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//v -7/ 15_, 3// NASA

Contractor

Report

191110

.

_z7

An Asymptotic Theory of Supersonic Propeller Noise

Edmane

Envia

Sverdrup Technology, Inc. Lewis Research Center Group Brook

Park,

Ohio

May 1992

Prepared for Lewis Research

Center

Under

NAS3-25266

Contract

(NASA-CR-191110) AN THEORY OF SUPERSONIC

IXl/ A

NOISE

Final

£echnology)

Report 20

N93-24070

ASYMPTOTIC PROPELLER (Sverdrup

Unclas

p

National Aeronautics and Space Administration G3/71

0156314

t

-

"

AN

ASYMPTOTIC

THEORY

OF SUPERSONIC Edmane

Sverdrup Lewis

NOISE

Envia

Technology,

Research

Brook

PROPELLER

Inc.

Center

Park,

Ohio

Group

44142

Abstract A theory geometries

for

predicting

the

is presented.

provides

an e_eient

of propeller

formula

noise.

predicted

by this

propellers,

with

a preliminary supersonic

propeller

eventually

reaches

qualitative

agreement

with

are

quite

of supersonic utilizes the

initially with

that

show

source peak

all

that

is rather

tip

helical The

sources the

for high

levels speed

substantial

sources.

noise

further.

blade

three

that,

and loading the

increase

the experimental

realistic

approximation,

indicate

also

increasing

not

with from

integration

thickness

with

anddoes

of sound

quadrupole

indicate

increases

a plateau

radiation

Calculations

by the blade

of the theory

"propellers

a large-blade-count

a full numerical

aceurat:e.

by the Lighthill

radiated

application

field which

for predicting

radiated

the noise

noise

theory,

Comparisons formula

the noise

compared

The

level

when

Results

from

generated

Mar.h

by

number,

predicted

but

trend

a it

shows

observations.

Introduction Over

the

last

dozen

technology

on the

one

regulations

on the

other,

activity On

in the the

equation, propeller by

of accurate

almost

the

all

of these

which

provides

blade

surfaces

flow

field

in the

area

and

drive

spurred

speed

theories theories (i.e.,

interest to meet

a great

rotor

in supersonic stringent

significant

strides

supersonic Ffowcs

the

and

(propfan)

and

cabin

experimental

noise

research

noise.

for predicting been basis

thickness

propeller

community

deal of theoretical

has

a theoretical

surrounding

of propeller

Hanson

renewed

the

side in particular

noise generated by thickness exemplified by the theories example,

the

and

have

of high

theoretical

development for

area

years,

hand,

and

propeller

the

propeller Williams

for calculating loading disc

been

made

noise. and

as well

Lighthill

towards

as the

the

The foundation

Hawkings

the radiation

noise) (i.e.,

have

1 (FW-H)

of sound

from

sound

generated

quadrupole

noise).

the The

and loading sources has, received a great deal of attention as of Hanson 2 and Farassat. 3 In contrast, most of the research

quadrupole Fink 4) even

noise though

been

of an

conclusions

has

from

exploratory these

efforts

nature have

(see,

for

suggested

that

quadrupole

work

by Peake

must

radiation and

Crighton,

be accounted b'_om

of the

multiple

for in predicting

but,

of numerical

integration.

source

to produce

for such

a representation

by

two

these

quadrupole

types

noise

context

approximation fective

For

alternative

demonstrated thickness Ref.

to direct

by Crighton and

results

noise

importance

Therefore,

as it stands,

integration

and

calculating

noise

which

is quite

can

part,

have

there

been

radiation

this

limited

realistic

from

the

the

of

measure

of

expensive.

in FW-H

and

theories

of this used

which

the

was

by the

authors

however,

the

utilize

in pub-

geometries.

full

rather

asymptotic

but,

ef-

for propeller

simple

are

a very

approach

theory

fairly

bug

equation

offers

To date,

with

and

disc

by I-Iawkings

approach

was also noise.

calculations

of

application

some

suggested

geometries

sources,

noise

number

use of the large-blade-count

effectiveness

to propellers blade

quadrupole

for quadrupole

of radiation

computationally

this

theory

of

a propeller

yielded

an asymptotic

quadrupole

is a gap between

accommodate

efficient

The

number necessary

larger

appearing

designs

use

sources

While

has

the

as well as the

computation

is the

developed

In fact,

geometry

viable.

integrals

propfan

of propeller

of a large

the

first

analysis,

r who

requires

loading

remained

integration.

