Numerical Method for the

Convected Lighthills Equation

Renzo Arina

Politecnico di Torino, 10129 Torino, Italy

The aim of the present work is to extend a numerical technique, developed for thesolution of the original Lighthills wave equation, to the convected wave equation validfor nonuniform mean flows. Assuming harmonic time dependence, the convected waveequation is solved in the frequency domain on unstructured mesh. For each wave number,an associated convected Helmholtz problem, with the corresponding inhomogeneous forcingterm, is solved. The main advantages with respect to existing techniques (such as GFD) isthe ability of dealing with generally shaped boundaries. Moreover, the frequency-domaindiscretization approach makes possible to treat each wave number separately. This propertyis particularly useful for design optimization calculation. Preliminary results for a testmodel and a turbofan engine configuration are presented.

I. Introduction

Numerical simulation of sound generation and propagation has received in the last years a constantincreasing attention. The most common approach is based on the Lighthills acoustic analogy, describingthe flow noise in terms of a wave equation, the Lighthills wave equation, for a homogeneous medium atrest12. The actual flow is effectively incorporated into the source terms, interpreted as a spatial distributionof sources embedded in a virtual medium at rest. Although the equation is exact, approximations to thesesource terms have the effect of suppressing sound convection and refraction by mean flow. The perceivedinability to calculate sound-flow interaction via Lighthills equation has been regarded as a serious flaw. Flownonuniformities play an important role in many practical situations. For example, in the design optimization(noise reduction) of inlet and outlet geometries of aircraft turbofan engines.

There have been different approaches to resolve this difficulty. Bogey et al.4 retain the original Lighthillsequation and evaluate the right-hand side more exactly via the flow field computation, evaluating the Lighthilltensor terms, linear in acoustics fluctuations, in the entire region encompassing all noise sources and sound-flow interactions. Another possibility is to discard the Lighthills equation, and to solve the Linearized EulerEquations, valid for general mean flow. In the third approach, which may be viewed as an intermediatesolution amongst the two previous strategies, the original Lighthills equation is reformulated into the formof a moving medium wave equation, termed convected wave equation9,11,13,15. Spatially varying mean flowproperties produce non-constant coefficients of the wave equation, which usually make analytical solutionsextremely cumbersome. As an example, for duct aeroacoustics, Rienstra14 extended the modal expansionanalysis, valid for the Lighthills equation, to the case of irrotational nonswirling flows in ducts with slowlyvarying properties, introducing a multiscale solution. Golubev and Atassi3 investigated the effects of theswirling mean motion on the duct modal behavior, and Cooper and Peake8 extended the multiscale solutionof Rienstra to the case of rotational swirl.

The investigation of possible numerical implementations of the convected wave equation seems reasonable,in view of the difficulties encountered in the development of analytical solutions. Some attempts in thesedirections have been undertaken under specific assumptions on the mean flow. More recently, a numericalmethod for the convected wave equation, written in the frequency domain, and based on Generalized GreensFunctions, termed GFD method, has been proposed5. No specific assumptions on the form of the potential

Associate Professor, Dipartimento di Ingegneria Aeronautica e Spaziale, Corso Duca degli Abruzzi 24, 10129-Torino, Italy,renzo.arina@polito.it, Member AIAA.

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mean flow are introduced. The method has been applied to the solution of exterior aeroacoustic problemsas well as to the case of wave propagation in annular ducts6,7.

It is important to point out that a great effort is presently made for the development of efficient numericalmethods for the solution of the Linearized Euler Equations with nonlinear source terms. The Linearized EulerEquations describe the propagation of acoustic, vorticity and entropy waves, therefore unstable solutions, ina globally unstable mean flow, might occur. To overcome this serious flaw, alternative acoustic perturbationequations, hydrodynamically stable, may be derived10, or it is possible to revert back to one of the equivalentforms of the convected wave equation, proposed in the past, and to develop efficient numerical methods fora general class of mean flows.

The aim of the present work is to investigate this last strategy. A numerical technique, developedfor the solution of the original Lighthills wave equation1,2 , is extended to the convected wave equationvalid for nonuniform mean flows. Assuming harmonic time dependence, the wave equation is solved inthe frequency domain on unstructured mesh. For each wave number, an associated convected Helmholtzproblem, with the corresponding inhomogeneous forcing term, is solved. The main advantages with respectto existing techniques (such as GFD) is the ability of dealing with generally shaped boundaries. Moreover,the frequency-domain discretization approach makes possible to treat each wave number separately. Thisproperty is particularly useful for design optimization calculation of the cuton/cutoff spinning modes inaircraft turbofan aeroacoustic analysis.

