REFRACTION CORRECTIONS FOR SURFACE INTEGRAL …lyrintzi/presentations/Panaiaa2873.pdf · acoustic sources to the linear far-ﬂeld sound through the traditional Lighthill’s acoustic

REFRACTION CORRECTIONS FOR

SURFACE INTEGRAL METHODS IN

JET AEROACOUSTICS∗

F. L. Pan†

Purdue University, West Lafayette, IN, 47907-2023

A. Uzun‡

Florida State University, Tallahassee, FL, 32306-4120

and

A. S. Lyrintzis§

Purdue University, West Lafayette, IN, 47907-2023

One of the major challenges in computational jet aeroacoustics is the ac-curate modelling and prediction of acoustic fields in order to reduce and

control the jet noise. Surface integral methods (e.g. Kirchhoff method andFfowcs Williams - Hawkings (FW-H) method) can be used in Computa-

tional Aeroacoustics (CAA) in order to extend the near-field CFD results tofar-field. The surface integral methods can efficiently and accurately predict

aerodynamically generated noise provided the control surface surrounds theentire source region. However, for jet noise prediction, shear mean flow ex-ists outside the control surface that causes refraction. The mean flow refrac-

tion corrections had been done in the past using simple geometric acoustics(GA). A more rigorous method based on Lilley’s equation is investigated

here due to its more realistic assumption of the acoustic propagation. Jetnoise computational results based on Large Eddy Simulation (LES) for the

near-field for an isothermal jet at Reynolds number 400,000 are used forthe evaluation of the solution on the FW-H control surface. The proposed

methodology for prediction of far-field sound pressure level shows that theacoustic field within the zone of silence is consistent with the experimentalresults.

Nomenclature

a = local sound speed normalized by its ambient value, a ≡ aao

Ai = Airy functionGω = free-space Green’s function in time domainGω = fundamental solution of Fourier transformed wave equation defined by

Eq. (21)

∗AIAA 2004-2873, Presented at the 10th AIAA/CEAS Aeroacoustics Conference, May 2004,Manchester United, United Kingdom.

†Graduate Student, School of Aeronautics and Astronautics, Student Member AIAA.‡Post-Doctoral Research Associate, School of Computational Science and Information Technol-

ogy, Member AIAA.§Professor, School of Aeronautics and Astronautics, Associate Fellow AIAA.

1American Institute of Aeronautics and Astronautics

Gω = fundamental solution of Fourier transformed wave equation defined byEq. (22)

ko = streamwise wavenumber, ko = ωao

L = wave operatorM = Mach number or mean Mach number profilen = critical azimuthal wavenumbernj = outward directed unit normal vectorp = local pressureq = arbitrary function defined by Eq. (23)Q = arbitrary function defined by Eq. (23)r = magnitude of radiation vector, |rj |rδ = turning point defined by Eq. (23)R = distance between observer and source pointReD = Reynods number based on jet diameterRω = arbitrary function defined by Eq. (23) or (24)S = closed surface defined by f(y, τ)St = Strouhal numbert = time or observer reception timeui = Cartesian surface velocity componentsPij = compressive stress tensor with the constant poδij subtratedU = mean velocity or mean velocity profile(~x, t) = observer coordinates and time<> = root mean square evaluation

Greek

φ = acoustic variableω = cyclic frequencyδij = Kronecker deltaρ = fluid densityθ = source emission angleϑ = sound propagation angle∆ϕ = difference in azimuthal angles of the observer and source point coordinatesε = arbitrary function defined by Eq. (23)η = arbitrary function defined by Eq. (23)Γ = acoustic source distributionΠ = acoustic pressure fluctuation normalized by ρa2

τ = retarded or emission time, t − R/ao

ξ = arbitrary function defined by Eq. (24)ζ = arbitrary function defined by Eq. (18)

Superscript

′ = pertubation quantity (e.g.ρ′ = ρ − ρo)· = source time derivative¯ = mean flow quantity = Fourier transformed quantityˆ = unit vector

Subscript

av = average parameterc = corrected parametercr = critical angle parameterJ = quantity on jet centerlinem = measured parametero = ambient or freestream quantity


r = inner product with rj

ref = reference parameters = quantity evaluated at source positionret = quantity evaluated at retarded time τn = inner product with nj

I. Introduction

Computational aeroacoustics (CAA) is concerned with the prediction ofthe aerodynamic sound source and the propagation of the generated sound.The sound is generated by nonlinear process, which is governed by the full,time-dependent, and compressible Navier-Stokes equations. Indeed, the acousticfar-field is linear and irrotational, one can consider two distinct characteristicsof sound, i.e. the nonlinear generation of sound, and the linear propagation ofsound. This implies that instead of solving the full nonlinear flow equations outin the far-field for sound propagation, one can extend the nonlinear near-fieldacoustic sources to the linear far-field sound through the traditional Lighthill’sacoustic analogy1 or surface integral methods, i.e. Kirchhoff’s method2 andFfowcs Williams - Hawkings (FW-H) method.3,4 The latter methods are moreattractive as the volume integration is avoided. The surface integral methodsassume that the sound propagation is governed by the simple wave equation,and can efficiently and accurately predict aerodynamically generated soundprovided the control surface surrounds the entire nonlinear source region.5,6

However, in jet noise prediction, the control surface cannot enclose the entireaerodynamic generated source region. Thus, the linear wave equation is invalidwhen the sound propagates through the region near the jet axis, downstreamof the control surface.

