled with the measured boundary layerdisplacement thickness δ1 is shown in Fig.(15)[right]. The most commonly assumed dependency of radiatedTBL-TE sound pressures in the acoustically non-compact frequency range, p2 ∼ U5δ1, has been applied,with δ1,ref taken for the 0.4m model at 60 m/s. BTE noise contributions again should be ignored. Theagreement in the range of the maximum SPL is not particularly good, the spectra seem over-corrected for thedifferent chords.24 However it cannot be expected, that this approach, neglecting detailed information on thedistribution of turbulent fluctuations in the boundary layer profiles, is very accurate. It should be noted thatwith XEnoise and Xnoise spectra a 10dB extra value is added and, the Rnoise results as depicted in Figure(15) is with isotropic assumption at Re = 1.55e6, Ma = 0.178, U = 60m/s. But employing anisotropic factorsignificantly improve the predicted spectra as observed in previous plots e.g. Figs. (11, 12 and 13). Therefore,it can be concluded that, in a RANS based noise prediction method one can consider the anisotropy effectfrom isotropic turbulence data.

V. Conclusions

A RANS based TBL-TE far-field noise prediction scheme Rnoise is developed. The RANS solution iscoupled with the noise prediction scheme in order to derive all the necessary turbulence noise source data forthe noise prediction scheme. The anisotropic feature of turbulence is accounted for two equation isotropicturbulence data by an anisotropic factor, which is a modification to the vertical Reynolds stress tensor andintegral correlation length scale and can be derived from two different methods, either by linear-nonlinearconstitutive relation or by a semi-empirical function. Predicted noise spectra by linear-nonlinear constitutiverelation based method agrees well with the measurement results, whereas semi-empirical function basedderivation need further improvement.

VI. Acknowledgments

A part of the present work is conducted by the finance of EU project UpWind. The authors would liketo acknowledge the European Commission (Sixth Framework Program (FP6)), and Werner Wurz and A.Ivanov for useful discussion on the experimental results.

10 of 15

American Institute of Aeronautics and Astronautics

VII. Figures

y/δ1

f anis

o

0 2 4 60

0.5

1

1.5

2

Exp.smoothe curvefit

y/δ1

f anis

o

0 2 4 60

0.5

1

1.5

2Rnoise: k- ω (Λ2|Bo)Rnoise: SST ( Λ2|Bo)Rnoise: EARSM ( Λ2|Bo)

Figure 1. The function for anisotropic factor from experimental data (Left)

f_cen [Hz]

L p[d

B/m

],r=

1m

2000 4000600080001000030

40

50

60

70

Exp. LWT, IAG (Array)Exp. LWT, IAG (CPV)RANS, k-ω (Iso~2/3k T)RANS, k-ω (Iso+scaled)RANS, EARSM (Iso+scale)

BTE Noise

Λ2 [mm]

y[m

m]

0 5 100

5

10

15

20

25

30

35

40 Exp., LWT (Aniso)RANS, k-ω (Iso)RANS, k-ω (Iso+scaled)RANS, EARSM (Iso+scale)

<u2u2> [m 2/s2]

y[m

m]

0 10 20 300

5

10

15

20

25

30

35

40 Exp. LWT (x/c=1.0025)RANS, k-ω (Iso.~2/3k T)RANS, EARSM (iso.~2/3k T)RANS, EARSM (Aniso)

Figure 2. Turbulence Properties: VTE kav, Re = 3.093 × 106, cl = 0.7, U∞ = 60 m/s

NACA0012: Re=1.5e 6, Ma = 0.166, c=0.4m, α=0°, C l=0.0

<ui2> [m 2/s2]

y[m

m]

0 5 10 150

5

10

15 Rnoise: k- ω (<u12>|Bo)

Rnoise: k- ω (<u22>|Bo)

Rnoise: k- ω (<u32>|Bo)

Exp. LWT (x/c=1.005)Exp. LWT (x/c=1.005)Exp. LWT (x/c=1.005)

<ui2> [m 2/s2]

y[m

m]

