facing rim cavities fluctuation modes

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Journal of Sound and Vibration

Journal of Sound and Vibration 333 (2014) 2812–2830

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n CorrE-m

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Facing rim cavities fluctuation modes

Damiano Casalino n, André F.P. Ribeiro, Ehab FaresExa GmbH, Curiestrasse 4, Stuttgart 70563, Germany

a r t i c l e i n f o

Article history:Received 5 August 2013Received in revised form19 January 2014Accepted 29 January 2014
Handling Editor: J. Astley
a coupling between shear-layer vortical fluctuations and acoustic modes resulting from

Available online 4 March 2014

x.doi.org/10.1016/j.jsv.2014.01.0280X & 2014 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ49 711 68 70 32 4ail address: [email protected] (D. Casalino).

a b s t r a c t

Cavity modes taking place in the rims of two opposite wheels are investigated throughLattice-Boltzmann CFD simulations. Based on previous observations carried out by theauthors during the BANC-II/LAGOON landing gear aeroacoustic study, a resonance modecan take place in the volume between the wheels of a two-wheel landing gear, involving

the combination of round cavity modes and wheel-to-wheel transversal acoustic modes.As a result, side force fluctuations and tonal noise side radiation take place. A parametricstudy of the cavity mode properties is carried out in the present work by varying thedistance between the wheels. Moreover, the effects due to the presence of the axle areinvestigated by removing the axle from the two-wheel assembly. The azimuthal proper-ties of the modes are scrutinized by filtering the unsteady flow in narrow bands aroundthe tonal frequencies and investigating the azimuthal structure of the filtered fluctuationmodes. Estimation of the tone frequencies with an ad hoc proposed analytical formulaconfirms the observed modal properties of the filtered unsteady flow solutions. Thepresent study constitutes a primary step in the description of facing rim cavity modes asa possible source of landing gear tonal noise.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The landing-gear and the flap side-edge are major sources of airframe noise of a civil aircraft in approach conditions.Reducing the acoustic efficiency of these two sources is a very difficult task and requires a detailed knowledge of theunsteady flow mechanics and how these are affected by design variations. Since it is not uncommon that low-noise conceptsderived from a purely heuristic intuition can lead to noisier designs, the use of numerical simulations to rank conceptsduring low technology readiness level design stages is becoming a common practice to reduce the development time andcosts of new low-noise concepts, like side-edge treatments [1,2] and landing-gear fairings [3]. Among other approaches, theLattice-Boltzmann Method (LBM) implemented in the software PowerFLOW provides a satisfactory combination ofreliability, simulation time and usage flexibility. Best practices have been developed and improved in the past years fora wide class of aeroacoustic problems [1,4–7]. In the field of airframe noise prediction in particular, important efforts havebeen focused on the prediction of noise from deployed landing gears [8,9] and, more recently, High Lift Devices [10,11,1](HLD). These two categories of problems face different challenges. The capability to manage highly complex geometries andto resolve one- and two-point statistics in the turbulent separated region are the driving requirements for accurate landing-gear noise and unsteady load predictions. Both requirements constitute competitive advantages of LBM compared to standard

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Nomenclature

c sound speedD wheel diameterf particle distribution functioni transversal cavity mode orderk modelled turbulent kinetic energyK turbulent kinetic energy (resolved þ modelled):

ðu2stdþv2stdþw2

stdÞ=2þklEE edge-to-edge rim distancelFF floor-to-floor rim distancem azimuthal cavity mode orderM Mach numbern Rossiter mode orderna number of acoustic wavelengths in Rossiter

loopnv number of vortical wavelengths in Rossiter

loopp pressurer rim cavity outer radiusRuu normalized two-point correlation of the

streamwise velocity fluctuationsRe Reynolds numberu flow velocityu; v;w flow velocity components

U1 free-stream velocityv particle velocityyþ distance from the wall in wall units: y

ffiffiffiffiffiffiffiffiffiffiτw=ρ

p=ν

⟨ � ⟩ time-averaged flow quantities

Symbols

Δt time stepΔx voxel sizeϵ turbulent dissipation rateλ annular duct mode eigenvalueμ dynamic fluid viscosityν kinematic fluid viscosityρ densityτw wall shear stress

Subscripts

E cavity edgeF cavity floorj discrete particle velocity index1 free-stream conditionsstd standard deviation

D. Casalino et al. / Journal of Sound and Vibration 333 (2014) 2812–2830 2813

CFD methods. The capability to model boundary-layer dominated flows in transitional regimes, and to resolve the shear-layerturbulent fluctuations in the slat cove and vortex rolling past the flap side edges, is the main challenges for accurate HLD noisepredictions through methods based on isotropically wall-refined meshes and wall function treatments. Notable progresses inthe readiness level of LBM for HLD aerodynamics have been demonstrated in the last years in the framework of the 1st and2nd AIAA CFD High Lift Prediction Workshop [12,13]. The main outcome of these efforts is that, in the case of clean wingconfigurations, the accuracy of LBM is comparable with that of standard CFD methods, provided that the location of theboundary-layer transition on the wing is accurately prescribed; in the case of realistic wing configurations, including bracketsand track systems, the LBM is in general more reliable than standard CFD methods. The knowledge acquired by Exa in the fieldof HLD aerodynamics is continuously translated into HLD noise best practices. However, different readiness levels have beenachieved by the landing-gear and HLD noise best practices, not only because of the different challenges, but also because of theseveral uncertainties that affect the available HLD experimental data sets, which make the method validation process difficult.

Recently, the landing-gear best practice setup has been applied to the LAGOON configuration [14,15] in the framework ofthe second workshop on Benchmark problems for Airframe Noise Computations (BANC-II) [16]. The outcome of thisbenchmark study is summarized in a recent paper by the authors [17]. One of the interesting findings of the LAGOON studywas the prediction of tonal peaks away from the flyover plane, which are due to acoustic modes taking place in the circularrim cavities of the two facing wheels. More precisely, the shear-layer instability modes are coupled with acoustic modes,which are responsible for side noise radiation and side force oscillations. The aim of the present work is to investigate inmore detail the structure of these rim-cavity modes and how these are affected by the presence of the axle and by thedistance between the wheels. Simulations with and without axle allow to understand whether a Kármán vortex sheddingplays any role in the generation of the sound tones; whereas simulations carried out for different wheel distances clarify therelationship between the frequencies of the tones and the structure of the fluctuation modes. In order to better highlight thecavity modal behaviour, all the structural components have been removed from the LAGOON configuration and the axle hasbeen replaced by a cylinder of equivalent diameter.

The generation of tones from flows past rectangular cavities has been the object of many experimental studies since1950s [18]. Rossiter [19] was the first to explain the occurrence of tones as a consequence of a phase compatibility betweenshear-layer vortical instabilities and the acoustic waves generated by the impact of these structures on the downstreamedge. The back-radiated waves enforce the generation of shear-layer fluctuations, thus resulting in a self-sustained feedbackloop mechanism, similar to the edge-tone [20] and to the screech tone [21] phenomena. Among several minor modificationsof the original Rossiter's model, the idea put-forward by Block [22,23] of including the depth cavity mode in the feedbackloop process was crucial to predict the tonal frequencies of cavities with high depth-to-length ratios.

