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Surface integral methods in computational aeroacoustics—From the (CFD) near-field to the (Acoustic) far-field * Anastasios S. Lyrintzis School of Aeronautics and Astronautics Purdue University W. Lafayette, IN 47907-2023 ABSTRACT A review of recent advances in the use of surface integral methods in Computational AeroAcoustics (CAA) for the extension of near-field CFD results to the acoustic far-field is given. These integral formulations (i.e. Kirchhoff’s method, permeable (porous) surface Ffowcs- Williams Hawkings (FW-H) equation) allow the radiating sound to be evaluated based on quantities on an arbitrary control surface if the wave equation is assumed outside. Thus only surface integrals are needed for the calculation of the far-field sound, instead of the volume integrals required by the traditional acoustic analogy method (i.e. Lighthill, rigid body FW-H equation). A numerical CFD method is used for the evaluation of the flow-field solution in the near field and thus on the control surface. Diffusion and dispersion errors associated with wave propagation in the far-field are avoided. The surface integrals and the first derivatives needed can be easily evaluated from the near-field CFD data. Both methods can be extended in order to include refraction effects outside the control surface. The methods have been applied to helicopter noise, jet noise, propeller noise, ducted fan noise, etc. A simple set of portable Kirchhoff/FW-H subroutines can be developed to calculate the far-field noise from inputs supplied by any aerodynamic near/mid-field CFD code. 1 BACKGROUND—AEROACOUSTIC METHODS For an airplane or a helicopter, aerodynamic noise generated from fluids is usually very important. There are many kinds of aerodynamic noise including turbine jet noise, impulsive noise due to unsteady flow around wings and rotors, broadband noise due to inflow turbulence and boundary layer separated flow, etc. (e.g. Lighthill 1 ). Accurate prediction of noise mechanisms is essential in order to be able to control or modify aeroacoustics volume 2 · number 2 · 2003 – pages 95 – 128 95 *Presented at the CEAS Workshop “From CFD to CAA” Athens, Greece, Nov. 2002. Professor, e-mail: [email protected]. Aero 2-2_Paper 1 28/10/03 5:37 PM Page 95

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Page 1: Surface integral methods in computational aeroacoustics ...lyrintzi/JAAfinal.pdf · The methods have been applied to helicopter noise, jet noise, propeller noise, ducted fan noise,

Surface integral methods incomputational aeroacoustics—From the

(CFD) near-field tothe (Acoustic) far-field *

Anastasios S. Lyrintzis†

School of Aeronautics and Astronautics

Purdue University

W. Lafayette, IN 47907-2023

ABSTRACTA review of recent advances in the use of surface integral methods in ComputationalAeroAcoustics (CAA) for the extension of near-field CFD results to the acoustic far-field isgiven. These integral formulations (i.e. Kirchhoff’s method, permeable (porous) surface Ffowcs-Williams Hawkings (FW-H) equation) allow the radiating sound to be evaluated based onquantities on an arbitrary control surface if the wave equation is assumed outside. Thus onlysurface integrals are needed for the calculation of the far-field sound, instead of the volumeintegrals required by the traditional acoustic analogy method (i.e. Lighthill, rigid body FW-Hequation). A numerical CFD method is used for the evaluation of the flow-field solution in thenear field and thus on the control surface. Diffusion and dispersion errors associated with wavepropagation in the far-field are avoided. The surface integrals and the first derivatives needed canbe easily evaluated from the near-field CFD data. Both methods can be extended in order toinclude refraction effects outside the control surface. The methods have been applied to helicopternoise, jet noise, propeller noise, ducted fan noise, etc. A simple set of portable Kirchhoff/FW-Hsubroutines can be developed to calculate the far-field noise from inputs supplied by anyaerodynamic near/mid-field CFD code.

1 BACKGROUND—AEROACOUSTIC METHODSFor an airplane or a helicopter, aerodynamic noise generated from fluids is usually veryimportant. There are many kinds of aerodynamic noise including turbine jet noise,impulsive noise due to unsteady flow around wings and rotors, broadband noise due toinflow turbulence and boundary layer separated flow, etc. (e.g. Lighthill1). Accurateprediction of noise mechanisms is essential in order to be able to control or modify

aeroacoustics volume 2 · number 2 · 2003 – pages 95 – 128 95

*Presented at the CEAS Workshop “From CFD to CAA” Athens, Greece, Nov. 2002.†Professor, e-mail: [email protected].

Aero 2-2_Paper 1 28/10/03 5:37 PM Page 95

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them to comply with noise regulations, i.e. Federal Aviation Regulations (FAR) part 36,and achieve noise reductions. Both theoretical and experimental studies are beingconducted to understand the basic noise mechanisms. Flight-test or wind-tunnel testprograms can be used, but in either case difficulties are encounted such as high expense,safety risks, and atmospheric variability, as well as reflection problems for wind-tunneltests. As the available computational power increases numerical techniques arebecoming more and more appealing. Although complete noise models have not yetbeen developed, numerical simulations with a proper model are increasingly beingemployed for the prediction of aerodynamic noise because they are low-cost andefficient. This research has led to the emergence of a relatively new field: ComputationalAeroAcoustics (CAA).

CAA is concerned with the prediction of the aerodynamic sound source and thetransmission of the generated sound starting from the time-dependent governingequations. The full, time-dependent, compressible Navier-Stokes equations describethese phenomena. Although recent advances in Computational Fluid Dynamics (CFD)and in computer technology have made first-principle CAA plausible, direct extensionof current CFD technology to CAA requires addressing several technical difficultiesin the prediction of both the sound generation and its transmission.2–3 A review ofaerospace application of CAA methods was given by Long et al.4

Aerodynamically generated sound is governed by a nonlinear process. One class ofproblems is turbulence generated noise (e.g. jet noise). An accurate turbulence model isusually needed in this case. A second class of problems involves impulsive noise due tomoving surfaces (e.g. helicopter rotor noise, propeller noise, fan noise etc.). In thesecases an Euler/Navier-Stokes model or even a full potential model is adequate, becauseturbulence is not important.

Once the sound source is predicted, several approaches can be used to describeits propagation. The obvious strategy is to extend the computational domain for thefull, nonlinear Navier-Stokes equations far enough to encompass the location wherethe sound is to be calculated. However, if the objective is to calculate the far-fieldsound, this direct approach requires prohibitive computer storage and leads tounrealistic turnaround time. The impracticality of straight CFD calculations forsupersonic jet aeroacoustics was pointed out by Mankbadi et al.5 Furthermore, becausethe acoustic fluctuations are usually quite small (about three orders of magnitude lessthan the flow fluctuations), the use of nonlinear equations (whether Navier-Stokes orEuler) could result in errors, as pointed out by Stoker and Smith.6 One usually has nochoice but to separate the computation into two domains, one describing the nonlineargeneration of sound, the other describing the propagation of sound. There are severalalternatives to describing the sound propagation once the source has been identified.