In the past, researchers utilizing surface integral methods to predict jet noisehad ignored sound generated at and outside of the downstream end of the controlsurface.5 When acoustic waves propagate through the jet mean flow downstreamof the control surface, it crosses velocity-shear interfaces, which deflects the ra-diated sound away from the mean flow direction. This effect is called refraction.7

For isothermal or hot jets, the effect of refraction gives rise to a “zone of silence”,a region where a substantial reduction of sound occurs. Early investigations7–9

found that the waves propagating in the zone of silence initially start as evanes-cent waves near the source. The presence of the zone of silence was also measuredexperimentally.10–12 Surface integral methods are not able to predict the refrac-tion outside the control surface and the resulting zone of silence. The mean flowrefraction corrections to the surface integral method have been investigated us-ing simple geometric acoustics (GA) approximation.6,13 This approximation isvalid only at high-frequency region. However, azimuthal variations between thesources in the sheared layer and the observers were not taken into account andhence the accuracy of this correction may be limited. It is worthwhile to at-tempt a more rigorous method in order to achieve more reliable prediction. Alinearized homogeneous third-order convective wave equation, namely Lilley’sequation,14 has been solved asymptotically using the stationary phase method


in the far-field.7 The high-frequency limit solution15 of Lilley’s equation is re-visited and applied to refraction corrections. The frequency range in which theapproximation solution of Lilley’s equation holds, is also studied.

The remainder of the paper describes the surface integral methodologies,i.e. the Kirchhoff and FW-H methods, both in time and frequency domain.Then, the two refraction corrections approaches, i.e. the simple GA method andLilley’s equation method are discussed. An example of a monopole source is usedto demonstrate the utility and efficienty of the refraction corrections schemes.An application involving an overall sound pressure level (OASPL) calculationbased on a LES computation of an isothemal jet at ReD = 400,000 is shownfollowed by the concluding remarks.

II. Surface Integral Methodologies

Kirchhoff’s Method. Kirchhoff’s formula was first published in 1882.2 It isan integral representation (i.e. surface integral around a control surface) of thesolution to the wave equation. Kirchhoff’s formula, although primarily used inthe theory of diffraction of light and in other electromagnetic problems, hasalso many applications in studies of acoustic wave propagation. Farassat andMyers16 derived a Kirchhoff formulation for a moving, deformable, piecewisesmooth surface.

Assume the linear, homogeneous wave equation

1

a2◦

∂2φ

∂t2−

∂2φ

∂xi∂xi= 0 (1)

is valid for some acoustic variable, φ, and sound speed, ao in the entire regionoutside of a closed and bounded smooth surface. For stationary surfaces, theKirchhoff formula is

4πφ(~x, t) =

∫

S

1

R

[1

a◦

φ cos θ −∂φ

∂n

]

ret

dS +

∫

S

[φ]ret

R2dS (2)

where cos θ = R · n, θ is the source emission angle, n is the outward directedunit normal vector of the closed control surface S, R is the distance betweenthe observer and the surface source. The surface integrals are over the controlKirchhoff surface, S, and evaluated at retarded or emission time, τ describedby the following equation

τ = t −R

ao(3)

where t is observer or reception time. The dependent variable φ is normallytaken to be the disturbance pressure, but can be any quantity which satisfiesthe linear wave equation.

With the use of a Fourier transform,

F{f(t)} = f(ω) =

∫ ∞

−∞

f(t)e−iωtdt


F−1{f(ω)} = f(t) =1

2π

∫ ∞

−∞

f(ω)eiωtdt (4)

equation (2) can be expressed in the frequency domain (i.e, starting fromHelmholtz equation) as

4πφ(~x, ω) =

∫

S

e−iωR/a◦

[1

R

(iω

a◦

cos θ φ −∂φ

∂n

)+

φ cos θ

R2

]dS (5)

where φ is the Fourier transform of φ, and ω is the cyclic frequency. Bothtime and frequency expressions can be extended for a uniform rectilinearmotion.17 Equations (2) and (5) work well for aeroacoustic predictions whenthe control surface is placed in a region of the flow field where the linear waveequation is valid. However, this might not be possible for some cases. Additionalnonlinearities (i.e. quadrupoles) can be added outside the control Kirchhoffsurface.18 Kirchhoff’s method has also been applied for open surfaces19 and canbe extended to include mean flow refraction effects based on the GA method.13

Porous FW-H Method. A general version of the FW-H equation is neededbecause the usual practice is to assume that the FW-H integration surface cor-responds to a solid body and is impenetrable. For the simple case of a stationarysurface we can define (following Brentner and Farassat20 and Francescantonio21)new variables Ui and Li as