Parry,

radiation.

theory

less

evaluation

be accomplished

and

approximation

distribution

numerical and

the of this

source

can

a substantially

this problem,

many-bladed

loading

5 to assess

lished

the

propellers.

of flow field around

considerably most

task

propeller

Recent radiation

involves

prediction

so as to make

representation

for the

speed

utilization

In contrast,

for circumventing

modem,

the of the

like the far-field has,

This

rotors.

quadrupole

noise

detailed

of thickness

of a frequency-domain

to calculate

asymptotically.

small

feasible.

computations

method

6 in the

and

number

for a faithful

calculation

An appealing Lawson

quite

schemes

a reallstie

the

that

by high

equation.

representation

In practice,

supersonic

of propeller

necessitates

is sufficiently

simplification

success,

an accurate

needed

quadrupole

various

in FW-I-I

of sources

sources

generated

appearing

in turn,

for

the notion

prediction

in general, a realistic

it.

noise

of view,

That,

flow field surrounding

significant

has reinforced the

point

integrals

methods

points

be potentially

5 however,

a computational

by various

makes

could

numerical

unwieldy

theory

for

of Ref.

so far, is restricted

5,

to simple

geometries. The

theory

presented

approximation asymptotics

is applied

integrals.

This

keeping

the

loading

noise

sources

direct

and,

therefore,

numerical

with

of the

approach

the

flexibility small.

in Envia

integration.

numerical

of completeness problem

gap.

It utilizes But,

integrals of using shown

is now

realistic

will then of interest

be

5 and

of source blade

to provide

shown

Refs.

analysis

extended

the full analysis

a large-blade-count

unlike

instead

The asymptotic

s and

The theory

integration

to a practical

that equation.

efficiency

requirements

was presented sake

bridges

in FW-I-I

radiation

affords

time

for the

a full

theory

paper

integrals

to the

computer

with isons

in this

to evaluate

a very

distribution

geometries

while

for thickness good

quadrupole

will be detailed

here.

a preliminary

and

agreement

to include and

7, the

noise Compar-

application

will be presented.

Analysis

The acoustic

starting pressure

point p(x,

for the t) written

analysis

is Goldstein's

in the propeller-fixed 2

9 version (wind

of FW-H tunnel)

equation

coordinate

for the system:

p(x,t)

= -

@)

+

pov,

_

ds(y)

Tjk

/-_/v oo

(la)

dy dr

02G OyiOyk

(_)

dr +

Tj_= pu_ + _j_[@- po)- c_(p- po)] Do

where

v, is the normal

component

0

Ob)

0

of the blade surface velocity, f is the amplitude of

the aerodynamic loading on the blade and Tjk is the Lighthill stress tensor, p, p and u s are the fluid density,pressure and velocitywith the subscript %" denoting their ambient values. The velocitiesu i and v, axe given relativeto a medium-fixed even

though

represent and

the

nj

the

Eq.

is expressed

propeller

is the

blade

component

convention

represent

(la)

for the

the

source

in a propeller-fixed

surfaces

of the direction

and

and

of unit

observer

represent

sources

contributions

of propeller

G = G(x, which

can

t/y,

from

noise,

the

normal

(la)

denotes

to have

the

following

_.

4_r_R 1

9c(r)= _(,_ M0_@) = M0_ej,

direction, the

observer

and

effects

spectively. where

is the medium Co is the

of medium The

explicit

blades, should

loading,

and

and noted

of Goldstein's.