1.1 Field solution of simpler equationsLinearized Euler Equations (LEE) The first alternative is to use simpler equationsin the acoustic far-field. The Linearized Euler Equations (LEE) have been used in orderto extend the CFD solutions to the far-field (e.g. Lim et al.7, Viswanathan and Sankar8,Shih et al.9). The LEE equations employ a division of the flow field into a

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time-averaged flow and a time-dependent disturbance which is assumed to be small.The hybrid (zonal) approach consists of the near-field evaluation using an accurateCFD code (e.g. for jet noise the code is usually based on Large Eddy Simulations: LES)and the extension of the solution to the mid-field using LEE. Considerable CPU savingscan be realized, since the LEE calculations are much cheaper than the CFD calculations.This approach is very promising, because it accounts for a variable sound velocityoutside the near-field where usually an LES model is applied. This method may also beappropriate for an intermediate region in some problems, outside from the reactivenear-field where the speed of sound is still not constant, before moving to anotherintegral method for the far-field.

Other Equations Hardin and Pope10 have proposed a decoupling of the time-dependent incompressible flow and the compressible aspects (acoustics) of the flow.This technique was used successfully to predict the flow over a two-dimensional cavity.A field solution of the wave equation can also be used (e.g. Freund11). Freund11 claimsthat the field solution of the wave equation is cheaper than the surface integral solutions(see section 1.2.2), when the solution everywhere in the field is sought. However, inmost applications only a few locations are needed to study directivity and compare withmicrophone measurements. Also, for any numerical solution of field equationsdissipation and dispersion errors still exist and an accurate description of propagatingfar-field waves is compromised.

1.2 Integral methods1.2.1 Volume integral methodsTraditional Acoustic Analogy The first integral approach for acoustic propagationis the acoustic analogy.12 In the acoustic analogy, the governing Navier-Stokesequations are rearranged to be in wave-type form. There is some question as to whichterms should be identified as part of the sound source and retained in the right-handside of the equation and which terms should be in the left-hand side as part of theoperator (e.g., Lilley13). The far-field sound pressure is then given in terms of a volumeintegral over the domain containing the sound source. Several modifications toLighthill’s original theory have been proposed to account for the sound-flowinteraction or other effects. The major difficulty with the acoustic analogy, however, isthat the sound source is not compact in supersonic flows. Errors could be encounteredin calculating the sound field, unless the computational domain could be extended inthe downstream direction beyond the location where the sound source has completelydecayed. Furthermore, an accurate account of the retarded time-effect requires keepinga long record of the time-history of the converged solution of the sound source, whichagain represents a storage problem. The Ffowcs Williams and Hawkings (FW-H)equation14 was introduced to extend acoustic analogy in the case of solid surfaces.However, when acoustic sources (i.e., quadrupoles) are present in the flowfield avolume integration is needed. This volume integration of the quadrupole source termis difficult to compute and is usually neglected in most acoustic analogy codes(e.g. WOPWOP15). Recently, there have been some successful attempts in evaluatingthis term (e.g. WOPWOP+16,17).

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1.2.2 Surface integral methodsKirchhoff Method Another alternative is the Kirchhoff method which assumes thatthe sound transmission is governed by the simple wave equation. Kirchhoff’s methodconsists of the calculation of the nonlinear near- and mid-field, usually numerically,with the far-field solutions found from a linear Kirchhoff formulation evaluated on acontrol surface surrounding the nonlinear-field. The control surface is assumed toenclose all the nonlinear flow effects and noise sources. The sound pressure can beobtained in terms of a surface integral of the surface pressure and its normal and timederivatives. This approach has the potential to overcome some of the difficultiesassociated with the traditional acoustic analogy approach. The method is simple andaccurate and accounts for the nonlinear quadrupole noise in the far-field. Fulldiffraction and focusing effects are included while eliminating the propagation of thereactive near-field.

This idea of matching between a nonlinear aerodynamic near-field and a linearacoustic far-field was first proposed by Hawkings.18 The separation of the problem intolinear and nonlinear regions allows the use of the most appropriate numericalmethodology for each. The terminology “Kirchhoff method” was introduced by Georgeand Lyrintzis.19 It has been used to study various aeroacoustic problems, such aspropeller noise, high-speed compressibility noise, blade-vortex interactions, jet noise,ducted fan noise, etc. The use of Kirchhoff’s method has increased substantially the last10 years, because of the development of reliable CFD methods that can be used for theevaluation of the near-field. An earlier review on the use of Kirchhoff’s method wasgiven by Lyrintzis.20

Porous FW-H equation A final alternative is the use of permeable (porous)surface FW-H equation. The usual practice is to assume that the FW-H integrationsurface corresponds to a solid body and is impenetrable. However, if the surface isassumed to be porous, a general equation can be derived (as shown in the originalreference 14 and in reference 21). The porous surface can be used as a control surfacein a similar fashion as the Kirchhoff method explained above. Thus the pressure signalin the far-field can be found based on quantities on the control surface provided by aCFD code.

Farassat in a recent review article22 reviewed all the available FW-H and Kirchhoffequations for application to noise evaluation from rotating blades. The current articlefocuses only on control surface methods (i.e. Kirchhoff, porous FW-H) and discussesissues with their application in various types of aerocoustic problems including rotornoise, jet noise, ducted fan noise, airfoil noise etc.). At first the main formulations willbe reviewed, advantages and disadvantages of each method will be discussed. Then wewill present several algorithmic issues and various application examples.

2 SURFACE INTEGRAL FORMULATIONS2.1 Kirchhoff’s method formulationsKirchhoff’s method is an innovative approach to noise problems which takes advantageof the mathematical similarities between the aeroacoustic and electrodynamicequations. The considerable body of theoretical knowledge regarding electrodynamic

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field solutions can be utilized to arrive at the solution of difficult noise problems.Kirchhoff’s formula was first published in 1882.23 It is an integral representation(i.e. surface integral around a control surface) of the solution to the wave equation.Kirchhoff’s formula, although primarily used in the theory of diffraction of light and inother electromagnetic problems, it has also many applications in studies of acousticwave propagation.

The classical Kirchhoff formulation is limited to a stationary surface. Morgans24

derived a formula for a moving control surface using Green’s functions. Generalizedfunctions can also be used for the derivation of an extended Kirchhoff formulation.A field function is defined to be identical to the real flow quantity outside a controlsurface S and zero inside. The discontinuities of the field function across the controlsurface S are taken as acoustic sources, represented by generalized functions. Ffowcs-Williams and Hawkings14 derived an extended Kirchhoff formulation for soundgeneration from a vibrating surface in arbitrary motion. However, in their formulationthe partial derivatives were taken with respect to the observation coordinates and timeand that is difficult to use in numerical computations. Farassat and Myers25 deriveda Kirchhoff formulation for a moving, deformable, piecewise smooth surface. The samepartial derivatives were taken with respect to the source coordinates and time. Thus theirformulation is easier to use in numerical computations and their relatively simplederivation shows the power of generalized function analysis.

It should be noted that Morino and his co-workers26–30 have developed severalformulations for boundary element methods using the Green’s function approach. Theseare based on the solution of the wave equation and hence, the integral expressions arethe same as in Kirchhoff’s method. However, the formulation in terms of the velocitypotential. This has advantages (e.g., the boundary condition is simple) as well asdisadvantages ( e.g., the pressure of the wake). Morino’s formulations were derived withaerodynamic applications in mind, so the observer is in the moving coordinate system.However, they can be used for aeroacoustics, for example when both the control surfaceand the observer move with a constant speed (e.g., wind tunnel experiments), asmentioned in reference 20. Their latest formulation29 appears to provide an integratedboundary element framework for Aerodynamics and Aeroacoustics. A detaileddiscussion about the differences in the aerodynamics and the aeroacoustics of their variousformulations can be found in reference 30.

