A numerical method is presented for the prediction of propagation and radiation of aftfan noise. The model is based on the convected Lighthill equation valid for nonuniformirrotational mean flows. Assuming harmonic time dependence, the convected wave equationis solved in the frequency domain on unstructured mesh. For each wave number, anassociated convected Helmholtz problem, with the corresponding inhomogeneous forcingterm, is solved. The numerical algorithm has proved to be efficient also for high Helmholtznumbers. The numerical procedure is able to deal with generally shaped boundaries.Moreover, the frequency-domain discretization approach makes possible to treat each wavenumber separately. This property is particularly useful for design optimization calculations.Numerical tests of propagation of cut-on modes inside annular ducts with variable sectionare presented. SPL directivity patterns of the radiated sound out of the annular pipe arealso described and compared with analytical results.
I. Introduction
The numerical simulation of sound propagation and radiation is becoming an effective tool for noiseprediction. Altough modern aircrafts have considerably reduced their noise level emissions, the demand forfurther noise reduction is demanding a further effort in the development of more efficient numerical methodsfor the solution of the governing equations of aeroacoustics.
While significant progress has been made in the reduction of intake noise radiation, little work has beenconducted on reducing the radiation noise from the aft fan duct and core nozzle of modern high bypassratio turbofan, in which sound propagates through the shear layers of high speed (hot) jets. In approachand cutback conditions, turbomachinery noise radiating from the bypass and core nozzles is becoming thedominant noise source. Accurate predictions of exhaust noise and aft fan noise are required to guide thedevelopment of innovative noise reduction solutions.
This problem represents a challenging task for CAA, due to the presence of the shear layers separatingthe core, bypass and free-stream fields. While the future probable solution of this problem lies in thedevelopment of appropriate numerical models, based on the Linearized Euler Equations (LEE) which candeal accurately not only with the shear layers but also with non-uniform mean flows, simplified models basedon the convected wave equation, such as the one proposed here, are useful both for parametric studies andas benchmark solutions for the next generation of numerical models. Moreover, in view of the difficultiesencountered in the development of stable numerical methods for the solution of the LEE6, the developmentof stable methods based on the convected wave equation is presently justified.
Some attempts in this direction have been recently undertaken. In Ref.[11] and Ref.[12], a hybrid nu-merical procedure is developed for the prediction of propagation and radiation from bypass ducts in thecase of simplified geometries. An in-duct finite element model for the modal propagation within the duct iscoupled to an approximate radiation model in the external domain in the presence of mean flow and a bypassshear layer. In Ref.[5], a finite element radiation model based on the assumption of irrotational acousticperturbation on an irrotational steady mean flow has been proposed for realistic geometries.
The aim of the present work is to extend a finite element numerical technique, developed for the solutionof the convected wave equation1,2,3, to the prediction of the propagation and radiation of aft fan noise andcore noise including the effect of forward flight.
∗Associate Professor, Dipartimento di Ingegneria Aeronautica e Spaziale, Corso Duca degli Abruzzi 24, 10129-Torino, Italy,[email protected], Member AIAA.
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Assuming harmonic time dependence, the convected wave equation is solved in the frequency domainon unstructured mesh. For each wave number, an associated convected Helmholtz problem is solved witha domain decomposition technique. The main advantage is the ability of dealing with generally shapedboundaries. Moreover, the frequency-domain approach makes possible to treat each wave number separately.This property is particularly useful for design optimization calculations of the cuton/cutoff spinning modes.In Section II the governing equations and the numerical model are introduced, and in Sections III and IV, theproblem of the sound propagation inside a rectilinear annular duct and the radiation out of it are investigatedfor different flow conditions.
II. Governing Equations and Numerical Model
The problem is that of propagation of sound waves, in the form of spinning modes, inside an annular ductwith hard walls. Depending upon the mode time frequency, the transmission varies in one of two distinctlydifferent ways. The pressure fluctuations are subjected to an exponential decay if the frequency is lower thana cut-off frequency. Usually the rate of decay is large enough to reduce the pressure intensity to a negligiblevalue in a short distance, compared to the duct radius. When the spinning mode frequency is higher enough,the pressure fluctuations propagate inside the duct without attenuation. In this last case, the pressure modewill propagate till the exit of the duct, and will radiate outside in the free field.
