AIAA 2001–2196 AERODYNAMIC NOISE PREDICTION USING PARALLEL METHODS ON UNSTRUCTURED GRIDS

AIAA 2001–2196

AERODYNAMIC NOISE PREDICTION USING PARALLEL METHODS ON UNSTRUCTURED GRIDS

Lyle N. Long, Frederic Souliez and Anupam Sharma Department of Aerospace Engineering

The Pennsylvania State University, PA-16802

X

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Ω (sec ^-1 ) 14274.9 12723.3 11171.7 9620.06 8068.45 6516.83 4965.21 3413.59 1861.98 310.359

7th AIAA/CEAS Aeroacoustics Conference May 28–30, 2001/Maastricht, The Netherlands

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

USING PARALLEL METHODS ON

UNSTRUCTURED GRIDS

Lyle N. Long



, Frederic Souliez y

and Anupam Sharma y

Department of Aerospace Engineering

The Pennsylvania State University, PA-16802

Aerodynamic noise from a cone in a uniform ow is computed using the Ffowcs

Williams-Hawkings(FW-H) equation. The time accurate ow datais obtained using a

nitevolume owsolveronanunstructuredgrid. TheFW-Hequationissolvedforsurface

integralsoverapermeablesurfaceawayfromthecone. PredictionsfromtheFW-Hcode

arecomparedwithdirectcalculationsbythe owsolveratafewobserverlocationsinside

thecomputationaldomain. Averygoodqualitativematchisobtained. Sounddirectivity

patterns in theazimuthal and in thelongitudinaldirectionsare presented. TheFW-H

codeisalsovalidatedagainstamodelproblemofamonopoleinauniformmean ow.

Nomenclature

C

p

coeÆcientofpressure

C

s

sub-gridscaleconstantin Smagorinskymodel

c soundspeedin quiescentmedium

d basediameterofthecone

f

s

vortexsheddingfrequency(Hz)

H(f) Heavisidefunction,H(f)=0forf <0and

H(f)=1forf >0

L

i

referEq. 2

L

M

L

i M

i

L

r

L

i

^ r

i

_

L

r

_

L

r

^ r

i

M localMachnumbervectorofthesource

M jMj

M

n

M

i

^ n

i

M

0

U

0

=c

M

r

M

i

^ r

i

_

M

r

_

M

i

^ r

i

^

n unitnormalvectortothesurface,n

i

P

ij

compressivestresstensor

withp

o Æ

ij

subtracted

p pressure

p

0

freestreampressure

p 0

acousticpressure,p p

o

p'

rms

rootmeansquaredpressureperturbation

ret retartedtime

T

ij

Lighthillstresstensor

t observertime

 angularlocationoftheobserver

U averagedstreamwisevelocity

U

0

;U

1

freestreamvelocity



Professor,Assoc. Fellow,[email protected].

y

Graduate Research Assistant, Pennsylvania State Univer-

sity

Copyright c 2001byLyle N. Long, ThePennsylvaniaState

University.PublishedbytheAmericanInstituteofAeronauticsand

Astronautics,Inc.withpermission.

U

i

referEq. 2

U

n

U

i

^ n

i

_

U

n

_

U

i

^ n

i

U

_ n

U

i _

^ n

i

u

i

componentsoflocal uidvelocity

u 0

avg

averagedstreamwiseperturbationvelocity

u

n

u

i

^ n

i

v

n

localnormalvelocityofthesourcesurface

Æ(f) Diracdelta function

Æ(f)=1forf =0,otherwiseÆ(f)=0

Æ

ij

Kroneckerdeltafunction,

Æ

ij

=1fori=j, otherwiseÆ

ij

=0

 densityof the uid



0

freestreamdensityofthe uid

 0

densityperturbation, 

o

vorticity(s 1

)

! angularfrequencyofthemonopolesource

 2

waveoperator, 2

( 1

c 2

@ 2

@t 2

r 2

)

Introduction

R

ECENTLY, the Ffowcs Williams-Hawkings

(FW-H)equationhasbeenusedwithpermeable

surfaces for predicting aerodynamic noise. The

applicationofFW-H inthis mannereectivelyallows

fortheinclusionofthequadrupolesourcetermsinside

the surface without performing volume integrations.