0 5 10 150

5

10

15 Rnoise: SST (<u 12>|Bo)

Rnoise: SST (<u 22>|Bo)

Rnoise: SST (<u 32>|Bo)

Exp. LWT (x/c=1.005)Exp. LWT (x/c=1.005)Exp. LWT (x/c=1.005)

<ui2> [m 2/s2]

y[m

m]

0 5 10 150

5

10

15 Rnoise: EARSM (<u 12>|Bo)

Rnoise: EARSM (<u 22>|Bo)

Rnoise: EARSM (<u 32>|Bo)

Exp. LWT (x/c=1.005)Exp. LWT (x/c=1.005)Exp. LWT (x/c=1.005)

<ui2> [m 2/s2]

y[m

m]

0 5 10 150

5

10

15 Exp. LWT (x/c=1.005)Rnoise: k- ω (<u2

2>|Bo)Rnoise: SST (<u 2>

2>|Bo)Rnoise: EARSM (<u 2

2>|Bo)

Figure 3. Reynolds Stresses computed by Boussinesq hypothesis & Non-linear constitutive relation

<u22> [m 2/s2]

y[m

m]

0 5 10 150

5

10

15Exp. LWT (x/c=1.005)Rnoise: SST (<u 2

2>|kT)Rnoise: SST (<u 2>

2>|bl)Rnoise: SST (<u 2>

2>|sc)Rnoise: SST (<u 2>

2>|Bo)

<u22> [m 2/s2]

y[m

m]

0 5 10 150

5

10

15Exp. LWT (x/c=1.005)Rnoise: EARSM (<u 2

2>|kT)Rnoise: EARSM (<u 2

2>|bl)Rnoise: EARSM (<u 2

2>|sc)Rnoise: EARSM (<u 2

2>|Bo)

NACA0012: Re=1.5e 6, Ma = 0.166, c=0.4m, α=0°, C l=0

<u22> [m 2/s2]

y[m

m]

0 5 10 150

5

10

15 Exp. LWT (x/c=1.005)Rnoise: k- ω (<u2

2|kT)Rnoise: k- ω (<u2

2>|bl)Rnoise: k- ω (<u2

2>|sc)Rnoise: k- ω (<u2

2>|Bo)

<u22> [m 2/s2]

y[m

m]

0 5 10 150

5

10

15 Exp. LWT (x/c=1.005)Rnoise: k- ω (<u2

2>|Bo)Rnoise: SST (<u 2>

2>|Bo)Rnoise: EARSM (<u 2

2>|Bo)

Figure 4. Vertical velocity fluctuation at different method & turbulence model

11 of 15

American Institute of Aeronautics and Astronautics

Λ2 [mm]

y[m

m]

0 1 2 3 40

5

10

15Exp. LWT (x/c=1.005)Rnoise: EARSM ( Λ2|iso)Rnoise: EARSM ( Λ2|iso2 )Rnoise: EARSM ( Λ2|sc)Rnoise: EARSM ( Λ2|Bo)

NACA0012: Re=1.5e 6, Ma = 0.166, c=0.4m, α=0°, C l=0

Λ2 [mm]y

[mm

]

0 1 2 3 40

5

10

15Exp. LWT (x/c=1.005)Rnoise: SST ( Λ2|iso)Rnoise: SST ( Λ2|iso2 )Rnoise: SST ( Λ2|sc)Rnoise: SST ( Λ2|Bo)

Λ2 [mm]

y[m

m]

0 1 2 3 40

5

10

15 Exp. LWT (x/c=1.005)Rnoise: k- ω (Λ2|iso)Rnoise: k- ω (Λ2|iso2 )Rnoise: k- ω (Λ2|sc)Rnoise: k- ω (Λ2|Bo)

Λ2 [mm]

y[m

m]

0 1 2 3 40

5

10

15 Exp. LWT (x/c=1.005)Rnoise: k- ω (Λ2|Bo)Rnoise: SST ( Λ2|Bo)Rnoise: EARSM ( Λ2|Bo)

Figure 5. Vertical integral length scale at different method & turbulence model

NACA0012: Re=1.5e 6, Ma = 0.166, c=0.4m, α=2°, C l=0.21

<u22> [m 2/s2]

y[m

m]

