[2014] Towards Lattice-Boltzmann Prediction of Turbofan.pdf

8/10/2019 [2014] Towards Lattice-Boltzmann Prediction of Turbofan.pdf

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Towards Lattice-Boltzmann Prediction of Turbofan

Engine Noise

D. Casalino A. F. P. Ribeiro E. Fares S. Nolting

Exa GmbH, Curiestrasse 4, Stuttgart, 70563, Germany

A. Mann F. Perot

Exa Corporation, 150 North Hill Drive, Brisbane, CA, 94005, USA

Y. Li P.-T. Lew C. Sun P. Gopalakrishnan R. Zhang H. Chen

Exa Corporation, 55 Network Dr, Burlington, MA,01803, USA

K. Habibi

Department of Mechanical Engineering, McGill University, Montreal, QC H3A 0C3, Canada

The goal of the present paper is to report verification and validation studies carriedout by Exa Corporation in the framework of turbofan engine noise prediction through thehybrid Lattice-Boltzmann/Ffowcs-Williams & Hawkings approach (LB)-(FW-H). The un-derlying noise generation and propagation mechanisms related to the jet flow field and thefan are addressed separately by considering a series of elementary numerical experiments.As far as fan and jet noise generation is concerned, validation studies are performed bycomparing the LB solutions with literature experimental data, whereas, for the fan noisetransmission through and radiation from the engine intake and bypass ducts, LB solutionsare compared with finite element solutions of convected wave equations. In particular, for

the fan noise propagation, specific verification analyses are carried out by considering tonalspinning duct modes in the presence of a liner, which is modelled as an equivalent acousticporous medium. Finally, a capability overview is presented for a comprehensive turbofanengine noise prediction, by performing LB simulation for a generic but realistic turbofanengine configuration.

Technical Director, Aeroacoustics, Aerospace, AIAA member.Team leader, Aerospace.Technical Director, Aerospace, Senior AIAA member.Vice President, Aerospace, AIAA member.Senior Aeroacoustics Engineer, Aeroacoustics, AIAA member.Senior Technical Director, Aeroacoustics, AIAA member.

Consulting Scientist, Physics, AIAA member.Senior Physics Validation Engineer, Physics, AIAA member.Consulting Scientist, Physics.Principle Scientist, Physics.Senior Director, Physics.Chief Scientific Officer, Physics.Research Assistant, AIAA student member.

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American Institute of Aeronautics and Astronautics

AIAA/CEAS Aeroacoustics Conference

June 2014, Atlanta, GA

AIAA

A Aviation


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other side, the computational resources required by space-/time-resolved turbulent flow simulations are stillprohibitively large for most cases of practical relevance.

Recently, a new CFD/CAA technique based on a Lattice-Boltzmann flow Model (LBM) has been em-ployed to tackle airframe noise problems, both at a system level1, 2,3, 4,5 and at full aircraft level.6,7 Sucha technique has achieved the required readiness level to become a viable alternative to WT experiments,both in terms of costs for a target accuracy level of about 1 dB, and turnaround time for down-selectinglow-noise designs and concepts. With the goal of extending the application of PowerFLOW to aero-engineaeroacoustics, towards a comprehensive CFD/CAA prediction of aircraft community noise, including engine

installation effects, the present effort addresses the prediction of the main turbofan engine noise components.Increasing the readiness level of PowerFLOW in this new application field through benchmark studies andthe development of simulation best practices is a necessary condition for achieving the ultimate goal of avirtual aircraft noise certification process to be executed at different phases of the aircraft design.

In the present study, the LBM is used to compute the generation and propagation of acoustic fluctuationsfrom the fan/Outlet Guide Vane (OGV) system and the jet of a turbofan engine. The turbulent flowfluctuations are resolved up to a certain scale using a Lattice-Boltzmann Very Large Eddy Simulation(LBM-VLES) approach. One original aspect is the use a new non-isothermal version of the LBM, whichallows extending the Mach number range of the standard LBM scheme up to about 0 .95. Such a capability isvalidated by computing the flow field and the associated noise of a single hot subsonic jet. Another originalaspect of the present study is the use of an Acoustic Porous Medium (APM) included in the flow simulationdomain to mimic the effects due of a honeycomb liner installed on the walls of the engine intake. Thephysical properties of the APM (thickness, resistivity and porosity) are tuned against a target impedance-frequency law using a genetic algorithm minimization technique. The liner simulation capability is verifiedby computing the propagation of spinning duct modes through an aero-engine intake and comparing the far-field noise directivity to the one computed using a Finite Element Method (FEM), which solves a convectedwave equation in the frequency domain.

The paper is organized in the following way. The numerical method is presented in sectionII,includingthe newly developed high-speed LB formulation and the acoustic porous medium. Section III presentselementary verification and validation studies for the different source components of an aero-engine. Ademonstration of a comprehensive turbofan noise prediction is provided in section IV.The main outcomesof the present effort are finally summarized in the conclusion section V. The derivation of the impedanceboundary condition used for the FEM computations of sectionIIIis reported in the Appendix.

II. Numerical approach

The LBM-based CFD/CAA solver PowerFLOW developed and distributed by Exa Corporation is usedto compute unsteady flow physics and the resulting generation and propagation of acoustics waves. A FW-Happroach is then used to extrapolate the near field solution sampled on a permeable surface to the far-field.A standard D3Q19 lattice scheme (3 dimensions, 19 velocity states per mesh node) is employed, but theMach number range is extended beyond the typical value of 0.4 by relaxing the iso-thermal assumptionof the standard LB model. The effects due to the presence of a liner are modelled using an APM 8 directlyimplemented in the LB scheme. Details about the standard LB formulation, its high-speed extension andthe APM are reported in the following subsections.

II.A. Standard LB model

The standard LBM implemented in PowerFLOW has the following form:

fi(x + vit, t+ t) fi(x, t) = Ci(x, t) (1)

wheref is the particle density function, which represents the probability for particles to travel with speed vfrom the positionx at timet in the discrete direction i. The collision termCis modeled with the well-knownBhatnagar-Gross-Krook (BGK) approximation9,10 as follows:

Ci(x, t) = 1

[fi(x, t) f

eqi (x, t)] , (2)

where is the relaxation time, which is related to the fluid viscosity, and feqi is the equilibrium distribu-tion, which is approximated by a third order expansion with constant temperature.11 Once the distribution

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function is computed, the density and linear momentum are simply determined through discrete integra-tion: (x, t) =

i fi(x, t) and u(x, t) =

i fi(x, t) vi. All the other quantities are determined through

thermodynamic relationships for an ideal gas.With the Chapman-Enskog expansion12 expansion it is possible to retrieve the compressible Navier-Stokes

equations from LBM.10,13 However, in contrast to the Navier-Stokes equations, the current method is basedon linear formulation which relies on simple computational operations, allowing for efficient, accurate, andhighly scalable implementations. An explicit time advancement scheme is employed, which also facilitatesmassively parallel simulations and the use of a very small time step, which is valuable when attempting to

simulate high frequency noise.PowerFLOW solves the D3Q19 formulation of the Lattice-Boltzmann equation for direct numerical sim-

ulations. For high Reynolds flows, turbulence modeling is introduced14 by solving a variant of the RNGk model15,16 on the unresolved scales,17 selected via a swirl model,18 a method referred to as LBM VeryLarge Eddy Simulation (LBM-VLES). An extended wall model including pressure gradient effects is used inthe near-wall region.19

The LBM scheme is solved on a grid composed of cubic volumetric elements (voxels). A Variable Reso-lution (VR) by a factor of two is allowed between adjacent regions. Consistently, the time step is varied bya factor of two between two adjacent resolution regions. Solid surfaces are automatically facetized withineach voxel intersecting the wall geometry using planar surface elements20,21 (surfels).