This hassignicantlyimprovedtheaccuracy of noise

prediction for cases where the contribution from

nonlinear interactions in the ow cannot be ignored.

Thisis typicalof highlyturbulent ows,forexample,

highReynoldsnumberjetsandwakes.

TheFW-Hequationrequirestimeaccuratedataon,

andin thevolumeinside thepermeablesurface. This

datais usuallyobtainedbysolvingtheEuler/Navier-

Stokesequationsaccuratelyin time. SincetheFW-H

equationusesdatafromwithintheFW-Hsurface,the

accuracy. Unstructured gridsprovidegreat exibility

in distributingthegridin thedomain, andhencecan

beused to cluster thecellsinside theFW-H surface.

This featurecanbeexploited tosignicantlyincrease

thecomputationspeedwhilekeepingalmostthesame

accuracy in predicting aerodynamic noise. This will

also permit themodeling of complexgeometriessuch

ashelicopterfuselages,landing gear,and aps.

Thegoalhereistotestthecombinationofunstruc-

turedgridswiththeFW-Hequationinpredictingthe

aerodynamic noise. The test case is chosen to be

the ow over acone. A cone has sharp edges which

xestheseparationpoint. This makesthe owfairly

Reynold'snumberindependent.

We use the Parallel Unstructured Maritime Aero-

dynamics (PUMA) 1

code for generating the time-

accurate ow data. PUMA has been validated for

time-accurate computations.

2{4

The ultimate aim is

topredicttheairframenoisefromcomplexgeometries

suchaslanding gear,slats, and aps. This conecase

maybeconsideredasabenchmarkproblem.

The Grid

Thegrid usedfor thesimulationof the owovera

coneofvertexangle60 o

wasgeneratedusingGridgen.

Figure1showsanoverallviewofthemeshconsisting

of approximately280,000 tetrahedra. The clustering

wasdonearoundtheconeandinthewakeregionwith

increasingcellsizetowardstheouterboundariesofthe

computationaldomain. Thereasonforusing Gridgen

comes from one interesting feature of this commer-

cialsoftware: arbitrarysurfacescanbecreatedaround

thecone(onewithintheCFDdomainboundariesand

the other beingthe CFD domain boundary)and are

sources for the meshing algorithm. It is possible to

exportseparatelyanyoftheseclosedsurfacesinasep-

aratele,providingameanstoextract owdataonthe

surfaceusing aFW-H module that wasaddedto the

unstructuredsolver. Thesmallestcylinderwasusedas

aporousFW-H surface. Attheboundingfacesofthe

CFDdomain, Riemann boundaryconditionswere as-

signedateachfacecenter,henceminimizingre ections

from the boundaries into the computational domain.

Thelarge cellsin thefar-eld also help dissipateany

re ections. A no-slip conditionwas used at thesolid

surface, even though the boundary layerwas not re-

solveddueto computerlimitations.

Byusingasetoffacesthatareactuallyusedbythe

owsolverduring the computation, there isno addi-

tional work required to extract the data needed for

thefar-eldnoise. ThistypeofFW-Hsurfacealsore-

ects thetruemesh clusteringpresentwherethe ow

variables arelocally being computed: there isno loss

in accuracy due to the interpolation onto a surface

whose renementmightnot be that of the computa-

tionalgrid. Sinceonlythesurfacetermsareevaluated

Fig.1 Overall viewofthe280,000 cellmesh.

duringtheacousticpredictionprocedure,onedoesnot

haveto takeinto accountanyphenomenon occurring

outsidethe integration surface. The surfacecanalso

crossregionsdominatedbynonlineareects.

