application of a high-order fem solver to aeroengine ...€¦ · helmholtz instabilities in real...

Application of a high-order FEM solver toaeroengine exhaust noise radiation

Karim Hamiche, Hadrien BériotSiemens Industry Software NV, Interleuvenlaan 68, B-3001 Leuven, Belgium.

Gwénaël GabardISVR, University of Southampton, SO17 1BJ, Southampton, United Kingdom.

Summary

A stabilised, high-order finite element model with unstructured mesh is proposed for solving theLinearised Euler Equations in the frequency domain, in the presence of a mean flow. The method issuccessfully applied to sound radiation from a straight semi-infinite duct and a realistic aeroengineexhaust.

PACS no. xx.xx.Nn, xx.xx.Nn

1. Introduction

The propagation of sound in complex flows is a crit-ical issue for many industries, especially when deal-ing with the sound generated by turbofan engines [1].Different models are available for solving the acousticpropagation in the presence of mean flows: the fullpotential formulation for irrotational flows [2], thePridmore-Brown equation for parallel sheared flows[3], the convected Helmholtz wave equation for uni-form mean flows [4], etc. The Linearised Euler Equa-tions (LEE) describe the effects of complex multi-directional sheared flows and the strong refractionthrough shear layers, crucial when dealing with turbo-machinery noise radiated through the engine exhaust.Another advantage of the LEE is that they supportthe acoustic, entropy and hydrodynamic waves, andtheir interactions.

The LEE can be solved either in the time domainor in the frequency domain [5]. Time domain solverssuffer from several shortcomings for industrial appli-cations, such as complexity in modeling impedanceboundary conditions [6] and flow instabilities. TheLEE do not take into account the non-linear andviscous terms that normally attenuate the Kelvin-Helmholtz instabilities in real flows. These growinginstabilities may be triggered in time domain solvers.These issues may be avoided by solving the LEE inthe frequency domain [7].

A technique for solving the LEE on unstructuredgrids is the Finite Element Method (FEM). But, theuse of a direct solver in the frequency domain can lead

(c) European Acoustics Association

to serious constraints in terms of computing time andmemory. Those can be reduced by using a high-orderFEM (P -FEM) [8]. The FEM is known to present de-ficiencies in convection-dominated problems: the con-ventional Galerkin formulation experiences a lack ofdiffusion [10]. This can be corrected by adding ar-tificial diffusion in the formulation, such as in theGalerkin/Least-Squares (GLS) method [11]. The sta-bilisation schemes developed for steady advective dif-fusive problems have already been successfully appliedto solving the LEE in the frequency domain using lin-ear FEM [12].

One crucial aspect when designing a computationalscheme for exterior propagation is the choice of non-reflecting boundary conditions (NRBC). The govern-ing equations are solved within a finite domain, andappropriate boundary conditions must be imposed onits borders, in order to avoid reflections of the out-going waves. Different types of NRBC have been de-veloped. Amongst them, the Perfectly Matched Layer(PML) technique, firstly introduced by Bérenger [13]for the absorption of electromagnetic waves, is widelyused.

A stabilised discretised axisymmetric form of theLEE is developed in the frequency domain, in con-junction with PML equations at the inlet and at thefar-field boundaries. The solver is applied to the prop-agation of duct modes from a turbofan engine ex-haust, with complex geometry and non-uniform meanflow. This paper is organized as follow: the governingequations and the numerical model are explained inSections 2 and 3. The results are then presented inSection 4.

Copyright© (2015) by EAA-NAG-ABAV, ISSN 2226-5147All rights reserved

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2. Linearised Euler Equations

The governing equations are written in cylindrical co-ordinates (r, θ, z) for a perfect gas, with isentropicdisturbances, no viscous effect (inviscid fluid), noheat transfer and no external source. The variablesare the mass density ρ, the momentum vector ρuand the non-dimensional pressure defined by Gold-stein [14], π = (p/p∞)1/γ , with p the pressure, p∞a reference pressure and γ the ratio of the specificheats. After linearisation, the LEE are written forthe time-harmonic (e+jωt dependence) perturbationsq′ = {ρ′; (ρur)

′; (ρuθ)′; (ρuz)

′;π′}t. The square matri-ces Ar, Aθ, Az and Ac contain the coefficients of theequations. The LEE read:

jωq′ +1

rAcq

′ +1

r

∂ (rArq′)

∂r

+1

r

∂ (Aθq′)

∂θ+

∂ (Azq′)

∂z= 0 , (1)

