Jet NoIse

AIAAJournai A Publication of the American
Institute of Aeronautics and Astronautics
JULY 1963
VOLUME 1
NUMBER 7
~\
Jet NoIse
M. J. LIGHTRILL
Royal Aircraft Establishment, Farnborough, England
1. Introduction
ii
I
.UI honored that this great association, devoted to the
sciences that permit vehicles heavier than air to follow
controlled paths away from the earth's surface, has asked me
to visit the countr)' of the Wright Brothers, who first achieved
this aim, and to speak in their memory. i am for a second
time so speaking, having given the Wilbur Wright :\Iemorial
Lecture in 1960, on the subject of ":\Iathematics and Aeronautics," to the Royal Aeronautical Society.! i spoke then,
\'o'ithenthusiasm, of the many-sided abilities revealed by the
documents of the Wright Brothers' career, a man)'-sidedness
that made them, in a \Va)',typical of the greatest men of this
cE'ntury. :Morerecent1)', in Michigan, visiting Henry Ford's
admirable reconstruction of the Wright Cycle Shop, with it:;
u,-;efuiwind tunnel in the smaIl back room, i was reminded
how their career contains alsa, for this century, a lesson-that
individua! initiative in modest surroundings can spark off a
transformatian of the human condition.
This lecture is also related to yet another that i gave in my
own country, the Bakerian Lecture2 of 1961, to the Royal
Society of London, on the subject of "Sound Generated
Aerodj'Ilamically." In that lecture, to an audiencerepresenting almost all the sciences, 1 attempted to sketch in its broadest features a new science of the middle of this century-that
of sound radiation fields that are by-products of airflows.
Having given such a broad outline of that new science to a
general scientific audience, i am glad to be able to deseribe in
more detai! the part of it which i have explored most myself,
namely, its application to the problem of the noise of jets.
This topic alone is a very complexone and deserves,in iia 0"'11
right, the extended treatment that can be given appropriately
to an audience concerned professionallywith the cacophon~-
i
f
i,
that is, to a greater or lesserextent, inseparable from turbojet
and rocket propulsion.
The noiseof a jet can be regarded as a separn.teentity, e\'en
whennoise emergesalsa from the exhaust nozzle, arising from
conditions in the nozzle or upstream of it. For, whate\'er
those conditions may be, a new, very intense turbulence is
created in the shear myer wherethe the jet fluid, after leaving
the nozzle,exchanges momentum with the atmasphere. The
noise that this generates can be thought of as, to ii.large extent, addith'e to combustion noise or turbomn.chinery naise.
Furthermore, to a good approximation, it is uninfluenced by
the presence of the nozzle and can be visualized DOS
souna
radiated into free space b)' a limited extent of turbulent gas
ftow.
lt is simply the radiated sound,representing energy actuall)'
extracted from the jet and propagated away through the atmosphere, that I shall discuss. My concern, in other words.
will be with the atmasphere pressure ftuctuations in the
"radiation field" or "far field," where those pressure ftuctuations fall offin proportion to the inverse first powerof the distance. Within a particu1ar narrow band of 'frequencies,J;h.e.
far field of an}' noi~p~Ol1r,.p
i" t.hpreKionwhose ilistanci' rro"l
Lt substantiallv e""pprl" onp "',,,'plenii:th.2 Bv contrast. the
near field "ithin a wavelen th or so of the
'ncludes
not nlv outwardly propaii:atinii:WM'es hnt alsa ]01,,,1rp.
'ciprocating motions and pressure fluctuations, such as mav be
inilnced ilirectly bv ftuctuatiniz:vortex movements in the
~urbulent ii:asflow. This near sound field is of concern to the
designer of struetures to be placed near jet orinces, but the
body of knowledge whieh has been sought after most by aireraft and roeket engineers comprises the acoustic power ragiated br a jet, its freQuencvspectrum. and its ilirp~tionaI
distribution.
Dr. )-lichael James Lighthill is the Director of the Royal Aircraft Establishment,
the premier
establishment
of its kind in Europe.
Born in January
1924, Dr. Lighthill
was educated
at
Winchester
College and at Trinity
College,
Cambridge
University.
He joined
the Aerodynamics
Diviiiion of the National
Physical
Laboratory
in 1943. From 1946 to 1950, he was
Senior Lecturer in l\ilathematics
at the University
of Manchester,
where, in 1950, he was named
Beyer Professor of Mathematics,
a position he held until hiii present appointment.
E1ected II.
Fellowof
Trinity College, Cambridge
Unh'ersity,
in 1945, Dr. Lighthill
also waii honored with
a Fellowiihip
in the Royal Society in 1953. The Royal Aeronautical
Society awarded
him its
Bronze :\ledal for work leading to advances in AerOOynamics in 1955, "for his contributions
to
theoretical
high speed aerodynamics
which recently
had included
outstanding
work on the
source of noise, from jets particularly."
Dr. Lighthill
was made a Foreign Member
af the
American
Academy of Arts and Sciences in 1958. He was elected Fellow of the Ro~'al Aeronautical
Society in 1961 and is an honorar~' Fellow Member of AIAA.
Received April25, 1963. In essence, this is the Wright Brothers Lecture presented at the 31st Annual ~Ieeting of the Institute of
Aerosp~ce Sciences, January 21, 1963. The author is deeply grateful to P. O. A. L. Davies, J. E. Ffowcs Williams, and Alan PoweJ[
fo.r their kindness in putting unpublished material at his disposal and drawing his attention to numerous published reports in connection
'\10'
.s preparations for this lecture.
fo11ow8from integrating the meiisured fluxes of meo.n-fiowand turbulent-flow energy.
.
.-
i.)
1
~
150i
". .0
~.--_......-
\
i
M. J. LIGHTHILL
ALU. JOURKAL
JULY 191
includingthe much higher speeds of rockets exhausts~-io
I.,
!":,,,
.
the aco.ustic efficienc:rBattens ofi to a maximum betwe~n 5
and 7 times 10-3. These are still "alues so law in relation to
viscous sources of dissipation that the efiect of acoustic
ener~'loss on ~urb~lence~evelopment is probably negligible.
Admittedly, with mcrea.sing}.!ach number, compressibilit~"
. .
Fig. 1 SubsoDlc
Jet
".;,..c.
''-\.t:~
e1Iectsare found to alter the turbulence lowermgits intensit"
<:..}o~ :-,..~...,...
~nd
the con~equeIitrate öf sprescl oJ th~ turbuIent shear Isyir
(1..v
~
\q!.:
~compsre Figs. 1 and 3). Biit the radiation of scoustic energy
~
~
()'\\~
~icr-i.\:::-~
18too. small to acco~t for this efiect, which perhaps should be
'i.,
-:iit-cx'"'-n"""
~
.y
c,;$
a.ssociated,rather,
with energy redistributions within the near
.. '\r.
\ ~
,- L
'go
dfi Id
\( ~
O
~4:.'" 4.-:.'(\
soun e .
.
\ '> 't"~ ,."" ~~~.(Zi
We have, th~, s. continuous.de~"elopmentin acous~ic
Now, i propose to regsrd this acoustic power ont.put. R
power output of Je~,. fro~ subs~>nicJets .through the undI5by- roduc f the main .et Bow not a henomenon thst has
m~grated supersonicJe~ (mcludmg turboJet engine exhausts)
ç.,-
.
"e o men .
~~elfany markesl.
18
"iew is wellSübstantiated for subsonic jets ig. ,which
radiate, at most, 10-4 of their power as souna, mostly from a
region within 4 or 5 diam of the orifice,namely, the intensely
turbulent shear la~'er between the central jet core and the atmosphere. Yet YlSCOUS
damping of the turbulence in that
short lengthof jet e~-iractsalready over a quarter ofthe normal
jet power,. so that e\ident1y the additional damping of less
han 0.0001 by sound radiation is negligible by comparison.
A-dmittedly,c~
jP.t.c;
at ver" lowRe"nolds nUm1)ers,which
!ileonh' maridiialh"unstable. showa sensith"einteraction with
ui
" soundi; 2 ut' ts
the much hi her Reynolds
:iumbers chara.cteristic of enidiieermii:app cations are fullv
tQrhiili:nt.indp.pp.ndently of an)" acoustic stimulation.
:E'9rsunersoniciets. this distinction is somewhat less clear:i.it. since jets can be generated in the laboratory (Fig. 2) at
"espectableReynolds numbers (though usuallr less than 108)
:hat
.
edal kind of ins . 't" f acoustic origin.