II. Convected Lighthills Equation

Lighthills theory treats the sound sources as if they were moving through a steady medium. In thisway convective amplification can be recovered, but it is not possible to account for wave convection andrefraction. In order to include these effects in Lighthills theory, it is possible to adjust the source terms insome manner. This approach requires a prior knowledge of the sound field.

A more appropriate approach is to formulate a moving medium wave equation. The first step, is toexpand the nonlinear source terms of the wave equation derived from the conservation equations15,

1

c20

2p

t2

2p

xixj=

2

xixj(vivj) + ij

(2p

xixj c20

2

xixj

); (1)

p is the acoustic pressure, c0 the sound speed, and vi the instantaneous velocity.Following Ribner15, it is possible to express the instantaneous local velocity and density as the mean plus

a perturbation vi = Ui +ui, = +, and to retain the compressibility effects where acoustically necessary.

Expanding the nonlinear source terms of Eq.(1)9,15, the resulting expression contains a number of termswhich do not depend on turbulent velocities, but only on the mean flow and density gradients. Such termsexpress the interaction of sound waves with the mean field, contributing more likely to sound propagationthan its generation. Equation (1) can be recast as

1

c2

[2p

t2+ 2Ui

2p

txi+ UiUj

2p

xixj

]

2p

xixj 2Ui

xj

(uj)acxi

= Qac , (2)

where c is a local time averaged speed of sound, ad Qac contains all the source terms. The last term of theleft-hand side represents the part of the nonlinear source terms of Eq.(1) participating in wave propagation.It accounts only for a small acoustic (compressible) component associated with wave propagation. For asubsonic flow, ui is mainly dominated by turbulent vorticity assumed incompressible. Splitting off this part,(ui)ac accounts for the small acoustic component and it is defined by the momentum equation.

Applying the Lighthill approximation = 0, Eq.(2) takes the form

1

c2

[2p

t2+ 2Ui

2p

txi+ UiUj

2p

xixj

]

2p

xixj 20

Uixj

(uj)acxi

= Qac . (3)

Equation (3), and the momentum equation for (ui)ac, generalizes the unconvected wave equation of Lighthillto allow explicitly for a mean flow.

In order to obtain an equation in which all the propagation effects are accounted for in the wave operatorpart of the equation, Lilley13 derived from Eq.(3) and the momentum equation, a third order wave equation.

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To devise an efficient numerical method for the solution of the third order Lilleys equation is not an easytask, therefore it may be worthwhile to recast Eq.(2) in the equivalent form proposed by Csanady9

1

c2

[2p

t2+ 2Ui

2p

txi+ UiUj

2p

xixj

]

2p

xixj

(Ujxi

Uixj

+Uixi

Ujxj

)= Qac , (4)

where Qac contains the source terms. In the case of an oscillatory point source

Qac = (x y) eIt .

For a parallel mean shear flow (U1 = U(y), 0, 0), the last term of the left-hand side vanishes.

III. Numerical Method

Considering Eq.(4), and setting c = c0 , being p = c20, it follows, with Mi = Ui/c0,

1

c20

2p

t2+ 2

Mic0

2p

txi (ij MiMj)

2p

xixj p

(Mjxi

Mixj

+Mixi

Mjxj

)= Qac . (5)

Assuming harmonic time dependence for the acoustic fluctuations

p = p eIt ,

with I =1 and being the angular frequency, Eq.(5) transforms in

(ij MiMj)2p

xixj k2p+ 2IkMi

p

xi p

(Mjxi

Mixj

+Mixi

Mjxj

)= Qac , (6)

with the wave number k = /c0. For a medium at rest (Mi = 0) Eq.(6) reduces to the Helmholtz

equation ( p+ k2p) = Qac.The acoustic pressure can be computed after the mean flow field calculations are completed, the inhomo-

geneous term Qac of Eq.(6) has been evaluated and a Discrete Fourier Transform (DFT) has been performed.In this way the range of the wave number k [0, kmax] is illustrated. For each value of k we must solve theassociated convected Helmholtz problem (Eq.(6)) with the corresponding inhomogeneous forcing term Qac.The acoustic pressure field is recovered performing an inverse DFT.

Along artificial boundaries, to avoid incoming spurious reflections, appropriate non-reflecting boundaryconditions must be imposed. Considering the one-dimensional analog of Eq.(5), its homogeneous part,neglecting the last term of the left-hand side, reads

D2p

Dt2 c20

p

x= 0 ,

withD

Dt t

+ U

x,

the wave operator can be factored as follows(D

Dt+ c0

x

) (D

Dt c0

x

)p = 0 .

