# Computational Aeroacoustics for Aeroengine Inlets

Mechanics

Feb 22, 2014 (4 years and 4 months ago)

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ComputationalAeroacousticsfor
AeroengineInlets
MikeGiles
MihaiDuta
AlistairLaird
LorenzoLafronza
giles@comlab.ox.ac.uk
OxfordUniversityComputingLaboratory
ComputationalAeroacoustics–p.1/33
Introduction
ComputationalAeroacoustics–p.2/33
Introduction
ComputationalAeroacoustics–p.3/33
Acoustics
Motivation:
engineperformancebecominglesscriticaldueto
relativelylowfuelcosts
strongpushtowardsfasterdesignmethods,better
futurepushtowardsmodellinguncertaintydueto
manufacturingtolerances
stillveryimportanttoavoidunpleasantproblems
noisebecomingacriticalissueforairlines,airports
andgovernments
ComputationalAeroacoustics–p.4/33
Acoustics
ComputationalAeroacoustics–p.5/33
PotentialFlowModelling
assumeirrotational,homentropicﬂow.
r,
whilethedensityandspeedofsoundaregivenby

1
 1
=
c2
c2
1
=1( 1)
qq1
c2
1
;
where
q=
1
2

r
2
+
@
@t
:
ComputationalAeroacoustics–p.6/33
Themassequationgivesthep.d.e.
r(r)=0;
indomain
V
,withDirichletb.c.’sonthefar-ﬁeldpartofthe
boundary
@V0
,andNeumannb.c.’s

@
@n
=;
onthewalls(=0)andthefanboundary(6=0).
Integrationbypartsthengivestheweakform
Z V
rrwdV
Z @V
wdS=0;8w2H
1
0
(V):
ComputationalAeroacoustics–p.7/33
singleharmoniccomponent
b
ei!t
,theweakformofthe
Z V
r
b
rw

c2
rr
b
+i!
b

(rrwi!w)dV

Z @V
b
wdS=0;
where
b
=
@
b

@n

c2
rr
b
+i!
b

@
@n
:
ComputationalAeroacoustics–p.8/33
Therearedifferentmodelsfor
b

onthedifferentpartsofthe
boundary:
ofincomingandoutgoingeigenmodes;
far-ﬁeldboundary:high-frequencyraytheory
determinestheangleatwhichtheacousticwavescross
theboundary;
walls:
b
=0
onsolidwalls,butwithacousticliners(lots
givesamodiﬁedweakformboundaryintegral.
ComputationalAeroacoustics–p.9/33
AxisymmetricGeometry
2Daxisymmetricgrid
witheachcellmappedtoaunitsquare
-
6


A
A
A

x
r
=)
r
r
r
r
rr
rr
-
6

1
1
ComputationalAeroacoustics–p.10/33
AxisymmetricGeometry
Thecoordinatesandﬂowpotentialareinterpolatedfrom
nodalvalues
x(;)=
X
n
xn
Nn
(;);
r(;)=
X
n
rn
Nn
(;);
(;)=
X
n
n
Nn
(;);
R()=0
whichissolvedbyNewtoniteration.
ComputationalAeroacoustics–p.11/33
AxisymmetricGeometry
modenumber
exp(i!t+i)
b
(x;r)
with
b
(;)=
X
n
b
n
Nn
(;);
b
L
b
=
b
f
withtheforcingterm
b
f
comingfromthemodalboundary
conditiononthefanface.
ComputationalAeroacoustics–p.12/33
AxisymmetricGeometry
Cylindricalductgridconvergencevalidationagainstanalytic
M=0:4;!=6;=2.
1.5
2
2.5
6
5
4
3
2
1
0
log10
(nJ)
log
10
(error)
bilinear elements
ComputationalAeroacoustics–p.13/33
AxisymmetricGeometry
EnginebypassductvalidationagainstACTRANcode,
withoutandwithacousticliner
ACTRAN
mode1
0.959
0.954
mode2
0.134
0.132
mode3
0.054
0.055
ACTRAN
mode1
0.0455
0.0450
mode2
0.1034
0.1029
mode3
0.0093
0.0094
ComputationalAeroacoustics–p.14/33
Non-axisymmetricGeometry
Threewaysofhandlingnon-axisymmetry:
3Dﬁniteelements,withhexahedral
elementsmappedtounitcubes
(;;)=
X
n
n
Nn
(;;)
spectralelementswith
x;r;;
b

expressedasFourierseries
(;;)=
X
m;n
m;n
exp(im)Nn
(;)
asymptoticanalysis,likethespectralapproachbut
assumingsmallasymmetry
ComputationalAeroacoustics–p.15/33
Non-axisymmetricGeometry
1.5
1
0.5
0
0.5
1
1.5
2
nacelle sections
x
r
=0 (top)
=45
=90
=135
=180 (bottom)
FFTofcorrespondingpointsoneachsectiongives
spectralgeometricrepresentationonnacelle
thisisextendedtotheinteriorbyinterpolation
ComputationalAeroacoustics–p.16/33
Non-axisymmetricGeometry
Prosandcons:
3DFEanalysisneedsaveryﬁnegrid,sodirectsolution
istoocostly(bothCPUandmemory)
spectralelementsrequiremanyfewerunknowns,but
trickytosolvetheequations,andstilltooexpensivefor
directsolution.
asymptoticanalysisuses2Dcalculationssocheap,but
canithandlelargeperturbations?
ComputationalAeroacoustics–p.17/33
SpectralAnalysis
ifgeometryisaxisymmetric,allofthecircumferential
modesareuncoupled—eachrequiresa2D
calculation;
ifthegeometryisnotaxisymmetric,allmodesare
coupled—for
M
modes,
O(M
3
)
directsolutioncost,
and
O(M
2
)
memoryrequired;
goodpreconditioner,

