31st Symposium on Naval HydrodynamicsMonterey, California, 11-16 September 2016

Computational Noise Prediction from Trailing EdgeFlows

Zane Nitzkorski and Krishnan Mahesh(University of Minnesota, USA)

ABSTRACT

We characterize the impact of near-field turbulenceand boundary layer thickness for a 45◦ beveled trail-ing edge by investigating two different inflow bound-ary layer thicknesses. Noise from this type of sce-nario is important for propellers, airfoils, fans andmarine propulsors. This geometry has been investi-gated experimentally by Olson and Mueller (2004)and Shannon and Morris (2006), Shannon et al.(2006) and computationally by Wang (2005) butboundary layer sensitivity has not been studied pre-viously. Large-eddy simulation (LES) is performedat the experimental boundary layer conditions aswell as a boundary layer momentum thickness ξwhich is eighteen times smaller. A porous Ffowcs-Williams and Hawkings (FWH) Ffowcs-Williamsand Hawkings (1969) acoustic analogy is used tosolve for the far-field noise and shows good agree-ment with experimental results. The acoustic anal-ogy is compared to a Curle based formulation wherethe surface terms are used as the sources in conjunc-tion with a specialized half-plane Green’s function.The scaling of noise production with the physicalproperties which produce acoustic sources are dis-cussed. Comparing and contrasting the two bound-ary layer thicknesses demonstrates that the noiseproduced in the lower half plane is fairly insensi-tive to boundary layer thickness because of the shed-ding vortex. The noise produced in the upper halfplane however is strongly related to the boundarylayer thickness. The separation point is similar be-tween both cases, but the near field turbulent inten-sities contribute to the increased noise for the thickerboundary layer case.

INTRODUCTION

Trailing edges of airfoil sections are ubiquitous inaeronautical and hydrodynamic applications, andunderstanding the noise generation process for thisconfiguration is therefore important. The produc-tion of sound depends on the geometry of the body

as well as the boundary layer. These two factorsinteract as the body modifies the near field turbu-lence and provides the mechanism to scatter the flowenergy and radiate it efficiently as acoustic energy.In order to characterize the noise production depen-dence on the boundary layer, LES computations areperformed for a trailing edge geometry with two dif-ferent boundary layer thicknesses. The flowfield andsound field is dominated by the separation of theboundary layer. As the flow accelerates over the topof the body it sets up a periodic shedding charac-terized by a fluctuating pressure and velocity nearthe airfoil which subsequently radiates to the soundfield. The fluctuations near the trailing edge tip arerelated to the boundary layer thickness and accountfor the sound intensity.

Motivation for the current investigation comesfrom two similar trailing edge configurations whichhave been studied experimentally and computation-ally. The first configuration was largely studied byBlake (1975) where trailing edges of 25◦ and 45◦

were studied. The 25◦ case produced tonal non-fixedcoherent shedding and lacked acoustic comparisondata, so the second beveled trailing edge at 45◦ wasstudied in more depth by Blake (1986), Gershfeldet al. (1988), and summarized in Blake and Gersh-feld (1989). Additional experiments were conductedmore recently to examine the sound field by Olsonand Mueller (2004), Shannon and Morris (2006), andShannon et al. (2006). These found a strong phasebehavior to the shedding that directly relates to therecovered sound.

The shedding behavior described in Blake (1975)provides a detailed look at the separation mecha-nism, pressure correlation, and velocities in the sepa-rated wakes. The frequency content associated withthe flow near the trailing edge as well as the rootmean square of the pressure at the surface providethe scaling associated with received sound to thesequantities. Blake and Gershfeld (1989) use thesecorrelations combined with traditional dipole scat-tering relations to predict acoustic pressure and theratio of scattered noise to that if directly radiated.

We use these relations to investigate the effect of thechange in boundary layer thickness.