The second order operator is split in two first order operators, with solutions p,(D

Dt c0

x

)p =

( t

+ c0(M 1)

x

)p = 0 . (7)

Consequently the solution p is the sum of two waves traveling from left to right (p+) and from right to left(p). Assuming harmonic time dependence Eq.(7) can be transformed as follows

(1M) dpdx ikp = 0 . (8)

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For M = 0, Eq.(8) corresponds to to the Sommerfeld radiation condition. Then Eq.(8) may be consideredas an approximate (because of the simplifying assumptions) non reflecting boundary condition for the generalcase with flow.

From Eq.(6) the following Galerkin variational formulation is then obtained

(ij MiMj)

xi

p

xjd k2

pd + 2Ik

Mip

xjd

p

(Mjxi

Mixj

+Mixi

Mjxj

)d =

Qacd +

p

nd , (9)

where denotes the domain of definition and its boundary. The Fourier coefficient p of the acousticpressure is interpolated by the nodal values so that

p(x) =

Nl

l(x)dl ,

where l(x) are linear independent basis (shape) functions with N unknown complex coefficients dl. Definingthe test functions as the linear span of basis functions l(x) (l = 1, N), the variational problem results ina complex matrix equation.

Two-dimensional geometrical domains are subdivided into non overlapping triangular and quadrilateralelements. The basis functions l(x) associated with element nodes are C

0 continuous interpolation func-tions with compact support such that the unknown amplitudes take on the point values dl = p(xl). Theinterpolation functions l(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 k, onboth coarse and fine grids.

IV. Numerical Results

To assess the effectiveness of the numerical method two preliminary test cases have been studied. Thefirst numerical example is a simple model problem represented by the acoustic propagation of the sound

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Figure 1. Instantaneous pressure countours (left) and Sound Pressure Level countours: medium at rest.

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Figure 2. Instantaneous pressure countours (left) and Sound Pressure Level countours: medium with uniformvelocity M1 = 0.25.

emitted by a circular cylinder with and without a mean flow. The computational domain is bounded bythe circular cylinder, of unitary diameter, and a circular far field boundary located at a distance equal to 5diameters. The domain is discretized with 2051 nodes, and 4100 triangular elements. The cylinder surfaceis harmonically emitting a pressure signal, representing a periodic forcing for the acoustic field, with wavenumber k = 4. In this case the mesh size is sufficiently refined to avoid pollution effects due to unresolvedhigh frequencies. Non reflecting boundary conditions (Eq.(8)) are imposed on the exterior boundary. Allthe computations are performed assuming a time harmonic behavior with 64 modes. In Fig.1 we have asnapshot of the acoustic pressure field and the Sound Pressure Level countours in the case of a medium atrest. Fig. 2 corresponds to the acoustic propagation within a medium with uniform velocity M1 = 0.25

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Figure 3. Instantaneous pressure countours (left) and Sound Pressure Level countours: medium with shearedvelocity profile M1(y) = 0.05 y.

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XY

Z

Figure 4. Geometry of turbofan engine exhaust. Unstructured grid with triangular and quadrilateral elements.

(flow from left to right). Finally, Fig.3 displays the acoustic field in a medium with a sheared velocity profileM1(y), ranging linearly from M1 = 0.25 at y = 5, to M1 = 0.25 at y = 5. Effects of the mean flow onsound propagation are clearly visible. Both convection and refraction effects occur.

A more realistic case is considered in the second numerical test. The geometry corresponding to therear part of a turbofan engine is considered. The acoustic field is forced by imposing a plane wave mode(with wave number k = 4) at the boundaries inside the fan duct and the nozzle. Both signals are in phase.The geometry is shown in Fig.4. The grid is formed by 8019 nodes and 9246 triangular and quadrilateralelements. Fig.5 reports the Sound Pressure Level countours in the case of propagation in a medium at rest(left) and in a medium with a uniform velocity (M1 = 0.25). The effects of the mean flow field are evenmore evident comparing Fig.6 (medium at rest) and Fig.7 (M1 = 0.25), reporting snapshots of the acousticpressure countours. The case with zero mean flow has been discretized in time with 64 modes, while 32modes have been employed in the uniform flow case. A more interesting comparison between the two flowfields is reported in Fig.9, where the time histories of the acoustic pressure in a control point located behindthe turbofan (dot position in Fig.8), are plotted. It is interesting to remark that, in spite of the convectionand refraction effects, the frequency is the same, as expected.

