AIAA-2002-2586

LARGE SCALE FREQUENCY DOMAIN NUMERICAL

SIMULATION OF AIRCRAFT ENGINE TONE NOISE

RADIATION AND SCATTERING

D. Stanescu,† M.Y. Hussaini† and F. Farassat‡

† CSIT, Florida State University, Tallahassee, Florida, 32306.‡ NASA Langley Research Center, Hampton, Virginia 23681.

INTRODUCTION

The engines represent a major source of aircraftnoise both at take off and landing. For the largeturbofan engines used nowadays on commercialaircraft, fan noise dominates in particular dur-ing approach flight and landing, when the thrustforce necessary from the jet is relatively small.The tonal noise generated by fan rotor-stator in-teraction as cylindrical duct modes is radiatedfrom the fan inlet and exhaust and is scatteredby the various surfaces surrounding the nacelle,such as fuselage and wing. Eventually both thedirect and the scattered field propagate to thefar field through the complex mean flow aroundthe aircraft. Earlier studies of fan inlet noisepropagation either incorporate only the effectsof the nacelle boundary layer [4] or consider theflow around the nacelle axisymmetric and in-viscid [10, 8]. However, the non-uniform flowaround the nacelle and the relative positions ofthe nacelle, fuselage and wing offer the designera range of options for reducing the noise foot-print of an aircraft. A numerical simulation ofthe phenomenon, which is a very convenient al-ternative to experiments at least in the initialdesign stage, represents a large-scale computa-tional problem which becomes feasible using therecent developments in computational methodsfor advanced architecture computers.

Time-domain methods have been proposedby various authors, including the presentones [9, 12, 14], to address this problem. Un-der that approach, it is straightforward to havea full nonlinear model for the sound propaga-tion, based on the inviscid flow (Euler) equa-tions, that applies both for take-off and landing.The use of an explicit low-storage time integra-tion method avoids the need to store and solve amatrix at each time step, thus keeping the com-puter memory requirements relatively low. Inthis paper we present a different and comple-mentary methodology based on the frequencydomain solution of the linearized full potentialequation. The possibility to solve for the acous-tic field in one step as a boundary value problemmust in this case be weighed against the diffi-culty of finding the solution of a system of equa-tions with complex coefficients. As the size ofthis system increases rapidly with the frequency,a direct solution even using sparse solvers is pre-cluded by the large memory requirements. Anefficient method for its iterative solution mustbe devised.

The paper first introduces the governingequations for the sound propagation problem.A new treatment of the radiation boundary con-ditions in the form of an augmented, perfectlymatched layer (PML) equation, is then pre-sented. This new equation allows us to have

Copyright c©2002 by D. Stanescu. Published by the

American Institute of Aeronautics and Astronautics, Inc.

with permission.

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8th AIAA/CEAS Aeroacoustics Conference & Exhibit<br> <font color="green">Fire17-19 June 2002, Breckenridge, Colorado

AIAA 2002-2586

Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

the radiation boundaries relatively close to suchscattering surfaces as the tip of the wing, thusmaking the study of actual three-dimensionalconfigurations feasible. After a brief descriptionof the spectral element discretization that yieldsthe linear system of equations for the degreesof freedom of the problem, the paper discussesthe methodology used to solve this system iter-atively on distributed-memory computers. Nu-merical results obtained for two generic config-urations are then presented, and some conclu-sions are provided at the end of the paper.

GOVERNING EQUATIONS

For the short times of propagation that we areinterested in, we assume that there is no dis-sipation of acoustic energy, so that the inviscidflow equations can be used to model the process.For an irrotational flow the continuity equationcan be written as

∂ρ

∂t+ ∇ · (ρ∇Φ) = 0 (1)

where ρ is the fluid density, and the poten-tial Φ is related to the velocity by V = ∇Φ.For convenience, the flow variables are non-dimensionalized with respect to ρ∞ and c∞,the far field values of the density and speedof sound, respectively, and dimensions with re-spect to R, the fan blade tip radius. Using theisentropic assumption, the momentum equationcan be reduced to an algebraic equation relatingthe density and the velocity potential,

ρ =

[

1 − (γ − 1)

(

∂Φ

∂t+

(∇Φ)2 −M2

2

)]

1

γ − 1

(2)where M is the far field Mach number andγ the specific heats ratio. Consider the un-steady flow field resulting from the superposi-tion of small acoustic perturbations, denoted bya prime, on a steady mean flow denoted by anoverbar: ρ = ρ + ρ′ and Φ = Φ + Φ′. The par-tial differential equation governing the acoustic

perturbations is

∂ρ′

∂t+ ∇ · (ρ∇Φ′ + ρ′∇Φ) = 0 (3)

where

ρ′ = −ρ

c2

[

∂Φ′

∂t+ ∇Φ · ∇Φ′

]