?owell first described and explained this usually asymme ric
lisintegration (see the Bakerian Lecture,2 Sec. 4.5 and ref~rencesthere giyen, together with arecent major study by
)ayies and Oldfield3)in terms of a mechanism involving the
Jattem of shock ~'a"es in a supersonic jet. Whfm an eddy
J"
shock waves an acoustic UISeis
tional ea' in t e ups ream direciw:i.
vorable conditions
a" on a.ssin t e
,rifice.e:eneratea new eddy that can extract sufficientener~
it "~Q"n""the shock, to rega.inthe am~li-
ude
. .
nd so mainta.inthe cycle. TheJet
to Jetsof speedsmanytimesthe atmosphericspeedof sound
~
(such as rocket exha.usts),and all of these it remains a minor
by-pro~uct of the powe! of the Jet. )ly main purpose in this
lecture is to show ho.w.one~heoreticalapproach, developedrecent1yby F~~,,;~s~illia.ms~n the basis o.ften-year-old work
?f. ~y ~wn!'
giv;s a g?od understanding not only of the
initial rise m aco,!stic efficienc)'with jet speed, but also of its
B';1bse9uent~ev~lin~ofi and of the associated changes in
directional distribution and frequency spectrum.
2.
Parts
of General
Aerodynamic
Noise, Theory
Relevant
to Jet Xoise
The theory begins2. 12 (as a well-beha,"ed theory should!)
with the fundamental equations of motion, e>...pressedin terms
of the ma.ss density p and the momentum density PV" where Vi
sta.nds for the velocity vector (Vi,t'2,V3).The equations are
C>p+
c>t
C>(pv,)
c>t
t
i- 1
t
+ j-l
=
C>(pv,)
C>x,
(1)
O
+ P'j) = O
C>(PV;t'i
(2)
and they relate rate of change of mass or momentum in a
small volume to the total mass transport or momentum
transport out of the volum~ (PViappears twice because it is
both the mass transport density and the momentum density).
The I?omentumtransportappearsin twoparts:
PV'Vi repre-
~
C>(pv,)/C>t
+
CliI2(C>P/C>Xi)
=
O
(3)
so that, from (1), the density p satisfies the classical \\"ave
equation
- - i:_ -
c>2p
3
C>2p
CliI2
C>t2
i
i
C>Xi2
=O
(4)
However, without making an)" approximation, one can
divide the momentum transport into two parts: first, this
Hacoustic" approximation, CliI2p~ii,and, secondly, the remainderi which can be written as
8upersonic jet
PhY$ic:
llSil turb
:;:imcftuc
cal s/a/ioi
plicd st re
atmosph,
he'suitabl
hy-produ
nificantl~.
fraction i
!;eDeratio
lcnt appH
separatel:
On elir.
and (i), (
C>Xi
. " an abnormall hi h uanti 'o acoustic ower
sents direct convection of the momentum component Pv,by
.ith a spectral pea" at a certain cha.racteristic frequency.
th~ velocity component Vi, whereas Pii is the stress between
.owever, any substantial irregularities created at the orificeadJ~cent el~ments of gas, which equally transfers momentum.
;ere found by Powell to reduce the amplitude gain in the
:rhis stress is the sum of a.pure pressure term P~ii (where ~ii
hear la~'ersufficientlyto destroy the e1Iectand to restore the
').~ 1 w:heni = and othe",ise is zero) and a viscous stress;
arrow sprea.d characteristic of most practical superson~~in
an ideal ~as, it represents simply cOD\'ectionof momentum
~ts..
..
.. ..
.
L- ~~.o-~
by the motion of.the molec~es relath'e to the gas velocity v,.
These
inte rated supersonicJets (Fig. 3), with ",hicn
No~, one special.approximate form for momentum translone i shaH be concerne in tim ecture, a\"e acous lc e _
port yi?lds the classical equations of acoustics; this is a pure
ciencies (that is, acoustic power output di"ided by jet
i~otr~picpressure, the variations in which bear to the variaower) rising rapidly from about 10-4 for jet speeds equal
tion in ?ensity a constant ratio CliI2(the square of the at) the atmospheric speed of sound to 2 or 3 times 10-3for jets
mosphenc sound speed). if this "acoustic" approximation to
f t~ice that speed (Fig. 4).i-~ At higher speeds, ho\\"ever,
the momentum transport is made, Eq. (2) takes the simple
form
Fig. 2 Disintegrated
ii in the
:in' pl:ice(
(5)
Xo\\", be(
the soune
mospheri
continu01
Tii per u:
quadrup(
(Fig. 5)
equal ani
acting OJ:
and so =
l'qui"aler
quadrup(
dina)" qi
poles; F
amplitlid
Xo\\",
portant,
their soll
.though t
composiJ:
the dista
of distan
radiation
square.
radiation
complete
times
if t
the radia
thequad
is propor
.L _ i-
,.
-LU
m.')
ci to
i:;tic
,ble.
ility
sity
iyer
rgy
d be
ciear
istic
iclis-
ists)
!Und
L(f'
Fig. " Acoustic eJl'iciency oC jets (excluding
disintegrated
8upersonic
Fig. 3 UndiBintegrated
i
i
d re.york
f
:s in
Icf'
-I
jets)
1i
)f its
LO
LO"
8uper80nic jet
ii in the momentum equation the terms involving this part
:ire placedon the right-hand side, one obtains these equations:
,
.
t
()t
+ i-I
()(PVi) +
ao2 ()p
()p
()t
()Xi
()(PVi)
()Xi
=
=O
- "/.
~
'"
(6)
(7)
t
-.
.,.
.
Jet V~
Atmosphct1c:
i
Spccd
2
3'"
-
Po) =
3
3
()2
[Til]
LL
-dr
f
i-ij-l ()xi ()Xi v 41M'
(9)
where V stands for the region of turbulent flow, and r represents the distance from a general point in V to the point
(Xi,%2'X,)where ao2(p Po),which is approximately the pres-
i
i
-
sure fluctuation, is being mea~ured. T-he brackets around Tif
ean that it is evaIuated at the retarded time t
(1)
(2)
,
O.,
th~ quadrupole, small compared ""ith an acoustic ,,'aY.rlength
js far greater, and it is this property of quadrupo~e radiation=
that local fluctuations are so much in exeess of fluctuations iD
.tion field-that
is the fundamental explanation o
ustic efticiencv of jets.
The actual solution of tq. (8) can be written as
Jory
uld!)
H
Lif'
0'3
ao2(p
.erms
re Vi
re
-
ACCllsticEfficicncy"
iinor
this
: the
Acoustie Power Output
Jet power
(8)
in a
ntum
e it is
.sity).
repre)fJiby
tween
citum.
-
that
~athadt to
e mstant
when
a waye travelin~
speed
sound
leave the
quadrupole
to reac at
thethe
Doini
of of
observ~
,tion at the current time t. The integral in (9) represents what
the solution would be if the right-hand side of Eq. (8) for the
density were simply Ti; itself; and, from this, it evident1y
followsthat the full Eq. (8) has the solution (9). i have said
that V stands for the region of turbulent flow; admittedly, it
would be more exact to take it as the ",hole of space, but outside the jet both terms in Til are smaIl. (pvt1/jfalls ofi like the
square
of the disturbances,
and so does Pil
- Q02pOis.)
Ac-
cordingly,in this formulatian it is a good approximation to integrate onlyover the region of inten...c:e
turbulent jet flowj
indeed, by neglecting contributions from outside it, one is
neglecting only how the sound emitted decays by attenuation
and finite-amplitude propagation efiects (combined), which
are better left to be estImated as a separate exercise.
Now, in the radiation field, the difierentiations in Eq. (9)
can be carried out very simply; no terms due to difierentiating the reciprocal of r with respect to Xi or X; are present,
since they faIl ofi at least !ike 1/r2. However, [Ti;]represents
the value of Ti; at a time t
r/ao, which depends on r and
hence alsa on the Xi; indeed, the rate of change of retarded
time with x. is
e Oil
;tress;
mtum
:itYVi.
trans-
-
3.pure
variahe at;ionto
3imple
~
GlXi
(t _ !..) = _
Qo
Xi
aor
(10)
Using this in (9), one obtaIns in the radiation field
(3)
, W:l\'e
STRESS
Tii--GTiL
is cquivalcnt ta
(4)
Fig. 5 Equivalence oC
appIied stress to acoustic quadrupole
ie can
;t, this
.he re(5)
-.'
i
-- - -
-
QUADRUPOLE
-++DIR~
STRESS
T1ilOl
T1i
is cquivalcnt to
QUADRUPOLE
... :!:.
-
J!.ISTRIBUTIONS
--- .