Tones from cylindrical cavities of high depth-to-diameter ratios like organ pipes and side branch pipes in hydraulic systemshave also been explained as a coupling between shear-layer instabilities and a depth-mode excited at resonance [24].

D. Casalino et al. / Journal of Sound and Vibration 333 (2014) 2812–28302814

More recently, cylindrical cavities of depth-to-diameter ratios of the order of one have been the object of both experimental[25] and numerical [26,27] studies. Marsden et al. [25], in particular, performed an exhaustive experimental and theoreticalanalysis for cavities of depth-to-diameter ratios from 1 to 1.5 and velocities from 50 to 110 m s�1. They showed that the tonesare associated with cylindrical pipe modes resulting from the combination of the first depth mode (wavelength slightly largerthan a quarter of the cavity depth) with the first three azimuthal modes, i.e. expðimθÞ with (m¼0,1,2), and the first radial mode(zero nodes along the radius). In the present study we consider two facing shallower cavities of depth-to-diameter ratio � 0:23and a grazing flow velocity �79 m s�1, with and without a coaxial axle, and we observe that the noise tones are associatedwith acoustic modes that result from the combination of azimuthal/radial modes and two transversal modes that play thesame role as the depth mode for a single cavity configuration. The acoustic modes are excited at resonance by shear-layerinstabilities, but a Rossiter-type retroaction mechanism can also concur to the tone selection and strengthening.

The paper is organized in the following way. The numerical method is briefly presented in Section 2. Section 3 describesthe different wheel assembly geometries employed in the present study and the setup of the LBM simulation. The mainnear- and far-field results are reported in Section 4, with emphasis given to the modes and how they are affected by thepresence of the axle and by variation of the distance between the wheels. The main finding of the present effort anddirections for future investigations are finally summarized in Section 5.

2. Numerical method

A hybrid LBM/Ffowcs-Williams & Hawkings (FW-H) approach [28] is used in this work to compute the unsteady flow andthe resulting noise for a set of facing wheels configurations. Although the focus of the present work is on the investigation ofthe rim cavity modes and not on the numerical methodology, the employed method is briefly described in this section forthe sake of completeness.

2.1. Flow solver

The software PowerFLOW 4.4d is used in the present study. The physical core of the software solves the Boltzmannequation for the distribution function f ðx; t; vÞ on a hexahedral mesh automatically generated around bodies consisting ofone or more connected solid parts. The function f represents the probability to find, in the elementary volume dx around xand in the infinitesimal time interval ðt; tþdtÞ, a number of fluid particles with velocity in the interval ðv; vþdvÞ. TheBoltzmann equation is solved by discretizing the particle velocity space into a prescribed number of values, in magnitudeand direction. These discrete velocity vectors are such that, in a prescribed time step, one particle is advected from one pointof the mesh to 19 neighbouring points, including the point itself, which constitute the computational stencil of the so-calledD3Q19 scheme (three-dimensional 19 states model). It can be demonstrated that using 19 particle velocity states ensuressufficient lattice symmetry to recover the Navier–Stokes equations for an isentropic flow [29].

Once the distribution function is computed, the macroscopic flow quantities, density and linear momentum, are simplydetermined through discrete integration: ρðx; tÞ ¼∑jf jðx; tÞ and ρuðx; tÞ ¼∑jf jðx; tÞvj. All the other quantities are determinedthrough thermodynamic relationships for an ideal gas.

Solving the lattice Boltzmann equation is equivalent to performing a Direct Numerical Simulation (DNS) of the Navier–Stokes equations in the limits of the dynamic range (Mach number) that can be accurately covered by the number of discreteparticle velocity vectors, and in the limits of the lattice resolution required to capture the smallest scales of turbulence. Forhigh Reynolds number flows, turbulence modelling can be incorporated into the LBM scheme by changing the relaxationtime in the collision operator that is computed according to a Bhatnagar–Gross–Krook (BGK) approximation [29]. Theturbulent kinetic energy and the turbulent dissipation are obtained by solving a variant of the RNG k–ϵ model for theunresolved scales [30]. This approach is referred to as LBM Very Large Eddy Simulation (LBM-VLES).

Since it is prohibitive to resolve the wall boundary layer using a Cartesian mesh approach down to the viscous sub-layerin high Reynolds number applications, a wall function approach is used in PowerFLOW to model boundary layers on solidsurfaces. The wall function model is an extension of the standard formulation [31], but it includes the effects of favourableand adverse pressure gradients, and accounts for surface roughness through a length parameter [12].

The LBM scheme is solved on a grid composed of cubic volumetric elements (voxels). A variable resolution by a factor twois allowed between adjacent regions. Consistently, the time step is varied by a factor two between two adjacent resolutionregions. Solid surfaces are automatically facetized within each voxel intersecting the wall geometry using planar surfaceelements (surfels). For the no-slip and slip wall boundary conditions at each of these elements, a boundary scheme [32]is implemented, based on a particle bounce-back process and a specular reflection process, respectively. Therefore verycomplex arbitrary geometries can be treated automatically by the LBM solver.

The local character of the LBM scheme allows an efficient parallelization of the solver. Moreover, thanks to the intrinsiclow dissipative properties of the scheme [33], LBM is particularly suited for aeroacoustic simulations and, in particular, forairframe noise applications [4,5,9,8].


2.2. Noise propagation solver

The far-field noise is computed through an integral extrapolation based on a solid FW-H acoustic analogy formulation.A forward-time solution [34] of the FW-H equation based on Farassat's formulation 1A [35], generalized to a permeableintegration surface [36], is used. The free-stream convective effects are taken into account directly in the integralformulation, as described in Refs. [37,28]. The convective permeable FW-H formulation developed by Exa has been appliedto investigate the wind-tunnel installation effects in a tandem cylinder configuration [5]. In the present study, nonlinearvolume sources are neglected and integrations are carried out on the solid surface of the wheel assembly. This approach isusually preferred for landing-gear noise computations for two reasons: the space–time resolution on the body surface ishigher than in the field, thus allowing higher frequency noise predictions, and no spurious effects are introduced due tostrong coherent turbulent fluctuations convected through a permeable integration surface.

The FW-H solver is embedded in the post-processing tool PowerACOUSTICS 2.0c, which is also used to perform statisticaland spectral analyses of all the solutions generated by PowerFLOW (volume fields, surface fields, and probe signals).

3. Geometry and computational setup

The two-wheel configuration extracted from the LAGOON gear model and the PowerFLOW computational setup aredescribed in this section.