Although PowerFLOW has intrinsic CAA capabilities and can compute the noise propagation directlyfrom the unsteady flow simulations, this is generally computationally expensive and limited to the near field.Therefore, in order to compute the far field noise, an integral extrapolation based on the FW-H acousticanalogy22 is used, both using solid and permeable23 surface formulations. A forward-time solution24 of theFW-H equation based on Farassats formulation 1A25 is employed. The solver is embedded in the post-processing tool PowerACOUSTICS, which is also used to perform statistical and spectral analysis of anyunsteady solutions generated by PowerFLOW (volume fields, surface fields, and probe signals).

The numerical methods herein described were extensively validated for a wide variety of applicationsranging from academic cases using direct numerical simulation26 to industrial flow problems in the fields ofaerodynamics,27 thermal management,28 and aeroacoustics.2,1,4,6,7

II.B. High-speed non-isothermal LB model

For the solution in the high subsonic Mach number range, e.g, flows with local Mach number greater than0.5, a standard D3Q19 LBM is applied. The BGK collision operator in Eq.2was replaced by a regularizedcollision operator which can significantly increase both numerical stability and accuracy when local flowMach number is high. An interaction force was introduced in Eq.1,29 which can modify the equation ofstate thus the speed of sound so that high Mach number flows can be simulated by a low order LB scheme.Moreover, in order to take into account the flow heating due to compression work and viscous dissipation,a hybrid approach was applied for the thermodynamics of energy field, by solving the entropy equationthrough a Lax-Wendroff finite difference scheme on the Cartesian LB mesh.

II.C. Acoustic porous medium model

In the LB method, external forces can be included in the fluid dynamics by altering the local-instantaneousparticle distributions during the collision step. This technique can be used to model, for example, buoyancyeffects due to gravity. The solver used in the present study implements a porous medium model by applyingan external force driven by the flow resistivity and as function of the local flow velocity.30 This model canbe used to predict pressure losses that affect the time-averaged flow field solution and, at the same time, theinstantaneous acoustic fluctuations.

The APM model in PowerFLOW, a recently patented technology by Exa Corporation,31 is characterizedby three main parameters: the viscous resistance R, the porosity and the APM thickness d. The char-acteristic surface impedance of the APM, say ZAPM(f,R, , d), can be fully determined with these threevariables via an analytical model, where f is the frequency in Hertz; ZAPM is also parameterized throughquantities that are directly related to the LB formulation, but these details are beyond the present scope.The characteristic surface impedance Z(f) of an acoustic liner can be obtained via experiments, or via nu-merical simulations of the complete geometry, in a normal or grazing impedance tube setup.32 The three

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parameters are then educed by minimization of the cost functionJ(R, , d):

J(R, , d) = maxf

(|Z(f) ZAPM(f,R, , d)|) (3)

The cost function is minimized using a genetic algorithm for (R, , d) constrained on specific variationranges. The optimal results (R0, 0, d0) are then validated with the simulation of the resulting APM in anormal impedance tube and the computation of its numerical characteristic impedance. The target acousticresistance and reactance curves, their analytical fitting of the LBM educed impedance curves are then plottedfor visual verification, as shown in subsectionIII.A.

III. Verification and validation

This section deals with the unitary verification and validation studies carried out to build up therequired knowledge in terms of flow physics and simulation setup towards a comprehensive turbofan engineaeroacoustic analysis. Fan and jet noise sources are addressed, whereas the internal turbomachinery flowfeatures (compressor, combustion chamber and turbine) are not in the present scope.

SubsectionIII.A deals with the propagation of a spinning duct mode through an acoustically treatedintake. LBM results are compared with a FEM solution of a convected 2nd order wave equation in thefrequency domain.33,34 The main goal of this study is to verify the LBM propagation capabilities in thepresence of an impedance wall modelled via an APM.

SubsectionIII.Bdeals with the propagation of a spinning duct mode through a bypass duct. LBM resultsare compared with a FEM solution of a convected 3rd order wave equation in the frequency domain.35 Themain goal of this study is to perform a cross validation of sound propagation in the presence of a shear layer.

Subsection III.C deals with the problem of tonal and broadband noise generation from rotating fanblades. The Advanced Noise Control Fan (ANCF) configuration36 is used as benchmark validation case.The complete 3-D in-duct rotor/stator model and the rotating rotor is simulated for the nominal value ofMach rotor tip of 0.33 used in the experiments. Far-field measurements conducted at the NASA Glennresearch center are used to validate the simulation results.

Finally, subsectionIII.D focuses on the prediction of single stream jet flow noise. The NASA Glennresearch center hot jet experiments37 carried out with the SMC000 nozzle geometry are used to validate theLBM solution for high speed cold and hot jets.

Throughout, the size of the simulation problems is reported in terms of equivalent number of voxels andsurfels as the solution in the whole simulation domain would be updated every time step. In other words,

one voxel or surfel belonging to the second finest mesh resolution level, for instance, would count as a halfin the overall count of Fine Equivalent Voxels (FEV) and Fine Equivalent Surfels (FES). Simulation time isreported in terms of overall computational hours (CPUh) on a cluster of Intel Xeon X5570 2.93GHz CPUsconnected by a Mellanox FDR Infiniband 56Gb/s network.

All simulations presented in this section were performed using the recently released PowerFLOW version5.0c, with the exception of the jet flow simulations presented in subsections III.B and III.D, which wereperformed using a beta release of PowerFLOW 5.1, providing the high-speed non-isothermal functionality.

III.A. Intake fan noise radiation

In this subsection we show results for the transmission of a spinning mode through an idealized aero-engineintake. The Helmholtz number, based on the fan plane outer radius, is kRf= 5.257. The acoustic mode (6, 1)is considered, with a magnitude corresponding to an overall inlet acoustic power of 130 dB. The acoustic

mode is introduced in the LBM simulation through a time-varying pressure/velocity boundary condition onthe fan plane. LBM results without and with mean flow convective effects and without and with an acousticliner are compared against a frequency-domain FEM solution of a convected wave equation for the acousticpotential34,38,39 put forward by Pierce.40

Simulations are carried out for a three-dimensional rigid and acoustically treated (soft) nacelle. A zero-splice liner is considered, located on the outer wall of the intake, as shown in Fig.1. Both for the LBM andthe FEM computations, the near-field solution is extrapolated to the far field through a permeable FW-Happroach. In order to reduce the impact of the FW-H extrapolation to the far-field noise prediction, thesame integration surface is used for the two simulations, as depicted in Fig. 1.

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Figure 1. Nacelle intake simulation setup.

The LBM simulations are performed using a mesh resolution that guarantees 14 points per acousticwavelength (without convective effects) in the near field volume inside the nacelle up to the FW-H integrationsurface. Views of the mesh for the rigid and soft cases are shown in Fig.2. The volume occupied by theAPM is also meshed, as this is treated as a fluid material with different properties. Quantities that are usefulto estimate the computational effort for the present case are reported in Table1.

Figure 2. Nacelle intake simulation mesh. Rigid case (left), soft case (right).

VRs x FEV (106) FES (106) Tt Ts CPUh

9 0/56 4.999 2.222 377/f0 10/f0 430

Table 1. Nacelle intake simulation properties. VRs denotes the number of variable mesh resolution levels.