Duringarun,thefaces(trianglesinthiscase)would

be identied and agged on each CPU, so that face

data would be output at a prescribed sampling rate

(around 50 kHz in the present case): the sampling

wasdone in such amanner that one had at least 20

data points per wavelength, the shortest wavelength

being 10 times that of the simulated shedding fre-

quency. To avoid any redundant data, faces shared

between two adjacent CPUs had to be identied at

thebeginningofeachrun,sothatthenumberoffaces

whose data are output is identical to the number of

triangles onthe actualFW-H surface. The gridpar-

titioning being done dynamically each time a run is

initialized,theglobalcellindexingchangesfromrunto

run,makingitnecessarytoruntheabove aggingpro-

cedureanytimetheprogramisrestarted. Thismakes

theroutineindependentofthenumberofCPUsbeing

used. Figure 2 illustrates the regions on the surface

shared between 8 processors using the Gibbs-Poole-

Stockmeyerreorderingalgorithm.

5

As expected,each

region is a neighborto at mosttwo other partitions,

minimizingtheamountofinter-processorcommunica-

tion.

Thetimestepneededforatime-accuratesolutionis

determined by the smallestcellcharacteristiclength.

This is estimated to be one third of the cellvolume

divided by the maximum face area. For the grid

described above, this yields a time step of 9.45E-08

second at a CFL number of 0.9. The shedding fre-

quencyfoundduringtheexperimentalinvestigationof

the owis36Hz,foraStrouhalnumberequalto0.171.

X Y Z

CPU 8 7 6 5 4 3 2 1

Fig.2 Partitioning ofthe FW-H surfaceacross8

processors.

TheStrouhalnumberwasdenedbasedontheconedi-

ameterasSt=f

s d=U

1

. Thenumericalsimulationis

performedatMach 0.2atstandardatmosphericpres-

sure and temperature conditions, with an increased

viscosityto matchtheexperiment'sReynoldsnumber

(50,000). Scaling the Strouhal number to the simu-

lation'sMach number yields ashedding frequency of

230Hz. Thecomputationofacompletesheddingcycle

requiresroughly46,000iterations.

The Flow Solver - PUMA

PUMA is a computer program, written in C, for

the analysis of internal and external non-reacting

compressible owsoverarbitrarycomplexgeometries.

PUMA uses the Message Passing Interface (MPI) to

run the code in parallel. It can berun on arbitrary

number of processors with very good scaling perfor-

mance. Several papers 2,4

detailthebenchmarking of

theperformance,and validationofPUMA.

PUMAisbasedonnitevolumemethodsand sup-

portsmixedtopologyunstructuredgridscomposedof

tetrahedra, wedges, pyramids and hexahedra. The

code may be run to preserve time accuracy for un-

steady problems, or may be run using a pseudo-

unsteady formulation to enhance the convergence to

the steady state. Primitive ow quantities are com-

puted at thecell centers. The code can be restarted

from anypointoftime at which thesolutionisavail-

able from previous computations. All ow variables

arestoredwithdoubleprecision,butmaybeoptionally

storedassingleprecisiontosavememoryandcommu-

nicationtimeat thecostofreducedprecision.

Parallel Machines

ComputationalAeroacoustics(CAA)codesareusu-

ally verycomputationally intensive. Even with very

months togiveresults. Parallelcomputing using Be-

owulfclustersoersaninexpensivewaytohandlesuch

time-consuming simulations in reasonableamount of

time.

Three facilities oering parallel computational

power at Penn Statewere used forthe computations

- COst eective COmputing Array (COCOA), 2

CO-

COA2andLionX.

6

COCOAisaBeowulfclustercom-

prising of 25 machines each having dual 400 MHz

PentiumII processor. This facilitywasassembledby

theauthorsandtheircolleaguesin theDepartmentof

Aerospace Engineeringat Penn State. The machines

are connected via fast-Ethernet network which can

supportupto100Mbpsbandwidth. AsingleBaynet-

works 24-port fast-Ethernet switch with abackplane

bandwidthof2:5Gbpsisusedforthenetworking.All

theprocessorsarededicatedto runparalleljobs. The

operating system is Red Hat Linux. Message Pass-

ing Interface(MPI) is used for parallel programming

andtheGnuCcompilerisusedforcompilingPUMA.