Computational studies have been performed toinvestigate the noise from trailing edges. These in-clude Wang and Moin (2000) who studied the 25◦

trailing edge corresponding to the work of Blake(1975) and found comparable trends to the work ofGershfeld et al. (1988). Manoha et al. (2000) in-vestigated noise from a flat plane and demonstratedthe need for a specific Green’s function if surfaceterms are used for a Curle type acoustic analogy.Oberai et al. (2002) studied scattered noise fromairfoils by comparing the scattered noise from pre-scribed quadrupoles near the trailing edge to a LESperformed for the airfoil geometry. A much thinnertrailing edge that was excited by a vortex genera-tor slightly upstream was computed using a porousFWH method by Singer et al. (1999). Marsden et al.(2007) investigated the optimal shape change for the45◦ configuration to reduce the shed vorticity and asa result the noise.

PROBLEM DESCRIPTION

We simulate, using LES, the experiment by Olsonand Mueller (2004) where a 45◦ beveled airfoil ofheight h and chord c = 18h is placed halfway intothe exit nozzle of a low speed M = 0.1 flow at aReynolds number Rec = ρUc

µ = 1.9x106, Reh =ρUhµ = 105, 555 based on height, such that 9h is

outside of the nozzle. Half of the airfoil is simulatedas well as an extended wake region. In the experi-ment a microphone array is positioned at a distanceof 20h away and is arranged to capture a correctednoise component roughly perpendicular to the trail-ing edge point. The LES simulation solves the in-compressible LES equations by applying the filteroperation to the Navier-Stokes (N–S) equations andwith the closure terms modeled using the surrogatecorrelation time scale Lagrangian averaging schemefor the dynamic Smagorinsky model developed byVerma and Mahesh (2012). An example flow visual-ization is shown in figure 1 which shows the turbu-lent wake.

Figure 1: Trailing edge problem visualization usingthe instantaneous u-velocity.

Computational Solution Methodology

The governing equations for a fluid are the Navier–Stokes (N–S) equations:

∂ui

∂xi= 0,

∂ui

∂t+

∂uiuj

∂xj= −∂p/ρ

∂xi+

∂τij∂xj

,

(1)

where the density has been absorbed into the pres-sure term.

In LES, the large energy carrying scales of tur-bulence are directly resolved on the computationalgrid and the effect of the smaller scales is modeled.The flow variables are decomposed into large scales(denoted by (·)) and small scales (denoted by (·)′).Applying the filter operation to the N–S equationsand assuming commutation with the spatial deriva-tives yields the filtered N–S equations:

∂ui

∂xi= 0,

∂ui

∂t+

∂

xj(uiuj) = − ∂p

∂xi+ ν

∂2ui

∂xj∂xj− ∂τij

∂xj,

(2)

where (·) denotes the filtered flow variable at scale∆ and τij = uiuj −uiuj is the sub–filter scale (SFS)stress which in our implementation uses grid–filteredLES, meaning SFS = SGS. The SGS stress τijwhich are needed close eq. 2 are modeled using thesurrogate correlation time scale Lagrangian averag-ing scheme for the dynamic Smagorinsky model de-veloped by Verma and Mahesh (2012).

Acoustic Analogy Methodology

A porous FWH acoustic analogy Ffowcs-Williamsand Hawkings (1969) is used to obtain the far-fieldnoise at x from the near field, y. The porous FWH

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analogy works by aggregating sources derived fromflow quantities on a tagged data surface with out-ward pointing normal n and uses a Green’s func-tion to propagate that effect to the far-field. TheNavier-Stokes (N–S) equations are rearranged intoan inhomogeneous wave equation for the fluctuatingpressure as

p′(x, t) =1

4π|x|∂

∂t

∫Qi (y, tret)nidS

− xi

4πc0|x|2∂

∂t

∫Lij (y, tret)njdS

+xixj

4πc20|x|3∂2

∂t2

∫Vext

Tij (y, tret) dV.

(3)

where the emission time is tret = t− |x|c0

+ x·yc0|x|

The thickness, loading, and volume terms areexpressed as

Qj = ρ0vj + ρ(uj − vj), (4)Lij = ρui(uj − vj) + ∆Pij , (5)Tij = ρuiuj + Pij − c20(ρ− ρ0)δij , (6)

where Tij is the Lighthill stress tensor, e.g. Lighthill(1952), un = uini and vn = vini are the velocityof the fluid and the surface respectively, and Pij =(p−p0)δij−τij is the compressive and viscous stresstensor. The terms are computed either at the FWHsurface or in the exterior volume.