3.1. Wheel assembly

The wheel assembly geometry is sketched in Fig. 1. Fig. 1(a) shows line cuts of the two-wheel assembly for three values ofthe distance between the wheels. The shape of the wheels is exactly the same as the LAGOON configuration [14,15]: thewheel diameter is 0:3 m, the rim cavity diameter is 0:162 m, the axle diameter is 0:044 m and the rim cavity depth is0:037 m, corresponding to a depth-to-diameter ratio of about 0.23. In order to understand whether a Kármán vortexshedding from the axle plays any role in the rim cavity mode behaviour, the original axle of the LAGOON gear has beenreplaced by a cylinder of the same diameter. A shaded view of the assembly is shown in Fig. 1(b). One of the wheels isoutlined in order to show the rim cavity. The wheels and the axle are imported as separate solid entities by PowerFLOW andthis is the reason why the actual axial extension of the axle is greater than the distance between the floor of the twoopposite rim cavities. In other words, the geometry used by PowerFLOW to generate the volume mesh can be composed bydifferent overlapping parts, thus resulting in a high user flexibility.

Six configurations are considered in this study, resulting from the combination of three values of the wheel distance andthe inclusion/exclusion of the axle in the simulated model. The list of the configurations is reported in Table 1 where themain geometrical parameters affecting the rim cavity modes, say the floor-to-floor lFF and edge-to-edge lEE rim cavitydistances, are reported; Conf. R-Y is equivalent to the original LAGOON geometry. For the sake of clarity, the six cases arenamed using a combination of two letters, one for the rim distance, namely Reference, Wide or Narrow, and one for thepresence of the axle, Yes or No.

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-200 -150 -100 -50 0 50 100 150 200

y [m

m]

x [mm]

Conf. R-Y - shift 0 mm (LAGOON)Conf. N-Y - shift -10 mmConf. W-Y - shift +10 mm

Fig. 1. Wheel assembly for configurations with axle (Conf. R-Y, N-Y and W-Y in Table 1). (a) Line cut in the plane z¼ 0. (b) Shaded view and outline showingthe rim cavity.

Table 1Simulation matrix.

Conf. # Axle Shift (mm) lFF (mm) lEE (mm)

R-Y/N Yes/no 0 200 125.5N-Y/N Yes/no �10 180 105.5W-Y/N Yes/no 10 220 145.5

Table 2Inlet/free-stream conditions; all quantities in SI units.

p1 ρ1 U1 Turb. length (m) Turb. intensity (%) M1 ReD ð106Þ μ ð10�5Þ

99 447.7 1.18036 78.9917 5�10�3 0.3 0.23 1.5409 1.5379

Fig. 2. Transition triggering devices; experimental setup (a), numerical setup (b).


3.2. PowerFLOW setup and flow conditions

The computational setup used in this study is similar to the one used for the LAGOON gear Ref. [17]. The computationaldomain extends over 80:75 m along the x (streamwise) direction, 52:67 m along the y (side, cross-flow) direction and56:17 m along the z direction. The resulting computational mesh is perfectly symmetric along the three Cartesian directionswith respect to the axle mid-point.

The computational free-stream conditions are listed in Table 2; the fluid viscosity is evaluated using the Sutherland law.No-slip wall boundary conditions are imposed on the wheels and the axle. In order to account for the presence of

transition triggering devices in the LAGOON experiments, patches of rough walls have been used in Ref. [4]. These patchesare delimited by the actual location of the strips in the experiments and their effect is to produce a faster boundary layerthickening and therefore reproduce the effect of the strips in the experiments. The same setup is used in this work, assketched in Fig. 2, in order to generate results that can be compared with the LAGOON measurements, thus allowing tohighlight the effects due to the structural components removed from the present configuration. Slip wall conditions areimposed on the four far-field side walls, and a pressure-based outlet condition is imposed at the far-field outlet boundary.

Fig. 2 also shows pressure probes on the surface of the numerical model distributed along four circles on the two wheelsevery 101, as well as two disks and one rectangle used to extract the flow solution for analysis purposes.

Based on the resolution study carried out in Ref. [17], a medium resolution is considered adequate to the goal of thepresent study. The frequency of the rim cavity tones, in fact, did not exhibit a significant dependency on the mesh resolution.For the first case in Table 1, the number of volume (voxels) and surface elements (surfels) is 37.8 and 3 million, respectively,the total distributed memory required is 67 GB, and the minimum voxel size in VR9 is 0:8 mm, providing a yþ value lowerthan 200 along a wheel circumferential section, as shown in Fig. 3. Based on extensive studies carried out by Exa in the fieldof wing HLD aerodynamics, an average value of about 100 provides the better trade-off between accuracy andcomputational time for time-averaged lift and drag prediction. In the present case, flow unsteadiness is the interest andthe grid resolution is about 2.5 times lower than what recommended by Exa for a partially dressed landing gear, and about 5times lower than what recommended for a fully dressed gear with no simplification made on the production CAD geometry.

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Fig. 4. Time history of the aerodynamic forces on one wheel along the three coordinate directions: Conf. R-Y (a) and Conf. R-N (b).


Such a high level of mesh coarsening is acceptable for the present simulations, due to the fact that the main goal of the studyis to investigate the nature of the facing rim cavity modes and their dependence on large-scale geometrical modifications.

The simulation is initialized with the free-stream conditions and after a transient time of 0:4 s, corresponding toa convective time of about 105 wheel diameters, the sampling of all the unsteady measurement files is started. Sucha duration of the initial transient flow simulation is the outcome of studies conducted on realistic gears in the presence ofa bay, which require a quite long simulation time to converge the low-frequency flow statistics. In the present case, afterexamination of the time history of the unsteady loads plotted in Fig. 4, one can argue that a shorter simulation time wouldhave been acceptable. However, for the sake of consistency with the previous study conducted on the full LAGOONconfiguration [17], the same duration of the transient time has been used. Fig. 4 shows the aerodynamic force componentsacting on one wheel of the gear for two configurations. The side force along y is characterized by the highest fluctuation


levels and its time-averaged value is significantly affected by the presence of the axle. Interestingly, results for Conf. R-Y(with axle) exhibit a higher level of statistical convergence than results for Conf. R-N (without axle). This is due to the factthat, as discussed in Section 4.4, the flow is instantaneously asymmetric because of the wheel rim modal dynamics and aleft/right inversion takes place every now and then. The presence of the axle has the effect of mitigating the flow asymmetryand consequently the side force acting on one wheel occurring at every left/right inversion. Besides the low frequencymodulation of the forces due to the flow asymmetry, the higher Fourier components of the flow fluctuations are expected tohave achieved an adequate level of statistical convergence throughout the sampling period; this is set to 0:4 s for all thesimulations (second half of the simulation period). The sampling time of 0:4 s is such that, for instance, a minimumfrequency of 10 Hz can be achieved by the spectral analyses with a hypothetical number of averages equal to 8 and anoverlapping coefficient of 0.5. A bandwidth of 10 Hz is used for all the reported spectral analyses.

The time step used for the present simulations is 2:33� 10�6 s, corresponding to 343 348 time steps per simulation. Therequired number of CPU hours per simulation was about 4250 on a cluster of Intel Xeon X5570 2.93 GHz CPUs connected bya Mellanox FDR Infiniband 56 Gb/s network. The wall-clock time was about 17 h per simulation on a cluster of 256 cores.

The FW-H computation is carried out by integrating the unsteady pressure field on the solid surfaces. A convective FW-Hformulation is used, with uniform stream velocity equal to the nominal free-stream velocity. The sampling frequency of thewall pressure field is about 110 kHz.