2.1.1 Farassat’s formulationFarassat’s Kirchhoff formulation gives the far-field signal, due to sources containedwithin the Kirchhoff surface. Assume the linear, homogeneous wave equation,

�2φ = 1

a2◦

∂2φ

∂t2− ∂2φ

∂xi∂xi= 0 (1)

is valid for some acoustic variable φ, and sound speed a◦, in the entire region outsideof a closed and bounded smooth surface, S.

The signal, in the stationary coordinate system, is evaluated with a surface integralover the control surface, S, of the dependent variable, its normal derivative, and its time

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100 Surface integral methods in computational aeroacoustics

derivative (Figure 1). S is allowed to move in an arbitrary rigid-body fashion. Thedependent variable φ is normally taken to be the disturbance pressure, but can be anyquantity which satisfies the linear wave equation.

4πφ(�x, t) =∫

S

[E1

r (1 − Mr )

]ret

dS +∫

S

[φE2

r2 (1 − Mr )

]ret

dS (2)

where

E1 = (M2

n − 1) ∂φ

∂n+ Mn �Mt · ∇2φ − Mn

a◦φ + 1

a◦(1 − Mr )2

[Mr (cos θ − Mn) φ

]

+ 1

a◦(1 − Mr )

[(nr − Mn − nM

)φ + (cos θ − Mn) φ + (cos θ − Mn) φ

](3)

E2 = (1 − M2)

(1 − Mr )2(cos θ − Mn) (4)

Here (�x, t) are the observer coordinates and time, and (�y, τ ) are the source (surface)coordinates and time. Mi is the Mach number vector of the surface, r is the distancefrom source to observer, θ is the source emission angle, and n is the control surface unitnormal vector (cos θ = r · n). �Mt is the Mach number vector tangent to the surface, and∇2 is the surface gradient operator. A dot indicates a source time derivative, with theposition on the surface kept fixed. Also,

Mr = Mi ri nr = ni ri Mn = Mi ni nM = ni Mi (5)

(y, τ′)

Observer (x, t)

n

r

Control surface S

Figure 1. Kirchhoff’s surface S and notation.

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The form of Equation (2) and E1, E2 were given by Farassat and Myers.25 E2 waspresented in the simplified form shown here by Myers and Hausmann.31 The surfaceintegrals are over the control surface S, subscript ret indicates evaluation of theintegrands at the emission (retarded) time, which is the root of

g = τ − t + |�x − �y|a◦

= 0 (6)

If the frame velocity is subsonic at the surface, then equation (6) has a unique solution.However, equation (2) is still valid for supersonically moving surfaces. As we can seefrom equations 2 through 5, the (1 − Mr ) term can produce a singularity in the casewhere the Mach number in the radiation direction reaches the sonic point. This is amajor limitation of the retarded time formulation. Farassat and co-workers29,30 haverecently presented a formulation that is appropriate for supersonically moving surfaces(i.e. formulation 4) and verified by application to benchmark problems. Since, thissupersonic formulation has not yet been applied to practical problems it will not bepresented here in the interest of brevity.

The above formulation is valid when the observer is stationary and the surface ismoving at an arbitrary speed. However, for the case of an advancing blade the observer isusually moving with the free-flow speed (e.g. rotor in a wind tunnel with a free stream notequal to zero). The formulation can be adjusted for this case by allowing x(t) to movewith the free stream instead of being stationary in equation (6) for the retarded time.

It is possible to write equation (2) in a simple form valid for stationary surfaces. TheKirchhoff formula is then

4πφ(�x, t) =∫

S

1

r

[1

a◦φ cos θ − ∂φ

∂n

]ret

dS +∫

S

[φ]ret

r2dS (7)

The retarded time for this case is t − r/c. With the use of a Fourier transformation,equation (7) can be expressed in the frequency domain (i.e. starting from Helmholtzequation) as

4πφ(�x, ω) =∫

Seiωr/a◦

[1

r

(− iω

a◦cos θ φ − ∂φ

∂n

)+ φ cos θ

r2

]dS (8)

φ =∫ ∞

−∞φeiωt dt, φ = 1

∫ ∞

−∞φe

−iωtdω

where φ is the Fourier transform of φ, and ω is the cyclic frequency. An equivalent toequation (8), valid for surfaces and observers in rectilinear motion was presented byLyrintzis and Mankbadi34 and Pilon.35

Two-dimensional formulations can also be developed (Pilon,35 Scott et al.36). Atassiet al.37 developed a two-dimensional frequency domain formulation that uses amodified Green’s function in order to avoid the evaluation of normal derivatives.Mankbadi et al.38 developed a modified Green’s function for a cylinder control surfacethat was applied in jet noise predictions. Hariharan et al.39 developed a framework forKirchhoff’s formulations without the use of normal derivatives; the method was appliedinitially for two-dimensional problems.

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For completeness we should mention that for the case where the Kirchhoff controlsurface S coincides with the body surface, (BIE-Boundary Integral Equations) there aresome non-uniqueness difficulties in the prediction of the radiated acoustic sound in theexterior region whenever the frequency coincides with one of the Dirichleteigenfrequencies. These problems where analyzed for the stationary Kirchhoff surfaceby Wu and Pierce40 and for moving Kirchhoff surfaces by Wu.41 Finally, Dowling andFfowcs Williams42 included the effects of infinite plane walls in the stationaryKirchhoff formulation. However, in this paper we are reviewing the use of Kirchhoff’sequation for extenting near-field results in the far-field (BIR Boundary IntegralRepresentation), so the issues mentioned in this paragraph are not relevant.

2.1.2 The extended Kirchhoff methodEquation (2) works well for aeroacoustic predictions when the control surface is placedin a region of the flow field where the linear wave equation is valid. However, thismight not be possible for some cases. Therefore, additional nonlinearities can be addedoutside the control surface.43–48 The modifications to the traditional Kirchhoff methodconsist of an additional volume integral. Thus equation (2) now becomes:46 (the pressureperturbation p′ is used here as the dependent variable)

4πp′(�x, t) =∫

S

[E1

r (1 − Mr )

]ret

dS +∫

S

[p′E2

r2 (1 − Mr )

]ret

dS +∫

V

[1

r (1 − Mr )

∂2Ti j

∂yi∂yj

]ret

dV(9)

where

Ti j = ρui uj − σi j + ((p − po) − a2

◦ρ′) δi j (10)

where ui is the fluid velocity, ρ is the density, ρ ′ the density perturbation, and σi j isthe viscous stress tensor. It is easy to show that this equation reduces to the traditionalKirchhoff integral if the control surface is placed in a fully linear region, as Ti j becomeszero. Through the use of Fourier transforms, equation (9) can also be expressed in thefrequency domain.

Isom et al.49 developed a nonlinear Kirchhoff formulation (Isom’s formulation) forsome special cases (i.e. stationary surface at the sonic cylinder of a rotor, highfrequency approximation and observer on the rotation plane). They have included intheir formulation some nonlinear effects using the transonic small disturbance equation.The nonlinear effects are generally accounted for with a volume integral, as shownabove. However, they showed that for the above special cases the nonlinear effects canbe reduced to a surface integral.