Ui =ρui

ρo(6)

andLi = Pij nj + ρuiun (7)

where subscript o implies ambient conditions, superscript ′ implies disturbances(e.g. ρ = ρ′ + ρo), ρ is the density, u is the velocity, and Pij is the compressivestress tensor with the constant poδij subtracted. Now by taking the time deriva-tive of the continuity equation and subtracting the divergence of momentumequation, followed with some rearranging, the integral form of FW-H equationcan be written as (Formulation I)

p′(~x, t) = p′T (~x, t) + p′L(~x, t) + p′Q(~x, t) (8)

where

4πp′T (~x, t) =∂

∂t

∫

S

[ρoUn

R

]

ret

dS (9)

4πp′L(~x, t) =1

a◦

∂

∂t

∫

S

[Lr

R

]

ret

dS +

∫

S

[Lr

R2

]

ret

dS (10)

and the quadrupole noise pressure p′Q(~x, t) can be determined by any methodcurrently available. The subscript r or n indicates a dot product of the vectorwith the unit vector in the radiation direction r or the unit vector in the surfacenormal direction ni, respectively. It should be noted that the three pressureterms have a physical meaning for rigid surfaces: p′

T (~x, t) is known as thickness


noise, p′L(~x, t) is called loading noise and p′Q(~x, t) is called quadrupole noise.For a porous surface the terms lose their physical meaning, but the last termp′Q(~x, t) still denotes the quadrupoles outside surface S.

An alternative way21 is to move the time derivative inside the integral: (For-mulation II)

4πp′T (~x, t) =

∫

S

[ρoUn

R

]

ret

dS (11)

4πp′L(~x, t) =1

a◦

∫

S

[Lr

R

]

ret

dS +

∫

S

[Lr

R2

]

ret

dS (12)

The dot over a variable implies source-time differentiation of that variable. Itappears that Formulation I (Eqs. 8-10) has less memory requirements, because itdoes not require storage of the time derivatives, and requires less operations perintegral evaluation. For our case of a stationary Kirchhoff surface the integralcan be reused at the next time step. Formulation I was used by Strawn et al.22

for rotorcraft noise predictions using a nonrotating control surface with verygood results. On the other hand taking the time derivative inside could preventsome instabilities for moving control surfaces. Thus Formulation II (Eqs. 11-12)might be more robust for a moving control surface, Formulation II was used forrotorcraft noise prediction by Brentner and Farassat20 with a rotating controlsurface with very good results. However, a more detailed comparison of the twoformulations would be very helpful.

With the use of Fourier transform pair in Eq. (4), both formulations can bewritten in the frequency domain for a stationary surface as

4πp′T (~x, ω) = iω

∫

S

e−iωR/a◦ρoUn

RdS (13)

4πp′L(~x, ω) =iω

a◦

∫

S

e−iωR/a◦Lr

RdS +

∫

S

Lr

R2dS (14)

where p′, Un, and Lr are the Fourier transforms of p′, Un, and Lr, respectively.and ω is the cyclic frequency. Both the above formulations provide a Kirchhoff-like formulation if the quadrupoles outside the control surface (p′

Q(~x, t) term)are ignored. The equivalence of the porous FW-H equation and Kirchhoff for-mulation was proven by Pilon & Lyrintzis18 and Brentner & Farassat.20 Thereader is referred to the above references for the details. The major differencebetween Kirchhoff’s and FW-H formulation is that Kirchhoff’s method needsp′, ∂p

∂n , ∂p∂t as input, whereas the porous FW-H needs p′, ρ, ρui. Also, the porous

FW-H method puts less stringent requirements on the placement of surface inthe linear region than Kirchhoff’s method, as shown in Ref. 20 for a rotorcraftnoise problem. The method has been extended to include refraction correctionsbased on GA.6


source

Uθ

ϑο

Fig. 1. Sound emission and propagation angle of a source due torefraction.

III. Refraction Corrections

Geometric Acoustics. The refraction problem has drawn attention, becauseof its importance in the open jet wind tunnels experiments. Early developmentof solution of the refraction by a thick cylindrical shear layer with a source onthe jet axis was given by Tester and Morfey,8 and for a plane shear layer byAmiet.23 Recent development on the source off the jet axis was given by Morfeyand Joseph.24 These refraction corrections approaches are geometrical algebrain close form solutions.

The application of simple geometric acoustics (GA) closed-form approxima-tion used in this work is based on the thick cylindrical shear layer refraction cor-rection.23 The simple GA approximation is valid only when the acoustic wave-length, λ is small compared with the shear layer thickness, δ. It is assumed herethat the downstream end of the cylindrical control surface is located far enoughdownstream of the jet potential core that the shear layer thickness is large com-pared with the acoustic wavelength. Regardless, Morfey and Szewczyk25 haveshown that jet mixing noise can be effectively modeled with the simple GAmethod even when δ/λ < 1. One of the advantages of utilizing their method isthat it can be applied to a general source. This flexibility was arrived due to theseparate corrections of the angle and amplitude.