1\1. J, LIGHTffiLL
J
Measurements ai Mai:h number0.3 and
mately as v/bi (JoO.o'r',
where Tb is a typical mean square
Buctuation of ij'i;,which also can be written w4T', where w is
a typical radian frequency of fluctuation, and T' is a typical
mean square value of the quadrupole strength Tii. This gives
the useful approxiIDateformula that the total acoustic power
output of one eddy divided by its yolume V. (in other words,
the acoustic power output per unit yolume of turbulence) is
proportional to
Reynolds number 6 x LO' (Laureni:e 1956)
_---- --
o-IX
X~'"
~.OF
-=~~'l4X
ADJUSTMtIT Tunui,UT
-u -- -
UGlOII
DttA'
I
(vtLDtmu
.. SC
)
x-o .Ji.!!II!:..~.
Xa4r--____
X-.d --.
{
mj~~ngrm
i veloi:iiy
reai:hCl.
maximum
== o.i,ur
region where mcan veloi:/iy== 0.5U
f
~
o"rOL conitoii'
btbltot.d ocoulUc
!
per unlt len,'h
V.w4T'/pgaol
,., po..' output
~o
Fig,6
P
-
Stational')'
3
PO
3
= i-ij-i
L: L:
cold jet
~
. [2"ij]dr
411"ao2rJT
X'X .
(ll)
and one recognizesthe appearance of the second time deriva'iye 'i'ij, resulting in this calculation from the yariabi1ity of
retarded time, a dependence on rapidity of variation of
quadrupole strength to be ~xpected from the physical argumentI ga\'e earlier.
Equation (ll) shows that the coefficients of the characteristic quadrupole terms in the directional distribution of
pressure amplitude are obtained by integrating the quadrupole
strength Tij oyer the jet volume V, after double differentiation
and e\'alua.tionat retarded times. In order to estimate this
integral contribution from the wholejet, it is essential to take
into account the statistical randomness of turbulent Bo,,"."li
It is a general principle of wave theolJ' that, with well-correlated sources, amplitudes combine linearly, but that, with
uncorrelated sources, energy intensities combine linearly,
?Ço",",tl.lrbulent Bow measurements taken simultaneously at
nearby points are well correlated, but those at points more
remote from one another are almost uncorrelated, and, in the
simplest form of the theory, one merel~'-uses:tlüSresult, neglecting that the [1'ii] are not exactly simultaneous ,'alues.
This yields a pil'tllTPof a wrbulent Bowdh;ded into reinons,
such that strenltths of quadrupoles mthin am' one reinon are
cor
.
stren
at oints in different re-,ons are uncorrelated, the exten~
eac ID epen ent
IRdMlpolpdistrihution beinl/:rouii:hlythe size of a typicBI'
pnpT'lQ,_hpiiT';n~
eddy,
For the output of a single region o
kind, oh'olume V.,
(12)
since mtliin r. the yalues of the integrand can be taken equal
to their value at the center; on the other hand, the outputs of
different regions are perlectlr uncorrelated, so that their
energy intensities, < (p - 1'0)'>/poao, must be added to gi,'e the
int~nsit" in the radiation field,
From'Eq. (12) one can get a good idea of ho,," the acoustic
energy intt'nsity generated by one such eddy at distance r
yanes with the main parameters in"oh'ed. it yanes approxiWotU
rodlotcd
i 'per holl-octo.c
0'11,
~
Vorlotlan os .....
'
~~
~/CYCJC
Vorfotlonas ",i
,
(iiiconswotti/cyclc _..)
.
.
Frcqo.i.:~cy
(ci:c1cs/~cc)
Z50 500 1000 2000 4000 8000 l6ÜOO
0'01 i
Fig. 7
aeousUc
from an
3.9 cm
m/see
AlA.o\. JOURNAL
Spectl'um
of total
power
output
ajr jet of diameter
and velocih'
290
(Fitzpatrick.
and
ueli)
(13)
StrictIy speaking, this estimation should be done separately
for different parts of the turbulent frequency spectrum, but in
practice the '\"ariationof the product Y.w4 over the spectrum
is rather small, because high frequencies go "ith smaIl eddy
volumes.
Evidently, an essential assumption in this estimation was
that variations of retarded time "ithin an edd~'of size Lcoiild
be neglectedj this, for Buctuations at radian frequency w,
wouldrequire wl/aoto be smail, so that the eddy sizeLis small
compared ",ith the wavelength of the sound it generates,
This assumption, that wl/aobe smail, is satisfied at low Mach
numbers, since the product wl of a characteristic frequency
and eddy size is a characteristic \'elocitr. Furthermore, i
shall show that the Mach number range \dthin which the
assumption applies can be extended at least ta unity by a
modification insuring that w is a frequency in a frame of reference moving with the eddies, for then wl is approximately a
root-mean-square velocity Buctuation, which is a mere fraction of the jet velocity.
JUI
E
par
fluc
leni
x-i
fall
pro
x "'
uni
tat
1
bay
fre,
hs:
wa~
st::.
cc
fr
".
Ct.
pn
to
ine
To
3. Turbulent Jet Aerodynamics
Fi
BD
1
dji
i
fre
tu:
ed
pe
be
tb,
tli
ftx
qu
dc
cb
ta
D
to
\'f
T:
in
di
ix
at
ra
in
tl
La
JET ~OISE
JCL Y 1963
~~~
~rew is
typical I
.
is gI\'es
: power
words,
!nce) is
Be\'ond :i: = 4<1,similar dimensiona! considerations apply,
Dsrti~ularly to the total sound emitted, a!though,
~
lik~ T.-ei-i, where the cprrelation
i
.
~ of ref-
a
,
.
're
frac- I
i
ribe the
ent jets
Ayer, or
Jecomes
blstanc~ from nozzl~
ii -"Sd
Dlstan« from j~t aiiis
r O'Sd
.
...J
ZOO
~ orifice.
perimenh should
t \'olume
e mixing
adrupole
e ,yith :i:,
~reasthe
~eshould
t ...olume
mgth inet should
4d being
.
1'5
radi~L..Jh.Q!!gll'
!
n ceases
et enters
= Bd,it
!s faliing
.!..
d
i
~:'Iroportionalto x for large en~.!!!~arieiu:ath~r
.
rly with
reen 0.2
;ion, the
.mum of
ls.tionof
;ream of
1 such a
:Telatian
~relation
it O,04x,
result of
.2 o.e
fiuctuations in Til fall off, so doe§_~.!t§'Q:!!.nd
emitte~ir~ ~t
Teng!ClJidee~Wld
x = Bd where velodties
f o-1[e
r':i the..§m!!l~mission pei::Jmit leQ.~ would be expected to
little..!!!>to
~l""lUs
suggests a distribution of sound emissionper:init length as in the lo~'er half of Fig. 6, with about half the
(L3)
total comingfrom the mixing region.
Different parts of the jet emit sound in di1Ierent frequency
-arately. hands; in particulB.r,in the mixing region, where a typical
, but in : irequency w a: x-i, the sound emitted uniformly in:i: should
,ectrum i ha\'e a frequency spectr,um per cycle !ike w-i, since ek a:
.11eddy ; ..,;-~.
the region x > Bd, one gets a spectrum varying as
i
, :is w2or In
w. accordingly as Lincreases !ike x or remains con,on was ~ ,;unt. E:\"Perimentalspectra (see, for example, Fig. t) are
1coiild
,:nn:;i~tentwith the view that about half the sound comes
mcy w, ~ ::.)111the mixing region and has the w-2 type of spectrum,
',\'hereasmost of the rest has a spectrum most !ike w thAn!ike
i,;sm:i.ll
:ierates. i ...;&
:ind probably comesfrom the adjustment region.
\' ~Iach
Experiments agree well, alse, as will be shawn, with the
~<:uency i
prediction that acoustic power output should be proportiona!
more, i
to {Joeld2/ail,although i must emphasize that i so far have
.ch the
indic:ited how it can be derived only for law :\Iach numbers.
;y by a I To understand why it is also a good approximation at higher
1ately
15i1
rii:. 8 Curves of constant
correlation
(Davies, Fisher, and
Boirr.att%O)between velodties
measured
at two wires, ODe
.ifi,'ectl~. dowDstreain
of the other, in a l-in.-diiun
jet at
M"" 0.45
~I:ichnumbers, or to prediet directional distributions or peak
irequencies,one must, as i mentioned earlier, consider the
turbulent fiuctuations in a frame of reference moving with the
edrlies.