The implementation of equation 3 depends onspecifying integration surfaces and volumes. TheFWH surfaces are prescribed on background un-structured grids, which makes arbitrary surface ex-traction within the computational domain challeng-ing. We define arbitrary surfaces and discretize thesurface in a manner that reflects the volume grid inthe interior. We accomplish this by projecting thecentroids of the volume grid on the FWH surfaceand then generate a constrained Delaunay triangu-lation for the connectivity while its mesh dual, theVoronoi diagram, provides the projected face areas.Finally, we establish exterior and interior volumeswhich allow for surface integration to be handled intandem with consistent volume integration.

In addition, a dynamic endcap procedure is uti-lized to get the flux corrected surface terms as de-scribed in Nitzkorski and Mahesh (2014). Here theseprojected partner pairs are correlated to establish aconvection velocity which can be a function of thespatial extent. This approach takes the conceptssuggested by Wang et al. (1996) in terms of vol-ume fluxes and applies them to surfaces terms byassuming that the primary cause of the error is a

truncation error due to a finite domain which canbe corrected on the porous FWH surface. Trunca-tion errors and the corrections within this frameworkwere also investigated by Sinayoko et al. (2015). Theflux correction function is constructed as,

T̈ij(t) =∂2

∂t2

∫V0

Tij(y, tret)dy3

+∂

∂t

∫Sext

UcTij(y, tret)dy2.

(7)

The corrective flux ∂∂t

∫Sext

UcTij(y, tret)dy2 is ap-

plied at the exit surface of the bounding volume inorder to approximate the missing sources exterior tothe surface of integration.

We dynamically calculate the model constant,Uc, by correlating the the correction term over sub-sequent exit planes with the projected pair partnerswith the element normal n̂k. The distance betweenthe elements dxj,k establishes a correlation functionbased on the corrective flux to compute Uc as

Uc =

Nj ,Nk∑j=0,k=0

dxj,k · n̂k

max(β(t)R̂xy(m)

N )dt, (8)

with correlation function

R̂xy(m) =

∑N−m−1

n=0 αxn+m y∗n m ≥ 0

R̂∗xy(−m) m < 0

. (9)

The multiplying factor β(t) is a windowing func-tion in time which ensures only one maximum cor-relation over any given averaging period and an op-tional multiplying term α allows to normalize thesignal’s strength before the correlation to account fordecay in the signal over longer separation distances.Obtaining the convection velocity via correlation asopposed to from the time–averaged flow–field en-sures its application to non–stationary problems andsituations such as gusting inflow and curved bound-ing surfaces.

Acoustic Analogy Validation

In order to validate the FWH acoustic analogy wecomputed noise from a Re = 89000 cylinder andcompared the results to experimental data of Revellet al. (1977) who placed a cylinder into the exhaustof a nozzle and captured the noise with a microphonearray and the unsteady Reynolds–averaged Navier–Stokes(URANS) computation of Cox et al. (1998).The noise data was computed using 180D/U∞ units

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of time and the Nyquist frequency was 500 Hz. Thenoise at θ = π/2 and r = 128D as shown in figure 2shows good agreement at the fundamental frequencyas well as the drag and first lift harmonic. The firstlift harmonic for the current results show some shiftrelative to the experiment but our results show thefrequency at f ≈ 3f0 where the first overtone of liftshould reside. In contrast, the URANS predicts noharmonics of lift and drag since it is a 2D computa-tion and has a larger shift in the Strouhal frequency,St = f0U/h, relative to current results. The soundfield is visualized by solving the FWH equations ata fixed time throughout the domain and the wake isvisualized by λ2 in figure 3. It is clear that the prin-cipal source of sound is associated with the cylinderbody and that the wake provides higher frequencycontent. Overall the agreement with experiment isquite good and significantly better than the URANS.

