TUE CASA Day 22 November 2006

Wiener-Hopf Solutions of Aircraft

Engine Noise Models

Ahmet Demir

Generalisation and Implementation of Munt’s Model

Munt’s Model

For over 25 years (1977-2003) there was “asleep” the very advanced but technically complicated solution by Munt for sound radiation from a straight hard-walled hollow duct with piecewise uniform mean flow.

It was proposed for TURNEX to take this as a starting point for similar, but extended and more advanced models.

New Models based on Munt Model

0000 ,c , U,M

jjjj ,c , U,M

1R

0R

0000 ,c , U,M

jjjj ,c , U,M

1R

0R

0000 ,c , U,M

jjjj ,c , U,M

1R

0R

0000 ,c , U,M

jjjj ,c , U,M 0R

Case 1 : Hollow duct

Case 1b : Lined Centerbody Case 1c : Lined Afterbody

Case 1a : Annular duct

lininglining

Coplanar Exhaust and Burried Nozzle

More advanced generalisations:

R0

R1

0000 ,c , U,M

jjjj ,c , U,M

2222 ,c , U,M

R0

R1

0000 ,c , U,M

jjjj ,c , U,M

2222 ,c , U,M

Analytical Solution

Solution is in the form of a complex Fourier integral

(where P is also defined by a complex integral) constructed by subtle complex-functional methods, variations of the Wiener-Hopf method.

Important issues, which can now be studied more exactly : - Vortex shedding from trailing edge - Kutta condition at trailing edge - Instability of the jet (vortex sheet unstable for all frequencies; finite

shear layer not)

dueurPetrzp uziimti ,,,,

Formulation of Lined afterbody

0M)(1

1

2

012

2

21

21

zi

r

m

rrr

rr

0MC)(1

2

2

12122

2

22

22

zi

r

m

rrr

rr

convected wave equations

0000 ,c , U,M

jjjj ,c , U,M

1R

0R

1exp,exp,

1exp,,,,

2

1

rhtiimzrtiimzr

rtiimzrtzr

i

1rhpC , , MD

1rp , , M

212211

110

v

v

zip

zip

scattered field:

mnmnmnmnmmnmnm mmYhJJhY 1

'''' 0 ,

mn

mn

tiimzrtzr ii exp,,,,

rYhJrJhYr mnmmnmmnmmnmmn ''

zirzr mnmni exp,

is axial wave number, is the root of equation

where:

Incident wave (hard-walled mode):

0j1j010j100001 /D , c/cC , c/UM , c/UM , R/R h

Dimensionless parameters:

0M

00,1

01 zz

zi

zz

r

0M

00,1

12 zz

zi

zz

r

0z , ,1M,1,1MD 10211

zz

izzz

i i

boundaryconditionsat hub andr = 1

continuity of pressure at vortex sheet

continuity of displacement

soft wall

00,2

zzhr

,

0,,,, 2

2

12

zzhzhz

MiZi

Dzhzh

r ii , 1

0000 ,c , U,M

jjjj ,c , U,M

1R

0R

hard wall

Kutta condition and instability: trailing edge behaviour, vortex shedding, excitation of instability

singular, stable

smooth, unstable

singular, unstable

Fourier Integral Representation of Velocity Potentials

duerHuAzr uzim

0

)2(1 )(

2,

duerYuCrJuBzr uzimm

112 )()(2

,

0Im , M1)( 022

00 uuu 0Im , M1C)( 122

1211 uuu

0

)'2(0

0M1)(

mH

uFuiuA

1

'1

'1 ,,

1,)(

mmYuZJuZ

uFuMuZiuB

mm

m

JYY 1

uBuZ

uZuC

m

m

,

,)(

YJ

Simultaneous Wiener-Hopf Equations (results from B.C. r=1)

1'

1'

1

31

2

1111 ,,

)1(21)()(

mmYuZJuZhZ

uFuMD

uZ

MhDuuNu

mmmn

mnmn JY

1

)(

)()(

uK

uKuK

mn

mnmn

mn

mnmn

mm

u

MDuG

uZ

MhDu

YuZJuZ

uMDiuKuF

mm

)1(1)(

1)(

,,

)1(22

111

1'

1'

1

1

11

1

JY

)(

)()(

uN

uNuN

note : B.C. at hub yields relation between B(u) and C(u).