LighthiU

V(r)

respectively, be

r is the source

that

y and

time

and

x t is

velocity which is three terms in Eq. stress

(quadrupole)

moving-medium

Mach

source,

medium convection dependence

on

the

function

,_(t- _"- gcR/Oo)

(2a)

_(_) = (M2o_+ _)

- YJ) R ' number

e i is the ambient

Green's

form:

) = (zi

convection and

the free-space,

_0 -- (1 - M_i )1/2 ei(r

between

that

respectively,

MoR),

P6

M0_

is opposite

It

denotes the medium convection forward flight velocity. The

1

tance

the S.

S(r)

respectively.

G

Here

to

coordinate system

system.

surrounding

normal

thickness,

r) in Eq.

be shown

unit

coordinates,

the observer (i.e., retarded) time. U0j equal to, and negative of, the propeller (la)

volume

outward

coordinate

(sum R(r)

speed

of various

necessary. 3

over

of sound. time parameters

(2b)

(2c)

j)

= Ix - y(r)l.

in the radiation

component

retarded

1/2

of the

unit

direction, vector

Parameters and

spherical on

(2d)

source

R is the disalong

gc and spreading time

radiation i¢ represent rate,

r is indicated

re-

In flow, the

general,

the

because

1-axis

is chosen

description

transverse

propeller

of the

planes

may

to coincide source

be operating

with

motion.

described

the propeller

As a result,

by the

2-

and

M01 = M0 cos a, where

a is the

propeller

angle

at an angle shaft

with

of the

to the

oncoming

1) in order

sources

Therefore,

M02 = 0,

of attack

(see Fig.

motion

3-axes.

of attack

to simplify

is confined

it is readily

seen

to the

that

(3)

M08 = M0 sin a

respect

to oncoming

flow

as depicted

in Fig.

(1). Provided are

known

(la)

for

which use

that

geometry,

in sufficient any

detail,

observer

entails

aerodynamic

acoustic

location.

solving

the frequency

harmonic

blade

pressure

This,

transcendental

domain

components

p(x,

of course, equations

approach,

pl(x),

loading,

which

and

t) may is the

for source

involves

flow field about

be computed

so-called time

directly

time

v.

expanding

the propeller from

domain

Alternatively,

p(x,

t) in terms

Eq.

approach one

could

of its Fourier

i.e., -[-00

p(x,t)=

_ l_

with

the

individual

harmonic

components

p,(x)

e -imt

(4a)

_OO

given

by:

p_(x)= pT,(x) + p_,(x) + p_(x)

(4b)

where

pT)(x)

in which

=

_

P_:t(x) =

_

p_(x)=

_,o

f_ is the

harmonic In writing

Eqs.

(la) have been of the relations

.,o

the

the

the usual phase only harmonic

_xj

e ira'

c_) Tjk

O_O-zk

speed,

thickness,

derivatives

p_,

relationship components

Pcl

loading,

of G with with

OG -

ds(y)

Dt

(,-) fnj

by its derivatives

OG

Using blades,

povn

e imt

angular

Pl from

(4c-4e) replaced

(,.)

.r-,o °r ''° [i;:z

propeller

component

e imt

.Io

DoG

--

Ozj '

Dr

arguments for which

and

Pet

and

quadrupole

respect

respect

ds(y)

dr

dy

denote

to source

to observer

dr

dr

dt

] ]

dt

(4d)

dt

(4e)

contributions sources,

from through

DoG

= -_

to the

respectively.

coordinates coordinates

Eq. use

(5)

Dt

it can be shown that, given l = mB (where m is an integer) 4

(4c)

B

identical contribute

to the infinite sum in Eq. (4a). This contribution is simply a single

blade.

harmonic

Therefore,

components

will denote

surface

To further

from

of, and

simplify

may be rewritten rule:

here

P_',,,B, P_:,,B, volume

Eqs.

in terms

on and

B

around,

(4c-4e)

of the

will

pQ,,,_.