We consider only mean flow effects on the sound radiation and model theouter shear layer as a stratified flow.23 The stratified flow is described in Eq.(15),

ao

cos ϑo= U +

ao

cos θ(15)

where U is the steady velocity at the downstream end of the control surface, θ isthe sound emission angle with respect to the jet axis, and ϑo is the propagationangle in the stagnant, ambient air, as shown in Fig. 1.

On reaching the shear layer, the sound is refracted and hence both amplitudeand angle change. The refraction consists of a collection of point acoustic sourcesacting at radial location, and the parallel sheared mean flow is determined ateach location. Equation (15) can be rearranged to show that the critical angleis defined by


cos θcr =1

1 + M(16)

Below the critical angle, θcr, no sound can reach the observer and this gives thezone of silence.

This result is purely kinematic and a thick cylindrical shear layer is consid-ered here to capture more detailed information of the refraction. The amplitudecorrection can be found from the following equation,

∣∣∣∣Pc

Pm

∣∣∣∣ = (1 − M cosϑ)2 (17)

where Pc is the corrected pressure, and Pm is the measured pressure. Note thatthe Mach number of the mean sheared flow, M varies along the control surfacein radial direction. The correction should be valid provided the radial distanceis large compared to the shear layer’s wavelength and the source dimensions.(18)

An additional correction is needed, which converts the present source posi-tion to the retarded source position; it can be done by a factor multiplication,

rret

rc=

(1 − M cos ϑ)√(1 + M2ζ2)

(18)

where ζ =√

(1 − M cos ϑ)2 − cos2 ϑA final correction is necessary to accurately account for the mean flow re-

fraction. Imposing the local zone of silence condition described earlier allowsa surface source at a relatively large radial location to radiate sound into andthrough the shear flow. This is because the local zone of silence decreases in sizewith the radial location of the source. The simple correction is to set the sourcestrength to zero if the observation point is located closer to thet jet axis thanthe source point on the FW-H surface. It should be noted that it is question-able that the energy argument used to derive the result for the large thicknesscase is still feasible for zero thickness case. More details can be found in Ref.23. The GA corrections can be applied to surface integral methods6,13 for thedownstream portion of the control surface using the local velocity distribution.The velocity is assumed to be parallel to the axis.

Another issue is the azimuthal variations between observer point and source,which is not accounted using this simplified approach. This azimuthal variationparameters will be introduced by using the alternative correction method inthe following section.

Lilley’s Equation. Lilley’s equation14 is derived from the inhomogeneous waveequation. The equation describes explicitly the combined effects of sound convec-tion and refraction by a unidirectional transversely sheared mean flow. Lilley’sequation can be written as

LΠ = Γ (19)


where Π denotes the acoustic pressure fluctuation normalized by ρa2, ρ is themean flow density, a is the mean speed of sound, Γ represents the acousticsource distribution. L is the wave operater originally derived by Pridmore-Brown(1958).

L =D

Dt

(D2

Dt2−∇a2∇

)+ 2a2∇u∇

∂

∂x(20)

where

D

Dt=

∂

∂t+ u

∂

∂x

is the convective derivative relative to mean flow. u is the mean flow velocity.In the past, many researchers7,9, 26 have tried to obtain the Greens’ func-

tion expression of Lilley’s equation using high-frequency, far-field asymptoticapproximation. This approximation is valid only when the acoustics wavelengthof the aerodynamic noise is much shorter than the characteristic length scale ofthe mean flow, and also the mean velocity profile is parallelly well-defined. Atthe end of the jet potential core, for example, the mean velocity profiles are sig-nificantly distorded, and the parallel mean flow assumption would fail. Based onthis assumption, three different closed-forms of Green’s function solution withlocally parallel and axisymmetric mean flow were obtained. The first expressionwas developed by Goldstein7 in which the source was placed on several wave-lengths off the jet axis. At the same time, Balsa9 developed an expression withthe source located near the jet centerline axis. The third form was derived byGoldstein27 with the implementation of ray-theory for parallel round jets. If amonople source distribution is considered, the Green’s function associated tothe Fourier transformed solution of Lilley’s wave equation, Gω(x|xs) satisfiesthe following equation

L[Gω(x|xs)e−iωt] =

D

Dt[a2δ(x − xs)e

−iωt] (21)

where ω is source frequency, xs is source position, δ is the Kronecker delta.The fundamental solution of Fourier transformed Lilley’s equation, which is

mathematically called the reduced Green’s function, describes the effect of thesurrounding mean flow on radiated sound. Mean velocity gradients and temper-ature gradients are known to refract the acoustic energy within the mean shearlayer. For simplicity, an isothermal jet condition is considered with no tem-perature gradients. Hence, the refraction is caused only by non-uniform meanflow whithin the shear layer, which varies radially at the outflow of the con-trol surface. One can solve the reduced Green’s function using a high-frequencyasymptotic approximation.