SPL(

dB)

St

Figure 2: The frequency content of the generatednoise, SPL(dB) vs frequency, at the location θ =π/2 and L=128D is compared to the experimentsof Revell et al. (1977) and the computations of Coxet al. (1998).

Computational Setup

The simulation domain was x/h = [−9, 35], y/h =±20, and z/h = 20 with 150 points discretizing thespan and approximately 72 million total control vol-umes. The wall spacing was ∆n = 2.5x10−5 nearthe trailing edge and ∆n = 5x10−4 in the bound-ary layer. A timestep of ∆t = 5x10−5 was used.The trailing edge tip was rounded with a radius of0.0005h and an extra dense region within a normaldistance of 2h from the wall was used to resolve the

Figure 3: An iso-surface of λ2 colored by u-velocityand the sound field is visualized by solving the FWHequations for a Re = 89, 000 cylinder.

separation and wake interaction. The acoustic calcu-lations have been performed with a porous FWH im-plementation with the porous surfaces located 1.5haway from upper and lower surfaces and 1.5h down-stream of the trailing edge point. The grid refine-ment in this area provides high quality resolution re-sulting in almost a third of all processors having con-trol volumes cut by these porous surfaces. A secondseries of FWH surfaces were placed 5h downstreamto ensure convergence and the computed noise variednegligibly so the nearer FWH surfaces are used forreported data. A grid schematic is shown in figure4 which also shows the FWH planes.

Rec

ycle

Res

cale

Inflo

w

Bottom oversized BC’s

Top oversized BC’s

Figure 4: A schematic of the grid with FWH planesshown in yellow.

Freestream boundary conditions when appliedto the top and bottom were found to not producethe correct mean pressure distribution along the air-foil for this computational domain. Therefore anoversized domain, including the upstream section,

4

of (x/h, y/h) = ±100 was used and a semi-cylinderfore-body for the airfoil was simulated. Since theentire airfoil was simulated and the domain was sig-nificantly larger than the body a far-field conditionof u = u∞ was applied on the inflow as well as thetop and bottom. An outflow Neumann conditionwas applied at the exit and periodicity enforced inthe span. The pressure distribution along the airfoilsignificantly improved with this simulation, so thetime averaged mean velocity at the location of thetop and bottom of the smaller domain was extractedand applied as the far-field boundary condition forthe finer calculation. The u-velocity on the upperboundary was represented by a fifth order polyno-mial whereas v-velocity on the top and u, v-velocitieson the bottom were solved with a ninth order poly-nomial. This procedure is similar to that of Wang(2005) and Marsden et al. (2007) who used a RANSsimulation performed on a much larger domain andalso found the pressure distribution to be in closeragreement. The effect of this approach on Cp alongthe airfoil is shown in figure 5.

Cp

x/h

Figure 5: Coefficient of pressure from all the meth-ods: (–) current with adjusted BCs, (-.) currentwith freestream BCs, (x) Olson and Mueller (2004),(− −) Wang (2005) with adjusted BCs, (. . . ) Wang(2005) freestream BC.

The recycle-rescale methodology of Lund et al.(1998) with momentum height ξ was employed onboth the upper and lower surfaces of the trailingedge with a prescribed Reξ = ρUξ

µ = 1895 and 1760

respectively, consistent with Wang (2005) for the ex-perimental thickness case. For the thinner case theinflow Reξ was eighteen times smaller. The recycleplane was set to be x/h = 1.75 or approximately

10δupper downstream of the inflow condition. Thevertical extent of the recycle domain is 1.5δ, whichis smaller than the vertical extent of the computa-tional domain. Therefore, at the inflow plane, therecycle values are prescribed at y-locations smallerthan the vertical extent of the recycle domain. Atlarger y locations, the streamwise velocity is set toits freestream value and the vertical velocity is ob-tained from a quadratic curve constrained by thevertical velocities at the top of the recycle domainand the computational domain respectively.