For axisymmetric mean flows and geometries, aFourier decomposition of the fields yields azimuthalcontributions of the form:

q′(r, θ, z, t) = q′(r, z)e−jmθe+jωt , (2)

where m is the azimuthal mode. The differential op-erator for the axisymmetric LEE, written for the vari-able q′(r, z), reads:

L(q′) = jωq′ − jm

rAθq

′ +1

rAcq

′

+1

r

∂ (rArq′)

∂r+

∂ (Azq′)

∂z= 0 . (3)

3. High-order Finite Element Method

The weak form of the weighted residual formulation isderived from Eq. (3). The domain Ω is partitioned intotriangular non-overlapping finite elements Ωi. Thestabilised formulation reads:

∫Ω

(jωrwTq′ − jmwTAθq

′ − r∂wT

∂rArq

′

−r∂wT

∂zAzq

′ +wTAcq′

)drdz

+∑i

∫Ωi

D(w)T τiL(q′)dΩi

= −

∫Γ

rwT (nrAr + nzAz)q′dΓ, ∀w ∈ W , (4)

with w the weighting functions defined in the testfunction vector space W ⊂ H1(Ω), and nr and nz re-spectively the r-component and the z-component ofthe unit normal vector n to the boundary line Γ. Thesuperscript T denotes the Hermitian transpose. The

matrix F = nrAr + nzAz represents the fluxes ofq′ along the n-direction. In the left-hand side of theformulation the stabilisation term is evaluated withineach element and depends on two main features: thestabilisation scheme that is defined by the stabilisa-tion operator D(w) and the stabilisation parameterτi. In this paper the Galerkin/Least-Squares (GLS)stabilisation operator is used. For the axisymmetricLEE, this operator reads:

D(w) = jωw − jm

rAT

θ w +1

rAT

c w

+1

r

∂(rAT

r w)

∂r+

∂(AT

z w)

∂z. (5)

The stabilisation parameter τi within each elementof the discretisation is the one used for instance byRao and Morris [12]: τi = max (αhi,l/λl)I, with αa stabilisation coefficient, hi,l the element size of theith-element in the lth-direction (z or r). λl is the spec-tral radius of the coefficient matrix Al and I is the5× 5 identity matrix. In the numerical computationsα = 1/(2P ): the coefficient 1/2 gives a value of τianalogous to the steady convective-diffusive equation,and the factor 1/P accounts for the use of high-ordershape functions [10].

A linear system Ks = f is obtained, where thecomplex-valued sparse matrix K contains the differ-ent contributions of the physical integrals, f is thesource vector and s is the solution vector. In the P -FEM, the space basis is enriched with edge and bubbleshape functions. Because of their hierarchic propertyand good conditioning, the Lobatto shape functionsare used in the numerical simulations [8].

Specific boundary conditions are applied: an axialsymmetry condition is enforced along the boundary atr = 0 and hard-wall conditions (u′ · n = 0) are speci-fied at the duct walls. PML are applied to inject theduct modes and to absorb the outgoing waves. Moredetails on the application of the boundary conditionscan be found in [15].

4. Numerical results

In this section, two test cases are considered. First,the effects of a uniform mean flow are investigatedfor the propagation of an acoustic wave in a straightcircular duct. Second, a realistic exhaust geometry isstudied for the static-approach condition with non-uniform mean flow inside the duct and fluid at restoutside the duct. The second test case has been de-veloped within the European project TURNEX [16]and used by several authors to benchmark Compu-tational Aero Acoustics (CAA) methods [18, 17]. Inboth cases the mesh is composed of unstructured tri-angle elements.

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K. Hamiche et al.: Application of...

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0 1 2 3 4 50

0.5

1

1.5

2

2.5

z

r

Figure 1. Geometry of the Munt problem with unstruc-tured mesh (h = 0.4m, hvs = 0.02m) and positions of thecontrol points.

4.1. Propagation in circular duct

Let us first examine the propagation of an acousticwave inside a semi-infinite straight duct, in the pres-ence of a uniform mean flow. This test case is alsoknown as the Munt problem [19]. An active PML isused for injecting acoustic modes (m,n) of unit pres-sure amplitude inside the duct, with m the azimuthalmode and n the radial mode. The PML also absorbsthe outgoing waves at the outer boundaries of thecomputational domain, which extends from z = 0 to5m and from r = 0 to 2.5m. The length of the duct isL0 = 2.5m and its radius is R0 = 1m. Control pointsare placed in the computational domain in order toevaluate the solution: along the circle of radius 2mand centred on the point (L0, 0), as well as on the con-tinuation of the duct wall. The angle Φ defined alongthe circle is measured from the positive z-direction.The geometry is shown in Figure 1, as well as an ex-ample of unstructured mesh: the characteristic meshsize is h = 0.4m with refinement hvs = 0.02m alongthe vortex sheet developing at the continuation of theduct wall. The element size is deliberately relaxedclose to the PML in order to have an homogeneousmesh inside the absorbing layer.