As mentioned in Ref. 15, if the distance between the source and the jetcenterline axis is sufficiently large (several factors of 1/ko, where ko is stream-wise wavenumber, ko = ω/ao), then the critical azimuthal wavenumber, n canbe scaled to the order of ko. This scaling was considered by Goldstein7 andthe resulting acoustic field was referred to as the asymmetric, high-frequency


approximation.

n = O(ko) as ko → ∞

On the other hand, as the source moves closer to the jet centerline axis, thescaling becomes

n = O(1) as ko → ∞

Here, the critical wavenumber, n scales like 1/rJ , where rJ is the jet nozzleradius. This near-axis source problem was studied by Balsa9 and the resultingacoustic field was referred to as the quasi-symmetric, high-frequency approxi-mation.

The following Eq. (22) gives the ratio of the reduced Green’s function to thefree-space Green’s function, which was implemented in the refraction correctionscodes developed in this research.

Gω(x |xs)

Gω(x |xs)∼

Rω(x |xs)

a(rs)(1 − M(rs) cos θ)(22)

It should be noted that this equation is valid only when both ko and R → ∞.The former assumption needs to be well defined and the lowest wavenumber, ko

value should be correctly determined. An analysis was done for this evaluationin Ref. 15. The same results are obtained here, which show the asymptotic high-frequency approximation for solutions of Lilley’s equation remains accurate withStrouhal number as small as 1/2.

Based on the asymmetric and quasi-symmetric behaviors of acoustic field,two distinct Rω(x|xs) functions from Eq. (22) are derived in separate approachesand are used to describe the acoustic field accordingly. The asymmetric, high-frequency solutions are used for off-axis source locations, whereas the quasi-symmetric solutions are applied to near-axis locations.

For the asymmetric, far-field approximation, Rω(x|xs) is defined by the fol-lowing equation.

Rω(x|xs) ∼

n=∞∑

n=−∞

[2

ko

√−ηn(rs)

rsQn(rs)

] 1

2

Ai[ηn(rs)]ε (23)

whereε = ein∆ϕ+iko(ζn(r)−R sin2 θ)

Qn(r) =

√

q2(r) −

(nko

r

)2

q(r) =

√[1 − M(r) cos θ

a(r)

]2

− cos2 θ

ηn(r) = −

[3

2koζn(r)

] 2

3


ζn(r) =

∫ r

rδ

Qn(r)dr, Q2n(rδ) = 0

where a ≡ a/ao is the local sound speed normalized by its ambient value, M =U/ao is the Mach number based on the ambiet speed of sound, and ∆ϕ is theazimuthal difference between observer and source point, rs is the source location,Ai is the Airy function. Note that cube root in the definition of ηn in Eq. (23)is taken such that ηn < 0 for r > rδ and ηn > 0 for r < rδ,

For the quasi-symmetric, far-field approximation, Rω(x|xs) is defined by thefollowing equation.

Rω(x|xs) ∼ eiko(ξ(r)−R sin2 θ−ξ(rs) cos ∆ϕ) (24)

where

ξ(r) =

∫ r

0

q(r)dr

It should be noted that rδ defined in Eq. (23) does not appear in Eq. (24). rδ

is the solution for which the nonlinear equation, Qn is zero, and is called turn-ing point. rδ is determined by solving Qn numerically. Due to its dependencyon Mach number, which varies along the radial direction, the use of asymmet-ric approximation turns out more computationally expensive compared to thequasi-symmetric approximation.

Figures 2 (a)-(b) show the comparisons of |Gω/Gω| at various azimuthal-angle parameters, ∆ϕ for both asymmtric and quasi-symmetric approximations.The asymmetric and quasi-symmetric approximations are indicated by the solidand dashed lines, respectively. We used the case suggested in Ref. 15, i.e. a pointsource at rs = 0.75 using a mean flow profile given by equation

u(r)

a(r)=

M(r)

a(r)= MJsech2(2r) (25)

The Strouhal number is 0.5 and can be calculated as

St =ko

π

rJ

MJ(26)

where rJ = 0.5 is the jet radius, MJ = 0.9 is the Mach number at jet centerline.The results are identical to Ref. 15.

One can see how |Gω/Gω| varies with different azimuthal angle parameters,∆ϕ. This implies that simply ignoring the azimuthal variations between thesource and the observer points will result to inaccurate predictions.

IV. Results and Discussion

Point Source. A simple monopole is used to demonstrate the application ofrefraction corrections in jet noise prediction. The frequency domain version ofthe Ffowcs Williams-Hawkings (FW-H) integral was used to calculate the acous-tic far-field of the monopole. A cylindrical control surface (Lk = 40, rk = 5)as shown in Fig. 4., was used in this test case. The monopole is placed at themiddle of the surface and the surface pressure is defined as


θ30 60 90 120 150

0

0.0003

0.0006

0.0009

0.0012

0.0015

0.0018asymmetricquasi-symmetric

(degree)b)

θ30 60 90 120 150

0

0.0003

0.0006

0.0009

0.0012

0.0015

0.0018

asymmetricquasi-symmetric

(degree)d)θ30 60 90 120 150

0

0.0003

0.0006

0.0009

0.0012

0.0015

0.0018


(degree)c)

θ30 60 90 120 150

0

0.0003

0.0006

0.0009

0.0012

0.0015

0.0018


(degree)a)

Fig. 2. |Gω/Gω| at (a) ∆ϕ = 0; (b) ∆ϕ = 30; (c) ∆ϕ = 90; (d) ∆ϕ = 120.