As already mentioned, the properties of the APM are defined by using an analytical expression that relatesthe surface liner impedance to resistivity, porosity and thickness of the APM. Then, a genetic algorithm isused to minimize the difference between a given (target) impedance and the one corresponding to a prescribedset of APM parameters. Finally, an LBM simulation is performed for the optimal APM in order to educethe effective value of the impedance to be compared with the target one and thus verify the accuracy of thebest fit procedure. As an example, in this study we have fitted the impedance of the well-known CeramicTubular liner (CT57) measured by NASA.41 Fig.3 shows the comparison between the measured impedance,its analytical fitting, and the educed impedance using an APM with the optimal parameters and a simplenumerical setup similar to a Kundts tube. The comparison between the analytical fitting and the educedvalues is quite good, meaning that the use of the analytical impedance model is appropriate for a searchinvolving thousands of designs. More interestingly, the agreement between the educed impedance and thetarget experimental impedance is satisfactory, as resulting from a best fit over a wide frequency range. Theproperties of the equivalent APM made dimensionless by the acoustic wave properties are: R0= 11.16/f0,0 = 0.5742 andd0 = 0.1320.

The equivalent surface liner impedance corresponding to the reference frequency of the present study is

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Figure 3. CT73 liner impedance fitting with APM. Resistance (left), reactance (right). Experiments (red), APManalytical fitting (blue), LBM educed impedance (black).

extracted from the best fit curve plotted in Fig.3 and used to perform FEM computations in axi-symmetricmodality. The mesh used for the FEM simulation is the one required to run a case at a Helmholtz numberabout 12 times higher than the present one, thus guaranteeing a mesh independent reference solution. The

time-averaged flow solution used for the FEM computations is computed by setting slip wall conditions inthe same setup used to perform the LBM acoustic simulations. The free-stream Mach number for the caseswith flow is M= 0.2.

Fig.4 shows the comparison between the near-field acoustic pressure at zero cycle computed using LBMand the reference FEM solution for cases in the absence of convective effects (M= 0). The agreementbetween the two solutions is quite satisfactory. Discrepancies are mainly due to a lack of accuracy in thespace/time synthesis of the acoustic mode prescribed on the fan plane in the LBM simulation. Interestingly,the accuracy of the LBM solution is preserved up to the FW-H surface, meaning that the employed resolutionis adequate for this type of verification study. It is worthwhile to mention that two different visualizationsoftware were used to generate the images for the two solutions, and this can be also responsible for somediscrepancies.

Figure 4. Acoustic pressure. Comparison between LBM solution (top) and FEM solution (bottom). Rigid case (left),soft case (right).

More quantitative comparisons between LBM and FEM results, both without and with convective effects,

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are shown in Fig. 5, where the noise directivity for both hard and soft wall cases are plotted. Noise wascomputed using a FW-H extrapolations to microphones located along an arc of radius equal to 15 .60,centered in the fan plane midpoint and covering an angular sector from 15 deg (forward) to 165 deg (aft). In

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Figure 5. Nacelle intake noise directivity. Comparison between LBM and FEM results without (left) and with (right)convective effects.

the presence of mean flow, two sets of FEM results have been reported, corresponding to the GeneralizedImpedance Condition (GIC) of Eq. 29with blending factor = 1 and = 0, respectively. As discussed inmore detail in the Appendix, the first condition corresponds to neglecting any effect due to the presence ofa boundary layer in the FEM computation, whereas the second one corresponds to the well-known Myersimpedance boundary condition,42 which takes into account the effects due to a boundary layer on the acousticpropagation past an impedance wall under the assumption of zero-thickness boundary layer.

In the absence of mean flow, discrepancies occur only in the shallow radiation angles where noise isexpected to vanish and where the LBM solution is contaminated by background noise, which is inherent theLB unsteady solution and prevents to reach the round-off accuracy. A dynamic range of more than 40 dBis however predicted, and this is quite high for a non-linear CFD solution. Small differences can be alsoobserved in the forward interference lobes that might be due to a lack of resolution in the LBM simulation.Overall the agreement between FEM and LBM solution is good.

In the presence of mean flow, the agreement between LBM and FEM solution for the hard wall case is

again quite satisfactory. In the presence of a lined wall, the LBM solution is in better agreement with theFEM solution obtained by using a standard impedance condition un=p/Zcorresponding to a GIC = 1(Eq.28). Without entering in the debate of the most appropriate impedance boundary condition for a FEMsolution based on a linearized potential flow model, we can argue that, for the addressed acoustic frequency,using the Myers boundary condition in the FEM model yields to larger discrepancies between LBM and FEMresults. This trend is confirmed by the velocity and pressure fluctuations extracted along a line parallel tothe outer nacelle wall, at a distance of 4.47 102 Rf from the wall, which is in the asymptotic boundarylayer region. The comparisons between the LBM and the two FEM solutions are shown in Fig. 6.

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Figure 6. Pressure and velocity fluctuation magnitudes along the line y=4.47 102Rf parallel to the intake outer wall.Comparison between LBM and FEM results in the presence of mean flow.

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III.B. Exhaust fan noise radiation

This subsection is focused on the prediction of noise transmission through a turbofan bypass duct and thepropagation through the shear layer. The LBM solution is compared with frequency domain FEM solutionof a 2nd order wave equation for the acoustic potential,40 the same wave model used in subsectionIII.A,andto the solution of a 3rd order wave equation35 derived from the Goldstein-Lilley43 acoustic analogy for thelogarithmic pressure perturbation. It is important to mention that, both FEM solutions are not theoreticallysuited to address the present model problem: the acoustic potential equation does not allow to take into

account the Kutta condition at the bypass nozzle edge, whereas the Goldstein-Lilley equation is strictly validonly for a unidirectional transversely sheared mean flow, which is not the case of the present problem. Thecomparison between LBM and FEM results is therefore only made for the sake of completeness, keeping inmind that the LBM solution is the one without theoretical limitations.

The three-dimensional turbofan bypass geometry considered for both the LBM and FEM computationsis shown in Fig. 7(a). The LBM mesh is shown in Fig. 7(b). The mesh resolution in the jet shear layer issufficiently fine to resolve the time-averaged shear layer flow, but not enough to promote the occurrence oflarge scale hydrodynamic instabilities that would result in high jet noise contributions, thus making difficultto isolate the tonal noise radiation from the bypass exhaust. The acoustic mode (3, 1) is considered, witha magnitude corresponding to an overall inlet acoustic power of 140 dB. The acoustic mode is introducedin the LBM simulation through a time-varying pressure/velocity boundary condition on the fan plane.The choice of this mode is due to the fact that, as discussed in Ref., 44 this is the most energetic modefor the fundamental Blade Passing Frequency (BPF) for the ANCF fan rotor-stator configuration used

in subsection III.C. Moreover, since the turbofan configuration simulated in section IV was derived byinserting the ANCF fan system into a realistic nacelle whose exhaust profile is the same as the one used inthis subsection, a certain consistency among all cases can be preserved by considering the same dominantspinning mode as for the ANCF configuration at the 1 st BPF. The Helmholtz number, based on the fanplane outer radius, is kRf= 10.37.

(a) Geometry sketch (b) LBM mesh

Figure 7. Turbofan exhaust configuration.