0 5 10 150

5

10

15Exp. LWT (x/c=1.005)Rnoise: SST (<u 2

2>|kT)Rnoise: SST (<u 2>

2>|bl)Rnoise: SST (<u 2>

2>|sc)Rnoise: SST (<u 2>

2>|Bo)

<u22> [m 2/s2]

y[m

m]

0 5 10 150

5

10

15Exp. LWT (x/c=1.005)Rnoise: EARSM (<u 2

2>|kT)Rnoise: EARSM (<u 2

2>|bl)Rnoise: EARSM (<u 2

2>|sc)Rnoise: EARSM (<u 2

2>|Bo)

<u22> [m 2/s2]

y[m

m]

0 5 10 150

5

10

15 Exp. LWT (x/c=1.005)Rnoise: k- ω (<u2

2|kT)Rnoise: k- ω (<u2

2>|bl)Rnoise: k- ω (<u2

2>|sc)Rnoise: k- ω (<u2

2>|Bo)

<u22> [m 2/s2]

y[m

m]

0 5 10 150

5

10

15 Exp. LWT (x/c=1.005)Rnoise: k- ω (<u2

2>|Bo)Rnoise: SST (<u 2>

2>|Bo)Rnoise: EARSM (<u 2

2>|Bo)

Figure 6. Vertical velocity fluctuation at different method & turbulence model

Λ2 [mm]

y[m

m]

0 1 2 3 40

5

10

15Exp. LWT (x/c=1.005)Rnoise: SST ( Λ2|iso)Rnoise: SST ( Λ2|iso2 )Rnoise: SST ( Λ2|sc)Rnoise: SST ( Λ2|Bo)

Λ2 [mm]

y[m

m]

0 1 2 3 40

5

10

15Exp. LWT (x/c=1.005)Rnoise: EARSM ( Λ2|iso)Rnoise: EARSM ( Λ2|iso2 )Rnoise: EARSM ( Λ2|sc)Rnoise: EARSM ( Λ2|Bo)

NACA0012: Re=1.5e 6, Ma = 0.166, c=0.4m, α=2°, C l=0.21

Λ2 [mm]

y[m

m]

0 1 2 3 40

5

10

15 Exp. LWT (x/c=1.005)Rnoise: k- ω (Λ2|iso)Rnoise: k- ω (Λ2|iso2 )Rnoise: k- ω (Λ2|sc)Rnoise: k- ω (Λ2|Bo)

Λ2 [mm]

y[m

m]

0 1 2 3 40

5

10

15 Exp. LWT (x/c=1.005)Rnoise: k- ω (Λ2|Bo)Rnoise: SST ( Λ2|Bo)Rnoise: EARSM ( Λ2|Bo)

Figure 7. Vertical integral length scale at different method & turbulence model

<u22> [m 2/s2]

y[m

m]

0 5 10 15 200

5

10

15

20 Exp. LWT (x/c=1.005)Rnoise: k- ω (<u2

2|kT)Rnoise: k- ω (<u2

2>|bl)Rnoise: k- ω (<u2

2>|sc)Rnoise: k- ω (<u2

2>|Bo)

<u22> [m 2/s2]

y[m

m]

0 5 10 15 200

5

10

15

20Exp. LWT (x/c=1.005)Rnoise: SST (<u 2

2>|kT)Rnoise: SST (<u 2>

2>|bl)Rnoise: SST (<u 2>

2>|sc)Rnoise: SST (<u 2>

2>|Bo)

<u22> [m 2/s2]

y[m

m]

0 5 10 15 200

5

10

15

20Exp. LWT (x/c=1.005)Rnoise: EARSM (<u 2

2>|kT)Rnoise: EARSM (<u 2

2>|bl)Rnoise: EARSM (<u 2

2>|sc)Rnoise: EARSM (<u 2

2>|Bo)

NACA0012: Re=1.5e 6, Ma = 0.166, c=0.4m, α=4°, C l=0.44

<u22> [m 2/s2]

y[m

m]