θ (deg)

norm

aliz

edam

plitu

de

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

with no refractionsimple GA methodLilley’s equation, ko=1.0Lilley’s equation, ko=1.5

Fig. 3. Refraction effects.

p(xs, ω) =1

rse−iω rs

ao (27)

The acoustics wavenumber, ko of 1.0 and 1.5 are considered. We are using 20points per wavelength. The distance between the midpoint of the cylinder’s axisand the observer is taken to be 60rJ , where rJ is 1. The result with no refractioncorrection is shown in Fig. 3. A normalized amplitude of 1 is expected with nomean flow outside the FW-H surface.

The refraction corrections are then applied to the right base of the cylinderusing the simple GA method and Lilley’s equation. The shear flow is modelledwith velocity profile shown in Fig. 5. The mean flow velocity profile is onlyapplied to the right base of the cylindrical control surface. At surface radius of5, the corresponding Mach number is 0.002 while the centerline Mach numberis 0.5. This is a reasonable distribution for jet velocites.6 Since velocity becomesnegligible near the radial distance of 5, the refraction correction is applied onlyfor r < 5.

When solving asymptotic solutions for the Lilley’s equation, which is basedon the high-frequency assumption, a minimum Strouhal number must be consid-ered. The Strouhal number can be determined from Eq. (26). As demonstratedin the last section, at Strouhal number of 1/2, the solutions are still very closeto the exact solutions of Lilley’s equation. Below this Strouhal number limit,asymptotic approximation does not apply and the refraction correction mustbe done by an alternative way, which we choose to be the simple GA method.Since the asymmetric solution is more accurate, it is used for all source elementsexcept for a near-axis source, where the asymmetric solution breaks down anda quasi-symmetric solution is used.

As shown in Fig. 3, the zone of silence is predicted in a region where there isa substantial reduction in the normalized pressure amplitude. For the monopole


rk

x

r

Observer(X,t)

Point Source

^nR

θ

S

Lk

Fig. 4. Point source location and control surface geometry.

case, the Lilley’s equation results a smooth amplitude decrease in the zone ofsilence, wheareas the simple GA method shows an abrupt decrease at this region.A similar trend was obtained from the Kirchhoff method, and hence only theFW-H result is shown here.

The azimuthal variation between the source and observer points has beenignored in the simple GA method, whereas the variation parameters can be eval-uated with Lilley’s equation, which has been discussed in the previous section.The azimuthal variations between the sources and observer points are importantbecause the FW-H control surface utilized is three-dimensional.

It is worthwhile to note that the limitation of the high-frequency appli-cations of Lilley’s equation to jet noise can be overcome by adding the axiallength of the control surface to capture more of the low frequency noise. Also,the refraction effects inside the control surface are captured better with a longersurface. However, higher computational cost will result from the increase of thesurface length.

Jet Noise Prediction Using LES. The surface integral methods can extendthe near-field flow data to far-field acoustic properties, e.g. acoustic pressure.The near-field input for the surface integral methods will be obtained from aLarge Eddy Simulation (LES) calculation.28,29

In this study, the LES calculation is peformed for a turbulent isothermalround jet at a Mach number of 0.9 and Reynolds number of 400,000. The LEScalculation employs an implicit sixth-order tri-diagonal spatial filter30 as theimplicit subgrid-scale (SGS) model. The jet centerline temperature was chosento be the same as the ambient temperature, To and set to 286K. A fully curvi-linear grid consisting of approximately 16 million grid points was used in thesimulation. The grid had 390 points along the streamwise x direction and 200points along both the y and z directions. The physical portion of the domainin this simulation is extended to 35 jet radii in the streamwise direction and 30jet radii in the transverse directions. The coarsest resolution in this simulationis estimated to be about 400 times the local Kolmogorov length scale.

The boundaries of the computational domain are applied using the Tam and


Mach number

r

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

Fig. 5. Mean-flow profile.

Dong’s 3-D radiation and outflow boundary conditions,31 as illustrated in Fig.6. A sponge zone32 was used in which grid stretching and artificial dampingare applied to suppress the unwanted reflections from the outflow boundary.Since the actual nozzle geometry is not included in the present calculations,randomized perturbations in the form of a vortex ring are added to the jetvelocity profile at a short distance downstream of the inflow boundary in orderto excite the 3-D instabilities in the jet and cause the potential core of the jet tobreak up at a reasonable distance downstream of the inflow boundary. The infowforcing used here was proposed by Bogey and Bailly.33 There are 16 azimuthalmodes in total and the first 4 modes are not included in the vortex ring forcing.A more detailed study of the inflow vortex ring forcing can be found in Ref. 34.The mean shear layer imposed on the inflow boundary has a streamwise velocityprofile

u(r) =UJ

2

[1 + tanh

[10

(1 −

r

rJ

)]](28)

where UJ is the jet centerline velocity at nozzle exit, rJ is the jet nozzle radius.There are about 14 grid points in the initial jet shear layer. The standard fourth-order explicit Runge-Kutta scheme is used for time advancement.