DetailsregardingsettingupandbenchmarkingofCO-

COA may be obtained from Modi and Long 2

and

COCOA'swebsite.

7

COCOAwasprimarilysetuptomakeparallelcom-

puting facility readilyavailable to the CFD groupof

theAerospaceEngineeringDepartmentat Pennsylva-

niaStateUniversity. Thetotalcostoftheclusterwas

just $80;000 in the year 1998, when it was set up.

Since then this facility has been intensively used for

variousCFDsimulations. COCOA2isanewlyassem-

bled Beowulf cluster at Penn State. It has 21 nodes

eachhavingdual800MHzPentiumIIIprocessorsand

1GB RAMeach. Thecluster hasdual fast-Ethernet

per node and all the nodes are connected using two

HP2524switcheswithchannel bonding.

Figure4plotstheparallelspeedupforCOCOAand

COCOA2(1M op=onemillion oatingpointopera-

tionspersecond). Fairlygoodperformanceisobtained

considering the small size of the problem. Figure 4

showsthereduction in the op rateperprocessoras

the grid points are distributed over a larger number

of processors. This trend is typical of Beowulf clus-

ters asthe ratio of computation overcommunication

decreases.

LionX is also a Beowulf cluster with 32 machines

(each having dual 400 MHz Intel Xeon processors).

These machines are connected via Myricom Myrinet

withwirespeed1:28Gbps. LionXalsousesLinuxwith

MPI for parallel programming. Performance com-

parison and benchmarking results for LionX can be

obtainedfrom itswebsite.

6

CFD Results

Afterinitializingallvariablestothefreestreamval-

ues,local timesteppingis usedto acceleratethecon-

vergence towards a physically realistic ow. This is

No. of processors

M F lo p s

0 5 10 15 20 25

0 100 200 300 400 500 600 700 800 900 1000

COCOA COCOA ideal COCOA2 COCOA2 ideal

Fig.3 ParallelspeedupforCOCOAandCOCOA2

No. of processors

M F lo p s p e r p ro c e s s o r

0 5 10 15 20 25

15 20 25 30 35 40 45

COCOA COCOA2

Fig.4

donebyassigningtoeachcellthemaximumallowable

time stepforagivenCFLnumberbasedon each cell

characteristiccelllength. Globaltimesteppingisthen

turned on forseveral cycles before data are sampled,

to ensure that the data on the FW-H surface follow

theequations ofmotion. Figure 5illustrates thevor-

ticitypatternsinthewakeofthecone,showingstrong

recirculation phenomena. The noise from this recir-

culation is predictedby theFW-H module. Figure 6

is theaveragedstreamlinecontouroveroneshedding

period, illustratingtheaxisymmetric bubblethatwas

observedduringCalvert'sexperimentalstudy.

8

In orderto validatethe solution, multiple compar-

isonsweremadebetweenthesimulationandtheexper-

imental measurements. AbasicSmagorinksysub-grid

scaleturbulence model 9

wasadded to the ow solver

inordertoimprovethepredictions,sincealarge-eddy

simulationshouldyieldbetterturbulentquantities.

X

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Ω (sec ^-1 ) 14274.9 12723.3 11171.7 9620.06 8068.45 6516.83 4965.21 3413.59 1861.98 310.359

Fig.5 Instantaneous vorticitycontours.

X

Y Z

Fig.6 Averagestreamlinesoveronesheddingcy-

cle.

Figure 7 shows the averaged streamwise velocity

prolescomputedbytheoriginal owsolverandthose

computedbythesamesolvercombinedwithanLES.

In all three cases the magnitude of the reverse ow

velocity is under predicted when compared with ex-

perimentalmeasurements.Thepredictionsagreefairly

wellwithCalvert's datain termsof thelengthofthe

recirculationzone. Pastthe stagnationpoint,the re-

sultsincludingLESmodeling followtheexperimental

curvemorecloselythanthose computed withoutany

turbulencemodel.