b
P
1
b
L
b
=
b
P
1
b
f;
with
b
P
b
L
andeasyto“invert”.
ComputationalAeroacoustics–p.18/33
SpectralAnalysis
novelideaisusing
b
P=
b
Laxi
where
b
Laxi
isthematrix
ﬂow.
solving
b
Pbv=
b
b
requiresaseparate2Dcalculationfor
eachcircumferentialmode.
memoryrequirementisminimisedbyusingstandard
sparsematrixsolution;CPUcostisminimisedby
performingLUfactorisationonceforeach
circumferentialmode,thenjustback-solveateach
iterationstep.
ComputationalAeroacoustics–p.19/33
SpectralAnalysis
Inmoredetail,oneofthecontributionstothematrix
b
Laxi
fromasinglecellis
Z
1
0
Z
1
0
nrNn
rNn0

c2
(rrNn
)(rrNn0
)o
rjJjdd;
X
i
wi
f(i
;i
)
where
f(i
;i
)
istheintegrandevaluatedattheGauss
pointsand
wi
istheappropriateweight.
ComputationalAeroacoustics–p.20/33
SpectralAnalysis
Similarly,partofthecontributiontotheproduct
b
L
b

froma
singlecellis
1
2
Z
2
0
Z
1
0
Z
1
0
r
b
r
ei(+m)
Nn

!
2
c2
b
ei(+m)
Nn

jJjrddd:
;;
thisbecomes
1
2
Z
2
0
eim
X
i
wi
f(i
;i
;)img(i
;
i
;)
d;
andthe

integrationisapproximatedbyanFFT.
ComputationalAeroacoustics–p.21/33
AsymptoticAnalysis
InitialideawastodoasymptoticanalysisatPDElevel,
andthendiscretise.However,asymptoticformofwall
boundaryconditionsbecomescomplex:
nr=0
=)
e
nr+n
r
e
+(
e
xr)r
=0
Founditmuchsimplertostartwithspectraldiscretisation
(withb.c.’shandledinweakform)andthendoasymptotic
analysis.
ComputationalAeroacoustics–p.22/33
AsymptoticAnalysis
Stepsinasymptoticanalysis:
startwithdecompositionofgeometryintoaxisymmetric
averageplusperturbation
compute
0
,axisymmetricaverageﬂowﬁeld
solvealinearperturbationequationforeach
circumferentialﬂowﬁeldperturbationmode
Lm
m
+Am
xm
+Bm
rm
=0;
compute
b
0
solution
solveseparateequationsfortheothermodes
b
L
m
b
m
+
b
Am
xm
+
b
Bm
rm
+
b
Cm
m
=0
ComputationalAeroacoustics–p.23/33
EngineNacelle
M1
=0:3
Mfan
=0:4
=26
!R
c1
=30
17,000gridnodes
(equivalentto3.5Min3D)
ComputationalAeroacoustics–p.24/33
Spectralconvergence
10
15
20
25
30
35
40
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
spectral mode no. (=26)
modal energy
4 modes
6 modes
8 modes
12 modes
ComputationalAeroacoustics–p.25/33
Iterativeconvergence
0
5
10
15
20
25
30
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
iterations
log (rms)
1o scarfing
2o scarfing
5o scarfing
ComputationalAeroacoustics–p.26/33
Spectral/asymptoticcomparison:1o
0
10
20
30
40
50
60
70
80
90
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
angle from axis [deg]
Re[ ]
  1
 asymptotic
 + 1
  1
 spectral
 + 1
ComputationalAeroacoustics–p.27/33
Spectral/asymptoticcomparison:5o
0
10
20
30
40
50
60
70
80
90
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
angle from axis [deg]
Re[ ]
  1
 asymptotic
 + 1
  1
 spectral
 + 1
ComputationalAeroacoustics–p.28/33
0
20
40
60
80
100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
angle from axis [deg]
abs[]
0
90
180
270
ComputationalAeroacoustics–p.29/33
0
1
2
3
4
5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
scarfing [deg]
10 log (J / J
0
)
asymptotic
spectral
ComputationalAeroacoustics–p.30/33
Spectral/AsymptoticComparison
Whatisasmallgeometricperturbation?
ex;erR
,assumingshapeoflip
doesn’tchangemuch.
ex;erx
forﬂowdetails,but
ex;erR
functionals.
Otherexamplesofsimilarasymptoticbehaviour?
ComputationalAeroacoustics–p.31/33
Conclusions
Canexploitweaknon-axisymmetryintwoways:
axisymmetricpreconditioningofspectralequations
asymptoticanalysisofspectralequations
Bothlikelytobeusefulengineeringtools:
asymptoticanalysisforfar-ﬁeldintegrals