4. Numerical results

Numerical results for the six wheel assembly configurations are illustrated in this section. Results are gathered in twosubsections for near- and far-field results. Furthermore, two subsections are focused on the characterization of the facingrim cavity modes.

4.1. Near-field results

This subsection is focused on time-average and unsteady near-field results. Fig. 5 shows the time-averaged streamwisevelocity and turbulent kinetic energy K extracted along two line cuts (spanwise and lift direction) located one wheeldiameter downstream of the wheel axis. It is interesting to observe that the expected flow symmetry is not perfectlyrecovered. This means that the simulation time of 0:4 s was not long enough to achieve a definite statistical convergence. Asalready pointed out in Ref. [17], a longer simulation time is required for the present case compared to standard landing gearsimulations. This is due to the occurrence of strong side flow oscillations induced by the rim cavity modes that cause asignificant flow asymmetry, with a left/right inversion occurring with a very low frequency (order of 10 Hz). Interestingly,the presence of the axle has the effect of reducing the flow asymmetry, and this is likely to be due to a slight increase of theside inversion frequency. Besides the flow asymmetry, the profiles in Fig. 5 show that the presence of the axle mitigates theeffect of the wheel distance on the flow.

Fig. 6 shows contour plots of the time-averaged streamwise velocity component u and turbulent kinetic energy K(predicted þ modelled) on a plane cutting through the centre of the rim cavity. Only results for Conf. R-Y and Conf. R-N arereported. The velocity field shows that the shear layer starts developing from the rim bump where a flow separation takesplace. The presence of the axle, as expected, has a strong influence on the velocity field. The profiles plotted in the figureshave been extracted at a distance of 30 mm downstream of the wheel axis; they clearly show the effect of the axle wake. Theturbulent kinetic energy fields show that, without the axle, the maximum fluctuation levels take place inside the cavity,below the shear layer, and on the downstream edge; in the presence of the axle, the maximum levels take placedownstream of the axle, where the wake from the axle interacts with the shear layer of the cavity. From these observationsone would expect that the presence of the axle has a strong influence on the acoustic cavity mode behaviour.

In order to better understand the effect of the axle on the development of the shear layer, contour plots of the two-pointvelocity correlation on a plane cutting through the rim cavity are shown in Figs. 7–9. These quantities are defined as follows:

Ruu ¼⟨u0ðx; tÞu0ðx0; tÞ⟩ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi⟨u02ðx; tÞ⟩⟨u02ðx0; tÞ⟩

p ; Rvv ¼⟨v0ðx; tÞv0ðx0; tÞ⟩ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi⟨v02ðx; tÞ⟩⟨v02ðx0; tÞ⟩

p ; Rww ¼ ⟨w0ðx; tÞw0ðx0; tÞ⟩ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi⟨w02ðx; tÞ⟩⟨w02ðx0; tÞ⟩

p ; (1)

where x0 denotes the reference point. Two reference points are considered, located at different streamwise locationsx0 ¼ 740 mm along the line z¼ 0 \ y¼ yE. Starting from Ruu in Fig. 7, we can observe that the axle corrupts the coherenceof the flow structures developing within the shear layer: a high correlation spot of increasing size along the flow path can beobserved in the absence of the axle, similar to what reported by other authors [25], whereas, in the presence of the axle, thesize of the high correlation spot is significantly smaller and almost constant along the flow path. In the absence of the axle,we can also observe that the main shear-layer mode has a wavelength equal to about the cavity diameter (twice the distancebetween two consecutive positive/negative peaks). This wavelength can be associated with the simultaneous presence oftwo large turbulent structures in the shear at any time, as argued by Marsden et al. [25]. Fig. 8 shows the correlation of thevelocity fluctuation component normal to the cavity opening (depth direction). Again, the development of the turbulentstructures in the shear layer is well evident only in the absence of the axle. Finally, Fig. 9 shows the correlation of thevelocity fluctuation component w. In this case, since the coherent turbulent eddies in the shear layer do not induce strong

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Turbulent kinetic energyConf. R-YConf. N-YConf. W-YConf. R-NConf. N-NConf. W-N

Fig. 5. Time-averaged solution along the line x=D¼ 1 \ z=D¼ 0 (left) and x=D¼ 1 \ y=D¼ 0 (right).

Fig. 6. Time-averaged streamwise velocity on Conf. R-Y (a) and Conf. R-N (b), and turbulent kinetic energy on Conf. R-Y (c) and Conf. R-N (d), on the planez¼ 0.


Fig. 7. Correlation field of the x velocity fluctuations (Ruu). Reference point (red dots) located on the plane z¼ 0 at the same y as the cavity edge and twostreamwise locations: x=D¼ �0:25 (left) and x=D¼ 0:25 (right). Conf. R-Y on the top vs Conf. R-N on the bottom.

Fig. 8. Correlation field of the y velocity fluctuations (Rvv). Reference point (red dots) located on the plane z¼ 0 at the same y as the cavity edge and twostreamwise locations: x=D¼ �0:25 (left) and x=D¼ 0:25 (right). Conf. R-Y on the top vs Conf. R-N on the bottom.

Fig. 9. Correlation field of the z velocity fluctuations (Rww). Reference point (red dots) located on the plane z¼ 0 at the same y as the cavity edge and twostreamwise locations: x=D¼ �0:25 (left) and x=D¼ 0:25 (right). Conf. R-Y on the top vs Conf. R-N on the bottom.


velocity fluctuations in the direction normal to the plane z¼ 0, the high correlation spots have a very small size forboth cases.

Wall pressure spectra computed at three probe locations on the outboard arc A depicted in Fig. 2 are plotted in Figs. 10–12for all wheel assembly configurations. The 01 probe is the one located at the upstream angular location. Fig. 10 shows results forconfigurations with axle. Two tones at about 1 and 1:5 kHz emerging by about 10 dB from the broadband levels can beobserved at the 901 probe. The levels of these tones are almost identical at the 01 probe, but the higher broadband levelsoverwhelm the tonal peaks. At the downstream probe (1801), the broadband levels induced by the turbulent fluctuations in thewake of the wheels overwhelm the tones completely. The tone magnitudes for the different wheel distances differ by less than2 dB. Furthermore, the tone frequencies decrease as the distance between the wheels increases.

Fig. 11 shows pressure spectra for configurations without axle. The same observations as for configurations with axle canbe made. For these configurations, however, the absence of the axle results in slightly lower broadband levels and,interestingly, the tones magnitude exhibits a stronger dependence on the wheel distance; the first tone for Conf. W-N at the901 probe, for instance, is about 7 dB higher than for the other configurations.

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Conf. R-NConf. N-NConf. W-N

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[dB/

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Fig. 11. Wall pressure spectrum for configurations without axle at probes on arc A; 01 upstream (a), 901 (b), 1801 downstream (c).

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Fig. 12. Effect of the axle on the wall pressure spectrum; probe at 901 on arc A. Conf. R-Y and R-N (a), N-Y and N-N (b), W-Y and W-N (c).