. (4)

For a frequency domain approach, the acous-tic potential is considered to be of the formΦ′ = φ(x, y, z) exp(iωt). Denoting by f = ρ/c,it can be found that the equation governing φ,written in a form that will be useful for thederivation of the perfectly matched layer equa-tion later on, is

φ+∇Φ · ∇φ

iω−

∇ · (ρ∇φ)

f(iω)2+

∇ · (fφ∇Φ)

f+

∇ · (f∇Φ · ∇φ∇Φ)

f(iω)2= 0

(5)

RADIATION BOUNDARY

CONDITIONS

To obtain solutions in a relatively small compu-tational domain, when sources may be placedrelatively close to the domain boundaries, weconstruct a PML for the equation governingthe Fourier transform Φ of the time dependentacoustic potential. Since its introduction byBerenger[2], the PML method has been increas-ingly used by researchers (e.g. Hayder, Hu andHussaini [6]), as it allows reflections from the ar-tificial boundaries to be decreased by orders ofmagnitude compared to other techniques. Forprevious work on the Euler equations in thetime domain, on which the subsequent devel-opment is based, we refer to Hu [7]. To avoida lengthy presentation, the construction of thePML method for the wave equation and its re-duced form, the Helmholtz equation, are pre-sented in the one-dimensional setting. Then weaddress the three-dimensional case for both theadvective and non-advective acoustics case.

Let us consider the one-dimensional wave

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equation in the form,

∂2Φ′

∂t2−∂2Φ′

∂x2 = 0, (6)

which can also be written as a system of first-order partial differential equations,

∂q

∂t−∂u

∂x= 0,

∂u

∂t−∂q

∂x= 0,

(7)

where q = ∂Φ′/∂t and u = ∂Φ′/∂x. Non-trivial wave solutions of the form Q = [q, u]T =Q0e

i(ωt−kx) exist only for ω2 − k2 = 0, which isthe dispersion relation for this system. In thePML method, the system is replaced with thefollowing damped system

∂q

∂t−∂u

∂x+ σxq = 0,

∂u

∂t−∂q

∂x+ σxu = 0,

(8)

which has the dispersion relation (ω − iσx)2 −k2 = 0. The wave number k will now have animaginary part for any driving frequency ω, sothat the waves will become evanescent. We notehowever that, in order to obtain the dispersionrelation for the new system, σx was supposedto be constant, a condition never satisfied inactual applications of the PML technique (it isat most piecewise constant). Taking the Fouriertransform of the damped system, one obtains

iωq −∂u

∂x+ σxq = 0,

iωu−∂q

∂x+ σxu = 0.

(9)

One can now obtain u from the second equationand substitute it in the first, to get

q −1

G2x

∂2q

∂x2 = 0 (10)

where Gx = iω+σx is a constant. It can be eas-ily seen that Φ′ satisfies a similar equation, andthe dispersion relation for this scalar equationis the same as that of the damped system.

Let us now turn to the three-dimensional

case and suppose that the PML can be situ-ated in a region where the mean flow is uni-form and aligned for convenience with the x-axis. Since f = 1 and ∇Φ = Mx (due to non-dimensionalization) in this region, the time de-pendent acoustic potential is governed by theconvected wave equation, obtained by usingEq. (4) in Eq. (3),

∂2Φ′

∂t2+ 2M

∂2Φ′

∂x∂t−

(1 −M2)∂2Φ′

∂x2 −∂2Φ′

∂y2 −∂2Φ′

∂z2 = 0.(11)

Using a Prandtl-Glauert transformation of theform

τ = t+Mβ2x; ξ = β2x;η = βy; ζ = βz,

(12)

where β =√

1/(1 −M 2), the equation is fur-ther reduced to the three dimensional waveequation,

∂2Φ′

∂t2−∂2Φ′

∂ξ2−∂2Φ′

∂η2 −∂2Φ′

∂ζ2 = 0. (13)

Since ∂/∂t = ∂/∂τ , we can conveniently setΦ′(τ, ξ, η, ζ) = φ(ξ, η, ζ) exp(iωτ) and get theHelmholtz equation for φ,

φ−1

(iω)2∂2φ

∂ξ2−

1

(iω)2∂2φ

∂η2 −1

(iω)2∂2φ

∂ζ2 = 0. (14)

The PML equation in the new coordinatesystem can now be developed using the sametechnique as in the one-dimensional case, and isfound to be

φ−1

G2ξ

∂2φ

∂ξ2−

1

G2η

∂2φ

∂η2 −1

G2ζ

∂2φ

∂ζ2 = 0, (15)

with Gξ = iω+σx, etc. Considering plane wavesolutions of the form φ = φ0e

−i(kξξ−kηη−kζζ), itcan be verified that the solution to the disper-sion relation now reads

kξ = (ω − iσξ) cos(θξ),kη = (ω − iση) cos(θη),kζ = (ω − iσζ) cos(θζ),cos2(θξ) + cos2(θη) + cos2(θζ) = 1.