The study of sound propagation in ducts, as well as the related problem of reflection and radiation froman open end are classical problems of acoustics.8,7,10However only idealized cases can be solved.
In the present case the study of the in-duct propagation and the angular directivity pattern due toradiation, as a function of geometrical and operating parameters, is extended to the case of a annular duct,with generally varying cross section , in the presence of potential mean flows in the outside free stream andinside the duct.
A. Convected Lighthill Equation
The propagation and radiation in a medium with mean irrotational velocity Ui(xj , t) is governed by the con-vected wave equation9, which can be recast in the following non dimensional form3, in Cartesian coordinates,
∂2p
∂t2+ 2Mi
∂2p
∂t∂xi− (δij −MiMj)
∂2p
∂xi∂xj− p
(∂Mj
∂xi
∂Mi
∂xj+∂Mi
∂xi
∂Mj
∂xj
)= Qac , (1)
with Mi = Ui/c0, and the variables are made non dimensional with respect to a reference speed of sound c0and a reference length L. The term Qac contains the source terms.
In the case of a parallel mean shear flow (U1 = U(x2), 0, 0), the last term of the left-hand side vanishes.In the case of axisymmetric geometries, and assuming that the mean flow is aligned with the axis coor-
dinate x, for a cylindrical reference system x, r, θ, Eq. (1) reads
(∂
∂t+M
∂
∂x
)2
p−[∂2
∂x2+
1r
∂
∂r
(r∂
∂r
)+
1r2
∂
∂θ
]p = Qac . (2)
Assuming that the pressure fluctuations have a harmonic time dependence and a m-lobe patter along thecircumferential direction,
p = p(x, r) eI(Kt+mθ) ,
with I =√−1, K = ωL/c0 is the non dimensional wave number or Helmholtz number, and ω the angular
frequency, Eq. (2) transforms in
−[(1−M)
∂2
∂x2+
1r
∂
∂r
(r∂
∂r
)]p+ 2IKM
∂p
∂x−
(K2 − m2
r2
)p = Qac . (3)
The acoustic pressure can be computed after the mean flow field Mach number M is specified. In thisway the range of the wave number K ∈ [0,Kmax] is illustrated. For each value of K we must solve theassociated convected Helmholtz problem (Eq. (3)). The acoustic pressure field is recovered performing aninverse DFT.
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Along artificial boundaries, to avoid incoming spurious reflections, appropriate non-reflecting boundaryconditions must be imposed. In our case, considering radial propagation along the direction η from thesource location to the boundary point, the transformed wave operator for p can be factorized with the terms
∂p
∂η±
(k − m
r
)p = 0 .
for the incoming (p−) and outgoing (p+) waves. Projecting the above condition along the normal to theboundary, positive when pointing inside the domain, denoting with α the angle formed between the twodirections, the non reflecting boundary condition can be written as
∂p
∂n= I
(k − m
r
)cosβ p . (4)
At the duct inlet an incoming wave must be specified, while the duct walls are considered as rigid andperfectly reflecting.
B. Numerical Method
Equation(3) is approximated by a Finite Element method. The Fourier coefficient p of the acoustic pressureis interpolated by the nodal values so that
p(x) =N∑
l
ϕl(x)dl ,
where ϕl(x) are linear independent basis (shape) functions with N unknown complex coefficients dl. Definingthe test functions ψ as the linear span of basis functions ϕl(x) (l = 1, N), the Galerkin variational formulationof Eq. (3) results in a complex matrix equation
(Kij +M2Dij
)pj + 2IKMEij = Fi +Qi , (5)
with
Kij =∫
Ωe
[(1−M2
) (∂ψi
∂x
∂ψj
∂x+∂ψi
∂r
∂ψj
∂r
)−
(K2 − m2
r2
)ψiψj
]rdΩe ,
Dij =∫
Ωe
∂
∂x
(ψi∂ψj
∂x
)rdΩe ,
Eij =∫
Ωe
ψi∂ψj
∂xrdΩe ,
Fi =[∫
Ωe
ψi∂ψj
∂xrdΩe
](Qac)j ,
Qi =∮
Γe
ψi∂p
∂nrdΩe ,
with sum over repeated indices. Ωe is the element surface and Γe its boundary where Neumann boundaryconditions are imposed.