The mean flow solution used for the FEM simulations is obtained by time-averaging the LBM solutioncomputed using the non-isothermal solver. LBM simulations are performed by prescribing a Mach numberof 0.3 both on the fan plane inlet boundary and on the core jet inlet boundary. The flow temperature onthe fan plane is equal to the free-field temperature, whereas a temperature ratio of 2.5 is prescribed on thecore jet inlet boundary. A quiescent free-field is assumed. Fig.8 shows the contour levels of the velocitymagnitude made dimensionless by the ambient speed of sound (acoustic Mach number) and the temperaturemade dimensionless by the ambient temperature (temperature ratio). Quantities useful to estimate thecomputational effort for the present case are reported in Table2.

Fig.9show contour levels of the real part of the Fourier component at the fan plane excitation frequency.A qualitative comparison is made with the two FEM solutions. Both the FEM solution of the Lilley-Goldstein equation and the LBM solution exhibit hydrodynamic perturbations in the shear layer. However,

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Figure 8. Turbofan exhaust LBM time-averaged flow solution: acoustic Mach number (left) and temperature ratio(right).

VRs x FEV (106) FES (106) Tt Ts CPUh

9 0/80 13.3 3.2 528/f0 10/f0 3563

Table 2. Nacelle exhaust simulation properties.

as expected for a linear wave model, the correctness of the perturbations taking place in the FEM solutionis quite questionable. The fluctuations levels in the shear layer for the LBM solution are significantlyhigher and cause a more pronounced interference pattern with the waves radiated from the bypass duct.Interestingly, the FEM solution of the Pierce equation exhibits a weak discontinuity of the wave fronts alonga direction almost normal to the axial direction, in correspondence of the bypass duct nozzle. Such a featureis a consequence of the fact that a potential wave model is not able to satisfy the proper condition for theacoustic pressure at the edge in the presence of a discontinuous mean flow about the edge.

Figure 9. Real part of the pressure fluctuation: FEM solution of Pierce equation (left), FEM solution of Lilley-Goldsteinequation (middle) and LBM solution (right).

A more quantitative comparison between LBM and FEM solutions is made in Fig. 10,where the noiseSound Pressure Level (SPL) directivity plots are shown. For all solutions, the SPL was computed by usingpressure fluctuations directly extracted from the near-field solution, without any integral extrapolation. Forthe LBM solution, a very narrow band integration around the tonal frequency was performed to filter out the

jet noise contribution. As depicted in Fig.7(a), the radiation arc is centered in the midpoint of the bypassduct outlet section and has a radius of 2.06 Rf. Interestingly, both for the LBM and the Lilley-GoldsteinFEM solutions, the sound pressure levels up to about 40 deg are dominated by the jet perturbations. Awayfrom the jet influence, the LBM and the Lilley-Goldstein FEM radiation pattern are in fairly good agreement,with the largest discrepancies of about 4 dB taking place in the upward radiation arc. Conversely, the PierceFEM solution is in fairly good agreement with the other solutions only in the angular sector between 45 degand 75 deg; furthermore, the upward radiation levels are underestimated by about 20 dB, and this is related

to the incorrect treatment of the edge condition in the presence of a discontinuous mean flow about the edge.This pitfall of an acoustic potential wave model is well known in the literature and inferred to the violationof the Kutta condition at the nozzle edge.45

Having stated the physical reliability of the fan noise radiation from a turbofan bypass duct in the presenceof a complex non-isothermal sheared mean flow, it is worthwhile to highlight some interesting feature of theLBM solution. Although the mesh resolution both in the primary and secondary jets is not fine enough toaccurately predict the jet noise contribution, which is indeed in the scope of subsection III.D, it is interestingto investigate the effects of the tonal excitation on the shear-layer hydrodynamic instabilities. Excitationlevels of 140 dB are strong enough to promote non-linear effects that can be investigated by computing the

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Figure 10. Exhaust fan noise directivity. Comparison b etween FEM and LBM solutions.

cross-bicoherence between a reference pressure signal (#1) extracted by a probe located in the laminar core ofthe secondary jet, and a pressure signal (#2) extracted at three angular locations along the noise directivity

arc depicted in Fig. 7(a),say 45 deg (downstream), 90 deg and 135 deg. The following formula was used tocompute the cross-bicoherence:

12(fi, fj) = Nw |

p1(fi) p2(fj) p

2(fi+fj)|2

p1(fi) p1(fi)

p2(fj) p2(fj)

p2(fi+fj) p2(fi+fj), (4)

where summations are carried out overNw =68 windows with an overlapping coefficient of 0.5 and by using aspectral bandwidth off0/10. Results are plotted in Fig.11as function of the harmonic counts. Interestingly,significantly high non-linear coherence levels can be observed for the microphone at 45 deg, which is morestrongly affected by the near-field noise induced by the shear-layer hydrodynamic wave packets. Non-linearcoherence can be also observed in the very low frequency range for the microphone at 90 deg.

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Figure 11. Cross-bicoherence 12 as a function of the first fi/f0 and second harmonic count fj/f0.

III.C. Fan noise generation and radiation

In this subsection we report some results of a validation study recently carried out by simulating the NASAGlenn ANCF36 fan configuration with PowerFLOW 5.0c. The simulation setup was derived from an existingone used in the past to perform pioneering fan noise simulations and analyses.44,46 Despite the low Machtip operating condition ( 0.33), which is not representative of a real turbofan engine, and the uncertaintiesassociated with some geometry detail in the experimental setup and to the installation of the so-called InletControl Device (ICD) around the nacelle intake, the ANCF constitutes a very useful benchmark experimentfor the validation of CFD/CAA tools.

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As shown in Fig.12,the full rotor/stator ANCF geometry, i.e. fan blades, guide vanes, hub and duct areincluded in the simulation. The main properties of the ANCF configuration are: a rotor diameter of 1.219 m,16 rotor blades, 13 guide vanes, 28 deg rotor pitch angle and stator located one stator chord downstream therotor. The rotor blades are fully embedded in a rotating mesh volume, as indicated in Fig.12.

Figure 12. Sketch of the ANCF computational setup.

Three main differences between the simulated geometry and the experimental configuration should bepointed out: no tip clearance is simulated, the outlet shaft system is not included in the simulation modeland the effects due to the presence of the ICD on both the flow field and the far-field noise are not modelled.The nominal operating condition is simulated, corresponding to 1800 rpm. The overall simulation size issummarized in Table.3.

VRs x[ m m] FEV (106) FES (106) Tt[ s] Ts[ s ] CPUh (103)

11 0.65 114.3 33.4 0.333 0.5 25

Table 3. ANCF simulation properties.

As depicted in Fig. 12, two permeable surfaces, which encompass the nacelle intake and exhaust, areused to perform far-field FW-H computations. These surface are fully enclosed in a mesh resolution levelVR8 (three coarsening levels with respect to the finest VR11 one) that guarantees a space/time resolutionof 20 points per acoustic wavelength and 35 samples per period, respectively, at the 7th BPF harmonic(3.36 kHz). Far-field noise was computed at 15 microphones located on a forward arc, which covers theangular sector [0 deg : 90 deg] (Mic. # 1 located at 0 deg), and at 15 microphones on an aft arc covering thesector [90 deg : 160deg]; the forward arc has a radius of 3.66 m and is centered in the midpoint of the intakesection, whereas the aft arc has a radius of 3.66 m and is centered in the midpoint of the exhaust section.

The time-averaged pressure and Mach number field solutions are shown in Fig. 13. The highest valuesof Mach number are achieved in proximity of the rotor blades. The flow compression induced by the fansystem is clearly evident in the pressure field. The flow field about the nacelle lip is very regular and doesnot reveal any flow separation.