(16)

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The PML boundary condition for advectiveacoustics is now simply obtained by transform-ing this equation back to (x, y, z) coordinates.

SPECTRAL ELEMENT

DISCRETIZATION

The spectral element method is used to dis-cretize equation (5). While it contains asparticular cases both the linear and the non-serendipity quadratic finite element methods, itoffers an advantage related to grid generation.Indeed, in our approach, once a linear finite el-ement grid is constructed around a configura-tion of interest, we can change the driving fre-quency and consequently change the polynomialdegree used in the approximation (in order tomeet the requirements in the minimum num-ber of points per wavelength) by simply spec-ifying it in the input file, without needing togenerate a different grid. The application ofthe spectral element method to the governingequation leads to a discrete system of the formA φ = b, where φ is the vector of pointvalues of the complex-valued acoustic potentialφ. Briefly stated, the computational domain Ωis partitioned into a set of non-overlapping hex-ahedra Ωe, e = 1, . . . , E. Within each element,a polynomial of degree N approximates the so-lution. The system is obtained from the varia-tional statement of equation (5) which can beexpressed as follows: find φ ∈ V such that

∫

ΩF (φ, φx, φy, phiz ;ψ,ψx, ψy, ψz) dΩ = 0 (17)

holds for each ψ ∈ V. Here V is the complexvector space of functions that are continuouson the closure of Ω and whose restriction toan element is a complex polynomial of degreeat most N in the three independent variables.The integral can be evaluated by summing upthe individual contributions of all the elements.For each element, the respective contributionis computed by mapping the element onto themaster element ΩM = [−1, 1]3 and expressingthe integrands in terms of Chebyshev polynomi-als [13] through the Lobatto quadrature points.

LINEAR SOLVER

The resulting algebraic system is of apprecia-ble size even for low values of the driving fre-quency ω. Since the memory usually availableeven on the most performant CPUs is far fromsufficient for storing the matrix in sparse ma-trix format, the computations have to use ei-ther a shared-memory or a distributed-memorymachine for storage and solution. We choseto use a distributed-memory model, as mostshared-memory machines can also be used un-der this model, with the code using the MessagePassing Interface (MPI) standard. Each pro-cessor stores in this case a number of lines inthe matrix. However, distribution of the matrixover several processors precludes constructionof powerful ILU-type preconditioners, as theyrequire massive communication among proces-sors. Without a powerful preconditioner, iter-ative methods for the solution of the systemare bound to fail, as the matrix usually has avery large condition number and is indefinite.To address this issue, we used a parallel Schur-complement [11] approach which is briefly de-scribed below. Let us consider the computa-tional domain Ω divided in a number P of sub-partitions, each of which is assigned to a proces-sor, and denote by B the union of all the sur-faces that have two neighboring subpartitions.In the most general case, for each p there willbe a number of unknown φ values located on B.The vector of unknowns is partitioned as

φ =

φ1I . . . φ

PI φB

(18)

where φpI denotes all the unknowns in subpar-

tition p not located on B. The right-hand sidevector b is partitioned accordingly. The ma-trix A can then be written in the form

A =

A1II 0 . . . A1

IB

0 A2II . . . A2

IB

. . . .A1

BI A2BI . . . ABB

(19)

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and straighforward elimination of the terms be-low the main diagonal leads to

A1II 0 . . . A1

IB

0 A2II . . . A2

IB

. . . .0 0 . . . S

φ1I

φ2I

.φB

=

b1Ib2I.bS

(20)

where bS = bB−∑

p

ApBI(A

pII)

−1bpI . The problem

has thus been reduced to solving a reduced sys-

tem with matrix S = ABB−∑

p

ApBI(A

pII)

−1ApIB

for the points on B only, followed by a solutionon each domain of the interior problem. Thematrix S is much denser than the original ma-trix A and its direct computation and storage isnot efficient or even possible. However, for aniterative method, only the action of S on a vec-tor is needed, and once the sparse, distributed,matrix ABB is formed, this action can be com-puted by matrix-vector multiplications and so-lutions with Ap

II which are local operations onprocessor p and do not require communications,followed by accumulation in the global vectorφB . All computations can be conveniently im-plemented by use of the high level primitives inthe PETSc [1] package for efficient solution ofpartial differential equations.