The noise was calculated at (x/h, y/h) = (3, 21)above the trailing edge for point SPL comparisonswith the experiment and at a 180 points with a ra-dius of 100h for directivity with total sampling timefrom start to finish of ∆T = 100 and a samplingtimestep of ∆t = 5e−5 giving a spectral discrimina-tion of 0.01Hz and a Nyquist frequency of 10, 000Hz.In the experiment a 10Hz pass band filter with aNyquist frequency of 4000Hz was used as opposedto our approach with a Hann filter using 50% over-lapping. Wake data as well as available noise datais compared to Wang (2005) and Olson and Mueller(2004). Additional noise calculations such as soundsources both on the trailing edge as well as withinthe wake are presented.

RESULTS

For each of the boundary layer thicknesses the flowfield is analyzed and in the case of the experimentalthickness, compared to computational and experi-mental data. After this, the acoustics for the exper-imental thickness are discussed and compared withavailable data. Then the smaller boundary layeracoustic data is compared and contrasted with thethicker boundary layer. The combined results areanalyzed together to establish the change in noiseproduction and the reason for this difference.

Flow-field validation

The experimental boundary layer thicknesses on thetop and bottom are δ99 ≈ 0.1712h and 0.1642h re-spectively. The boundary layer profile comparisonsalong the upper surface boundary are shown in fig-ure 6. A set of follow-on experiments to the work byOlson and Mueller (2004) was conducted by Shan-non and Morris (2006). They found that dominantphase behavior indicating a dominant frequency waspresent. The St number for the principal oscilla-tion of the wake was found by Shannon and Morris(2006) to be 0.42, Olson and Mueller (2004) found

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0.40, Wang (2005) to be 0.44, and we calculate it at0.41.

(y−y w

all)/h

u/U∞

Figure 6: Comparison of the boundary layer at x-locations of x/h = −7.5,−3.5,−2.5,−1.625,−1.25,and −0.875 shown with an offset, between (–) cur-rent, (- -) Wang (2005), (· · · ) Olson and Mueller(2004).

The mean streamwise wake profiles downstreamof the trailing edge are shown in figure 7 which showclose agreement between all methods. The evolutionof the rms values of u-velocity are shown in figure 8.It is clear that the wake and the fluctuations nearthe wake edge are accurately captured. This wouldsuggest that the flowfield fluctuations in the experi-ment and those in the simulation are directly relatedand since these fluctuations and pressure fields de-termine the noise that sound should be comparable.

There are no direct comparisons which can bemade to validate the flow-field for the thinner bound-ary layer. A flow visualization showing the u-velocity which clearly demonstrates the thinness ofthe boundary layer is shown in figure 9. This is fur-ther reinforced with the boundary layer profile com-parison in figure 10 which shows how much thinner.The momentum thickness θ is initialized to be 18times thinner but grows faster to be only 12 timesthinner at x/h = 3 before the acceleration over thetrailing edge thins the boundary layer again. Alsonote how the boundary layer at x/h = −1.625 closeto the beginning of the bevel thins much faster thanin the thicker case.

Acoustic sources

The acoustic sources can be decomposed for fur-ther investigation to their principal components of

y/h

u/U∞

Figure 7: Mean u-velocity wake profiles between (–) current, (- -) Wang (2005), (· · · ) Olson and Mueller(2004), shown with an offset at wake x-locationsx/h = 1, 1.5, 2, 3, 4, 6

the fluctuating surface pressure and the Lighthillstress tensor in the volume. The source terms inthe volume stress tensor are principally the Reynoldsstresses which are related to the Lamb vector, thecross product of vorticity and velocity, ω⃗ × u⃗. Theacoustic sources for both are examined using thetime average of their absolute values.

The surface terms are described by the load-ing term and are most strongly influenced by thepressure fluctuations dp/dt. In figure 11 the surfaceforces are approximated by the mean of |dp/dt| at ev-ery location dot multiplied with the local normal togive magnitude of the fluctuating pressure. It is clearthat the trailing edge point produces most of thepressure fluctuations and therefore is the strongestsource of sound, however, for the thicker boundarylayer a significant amount of fluctuating pressure oc-curs near the separation point. Related, the thinnerboundary layer with the later separation shows muchless source contributing over the top of the bevel.