the

explicitly

appear

spatial

the

contribution

in expressions

Correspondingly,

a single

temporal

B times

S and

V,

for

blade. derivatives

derivative

O/Ot

O/Ozj

and

through

the

02/OziOxk use

of G

of the

(6a)

o_e = t[0va(t) 0x_0_---S 0x_ + v_(t)vk(t)] a

(_b)

1 auxiliary

The

temporal

over

t, which

o

derivative

derivatives involves

chain

z=--_ = v_(t)a vx i

the

the

respectively,

OG

where

of

can a delta

operator

:Dj(t)

has been

now be removed function,

can

using then

pr_.(x) = _,0

_.) =

PL:,.B(X)=

(*)

_-_'r2j0

1 (_ge_+M0.M0_)] introduced

integration

be easily

(6c)

for notational by parts,

evaluated

and

brevity.

the

integral

to yield

(Ta)

=

Rt

/

ds(y)

(7b)

dr

(7c)

where and

Q_0 quadrupole

according on 2_/ft

Q(_)

the

and

Q_)

sources,

to which integration _e

given

the

respectively.

power over

in t is mapped

strengths

represent

Note

of radiation r

exactly

have

that

distance

been

into

expressions

changed

an interval

the

for the

strengths

Q!.t) s for each

R they

multiply.

to reflect

the

of size

2_r/_

source Note

fact

in r.

of thickness,

The

that

type also

loading

are organized that

an interval

expressions

the

limits of size

for source

by

Q_)

= -_

Q(_)

=

imBFt

1 ----_ pov.

p°v"

[ 1-1M°_(ei ]to

Mo¢ (_2oej + MoRMoj) 5

-gcM°_)

(8a)

(Sb)

0(2)=

1 imB f j( j-goMo )

(8c)

1

=

Ins

Q_)=

+ Mo,,Mo )

1 Tjk _F+2

A_t)_jj,

+ A_t)ejek

1)

0,

,A(2 1)

"-

1,

,_2)

:

3_02

+'"3 _(t)

"_o_'_'o. ,At ,_

t = 1,2,3

+ Mo,

+.4o A_

(8d)

A (') = g_, '

,A(3 2)

--

(8e)

A_ 1) -

-(1

-{- 2ioRgc)

-go, - fl_)gc,

A_ 2) = 2Mort

,

A 3)=

A_ 3) = 3fl0_Uon.

(sf) where

sums

over

normal surface on source time Once the

source

strength

the

is not

tractable

observer given

of the

gested

(or

in Ref.

more

6. In this

utilized

the

to show

surrounding points.

that,

sented

in Refs.

stationary

approach,

which

is particularly

harmonic

index

rnB)

from

the

pointed

out

5 and

7. The

phase

points

of the

neighborhood

of special

in introduction, utility

this

of this

- a difficult

useful

task

from points

approach

approach when

or fax-field The

the

blade

volume)

mentioned sug-

count

B (or

blade

is

(or volume

stationary

basis

blades

phase

surface the

the

is

manner

phase

of theory

on determining,

with

result

(or

as was

called

hinges

integral

of stationary

forms

dealing

acoustic

this

in the

when

is

of the

surface

method

out

equations

for near-

Alternatively,

radiation

is to carry

simplified.

remaining

the

Tit depend

strengths

asymptotically

is large,

tensor

efficiency

approximately

scheme.

in general,

in these

source

be significantly The

that,

remains

radiation

be calculated

most

stress

what

and

functions.

may

asymptotically,

it) comes

As was

the

a quadrature

noted

of integration

computed

Bessel

using

estimated,

may

be

LighthiU

geometries

integrand

integral

the

order

represents

it is usually

out

and

the

for general

the

volume)

appropriately

f,

It should

Q!.t.) are

r (which

appropriate

carried

surface

assumed.