Fig. 14 shows an instantaneous view of the time derivative pressure field, which is quite effective invisualizing the instantaneous acoustic waves. Very interestingly, a dominant wave front takes place in theacoustic field radiated from the nacelle intake, and the radiation direction seems to be related to the maintonal content (fundamental BPF). Conversely, the noise radiated from the nacelle exhaust exhibits at leasttwo dominant wave fronts, one associated with the fundamental BPF, the other one with 2nd BPF. This

is a clear diffraction effect from the sharp edge of the exhaust nozzle. Another interesting feature of theinstantaneous acoustic pressure field is the complex interference pattern taking place between the wavesradiated from the intake and exhaust openings.

Noise spectra at two angular positions are plotted in Fig. 15. Both for the measured and computedacoustic pressure signals, Fourier transforms were computed with a bandwidth equal to 30 Hz; spectra areplotted in terms of BPF harmonic count. Some discrepancies can be observed between numerical andexperimental results, both in the tonal and in the broadband spectral components. More precisely, thebroadband components exhibit a systematic underestimation ( 5 dB), which could be due to two reasons:a lack of mesh resolution in the rotor/stator region, resulting in more coherent turbulent structures in therotor wake, and the absence, in the simulation, of the rotor tip clearance, where interactions take place

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Figure 13. ANCF simulation: time-averaged pressure field (left) and Mach number (right). Front view on the bottomextracted from the rotating mesh, cutting through the rotor.

Figure 14. ANCF simulation: snapshot of pressure time derivative in one azimuthal plane.

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between the nacelle boundary layer turbulent structures and the rotor blades. It is important to mentionthat the ANCF is characterized by a predominant tonal character, with tones arising by about 30 40dBfrom the broadband levels, and that it is difficult to cover such a dynamic range in CFD simulations.

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Figure 15. ANCF simulation: narrow-band noise spectra at two angular locations: Mic. #11 (left) and Mic. #23(right). Comparison between measured and computed results.

Fig.16shows a comparison between measured and predicted noise directivity patterns for the first two

BPF harmonics and for the Overall SPL (OASPL) covering a broad frequency range. It is important tomention that, for the sake of computational efficiency, the noise signals along the forward and aft arcshave been computed by performing separate FW-H integrations on the two permeable surfaces depicted inFig.12,thus neglecting the interference effects between the intake and exhaust radiation that are expectedto have a non negligible effect only around 90 deg. The numerical predictions exhibit a systematic OASPLoverestimation that is due to an overestimation of the tonal peaks. This is confirmed by the BPF SPLdirectivity plots of Fig. 16. The overestimation of the tonal components can be also related to a lackof resolution in the rotor/startor region, which results in a too deterministic rotor-wake/stator-vanesinteraction. It is interesting to observe that the discrepancies between measurements and predictions arehigher in the forward direction ( 5 dB) compared to the aft direction ( 3 dB). This can be related to thepresence of the ICD in the experiments, which results in a noise transmission loss from the nacelle intake tothe microphones. Indeed, the measured forward noise levels are 2 3 dB lower than the measured aft levels.The reader is remanded to Ref.44 for other comparisons between measurements and predictions.

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Measurements - forward arcLBM simulation - forward arc

Measurements - aft arcLBM simulation - aft arc

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Figure 16. ANCF simulation: SPL noise directivity along forward and aft microphone arcs in different frequency bands.Comparison between measured and computed results.

III.D. Jet noise generation and radiation

The goal of this subsection is to validate the jet noise prediction capabilities of the high-speed non-isothermalLBM formulation implemented in a beta release version of PowerFLOW 5.1. The NASA Glenn researchcenter hot jet experiment37 performed by using the SMC000 nozzle was modelled for two flow conditions:setpoint 07, characterized by an acoustic Mach number Ma = Uj/c = 0.902 and a temperature ratioTr = Tj/T = 0.842 (cold jet), and setpoint 46, for which Ma = 0.901 and Tr = 2.702 (hot jet). Thecorresponding nominal centerline exit Mach numbers are 0.983 and 0.548, respectively.

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Fig. 17 illustrates the computational setup. A total pressure inlet boundary condition is imposed atthe nozzle inlet, with a prescribed value of the temperature computed from the experimental value at thenozzle exit centerline by using isentropic relationships. The external boundary of the nozzle is extended upto the far-field upstream boundary through a conic surface. The nozzle is embedded in a computationalanechoic environment consisting of three layers of acoustic sponge regions. The FW-H permeable surfaceis fully embedded in a VR10 mesh resolution level (two coarsening levels with respect to the finest VR12one), which has a grid cut-off frequency estimated around 20 kHz (20 voxels per wavelength). The cup of theFW-H surface is included in the sampled flow file, but it is excluded from the FW-H integration in order to

avoid the generation of fictitious noise fluctuations due to the strong vortical fluctuations passing throughthe surface. A view of the computational mesh is illustrated in Fig. 17, showing the different VR levelsaround the nozzle and in the jet plume. A VR11 offset region has been generated from all the internal wallsof the nozzle, whereas a VR12 (finest one) offset region has been generated from a small section of the nozzleclose to the exit. The overall simulation size is summarized in Table.4. Other details about the jet noisevalidation setup are reported in a companion paper.47

Figure 17. SMC000 simulation: computational setup (left) and mesh (right).


13 0.198 94.3 2.8 0.150 0.191 51

Table 4. SMC000 jet noise simulation properties.

Figs. 18and 19show time-averaged field results for setpoint 07 and 46, respectively. The streamwisevelocity standard deviation was computed by making an isotropic turbulence assumption for the smallunresolved turbulence scale, thus adding

2K/3 to the resolved fluctuation levels. All quantities, including

the velocity standard deviation exhibit a regular and symmetric pattern, and this is a qualitative indicationof simulation accuracy and statistical convergence.

Figure 18. SMC000 simulation: setpoint 07.37 Mach number (left), temperature ratio (middle) and dimensionlessstreamwise velocity standard deviation (right).

A quantitative analysis of the LBM solution accuracy is performed by comparing the time-averagedstreamwise velocity and its standard deviation levels levels along the jet centerline. Results made dimen-

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Figure 19. SMC000 simulation: setpoint 46.37 Mach number (left), temperature ratio (middle) and dimensionlessx-velocity standard deviation (right).

sionless by the exit centerline velocity Uj are plotted in Figs. 19 and 20 for setpoint 07 and 46, respectively.For both setpoints, the extension of the predicted laminar core length Xw and the turbulent mixing are invery good agreement with value estimated by using the formulas put forward by Witze48 and used in Ref.,49

i.e.:

Xw = 4.375 Dj(j/)

0.28

1 0.16 Mj, and

U

Uj= 1 e1,43/(1x/Xw). (5)

It is important to mention that no tuning of the turbulent levels prescribed at the inlet nozzle boundary

condition was performed to get the proper value of the laminar core length, which is a genuine resultof the simulations; moreover, no random forcing at the nozzle inlet was employed to seed the turbulencefluctuations in the nozzle boundary layer, which develops naturally along the nozzle walls, thus providing theproper boundary layer integral quantities at the nozzle exit. Therefore, the accuracy of the time-averagedcenterline velocity is a good indication of the proper bahavior of the turbulence model in the shear layer, aswell as in the wall region inside the nozzle. The streamwise velocity standard deviation velocity along the jetcenterline is compared with the curve plotted in Fig.(36) of Ref.49 The agreement is very good for setpoint46, whereas a significant overestimation of the fluctuation levels in the laminar core region is predictedfor setpoint 07. It should be mentioned that setpoint 07 is characterized by almost sonic conditions, thusconstituting a challenging test case for the high-speed D3Q19-based LBM flow solver. At the present stageof the research, it is not clear whether the overestimation of the fluctuation levels for setpoint 07 is due to thecomputational setup or to the fact that we approached the usage limits of the high-speed LB formulation.