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Frequency [Hz]


Fig. 10. Wall pressure spectrum for configurations with axle at probes on arc A; 01 upstream (a), 901 (b), 1801 downstream (c).


Fig. 12 shows comparisons between the pressure spectra at the 901 probe with and without the axle. It is interesting toobserve that the tone frequencies are weakly affected by the presence of the axle, whereas the tone magnitudes, for givenwheel distances, are strongly affected by the presence of the axle. For instance, comparing Conf. W-Y and Conf. W-N, thefirst tone for Conf. W-N (without axle) is about 7 dB higher, whereas the second tone for Conf. W-N is about 10 dB lower(almost absent).

As argued by the authors in Ref. [17], the tones taking place at about 1 and 1:5 kHz are not due to any vortex sheddingmechanism from the axle. This is definitively confirmed by the present results that reveal the occurrence of tones also whenthe axle is removed from the model. Therefore, the origin of these tones is to be found in some shear-layer vorticalinstability mode that excite at resonance a cavity mode, as also observed for simple round cavities by other authors [25].Indeed, the observed influence of the wheel distance on the tone frequency confirms the hypothesis put forward in Ref. [17]according to which shear-layer fluctuation modes are coupled with purely acoustic modes taking place between the twofacing wheels. Following the analysis made by Marsden et al. [25], who related the cavity tones for a single cavity to theacoustic depth mode, it is possible to relate the tonal frequencies to the modes of an annular duct of length L closed at thetwo extremities, i.e.:

f im ¼ c12π

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλmr2

þ i2π4L

� �2s

; (2)

where r is the outer radius of the annular duct (a half of the rim cavity diameter), λm is the eigenvalue for the annular ductmode of first radial order (no nodes along the radial direction) and azimuthal order m for a duct of inner radius equal to the

Table 3

Theoretical (f FF, f EE) vs predicted (f ð1Þ , f ð2Þ) rim cavity tonal frequencies; all quantities in SI units.

Conf. # f ð1Þ f ð2Þ λm ¼ 0 λm ¼ 1 λm ¼ 2 f FF0 f FF1 f FF2 f EE0 f EE1 f EE2 Err1 Err2

R-Y 1050 1500 0 1.62 2.99 1047 1354 1569 1233 1502 1698 0.3 0.2N-Y 1125 1600 0 1.62 2.99 1164 1446 1649 1467 1699 1875 3.4 6.0W-Y 1000 1425 0 1.62 2.99 952 1282 1507 1063 1367 1579 4.9 4.2R-N 1025 1575 0 1.84 3.00 1047 1391 1570 1233 1536 1699 2.1 2.5N-N 1125 1675 0 1.84 3.00 1164 1481 1650 1467 1729 1876 3.4 3.2W-N 975 1500 0 1.84 3.00 952 1321 1508 1063 1403 1581 2.4 6.7

Table 4Frequencies (Hz) of Rossiter cavity modes for an equivalent rectangular cavity.

n f

1 2532 5073 7604 10145 12676 15217 17748 20289 2281

10 253511 278812 304213 329514 354915 3802


axle radius (zero in the absence of the axle), i is the transversal mode order counting the number of quarter wavelengths ofthe transversal mode that takes place along the transversal distance L.

Table 3 summarizes the values of the eigenvalues and the tone frequencies computed for the azimuthal mode ordersm¼ 0, m¼ 1 and m¼ 2 and for the second transversal mode order i¼ 2. Since two transversal modes are expected, onerelated to the floor-to-floor rim distance lFF, the other one to the edge-to-edge rim distance lEE, two values of the transversaldistance L have been used: LFF ¼ 0:82 lFF and LEE ¼ 1:11 lEE, where the correction factors 0.82 and 1.11 account for thegeometrical differences between the wheel assembly and a pure annular cavity on a flat plate; the correction factor forthe first tone is the same as the one used in Ref. [25]. The relative percentage error is also reported in the table for all theconfigurations, both for the first and the second computed tones, by assuming that the first tone is related to the zeroth-

order azimuthal mode and second floor-to-floor transversal mode (f FF0 ), whereas the second tone is related to the first-order

azimuthal mode and second edge-to-edge transversal mode (f EE1 ), i.e. Err1 ¼ 200jf ð1Þ � f FF0 j=ðf ð1Þ þ f FF0 Þ and Err2 ¼ 200jf ð2Þ �f EE1 j=ðf ð2Þ þ f EE1 Þ. The corresponding average relative errors are gErr1 ¼ 2:7 percent and gErr2 ¼ 3:8 percent for the first and thesecond tone, respectively. This analysis could be easily extended to other weak tones that can be observed in the computedspectra both below 1 kHz and above 1:5 kHz, which are likely to be due to different combinations of azimuthal andtransversal modes for the two transversal lengths.

According to the modal analysis in the previous paragraph, shear-layer fluctuations can excite at resonance several tonesthat result from the mutual combination of annular duct modes with wheel-to-wheel modes. However, the reason whyamong other tones only few of them exhibit a strong emergency can be found in a retroaction mechanism of Rossiter type,that is, between the onset of a shear-layer vortical instability at the upstream edge and the acoustic wave generated by theimpact of the shear-layer vortices on the downstream edge. In order to validate this hypothesis, we can estimate thefrequencies of the Rossiter cavity tones for an equivalent rectangular cavity of width l¼ 0:162 m equal to the rim diameterand depth d¼ 0:03725 m equal to the rim cavity depth, by using the formula proposed by Block [22], which reads

f ¼ nU1=l1=kvþM1ð1þ0:514 d=lÞ with n¼ naþnv (3)

where n is the Rossiter mode order, resulting from the combination of na acoustic and nv vortical wavelengths, and kvC0:6is the ratio of average vortex convection velocity across cavity length to free-stream velocity. The frequencies for the first15 modes are reported in Table 4. According to Block's formula, the modes n¼ 4 and n¼ 6 provide frequencies that are veryclose to the expected tones.


The occurrence of a Rossiter retroaction mechanism in a cylindrical cavity for which the distance between the upstreamand downstream edges is not constant could appear weakly relevant. However, according to the experimental observationsmade by Dybenko and Savory [38], a cylindrical cavity of depth-to-diameter ratio 0.48 in a grazing flow velocity of 27 m s�1

generates a tone whose frequency is very close to the one predicted for an equivalent rectangular cavity of width equal tothe rim diameter by using a Rossiter formula. Therefore, they concluded that, somehow, a Rossiter-type feedback loopmechanism can take place in such a cavity configuration. Moreover, they argued that the feedback loop mechanism could bealso responsible for the asymmetric flow behaviour and the bistable fluctuation regime observed for that specific value ofthe depth-to-diameter ratio. Very interestingly, a bistable behaviour has been also observed for the present wheelconfiguration [17], and this supports the hypothesis of the existence of a Rossiter-type feedback loop at the origin of thetones predicted for the rim cavities.

4.2. Far-field results

This subsection is focused on far-field noise spectra. For the sake of comparing the present results with the ones of thefull LAGOON gear configuration, the same microphone locations as for the LAGOON gear analysis [17] have been used.