In my paperu of i954, i assumed, without conclusive ex;>eriment:ile\idence, that at each point of the jet there would
be such n frame of reference, roughly speaking ..mo ing with
theeddies,"in whichfiuctuations wouldbe considerablyslower
than in a fixed frame. in other words, time variation at a
fi..wdpoint would giye a misleaoing impression of high frequencr, refiecting merely the speed of comoectionof the randam space pattem past the point, rather than the speed of
changesin that pattem as it is swept dow1lStream.
~n the past two :rears this assumption has been given det.'iiledexperimental erifieation, particularly in the work of
Da\ies, Fisher, and Barratt of the University of Southampton%O
(but see their later paperu for corrected numencal
o;aluesof certain expressions caleulated 'from their data).
:nieycarried out a systemAticseries of hot-"ire measurements
~ the
O'Z
o
-o-t
o-Z
0'1
iladial distancc
Fig. 9 Radial distribution
of meaD veloeity ii, and
convection
velocity U. in the mixing region of a jet
veloeit~' U (Davies, Fisber, and BarraU20)
of
of
variable time delay in the secondhot-~ire circuit. By these
means they were able to infer how, in a frame of reference
movin at any chosen s eed e ual to hot-wire separation
di\ided y time e y, t e correlation etween one-turbulen~
velociw value and alater onefalls oif ,,"ith time.
Their results are plotted best as cur es of constant correlation (Fig. B), that is, eurve~in a diagram of wire separation
against time delay on which the correlation between the
measured velocities takes constant values. These show thAt
the frame of reference in which the eorrelation falls ofi most
slowly with time is rather well defined, and that the rate of
fall in this moving frame is onlyabout a quarter of that measured in a fi.."edframe. This impIies that typical frequencies
in the frame moving at the eddy eonvection velocity U. are
about a quarter of those that would be measured at a fixed
point.
Davies, Fisher, and Barratt20made similar me:isurements at
sever:ilstations in the mixingregion and at variOU5jet velocities. Theyfoundthat the ,'elodtyof eddycom'ectionvaries
across the mL'\:Ingregion (Fig. 9) but not nearly so widely as
does the mean velocity in the jet; values of U. from 0.2U to
0.7U were observed, an ayerage value being 0.5U. As a
measure of rates of change in a frame mm'ing with the eddies,
they used the time scale T, defined as the time t:iken for the
correlation in that fra.meto fall to a :ilue ile; the reciprocal
of T would be a typica.l radi:in frequency of turbulent fiuctuations in that fra.me. By plotting the time sca!e T against the
reciprocal of the loc:il mean shear (Fig. iO), they discovered
the interesting approximate relationship
T ::: 4, mean shear
(15)
at eii.chpoint of the jet, This means that correl:ition falls to
the alue ii e in the time t:iken for a sphere of fiuid to be
stretched br the me:in she:ir into an ellipsoid with ::i.xesin the
ratios 1:4t :20.
Dayies, Fisher, and Barr:itt%O
also obt:iin a number of other
Ha_ard 11- 0.34
+
...
, u ISO
.- H-0045
H-004s" .-I'Sd
.
}H- 0.Z2or-0'5d
_~ 1001
.~
~
mi"<Ing region of a jet, using two wires, one placed
directlydownstream of the other, the distance between them
being ariable. B:r correlating simultaneous va!uesofvelocity
at ~hetwo wires, they evaluated the longitudinal correlation
~3diusof an eddy, obtaining good agreement with previous
I!l,oestigators.However, theyalsa eorrelated the velocity at
the upstream point \\ith the velodty observed a short time
!:iterat
the downstream point, by ineorporating a sman and
'1
o
-11
50~
/:
oV
Fig.
io
/
/
/
/
/
200
400
600
Timescale .,. (ii see:)
Variation
of time scale with ioeal
(Davies, Fisher, and Barratt20)
800
rnean
sheiir
.-'
M. J. LIGRTHILL
,512
ALU JOURXAL
into which we earlier imagined the jet as divided must be
thought of as moving quadrupole sources of sound. But ac..
coring to well-established theory (see the Bakerian Lecture'
for full references), a moving volume 1'. in which the quadrupole strength per unit yolu me is Tij has the sound radiation
field
Fig. II Inerease oCemittil1g vohime and emission
time differenees, because
oC convection, b~' the CaetOI"1/(1 - M. eosB)
P
results of importance to turbulent-shear-flow theory as well
as to jet-noise theory. These include the follovringrelationvi)2>}1I2, the
ships between the rms velocity vi' = I«t'i
longitudina,l correlation radius l, the time scale T, and the
mean shear:
creased by the factor 1/(1
of Quadru-
poles which is moyini downstream.2.13 i hinted earlier at
one reason for this, namely, that, by confiningconsideration to
frequenciesin a frame of referencemoying with the eddies,the
range in , hich wl/ aois small is e~..tendedup to at least M = i.
This is necessary in order to justify the neglect of difierences
in retarded times within an eddy, and the results i have just
quoted on frequency in the moying frame show that in a sulisonic jet wl/aois indeed everywhereless than one-sLth.
A more basic reason, and one that applies even at low jet
Mach numbers, why in estimating
Furthermore, because
pooo
,,'here the e:
of 1954. F:
for a jet in
it i:; ,"er:
directional :
niade "alid
!'C\lnd spee<
t'<idies are'
b:isI5 of th,
,'cloeit:-', De
i:,. howe\'e:
fuii:-- what
ii:imdy, it
fur sma11a
of emi5sioi
jrl ,"clodt
<lirectiona:
300 m 'see
using :in ~
,'rlodt\' .
rraiiy goo
the orifief;
other so\l
region. (
('r:i.l eddi!
praeiical
intensit\'
that not t
-
Iv TijdT
over an eddy the time rate of change must be taken in a frame
of reference mo,'ing with the com'ection velocity, is as
follows. if instead one were to inflate that time rate of chan~e
by adding to it a space rate of change times a convection
velocity, the extra contribution to the volume integral from
the space rate of change could be transformed at once into a
surface integral oyer a surface outside the eddy, Its contribution to the intensity of aeoustic radiation would then be
zero, because values on that surface would be uncorrelated
with values within the eddy; so, in the end, one would be left
simply with the time rate of changein the moving frame.
These eonsiderations indicate that the eddy-sized regions
100
5db
o\lld 1
..
..
-
Lee(955)
LGilitir
and Hubbard (1952)
. '0.111 (1951)
O-Hc
cos.).,
QUadrUpaies convected
wlth veloclty component
ütent of the
Guaclrupoles
whose signols
arrlve togcther
X(I-M.cos e).
9
\
r aciM.cas_.,,,6~
.~~~
1'&
Number of
quadrupoles whoie
signols arrille togethcr
. '?;I has factor(I-H. cos 9)
TIME
-EXTENT-Intensity of eacn has
OFJET
factor(I-H.cos e)".
Distan«
X
U
'n direct!,>n
ol CINSlion
Stationar~'
- M. cos8).
«p
_ po)2)
;;;;.---
the radiation field of a quadrupole is due entirely to incomplete canceling of signals from posith'e and negative source8,
ovringto differences in times of emission, two ex'tra factors
1/(1 - M. cos8) appear, representing the increases in those
time differences, These are related closelyto the Doppler increasein the observed frequency.
No\\",this incorporation of a directional factor (1 - .lle' ity Ii\lt ii
witli jet'
cos8)-i, in the pressure amplitude generated by each eddy in
Indeed
its radiation field,callsfor a directional factor (1 M. cos8)-C
i-hould c
in the intensity of sound generated by the eddy. When one
(i - .11
considersthe sound intensity generated by the jet as a whole,
that, :<\11
bowe,'er, or e\'en by a gi\'en restricted extent of jet, this
ferred rir
direetional factor is modified, as Ffowcs Williams fir:"t
tutal rad
showed.n Figure 12 depicts the mo,"ingeddies within a giwD
kiiicl. 11\1
extent of jet X and showsthat this extent must be multipEe,d..
ari:lil'd i:
by (1 - M. cos/J)to inelude only that number whose radiasl1l'ar, tl
tion arri"es simultaneously. it foiio\\'s (multipIying this by
ilr~ wi
the directional factor on intensit)') that the aeoustic intensity
hi" fol
field per unit \'olume of jet takes the form
tt'-of-;
After this descriptionofjet aerodynamics,i canreturnnow
to the soundradiation fieldand explainwhy, for calculating
Fig.]2
(18)
-
Subsonic Jet Noise Theory
it, the tnrh111pnt prlr1ie!!ml1!!t hp viewed a!! a pattem
xixjTT.[T'i)
411'ao2r3(1
~ M. COS8)3
where M. is the velocit\' of convection U. di"idcd bv the atmospheric sound speed-ao. The effect of con"ectio~ appears
factor, in which 8 is
merelyin the additional (1 M. COS8)-3
the angle between the direction of emission and the jet
direction.