A Ffowcs William - Hawkings (FW-H) control surface was placed around thejet as illustrated in Fig. 7. The control surface starts 1 jet radius downstream ofthe inflow boundary and extends to 31 jet radii along the streamwise direction.Hence the total streamwise length of the control surface is 30rJ . The controlsurface is initially at a distance of 3.9 jet radii from the jet centerline, andopens up with downstream distance. The LES flow field data are gathered onthe control surface in time domain at every 5 times steps over a period of 25,000time steps during the LES run. The initial transients exited the domain over


Tam & Dong’s radiation boundary conditions

Tam & Dong’s radiation boundary conditions

Tam & Dong’soutflow boundaryconditions

Sponge zone

Tam &Dong’sradiationbcs

Vortex ring forcing

Fig. 6. Schemetic of the boundary conditions.

Fig. 7. Schematic of the control surface surrounding the jet flow.

the first 10,000 time steps. The total acoustic sampling period corresponds to atime scale in which an ambient sound wave travels about 14 times the domainlength in the streamwise direction. Assuming at least 6 points per wavelengthare needed to accurately resolve an acoustic wave, it is found that the maximumfrequency resolved with the grid spacing around the control surface correspondsto a Strouhal number of approximately 3.0. This is the grid cutoff frequencyused in computation domain. Based on the data sampling rate, the number oftemporal points per period in this maximum frequency is 8.

To investigate results of the refraction corrections, a frequency domain of theFW-H method is developed and has LES data as an input at outflow surface.A 4608-point Fast Fourier Transform (FFT) analysis is performed to obtain

the variables required in FW-H calculation, which are p′, Un, and Lr at eachfrequency. These terms are calculated in the retarded time algorithm using inEq. (3), which can be solved by performing simple linear interpolation.


Fig. 8. Schematic showing the center of the arc and how the angle θ ismeasured from the jet axis.

It should be noted that the refraction corrections evaluated on the down-stream control surface are using the frequency domain FW-H equation. Theseresults are added to the time domain FW-H results for an open surface.29 Thus,an inverse FFT is needed to recover the signal for the downstream refractionsurface in the time domain.

Numerical results of far-field overall sound pressure level (OASPL) are cal-culated using Eq. (29) along an arc of radius 60rJ from the jet nozzle.

OASPL = 20log10(E

Pref) (29)

where Pref is the non-dimensional reference pressure calculated from the am-bient temperature, To = 286K, ambient pressure, po = 1.01KPa, M = 0.9 atjet centerline nozzle exit, and dimensional reference pressure, 2x10−5Pa. E isdefined as <

∫Pav(f)df >, where Pav is calculated from the average acoustic

pressure signals over the equally spaced 36 azimuthal points on a full circle ata given θ location on far-field arc of radius 60rJ from the jet nozzle. The centerof the arc is chosen as the jet nozzle exit and the angle θ is measured from thejet axis as illustrated in Fig. 8.

Table 1. Some jet noise experimentswith Mach numbers similar to that our

LESM ReD Reference0.9 540,000 Mollo-Christensen et al.35

0.88 500,000 Lush11

0.9 3,600 Stromberg et al.36

0.9 400,000 SAE ARP 876C37

0.9 400,000 Our 3-D LES28


θ (deg)

OA

SPL

(dB

)

0 10 20 30 40 50 60 70 80 90 100 110 120100

102

104

106

108

110

112

114

116

118

120

SimpleGA refractionLilley’s refractionOpen control surfaceClosed control surfaceSAE ARP 876C predictionMollo-Christensen et al. data (cold jet)Lush data (cold jet)Stromberg et al. data (cold jet)

Fig. 9. Overall sound pressure levels at 60rJ from the nozzle exit.

One should remove the low frequencies resulting from the outflow boundarywhich may not be damped out effectively by the sponge zone. In our jet case, thisfrequency with the corresponding Strouhal number of 0.05 was found. Hence,the OASPL spectra integration only includes the resolved frequencies range of0.05 ≤ St ≤ 3.0.

The computations of surface integral method were performed on IBM Power3 machine at Purdue University. Advanced features offered by FORTRAN 90such as dynamics memory allocations are implemented in the code to enablethe computations. The codes are run in parallel using MPI implementation.The outflow surface comprises of a total of 135 x 135 patches. The walltimerequired for computing acoustic signals at each arc location (with 36 azimuthallocations) from the outflow FW-H surface is around 5.8 hours using eight 375MHz processors. The solutions of the rest of the FW-H surfaces are added tothe solutions of the outflow surface that includes the refraction corrections. InFig. 9, the solid line represents the OASPL result with no refraction correctionat observer location along a far-field arc of x = 60rJ . Table (1) summarizesthe Mach numbers and Reynolds numbers of the experiments and the ARPdatabase37 that we are comparing to. It should be noted that the experimentaljets were cold jets, whereas our jet is an isothermal jet. However, as indicatedin the ARP database, the effect of the temperature is very small, i.e. about 0.3dB.