0 5 10 15 200

5

10

15

20 Exp. LWT (x/c=1.005)Rnoise: k- ω (<u2

2>|Bo)Rnoise: SST (<u 2>

2>|Bo)Rnoise: EARSM (<u 2

2>|Bo)

Figure 8. Vertical velocity fluctuation at different method & turbulence model

12 of 15

American Institute of Aeronautics and Astronautics

Λ2 [mm]y

[mm

]

0 1 2 3 40

5

10

15

20Exp. LWT (x/c=1.005)Rnoise: SST ( Λ2|iso)Rnoise: SST ( Λ2|iso2 )Rnoise: SST ( Λ2|sc)Rnoise: SST ( Λ2|Bo)

Λ2 [mm]

y[m

m]

0 1 2 3 40

5

10

15

20Exp. LWT (x/c=1.005)Rnoise: EARSM ( Λ2|iso)Rnoise: EARSM ( Λ2|iso2 )Rnoise: EARSM ( Λ2|sc)Rnoise: EARSM ( Λ2|Bo)

NACA0012: Re=1.5e 6, Ma = 0.166, c=0.4m, α=4°, C l=0.44

Λ2 [mm]

y[m

m]

0 1 2 3 40

5

10

15

20 Exp. LWT (x/c=1.005)Rnoise: k- ω (Λ2|iso)Rnoise: k- ω (Λ2|iso2 )Rnoise: k- ω (Λ2|sc)Rnoise: k- ω (Λ2|Bo)

Λ2 [mm]

y[m

m]

0 1 2 3 40

5

10

15

20 Exp. LWT (x/c=1.005)Rnoise: k- ω (Λ2|Bo)Rnoise: SST ( Λ2|Bo)Rnoise: EARSM ( Λ2|Bo)

Figure 9. Vertical integral length scale at different method & turbulence model

VTE_kav: Re=3.0e 6, Ma = 0.178, c=0.8m, C l=0.7

Λ2 [mm]

y[m

m]

0 5 100

5

10

15

20

25

30

35

40 Exp. LWT (x/c=1.005)Exp. LWT (x/c=1.005)Rnoise: k- ω (Λ2|iso)Rnoise: k- ω (Λ2|iso2 )Rnoise: k- ω (Λ2|sc)Rnoise: k- ω (Λ2|Bo)

Λ2 [mm]

y[m

m]

0 5 100

5

10

15

20

25

30

35

40Exp. LWT (x/c=1.005)Exp. LWT (x/c=1.005)Rnoise: EARSM ( Λ2|iso)Rnoise: EARSM ( Λ2|iso2 )Rnoise: EARSM ( Λ2|sc)Rnoise: EARSM ( Λ2|Bo)

Λ2 [mm]

y[m

m]

0 5 100

5

10

15

20

25

30

35

40Exp. LWT (x/c=1.005)Exp. LWT (x/c=1.005)Rnoise: SST ( Λ2|iso)Rnoise: SST ( Λ2|iso2 )Rnoise: SST ( Λ2|sc)Rnoise: SST ( Λ2|Bo)

Λ2 [mm]

y[m

m]

0 5 100

5

10

15

20

25

30

35

40 Exp. LWT (x/c=1.005)Exp. LWT (x/c=1.005)Rnoise: k- ω (Λ2|Bo)Rnoise: SST ( Λ2|Bo)Rnoise: EARSM ( Λ2|Bo)

Figure 10. Vertical integral length scale at different method & turbulence model:VTE kav

x x x x x x

fcen [Hz]

L p[d

B/m

]

5000 1000020

30

40

50

60

70

Rnoise: k- ω (Λ2|iso)Rnoise: k- ω (Λ2|iso2 )Rnoise: k- ω (Λ2|sc)Rnoise: k- ω (Λ2|Bo)LWT (CPV) [60m/s]LWT (CPV) [50m/s]LWT (CPV) [40m/s]x

NACA0012: Re=1.5e 6, Ma = 0.166, c=0.4m, α=0°, C l=0

x x x x x x

fcen/ [Hz]

L p[d

B/m

]