The volume sources are directly related to theReynolds stresses. The region of activity for bothcomponents in both extent and magnitude have di-minished for the smaller boundary layer case. Notethat the truncation which appears in the near-fieldis due to the extraordinarily fine region of resolutionwhich wraps body at a distance of 2h away from thesurface.

The time averaged volumetric source fields areshown in figures 13 and 14 and show similarities tothe Reynolds stress fields. These regions are associ-ated with the Reynolds stress component u′u′ in par-

6

y/h

urms/U∞

Figure 8: Mean u-velocity rms wake profiles be-tween (–) current, (- -) Wang (2005), (· · · ) Olsonand Mueller (2004), shown with a 0.2 offset at wakex-locations x/h = 1, 1.5, 2, 3, 4, 6

Figure 9: A visualization of the thin boundarylayer case of u-velocity.

ticular, as the vorticity is released from the wall andis accelerated in the downstream direction. There-fore the large magnitude vorticity vectors are nearlyperpendicular to the background velocity which pro-duce large amounts of noise. The sources of soundfor the thin boundary layer case are similar to thoseof the larger case, except decreased in amplitude andextent. The noise production associated with theshear layer on the upper surface is diminished dueto the thinner momentum thickness layer which in-teracts over a shorter length. The ejection of thethe lower boundary layer shows that the magnitudefor the thinner case is slightly diminished but is stillfairly similar to the thicker boundary layer in thisregion.

(y−

y wall)/h

u/U∞

Figure 10: Boundary layer profiles compared withthe thicker case.

Acoustic results and effect of boundary layerthickness

The noise production from both boundary layerthickness cases are evaluated by examining the pointspectra at a given location. The noise at a point(x/h, y/h) = (3, 21) in the is shown in figure 15.Also shown is the experimental measurement ob-tained using a two-point acoustic array. The Nyquistfrequency of the experiment is 4000 Hz and a 10Hz passband filter is applied to the data; in con-trast, the Nyquist frequency of the computations is100000 Hz and no passband filter is used. The thickboundary layer case shows good agreement with ex-periment. The mismatch at low frequencies is char-acteristic of such comparison across computationaldata, posited to be ambient low frequencies. At highfrequencies the simulations capture more sound dueto the higher Nyquist frequency which is also higherthan that of Wang (2005).

DISCUSSION

We have established that the fluctuating pressurescattered by the surface and the near field turbu-lence serve as sources the far-field acoustic pressure.The scaling laws which relate these quantities werethoroughly described by Blake (1975) and Blake andGershfeld (1989) associated with fluctuating pres-sure strength and pressure velocity correlations inthe turbulent wake and scattered to the far-field asdescribed by Williams and Hall (1970) for the trail-ing edge case.

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(a)

(b)

Figure 11: Major component of the surfacesources, |dp/dt|, plotted as the magnitude of the vec-tors for for (a) experimental boundary layer case and(b) thin boundary layer case.

It was shown by Blake (1975) for the 45◦ casethat the pressure frequencies found in the near fieldcontained similar content, but that at non-peak fre-quencies they quickly diminished even at x/h = 1.1away from the test body. The fluctuating pressurehad higher rms values near the body as well as moredistinguished overtones. We use this near field fluc-tuation for investigation by examining the normal-ized pressure in the near field (x, y) = (0.5, 0), shownin figure 16 and comparing it against the surfaceforces shown in figures 11 for their location inten-sities. The Blake experiments had two velocities,U∞ = 15.24, 30.48m/sec, and found that for thelower speed with the thicker boundary layer that av-erage prms was 2.3 times higher due to the increasein the broadband components. The tonal quality at

(a)

(b)

Figure 12: Reynolds stress components (a) u′u′

and (b) v′v′ for the thin boundary layer case areshown.

Figure 13: Mean of |∂2Tij/∂t2| for thin boundary

layer case.

this lower speed however was not as prominent andthis provided the explanation that the two principalsources were the periodic ejection of vorticity caus-ing the tonal pressure fluctuations associated with ascattered dipole and the setting up of the convectingHelmholtz vorticity in the near field which radiatesin a broadband fashion.