The fluid properties are: the ratio of specific heatsγ = 1.4, the density ρ0 = 1.225 kg/m3 and the speedof sound c0 = 340.27m/s. The mean flow Mach num-ber is M = 0.3 in the whole computational domain.The frequency of the wave is f = 920.65Hz, corre-sponding to the Helmholtz number k0R0 = ωR0/c0 =17. The pressure field of the cut-on mode (5, 1) isshown in Figure 2. It shows the propagation of theacoustic wave inside the duct and the radiation fromthe trailing edge. The absorption of the outgoingwaves is also visible inside the PML. Furthermore, thevortex shedding developing from the trailing edge isobserved on Figure 3, where the r-momentum field isrepresented. The hydrodynamic wavelength can alsobe seen on this figure: λhydro ≈ 0.11m. Figure 4 showsthe contours of the z-momentum in the computationaldomain. It can be seen that the gradient of the axialmomentum at crossing the vortex sheet is very high,which makes the approximation quite difficult.

Figure 2. Real part of the non-dimensional pressure π′ forthe Munt problem. Mode (5, 1). k0 = 17m−1, h = 0.4m,hvs = 0.02m, P = 6.

Figure 3. Real part of the r-momentum (ρur)′ for the

Munt problem. Mode (5, 1). k0 = 17m−1, h = 0.4m,hvs = 0.02m, P = 6.

Figure 4. Real part of the z-momentum (ρuz)′ for the

Munt problem. Mode (5, 1). k0 = 17m−1, h = 0.4m,hvs = 0.02m, P = 6.

The Sound Pressure Level (SPL) along the controlcircle in Figure 5 is compared to the reference solu-tion provided by the analytical model of Gabard andAstley [20], for sound radiation from a semi-infiniteduct. The reference pressure for the calculation of theSPL is taken as pref = 2 × 10−5Pa. The agreementbetween the numerical solution and the analytical so-lution is excellent: the behaviour in the duct is wellrepresented and the peaks due to the fluctuation ofthe momentum across the vortex sheet are reproducedby the model. The nodal errors on the control circleare: Eπ = 0.84%, Eρur

= 4% and Eρuz= 4.2%.

The agreement is excellent for the pressure. The de-velopment of vortices at the continuation of the ductwall makes the approximation more difficult for thecomponents of the momentum.



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30 60 90 120 150

60

80

100

120

Φ(◦)

SPL(dB)

Figure 5. SPL along the control circle for the Munt prob-lem. k0 = 17m−1. h = 0.4m, hvs = 0.02m, P = 6. Solidline: analytical solution. � marker: numerical solution.

−0.5 −0.3 −0.1 0.1 0.3 0.50

0.1

0.2

0.3

0.4

0.5

z

r

Figure 6. Geometry for the TURNEX test case with un-structured mesh (h = 0.04m, hwall = 0.02m, hlip =0.0025m) and positions of the control circle.

4.2. Aeroengine exhaust noise radiation

Let us now focus on the acoustic radiation fromthe aeroengine exhaust geometry developed withinthe TURNEX project. The computational domainstretches from z = −0.493m to 0.51m and from r = 0to 0.51m, and is surrounded by a PML. An acousticduct mode (m,n) of unit incident intensity is injectedthrough the active PML at the inlet of the bypassduct. A PML is also added at the inlet of the coreduct. In order to check the solution and its conver-gence, control points are located on a circle centredon (0, 0) and of radius R = 0.5m. The angle Φ de-fined along this circle is measured from the positivez-direction. Figure 6 shows the geometry of the ex-haust with an unstructured mesh. The characteristicelement size is h = 0.04m in the majority of the do-main, whereas the refinement is hwall = 0.02m alongthe duct walls and hlip = 0.0025m at the duct lips.

For the static-approach configuration, the meanflow properties at the ducts inlets are: Mbp = 0.447,c0bp = 347.19m/s, ρ0bp = 1.177 kg/m3 for the by-pass duct and Mc = 0.223, c0c = 527.62m/s, ρ0c =0.509 kg/m3 for the core duct. Far from the exhaustthe fluid is at rest (M∞ = 0). The ComputationalFluid Dynamics (CFD) data obtaiend by solving theReynolds Averaged Navier-Stokes (RANS) equations,within the TURNEX project, are linearly interpolatedon the nodes of the acoustic mesh. The Mach number

Figure 7. Mean flow Mach number for the static-approachcondition of the TURNEX test case.