2.2 The porous Ffowcs Williams—Hawkings equationA modified integral formulation for the porous surface FW-H equation18 is neededbecause the usual practice is to assume that the FW-H integration surface correspondsto the body and is impenetrable. A convenient way to formulate this is as an extension

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of Farassat’s formulation 150 which was originally developed for the rigid surfaceFW-H equation. Following Di Francescantonio45 we define new variables Ui and Li as

Ui =(

1 − ρ

ρo

)νi + ρui

ρo(11)

and

Li = Pi j n j + ρui (un − νn) (12)

where subscript o implies ambient conditions, superscript ′ implies disturbances(e.g. ρ = ρ ′ + ρo), ρ is the density, u is the fluid velocity, ν is the velocity of the controlsurface, and Pi j is the compressive stress tensor with the constant poδi j subtracted. Nowby taking the time derivative of the continuity equation and subtracting the divergenceof momentum equation, followed with some rearranging, the integral form of FW-Hequation can be written as (Formulation I)

p′(�x, t) = p′T (�x, t) + p′

L(�x, t) + p′Q(�x, t) (13)

where

4πp′T (�x, t) = ∂

∂t

∫S

[ρoUn

r |1 − Mr |]

ret

dS (14)

4πp′L(�x, t) = 1

a◦

∂t

∫S

[Lr

r |1 − Mr |]

ret

dS +∫

S

[Lr

r2|1 − Mr |]

ret

dS (15)

and p′Q(�x, t) can be determined by any method currently available (e.g. references

16, 17). In equations (14) and (15) a subscript r or n indicates a dot product of thevector with 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 pressure terms have a physical meaning for rigidsurfaces: 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 term p′Q(�x, t) still denotes the quadrupoles outside the control

(porous) surface S.An alternative way45 is to move the time derivative inside the integral: (Formulation II)

4πp′T (�x, t) =

∫S

[ρo(Un + Un)

r(1 − Mr )2

]ret

dS +∫

S

[ρoun(r Mr + c(Mr − M2))

r2(1 − Mr )3

]ret

dS (16)

4πp′L(�x, t) = 1

c

∫S

[Lr

r(1 − Mr )2

]ret

dS +∫

S

[Lr − L M

r2(1 − Mr )2

]ret

dS

+1

c

∫S

[Lr (r Mr + c(Mr − M2))

r2(1 − Mr )3

]ret

dS (17)

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This is now an extension of Farassat’s formulation 1A51 (also originally developed forthe rigid surface FW-H equation) where the dot over a variable implies source-timedifferentiation of that variable, L M = Li Mi , and a subscript r or n indicates a dotproduct of the vector with the unit vector in the radiation direction r or the unit vectorin the surface normal direction ni , respectively.

Comparing the two FW-H formulations, it appears that Formulation I (equations 14, 15)has less memory requirements, because it does not require storage of the time derivatives,and requires less operations per integral evaluation. However, in general, integrals haveto be evaluated twice in order to find the time derivative. In the special case of astationary control surface, or a fixed microphone location, i.e. “flyover,” the integral canbe reused at the next time step. Since memory appears to be more important for thesetype of calculations, Formulation I is a good choice for stationary surfaces. Formulation Iwas used by Strawn et al.52 for rotorcraft noise predictions using a non-rotating controlsurface with very good results. On the other hand, taking the time derivative inside couldprevent some instabilities. Thus Formulation II (equations 16, 17) might be more robustfor a moving control surface. Formulation II was used for rotorcraft noise prediction byBrentner and Farassat47 with a rotating control surface with very good results. However,a more detailed comparison of the two formulations would be very helpful.

For a stationary surface Formulation I reduces to:

4πp′T (�x, t) = ∂

∂t

∫S

[ρoUn

r

]ret

dS (18)

4πp′L(�x, t) = 1

a◦c

∂t

∫S

[Lr

r

]ret

dS +∫

S

[Lr

r2

]ret

dS (19)

and Formulation II becomes:

4πp′T (�x, t) =

∫S

[ρ◦Un

r

]ret

dS (20)

4πp′L(�x, t) = 1

a◦

∫S

[Lr

r

]ret

dS +∫

S

[Lr

r2

]ret

dS (21)

With the use of a Fourier transformation both formulations (for a stationary surface)can be written in the frequency domain as53

4π p′T (�x, ω) = −iω

∫S

eiωr/a◦ ρ◦Un

rdS (22)

4π p′L(�x, ω) = −iω

q◦

∫S

eiωr/a◦ Lr

rdS +

∫S

Lr

r2dS (23)

where p′,Un , and Lr are the Fourier transforms of p′, Un , and Lr , respectively and ω isthe cyclic frequency. It should be noted that both time formulations reduce to the samefrequency formulation for a stationary control surface.

104 Surface integral methods in computational aeroacoustics

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Time and frequency formulations for a uniform rectilinear motion can be found inreference 54. Two-dimensional formulations for a solid surface FW-H equation havealready been developed in the past (see, for example, references 55, 56) and can bereadily extended to a porous surface. Finally, a supersonic formulation can also befound in reference 33.

2.3 Comparison of Kirchhoff and FW-H methodsBoth the above formulations provide a Kirchhoff-like formulation if the quadrupolesoutside the control surface (p′

Q(�x, t) term) are ignored. The equivalence of the porousFW-H equation and Kirchhoff formulation was proven Pilon & Lyrintzis43 and Brentner& Farassat.44 They showed that, for a surface placed in a linear region, the poroussurface FW–H formulation is equivalent to the linear Kirchhoff formulation, plus avolume integral of quadrupoles (ρui uj ). (Pilon and Lyrintzis43 also claim that thecontrol surface need not be placed in an entirely linear region. The nonlinearities can beaccounted for with the use of φ = a2

◦ρ′ as the dependent variable, and the volume

integral of quadrupoles, Ti j .)One difference between Kirchhoff’s and FW-H formulation is that Kirchhoff’s

method needs p′, ∂p/∂n,∂p/∂t (3 variables) as input, whereas the porous FW-H needsp′, ρ, ρui (5 variables), or Un and Li (4 variables), or their time derivatives forformulation II. Thus the porous FW-H method requires more memory, which can besignificant for large LES runs. The CPU time is about the same. However, the majordifference is that the porous FW-H method allows for nonlinearities on the controlsurface, whereas the Kirchhoff method assumes a solution of the linear wave equationon the surface. Thus if the solution does not satisfy the linear wave equation on thecontrol surface the results from the Kirchhoff method change dramatically. This leadsto a higher sensitivity for the choice of the control surface for the Kirchhoff method inpractical cases when the wave equation is not satisfied on the control surface due tonumerical errors or non-uniform velocities outside the control surface. This was shownin reference 47 for a rotorcraft noise problem (see section 5.2). Another way to state thisdifference is to state that the Kirchhoff method puts more stringent requirements to theCFD method to reach to the linear acoustic field before dissipation and dispersion errorsdue to coarsening in the far-field take over.

The volume integral of quadrupole sources that arises in the non-linear region outsideof the control surface presents a challenge. A major motivation for the use ofKirchhoff/porous FW-H methods is the lack of volume integrations, which reducesnecessary calculations by an order of magnitude. However, the methods used inWOPWOP+16,17 provide an efficient means of accounting for the quadrupoles in FW-Hcalculations that may be used in both methods, because the quadrupole terms are similar.