Figure8illustratesthevariationofthepressureco-

eÆcient C

p

along the wake centerline. In this case,

the LES having the largest sub-grid scale constant

C

s

greatlyover-correctsthepressuredropin thenear

wake of the cone. The LES using a Smagorinsky

constantof 0.10matchesthe measuredpressuredata

x/d U /U ω

1 2 3

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

experiment numerical numerical - C _S = 0.10 numerical - C _S = 0.25

Fig. 7 Comparison of the averaged streamwise

velocity withexperiments for the owsolverwith

and withoutLES.

x/d

C p

0.4 0.8 1.2 1.6 2 2.4 2.8 3.2

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

experiment numerical numerical - Cs = 0.10 numerical - Cs = 0.25

Fig. 8 Comparison ofthe Cp coeÆcient withex-

perimentforthe owsolverwithandwithoutLES.

verywelluntilthestagnationpointisreached. These

results are consistent with those found in other re-

lated investigations, using either the k- turbulence

model 10

orthek--v 2

model.

11

Thesesimulationswere

compared against a set of experiments 12

at a lower

Reynolds number (42,000). Madabhushi 13

also used

an LES with as many as 850,000 mesh points, but

completely over-predicted thelength oftherecircula-

tionzone.

Figure 9 shows that the averaged streamwise per-

turbation velocity is not well predicted using any of

thesub-gridscaleconstants. Withthegridcoarsening

in thefar wake,the uctuatingvelocitiesaredamped

veryrapidly asonegoesawayfrom theconebase. It

is the ow solverwithout any turbulencemodel that

x/d u ’ a v g /U ω x 1 0 0

0 1 2 3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

experimental numerical numerical - C _S = 0.10 numerical - C _S = 0.25

Fig.9 Comparisonofaveragedstreamwisepertur-

bationvelocitywithexperimentswithandwithout

LES.

yieldsunsteadyvelocityvaluesthat areclosesttothe

experimentaldata. ThesolutionwithoutLESwasse-

lectedtotrytopredictthefar-eldnoise. Italsoleads

to the conclusion that a more advanced turbulence

model (dynamic LES, Detached Eddy Simulation) is

needed to simulate such separated ows, as found in

Strelet.

14

Far-Field Noise Prediction

Thetwocommonlyusedmethodsforfar-eld aero-

dynamic noisepredictionsusethe Kirchhoequation

or the Ffowcs Williams-Hawkings (FW-H) equation.

Whilethegoverningequationin the `moving surface'

Kirchhoformulation 15

isaconvectivewaveequation,

the FW-H equationis anexact rearrangementofthe

continuityandthemomentumequationsintotheform

of aninhomogeneouswaveequation. Thereinliesthe

strengthoftheFW-HequationovertheKirchhofor-

mulation. TheFW-H equation givesaccurateresults

even ifthesurfaceofintegration liesin thenonlinear

owregion. Thisistypicallythecaseinjetsandwakes

when the nonlinear region extends to large distances

downstream.

In the Kirchho formulation the source terms are

assumed to bedistributed over actitioussurface in

the ow. The nonlineareects(nonlinearwaveprop-

agation and steepening; variationsin the local sound

speed; and noise generated by shocks, vorticity, and

turbulence in the ow eld) happening within the

Kirchhosurfacearecaptured bythesurfaceintegra-

tionterms,buttheKirchhoformulationrequiresthe

integration surface to be placed in a linear ow re-

gion (i.e. far away from the body). This is diÆcult

to achieveasmostcomputational gridsaregenerated

withtheconcernofminimizingcomputations. Usually,

anequalitymeshisusednearthebodywithincreas-

ingcellsizetowardstheouter boundaries. Therefore,

thequalityofthesolutionavailable in thelinear ow

region is generallybad. TheFW-H equation, onthe

other hand,works ne evenif theintegration surface

isinthenonlinear owregion. Adetailedcomparison

of the Kirchhoand FW-H formulations is provided

in BrentnerandFarassat.

16

ThesolutionofthefullFW-Hequationrequiresthe

evaluation of two surface integrals and one volume

integral. The surface integrations correspond to the

\thickness"noise(monopole)andthe\loading"noise

(dipole). The volume integration corresponds to the

quadrupole term which accountsfor the nonlinearity

in the ow.