Figure 8. Mean flow density for the static-approach con-dition of the TURNEX test case.

and density mean flow fields are shown respectively inFigures 7 and 8. The acceleration of the flow by thenozzle exits is observed, along with the development ofthe shear layers from the trailing edges. The changesin mean flow density indicate a significant variationof temperature at the engine exhaust.

The frequency of the duct modes injected in thebypass duct is f = 7497Hz, corresponding to theHelmholtz number k0rfan = 17.5 with rfan = 0.13m.The larger elements of the mesh have a characteris-tic length h = 0.04m, which means that there is ap-proximately one element per wavelength in the regionwhere the fluid is at rest. The pressure fields of thecut-on modes (0, 1) and (10, 1) are shown respectivelyin Figures 9 and 10. They both show the wave prop-agation inside the bypass duct and the main radia-tion directions from the engine exhaust. The vorticityshedding develop from the trailing edges. These re-sults illustrate the complexity of the physics with thepresence of multiple-scale wavelengths. The acousticwaves are dominant in most of the domain. But the ef-fects of the hydrodynamic waves tend to prevail closeto the trailing edges of the ducts. An efficient modelshould be able to adjust the numerical effort to thelocal dominant wavelength. This suggests the imple-mentation of an (h, P )-adaptive scheme to optimisethe computational cost while guaranteeing that thesolution has converged in all the regions of interest.

The SPL measured along the control circle are alsoplotted in Figures 11 and 12, respectively for themodes (0, 1) and (10, 1). The directions of main in-tensity in the SPL directivity can be noticed. The P -convergence of the solution is also observed. With thechosen mesh, the order P = 4 already gives a good



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Figure 9. Real part of the non-dimensional pressure π forthe TURNEX test case. Mode (0, 1). h = 0.04m, hwall =0.02m, hlip = 0.0025m, P = 6.

Figure 10. Real part of the non-dimensional pressure π

for the TURNEX test case. Mode (10, 1). h = 0.04m,hwall = 0.02m, hlip = 0.0025m, P = 6.

0 30 60 90 120 150

80

90

100

110

Φ(◦)

SPL(dB)

Figure 11. SPL along the control circle for the TURNEXtest case. Mode (0, 1). � marker: P = 3. � marker: P = 4.+ marker: P = 5. Solid line: P = 6.

0 30 60 90 120 15050

60

70

80

90

100

110

Φ(◦)

SPL(dB)

Figure 12. SPL along the control circle for the TURNEXtest case. Mode (10, 1). � marker: P = 3. � marker: P =4. + marker: P = 5. Solid line: P = 6.

approximation of the solution in the near field, with130, 015 DOF.

5. Conclusions

A stabilised high-order finite element model with un-structured mesh is developed for solving the axisym-metric LEE in the frequency domain, in the presenceof complex sheared flow. The method is applied tothe propagation of duct modes from a realistic turbo-fan engine exhaust, with realistic geometry and non-uniform mean flow. The sound propagation and ra-diation are accurately described, as well as the inter-action between the acoustic waves and the hydrody-namic field resulting in the vorticity shedding fromthe duct lips. The application of the P -FEM allowsto use coarser elements than for the classical FEM,which translates into a lower number of degrees offreedom for an equivalent accuracy.

Several improvements can be identified. First, therepresentation of the geometry can be optimised byusing curved elements on the boundaries. This wouldallow to use larger elements in the vicinity of theduct walls. Second, the interpolation of the complexmean flow is a critical point, that needs to be fur-ther investigated, in particular in the regions of stronggradients. Third, the presence of multiple-scale wave-lengths highlights the complexity of the physics andindicates that an efficient model should account forthe local dominant wavelength. The implementationof an (h, P )-adaptive scheme appears to be necessaryfor optimising the computational cost across the do-main, while assuring that the solution has reachedconvergence everywhere.

Acknowledgement

The geometry/flow test cases used in this paper wereprovided by the European TURNEX project. The au-thors would like to thank the partners for this con-tribution, especially the industrial partners (Airbus,Rolls-Royce, Dassault and AVIO), who provided fi-nancial support for the experiment.

Some of the work presented in this paper wasalso supported by the Level 1 Collaborative ProjectIDEAL VENT coordinated by the von Karman Insti-tute for Fluid Dynamics.

The authors gratefully acknowledge the financialsupport from the European Commission throughthe FP7-PEOPLE-ITN-2011 FlowAirS project, grantagreement 289352, coordinated by Y. Auréguanfrom the Laboratoire d’Acoustique de l’Université duMaine (France).

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application of a high-order fem solver to aeroengine ...€¦ · helmholtz instabilities in real...

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