Ordinarily,

Since

for which

is then

earlier,

over

analytically,

in terms

integral

distributions

first.

locations

k are amplitude

(7a-7c).

integral

is computed

j and

vn, loading

in Eqs.

and

source)

indices

velocity r.

integrations

reversed

the

the

pre-

analytically,

having

complicated

geometries. This

difficulty,

proximation efficiency using

however,

to integral of each

source

a modification

advantage

of this

In this

paper

The

extension

sake

of brevity

approach

the

standard is that

derivation

to nonzero the

be circumvented

r instead.

asymptotically.

to the

can

over

final

angle formula

by

In other This

for the

zero

the

is accomplished

steepest

descent

it is applicable

of attack

applying

words,

angle

follows

will be given

to general of attack the

in terms 6

and

analysis

the idea

large-blade-count is to find

by evaluating saddle

point

ap-

the

radiation

the

r integral

methods.

The

geometries. case

(a

= 0) will be outlined.

described

of a generic

in Ref.

source

8. For

strength

the

Q and

a generic radiation distance "factor" 7_. The resulting expression is therefore applicable to any Q!.t.) and For practical

R t dependence. considerations

in Eqs (7a-7c) and carry by dividing blade surface volume) source

may

(or volume) the

integrand

be

assumed

integral

may

elemental

surface volume

to be area

at the geometric

by choosing Therefore,

generic

constant

over

the

by some If the

of element, Ns,

order

would

1

s indicates a particular size is sufficiently small,

appropriate mean

and

"mean

value

error

error the

surface

of integrand

to be the

value

of

in this approximation

could

effective

PamB (X) may

thus

value"

is chosen

Ym, the

reduce

of integration

To do so, we begin of small surface (or

of element

Of course,

amplitude

) _'

extent

designated

which

hasznonic

_Dsras(X

the

be Ns, where the subscript If the typical elemental

L = [y - Ys]max.

enough

to preserve

(or the volume) integral first. surrounding it) into a number

(or volume).

center

Ns where

a large the

convenient

be approximated

surface

is O(L 2) for a given small

out (or

elements. Let that number element under consideration.

integrand times

it is more

be made

arbitrarily

size of each

be written

element.

as:

N,

_

_

A s e -imB01c"

Is

(9a)

s=l

rs= fo2" Qs(O)

emB,_.(o) dO

(9b)

(9c) MtipMot

0 = f_r + Cs - ¢,

Mtip

x, = where

(zl,

parts

x2, zs)

(xl,

now

replaced

and

qc,

not the

by a new on r.

and

As

notation, element.

angle

of attack,

kinematic results observer Rtip.

Qs(0)

and

¢s(0)

factors. apply and

(Note

phase

the is the

Note or the

that

location

to arbitrary source that

M0t

0. Mtip

area

as and

have

factor

= M0

replaced

is the

are

the

for zero

and

for

a combination has

observer

made. locations.

are nondimensionalized angle

of attack 7

case.)

(i.e.,

which

radiation

operating

regarding In the

above

by the

centroid at

convective the

reason,

the

zero and

geometry

of

subsequent

expressions

propeller

do

to simplify

a propeller

this

is

distance

In order

of geometric,

r

R,pn/Co)

terms

on Ys, the

For

counter-

variable

variables

no assumption been

and

element.

of various function

number of phase

strength

sth

polar

integration

collection

source

for the

Os(0),

of observer

cylindrical

Mach

the

phase

in deriving

by their

tip rotational

dependence

canonical

(9f)

the

generic

bs representing

coordinates

(9e)

Furthermore,

volume)

explicit

geometries

spatial

been

representing

(or

2rrs

= (I-

respectively.

T_s(0)

is elemental

with

blade

y2o, ys.)

variable

we suppress

of sth

propeller

(Yl,,

bs =

--

(y_., rs, ¢s),

is a convective

depend

factor,

and

r, ¢) and

(9d)

/37,, (=, - u,.) + ¢. - ¢

both

tip radius

Now,

for a typical

propfan

(i.e.,

m = 1) the

integrand

tone

considered

only

gration

the

scheme an ever

integrand

and

evaluation

order

complex

and

Replacing is now ¢'s(u) saddle

increasing that

the

roB,

the

path

(i.e.,

a° and

be solved. saddle

b° were

Thus

points

component

in the of the

be quite

cost-effective.