15,17

Evaluating the volume integral can

be extremely computationally intensive and diÆcult

to implement. Fortunately, the quadrupole term can

besafelyignoredformostsubsonic owsasisthecase

in thepresentstudy.

OnlyrecentlyhastheFW-Hequationbeenusedon

actitious(i.e. notthesameasthebody)permeable

integration surface 18

- exactly like the Kirchho ap-

proach. diFrancescantonio 18

demonstratedthatwhen

the FW-H approach is applied on a Kirchho-type

surface, the quadrupole sources enclosed within the

surface are accounted for by the surface sources. It

should be noted that the \thickness" noise and the

\loading"noiseasobtainedfromsolvingFW-Hequa-

tiondonothaveanyphysicalsignicanceifthesurface

of integration is chosen to be permeable (ctitious).

However,when theintegrationsurfacecoincides with

the body, these termsprovidea physical insight into

thesourceofsoundgeneration.

TheFW-H equationis writtenin thestandarddif-

ferentialform as

 2

p 0

(x;t) =

@ 2

@x

i

@x

j [T

ij

H(f)] (1)

@

@x

i [L

i Æ(f)]+

@

@t [(

o U

n )Æ(f)]

whereL

i andU

n

aredenedas

U

n

=U

i

^ n

i :: U

i

= (1





o )v

i +

u

i



o

L

i

= P

ij

^ n

j +u

i (u

n v

n ) (2)

andT

ij

istheLighthillstresstensor. TheFW-Hequa-

tion canbesolvedusing the formulationin Brentner

andFarassat, 16

andthesolutioncan bewritten inan

integralform as

4p 0

(x;t)= Z

f=0

"



0 (

_

U

n +U

_ n

)+ _

L

r

=c

r(1 M

r )

2

#

ret dS

+ Z

f=0

"

(

0 U

n

+Lr=c)(r _

M

r +c(M

r M

2

))

r 2

(1 M 3

r )

#

ret dS

+ Z

f=0



L

r L

M

r 2

(1 M

r )

2



ret

dS (3)

-1e-07 -8e-08 -6e-08 -4e-08 -2e-08 0 2e-08 4e-08 6e-08 8e-08 1e-07

0 0.5 1 1.5 2 2.5 3 3.5 4

Pressure perturbation (in Pa)

time (in s)

FW-H prediction Analytical

Fig.10 Validation ofthe FW-H code against the

analytical solution for a stationary monopole in a

uniformmean ow.

Thequadrupoletermis ignoredinthepresentformu-

lation. The integrationsare performed onthe FW-H

surface at retarded time. Since the FW-H surfaceis

xedrelativetothebody(thecone)forthisstudy,and

the owMachnumberisconstant,thefollowingterms

in the aboveintegralsare zero: U

_ n

= _

M

r

= 0. The

standardtimebinningtechniquediscussedby



Ozyoruk

and Long 19

is used for obtaining pressureat the ob-

serverlocations.

The FW-Hcode and itsValidation

The FW-H code is written in Fortran 90. The

code was tested for a model problem - a stationary

monopolein auniformmean ow. TheFW-Hsurface

is chosento beaboxmade upof rectangularpanels.

Theanalyticalsolutionto themodel problemis eval-

uated at the center of each panel to obtainthe time

history of the primitive variables on the FW-H sur-

face. Theprediction from theFW-H code (usingthe

analyticaldata on thesurfaceasinput) is then com-

pared with the analytical pressure perturbation at a

point outside the surface. Figure 10 compares at an

arbitrary point (300 m, 0, 0) the pressure perturba-

tion predicted by the FW-H code and that obtained

analyticallyforastationarymonopolesourcewithan

amplitudeof0.01Pascalsandafrequencyof2.267Hz

placed in a uniform mean ow of 0.3 Mach number.

Theanalyticalsolutiontothisproblemis:

(x;t) =

exp(i!

