May 11, 1992 - 0156314 https://ntrs.nasa.gov/search.jsp? ... the peak noise level generated by a supersonic propeller initially ..... can be quite cost-effective.
//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
A03 17. SECURITY CLASSIFICATION OF REPORT Unclassified NSN 7540-01-2B0-5500
18. SECURITY CLASSIFICATION OF THIS PAGE
19. sEcURITY CLASSIFICATION OF ABSTRACT
Unclassified
Unclassified
OF
ABSTRACT
Standard Form 298 (Rev. 2-89) Prescribedby ANSI Std. Z39-1B 298-102