Fig. 13 shows a comparison between the measured noise spectra for the full LAGOON gear and the present prediction. Weshould mention that the prediction for the full gear reported in Ref. [17] shows an almost perfect agreement with themeasurements over the whole frequency range. Therefore, the difference between measurement and prediction below� 400 Hz and above � 2 kHz herein reported is likely to be due to the different geometry, that is, the absence of the leg inthe present analysis. As pointed out in Ref. [17], the tones related to the rim cavity modes are not effectively radiated in theflyover plane and can be clearly distinguished only at sideline microphone locations. This confirms the experimentalobservation made by Marsden et al. [25] for a single cavity, according to which there was no trace of the azimuthal cavitymode structure in the far field, while the tones were only related to the depth mode. In other words, we can argue that thefacing rim cavity modes are evanescent in the radial direction, and are effective only in the transversal direction.

Fig. 14 shows the computed spectra for configurations with axle. Interestingly, the second tonal peak at about 1:5 kHz atthe sideline microphone increases by about 10 dB when the wheel distance is decreased from the highest (Conf. W-Y) to thelowest (Conf. N-Y) addressed values. The first tonal peak at about 1 kHz can be clearly distinguished from the broadbandnoise only for Conf. W-Y. The flyover noise spectra exhibit bumps at the two tonal frequencies, but a net peak takes placeonly at about 3:3 kHz; the origin of this tone will be discussed later on.

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Conf. R-YLAGOON experiment

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[dB

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Conf. R-YLAGOON experiment

Fig. 13. Noise spectra at 401 (aft radiation) with respect to the flow direction on the sideline arc (a) and the flyover arc (b). Comparison betweenmeasurements for the LAGOON gear and the present prediction for Conf. R-Y.

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Fig. 14. Computed noise spectra at 401 (aft radiation) on the sideline (a) and flyover arc (b) for configurations with axle.

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[dB

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[dB

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]

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Fig. 15. Computed noise spectra at 401 (aft radiation) on the sideline (a) and flyover arc (b) for configurations without axle.

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Conf. W-YConf. W-N

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[dB

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]

Frequency [Hz]

Conf. W-YConf. W-N

Fig. 16. Computed noise spectra at 401 (aft radiation) on the sideline (left) and flyover arc (right). Comparison between configurations with andwithout axle.


Fig. 15 shows computed spectra for configurations without axle. For these configurations, the first tonal peaks at about1 kHz have a higher magnitude than the second tonal peaks at about 1:5 kHz. The highest first tone peak occurs for thelargest wheel distance (Conf. W-N).

The same noise spectra are plotted in Fig. 16 in order to better highlight the influence of the axle. As a general finding, thefirst tonal peak is higher in the absence of the axle, whereas the second tonal peak is higher in the presence of the axle.Therefore, we can argue that the axle has the effect of changing the acoustic modal behaviour and that the plane transversalmode is strengthened without axle, whereas the first azimuthal mode is strengthened with axle. Another interestingoutcome of this comparison is that a broad tone takes place around 3:3 kHz only for configurations with axle and that this


tone is not evanescent in the radial direction (flyover plane). The origin of this tone is expected to be related to the fifth-order transversal mode (i¼ 5 in Eq. (2)), as discussed later on. Interestingly, the Rossiter mode n¼ 13 for an equivalentrectangular cavity would provide a tone at 3:3 kHz, as reported in Table 4.

Finally, the overall sound pressure level directivity for all cases is compared in Fig. 17. Configurations with axle are noisierthan corresponding configurations without axle by less than � 2:5 dB. The wheel distance affect the overall noise levels byless than � 1 dB, the smaller the distance the higher the noise. The higher noise levels computed at 801 on the flyover arcare due to the smaller radial distance of the microphone at that angular location in the LAGOON measurements.

75 80 85OASPL (100-10000 Hz) [dB]

30deg

60deg90deg

120deg

150deg Conf. R-YConf. N-YConf. W-YConf. R-NConf. N-NConf. W-N

75 80 85OASPL (100-10000 Hz) [dB]

30deg

60deg90deg

120deg

150deg Conf. R-YConf. N-YConf. W-YConf. R-NConf. N-NConf. W-N

Fig. 17. Computed OASPL directivity in the frequency range ½100 Hz : 10 kHz� along the flyover (a) and sideline arc (b) for all configurations.

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5

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-0.1 -0.05 0 0.05 0.1

Pre

ssur

e [P

a]

y [m]

Conf. R-YConf. R-N

cosinus, i=5 (five quarter wavelengths)

Fig. 18. Band-filtered pressure fluctuation in the range ½3065 Hz : 3710 Hz� for Conf. R-Y (top) and Conf. R-N (middle) on plane z¼ 0 (a) and y¼ 0 (b); onthe bottom, filtered pressure along the line x=D¼ �0:133 \ z=D¼ 0 compared with a transversal mode i¼ 5.


4.3. Facing rims cavity modes

The goal of this subsection is to confirm the acoustic mode analysis made in Section 4.1 by performing narrow-bandfiltering of the unsteady pressure field in three regions between the wheels: the top/bottom symmetry plane z¼ 0 (greenoutlined rectangle in Fig. 2), the left/right symmetry plane y¼ 0 and a circle of diameter equal to 0:34 m located on theplane y¼ 0:05 m normal to the axis, about 10 mm away from the rim (one of the blue outlined disks in Fig. 2). One frame ofthe reconstructed (filtered) transient time solution is used to highlight the structure of the mode.

The first band filtering is performed around the frequency at about 3:3 kHz in order to highlight the structure of themode that takes place only for configurations with axle (tone at about 3:3 kHz in Fig. 16). Fig. 18 shows the filtered pressurein the band ½3065 Hz : 3710 Hz� for Conf. R-Y and, for the sake of comparison, Conf. R-N. Interestingly, the image in the planez¼ 0 and the line cut plot for the case with axle (R-Y) confirm that five quarter wavelengths of the transversal mode takeplace between the wheels (fifth transversal mode order). The corresponding time evolution of the band-filtered pressureshows that a feedback loop takes place between the shear-layer fluctuations and the acoustic waves generated by theirimpact on the axle. The field extracted in the side symmetry plane confirms that the mode for the case with axle is able topropagate away from the wheels, as shown by the noise spectra at the flyover microphones plotted in Fig. 16. As an oddorder mode (i¼ 5), in fact, the transversal standing wave has a peak value in the symmetry plane.

Further pressure band filtering is performed for all configurations around their respective first and second tonalfrequencies. Fig. 19 shows the phase of the transfer function between two facing probes located on the inboard arc, sketchedin Fig. 2, filtered around the tonal frequencies. A phase opposition is clearly evident at angular locations of the probes awayfrom the cavity/axle wake (o901). This confirms that transversal acoustic modes of even order (i¼ 2;4;…) take place forboth tones.