evaluation,

variable

u = 0 + ia leads

straightforward

are

"sonic

second

order

integral.

branch the

branch

cuts

the

cuts,

Note

that

greater

than

that

u-plane

steepest

descent

choice

of steepest

the

original

of integrand

the

of which

shown

only

that

the

is exponentially points. contours in this

contour

figure

[0, 27r] into

in 0, the

from

these

real

if Mr

with

points

of interest.

are Mr

contributions points,

shown is less

contours

contour(s). cancel

A

integral

saddle

descent

to

of branch

contour the

locations

to a

contributes

region

contours

> 1.

two saddle

appropriate

descent

the

is subsonic

number

on whether

simple

to give rise

to the

auxiliary

two

Mr,

point

an infinite

compared

steepest

contributions

of the

saddle

source

the

are

lie in the

depends

are

one

of the

typical

path

(i.e.,

on whether

merge

contributions

location

for

descent

has

small

The

which

may readily

in general,

< 1 and

that

four

in _ and

points

only

only also

of ¢o(u)

depending

if Mr

source,

_o(U)

points

of observer,

saddle

Hence

integrand

guarantees

of saddle

Also

path.

of ff,(u),

the

deforming

periodicity

For a subsonic

and unity.

forms

pair

contour.

(lo)

has,

direction

two

to be

function

quadratic

different

these

allowed

2 = 0

_(u)

conjugate

be

of the

asymptotic

The easiest way to find the rewrite '_'o(U) = 0 in terms of

(aobs/2)

in the

nature

appropriate

saddle

locations

have

an

inte-

harmonics

The

0 must

into

(10) is clearly

number

descent

cuts

in the

neighborhood

source

condition")

point.

those

of branch

and

a complex

be noted

with

choice

Eq.

Mach

s_eepest

It should

coinciding

along

saddle

appropria2e

(9d).

a numerical

oscillatory

to a phase

to find

_ b°_ + 1 -

[0, 27r] which

relative

the

lies on the

from

interval

source two

single,

points

observer

The

Mr

_2

in Eq.

= 1 (i.e.,

When

judicious

defined

for given

or supersonic.

for

the

et al, i0 Who

of higher

substantially.

= 0) and the appropriate steepest descent contours. points is to define an auxiliary variable _ = cos u and

where

2.

cost

[0, 27r]) be deformed

variable

It is fairly

(aob°/2)2

the

can

employed

computation

to capture

(BPF)

Nallasamy

contributions, accurate

asymptotic

blade-passing-frequency

oscillatory.

computational

however,

complex

complex.

noise resolution

the

the

integration

0 with

loading

for the

is highly

Unfortunately,

increase out

even

(9b)

quadrature

can

to carry

also

and I,.

of I, for large

In

in Eq.

thickness

to calculate

requires

B = 8. Thus

in Fig. than

needed

Due each

or

to the

other

out

exactly. For a given for a source easy

observer

with

to show

Mr

that

location

and

= 1 is different

the

operating

from

asymptotic

condition,

that

for a source

expansion

is given

the

asymptotic

with

Mr

in terms

structure

_ 1. In fact

of the

inverse

l/s-powers

parameter mB for a "sonic" source and in terms of the inverse 1/2-powers of rnB sonic" one. Since there is no simple way of constructing a composite expansion two

expansions

very

a uniform of the

to allow

convenient

asymptotic

methodology

for a smooth

to use.