This factor, whichincreasesthe sound emitted forwardsul:>staDtiallymore than it decreases that emitted backward, can
be understood from a diagram (Fig. 11) in which time is
plotted vertically and distance in the direction of emission
horizonta1ly; the figure shows asound signal being emitted
with velocity ao from unconvected eddies and from eddies
whoseconvection velocity component in this direction is ao.lf.
cos8, It shows that, for a given quadrupole strength per unit
volume, the emission is increased when this velocity component is positive, because the total emitting volume is in-
-
4.
- PO = i-1t j-1t
JtjL Y 1963
iHen«
factor
total
intenslty
(I-Hecos e)'>
jet; acoustie intensit)'
volume oCjet is
-50
has
-100
field per unit
Fig. 13 Direetional
distribution
of sound radiation
aIr jets at 300 m/see. in decibels relath'e to B =
-
---
--
from
75°
ri;;. ]
JET
JeL)"
. p - pop) = ~ «
.~t be
Lut ac~cture~
nadrn.
:liation
(i~)
':\UI;::L
1063
;
[:ofIo
poa{)
tt
i_1j
_1
2
x,x;[2\~]
411'ao2r
))
LO'
x
Ret.7 ~ol<l-oirJet lu - ---iici.26ond 27
.
(nozzle-exit
diometer._
turbojet
Dt3 ,".j68convergence
i
.engillos!~
i \ .
ongle)- ___ __ __n.,
Ret.7 cold oir jet
\
(O-Sin}------....
(0-3in)
, "
!
i
'
LO'
:: io'
(1 - M. cos8) -s
(19)
i
ioi
7i io'
where the exponent 5 replaces the value 6 given in my paper13
; L!)54 Ffowcs Williamsu has made a simils.r modification
~:'.r:i je~ in fiight which there is not time to discuss here.
,
1t i:; \'ery satisfactory that, by incorporating this simple
i
,:irectional factor, a basicaIly lowMach number theory can be
;~:~Je\'alid for jets at speeds at least up to the. atmospheric
..)und speed~, The reason, as has been shown, is that, when
the at~
.ppears 5 ,'<idicsare viewed in a mo\-ing frame, the assumption at the
ich8is o !135isof the approximations becomes that the rms turbulent
','l'lodh', not the mean \'elocity, is smaIl compared with ao. it
:he jet
~ j:,. ho\\:ever,stilI more satisfactory that this factor explains
rd sul). .!. :ulh' hat is one of the most noticeable features of jet noise,
:1:i~1l'h',its marked directionality, that is, greater intensity
rd,
. can.,
:ime15,
..
. :..r :;n;al1angles 8 between the jet direction and the direction
11l~~ion
.'
,ii l'mL--sion,
a variation that becomesmore pronounced as the
~mitted
'l't vclocitv increases. For example, in Fig. 13, showing the
eddies ' :!irectionaI distribution of sound for stationary cold jets at
i~110.1[,: :300m./sec,the cur\'e represents the. (1 - M. cos8)-S factor,
Jer unit ~ u:,ingan a\'erage \'elocity of com'ection equal to half the jet
\' com- ~ ','('Iocit\'. The measured directional distributions are in geni~ in- ~ "raiiy good agreement; they fnlI off somewhat simyerbehind
thc orificeth:in the formula.would predict, probably because
>eca
. u~e i
mcom- ~ other sources of sound are additionallv detectable in this
:ource~,t ~I'gion. (These include the sound fields~f the weaker periphfactors . ..ral eddies with lower nlocities of convection and also, in
n tho~e 1 :iractical experiments, some reverberation of the much higher
intensitv sound transmitted forwards.) i shaIl show later
that not only the directional distribution at a giyen jet velocpler init~' but also the measured change in directional distribution
with jet velocity are in good accord with theor)'.
2ddyin
Indeed, theory indicates only how directional distribution
cos8)-6 i
1enone i ~hould change with jet velocity, through changes in the
!1
.ll. cos/J)-6 factor, whileleavingopen the possibility
. whole,
that, superimposed on this factor, there may be some preet, this
ferredorienta.tionof the quadrupoles Tif. Experimentally, the
1~ first
total radiated sound does not exhibit any marked effectof this
a given
kind. Imt the high-frequency sound does; to explain this, i
iltH:"o
'f
:irgiiedin my paperu of 1954that, in any regionof large mean
if
~hear,there would be some predominance of lateral quadruif
poles with directional peaks at 45° to the direction of flow.
}a
Thi.i followedfrom an expression for Tr'i as the product of
Erin
r:ite-of-strain and pressure, plus other terms whose effect
~houldbe smaIler. This relation would lead one to expect a
ti!n
p.cz,
dominant lateral quadrupole associated with the mean rateof
oC-strain,with directional peaks at 45° to the flow direction
<'Ombinedwith randomly oriented quadrupoles (associaterl
}J('
be;es
\\;th fiiictuatingrates-of-strain :ind other terms in Tif).
!
!
1"1
o
Prolt
.~ 10-1
i°
~
, lItIitney
.d.iZ
~-Ref. 28
cold-oir jet
LO..
i
LO..,
i
LO
i
(
-
i
tli,?rs
tli" it
fj.,)ng
'lU!
d'J/"encha,uld
pme
tail(cing
Da \pole
ton::h i"
l'aM the
nip.ould
!1 iIlume
irNh in?in,10uld
; thheing
diu
,'e..,
~ U (14)
O~
_
sin i 28
(I-M~~ose)'
(pure lateral
\quadrupoie:t:j:
)
distribution of quodrupolu)
Sdb
o
ig. 14
Direetional
distributions
ol intensity
io'
W-IÖ'
i
j
LO.
lO
106
i
'
LO
11,u' "/0: . woits
Fig.15
Aeoustie
power
radiated
600 m/see
by jets
at speeds
up to
Figure 14showsthe type of directionaldistribution that would
be expected, ",ith a peak around 30° and a subsidiary peak
around 120°, .ind it is significant that. the upper-frequency
part of the jet noisespectrum, whichis regarded as emitted by
the region of high mean shear, does ha\'e this type of directional distribution. s. 24. 24
i comenow to the rate of increaseof acoustic inteusity with
jet velocity, \\'hich is expected to be in proportion to the
product of the factor r,w4T2/poaos( per unit yolume) which i
discussed earlier and the directional factor (1 - .ll. cos8)-5.
At lowMach numbers the formerfactor should, as i expIained,
increase as the eighth power of the jet velocity i;. However,
turbulence measurements at the higher subsonic ~Iach numbersu. is indicate a definite reduction in turbuIent intensit~-,
that is, in the ratio of rms velocity to mean velocit)', as the
Mach number increases. With this goes a small reduction in
rate of spread of the mixingregion (which, being much easier
to measure. is valuable collater:il evidence for the intensity
reduction), preparatory to the inuch greater reduction in rate
of spl'ead that occurs, as i mentioned earlier, at still higher
Mach numbers (compare Figs. 1 and 3). The work of Davies,
Fisher, and Barratt20suggests that a reduction in rms velocit)'
also should reduce the product wl of the eddy size L:ind the
frequency w in a moving frame and so reduce also the product
V,w4. if the rms velocity goes up as the tth power of the jet
velocity U, as the limited e\'idence at the higher subsonic
jet speeds seems to indicate, then V.w4p should be proportional to U6,rather than to Usas it is at lo\\' l',lach numbers,
Now, this checks well with measurements of acoustic intensity at /J = 90°, where the directional factor (1 - ii[.,
cos/J)-6has no effect because its value is unity; such measurementsS.25show close proportionalit)" to [-a between jet Mach
numbers of 0.5 and i. Furthermore, measurement;; at other
angular positions show dependence on i.' equal1y consistent
with the idea of a basic U6term. modified by the directional
factor (1
M. cos8)-&. For example, at 8 = 20°, measurements of acoustic intensity5. 25are roughly proportional to U'
between M = 0.5 and 1, where the factor (1 - .ll. cos8)-i
does vary in close proportion to U3. Similarly, the integral of
-
(purc loteral quodrupole.
plus randomly oricnted
10db
LO
i
(,,-oreo ot nozzle)
~
- .ll,
'-Rc!.26
rm-aiLJets
_--'
io-z
10-'
.'
r
hat-and (old'oir jets --~
;
.
oireratt
.
in deeibels
-
(1
M. cos/J)-6 over a sphere va.ries in close proportion to U2,
which superimposed on the ca variation at /J = 90° gives a
UBvariation for the total power output, as is observed. A
smaIl selection of the numerous data bearing on this eighthpower law for total power output is included in Fig. 15, taken
from a report7of the XASA LewisResearch Center.