2.4 Mean flow refraction corrections for jet noiseThe Kirchhoff and the FW-H formulas presented above can efficiently and accuratelypredict aerodynamically generated noise, as long as the control surface surrounds theentire source region. In jet noise predictions, however, it is usually impossible, withcurrent numerical methods, to determine the entire source region. This is due to time

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106 Surface integral methods in computational aeroacoustics

and memory limitations imposed by the computer architecture, as well as dispersionand dissipation constraints. Thus, a significant nonlinear source region, as well as asteady mean flow, will exist outside of the control surface. Even if the unsteady soundsources outside of the control surface can be ignored, there is still a substantial steadymean flow in the region near the jet axis, downstream of the control surface. Thus, somemeans of approximating the effects of this steady shear flow are required if an acousticprediction is desired for observer points lying near the jet axis.

A suitable approximation to the downstream shear flow is necessary, in order todetermine the refraction effects. In the past, several researchers have used anaxisymmetric parallel shear flow model to determine sound produced by point acousticsources within circular jets (e.g. Amiet57). This approach was adopted by Lyrintzis andco-workers53,58 and in order to account for refraction effects in the Kirchhoff and theporous FW-H method. A real jet has non–zero radial velocity, but the refracting effectof this component is minimal, and can safely be ignored. Also, the lack of azimuthalvariation in the parallel shear flow approximation has a very small effect. The value ofthe axial velocity to be used in the shear flow approximation can be taken directly fromthe CFD numerical simulation, at the downstream end of the control surface, as anaverage of the time dependent axial velocity at each radial grid point.

The refraction problem now consists of a collection of point acoustic sources (theintegrands of equations (8) and (22) acting at radial location R, and the parallel shear flowwith U determined at each R). If the acoustic wavelength, λ = 2πa◦/ω, is assumed to besmall compared to the shear layer thickness δ, then geometric acoustics principles hold.

If the steady velocity at the downstream end of the Kirchhoff surface is denoted Us ,the sound emission angle with respect to the jet axis ϑs , and the propagation angle inthe stagnant, ambient air is denoted ϑ◦, then the axial acoustic phase speeds arepreserved by the stratified flow

Pa◦

cos ϑ◦= Us + a◦

cos ϑs(24)

It is assumed that the speed of sound at the source is equivalent to that in the ambientair. This equation can be rearranged to show that there is a critical angle, ϑc defined by

ϑc = cos−1

(1

1 + Ms

)(25)

If the observer angle ϑ◦ is greater than ϑc then no sound emitted at the source on theKirchhoff surface can reach the observer. This criterion is easily added to a stationarysurface Kirchhoff program. (Note that Ms is the Mach number of the mean shear flow,and not the Kirchhoff surface, which is assumed stationary.)

An additional correction is necessary to accurately account for the mean flowrefraction. Imposing the local “zone of silence” condition described above can allow asurface source at a relatively large radial location to radiate sound into and through theshear flow. This is because the local “zone of silence” decreases in size with the radiallocation of the source, due to the decrease in source Mach number. The simple

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correction is to set the source strength to zero if the observation point is located closerto the jet axis than the source point on the Kirchhoff surface.

Finally, the geometric acoustics approximation is only valid for δ/λ > 1. It isassumed here that the downstream end of the cylindrical Kirchhoff surface is locatedfar enough downstream of the jet potential core that the shear layer thickness is largecompared with the acoustic wavelength.

In reference 58 the mean flow refraction corrections were applied to the frequencydomain version of the Kirchhoff method (equation 8). In reference 53 an amplitudecorrection as recommended by Amiet57 (but not included in reference 58) was addedand the methodology was applied to both Kirchhoff and FW-H methods (equations 8and 22-23).

2.5 Open control surfaceFreund et al.59 developed a way to improve the accuracy of Kirchhoff evaluations ofsound fields for an open Kirchhoff control surface. Asymptotic analysis was used toprovide correction terms which partially account for the missing portion of the integralsurface. It was shown that the major contribution comes from a point on the surface thatintersects the line between observer and source. A correction term was estimated toaccount for the missing parts of the Kirchhoff surface. The study is restricted to the casewhere the mean flow is parallel to the available surface, as happens for example, forjet noise problems when the downstream surface vertical to the jet axis is missing.The corrections are limited to observers away from the jet axis. More details can befound in the original reference. It appears that this study can be extended to the porousFW-H equation, as well.

3 ALGORITHMIC ISSUESSome algorithmic issues are discussed below. Additional information for numericalalgorithms for acoustic integrals, in general, is given by Brentner.60

3.1 Choice of control surfaceThe Kirchhoff scheme requires stored data for pressure and pressure derivatives on asurface. Since Kirchhoff’s method assumes that the linear wave equation is validoutside the closed control surface S, S must be chosen large enough to include theregion of all nonlinear behavior. However, the accuracy of the numerical solution islimited to the region immediately surrounding the moving blade because of the increaseof mesh spacing in CFD codes. Thus a judicious choice of S is required for theeffectiveness of the Kirchhoff method. For example, in the case of airfoil/rotor noise thecontrol surface is typically located a couple of chordlengths away from the airfoil/rotorsurface.

For a porous FW-H formulation no normal derivatives are required and (becausenonlinearities are allowed on the control surface) the results are less sensitive to thechoice of the control surface,47 as will be shown in section 5.2. Thus the CFDrequirements for the FW-H are less strigent, making the method more attractive. Singeret al.61 used a FW-H method for the analysis of slat trailing-edge flow. The interesting

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thing about this application is that part of the control surface is solid and another partis porous.

3.2 QuadratureFor sufficient accuracy in the far-field calculations, high order quadrature shouldbe used to solve the surface integrals in equation (2). The predicted surface quantities(p′, ∂p/∂n, ∂p/∂t, ρ, ρui ) should also be very accurate. This can be achieved throughthe use of a very fine mesh in the CFD calculations. However, memory and timeconstraints often make this impractical. Meadows and Atkins62 have shown that it ispossible to obtain highly accurate Kirchhoff predictions from relatively coarse–gridCFD solutions. Through an interpolation process, more spatial points are added to theKirchhoff quadrature calculations without additional effort in the CFD process. This hasthe effect of refining the CFD mesh with almost no additional cost. They refer to thisprocess as “enrichment”. High order quadrature, temporal interpolation, and enrichmentare important for accurate far-field noise predictions for both the Kirchhoff and theFW-H equation methods, especially if the CFD grid resolution is somewhat coarse.

3.3 Retarded or forward timeThe retarded time equation (5) has a unique solution when the surface movessubsonically. A Newton-Raphson (or divide and conquer) method can be used to solvethis nonlinear equation. This method has been the basis of several Kirchhoff codes(e.g. Lyrintzis & Mankbadi34, Strawn et al.63, Lyrintzis et al.64 Polacsec & Prier65). Thealgorithm can be easily parallelized (e.g. Wissink et al.66, Strawn et al.67) by partitioningthe control surface and distributing to different processors. Since the onlycommunication is the final global summation the parallel efficiency of the code is veryhigh. Lockard54 discussed parallelization of FW-H codes. Long and Brentner68

proposed a master-slave approach for load balancing.However, it is difficult to write a versatile code for various mesh topologies used by

current CFD codes, including unstructured grids, based on this approach. In addition,when these codes are extended to supersonically moving surfaces, the retarded timeequation will have multiple roots that will be difficult to evaluate. Also, the codessometimes require significant memory. Finally62, the variation of the source strengthover a surface element in the retarded time can be very high at certain observerlocations (r · n → 0) and near sonic velocities (Mr → 1) requiring a large number ofpoints per wavelength.