5000 1000020

30

40

50

60

70

Rnoise: SST ( Λ2|iso)Rnoise: SST ( Λ2|iso2 )Rnoise: SST ( Λ2|sc)Rnoise: SST ( Λ2|Bo)LWT (CPV) [60m/s]LWT (CPV) [50m/s]LWT (CPV) [40m/s]x

x x x x x x

fcen/ [Hz]

L p[d

B/m

]

5000 1000020

30

40

50

60

70

Rnoise: EARSM ( Λ2|iso)Rnoise: EARSM ( Λ2|iso2 )Rnoise: EARSM ( Λ2|sc)Rnoise: EARSM ( Λ2|Bo)LWT (CPV) [60m/s]LWT (CPV) [50m/s]LWT (CPV) [40m/s]x

Figure 11. Predicted (at U=56m/s) & Measured Noise Spectra at cl = 0: NACA0012

fcen [Hz]

L p[d

B/m

]

5000 1000020

30

40

50

60

70

Rnoise: k- ω (Λ2|iso)Rnoise: k- ω (Λ2|iso2 )Rnoise: k- ω (Λ2|sc)Rnoise: k- ω (Λ2|Bo)

fcen/ [Hz]

L p[d

B/m

]

5000 1000020

30

40

50

60

70

Rnoise: SST ( Λ2|iso)Rnoise: SST ( Λ2|iso2 )Rnoise: SST ( Λ2|sc)Rnoise: SST ( Λ2|Bo)

NACA0012: Re=1.5e 6, Ma = 0.166, c=0.4m, α=2°, C l=0.21

fcen/ [Hz]

L p[d

B/m

]

5000 1000020

30

40

50

60

70

Rnoise: EARSM ( Λ2|iso)Rnoise: EARSM ( Λ2|iso2 )Rnoise: EARSM ( Λ2|sc)Rnoise: EARSM ( Λ2|Bo)

Figure 12. Predicted Noise Spectra at cl = 0.209: NACA0012

13 of 15

American Institute of Aeronautics and Astronautics

NACA0012: Re=1.5e 6, Ma = 0.166, c=0.4m, α=4°, C l=0.44

fcen [Hz]

L p[d

B/m

]

5000 1000020

30

40

50

60

70

Rnoise: k- ω (Λ2|iso)Rnoise: k- ω (Λ2|iso2 )Rnoise: k- ω (Λ2|sc)Rnoise: k- ω (Λ2|Bo)

fcen/ [Hz]

L p[d

B/m

]

5000 1000020

30

40

50

60

70

Rnoise: SST ( Λ2|iso)Rnoise: SST ( Λ2|iso2 )Rnoise: SST ( Λ2|sc)Rnoise: SST ( Λ2|Bo)

fcen/ [Hz]

L p[d

B/m

]

5000 1000020

30

40

50

60

70

Rnoise: EARSM ( Λ2|iso)Rnoise: EARSM ( Λ2|iso2 )Rnoise: EARSM ( Λ2|sc)Rnoise: EARSM ( Λ2|Bo)

Figure 13. Predicted Noise Spectra at cl = 0.44: NACA0012

VTE_kav: Re=3.0e 6, Ma = 0.178, c=0.8m, C l=0.7

fcen [Hz]

L p[d

B/m

]

5000 1000020

30

40

50

60

70

Rnoise: k- ω (Λ2|iso)Rnoise: k- ω (Λ2|iso2 )Rnoise: k- ω (Λ2|sc)Rnoise: k- ω (Λ2|Bo)Exp. LWT (CPV)

fcen/ [Hz]

L p[d

B/m

]

5000 1000020

30

40

50

60

70

Rnoise: SST ( Λ2|iso)Rnoise: SST ( Λ2|iso2 )Rnoise: SST ( Λ2|sc)Rnoise: SST ( Λ2|Bo)Exp. LWT (CPV)

fcen/ [Hz]

L p[d

B/m

]

5000 1000020

30

40

50

60

70

Rnoise: EARSM ( Λ2|iso)Rnoise: EARSM ( Λ2|iso2 )Rnoise: EARSM ( Λ2|sc)Rnoise: EARSM ( Λ2|Bo)Exp. LWT (CPV)