People ah..-:iysfind it difficult, at first, to believe that the
fact that the total acoustic power output of jets varies in the
high subsonic region as Us. which happcn:; to be the same
power as is gi..-enby low Mach number theory, is beeause of
the canceling out of two corrections to that theory. However, both correetions are indisputably neces5arYi the directional factor is supported excellently by experiment, as well
.',
ra
~.-
M. J. LIGHTHILL
as being theoretically essential, and the decrease in turbulent
intensity with Mach number is also well substantiated.
AlAA JOUR~AL
when M. cos8 = 1 (so that the slope of the sound signal becomes equal to the slopeof the lines signifying the motion of
the eddy), it appears inevitable that the emitting volume and
emissiontime difierencesincrease to infinity. However, from
the experlmental work of Davies, Fisher, and Barratt,20 one
knows that this picture of a mo\'ing eddy is oorealistic, because even in the moving frame of reference it is changing
gradually, and velocity values are losing correlation with
earlier values. Figure 8 showed this elearly and suggests the
revised version of Fig. 11 wmch is shown as Fig. 16,still with
time plotted vertically and distance in the direction of emission horlzontally.
This represents properly the decay of an eddy with time, by
means of curves of constant correlation or, more precisel~'
(taking into account the three-dimensional character of the
correlation function), curves on whicb"its surface integral
over planes at right angles to the direction of emission re';
mains constant. In the case when the component aoM.co;;8
of the convection velocitv in that direction is considerabh'
less than the atmospheri~ sound speed Qo,one sees that th'e
emitting volume and emission time difierences are still to a
M. cos8).
good approximation increased by the factor 1/(1
However, in the case when the two speeds coincide, neither
the volume nor the time difierence becomes infinite; in particular,the emissiontimedifierence,whichwas formerlyl/Go,
becomes I/w, where w is a typical radian frequency in the
moving frame. As for the emitting \'olume, its dimensions
perpendicular to the direction of emission do not change, but
its dimension parallel to the direction of emission increa"es
from Lto ao/w. Thus, at the angular position 8 = cos-l(l/.lI,),
both the emitting volume and the emission time difIerences
increaseby the factor ao/wlinstead of b:r the factor 1/ (1 - .ll<'
cos8). The deduction of the acoustic radiation fieldfrom this
goes through exactly as before, with 1
li{. cos8 replaced
throughout by wl/ao, and, in particular, one finds (wl/ao)-'
appearing in place of (1
M. cos8) -I in the expression for
-
-
acoustic intensity
in the direction
8
= cos-i(l/M.).
FfowcsWilliamsu has carried out the calculation for a more
general case, when the two slopes do not coincide but may be
near each other in value, and he find" that (1
M, cos8) in
this more general case should be replaced by
-
{(I
- M.
COS8)2
+
(WZ'GO)2}
1/2
(20)
In particular, the acoustic intensity field of unit volume of
turbulent jet is proportional to
«p -=--x
- Po)2)
poaol
CONVECTED
WITH
VELOcin COMPONEMT
a M cosll
UNCONVECTED
!
Time
When fi
. cos., (I/Mc) slopu
emls~ion time dlHennce
dlmenilon of cmlttlng
iii
to
Fig. 16
dlrectlon
Wl
{ (1
o .i.c
DIstance
ol .minlon
Illustrating
ol emission
In rlght-hand diagram colnclde j
(i.
volume
In direction
In Icft-hand
diagraiii) becomes
( t In Idt-hand
the Ffowcs Williams
sonic edd" eonvection
diagram
) becomCl
tlieory
F.w4T2 1
Poao 411'T2
~
i
~
w'
of super-
2
-1/2
- M. CO;;8)2+ (ao) }
(21)
if one neglects for simplicity the additional directional efiects
arising from any preferred quadrupole orientations.
in estimating the terms in this, it is fair to assume as a
basic property of turbulence what Davies, Fisher, and Barratt
found in their jets, namely [see Eq. (1i»), that the product
wl of frequencyand length-scale in a frame of reference moying v.'Ith the eddies is approximately the rms turbulent Yelocity. Then both the peak intensity and the total power
output given by Eq. (21) must varyas va, once the Mach
number of comoectionis well supersonic, say for jet speeds
three or IDoretimes the atmospheric sound speed. Since the
jet power also varles as va, this means that the acoustic efficiency, that is, the ratio of acoustic power output to jet
power, becomes constant above about this speed, as I indicated earlier to have been found experimentally.
For subsonic jets, it has been show'n that acoustic power
output varies as UI, so that acoustic efficiency \'aries as ["6,
experimental values being close to 10-4(U /ao)5 for subsonic
jets (Fig. 4). In the upper half of the subsonic range, it was
shown that the correctness of this law was maintained by tw~
(lppO~iD~ .i
tcn::ity
\n'
inrn'~i::e in
number of
1,.,con1r:, \.
nrW tl'rm
forc tl. (01
mtlii':"
i...'i:::
:i;ii'rii:!diii
with it. T
tu
zi-rii.
:1'
n1t'nt~l\iy :
:itin(l~i'h('r
lin.' rnn
\i'iii:tli
oi
rii:ih'lY
ti.,ti
tlf
l'"
tn t:'1'
,\.i.(.~~.
~itni'
'-, .Ii..:
J!:,..t°::' t~ll":
th:it
tlll'
rOll'kl't
I'X!
J:,
i :ii:::rl'
:\ ft'.'r t i
)1:u'" nur
t ri Iiui
i:i\','
j..i lo
:i
r.'
'1111'11":,. \\"
r:ii li:iii i ri
nil':i-IIT\'c!
di~t:iiii'i' .
11('~lk in'o
of thi' ii;
t~'Jlic':i1 o
quitl. "i) i:
:i maximi
dillll'ii-ii,)
w:iy- iii t'
Iiidll'r (.x
IIl':iK ir.."
(.i\ily in:
~(liii..\\" Iia
frl'quI'I1/':
i;i-ti'nt \\.
:\i'xt.
:
('iiin.. int.
r.,~lI.i 011
ruiin' ra 1
infiiii'ii,';'
,...h
:il
ra:,i,lh'
t
,Ji-
!'iri,
iiii.r'.:!-....
tlll' 1"II~1
tli,. ~lii.;:
nt tlil' i'n
.
f:i('t..r~
n,,!';III'r'
Exi"'riii'i
di
in"!
:.i.pprll:lf'!
Fiiiali
~"iinr.l 1'1
qii('nr\' ,
- .ll; <:'
tin\!' difi
rddir~ r
:ilii,iit tl
:~o(' /. \\'
!Jr tlii'
li
(Jj tlil'
..,
bilt it i,
"_. ,._- ..__.__..
".
JET XOlSE
ian of
.e and
.Ifroni
20
r
i
one
c, he.nging t
with .
1
ts the
1with !
emisae, by
:ciel\'
of th
.tegr:il
:)11re-
r:ihly
at the
Lito a
eo;;8).
ieither
n pary l/Qo,
in the
'ni;ions
ie, but
:rea;;es
1,.ll.),
'renres
- .ll,'
imthis f
'PInced
-l/ LIO)
-I
t
ion for
i
a more
i
na\" be ..
,,8)(20)
in i.
::
ume of
i.
i
iii
(21)
i effeets
ne as a
Barratt
product
ce mo\'lent \'e.i power
.e Mach
t speeds
ince the
ustic ef.t to jet
s i indiie power
:s as Us,
subsonic
, it waS
:i by tw
.'
in.aL
"'IP"'~I D "" fnct01'3: the gradual reduction dinthturbul(,l1t
b
i.:n--ity \\;th incre~in~ ~Iach. nu~ber an.
e su stanti
: , "i...:ein forward emission with mcrease m JI., the Mach
.m~ber of com'ection. it is now clear that this latter effect
~U
further, owinii;to
the
n1~... weakened as .ll. increases
r..."1:'"
.
.
~
ft"
. w tNm (i.:l a~p in the denommator of (21). it fails theref:~re to iii;trun the reduction in turb~lent II!-tensity .~ut
r:itilt'r hrltin5. as .ll. approaches .1 (that is, for Jet \'elocit~es
a:,:irii:ii:hing twice the atmosphei:ic so~d speed), to conspire
;\'iih it. The rate of rise of acoustic efficiency then falls rapidly
i., zi'ro. and the efficiency levels out at a constant (experi:i1<'n~llIyabout 0,006) at jet speeds around three times the
:itll1.,:,pheric sound speed:
. eo:,8
!