 )

4[(x+U

0 (

 t))

2

+y 2

+z 2

] 1=2



1

1+

M0(x+U0( t))

[(x+U0( t)) 2

+y 2

+z 2

] 1=2

(4)

where



isgivenby





=t+ M

0 x



(x 2

+(1 M

0 2

)(y 2

+z 2

)



c(1 M

0 2

)

(5)

-2e-07 -1.5e-07 -1e-07 -5e-08 0 5e-08 1e-07 1.5e-07 2e-07

0 0.5 1 1.5 2 2.5 3 3.5 4

Pressure perturbation (in Pa)

time (in s)

FW-H prediction using unstructured surface grid Analytical

Fig. 11 Comparison of the FW-H prediction us-

ingunstructuredsurfacegridagainsttheanalytical

solution.

Theunstructuredgridovertheconeiscreatedsuch

that there is an unstructured cylindrical surface en-

closed in the computational domain (Fig. 1). This

surfaceis chosento bethe permeableFW-H surface.

Theelementsofthesurfacearefacesofthetetrahedra,

andtherefore,triangles. Sincethesetrianglesarecho-

senfromtheunstructuredmesh,theareaandnormal

variesfrom elementto element. This,however,is not

a problem becausethe FW-H equation only requires

informationonaclosedsurface;itdoesnotdependon

thestructureoftheelementsconstitutingthesurface.

Clusteringofthesurfaceelementsisdesiredtoincrease

theresolutionofthesources. TheFW-H surfaceused

for the present computation is the inner cylinder in

Fig. 1. Thisgridwasusedwiththemodelproblemof

stationarymonopole inauniformmean owtotestif

the unstructuredgrid posesanyproblems. A perfect

match is observedbetweenthe FW-H predictionand

the analyticalsolution (Fig. 11). The comparison is

made at an aribtrary point (300 m, 0,0). This con-

rmsthat anunstructured-mesh surfacecan be used

asaFW-Hsurfacewithoutanylossofaccuracy. Note

that therstfewsecondswheretheFW-Hprediction

does not match the analyticalsolution is the time it

takesforthesound toreachtheobserver. Thisdelay

ismoreinFig.11thaninFig. 10becausetheunstruc-

tured FW-H surface is verysmall and hence, farther

awayfromtheobserverpointthanthestructuredsur-

faceused forFig. 10.

Results forthe Cone

PUMA is used to obtain time accurate data (the

primitive ow variables) on the FW-H surface. One

complete shedding cycle ofthe simulation isused for

far-eldnoiseprediction. Pressureatafewpointsout-

side theFW-H surface(in the near eld) is collected

to compare with the predictions of the FW-H code.

Fourpointsdistributedintheazimuthaldirectionnear

thebaseoftheconeandveryclosetobut outsidethe

FW-H surface were chosen for comparison. The co-

Point No. x (m) y(m) z(m)

1 0 -0.055 0

2 -0.025 0.055 0

3 0 0 0.05

4 0 0 -0.055

Table1 Coordinates oftheobserverlocations for

comparing FW-H predictions againstPUMA.

ordinatesof thepointsaretabulated in Table1. The

conehasabasediameterof0:02mandavertexangle

of60 o

. Thecenterofthebaseoftheconeisattheori-

ginand thevertexpointsupstream(positivex). The

FW-Hsurfaceisacylinderofradius0:05mandlength

0:175m,centeredattheorigin.

Figure12comparesthepressure uctuationsatthe

four points listed in Table 1. Note that the PUMA

pressurepredictions havebeenshifted upby 20 Pas-

cals. This is relatively a very small amount, about

0:02% of the mean pressure. We believe that this

under-prediction by PUMA may be due to the dis-

sipationcausedbyinadequate clusteringofgridcells.

It may also be due to the small sample size, and we

plan to do ensemble averaging. Note that this error

is ofthe orderof magnitude ofpressurepertubations

predicted bythe FW-H code at any point inside the

FW-H surface, which should actually be zero. How-

ever,thepredictionbytheFW-Hcodeagreesverywell

qualitativelywith thePUMAsolution.