Two-dimensional geometrical domains are subdivided into non overlapping triangular and quadrilateralelements. The basis functions ϕl(x) associated with element nodes are C0 continuous interpolation func-tions with compact support such that the unknown amplitudes take on the point values dl = p(xl). Theinterpolation functions ϕl(x) are assumed to be linear in the case of triangular elements and bilinear forquadrilateral elements.
The resulting sparse linear system, for the real and imaginary parts of p, are solved by an iterative method.We have applied the GMRES method with ILUT preconditioning, consisting of an ILU decomposition withthreshold and diagonal compensation. In previous work1,2 it has been shown that in the case of the Helmholtzproblem, the performances of this iterative method remain the same also for very high wave numbers Kprovided the grid is sufficiently refined to represent the solution oscillations.
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H
R
M
M0
1
Figure 1. Annular duct geometry
III. Propagation
The effectiveness of the numerical method in reproducing the propagation of the pressure fluctuationsinside the duct has been tested for the simple geometry shown in figure 1. It consists of a semi-infinite pipe,of radius R inside of which is a double infinite hub of radius H. It forms a simple model for the propagationand radiation of fan noise from the bypass duct of a turbofan engine. It is an idealized test case, but it offersthe possibility to compare the numerical results with analitycal ones.
The analitycal theory of sound propagation in infinite annular ducts can be built up by a Fourier-Besselsuperposition of (m,µ)-modes. For each single mode it is possible to evaluate the corresponding cut-offfrequency.10
Several test cases have been performed in absence of mean flow (M1 = 0) for various (m,µ)-modes. Inall cases no propagation occurred when the incoming wave frequency was lower than the predicted cut-offfrequency, and undamped propagation occurred for f > fcut−off . In figure 2 (upper), we report the caseof the (1, 3)-mode with a frequency f = 1100 Hz, corresponding to a Helmholtz number K = 20, havingassumed as reference length the pipe radius R, and co = 340 m/s. The hub diameter is H/R = 0.5 and thepredicted cut-off frequency is fcut−off = 1026.0 Hz. The propagation occurs without damping as expected.
In figure 2 (lower), the same cut-on (m,µ)-mode propagating in the annular part with H/R = 0.5, turnscut-off when the cross section is reduced by an enlargement of the hub diameter (H/R = 0.75), Moreoverthe mode is reflected back, forming a standing-wave pattern.4
In figure 3, the instantaneous pressure fluctuations are reported for the same conditions as in the previouscase, but now with a mean-flow inside the duct, M1 = 0.25. The frequency is still higher than the cut-offfrequency, which in this case is slightly lower with respect to the no mean-flow case, fcut−off = 993.5 Hz,
in the part with H/R = 0.5. However now the propagation is influenced by the convective influence of themain stream.
Again the mode turn to cut-off in the narrowed part, and a standing wave pattern appears.
IV. Radiation
The next numerical tests deal with the radiated sound out of the annular pipe. The computationaldomain is now composed by the duct geometry and a part of the free field, with non-reflecting boundaryconditions in the far field.
The directivity pattern far away from the duct termination is an appropriate way to describe the pressurefield radiated out of the pipe and propagating in the free space.
As first example, the propagation in absence of mean flow has been computed. In figure 4 the SoundPressure Level directivity is shown in the case of a (4, 0)-mode with K = 30. The directivity pattern displaysthe typical lobe pattern. In the calculation, no cone of silence is present, probably because of the presenceof the infinite hub. The SPL levels are compared with the analytical results,10 showing a good agreement,except for small angles, near the axis, where the theoretical model predicts a small silent region. In the partfar from the axis (higher angles), a wavy behavior occurs in the numerical solution, not predicted by thetheory. A similar behavior has been remarked in the same kind of comparisons.7
In figure 5 the effect of the wave number is investigated. The previous radiated field is compared withthe the sound field generated by the same (4-0)-mode, but at a lower frequency (K = 15). As expected,the pattern is smoother. It is possible to remark that a quasi-silent region near the axis is now present. Apossible explanation of the absence of the theoretical cone of silence, could be the imposition of perfectlyreflecting hub wall conditions. This condition is not taken into account in the theoretical far-field model.Again an oscillatory behavior is present for higher angles.