1515
. .
Oill' consequence of this for rocket noise is that a \'err great
','iii~th of rocket emaust mn be radiating sound at approxi~::Itt'h' cnn~tant acoustic efficiency, that is, in direct propor:;.'11t:, the jet energy dissipated. This length should erlend
~.. WIll'rl' the jet \"elocity has fallen below three times the
..t:lili:,uhC'ril' sound speed, in other words, to where the jet
O::L-":'tlil'Iu:,C'h'esare mo\ing subsonically, This conclusion,
t!i:it tl1l' noi:!e sources e"..tend over a substa,ntia,l length of
i' ,,'kC'ti'xliaust, perhaps as much as 30 nozzle diameters, is in
i::IKKIai;rcement ~ith obsen'ation.'
.\ftC'r thi:, reasoned account of the changes, throughout the
~I:idi number range, in acoustic efficiency and directional disirilnnion, 1 should !ike to point out that the theory seems to
;:h'C' :i rr:'onable e:..-planation also of changes in peak fre'IIII'II('Y",ith jet \'elocitr. At low Mac.h numbers, a typical
r:iili:iii fri'quencr w for a moving eddy in the mixing region is
:11t':L"'lircd
by Da\ies, Fisher, and Barratt20 as 1.3 i-/x at a.
di.-t:iiil'C x from the nozzle.
This rises to 0.3 U i d where the
jH':lk frequency is expected to be emitted, that is, at the end
IJi the mi:..ing region. On the other hand, this frequency
typii'al of the main energy-containing eddies mar not be
quite ~olarge as the frequency of the eddies that make V",,4T2
:i maximum. The eddy yolume V. (proportional to an eddy
.!imrn"ion cubed) and the frequency factor w4 pull different
w:iy:,in this expression, but the frequency factor (carI'ying the
iiii::hi'rcxponent) must be expected to win and to cause the
;1':lk frl'quency to be a multiple greater than 1 of the typical
'1111Yfrequency 0.3 i- Id. E,''(perlmenta11y, the peaks are
"JI11l'",hatfi:i.t, as i ha\'e already shown, but they occur at a.
i'n.'(iucncyusually between 1 and 1.5 times Uld, which is con,i."'h'nt ",ith the argument i have just giyen.
Xl'xt, as the jet yelocity increases, two opposing influences
romc into play, of ,,'hich one is the Dopplerfactor
1/(1
- M..
ro~8) on the peak sound frequency, tending to ms.ke it rise
~ore rapi?ly than in proportion to U. The other opposing
int1l1l'nreIS that the t)-pical frequency w of the eddies them>t.i\"('",at the end of the mbcing region tends to rise less
r:ipi.iiy than in proportion to U. For it has been shownthat
,:"ii~ "ruportional to the rms turbulent velocity, which itself
ini'ri':!:,c"les;; rapidly than U. Furthermore, as U inereases,
tlil' Ii'iigth of the mi'.:Ing region increases, and so stretching in
ilii' ::hl'ar layer gi\'es the longitudinal dimension Lof an eddy
:si thl' l'nd of it an opportunity to become greater. Both these
!:ictors reduel' w and may be expected to outweigh the
JI. cos8) in the radiated frequency.
D(,p!>l~rfactor 1/(1
E.'<Pl'!1ment:il1r,S'
7. 11.~~-:!5one finds that the peak frequency
dOl':!merease less than in proportion to jet speed as the latter
:ipp~oaehesthe speed of sound.
Fin:illy, for yery high speeds, the directional peak of the
:'l)iind emission b:r eddies of frequency w should haye a fre'1l1l'ne)'equal to w multiplied not by the Doppler factor 1/(1
-:- .ll. cos8) but by the corrected factor am p lifvin g emission
-
ti
d'ff.
J
~1~ i erenees, that is, aolwl. if one assumes again that the
ilics most effecth'e for sound emission have a frequency
~~t t~ree times as great, then the peak frequenc)' should b"e
f ,wi~~ Lequal to the longitudinallength-scale at the end
o the mixing region. Experimentally,S-IO the peak frequency
'bf ~~~~ound.radiation field of a. rocket exhaust is about 0.01d,
u i is not impossible that in 30 diam or so the longitudiniil
a)
b)
Fig.17
Noise suppressors
on the Boeing 707; a) wiih Pratt
& Whitney 3C-6 engine, h) ~ith Rol1s Royce Conway engine
length-scale Lmay have risen as high as 3d, which would explain the observed figure.
6. Conclusion
To sum up, the theory gi\'es a reasonably clear explanation
of the sound radiation fieldsof jets in terms of observations on
the turbull'nce in those jets (although not pretending to deal
with other noise sourees that may be present along with the
jet or with losses associated with propagation of the jet
noise), In particular, the observed yariations in acoustic
efficiency, directional distribution, and frequency spectrum,
with increasing jet velocit)" are accounted for adequately by
the re resentation of a .et as a distribution of acoustic
~uacirupoes ID an ot ennse undistur ed atmos ere, e
uadru ole stren
s mg wl' corre te o y wit
conv,e
eddies of limite
exten an
i e
e ime.
.
To go beyon the matter of physical understanding, and
inquire concerning means for reducing the noise, elicits from
the theory a somewhat pessimistic answer, namely, that the
one parameter that affeets total acoustic power output to ani
a,IDJreciablee:<tent is rms turbulent ,,~ity.
which in turn
depends principally on jet speed. By far the best way to reduce noise output is, therefore, to reduel' jet speed; and, indeed, if this is aIready less than twice the atmospheric sound
speed, the factor by which noise output is redueed is the sixth
powerof the factor by whichjet speed is reduced, for a constant
value of the jet thrust. This has been one of the motive forces
behind the trend to engines of high b:rpass ratio.
Apart from this, only two successful methods of reducing
..jet noise radiation for a. modest weight penalty (with no reduction in jet speed) have been found (see the Bakenan Lecture2 for full references). One of these contrives to ~e
rms turbulent velocity by diminishing thjLre1ii.tivevelocitv of
the jet an4 the air adjacent to it, as a result of nozzle shaping
t.hl'lt.indtlC'e!!fQIJYardmotion of much of that adiiicent air.
The other method makes u..i.eof the ma.rked directionality or
jet noise, from which it follows that the peak noise from a
cluster of nozzles "iii be less than the sum of the peak noise
from each separately. This is because noise in the peak direction from each nozzle will be redirected in some different
directionon encounteringone of the others. Several good
designs of noise suppressors for aircraft (see, for ex:i.mple,
Fig. 17) h::l.\'emanaged to make use of both these principles,
.__._---
M. J. LlGRTRlLL
and reductions of up to 10db (in peak sound) can be made with
an overall performance loss, which, although severe, is not
crippling economically. in the long run, however, to bring
down the jet velocity is the only satisfactory solution. In
contrast with all this, hopes of reducing the noise ofrockets, at
any rate after theyare airbome, when even moderate penalties on thrust and weight are likely to be quite unacceptable,
appear very slender indeed.
I have tried in this lecture to build up a picture, by a combination of theoretical analysis and experimental evidence,of
how the shearing motions in a turbulent jet manage to shed
same of their energy as sound radia.tion. I hope that a fe\\"of
the ideas that I have presented may prove useful to those who
either may be faced with practical jet-noise problems or who
have in mind e=-.-perIments
to throw new light on this important and difficultsubject.
Appendix
A: Infiuence
of the
Ratio
Density
to Atmospheric
Density
of
Jet
A matter, not discussed in the lecture, on which experimental data have not reached complete agreement is the
efiect of dift'erencesbetween jet density
pJ
(defined as the
average density across the oMce) and atmospheric densit:r
Po. Same good correlations of power output with PoU8A/{MJI,
such as Fig. 15, have been obtained.7.32 However, other
work, i. i. 23 using values of pJ considerably smal1er than Po,
has demonstrated a resulting reduction in the acoustic power.
This might at first sight, through its dependence on T'/ po
[Eq. (13)], be expected to vary as PJ2/po rather than as Po.
However, the peak sound is believed to originate in the center
of the mi,dng region,wherea typical density Pi shouldbe
intermediate between po and PJ, suggesting dependence on
Pi'/ Po,wIiich does not fall ofi with decreasing pJ so rapidly as
PJ'/Po. The situation is complica.ted further when, as often
happens, the velocity of sound ai in the jet difi'ers considerably
fromaa. Thenthe termp,! -
{MJ'P~'iin
T'I [Eq. (5)] represents sound generation approximatel;}'equivalent [by arguments like those leading to (B3) below] to that froma source
distribution of strength (1 Qo'/aJ')o'p/()t' per unit volume.