Sound Directivity

The directivityof the noise from the cone wasob-

tained by calculatingthe root mean squared (r.m.s.)

pressureperturbationforonesheddingcycleatdier-

ent observer locations in azimuthal and longitudinal

directions. Sincethecalculationforoneobserverloca-

tioniscompletelyindependentofanyotherlocation,it

isaperfectproblemtoruninparallel. LongandBrent-

ner 20

suggestedsomeself-schedulingparallelmethods

for multiple serial codes. However, noparallalization

was done for the noise prediction results presented

here.

Figure 13 plots the directivity pattern in the az-

imuthal direction on theplane x = 0:1 m,which is

rightbehindthebaseofthecone. Thepatternin Fig.

13is symmetricbecauseofthesymmetryof thecone

aboutits axis. Since theFW-Hequation cannotpre-

dictthepressure uctuationinsidetheFW-Hsurface,

wecancomputethenoiseonlyoutsidetheFW-Hsur-

face. Therefore,thedirectivitypatternsareplottedin

anannularregionoutsidetheFW-H surface.

Figure14 plotsthedirectivity patternin thelongi-

tudinal direction onthe z=0plane. Sincethe noise

is caused byboth turbulence and uctuating surface

forces,thedirectivityshowsseverallobes.

Aconventionalpolardirectivitypattern inthelon-

gitudinal direction(z =0plane)isplotted in Fig.15

(1)

-155 -150 -145 -140 -135 -130 -125

0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004

Pressure Perturbation (P - P inf ) (in Pa)

time (in secs)

FWH prediction PUMA calculations

(2)

-445 -440 -435 -430 -425 -420 -415 -410 -405 -400 -395

0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004

Pressure Perturbation (P - P inf ) (in Pa)

time (in secs)

FWH prediction PUMA calculations

(3)

-180 -175 -170 -165 -160 -155 -150 -145 -140 -135 -130

0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004

Pressure Perturbation (P - P inf ) (in Pa)

time (in secs)

FWH prediction PUMA calculations

(4)

-170 -160 -150 -140 -130 -120 -110 -100

0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004

Pressure Perturbation (P - P inf ) (in Pa)

time (in secs)

FWH prediction PUMA calculations

Fig.12 Comparisonofpressure uctuation,p p1

aspredicted byPUMA andFW-Hcode atvarious

X Y

Z

214.339 184.91 155.48 126.051 96.6213 p’ rms

Fig. 13 Directivity of the noise in the azimuthal

direction behind the baseofthe cone(x= 0:1).

X Y

Z 210.996

162.304 113.613 64.9217 16.2304 p’ rms

Fig.14 Directivity ofthenoiseinthelongitudinal

direction in theplane z=0.

forobserversat 10dierentradiallocations(r=0:15

-0:24m). InFig.15,theconeispointingtotheright;

theradialdistancefromtheoriginisequaltother.m.s.

pressureandtheangle(theta)illustratesthelocation

oftheobserverpointinthedomain.

Conclusions

Aerodynamic noise from a cone has been studied

as a model problem to test the possibility of using

unstructured grids for noise prediction from compli-

cated bodies like landing gears, slats etc. A nite

volume ow solver, PUMA has been used to obtain

time-accurate ow data on a permeable FW-H sur-

face. The FW-H code wasvalidated against amodel

problem of amonopole in auniform mean ow. The

predictionsfromtheFW-Hcodehavebeencompared

rms p’ sin θ

p’ _rms cos θ (Pa)

(Pa) θ

increasing r

−50

−40

−30

−20

−10 0 10 20 30 40 50

−300 −250 −200 −150 −100 −50 0 50 10

Fig. 15 Polar plot of sound directivity in z = 0

plane atafewradial locations.

atfourobserverlocationsin theneareld withdirect

calculationsfromPUMA. Noise predictionsaremade

for a period of one shedding cycle. The comparison

is fairly accurate with only a small D.C shift error.

The directivity patterns of the noise from the cone

are plotted in azimuthal and longitudinal directions.

The sound directivity pattern has been shown to be

fairly complicated due to the complex physics inside

theFW-Hsurface.

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