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Figure 20. SMC000 simulation: setpoint 07.37 Dimensionless centerline time-averaged (left) and standard deviationx-velocity (right).

Finally, noise results are reported in Figs.22 and 23 for setpoint 07 and 46, respectively. The computedthird-octave SPL spectra at 90 deg and 135 deg (downstream) are compared with measurements by Brown& Bridges.37 The estimated grid cutoff frequency for the employed FW-H surface is about 20 kHz, and thisseems to be confirmed by the rapid drop-off of the noise levels around that frequency for both cases. Thelow-frequency part of the predicted noise spectra exhibit a certain lack of statistical convergence. The noisedirectivity for setpoint 07 is predicted within 1 dB accuracy up to 135 dB. For higher value (downstream)

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Figure 23. SMC000 simulation: setpoint 46.37 One-third octave band SPL (left) and OASPL directivity (right).

IV.A. Geometry and flow conditions

The geometry shown in Fig.24was designed by considering a generic shape for the nacelle with a realistic fullscale outer diameter of about 3.5 m and a length of about 5.2 m. The core nozzle and the bypass diameters

were chosen to be about 1.5m and 2.9 m, respectively. The fan and stator geometries are taken from theANCF simulations described previously in subsectionIII.Cwith a geometric scale factor of 2.3 to match theoverall geometrical size of the chosen turbofan.

Figure 24. Geometry of the complete turbofan demonstrator.

The simulations were conducted at a free-stream Mach number M = 0.1 and standard sea level atmo-spheric free-stream conditions for temperature, pressure, and molecular viscosity. The core jet was chosento be atMj = 0.424 and a temperature ratio of 2 compared to the ambient value. The fan RPM was chosenat around 1400 resulting in a nominal fan tip Mach number of about 0.6 and a BPF of 376.6Hz.

IV.B. Computational setup

The geometry was simulated in a free air condition with inflow/outflow BC located at 200 nacelle diametersaway. In addition, three sponge layers were included to damp out acoustic fluctuations and reflectionsfrom the far field boundary conditions. The rotating fan blades and hub were enclosed in a sliding trulyrotating mesh region, as described in subsection III.C. This demonstrator simulation was conducted at acomparatively coarse resolution in relation to the previous sections, but great care was taken in locallyresolving the fan blades, interaction regions between rotor and stator, as well as the two stream jet shearlayers. The automatically generated volume mesh is depicted in Fig.25and clearly highlights the regionsof local refinement and the extended far field fine resolution used to directly resolve the acoustic wave

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propagation with sufficient resolution up to the porous surface, where the flow data are collected for thesubsequent FW-H computations.

Figure 25. Volume mesh, near field (left) and far field (right) with every second line shown

The overall simulation size is summarized in Table. 5. The simulated physical time corresponds to 2 swith 0.2 s at the start used for the settling time for the flow after which the simulation data is used forstatistical analysis, far field propagation and further postprocessing. The history of the fan torque and the

pressure at a near field direct probe 25 nozzle diameters away in the downstream direction documented inFig.26 indicates that the choice of settling time is adequate for the statistical analysis. With the choice ofsimulation time and resolution the expected numerically resolved frequency range is from fmin 20Hz tofmax 1 kHz taking into account 20 spectral averages for the lowest frequency and 15 voxels to resolvethe propagation to the porous FW-H surface of the wave length at the high frequency.

Figure 26. Fan torque time history (left) and exemplary near field fluid probe pressure history (right)


14 2.98 168 10.5 0.2 1.8 54

Table 5. Turbofan simulation properties.

IV.C. Results and discussion

A qualitative view of the flow structures are depicted in Fig.27. The snapshot of the flow clearly highlightsthe small vortical structure emanating from the fan blade tips and their interactions with the stator aswell as the large and small structures generated in the highly unsteady region of the two stream jet shear

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layers. The large structures in the jets are quickly breaking up into smaller structures and indicate that thechosen resolution of the simulation was sufficient to capture the unsteady structures of the jet flow. This canalso be seen in the temperature field shown in Fig.28of the spreading of the hot core jet. Furthermore thedilatation field achieved by differentiating the pressure field over two subsequent unsteady time frames shownin Fig.28visually illustrates the sound propagation as well as the directivity of the acoustics in addition tothe hydrodynamic pressure fluctuations in the jet shear layers. The pressure field can be also filtered aroundthe BPF as illustrated in Fig. 29 to qualitatively visualize the shape of the pressure field at that specificfrequency and its dominant modal content. An azimuthal mode order equal to 3 is clearly visible.

Figure 27. Snapshot of2 iso-surfaces colored by velocity magnitude.

Figure 28. Snapshot of temperature field (left) and dilatation field (right).

A far field propagation using FW-H was carried out based on the data collected on the porous surfaceillustrated in Fig.30. The propagation is performed to several far field microphones located at 120 m distance,but only three located at 45 deg, 90 deg and 135 deg are discussed here as representative of forward, fly-overand backward directions. The FW-H surface is also subdivided into an intake region surrounding only theintake part of the turbofan and an exhaust region surrounding the complete exhaust and the large jet regionof the turbofan engine.

The total noise at the three microphone locations is documented in Fig.31. As expected the downstreamprobe at 135 deg has an increased level of broadband noise due to its proximity to the jet flow, whereas the

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Figure 29. Narrow bandpass filtered pressure around the1st BPF harmonic across the fan inlet face through the nacelle(left) and on the nacelle surface (right).

Figure 30. Permeable FW-H surface around the turbofan and far-field microphone locations.

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tonal noise contributions from the first 5 BPF harmonics are present at the expected frequencies in bothforward and backward microphone locations. The propagation of the tonal modes through the bypass flowat the exhaust part of the flow can also be seen in the Fig. 30.

Figure 31. Noise PSD at three far-field microphone locations.

It is interesting to investigate the breakdown of the contributions to the noise based only on the intakeand the exhaust FW-H surfaces as juxtaposed in Fig.32. The results follow the expectation that the intakepart is greatly contributing to the tonal parts of the noise, whereas the exhaust region is generally muchhigher in broadband contributions except for the backward microphone at 135 deg, where the contributionfrom the tonal part propagating through the nacelle is additionally observed. This comparison can bedone for each microphone separately as depicted in Fig. 33, again highlighting the same behavior for eachmicrophone. This type of analysis is very helpful to generally distinguish contributions and hence potentialnoise sources based on the geometrical region and type of noise generation mechanism in the far field. From

an engineering perspective, the presented tools are capable of not only predicting the noise generation of a fullturbofan engine, but also guide potential designers in better understanding of the different noise generationmechanisms and subsystem contributions.

Figure 32. Contribution break-down of the noise spectra at three microphones when using only the intake FW-H region(left) and the exhaust FW-H region (right).

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Figure 33. Comparison of intake and exhaust FW-H surface contributions to far field microphones at 45deg (left),90 deg (middle) and 135 deg (right).