Finally, figure 6 reports the effect of the duct mean flow on the SPL directivity pattern. The (4−0)-mode,with K = 15, is radiated under different values of M1, in a outer field at rest (M0 = 0.). For comparisonthe solution with M1 = 0. is also plotted. As expected the directivity patterns display an higher number oflobes. This behavior is consistent with the results reported in Ref.[7].
V. Conclusions
A numerical technique for the solution of the convected wave equation, valid for nonuniform mean flows,has been described.
Assuming harmonic time dependence, the convected wave equation is solved in the frequency domain
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on unstructured mesh. For each wave number, an associated convected Helmholtz problem, with the corre-sponding inhomogeneous forcing term, is solved. In order to deal with complex geometries a Finite Elementdiscretization has been adopted for solving the convected Helmholtz problem. The numerical algorithm hasproved to be efficient also for high Helmholtz numbers. The method can deal with generally shaped bound-aries. Moreover, the frequency-domain discretization approach makes possible to treat each wave numberseparately. This property is particularly useful for design optimization calculations.
Several test cases of wave propagation inside annular ducts have been performed for various (m,µ)-modes.In all cases no propagation occurred when the incoming wave frequency was lower than the predicted cut-offfrequency. Numerical tests of propagation of cut-on modes inside annular ducts with variable section havebeen presented, showing that the effects of section changes are correctly reproduced.
SPL directivity patterns of the radiated sound out of the annular pipe were also described and comparedwith analytical results. The effect of the pipe mean flow were analyzed.
Acknowledgments
The author gratefully acknowledge E. De Jaeghere, A. Iob and C. Schipani for the helpfull discussions.The development of the work has been founded by AVIOGroup, Torino, Italy.
References
1Arina R. and Ribaldone E., Aeroacoustic Modelling of Complex Flow Problems I - Domain Decomposition Method for theReduced Wave Equation, Comput. Visual. Sci., 4, pp.139-146, January 2002.
2Arina R. and Falossi M., Domain decomposition technique for aeroacoustic simulations, App. Num. Math., 49(3-4), pp.263-275, 2004.
3Arina R., Numerical Method for the Convected Lighthill’s Equation, AIAA paper 2005-2928, 11th AIAA/CEAS Aeroa-
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American Institute of Aeronautics and Astronautics
coustics Conference, May 2005, Monterey, CA, USA, 2005.4Casalino D., Roger M. and Jacob M., Prediction of Sound Propagation in Ducted Potential Flows Using Green’s Function
Discretization, AIAA J., 42-4, pp. 736-744, 2004.5Eversman W. and Okunbor D., Aft Fan Duct Acoustic Radiation, J. Sound Vibr., 213-2, pp. 235-257, 1998.6Ewert R. and Schroder W., Acoustic Perturbation Equations Based on Flow Decomposition via Source Filtering, J.
Comput. Phy., 188, pp. 365-398, 2003.7Gabard G., Astley R.J. , Theoretical Model for Sound Radiation from Annular Jet Pipes: Far- and Near-field Solutions,
J. Fluid Mech., 549, pp. 315-341, 2006.8Munt R.M., The interaction of sound with a subsonic jet issuing from a semi-infinite cylindrical pipe, J. Fluid Mech.,
83-4, pp. 609-640, 1977.9Ribner H.B., Effects of Jet Flow on Jet Noise via an Extension to the Lighthill model, J. Fluid Mech., 321, pp. 1-24,
1996.10Rienstra S.W., Acoustic Radiation from a Semi-infinite Annular Duct in a Uniform Subsonic Mean Flow, J. Sound
Vibr., 94-2, pp. 267-288, 1984.11Sugimoto R. and Astley R.J., Modelling of Flow effects on Propagation and Radiation from Bypass Ducts, AIAA paper
2005-3011, 11th AIAA/CEAS Aeroacoustics Conference, May 2005, Monterey, CA, USA, 2005.12R. Sugimoto et al., Prediction Methods for Propagation in Bypass Ducts and Comparison with Measured Data, AIAA
paper 2005-3059, 11th AIAA/CEAS Aeroacoustics Conference, May 2005, Monterey, CA, USA, 2005.
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