However, expansion

may

transition

this using

be found

difficulty a theorem

in Bleistein 8

through may

derived and

the

sonic

be avoided by

condition, altogether

Chester

Handelsman.

of I,_

it, is fairly

they by

are

basic

not

developing

et al. 11 The

1_ The

of

for a "nonfrom these

idea

details is to

map, conformally

(say,

cubic

polynomial)

which

region

of interest

descent

contours)

standard

The

The

function

the

v-plane

descent

and

definitions

is given

Os(V')

relevant (i.e.,

is correspondingly

key

cubic

_), phase

exhibits

in complex

steepest

(-plane. below.

v _

the

saddle

point

containing

into

simpler

along

in the applied

with

final

(i.e.,

function.

points

and

complex

axe then the

function

phase

saddle

a region

methods

parameters,

a much

of the original

region

mapped

and

into

features

steepest

(-plane.

to the result,

The

integral are

a

The

in the

summarized

by:

(::a)

where and

v + and 7,

are

t_:

parameters

on three

possible

branches

defines

for which and

desired

After

(::c)

locations the

With

a fair

cop.formal

the

first

carried

2

Q,(_(O)

coefficients the

in the

locations

roB.

provide

to the

source)

region

Ai and For

7,

_

is O((,nS)ve). After the

substituting Ns elements,

Airy

be written

d..

'

points

of blade

A' i are

O(1)

0 (i.e.,

a

and

Mr

(9a)

harmonic

it turns

as the

out

even

term

respectively,

observer than

component

(12a)

large and

PmB(X) 9

that

at BPF

is considered.

following

formula:

(12a)

(:2b)

mBAi

and

d, 0 and

:t=7,

corresponds

Airy

for which

d, t are

function Mr

1/s-powers

is O((mB)

-_/6)

of mB

be calculated.

and

a "sonic"

to inverse

the contributions

to

is less than

For 7, = 0 (i.e.,

1/2-power adding

can

(:2c)

location,

unity.

to inverse

Eq.

that

of blade

is proportional

and

I,

7,2 - (2

region

Is

source) from

the

is greater

Is is proportional

for Is in Eq.

the first

mentioning

For any

from

one

of integral

27,

its derivative,

consequently

"non-sonic"

is the

6¢ - _',(_(I

Contour

Steepest

u-Plane.

15

Descent

Contours

in the

1651

160

,I

_FTO_

_o,._.o._c... I!

_eedVans:eam_ch3"No06.=0.8 _

V1

......

_V3 ............ ll

SidelineDistance'xz/P_=l"6"

1,o F ?

I

U f

\\',.

i;21 (3)

I ,,

Axial Distance, Fig.

[1

Sideline

Directivity

as a Function (Asymptotic

of the

of the

Predicted

Computational

Calculations)

16

x_ / l_.p Quadrupole Volume

SPL Chosen.

165 Surface • Surface

& Volume

Sources

Sources (7" + L + Q)

160

& Volume (T + L + Q)

(asymptotic)

(Numerical)

155

150

Surface Sources (T +

(asymptotic) Surface Sources

145

(T + L)

(NamericaO

BPF 140

TONE

Advance Ratio = 3.06 Free Stream Mach No. - O.8 Sideline Distance,

x2 / R ue --- 1.6

135

FWD 130 -1.5

-1.0

-0.5

0.0

Axial Distance, Fig.

(4)

0.5

x 1 / R_.p

Sideline Directivity of Thickness + Loading + Quadrupole and Thickness + Loading SPLs. Comparison of Asymptotic and Numerical Results.

17

1.0

1.5

170 Measurements from Ref [14] ...tt"-m--w

170

-_

,_

16o

S SS

_ _

Advance



• 1,o0.8 " 0.9 " 1.0-

= 3.06

_

@

_0"-]_"" _ll_. _.'"

65.3 deg.

• 60,1 deg. .A SZ 7 des. _ 1.1 " 1.2 - 1.$ - 1.4_r

Tip HelicalMach No. /J,

...o.

o

o

eladeSemnean#e

,

Ratio



,_"'/_-_" • ", /.,'¢"

_lJO

160-

BPF TONE •

....

-

0 0

f.'-

sU,./

,,4.,," Asymptotic

/_,._" /_/

Surface

o

Calculations Sources

Only

sS_,-I

140

_

_ ,///'_ " I

13%. 8

• a

0.9

I



!

0.6

I

I

0.7

(5)

Predicted

Variations

I

.,

!

1.2

Mach

1.3

No.