V. Conclusion

A comprehensive verification and validation study of PowerFLOW CFD/CAA solver capabilities in thefield of turbofan noise prediction was carried out. Emphasis was given to the overall view of the problem,and not on the specific details of the different elementary case studies. The usage envelope of the LBM inthis new application field was outlined, highlighting both potentialities and areas of further improvement.The main outcomes of this effort are summarized hereafter.

- Fan noise transmission through and radiation from the nacelle intake and bypass duct, also in the presenceof an acoustic liner, can be accurately predicted. For the intake radiation problem, other techniques basedon the solution of simple wave models in the frequency-domain are certainly more convenient. However,whenever geometrical three-dimensionality and nacelle installation effects are in the scope of the analysis,integrated engine/airframe LBM simulations are one of the few exploitable computational technologies. Forthe bypass duct radiation problem, linear wave models have limited usage potentialities and the solutionof the full non-linear flow governing equations or, more continently, LBM simulations, are definitively morereliable.- The effect of noise absorption due to an acoustic treatment in the presence of grazing flow can be modelledby means of a specifically tuned APM. A primary validation of this new capability in PowerFLOW wasperformed. However, further studies are required to assess the capability of reproducing the absorptionproperties of a wider range of acoustic treatments, including honeycomb liners, as well as to understand therelationship between boundary layer properties and acoustic properties of the porous material. Space-/time-resolved flow/noise measurements are required to accomplish this goal.- Tonal and broadband fan noise generation can be predicted by LBM. However, the outlined uncertaintyrange of 3-to-5 dB compared to experimental results for the addressed NASA Glenn ANCF configuration,is not fully satisfactory. Further work is required to reduce this uncertainty range, by improving the com-putational setup on the simulation side, and by quantifying the uncertainties related to the test rig and therequired corrections on the experimental side.- Jet noise can be predicted by the non-isothermal version of the LBM up to almost sonic conditions, bytaking into account the primary jet heating effects. Further work is required to optimize the computationalsetup and reduce the simulation efforts for a prescribed accuracy level. Areas for further improvement werefound in the low-frequency range and in the downstream radiation; longer physical simulation times will betested in the future, as well as the inclusion of the permeable surface cup in the FW-H integration.A qualitative demonstration of PowerFLOW integrated fan/jet noise prediction capabilities was finally

achieved and constitutes a primary milestone towards future lattice-Boltzmann prediction of turbofan enginenoise.

Acknowledgments

The authors are grateful to James Bridges and to Daniel L. Sutliff at NASA Glenn Research Center forproviding the SMC000 nozzle geometry and the detailed ANCF experimental data needed for the validation.

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for Sound Propagation in Lined Flow Ducts, Journal of Fluid Mechanics , Vol. 437, 2001, pp. 367384.54Astley, R. J. and Gamallo, P., Partition of Unity Finite Element Method for Short Wave Acoustic Propagation on

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Brambley, E. J., Fundamental Problems with the Model of Uniform Flow over Acoustic Linings, Journal of Sound andVibration, Vol. 322, No. 4-5, 2009, pp. 10261037.

62Brambley, E. J., A Well-posed Modified Myers Boundary Condition, AIAA Paper 2010-3942, June 2010.63Rienstra, S. W. and Darau, M., Boundary-Layer Thickness Effects of the Hydrodynamic Instability Along an Impedance

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3241 , May 2009.66Renou, Y. and Auregan, Y., Failure of the IngardMyers Boundary Condition for a Lined Duct: An Experimental

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Appendix: generalized impedance boundary conditionIn this appendix the problem of an impedance boundary condition for an acoustic velocity potential

wave model in the presence of grazing flow is reviewed with the intent of highlighting the main assumptionsbehind the two boundary conditions employed for the FEM computations of sound transmission through anacoustically treated nacelle intake discussed in subsectionIII.A.For the sake of generality, a vibrating surfaceis considered in the following analysis, although only non vibrating surfaces were considered in present paper.

Historical note

During the Seventies, some discussion and controversy arose in the literature about the proper acousticboundary condition to be applied on a generic vibrating surface in the presence of a grazing flow. Thefirst formulation for an untreated vibrating surface was first suggested by Taylor, 50 and it was successivelyderived in a formal way by Myers.42 In the same paper, Myers also extended the formulation to the case ofan impedance steady surface. Myers Impedance Condition (MIC) can be seen as an extension of the wellknown Ingards condition51 that is valid for a flat plate in a uniform stream. The MIC has been widelyaccepted and used in a large number of numerical and analytical works. 52, 53,54,55,56, 33,57,34 In the last yearssome problems arose, in particular with time-domain numerical solvers using the MIC, that evidenced how,in the presence of impedance, not only the numerical solution diverges due to a numerical instability, but alsothe boundary condition itself is ill-posed and intrinsically unstable.58,59,60, 61 These results, however, seem tobe in contrast with the experiments, since evidence of this instability has been reported only rarely. Severalefforts have been therefore undertaken in order to eliminate or mitigate the instability, and the attention hasbeen recently focused in the direction of adding more physicsto the phenomenon, thus deriving modifiedboundary conditions that better reproduce the sound propagation in a very thin layer close to the surface. 62, 63

These attempts have succeeded in showing that in some case the boundary condition can become well-posedand stable.

In 2001 Aureganet al64

and more recently Brambley65

have argued that the viscous and turbulent effectsnear the wall can alter the MIC. In particular, they have demonstrated that the continuity of the normalparticle displacement across the vortex sheet, which is used to model the effect of the boundary layer in anotherwise inviscid medium, holds only when the thickness of the acoustic boundary layer is much smallerthan the thickness of the stationary boundary layer, i.e., typically in the high frequency range. At very lowfrequencies, continuity of the acoustic velocity normal across the vortex sheet must be applied instead. Veryrecently, Renou & Auregan66 have conducted some measurements in a duct with the aim of proving thefailure of the MIC and validate a modified model in which a factor v, varying from 0 to 1 and reproducingthe effect of transfer of momentum into the lined wall induced by molecular and turbulent viscosities, allowsto blend the MIC (v =0) and a condition reproducing the continuity of the acoustic velocity normal to thewall (v =1).

In the present Appendix the mathematical formalism introduced by Myers is used to derive a GeneralizedImpedance Condition (GIC) for a vibrating surface. In a first step the concept of acoustic impedance for

a vibrating surface is discussed, starting from the consolidated definition adopted in the vibro-acousticliterature that expresses the surface impedance as the ratio between the pressure and the relative velocitybetween fluid and surface. In a second step, two forms of the GIC are derived, depending on the hypothesismade to model the velocity of the vortex sheet. Finally, following Auregan et al,64 a blending formula isproposed in order to incorporate the two formulations into a unique GIC.

Impedance definition for a vibrating surface

The impedance definition that we are considering in the present work is the so called normal surfaceimpedance or boundary impedance that, in the hypothesis of locally reacting material, can fully charac-

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terize the absorbing properties of a liner placed over a solid surface. It is worthwhile to mention that thisimpedance is just a macroscopic quantity that can be used to obtain a macroscopic description of the complexfluid/structure interactions taking place at microscopic level in the sound absorbing material. The situationis illustrated in Fig. 34, where a porous liner is placed over a solid plate in the presence of a grazing flowand an incident acoustic field. It is evident that, at the microscopic level, the acoustic field at the interfacebetween the liner and the fluid greatly change locally, but once that one averages the pressure and thevelocity over a sufficiently large region some averaged quantities can be obtained that describe the effect ofthe liner at a macroscopic level. In the following we will focus on these macroscopic fluid quantities and we

will try to identify a proper definition of the surface impedance in terms of these macroscopic quantities forthe case of a liner placed on a vibrating surface.