Splitting first Equation: weak factorization in lined duct wave number

1

2

11

2

111

1

2

11

1'

1'

1

311

11)(

1

1

,,

)1(2)(

p mp

mp

mnmn

mnmn

mn

mnmn

p mp

mp

mnmn

mnmn

mm

u

a

NuZ

MhD

uZ

MhDu

uN

u

a

NuZ

MhD

uNYuZJuZhZ

uFuMD

uN

u

mm

JY1

'1

'1

'1

3

1

,,

)1(2

mpmm ummmpmp

mpmpmp

YuZJuZN

FM

hZ

Da

JY

1

note : no left running contribution in z>0

mp

Splitting second Equation: essentially with the same way:weak factorization in lined duct wave number

mp

1

2

111

1'

1'

1

1

1

)1(1)(

1)(

,,

)1(2

p mp

mp

mn

mnmn

mn

mnmn

mm

p mp

mp

u

buK

u

MDuGuK

uZ

MhDu

YuZJuZ

uKuMDi

u

buKuF

mm

11

1

JY

mnmp

mnmnmp

ummmp

mpmpmp Z

MhD

YuZJuZ

KMDib

mpmm

2

111'

1'

1'

1

1 1)(

,,

)1(2

JY

1

note : no right running soft wall modes from z<0.

Solution to second Equation: for far field we need only F+

mpp mp

mp

u

uu

u

buKuF 0

1 0

mnmnmr

mnmn

p mpmr

mp

ummmrmrmr

mrmr

NZ

MhD

a

YuZJuZNKMD

bimrmm

2

11

1

'

1'

1'

1

1

1

,,)1(2

1

JY1

mpmr

mr

p mp

mp

ummmr

mrmrmrmr

u

u

b

YuZJuZMD

NKhZa

mrmm

0

1 0

'

1'

1'

31

1 ,,)1(2

1

JY1

Coefficients amp and bmp are determined by the following infinite linear system

Complex Contour Integral

Careful management & bookkeeping of poles and other singularities is necessary for correct answer

Total field (double integral: takes more time)

iZ 2

w = 15, Mode(4,1)

w = 25, Mode(4,1)

975.0943.053.03.0667.0 1110 C DMMh

Lined Centerbody

Far Field

Far field pressure for ωr → ∞.

imtimiM

MMRi

R

Dzrp p

21

1

sin1cosexp

)(,

20

2/12200

'sin

')'1(

0)'2(

2

uH

uFMuD

m

p

0directivity

2

0

0

2/1220

M1

MsinM1cos'

u

Numerical Examples

Case 1

Approach Mode(0,1)

without mean flow with mean flow

0000 ,c , U,M

jjjj ,c , U,M 0R

Approach Mode(17,1)without mean flow with mean flow

Cutback Mode(23,1)without mean flow with mean flow

Case 1a : Effect of hub (Kutta on)

Approach Mode(17,1)Approach Mode(0,1)

0000 ,c , U,M

jjjj ,c , U,M

1R

0R

Cutback Mode(23,1)Cutback Mode(0,1)

Case 1b,1c : Effect of lining and semi-lining (Kutta on)

Approach Mode(9,1)Approach Mode(0,1)

0000 ,c , U,M

jjjj ,c , U,M

1R

0R

0000 ,c , U,M

jjjj ,c , U,M

1R

0R

Approach and Cutback mean : different flows inside and outside the duct (specific parameters for the project)

Cutback Mode(23,1)Cutback Mode(0,1)

Comparison with numerical models

Case 1: Flesturn (METU)

and Actran (FFT) results

Approach parameters

Modes (0,1),(10,1),(19,1)0000 ,c , U,M

jjjj ,c , U,M 0R

Zero flow Mode(0,1) Zero flow Mode(19,1)

Case 1a : METU, NLR and FFT results

Approach Mode(17,1)

0000 ,c , U,M

jjjj ,c , U,M

1R

0R

SPL (dB)

0

30

60

90

120

0 20 40 60 80 100

analytic (TUE)coarse gridmedium gridfine grid

SPL (dB)

0

30

60

90

120

0 20 40 60 80 100

analytic (TUE)coarse gridmedium gridfine grid

Cutback Mode(23,1)Cutback Mode(0,1)

Case 1c : METU and FFT results (liner impedance Z = 2 - i)

0000 ,c , U,M

jjjj ,c , U,M

1R

0R

FFT-Actran vs TUE results

Coplanar Exhaust : TUE and METU results R0

R1

0000 ,c , U,M

jjjj ,c , U,M

2222 ,c , U,M

Zero flow Mode(2,1) Zero flow Mode(10,1)

Approach Mode(10,1)

R0

R1

0000 ,c , U,M

jjjj ,c , U,M

2222 ,c , U,M

Conclusions A series of non-trivial extensions of the classical Munt problem have

been successfully solved and implemented.

Comparison with fully numerical solutions have been very favourable and encourages their trustful use in industrial applications.

Case 1b+1c (lined centerbody+lined afterbody) has been published: Sound Radiation from an Annular Duct with Jet Flow and a Lined Center Body, A. Demir and S.W. Rienstra, AIAA 2006-2718, 12th AIAA/CEAS Aeroacoustics Conference, 8-10 May 2006, Cambridge, MA, USA

First results show that lining of centerbody reduces sound field only in crosswise direction.

Effect of instability is for these high frequencies acoustically small in all cases considered