Figure 14. Predicted Noise Spectra at cl = 0.7: VTE kav

45

50

55

60

65

70

75

100 1000 10000

L p [d

B/m

]

frequency [Hz]

NACA 0012, c=0.4 m, trip 5%/10%

LWT, CPV, c=0.4 m, 60 m/sAWB, M. Herr, c=0.4 m, 60 m/s

Xnoise Swafford, c=0.4 m, 60 m/sXnoise Coles, c=0.4 m, 60 m/sXEnoise Mo, c=0.4 m, 60 m/s

Rnoise WKW, c=0.4 m, 60 m/sRnoise SST, c=0.4 m, 60 m/s

Rnoise EARSM, c=0.4 m, 60 m/s

45

50

55

60

65

70

75

0.1 1 10

L p +

10·

log1

0(δ 1

,ref

/δ1)

[dB

/m]

ω δ1 /U [-]

NACA 0012, c=[0.2,0.4,0.6] m, trip 5%/10%

LWT, CPV, c=0.2 m, 60 m/sLWT, CPV, c=0.4 m, 60 m/sLWT, CPV, c=0.6 m, 60 m/s

Rnoise EARSM, c=0.2 m, 60 m/sRnoise EARSM, c=0.4 m, 60 m/sRnoise EARSM, c=0.6 m, 60 m/s

BPM, c=0.2 m, 60 m/sBPM, c=0.4 m, 60 m/sBPM, c=0.6 m, 60 m/s

Figure 15. Comparison of Noise Spectra: U = 60m/s: NACA0012

References

1Kroll, N., Rossow, C. C., Schwmborn, D., Becker, K., and Heller, G., “MEGAFLOW – a Numerical Flow SimulationTool for Transport Aircraft Design,” Proceedings of the 23rd International Congress of Aeronautical Sciences, ICAS, 2002,Toronto.

2Blake, W., Mechanics of Flow-Induced Sound and Vibration Vol. I and II , Academic Press, INC., 1986.3Lutz, T., Herrig, A., Wurz, W., Kamruzzaman, M., and Kramer, E., “Design and Wind-Tunnel Verification of Low-Noise

Airfoils for Wind Turbines,” AIAA Journal , Vol. 45, No. 4, 2007, pp. 779–792.4Kamruzzaman, M., Lutz, T., and Kramer, E., “An Approach to RANS Based Prediction of Airfoil Trailing Edge Far-Field

Noise,” Conf. Proc. , Wind Turbine Noise-2007, Lyon, France, September 2007.5Oerlemans, S. and Lopez, B. M., “Localisation and Quantification of Noise Sources on a Wind Turbine,” Wind Turbine

Noise: Perspectives for Flow Control, Berlin, October 17-18 , 2005.6Brooks, T. F. and Hodgson, T. H., “Prediction and Comparison of Trailing Edge Noise Using Measured Surface Pressures,”

, No. 80-0977, June 1980, Presented at the AIAA 6th Aeroacoustics Conference, Hartford, CT.

14 of 15

American Institute of Aeronautics and Astronautics

7Brooks, T. F., Pope, D. S., and Marcolini, M. A., “Airfoil Self-Noise and Prediction,” Reference Publication 1218, NASA,1989.

8Singer, B. A., Brentner, K. S., Lockard, D. P., and Lilley, G. M., “Simulation of Acoustic Scattering from a TrailingEdge,” Vol. 1999, 1999.

9Ewert, R. and Schroder, W., “On the simulation of trailing edge noise with a hybrid LES/APE method,” Jornal ofSound and Vibration, Vol. 270, 2004, pp. 509–524.

10R. Ewert, M. M. and Schroder, W., “Computation of Trailing Edge Noise via LES and Acoustics Perturbation Equations,”Paper 2002-2467, AIAA, 2002.

11Marsden, A. L., Wang, M., Jr, J. E. D., and Moin, P., “Trailing-edge noise reduction using derivative-free optimizationand large-eddy simulation,” .