Flow

Sound wave

U(y)

pavun av

pun

Figure 34. Macroscopic versus microscopic flow behaviour at the interface between fluid and a sound absorbing materialplaced on a solid plate.

Looking at the literature we find substantially two definitions of the boundary impedance for an acous-tically treated surface. The first one that we denote as Za can be found in the acoustic and aeroacousticliterature,51,67,42 and defines the impedance as the ration between the acoustic pressure p and the acousticnormal velocity un:

Za = p/un, (6)

where un is the averaged acoustic velocity at the liner interface projected in the surface normal direction.Throughout, fluctuating quantities (fluid pressure and velocity, surface velocity) are represented by theirFourier transform counterpart denoted with the symbol. The second definition, denotes as Z

vand typically

used in the vibro-acoustic domain,68,69 states that the impedance is defined as the ratio between the acousticpressure p and the relative normal velocity ur of the fluid with respect to the surface:

Zv = p/ur. (7)

The surface velocity projected in the surface normal direction is denoted as Vn, as sketched in Fig.35. Noticethat in both the above impedance definitions the minus sign is due to the fact that we have used the typicalconvention for Zaccording to which the velocity is positive when entering the liner, while the unit normalvector points from the surface into the fluid. Obviously Za andZv are coincident when the surface is at rest

Vn un

Vibrating support surface

Vibrating

liner surface

n

Figure 35. Illustration of a liner placed on a vibrating surface; Vn is the velocity of the liner surface, while un is theacoustic velocity at the fluid/surface interface.

(Vn =0), but when the surface is vibrating the two expression are substantially different, and so we have totry to identify the most appropriate definition to be used. In order to do that, it is useful to focus on theexpected properties for the impedance. It is clear that one important property would be that the impedance

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of a given surface does not change with the surface motion. Let us look at Za and Zv for an untreatedsurface for which we expect to have Z. For an untreated vibrating surface we have ur= 0 and so weget Zv , but for Za instead we get Za and in addition Za is a function of the surface velocity.The definition in terms of the relative fluid/structural velocity seems therefore preferable for our scopes.This is further reinforced by considering other aspects, such as the physical mechanism of damping throughthe liner orifices that is driven by the relative velocity, and also the classical coupling of dynamical systemswhere the relative velocity is used.70 It can be therefore concluded that the more convenient definition forthe surface impedance is based on the relative velocity between fluid and surface and we will therefore adopt

this definition in the following.

Expression of the relative velocity normal to a vibrating surface

In this section the expression of the relative velocity ur between fluid and a vibrating surface is derived. Theexpressions is valid for a generic material surface, both solid or fluid (e.g. a vortex sheet).

Let us consider a generic curved surface defined by the equation f(x, t) = 0. The unit normal vector tothe surface is given by:

n= f/|f|, (8)

while the surface velocity V is such that df/ dt=0, i.e.:

f

t + V f=

f

t + |f|V n= 0, (9)

from which the normal surface velocity Vn can be derived, i.e.:

Vn= 1

|f|

f

t. (10)

Following Myers,42 the surfacef=0 can be modeled as a time-averaged shape(x)=0 with superimposedthe unsteady perturbationg(x, t), i.e. f=g, where the function g is of the same order as and 1.Hence the moving surface is defined by the equation (x) =g(x, t). Considering the Fourier componentwith angular frequency (+it convention) of the surface motion and using Eq. 10, we obtain:

Vn= ig

||+ O

2. (11)

It is also useful to derive an expression for the unsteady unit normal vector, which reads:

n(x, t) = g

|| +O

2

=

||

g eit

|| +O

2

; (12)

hence, making use of Eq.11and introducing the unit normal vector to the time-averaged surface n0, we canwrite:

n(x, t) = n0(x) Vn0(x)

i eit +O

2

on = g. (13)

Let us now suppose that the unsteady velocity field can be decomposed into a time-averaged field U(x)and a superimposed acoustic perturbation u(x, t), such that |u| |U|. Then, let us introduce the relativefluid/surface velocity projected on the normal direction and defined on the deformed surface = g, whichreads:

ur(x, t) = [u(x, t) + U(x) V(x, t)] n(x, t)

=

u(x) eit + U(x) V(x) eit

n0(x)

Vn0(x)

i eit +O

2

(14)

= u(x) n0(x) eit + U(x) n0(x) U(x)

Vn0(x)

i eit V(x) n0(x) e

it +O2.

By applying the perturbation analysis adopted by Myers,42 the relative velocity ur on the moving surfacecan be computed through a Taylor expansion applied to the time-averaged surface, i.e.:

ur(x, t) |=g=ur(x, t) |=0+gur(x, t)

|=0+O

2. (15)

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Hence, making use of Eq. 14,we can write:

ur(x, t) |=0 = u(x) n0(x) eit + U0(x) n0(x)U0(x)

Vn0(x)

i eit V(x) n0(x) e

it + O2, (16)

and:

gur

|=0 = g

||

urn|=0

= g|| (U0 n0) n0+O

2

(17)

=Vn0i

(U0 n0) n0eit +O

2,

where use of Eq. 11 has been made, only the zero-th order term in Eq. 16 has been retained (ur(x, t) |=0 =U0(x) n0(x) +O()), and the hypothesis of smoothly curved surface has been introduced, i.e.:

ur = [U0(x) n0(x)] +O() [U0(x)] n0(x) +O() . (18)

Finally, substituting Eqs.16 and 17 into Eq.15,and making use of the slip condition U0 n0 =0, we obtainthe expression of the harmonic component of the relative velocity on the deformed surface:

ur = un0 U0 Vn0

i

+Vn0

i

(U0 n0) n0 Vn0+ O2 (19)Impedance boundary condition for a vibrating surface

In this subsection, a GIC for a vibrating surface is derived, with reference to the conceptual scheme depictedin Fig. 36. The effect of the boundary layer is modeled through a vortex sheet that supports an idealizeddiscontinuous time-averaged flow that below the vortex sheet is forced to be at rest (U =0). The liner andvortex-sheet normal velocities are indicated as Vn and V

In , respectively.

liner

surfac

e

vortex

-sheet

VI

n

Vn

+

-

U=0

U 0

(x)=0

I(x)=0

nI

0

n0

Figure 36. Conceptual scheme for a moving surface in the presence of a discontinuous time-averaged velocity field Uand a fluctuating vortex sheet.

The derivation proceeds with the definition of the relative normal velocity in the different layers of thedomain indicated in Fig.36. Immediately above the liner surface, Eq. 19 yields:

uwr = uwn0 Vn0 . (20)

Hence, making use of the surface impedance definition, as given in Eq. 7,we obtain:

pw

Z = un0 +

Vn0 . (21)

Immediately below and above the vortex sheet, Eq. 19 yields:

ur = u

n0 VIn0 (22)

u+r = u+n0

VIn0 U0 VIn0i

+VIn0i

U0 n

I0

nI0, (23)

where U0 is the time-averaged velocity immediately above the vortex sheet. Since the relative velocityur iscontinuous across the vortex sheet, by subtraction of Eqs.22 and 23 we obtain:

u+n0 = u

n0 + U0 VIn0i

VIn0i

U0 n

I0

nI0. (24)

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Finally, making use of the hypothesis of continuity of pressure across the vortex sheet (pw = p = p+), andby setting, for simplicity of notation, p+ p and u+n0 un0 , the following condition relating the inviscidacoustic normal

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