In order to overcome the limitations stated above, another approach can bedeveloped which accumulates signals (with time moving forward) from each surfaceelement to an observer, thus it avoids the retarded time calculation. Computer memoryrequirements are reduced dramatically and the algorithm is inherently parallel. In thisapproach, the final overall observer acoustic signal is found from the summation of theacoustic signal radiated from each source element of control surface during the samesource time. The observer time is a straight-forward calculation using equation (6).For each surface element time is moved forward from the source (emission) to theobserver time. Since a different surface element will result in a different observer time,

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interpolation techniques are required when the integration is performed to obtain theoverall acoustic signal at the observer position. Both linear interpolation and splinesubroutines can be used. For high frequencies a digital filter may be used to increaseaccuracy (e.g. Glegg69). This method has been used by several investigators,e.g. Ozyoruk and Long,70–72 Lyrintzis and Xue,73 and Rahier and Prier,74 Algermissenand Wagner,75 Delriex et al.,76 and Kim et al.77 Finally, a marching-cubes algorithm78

can be used to provide an efficient algorithm that is easy to parallelize for the evaluationof the propagation from an emission surface.

3.4 Rotating or nonrotating control surfaceFor rotor applications both a rotating and a nonrotating formulation can be used.A non-rotating formulation uses a nonrotating control surface that encloses the entire rotor(e.g. Forsyth and Korkan,79 Strawn and Biswas,80 Baeder et al.81). A rotating Kirchhoffformulation allows the control surface to rotate with the blade aligning with the CFD lines,e.g. Xue and Lyrintzis,82 Lyrintzis et al.,64 Polacsec and Prier.65 No transformation of datais needed since the CFD input is also rotating. A comparison of the rotating and thenonrotating Kirchhoff methods showed that both methods are very accurate and efficient(Strawn et al.63). For the porous FWH method there are fewer applications. A rotatingmethod was used in references 45, 47, 76 and 83 and a nonrotating method in reference 52.

It should be noted that the nonrotating formulation requires reliable data out to anonrotating cylinder (i.e. the control surface) surface that is usually farther out than arotating surface. Therefore, more accuracy of the CFD results is needed. Thus thenonrotating method has been used in conjunction with Euler/Navier Stokes codes (e.g.,TURNS code,84,85 OVERFLOW code86) whereas the rotating Kirchhoff method hasbeen used with full potential codes (e.g. FPR code87,88), as well. However, a drawbackof the rotating method is that the rotating speed of the tip of the rotating surface needsto remain subsonic, to use the subsonic formulas shown in section 2, because ofthe singularities appearing at some terms. For high tip speeds (e.g. Mti p = 0.92) thesupersonic formulation of Farassat et al.32,33 can be employed. For forward timealgorithms the singularities disappear and a simple method for supersonic rotationspeeds has been developed by Delrieux et al.76

4 VALIDATION RESULTSBoth Kirchhoff and FWH formulations have been validated using model problems.The first thing to do is, of course, check that the signal becomes zero inside the controlsurface. The number of points per period and the number of points per wave lengthshould also be studied.34,53

A stationary or translating point source have been used by Lyrintzis et al.,34,53

Myers & Hausmann,31 and Lockard54 and a rotating point source by Lyrintzis et al.64

and Berezin et al.89 Exponential source distributions have been used by Pilon andLyrintzis.35,43,44,46 Hu et al.90 used a line monopole source and a Gaussian pressure andvorticity pulse (category 3 benchmark problem91) to verify their two-dimensionalFW-H formulation. Farassat and Farris33 used dipole distributions on a flat surface anda sphere to validate the supersonic formulation (i.e. formulation 4). Singer et al.92 used

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110 Surface integral methods in computational aeroacoustics

a line vortex around an edge. Meadows and Atkins62 used an oscillating sphere andstudied the effects of quadrature (see section 3). Ozyoruk and Long68 have used thescattering problem of sound by a sphere (Figure 2). The spherical sound waves aregenerated by a partially distributed Gaussian mass source. The results from an exactsolution and a direct Euler solver are also shown. Note that near 180° the Kirchhoffresults are better than the direct calculation, because of numerical dissipation as thewaves travel longer distances to arrive at the observer locations.

5 AEROACOUSTIC APPLICATIONSKirchhoff’s formula has been extensively used in light diffraction and otherelectromagnetic problems, aerodynamic problems, i.e. boundary-elements (e.g. Morinoet al.28), as well as in problems of wave propagation in acoustics (e.g. Pierce93).Kirchhoff’s integral formulation has been used extensively for the prediction ofacoustic radiation in terms of quantities on boundary surfaces (the Kirchhoff controlsurface coincides with the body). Kirchhoff’s method has also been used for thecomputation of acoustic scattering from rigid bodies using a boundary elementtechnique with the Galerkin method. The solid surface FW-H equation with its variousforms21 has been used in several problems including propeller and helicopter noise.Here we will concentrate on the use of “Kirchhoff,” and “porous” FW-H equationmethods, i.e. using a nonlinear CFD solver for the evaluation of acoustic sources inthe near-field and a Kirchhoff/porous FW-H formulation for the acoustic propagation.We will review some “real-life” aeroacoustic applications of both methods con-centrating in recent advances.

5.1 Propeller noiseHawkings18 suggested a stationary-surface Kirchhoff formula to predict the noisefrom high-speed propellers and helicopter rotors. Forsyth and Korkan79 calculated

DIRECTIVITY AT 4.5 SPHERE RADII

Exact solutionDirect CFD dataKirchhoff prediction using CFD data

Angle From Negative x-Axis, Degrees0

0.10

Nor

mal

ized

RM

S P

ress

ure

0.20

0.30

0.40

0.50

50 100 150

Figure 2. Sound scattering by a sphere. Comparison with exact solution (fromreference 70).

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high-speed propeller noise using the Kirchhoff formulation of Hawkings.18 Jaeger andKorkan94 used a special case of the Farassat and Myers25 formulation for a uniformlymoving surface to extend the calculation to advancing propellers. In the aboveapplications, the control surface S was chosen to be a cylinder enclosing the rotor.

5.2 Helicopter impulsive noiseKirchhoff’s method has been widely applied in the prediction of helicopter impulsivenoise.95 The Kirchhoff method for a uniformly moving surface was initially used intwo-dimensional transonic Blade-Vortex Interactions (BVI) to extend the numericallycalculated nonlinear aerodynamic BVI results to the linear acoustic far-field.96–98

Actually, the first application of Hawkings18 Kirchhoff Method was given by George andLyrintzis,19 where the terminology “Kirchhoff Method” was introduced. The Kirchhoffmethod was used to test ideas for BVI noise reduction (Xue and Lyrintzis99). The methodwas also extended to study noise due to other unsteady transonic flow phenomena(i.e. oscillating flaps, thickening-thinning airfoil) by Lyrintzis et al.100 Later, the methodwas used for the two-dimensional BVI problem by Lin and co-workers.101,102

Kirchhoff’s method has also been applied to three-dimensional High-SpeedImpulsive (HSI) noise. Baeder et al.81 and Strawn & Biswas80 used a nonrotatingcontrol Kirchhoff surface that encloses the entire rotor. The Transonic Unsteady RotorNavier Stokes (TURNS) code84,85 was used for the near-field CFD calculations.An unstructured grid was used by Strawn et al.103 and an overset grid code (OVERFLOW)86

by Ahmad et al.104 Kirchhoff’s method predicted the HSI hover noise very well using asmall fraction of CPU time of the straight CFD calculation.