Contour plots of the band filtered pressure on the circle at y¼ 0:05 m (side view) and on the plane z¼ 0 (top view) for allconfigurations are shown in Figs. 20–22. These images confirm that in all cases the mode for the first tone has a prevailingplane structure (m¼ 0), as clearly revealed by images on the plane z¼ 0, whereas the mode for the second tone hasa prevailing first-order azimuthal structure (m¼ 1), leading to phase opposition between diametrically opposite locations.Interestingly, the occurrence of a similar modal behaviour for configurations with and without axle confutes theexpectations according to which, based on the velocity correlation results reported in Figs. 6–8, the presence of the axlecan prevent a Rossiter-type feedback loop mechanism to take place. The top views for both tones clearly confirm that thesecond-order (i¼ 2) transversal mode takes place; since this mode has a node in the symmetry plane, the associatedacoustic radiation is evanescent in the flyover plane. Finally, it is interesting to count the number of wavelengths associatedwith the vortical fluctuation in the shear layer, say, 3 and 5 for the first and second tones, respectively; this finding supportsthe hypothesis according to which the two tones are associated with the Rossiter modes n¼ 4 and n¼ 6, say na ¼ 1 andnv ¼ 3 or nv ¼ 5 in Eq. (3).

0 20 40 60 80

100 120 140 160 180

0 20 40 60 80 100 120 140 160 180

TF p

hase

[deg

]

Probe angle [deg]

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hase

[deg

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[deg

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[deg

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[deg

]

Probe angle [deg]

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hase

[deg

]

Probe angle [deg]

Fig. 19. Phase of the transfer function of wall pressure signals at facing probes located on the inboard arc; Conf. R-Y (a), N-Y (b), W-Y (c), R-N (d), N-N (e),W-N (f); n: first tone, □: second tone.

Fig. 20. Case R-Y (left) and Case R-N (right): band-filtered pressure fluctuations around the first (top) and second tone (bottom). Side and top views on theleft and the right, respectively.

Fig. 21. Case N-Y (left) and Case N-N (right): band-filtered pressure fluctuations around the first (top) and second tone (bottom). Side and top views on theleft and the right, respectively.


4.4. Side asymmetry and inversion mechanisms

The near-field results extracted along the line cuts of Fig. 5 exhibit a significant side asymmetry that was attributed toa side asymmetry of the flow, characterized by a left/right inversion occurring with a frequency of the order of 10 Hz. It was

Fig. 22. Case W-Y (left) and Case W-N (right): band-filtered pressure fluctuations around the first (top) and second tone (bottom). Side and top views onthe left and the right, respectively.

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ssur

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Left wheelRight Wheel

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Fig. 23. Pressure signals at 901 away from the flow on two facing probes on the inboard arc filtered around the first tone (a) and the second tone (b).


also argued that the presence of the axle has the effect of reducing the asymmetry of the time-averaged flow by increasingthe side inversion frequency.

Another inversion mechanism that has been observed and that is probably related to the left/right inversion mechanismis that the two tones band filtered pressure signals at facing probes undergo, now and then, a change of the sign of the phaselag, passing through few instants of perfectly phased conditions. The phased condition occurs in time proximity ofconditions of minimal oscillation amplitude. Fig. 23 shows the full length filtered signals for one of the simulated cases;these undergo a strong amplitude modulation, but, examining the results for all cases, no generic rule was found betweenthe modulation pattern on the two wheels. Zooming into these signals, it is possible to find examples of phase lag inversion,as shown for instance in Fig. 24 for the same cases as in Fig. 23.

Finally, a further inversion mechanism was observed examining the band filtered pressure time evolution: forconfigurations with axle the second tone mode m¼ 1 was observed to change rotation direction (sign of the azimuthalmode). Indeed, the two modes m¼ 71 have the same probability to occur and therefore they both concur to the generationof the second tone. The same modal switching behaviour was observed for simple grazing cylindrical cavities by otherauthors [38,27] who related the bistable nature of the unsteady flow with the asymmetry of the flow with respect to thenominal flow direction. We believe that the three inversion mechanisms observed for the facing rim cavities are theconsequence of the dynamic triggering of three mechanisms: the occurrence of shear-layer instabilities, the excitation of

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ssur

e [P

a]

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-4

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0.52 0.525 0.53 0.535 0.54

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ssur

e [P

a]

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Fig. 24. Pressure signal at 901 away from the flow on two facing probes on the inboard arc filtered around the first tone (a) and the second tone (b).


facing rim acoustic modes, and the onset of a Rossiter-type feedback loop. There would exist perhaps geometry/flowconfigurations for which the cavity mode resonance is better locked with the feedback loop, thus leading to even highertone emergencies and lower asymmetry in the fluctuation levels taking place in the two rims.

5. Conclusion

Acoustic radiation from two facing round cavities has been studied numerically. The geometry was obtained by removingthe leg from a simplified two-wheel LAGOON landing gear configuration. A parametric study was performed in order toshow the influence of the wheel distance and the presence of the axle between the wheels, on the acoustic mode behaviour.The acoustic modes corresponding to two tones observed both in the wall pressure spectra and in the noise spectra awayfrom the flyover symmetry plane have been characterized with reference to their space structure. It was clearlydemonstrated that one tone was related to a plane mode (second-order transversally and zeroth-order circumferentially)for the floor-to-floor cavity distance, whereas the other tone was related to an azimuthal mode (second-order transversallyand first-order circumferentially) for the edge-to-edge cavity distance. The tonal frequencies predicted with a simpleanalytical formula involving the acoustic mode eigenvalues of an annular duct agree quite well with the CFD solutions.Estimation of the Rossiter frequencies for an equivalent rectangular cavity provided values close to the predicted tones, thussupporting the hypothesis of a Rossiter-type feedback loop mechanism at the base of the tone selection. The axle was shownto have a strong influence both on the near-field flow structures and on the noise radiation. In particular the axle enhancedthe occurrence of the first azimuthal mode, thus resulting in higher amplitude of the second tone peaks. This result was notobvious, since the presence of the axle was shown to corrupt the development of the shear-layer instabilities and possiblyprevent a Rossiter-type feedback loop. As a matter of fact, the axle was even shown to promote further feedback loopmechanisms, thus resulting in a higher tonal content of the radiated noise spectra. Finally, three inversion mechanismsresponsible for side flow asymmetries were reported, although two of them require inspections of flow movies to be fullydocumented. These mechanisms are likely to be mutually related, and, understanding how will be the object of furtherstudies. A fundamental question arises at the end of this study: is there any chance for facing rim cavity modes to take placein a real landing gear configuration? To the authors' opinion, this is likely to happen if the rim volume is not fully occupiedby the brake system, since a feedback mechanism is difficult to be suppressed in general, and could be even more difficult inthe present case for which the loop is driven by the transversal acoustic mode between the wheels. Indeed, the presence ofthe axle with its fully turbulent wake was shown to enhance instead of suppress the tones. The fact that these modes arealmost evanescent in the flyover plane has perhaps prevented this mechanism to be observed and documented so far.

References

[1] M. Murayama, Y. Yokokawa, T. Imamura, K. Yamamoto, H. Ura, T. Hirai, Numerical investigation on change of airframe noise by flap side-edge shape,AIAA Paper 2013-2067, 2013.