Finding a preconditioner for S is a difficulttask, as the entries of this matrix are never ex-plicitly computed, however we found that evena simple preconditioner based on the diagonalpart of ABB leads to much faster convergencethan iterating on the original matrix A with adiagonal preconditioner, as will be shown in theresults section. All cases presented herein usethis preconditioner.

NUMERICAL EXAMPLES

The computations presented here have beenperformed in order to study the parallel perfor-mance of the method as well as to gain phys-ical insight into the influence of the fuselageand/or other surfaces on the sound field of anacelle. We consider two cases of increasing ge-ometrical complexity. Lengths have been non-

dimensionalized in all cases using the radius ofthe inlet duct, R, as reference value. The sameamplitude of the incoming acoustic mode wasused for all computations of the propagation ofthat mode in different configurations. For allcases the flight Mach number M = 0.2 and thefan face Mach number is 0.35. The mean flowhas been computed using the same spectral el-ement method to solve the potential equation(incompressible flow model). The reduced fre-quency ωr = ωR/c∞ = 4, and we only showhere results for plane wave and mode (1,0) prop-agation.

Nacelle alone

The first computation was performed for thesound field radiated from an axisymmetric bell-mouth nacelle with an incoming plane wavespecified on the source plane. The radius ofthe leading edge of the bellmouth has been cho-sen to be 1/4 of the radius of the inlet duct,and no centerbody (hub) is present in the na-celle. The grid has E = 9068 elements for thiscase, and we used quadratic elements, for whichthe total number of points in the discretizationraises to M = 78130. Figure 1 presents theSPL contours computed in this case. As canbe noticed, the SPL contours do not have ax-ial symmetry. This is due to the fact that thecomputation was slightly underresolved (aboutfour points per wavelength) and performed ona non-axisymmetric grid (the computational do-main is a box in 3D space). SPL contours forthe mode (1,0) case are plotted in figure 2. Thiscase was run on the same grid but using quarticelements.

Fuselage-wing-nacelle configuration

The geometry used for the previous case wasaugmented with a wing of elliptic cross-section,slightly swept and mounted below the nacelle.The center of the ellipse defining the cross-section of the wing is situated in the planez = −3. A grid of E = 22843 elementswas generated in this case We present prelim-inary results for radiation of a plane wave atreduced frequency ωr = 4. Quartic elements

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have been used in this case, with the total num-ber of discretization points in the domain beingM = 1, 512, 216. The number of nonzero en-tries in the matrix is in this case 318, 540, 948,so that the storage required to store only thenonzero structure with double precision for thecomplex numbers is 6 Gigabytes. Fig. 3 showsthe Mach number distribution on the solid sur-faces of the configuration. Note that the sym-metry plane of the fuselage is fully equivalentto a solid surface both for the potential flowand the acoustic models. Fig. 4 shows the realacoustic potential contours on the surface. Theregion inside the PML is also included in thisfigure, thus damping of the waves can be no-ticed towards the limits of the computationaldomain (in that region the solution does notmake physical sense). The convergence historyof the linear solver using the Schur complementapproach is presented in figure 5. We used theTFQMR [5] iterative method in both cases, withthe diagonal part of A as preconditioner for thefull matrix and the diagonal of ABB as precon-ditioner for the Schur complement computation.As can be noticed, the use of the Schur comple-ment approach drastically reduces the numberof iterations, and consequently the computingtime, for this relatively large test case. Figure 6shows the SPL contours for the plane wave case,and figure 7 for the mode (1,0) case.

An useful way to visualize the effects of thefuselage and the wing on the sound field ra-diated by the nacelle alone is to extract theSPL data in specific regions of the computa-tional domain, not only on the solid surfaces.Such data is presented in figure 8, extractedalong the line y = 0, z = 0 in front of thenacelle. In the abcissa the non-dimensional dis-tance from the source plane (situated at x = 0)is used. While the SPL due to the nacelle alonedecreases monotonically along this line, the re-flection of the main lobe on the surface of thefuselage produces a significant increase in SPLstarting from d = |x| = 3. This increase is largerthan 10dB, and is rather impossible to computecorrectly without considering the exact geomet-ric configuration of the airplane.