Our new knowledge (See. 3) on frequencies in the mixingregion in a frame maving with the eddies indicates that, when
aJ'/{MJ'is large (which general1~'goes with smaIl PJ/Po),this
radiation will represent an addition t of around 20% to that
represented by the term pM'j and therefore wili help to mitigate the decrease of sound output with decreasing PJ/Po.
A bals.nced ,-iew of the existing data, and one reasonably
consistent with the preceding argument, seems to be that
some falling-ofi from proportionalit)" to POU'A/{MJ'
\\ith increasing PJ/po occurs, but that it is represented better by a
factor PJ/po than by (PI!po)2. This means that acoustic
power output for U/{MJ< 1.5 is about 1O-.(!PJ,U8A/{MJI)
in
agreement \\ith the law, stated several times in this paper,
that acoustic efficiencyis about 1O-.(['/ao)8.
-
Appendix
B: Simple-Source
Theories
of Jet Noise
Ribner3f and others (cited by him) have deseribed jet
noise and other aerodynamic-soundphenomena in an alternative manner using sirnple sources. The~-define a function </>
such that
V'</>
=
t t ox,ox,
02~i~.
(BL)
i-i i-i
in terms of which Eq. (8) becomes
o'P
- ()t'
o'P
i:3 ao' ---;;=
i_ i
v2</J
(B2)
C>Xi'
t The :iuthor's HJ,54paper,n on p. 22, estimated t.his proportion as.li- of the squ:ire of the ratio of radian frequencr to mean
shear (a mtio now known, from Fig. 10, tu be around 0.22).
AlAA
JOURNAL
This new equation is evidently the same as (8) but "ith T,i.
replaced by the isotropic tensor ipOi/. The solution [compare
Eq. (9)] is
{MJ2(p
- Po) = V' f
-
[</>]
~' 47!7'
= .!. ~,
{MJ'c>t'
.;,,-hai:
like r!'tep in
peneli ,
dr
[</>]dr =
f V 4'11'7
f V i4i/{MJ']dr
47!7'
(B3)
The equality of the second and third terms in (B3) follows
from the fact that the retarded potential is a solution of the
wave equation. The fourth term shows that the acoustic
radiation is equivalent to that generated by a distribution of
sources of strength 41/00'per unit volume. it should be remarked that </>can be regarded as a sart of incompressibleflowapproximatian to the pressure.
Some mathematica.l criticisms of these theories in their
earlier forms were made, particularly in relation to the treatment of source convection, but these criticisms appear to be
met fully in Ribner's latest paper.if The correct directional
factor [that is, for subsonic jets, (1
M. COS8)-I] is now dedueed. Furthermore, the author pro\"es conclusivelythat his
formal expressionsfor the acoustic radiation field are equivalent mathematically to the theory set out in this lecture.
The relative value of these two mathematica.lly equivalent
formulations can accordingly be assessed only on the basis of
whichgives more help in estimating how the noise generated
in Yarious regions of a jet depends upon all the relevant
variables, on the basis of approximations suggested by physical argument and experiments on jet turbulence. The
simple-source theory has three main difficulties from this
point of view:
1) One essential feature of Tij is that it is sina1l, of the
order of the square of the disturbances, outside the jet itself
(See.2), so that the integral for the sound field that has to be
estirnated is merely an integralayer the region of the jet.
This is not true of or of the pressure (to which it represents
an approximation), so that, in the simple-source theory, noise
generatian from outside the jet has, rather artificially, to be
postulated and, presumably,estimatedo
2) Inside turbulent jets extensive measurements of the
correlation of velocity ys.luesat difi'erent points, and of ho\\'
it faUs away ",ith increasing distance between the points,
have been made. These, ",ith further information on velocity
spectra and space-time correlations, were used in the estimation (Sees. 3 and 4) of the sound generation associated ",ith
the quadrupole distribution Ti;, with reason, because its
fluctuations arise mainl~'from \"elocityfluctuations. By contrast, little is known of pressure fluctuations inside turbulent
jets, whose correlations and spectra cannot be measured
directly. Estimation of fluctuations in </>though fluctuations
in the pressure (to which it is same sart of approximatian)
is therefore not feasible, whereas direct estimation through
Eq. (Bl) defining </>
leads right back to the problem offluctua-
tion. aJ
('()rrt'ct
L-ni'\t
rluil('~
tlir "'.:1
\\"hid:
l>('("(lmi
hilwr\-'
u-"t'\1 ii
n't:ir.1f
iii tlll.
,;{'n'~ ri
hi. Fi::
Ti",
i.. ii..t
-
</>
tions in TH itself.
_
3) Furthermore, yalues of Ti; are expected to be well
correlated only "ithin convected eddies of limited extent and
limited lifetirne, and, for at any rate subsonic jets, the eddy
size is very smaIl compared \\ith the wayelength of the sound
that it generates, whose estimation is made much easier by
this fact. By contrast, the correlation of ipat two points h
expected to fall off much more slowl~' ",ith the distance between them. Indeed, it appears doubtful whether carrelatian
ra.diIfor </>are smal1enough compared with the wa"elength,
for any jet Mach numbers, to permit the neglect of dift'erences
between retarded times where \"alues of </>are carrelated signmeantly, for this neglect would lead to a uniform directional distribution of Ii'coU$ticradiation (exeept for the canvecthoefactor), a serious errorin eyery case when T'j departs
signifieantly from isotropy. It i;;, in fact, easy to show (one
exampleis given in a footnote to the Bakerian Leeture) that a
localizedYariatian in Tij proom'e;; a ;;olution of Eq. (Bl) for
- - --
:'t,,\.,:~t'
hri,.fiy
ti..ii~ :i
:tcrrirti
, L.i:
,\I'r"D~
: 1.ii
nil", :;.,
in:
:ixisrm
'Lt:
lIIl'trk
~till air
'c"
Urui'lii:
'C,..
i i h:rii~
111,
TH 1:"
'Ci.
I'iivir..!
(i~I-T
) :\1
lil'\d :u
r:l1u:'.,
i.. ('.
ali.1 H
II"\".
i:
(
r:
\'l'i'i",d
;;in
l!
JET XOISE
Jt"I. Y 1\163
Til
pare
:B3)
10\Vs
the
istic
n of
~reible,heir
'eatJ be
Dna!
. de~his
.iva-
~nt
is of
ited
Tant
iysiThe
this
the
tseU
obe
jet.
ants
oise
>be
the
tio,,"
nts,
city
ma\ith
its
:onlent
ired
ions
ion)
ugh
~ua-
well
and
dd\'
und
br
;s is
be~ion
~th,
ices
sigrec:onirts
one
it il
for
12
~ who:,e correlation with that local variation falls ofI in general
:ike r-i. This ma.kes impossible in simple-source theory a
:n~p like th:it from Eq. (ll) :o Eq. (12), whose validity de:1C."nd:'
on 1r; being a volume mtegral of the relevmt correla~ion.:in integml that in this case diverges. Ribner points out
,-orrcoi.th'th!Lt the exact integral for the sound radiation field
:..~Mt di\'ergent, beeause for sound of each frequency it in,-lo.1,I.,:;
:i sinusoidally ,'al1ing term (with wavelength that of
t!:i' ,"",unditself) , !LSsociat~dwith difIerences in ret&rded times,
:\'hidi renders it con"ergent as the region of the integration
:,<"("()01C:;
l:irge compai-ed "ith the wavelength. This mea.ns,
howco\'cr, that ordinary eddy-size measurements cannot be
:1."4."1
in the estimation process, and also that difIerences in
n'urded times ne\'er can be neglected. it may be significant
in the fje:ht of the foregoing remarks, that Mollö-Christenn.:' nica,.urementsJ1 of pressures just outside a. jet show (see
~i,. Fig. 12) e~..tremely long tails to the correlation curve.
Th,' threedifficultiesnotedhereseemto indicatethat q,&ii
~" II'lt :i useful alternati\'e form of Ti;. There are not such
"\'rl.clitficultie:s
",ith the 'Pii-
ao2p&ifterm
in Tif, discussed
:1~id1r in Appendb.: _\, because outside the jet itself its varia.tiiin:' :ire negIigible, and the large correlation spheres reduel',
:ii'i'ordingly, to their intersections with the jet itself.
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.
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