From the OASPL plot for an open surface, it is shown that the predictionfrom ARP database for a cold jet at ReD of 400,000 is slightly lower (about1dB) than the LES calculations for high angles, i.e. θ ≥ 40o. Also the OASPLare comparable to the experimental values at these angles. However, at shallowangles, i.e. θ ≤ 40o, the OASPL from the LES drops significantly due to the


omission of the signal from the downstream surface. For the closed control sur-face, the OASPL is generally higher than that of the open control surface. Theresults increase significantly for lower angles. However, they become very highat low angles as refraction effects outside the control surface are not included.For higher angles, i.e. θ ≥ 70o, there is also an overprediction due to an spu-rious line of dipoles on the downstream surface area, as the quadrupoles movedownstream and exit the control surface.30

The implementation of the refraction correction begins with generating aparallel shear mean flow. The mean flow velocity profile generated from LESis shown in Fig. 10. To reduce the required computational time for Lilley’sequation, the mean flow velocity profile is modeled with the following equation

M(r) = MJeAr2

+ B (30)

where MJ = 0.46, A, B are constants, i.e. -0.14 and 0.0044, respectively. Equa-tion (30) is represented with dash-line in Fig. 10.

This profile is used for both the simple GA method and Lilley’s equationcalculations. This sheared mean flow has an influence on the signals emittedfrom the surface elements on the downstream outflow FW-H surface. Refrac-tion corrections are performed on the surface elements inside the sheared flow,i.e. on the downstream surface. The variation of the velocities across the ra-dial direction of the shear layer is taken into account at each surface elementlocation. The refraction corrections schemes result in the damping of far-fieldsignal contributed by the surface element. Within the high-frequency region of0.50 ≤ St ≤ 3.0, this refraction corrections are performed at each frequency andon each surface element.

As shown in Fig. 9, the OASPL with refraction corrections starts to decreaseas θ goes below 20o. This gives a zone of silence as expected from the experimen-tal results. The slight decrease of the results with no refraction corrections is dueto the existence of refraction effect inside the FW-H control surface. With therefraction corrections outside the control surface using GA method, the resultsshow a decrease on the OASPL making closer to the experiments. The resultsfrom the Lilley’s equation method exhibit a smoother decrease on the OASPLinside the zone of silence which is consistent with the results for a monopolesource shown in Fig. 3.

With the application of Lilley’s equation, the computational walltime re-quired increases to about 180 hours for each arc location inside the zone ofsilence using eight 375 Mhz processors run in parallel. The maximum time ratiofor Lilley’s equation method to GA method is 60. This significant increase incomputational time is expected when solving the Lilley’s equation because themean flow velocity dependent turning point, rδ in Eq. (23) needs to be evaluatednumerically. The scheme currently used in this research is Brent’s method.

The use of the simple GA method and Lilley’s equation have shown a suc-cessful prediction of the zone of silence, which cannot be done by the traditionalsurface integral methods.


M

r

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

LES dataAnalytical data

Fig. 10. Jet mean-flow profile.

V. Concluding Remarks

In the past the surface integral methods have proven to be an efficient andaccurate tool for the prediction of jet noise and other aeroacoustic phenomena.However, they fail to predict the zone of silence accurately. We have developedrefraction corrections and utilized them with the surface integral methods, i.e.Kirchhoff and porous Ffowcs Williams - Hawkings (FW-H) method. The re-fraction corrections are based on simple geometric acoustics (GA) and Lilley’sequation high-frequency, far-field asymptotic solutions. The methods were ap-plied to the noise calculation of an isothermal jet of Reynolds number 400,000.The solution on the FW-H control surface was provided using LES. The aboverefraction corrections were applied on the downstream control surface to accountfor the mean shear flow outside the control surface. Both refraction correctionsimprove the OASPL results dramatically at shallow angles near the axis.

The comparisons between the two methods were also made and it was shownthat the simple GA method is much simpler and requires less computational timecompared to Lilley’s equation. On the other hand, the high-frequency asymp-totic solutions for Lilley’s equation has the potential of being more realisticwhich allows azimuthal variations between source and observer.

Acknowledgments

The simulations were carried out on Purdue University’s 320 processor IBM-SP3 supercomputer. The third author acknowledges support by the Indiana21st Century Research and Technology Fund, and the Aeroacoustics ResearchConsortium (AARC), a government and industry consortium managed by theOhio Aerospace Institude (OAI). The third author would also like to thank Dr.Khavaran from NASA Glenn for suggesting the Lilley’s equation method forrefraction corrections. The first author is grateful to Dr. Gregory Blaisdell and


Phoi-Tak Lew for providing useful suggestions and discussions.

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REFRACTION CORRECTIONS FOR SURFACE INTEGRAL …lyrintzi/presentations/Panaiaa2873.pdf · acoustic sources to the linear far-ﬂeld sound through the traditional Lighthill’s acoustic