Conclusions

A method for computing sound radiation fromturbofan engines which accounts for mountingeffects has been developed. The method, basedon the linearized full-potential equation, can beused to obtain the sound field of this type of en-gines when in the proximity of the fuselage andwing, and thus study the effect that a changein the configuration has on the noise footprintof the aircraft. Further work will be devoted toimproving the efficiency of the complex Schur-complement matrix solution, as well as address-ing more complicated configurations and meanflow fields.

Acknowledgements

The support of the first author by NASA grantNAG-1-01031 is gratefully acknowledged.

References

[1] Balay, S., Gropp, W.D., McInnes, L.C.and Smith, B.F., “PETSc Users Manual”,ANL-95/11 Revision 2.1.1, Argonne Na-tional Laboratory, 2001.

[2] Berenger, J.-P., “A Perfectly MatchedLayer for the Absorbtion of Electromag-netic Waves”, J. Comp. Physics, Vol. 114,185-200 (1994).

[3] Canuto, C., Hussaini, M.Y., Quarteroni,A., and Zang, T.A., Spectral Methods in

Fluid Dynamics, Springer-Verlag (1988).

[4] Dougherty, R.P., “Nacelle Acoustic De-sign by Ray Tracing in Three Dimensions”,AIAA Paper 96-1773, State College, PA.

[5] Freund, R.W., “A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems”, SIAM J. Sci.

Comput., Vol. 14, 470-482 (1993).

[6] Hayder, H.E, Hu, F.Q., and Hus-saini, M.Y., “Towards perfectly absorbing

6American Institute of Aeronautics and Astronautics

boundary conditions for the Euler equa-tions”, AIAA Journal, Vol. 37, 912-918(1999).

[7] Hu, F.Q., “On Absorbing Boundary Con-ditions for Linearized Euler Equations bya Perfectly Matched Layer”, J. Comp.

Physics, Vol. 129, 201-219 (1996).

[8] Nallasamy, M., “Computation of Noise Ra-diation from Fan Inlet and Aft Ducts”, J.

of Aircraft, Vol. 34, No. 3, 387-393 (1997).

[9] Ozyoruk, Y. and Long, L.N., “Computa-tion of Sound Radiating from Engine In-lets”, AIAA Journal, Vol. 34, No. 5, 894-901 (1996).

[10] Parrett, A.V., and Eversman, W., “WaveEnvelope and Finite Element Approxi-mations for Turbofan Noise Radiation inFlight”, AIAA Journal, Vol. 24, No. 5, 753-760 (1986).

[11] Smith, B., Bjorstad, P., Gropp, W., “Do-main Decomposition: Parallel MultilevelMethods for Elliptic Partial DifferentialEquations”, Cambridge Univ. Press, 1996.

[12] Stanescu, D., Ait-Ali-Yahia, D., Habashi,W.G. and Robichaud, M., “MultidomainSpectral Computations of Sound Radiationfrom Ducted Fans”, AIAA Journal, Vol.37, 296-302 (1999).

[13] Stanescu, D., Ait-Ali-Yahia, D., Habashi,W.G. and Robichaud, M., “Galerkin Spec-tral Element Method for Fan Tone Radi-ation Computations”, AIAA Paper 2000-1912, Lahaina, HI.

[14] Stanescu, D., Hussaini, M.Y. and Faras-sat, F. “Aircraft Engine Noise Scattering- A Discontinuous Spectral Element Ap-proach”, AIAA Paper 2002-0800, Reno,NV.

7American Institute of Aeronautics and Astronautics

Fig. 1: SPL contours for the nacelle, plane wave radiated at ω = 4.0.

Fig. 2: SPL contours for the nacelle, mode (1,0) radiated at ω = 4.0.

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Fig. 3: Mach number contours for the fuselage-wing-nacelle case. 15 contours from 0.2 to 0.4 shown.

Fig. 4: Real acoustic potential contours for the fuselage-wing-nacelle case, plane wave radiated at ω = 4.0.

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1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

0 500 1000 1500 2000 2500 3000 3500 4000

||b

-Ax||

iteration

SA

Fig. 5: Convergence history for the fuselage-wing-nacelle case, Schur complement (S) versus iterations onthe full matrix (A).

X Y

Z

SPL212211911611311010710410198959289868380

Fig. 6: SPL contours for the fuselage-wing-nacelle case, plane wave radiated at ω = 4.0.

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Fig. 7: SPL contours for the fuselage-wing-nacelle case, mode (1,0) radiated at ω = 4.0.

70

75

80

85

90

95

100

105

2 2.5 3 3.5 4 4.5 5

SP

L

d=|x|

Nacelle aloneFull configuration

Fig. 8: SPL variation along the nacelle axis for mode (1,0) radiated at ω = 4.0. Nacelle alone versus fullconfiguration case.

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