12Billson, M., Eriksson, L.-E., Jordan, P., and Davidson, L., “Anisotropic Formulation of the Velocity Correlation Tensor,”Internal Report 04/02, Department of Mechanical Engineering, Chalmers University of Technology, Sweden, 2004.

13Drela, M., “XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils,” Low Reynolds Number Aero-dynamics (Conference Proceedings), edited by T. J. Mueller, University of Notre Dame, Paris, June 1989, pp. 1–12.

14Schereps, J., Curvers, A., Oerlemans, S., Braun, K., Lutz, T., Herrig, A., Wurz, W., and Lopez, B. M., “SIROCCO:SIlent ROtors by acoustic Optimisation,” Wind Turbine Noise: Perspectives for Flow Control, Berlin, October 17-18 , 2005.

15Schepers, J. G., Curvers, A., Oerlemans, S., Braun, K., Lutz, T., Herrig, A., and Wurz, W., “SIROCCO: SIlent ROtorsby aCoustiC Optimization,” Conf. Proc. , Wind Turbine Noise-2007, Lyon, France, September 2007.

16Herrig, A., Wurz, W., and Wagner, S., “SIROCCO report: 2D aerodynamic wind tunnel tests on Variable Trailing Edgemodel,” Report, Institute of Aerodynamics and Gas Dynamics, University of Stuttgart, December 2003.

17Wilcox, D. C., Turbulence Modeling for CFD , DCW Industries, 1994.18Kraichnan, R. A., “Pressure Fluctuations in Turbulent Flow over a Flat Plate,” Journal of Acoustics Soc. America,

Vol. 28, 1956, pp. 378–390.19Panton, R. T. and Linebarger, J. H., “Wall Pressure Spectra Calculations for Equilibrium Boundary Layers,” Journal of

Fluid Mechanics, Vol. 65, 1974, pp. 261–287.20Chandiramani, K., “Diffraction of evanescent wave with application to aerodynamically scattered sound and radiation

from unbaffled plates,” J. Acoust. Soc. Amer., Vol. 55, 1974, pp. 19–29.21Hinze, J. O., Turbulence, McGraw-Hill, 1975.22Wilson, D. K., “Anisotropic Turbulence Models for Acoustic Propagation Through the Neutral Atmospheric Surface

Layer,” Report: Arl-tr-1519, Army Research Laboratory, Adelphi, MD 20783-1197, US, February 1998.23Kristensen, L., Lenschow, D. H., Kirkegaard, P., and Courtney, M., “The spectral velocity tensor for homogeneous

boundary-layer turbulence,” Boundary Layer Meteor., Vol. 47, 1989, pp. 149–193.24Herrig, A., Wurz, W., Kramer, E., and Wegner, S., “Trailing-Edge Noise Measurements on NACA0012 Airfoils Using

the Coherent Particle Velocity Method,” Conf. Proc. , 15th International Congress on Sound and Vibration, 6-10 July 2008,Daejeon, Korea, July 2008.

25Herrig, A., Wurz, W., Lutz, T., and Kramer, E., “Trailing-Edge Noise Measurements using a Hot-Wire Based CoherentParticle Velocity Method,” AIAA, june 2006, Fluid Dynamics Conference and Exhibit.

26Wortmann, F. X. and Althaus, D., “Der Laminarwindkanal des Instituts fur Aero- und Gasdynamik an der TechnischenHochschule Stuttgart,” Zeitschrift fur Flugwissenschaften, Vol. 12, No. 4, April 1964, pp. 129–134.

27Lysak, P. and Brungart, T., “Velocity Spectrum Model for Turbulence Ingestion Noise from Computational-Fluid-Dynamics Calculation,” Vol. 41, No. 9, Sept. 2003, pp. 1827–1829.

28Guidati, G. and Wagner, S., “Design of Reduced Noise Airfoils for Wind Turbines,” ECCOMAS 2000: European Congresson Computational Methods in Applied Sciences and Engineering, September 11–14, 2000, Barcelona, Spain, 2000.

15 of 15

American Institute of Aeronautics and Astronautics