[2] M. Choudhari, D.P. Lockard, M.G. Macaraeg, B.A. Singer, C.L. Streett, G.R. Neubert, R.W. Stoker, J.R. Underbrink, M.E. Berkman, M.R. Khorrami, S.S.Sadowski, Aeroacoustic experiments in the langley low-turbulence pressure tunnel, NASA TM-2002-211432, 2002.

[3] W. Dobrzynski, L.C. Chow, P. Guion, D. Shiells, Research into landing gears airframe noise reduction, AIAA Paper 2002-2409, 2002.[4] D. Casalino, S. Noelting, E. Fares, T. Van de Ven, F. Pérot, G.A. Brès, Towards numerical aircraft noise certification: analysis of a full-scale landing gear in

fly-over configuration, AIAA Paper 2012-2235, 2012.[5] G.A. Brès, D.M. Freed, M. Wessels, S. Noelting, F. Pérot, Flow and noise predictions for the tandem cylinder aeroacoustic benchmark, Physics of Fluids 24

(3) (2012) 036101.[6] A. Mann, F. Pérot, M.-S. Kim, D. Casalino, E. Fares, Advanced noise control fan direct aeroacoustics predictions using a lattice-Boltzmann method, AIAA

Paper 2012-2287, 2012.[7] F. Pérot, M.-S. Kim, M. Meskine, D. Freed, NREL wind turbine aerodynamic validation and noise predictions using LBM, AIAA Paper 2012-2287, 2012.[8] A. Keating, P. Dethioux, R. Satti, S. Noelting, J. Louis, T. Van de Ven, R. Vieito, Computational aeroacoustics validation and analysis of a nose landing

gear, AIAA Paper 2009-3154, 2009.

http://refhub.elsevier.com/S0022-460X(14)00091-1/sbref5



[9] S. Noelting, G.A. Brès, P. Dethioux, T. Van de Ven, R. Vieito, A hybrid lattice-Boltzmann/FW-H method to predict sources and propagation of landinggear noise, AIAA Paper 2010-3976, 2010.

[10] L.G.C. Simoes, D.S. Souza, M.A.F. Medeiros, On the small effect of boundary layer thicknesses on slat noise, AIAA Paper 2011-2906, 2011.[11] P.-T. Lew, R. Shock, A. Najafi-Yazdi, L. Mongeau, S. Colavincenzo, R. Lapointe, G. Waller, Noise prediction from a partially closed slat junction, AIAA Paper

2013-2161, 2013.[12] E. Fares, S. Noelting, Unsteady flow simulation of a high-lift configuration using a lattice Boltzmann approach, AIAA Paper 2011-869, 2011.[13] B. Koenig, E. Fares, S. Noelting, Lattice-Boltzmann flow simulations for the HILIFTPW-2, AIAA Paper 2014-0911, 2014.[14] E. Manoha, J. Bulté, V. Ciobaca, B. Caruelle, Lagoon: further analysis of aerodynamic experiments and early aeroacoustic results, AIAA Paper 2009-3277,

2009.[15] E. Manoha, J. Bulté, B. Caruelle, Lagoon: an experimental database for the validation of CFD/CAA methods for landing gear noise prediction, AIAA Paper

2008-2816, 2008.[16] Lagoon simplified (2-wheel) nose landing gear configuration #1 – experimental database, Technical Report R12, Airbus, April 2011.[17] A.F.P. Ribeiro, D. Casalino, E. Fares, CFD/CAA analysis of the LAGOON landing gear configuration, AIAA Paper 2013-2256, 2013.[18] D. Rockwell, E. Naudascher, Review—self-sustaining oscillations of flow past cavities, Journal of Fluids Engineering 100 (2) (1978) 152–165.[19] J.E. Rossiter, Wind tunnel experiments of the flow over rectangular cavities at subsonic and transonic speeds, Aeronautical Research Council Reports

and Memoranda 3438, 1964.[20] C.K.W. Tam, Discrete tones of isolated airfoils, Journal of the Acoustical Society of America 55 (6) (1974) 1173–1177.[21] J. Berland, C. Bogey, C. Bailly, Large eddy simulation of screech tone generation in a planar underexpanded jet, AIAA Paper 2006-2496, 2006.[22] P.J.W. Block, Noise response of cavities of varying dimensions at subsonic speeds, NASA TN D-8351, 1976.[23] P.J.W. Block, H. Heller, H. Hanno, Measurements of farfield sound generation from a flow-excited cavity, NASA TM X-3292, 1975.[24] A. Powell, On the edgetone, Journal of the Acoustical Society of America 33 (4) (1961) 395–409.[25] O. Marsden, C. Bailly, C. Bogey, E. Jondeau, Investigation of flow features and acoustic radiation of a round cavity, Journal of Sound and Vibration 331

(15) (2012) 3521–3543.[26] O. Marsden, E. Jondeau, P. Souchotte, C. Bailly, C. Bogey, D. Juvé, Investigation of flow features and acoustic radiation of a round cavity, AIAA Paper

2008-2851, 2008.[27] C. Mincu, I. Mary, S. Redonnet, L. Larcheveque, J.-P. Dussauge, Numerical simulations of the unsteady flow and radiated noise over a cylindrical cavity,

AIAA Paper 2008-2917, 2008.[28] G.A. Brès, F. Pérot, D.M. Freed, A Ffowcs Williams–Hawkings solver for lattice-Boltzmann based computational aeroacoustics, AIAA Paper 2010-3711,

2010.[29] H. Chen, S. Chen, W. Matthaeus, Recovery of the Navier–Stokes equations using a lattice-gas Boltzmann method, Physical Review A 45 (8) (1992)

5339–5342.[30] V. Yakhot, S.A. Orszag, Renormalization group analysis of turbulence. i. Basic theory, Journal of Scientific Computing 1 (1) (1986) 3–51.[31] B. Launder, D. Spalding, The numerical computation of turbulent flows, Computer Methods in Applied Mechanics and Engineering 3 (1974) 269.[32] H. Chen, C. Teixeira, K. Molvig, Realization of fluid boundary conditions via discrete Boltzmann dynamics, International Journal of Modern Physics C 9

(8) (1998) 1281–1292.[33] G.A. Brès, F. Pérot, D.M. Freed, Properties of the lattice-Boltzmann method for acoustics, AIAA Paper 2009-3395, 2009.[34] D. Casalino, An advanced time approach for acoustic analogy predictions, Journal of Sound and Vibration 261 (4) (2003) 583–612.[35] F. Farassat, G.P. Succi, The prediction of helicopter discrete frequency noise, Vertica 7 (4) (1983) 309–320.[36] P. Di Francescantonio, A new boundary integral formulation for the prediction of sound radiation, Journal of Sound and Vibration 202 (4) (1997)

491–509.[37] A. Najafi-Yazdi, G.A. Brès, L. Mongeau, An acoustic analogy formulation for moving sources in uniformly moving media, Proceeding of the Royal Society

of London A 467 (2125) (2011) 144–165.[38] J. Dybenko, E. Savory, An experimental investigation of turbulent boundary layer flow over surface-mounted circular cavities, Proceedings of the

Institution of Mechanical Engineers Part G: Journal of Aerospace Engineering 222 (1) (2008) 109–125.




















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