I

I

I

0.8

Free Stream Fig.

"

1.1

1_'p Helical I

Volume(T&+Surface L + Q)Sources I

1.0

+ L)

of Peak

0.9

Mach No. Thickness

+ Loading

Quadrupole and Thickness + Loading SPLs Helical Mach Number. Inset: Experimental

18

with Data.

+ Tip

1.4

Form Approved

REPORT Public

reporling

gathering collection Davis

burden

tor this

and maintaining of information,

Highway,

Suite

DOCUMENTATION

collection

of informaiion

is

estimated

the data needed, and completing including suggestions for reducing

1204,

Arlington,

VA

222024302,

average

I hour

per

and renewing the cotlection this burden, to Washington

and

1. /_GENCY USE ol_Ly (Leave blank)

to

PAGE

to

the

Office

including

of information. Headquarters

of Management

and

the

Send SewIces,

Budget,

time

for

comments Directorate

Paperwork

reviewing

r_ _ding this for Information

Reduction

Project

Final

1992

4. TITLE AND SUBTITLE An Asymptotic

Instructions,

searchhlg

existing

data

sources,

burden estimate or any other aspect of this Operations and Reports, 1215 Jefferson

(0704-0188).

3. REPORTTYPEANDSATES

2. REPORT DATE May

OM8No.0Z0_-0188

response,

Washington,

DC

20503.

COVERED

Contractor

Report

s. FUNDING NUMBERS

Theory

of Supersonic

Propeller

Noise WU-535--03--10

2. AUTHORIS)

C-NAS3--25266

Edmane

Envia

S. PERFORMING ORGANIZATION REPORTNUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Sverdrup Lewis

Technology,

2001

Research Aerospace

Brook

Park,

Inc.

Center

Group E-7723

Parkway

Ohio

44142 10. SPO NSORING/M()NITORI'_IG AGENCY REPORT NUMBER

9. SPONSOBING/MONITORII_G AGENCY NAMES(S i AND ADDRESS(ES) National Aeronautics and Space Lewis Research Center Cleveland,

Ohio

Administration NASA

CR-191110

44135-3191

11. SUPPLEMENTARY NOTES Prepared May

for the

11-14,

Center,

14th Aeroacoustics

1992.

(216)

Project

Conference

Manager,

sponsored

J. Groeneweg,

by the DGLR

Propulsion

Systems

and AIAA, Division,

Aachen,

NASA

Research

433--6751.

12a. DISTRIBUTIONIAVAII._ABILI'FYSTATEI_ENT

12b. DISTRIBUTION CODE

- Unlimited

Unclassified Subject

Germany,

Lewis

Category

71

13. ABSTRACT (Msxlmum 200 words) A theory theory,

for predicting which

of sound levels

from

predicted

radiated

the noise

utilizes all three

sources

thickness

noise

level

eventually

and

generated reaches

the experimental

of supersonic

are quite

quadrupole

loading

sources.

by a supersonic a plateau

and does

propellers

approximation,

of propeller

by this formula

by the Lighthill

blade

field

a large-blade-count

realistic

blade

an efficient

geometries

formula

noise.

Comparisons

with

a full numerical

accurate.

Calculations

also

show

source

is rather

Results

from

propeller not

with

provides

substantial

initially

increase

when

a preliminary further.

that,

compared

application

increases The

with

speed

trend

indicate radiated

indicate

tip helical shows

The

the radiation that

propellers,

the noise

of the theory

increasing

predicted

integration

for high wth

is presented.

for predicting

that

Mach

qualitative

the

the noise by the the peak

number,

but

agreement

it

with

observations.

"14. SUBJECT TERMS

IS.NUMBER

OF PAGES

20 Supersonic

propeller

noise;

Uniform

asymptotic

theory;

Quadrupole

radiation

is.

PRICE CODE

20.

UMITAT_N

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18. SECURITY CLASSIFICATION OF THIS PAGE

19. sEcURITY CLASSIFICATION OF ABSTRACT

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Unclassified

OF

ABSTRACT

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