Another Kirchhoff method used in helicopter noise is the rotating Kirchhoff method(i.e. the surface rotates with the blade). The method was used for three-dimensionaltransonic BVI’s for a hovering rotor by Xue and Lyrintzis.82 The near-field wascalculated using the Full Potential Rotor (FPR) code.87,88 The rotating Kirchhoffformulation allows the Kirchhoff control surface to rotate with the blade; thus a smallercylinder surface around the blade can be used. No transformation of data is neededbecause the CFD input is also rotating. Since more detailed information is utilized forthe accurate prediction of the far-field noise this method is more efficient. Finally, themethod was extended for an advancing rotor and was applied to HSI noise105 and BVInoise.106,107 Berezin et al.89 showed that sometimes special care is needed for choosingthe CFD grids, because the highly stretched grids used for aerodynamic applicationsmay not provide accurate information on the control Kirchhoff surface.

A comparison63 of the rotating and the nonrotating Kirchhoff methods showed thatboth methods are very accurate and efficient. Figure 3 shows a comparison for anadvancing HSI noise case (1/7 scale AH-1 helicopter, hover tip Mach numberMH = 0.665, advance ratio µ = 0.258, which corresponds to an advancing tip Machnumber of Mat = 0.837). TURNS84,85 is used for the CFD calculations. We see thatboth methods compare very well with the experiments.108 Kirchhoff’s method hasbecome a standard tool for rotorcraft acoustic predictions. The method is currentlyimplemented in the TRAC (TiltRotor Aeroacoustic Codes) system developed by NASALangley (RKIR code, Lyrintzis et al.,64 Berezin et al.89) and is employed at NASAAmes

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112 Surface integral methods in computational aeroacoustics

AFDD (Strawn et al.63). In Europe, additional versions of rotating and nonrotatingKirchhoff codes have also been developed.65,74,75,76,109,110

Kirchhoff’s method results have also been compared with the traditional acousticanalogy (solid surface FW-H equation). A comparison with the acoustic analogy codeWOPWOP15 (WOPWOP uses the solid surface FW-H equation without accounting forquadrupoles) has shown that Kirchhoff method is superior when quadrupole sources arepresent (Lyrintzis et al.111) for advancing HSI cases. Baeder et al.81 also compared theresults with a linear (i.e. monopole plus dipole sources on the rotating blade) solidsurface FW-H equation method for hover HSI. The FW-H results were inaccurate fortip Mach numbers higher than 0.7, because of the omission of quadrupole sources.However, a further comparison of the rotating Kirchhoff method to WOPWOP+16,17

(WOPWOP+ is a solid surface FW-H equation method accounting also for quadrupoleswith a volume integral) has shown that the two methods give about the same results(Brentner et al.112), but Kirchhoff method uses only surface integrals and avoids thequadrupole volume integration. It should be noted that the robustness of the Kirchhoffmethod improves with the use of a less stretched grid (Berezin et al.89) or an Euler code,e.g. TURNS (Baeder et al.81).

Experiment

240�80

Pre

ssur

e (P

a)

�60

�40

�20

0

20

250 260 270 280

260�60

Pre

ssur

e (P

a)

�40

�20

0

20

270Blade azimuthal angle (deg)

280 290 300

(1)

210ω

220Blade azimuthal angle (deg)

230 240 250

(4)

(2)

210 220 230 240 250

(3)

Rotating KirchhoffNonrotating Kirchhoff

M∞

r /R�6.88 4

21 3

30°�30°

r /R�3.44

Figure 3. Comparison of acoustic pressures with experimental data102 at fourdifferent microphone locations for an AH-1 blade with Mat = 0.837. Allmicrophones are in the plane of the rotor (from reference 63).

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Isom et al.,49 and Purcell113,114 used a modified Kirchhoff method which alsoincluded some nonlinear effects for a stationary surface, to calculate hover HSI noise.Results (not shown here) show good agreement with experimental data.

A porous FW-H method based on Kirchhoff subroutines was also developed byBrentner & Farassat47 (FWH/RKIR code), Morgans et al.,83 Strawn et al.,52 and Delriexet al.76 These codes do not include quadrupoles outside the control surface, because itwas found to be of minor importance unless the Mach number is really high.115 Thusthe porous FW-H equation is also based on surface integrals. The porous FW-Hformalism is more robust than the traditional Kirchhoff method with regards to thechoice of the control surface, as shown in Figures 4 and 5 for a hover HSI noise case(1/4 model UH-1H model helicopter, hovering at MH = 0.88, experiments fromPurcell113). FPR87,88 was used for the CFD calculations.

5.3 AirfoilsAtassi and his co-workers37,116–118 have used Kirchhoff’s method for the evaluation ofacoustic radiation from airfoils in nonuniform subsonic flows. They employed rapiddistortion theory to calculate the near-field CFD. A sample comparison for the far-fielddirectivity of the acoustic pressure using the Kirchhoff method and the direct calculationmethod (i.e. rapid distortion theory119–121 is given in Figure 6 (from reference 37) fora 3% thick Joukowski airfoil in a transverse gust at k1 = (ωc/2V∞) = 1 and M = 0.1.The semi-analytical results for a flat plate encountering the same gust are also shownin Figure 6 and are very close to the results from Kirchhoff’s method. The figureindicates that the direct calculation method is not accurate in the far-field, as the direct

aeroacoustics volume 2 · number 2 · 2003 113

0.0�5000

�4000

�3000

�2000

�1000

0

0.5 1.0 1.5 2.0

time, msec

p′, P

a

data

k=21

k=18

k=12

k=7

k=2

Figure 4. Comparison of Kirchhoff acoustic pressures with experimental data108

for an observer in the plane of the rotor at 3, 4R from a UH-1H modelrotor hovering at MH = 0.88 (from reference 47).

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114 Surface integral methods in computational aeroacoustics

simulation results are very different from the semi-analytical and the Kirchhoffresults. This is due to discretization errors. However, this CFD code is accurate in thenear-field and the Kirchhoff method should be used instead in the far-field, as indicatedin Figure 6.

Singer et al.92,61 used a FW-H method for the evaluation of acoustic scattering froma trailing edge and slat trailing edge. The interesting thing about the slat trailing edgeapplication is that part of the control surface is solid and another part is porous.

5.4 Fan noiseKirchhoff’s method can also be applied to ducted fan noise. Very good results wereshown by Ozyoruk and Long70–72 for a control surface in rectilinear motion. A forwardtime parallel algorithm was used. A porous FWH method was used by Zhang122 withvery good results.