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7.000e+017.200e+017.400e+017.600e+017.800e+018.000e+018.200e+018.400e+018.600e+018.800e+019.000e+019.200e+019.400e+019.600e+019.800e+011.000e+021.020e+021.040e+021.060e+021.080e+021.100e+021.120e+021.140e+021.160e+021.180e+021.200e+02

SPL [dB] 7.000e+017.200e+017.400e+017.600e+017.800e+018.000e+018.200e+018.400e+018.600e+018.800e+019.000e+019.200e+019.400e+019.600e+019.800e+011.000e+021.020e+021.040e+021.060e+021.080e+021.100e+021.120e+021.140e+021.160e+021.180e+021.200e+02

SPL [dB]

Figure 5. Sound Pressure Level Countours: medium at rest (left), medium with uniform flow M1 = 0.25 (right).

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Figure 6. Snapshots of the instantaneous pressure countours: medium at rest.

V. Conclusion

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 thefrequency domain on unstructured mesh. For each wave number, an associated convected Helmholtz problem,with the corresponding inhomogeneous forcing term, is solved. In order to deal with complex geometriesa Finite Element discretization has been adopted for solving the convected Helmholtz problem. The mainadvantages with respect to existing techniques (such as GFD) is the ability of dealing with generally shapedboundaries. Moreover, the frequency-domain discretization approach makes possible to treat each wavenumber separately.

Preliminary results for a model problem and a geometry representing the rear part of a turbofan engine

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Figure 7. Snapshots of the instantaneous pressure countours: medium with uniform flow M1 = 0.25.

Figure 8. Position of the control point.

have been presented, showing the ability of the method to capture the convection and refraction phenomenaassociated with the mean flow.

As future developments of the current research, more precise comparisons with existing analytical and

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-30

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-10

0

10

20

30

0 1 2 3 4 5 6

pres

sure

time

medium at restuniform flow

Figure 9. Pressure time history in the control point of Fig.8.

numerical (Linearized Euler Equations) solutions, as well as some further improvements of the numericalalgorithm (high-order discretization) will be made.

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.

3Golubev V.V. and Atassi M., Acoustic-Vorticity Waves in Swirlig Flows, J. Sound Vibr., 209(2), pp.203-222, 1998.4Bogey C., Gloerfelt X. and Bailly C., Illustration of the Inclusion of Sound-Flow Interaction in Lighthills Equation,

AIAA Journal, 41-8, pp. 1604-1606, 2002.5Di Francescantonio P. and Casalino D., Greens Function Discretization Scheme for Sound Propagation in Nonuniform

Flows, AIAA J., 37-10, pp. 1161-1172, 1999.6Casalino D., Roger M. and Jacob M., Prediction of Sound Propagation in Ducted Potential Flows Using Greens Function

Discretization, AIAA J., 42-4, pp. 736-744, 2004.7Casalino D., Di Francescantonio P. and Druon Y., GFD Predictions of Fan Noise Propagation, AIAA Paper 2004-2989,

2004.8Cooper A.J. and Peake N., Propagation of Unsteady Disturbances in a Slowly Varying Duct with Mean Swirling Flow, J.

Fluid. Mech., 445, pp. 207-243, 2001.9Csanady G.T., The effect of mean velocity variations on jet noise, J. Fluid Mech., 26-1, pp.183-197, 1966.10Ewert R. and Schroder W., Acoustic Perturbation Equations Based on Flow Decomposition via Source Filtering, J.

Comput. Phy., 188, pp. 365-398, 2003.11Leib S.J. and Goldstein M.E., Sound from Turbulence Convected by a Parallel Flow within a Rectangular Duct, AIAA

Journal, 39-10, pp. 1875-1883, 2001.12Lighthill M.J., On Sound Generated Aerodynamically, I: General Theory, Proc. Royal Society, 211A-1107, pp. 564-587,

1952.13Lilley G.M., The generation and Radiation of Supersonic Jet Noise, in Theory of Turbulence Generated Noise, Vol. 4,

part II, Air Force Aero Propulsion Lab, AFAPL-TR-72-53, 197214Rienstra S.W., Sound Transmission in Slowly Varying Circular and Annular Lined Ducts with Flow, J. Fluid Mech.,

380, pp.279-296, 1999.15Ribner H.S., Effects of Jet Flow on Jet Noise via an Extension to the Lighthill model, J. Fluid Mech., 321, pp. 1-24,

1996.

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