Interestingly, in the experiments, doubling themean background velocity had an amplifying ef-fect on the fluctuating components, but the overallStrouhal frequency did not shift. This implies thatthe body shape was highly responsible for the typeof shedding for these geometries. Blake and Gersh-feld (1989) provide a diagram showing which char-acteristics govern source production for a range ofReynolds numbers and angles of attack conditioned

8

Figure 14: Volumetric sound sources for experi-mental boundary layer case.

SPL(dB)

fh/U∞

Figure 15: Comparison of noise from (–) currentthick, (- -) current thin, (-.) Olson and Mueller(2004), and (· · · ) Wang (2005)

on the body shape.

In the experiments of Blake (1975), the pressure-velocity correlations for the 45◦ case at U∞ =30.48m/sec and Re = 1.9x6 were fairly constant withsome subtle curvature variation only in the first 1.5h.This correlation in the upper shear layer was used toidentify the transition region based on the recoverydistance. The vortex was found to shed in the first1.38h with a recovery/pairing off zone that lastedtill 2.1h after which the correlation was monotonic.It is over this length scale in the setup region thatour results indicate Lighthill tensor based contribut-ing sources due to the wake velocity being differentfrom the convection velocity, acting on the vorticityreleased form the body as the the von Karman streetis established.

As formulated by Blake and Gershfeld (1989),the relation between the received (r) and surface

10Log10

ΦppU

∞/h

q2 ∞

fh/U∞

Figure 16: The pressure spectrum variation be-tween a point on surface x1/h = (−.375, .375) and apoint in the near-field x2 = (0.5, 0) referenced off thetrailing edge for both boundary layer thicknesses: (--) Thick x1, (-.) Thick x2, (–) Thin x1, (...) Thinx2.

,

source (s) fluctuating pressure is,

p2r(r⃗, ωs)

p2s(y1)≈ 1

2π2

U∞

c0

2L3

yf

|y1 − ys|Λ3

r2| sinϕ| sin2 θ

2,

(10)

where yf is the approximate thickness of the wakeas measured by fluctuating velocity, y1 − ys is thedistance between the fluctuation on the surface andthe separation point, Λ3 is the spanwise correlationlength and the angles θ for the in-plane angle andϕ for the out of plane angle. We use equation 10 toestimate the impact of the change in boundary layerthickness between the thick and thin cases. The ra-tio of wake thicknesses is yf,rat ≈ 0.68h/0.11h =6.2. By using the normalized fluctuation strengthalong the surface and averaging at the distancesfrom the trailing edge tip we estimate the ratio of|y1−ys|rat ≈ .63h/.27h = 2.33. Gershfeld calculatesthe spanwise correlation distance by using boundarylayer edge values, but we lacking that we estimatethe correlation from the surface pressures to roughlyobtain the ratio of Λ3,rat ≈ .8. Therefore the esti-mated difference in acoustic pressure is 5.22dB whichis comparable to the found 5.45dB difference. Thiscorresponds to the low frequency offset and it shouldbe noted in figure 15 that the noise does not have adistinct peak frequency for the thin boundary layercase and that the decibel level drops quite quickly tomuch lower level over a broad range of frequencies.

9

The broadband components are also discussedby Blake and Gershfeld (1989) associated with theanalytical expansion by Howe (1975) for the acous-tic pressure with regard to maintaining the Kuttacondition for a half-plane. The derived governingequation is,

pa(x, t) =ρ0Γ3Uc sin(θ/2)