5.5 Jet noiseKirchhoff’s method has also been applied in the estimation of jet noise. Soh,123 Mitchelet al.124 Zhao et al.,125 and Billson et al.126 used the stationary Kirchhoff method(equation 7) and Lyrintzis & Mankbadi,34 Chyczewski & Long,127 Morris et al.,128

Gamet and Estivalezes,129 Choi et al.,130 and Kandula and Caimi131 used the uniformlymoving formula. It should be noted that most of the above references use an LES code forthe CFD data. However, a RANS code can also be used, as shown in reference 131, whereOVERFLOW86 was used. Lyrintzis & Mankbadi34 also compared time and frequencydomain formulations. Mankbadi et al.38 applied a modified Green’s function to avoid

0.0�400

�300

�200

�100

0

100

0.5 1.0 1.5 2.0

time, msec

p′, P

a

data

k=21

k=18

k=12

k=7

k=2

Figure 5. Comparison of porous FW-H acoustic pressures with experimentaldata108 for an observer in the plane of the rotor at 3, 4R from a UH-1Hmodel rotor hovering at MH = 0.88 (from reference 47).

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the evaluation of normal derivatives. Balakumar132 and Yen133 used parabolized stabilityequations for the jet simulation and a cylindrical (i.e. two-dimensional) Kirchhoffformulation for the noise evaluation. Shih et al.134 compared several Kirchhoffformulations with the acoustic analogy, extending the LES calculations and using a zonalLES + LEE method. The results showed that the Kirchhoff method is much more accuratethan the acoustic analogy (for the compact source approximation used) and much cheaperthan extending the LES or performing a zonal LES + LEE.

The uses of FW-H method in jet noise have been sparse. Morris et al.135,136 and Uzunet al.137 used the method with good results and Hu et al.,90,138 used a two-dimensionalformulation of the porous FW-H equation to evaluate noise radiation from a planejet. Rahier et al.139 compared the methods for numerical acoustic predictions for hotjets. Both the Kirchhoff method based on pressure disturbance and the FW-H equationgave good results, whereas the Kirchhoff method based on density gave erroneousresults.

Most of the above approaches have used an open control surface (i.e. withoutthe downstream end) in order to avoid placing the surface in a nonlinear region. Freundet al.59 showed a means of correcting the results to account for an open control surface,for cases that the observer is close to the jet axis. Pilon and Lyrintzis43,44,47 developeda method to account for quadrupole sources outside the control surface. Thisapproximation is based on the assumption that all wave modes approximately decayin an exponential fashion. The volume integral is reduced to a surface integral for a far-field low frequency approximation and a Taylor series expansion for axisymmetricjets. However, a simpler method (suggested in reference 53) is to just use an existingempirical code (e.g. MGB140) to evaluate the noise using as inflow the CFD solution onthe right side of the control surface. Thus MGB can provide an estimate of the error ofignoring any sources outside the control surface of the Kirchhoff/porous FW-H method.

aeroacoustics volume 2 · number 2 · 2003 115

�0.15 �0.10 �0.05 00

0.05

0.10

0.15

0.20

0.05 0.10 0.15 0.20

Figure 6. Comparison between far-field directivity of acoustic pressure valuesusing the Kirchhoff method (- -) and the direct calculation method (-•-)for a 3% thick Joukowski airfoil in a transverse gust at k1 = 1.0, M = 0.1.The semi analytical results (–) for a flat plate encountering the same gustare also shown (from reference 37).

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116 Surface integral methods in computational aeroacoustics

An approximate way to account for refraction effects was developed by Lyrintzisand co-workers,53,58 as explained above in section 2.5. A typical result shownhere (Figure 7) shows the effects of refraction corrections for a supersonic Machnumber case (excited, Mach 2.1, unheated (T◦ = 294K ), round jet of Reynolds NumberRe = 70000; the jet exit variables were perturbed at a single axisymmetric modeat a Strouhal number of St = 0.20, the amplitude of the perturbation was 2% of themean). Further development of refraction corrections based on Lilley’s equation,(e.g. reference 141) is possible.

Finally, it should be noted that for some complicated noise problems (as, for example,in jet noise) several computational domains might be needed: a complicated near-field(e.g. using Large Eddy Simulations-LES), a simplified mid-field with some nonlineareffects, and a linear Kirchhoff’s method for the far-field. Kirchhoff’s formulation canbe the simplest region of a general zonal methodology. This idea has been proposed byLyrintzis,19 but it has not yet been implemented.

5.6 Other applicationsOther applications of these surface integral methods have also been attempted. Forexample, in reference 142 a Kirchhoff method was applied to a cavity problem, withapplication to vehicle noise predictions. Also the method is now part of the Fluent codeand has been applied to a wide range of problems77 and is currently implemented143 inthe Star-CD code.

0 10 20 30 40 50 60 70 80 90 100

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0

10

20

30

40

50

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j

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Figure 7. Instantaneous contours of a2◦ρ

′/p◦.R > 0: No refraction corrections.R < 0: Refraction corrections imposed (from reference 58).

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Finally, it should be noted that FW-H can not be used with incompressible CFD data.For low speeds (i.e. incompressible flow) and a stationary, impermeable surface theFW-H equation reduces to Curle’s integral (reference 144). Wang et al.145 used Curle’sintegral for the evaluation of trailing edge noise with an incompressible Navier Stokessolver. (They concluded that the volume integral of the equation, i.e. quadrupolesoutside the airfoil surface, is not important for low speeds.)

6 CONCLUDING REMARKSKirchhoff’s and porous FW-H methods consist of the calculation of the nonlinear near-and mid-field numerically with the far-field solutions found from a Kirchhoff/porousFW-H formulation evaluated on a control surface S surrounding the nonlinear-field.The surface S is assumed to include all the nonlinear flow effects and noise sources. Theseparation of the problem into linear and nonlinear regions allows the use of the mostappropriate numerical methodology for each. The advantage of these methods is thatthe surface integrals and the first derivatives needed can be evaluated more easily thanthe volume integrals and the second derivatives needed for the evaluation of thequadrupole terms when the traditional acoustic analogy is used.

The porous FW-H equation method is a newer idea and there fewer applications inthe literature. The method is equivalent to Kirchhoff’s method and has the sameadvantanges. In comparison, it is more robust with the choice of control surface anddoes not require normal derivatives. Since the method also requires a surface integral,it is very easy to modify existing Kirchhoff/solid surface FW-H codes. The methodrequires larger memory, because more quantities on the control surface are needed.However, we believe that the robustness is more important and thus the porous FW-His the method we recommend.

The use of both methods has increased substantially over the last 10 years, becauseof the development of reliable CFD methods that can be used for the evaluation of thenear-field. The methods can be used to study various acoustic problems, such aspropeller noise, high-speed compressibility noise, blade-vortex interactions, jet noise,ducted fan noise, etc. Some results indicative of the uses of both methods are shownhere, but the reader is referred to the original references for further details. The methodsare becoming more popular and have been coupled with production codes, such asOVERFLOW, Fluent and Star-CD. We believe that, a simple set of portableKirchhoff/FW-H subroutines can be developed to calculate the far-field noise frominputs supplied by any aerodynamic near/mid-field code.

ACKNOWLEDGEMENTSThe author was supported by the Indiana 21st Century Research and Technology Fund,and the Aeroacoustics Research Consortium (AARC), a government and industryconsortium mananged by the Ohio Aerospace Institute (OAI).

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