2π√r

(1− Uw

Uc

)[cos θ0/2

r1/20

]tret

,

(11)

where Γ3 is the circulation of the spanwise vortexcoming off the trailing edge and Uc and Uw are theconvection velocity and the wake velocity respec-tively. θ is directivity angle in the plane runningparallel to the inflow and θ0 is the polar angle. Ifthe wake and convection velocity are the same, nonoise would be produced, but as observed by Yu andTam (1978) usually Uw = 0.6Uc. Also of note is howthe broadband components are not influenced by thethickness of the wake, but by the velocity scales.Specifically, since we reduced the physical height ofthe boundary layer, but not the velocity, the changesin the strength of the vorticity Γ3 is nearly inverselyproportional to the difference in convection and wakevelocity Uc − Uw meaning that the broadband com-ponents between the two cases vary little. This isdifferent from the experiments of Blake and Gersh-feld (1989) where the velocity was varied and re-sults in a non-linear difference between the vorticalstrength and the wake velocity characteristics. Thismechanism explains the close agreement between thebroadband components for our two boundary layercases.

EFFECT OF GREEN’S FUNCTION

A significant motivation to develop the porous FWHmethodology is that it enables the use of the free-space Green’s function irrespective of geometricalcomplexity. In this section we demonstrate thereasonableness of this approach by comparing thesound prediction to that obtained using only thesurface terms and half-plane Green’s function. Thehalf-plane Green’s function provides a good repre-sentation for the trailing edge. It was describedby Williams and Hall (1970); from which Crighton(1972) then provided a two-dimensional compactGreen’s function representation. Howe (1975) thenrepresented the Green’s function in physical space

using the method of stationary phase to achieve,

G(x,y, t− tret) ≈φ(x)φ(y)

4π|x|δ(t− tret),

x = (rxcos(θx), rxsin(θx), x3,x),

y = (rycos(θy), rysin(θy), y3,y),

(12)

where φ(.) =√r∗sin(θ∗/2), for each location, (.)∗,

defined in polar coordinates.The acoustic calculations have been performed

with a porous FWH implementation with the poroussurfaces located 1.5h away from upper and lowersurfaces and 1.5h downstream of the trailing edgepoint as well implementing 12 using surface terms.The grid refinement in this area provides high qual-ity resolution resulting in almost a third of all pro-cessors having control volumes cut by these poroussurfaces. The point noise, figure 17, is computed us-ing both approaches for the experimental boundarylayer case. It is clear that the differences are slight.This points to the considerable attractiveness of ourporous FWH methodology and the effectiveness ofour dynamic endcap approach.

SPL

(dB

)

St = fh/U∞

Figure 17: Point sound comparison using the half-plane Green’s function with surface terms or theporous FWH approach.

CONCLUSION

We performed LES paired with our FWH methodfor a 45◦ beveled trailing edge. The flow fields showgood agreement with experiment. The approach forobtaining this agreement involved using an oversizedlower fidelity computation in order to obtain reason-able boundary conditions to achieve a comparable

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coefficient of pressure with experiment. The use ofthe recycle rescale methodology was also importantin order to get boundary layers that were in closeagreement with the experiment. These boundarylayers dictate both separation as well as fluctuat-ing Reynolds stress components which were in closeagreement with those previously observed.

Between the two boundary layer thicknesses, thevorticity shed into the wake is larger for the thin-ner boundary layer, but since it separates later, thevalues of ω⃗ × u⃗ are lower. Also, this thinner bound-ary layer results in a smaller region of source dis-tribution. On the lower surface of the trailing edgeboundary layer separation happens at roughly thesame location and ∂2u′u′/∂t2 is the dominant sourceterm which is produced over a thinner region but isalmost as intense and therefore produces comparablenoise. This accounts for the increased noise in theupper half plane for the thicker boundary layer rela-tive to the difference in the lower half plane betweenthe the two thicknesses.

Our porous FWH methodology is shown toagree well with sound predicted using the surfaceterms and a half-plane Green’s function. This pointsto the considerable promise of our methodology topredict the sound from turbulent flows and complexconfigurations.

ACKNOWLEDGMENTS

This work was supported by the United States Officeof Naval Research (ONR) under Grant No. N00014-14-1-0304 with Dr. Ki-Han Kim as technical moni-tor. Computing resources were provided by the DoDHPCMP Open Research Systems, the Minnesota Su-percomputing Institute, and in part by the NationalScience Foundation through XSEDE resources pro-vided by the XSEDE Science Gateways program.

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